Narrow Results

In this paper, we define $X$-convex and quasi-$X$-convex functions. This class of real functions is the generalization of a family of convex functions. Additionally, we provide a thorough analysis of the core characteristics of these functions in this paper, along with numerous examples that help to illustrate the ideas. Furthermore, quasi-$X$-convex, semistrictly quasi-$X$-convex, and strictly quasi-$X$-convex functions are used to study and show its applications on optimization problems.

In this paper, we establish a generalized version of Gibbons’ conjecture in the context of the master equation

This paper investigates the topological pressure of isotropic axial products of Markov subshifts on the $d$-tree. We show that the quantity increases with dimension $d$. To achieve this, we introduce the pattern distribution vectors and the associated transition matrices and partially transplant the large deviation theory to tree-shifts. Additionally, we apply our main result to a broader class of shift spaces, accompanied by numerical experiments for verification.

In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c² f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions.

In this paper, we consider fractional equations involving the logarithmic Laplacian with indefinite nonlinearities:

In this work, a nonlinear finite-difference scheme is provided to approximate the solutions of a hyperbolic generalization of the Burgers–Fisher equation from population dynamics. The model under study is a partial differential equation with nonlinear advection, reaction and damping terms. The existence of some traveling-wave solutions for this model has been established in the literature. In the present manuscript, we investigate the capability of our technique to preserve some of the most important features of those solutions, namely, the positivity, the boundedness and the monotonicity. The finite-difference approach followed in this work employs the exact solutions to prescribe the initial-boundary data. In addition to providing good approximations to the analytical solutions, our simulations suggest that the method is also capable of preserving the mathematical features of interest.

The work is devoted to the study and solution of one class of nonlinear integral equations with monotone and noncompact Hammerstein-type operators on whole line. Equations of this kind arise in many branches of natural science. In particular, such equations are applied in the dynamic theory of $p$-adic open–closed strings, in the kinetic theory of gases, in the mathematical theory of the spatial-temporal spread of pandemic. A constructive theorem for existence of non-negative nontrivial and essentially bounded solution is proved. The asymptotic behavior of constructed solution at $\pm \infty$ is studied. It is also proved that the difference between the limit in $+\infty$ and the solution represents summable function on $(-\infty ,+\infty ).$ Using results obtained a non-negative integrable and bounded solution for new class of integral equations on whole line with monotone and sign-alternating nonlinearity is constructed. Under additional restrictions on nonlinearity, a uniqueness theorem for a solution in a certain class of continuous and bounded functions is also proved. At the end of the work, the specific examples of above equations, satisfying all conditions of the formulated theorem are given.

Motivated by the research on social opinion dynamics over large and dense networks, a general framework for verifying the monotonicity property of multi-agent dynamics is introduced. This allows a derivation of sociologically meaningful sufficient conditions for monotonicity that are tailor-made for social opinion dynamics, which typically have high nonlinearity. A direct consequence of monotonicity is that social opinion dynamics is nonchaotic. A key part of this framework is the definition of a partial order relation that is suitable for a large class of social opinion dynamics such as the generalized naming games. Comparisons are made to previous techniques to verify monotonicity. Using the results obtained, we extend many of the consequences of monotonicity to this class of social dynamics, including several corollaries on their asymptotic behavior, such as global convergence to consensus and tipping points of a minority fraction of zealots or leaders.

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, $f$ and $G$, we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

A continuous map f from a metric space X to itself is said to contain a two-sided symbolic dynamical system if there exists an invariant set X₀ of f such that the subsystem f|X₀ is topologically conjugate to the shift map on a two-sided sequence space of some symbols. In this paper we show that, for any given integer n ≥ 2, there exists a Lipschitz continuous interval map which contains a two-sided symbolic dynamical system of n symbols. Furthermore, we investigate the effect of differentiability and monotonicity assumptions, and prove that neither piecewise monotonic nor piecewise continuously differentiable graph map can contain a two-sided symbolic dynamical system.

Let T be a competitive map on a rectangular region , and assume T is C¹ in a neighborhood of a fixed point . The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that is an increasing curve that separates into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

In this paper, we are concerned with the wave propagation for a system of 2-D lattice differential equations with delay. Under the monostable assumption, the asymptotic behavior, the monotonicity and uniqueness of traveling wave are established when the wave speed is greater than or equal to the minimal wave speed c_*(θ) > 0. In addition, the directional dependence of the minimal wave speed is analyzed numerically.

In this paper, we analyze the monotonicity and the number of critical periods of the period function associated with a family of piecewise planar continuous potential systems. In particular, we describe the bifurcation diagram of the period function, determining the regions of the parameter space where it is monotonically increasing or decreasing and where it has at most one simple critical period. In order to prove the main result, we rewrite such a function in terms of the two period functions of the uncoupled planar Hamiltonian systems matched by the separation line.

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct $\alpha$-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the $\alpha$-fractal rational quartic spline when the original function is in ${{𝒞}}^4(I)$. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the $\alpha$-fractal rational quartic spline to ${{𝒞}}^2$. The elements of the iterated function system are identified befittingly so that the class of $\alpha$-fractal function $Q^{\alpha }$ incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ $Q$. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

A set of perceived random events is given by a fuzzy random variable, and an estimation of real random variables is represented by a functional on real random variables. The perception-based extension of estimation regarding random events is introduced, extending the functional to a functional of fuzzy random variables. This paper discusses some conditions and various properties of the extended estimations, for example, monotonicity, continuity, linearity, sub-additivity/super-additivity, convexity/concavity. Several examples of the perception-based extended estimations are investigated. This paper analyzes the general cases, where the estimations do not have monotone properties, from the viewpoint of convexity/concavity. The results can be applicable to other estimations in engineering, economics and so on.

A similarity measure is a measure that indicates the degree of similarity between two objects. The purpose of this work is to investigate the monotonicity properties and applications of similarity measures on interval-valued fuzzy sets. Through analyzing the intuitions of similarities, three kinds of monotonic similarity measures are defined. Furthermore, their properties and relationships with entropy and inclusion measure are investigated and discussed. Finally, some applications of the proposed monotonic similarity measures, such as pattern recognition, medical recognition, and medical diagnosis are presented.

In this article, we study the price monotonicity in the parameters of the Heston model for a contract with a convex pay-off function; in particular we consider European put options. We show that the price is increasing in the constant term in the drift of the variance process and decreasing in the coefficient of the linear term in the drift of variance process. We also show that the price is increasing in the correlation for small values of the stock and decreasing for the large values.

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients.