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The notion ‘Convexity’ is applied in various areas of mathematics particularly in optimization techniques. It is known that this concept is applied in fuzzy sets, which is studied by many authors. This article deals with convexity utilized for neutrosophic and interval valued neutrosophic sets which is a generalization of intuitionistic fuzzy sets. Also some interesting preservation properties of convexity under intersection operators in interval valued neutrosophic sets are discussed. Adding to the discussion, preservation property of digital convexity under intersection using deformation and other techniques are touched upon. Eventually the application of convexity to decision making are illustrated using examples. [ABSTRACT FROM AUTHOR]

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to clarify a few important aspects about the aggregation issues in efficiency and productivity analysis. By doing so we also sketch a brief historical map on how the area of aggregation in efficiency and productivity analysis has developed to where it is now and its connection to some classic studies in economic theory. As a result, this paper also serves as a tribute to the great scholars I was lucky to learn from, Rolf Färe and Shawna Grosskopf, who made a profound impact on research in economics in general and the topic of aggregation in productivity and efficiency analysis in particular. [ABSTRACT FROM AUTHOR]

Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization. Random walks ensure uniform generation and adequate coverage. The Möbius representation of set functions is a valuable tool for establishing the sparse structure of fuzzy measures, with its non-negativity closely linked to monotonicity and convexity/supermodularity checking. We propose three efficient methods for monotonicity verification and convexity/supermodularity verification applicable to random walks in Möbius representations, specifically tailored for universal sets larger than ten inputs and k-order fuzzy measures. We first present the baseline methods by directly inspecting monotonicity and convexity constraints. Building on the observation that the majority of initially generated values exhibit non-negativity, we intentionally track negative Möbius values to enhance the computational performance of these baseline approaches. Further, we introduce the more agile methods that employ insertion and merge sorting techniques for both monotonicity and convexity checks in random walks that involve small perturbations of fuzzy measures. To address sparsity in large-scale scenarios, we focus on two major types of measures: k-additive and k-interactive measures, demonstrating their effectiveness through theoretical analysis and experimental results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the approximate controllability of control problems governed by evolution hemivariational inequalities in super-reflexive Banach spaces by preassuming the approximate controllability of the associated linear system. We first show that the original problem is connected with a differential inclusion problem involving the Clarke subdifferential operator, and then we prove the approximate controllability of the original problem via the approximate controllability of the differential inclusion problem. The paper offers a unique solution to a challenge introduced by assuming the underlying space X as a super-reflexive Banach space, which presents issues of convexity due to the nonlinear nature of the duality map involved in the expression of the control. Such issues are not present when X is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for approximate controllability of the evolution hemivariational inequality in which X is a super-reflexive Banach space. [ABSTRACT FROM AUTHOR]

The crystalline lens of the human eye possesses a distinct convex shape, a subject that has garnered significant attention in optical research. In our study, we delve into the methodologies for quantifying the convexity of these lenses. Utilizing lens maps as our foundational model, we demonstrate that these mappings exhibit convexity of order β > 0 and are starlike of order α > 0 . Furthermore, we establish a direct correlation between these orders and the opening angle of the lens map. [ABSTRACT FROM AUTHOR]

Numerical range is the subject of study for over a century. Its extension to matricial range is defined in terms of completely positive maps. A linear map ϕ between matrix algebras is completely positive if and only if its Choi matrix $ C(\phi) $ C (ϕ) is positive semidefinite. In this paper, we extend the study to some joint matricial ranges defined by those ϕ's where $ C(\phi) $ C (ϕ) is Hermitian with specified spectrum. We prove some convexity theorems which extend previous results on generalized joint numerical ranges and matricial ranges. We also extend Bohnenblust's result on joint positive definiteness of Hermitian matrices and Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of Hermitian matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal control problem for integro-differential equations on the semi-axis, which are non-linear with respect to the phase variables and linear with respect to the control. We obtained the sufficient conditions for existence of optimal control in terms of the right-hand side and the quality criterion. Also, we studied the relation between the solutions of the problems on infinite and finite intervals when the length of the interval goes to infinity. [ABSTRACT FROM AUTHOR]

When determining new equity grants, corporate boards face a tradeoff between the CEO's incentives generated from the grant's duration versus those arising from the convexity of the embedded equity risk. We hypothesize and find that boards lengthen the horizon of new compensation grants in the presence of greater pre-existing compensation sensitivity to stock return volatility (vega). In addition, consistent with our hypothesis, we find stronger results in the presence of greater left-tail risk. Further, employing two exogenous shocks to left-tail risk, we provide evidence consistent with our hypothesis that grant horizons are related to risk incentives. Our analysis of the interaction of these two incentive mechanisms provides new insights on compensation contracting. JEL Classifications: J33; M52. [ABSTRACT FROM AUTHOR]

We develop a continuous time framework for sequential goals-based wealth management. A stochastic factor process drives asset price dynamics and the client's goal amount and income. We prove the weak dynamic programming principle for the value function of our control problem, which we show to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We develop an equivalent and computationally efficient representation of the Hamiltonian, which yields the optimal portfolio within a factor-dependent opportunity set defined by the maximum and minimum variance hypersurfaces. Our analysis shows that it is optimal to fund an expiring goal up to the level where the marginal benefit of additional fundedness is exceeded by the opportunity cost of diverting wealth from future goals. An all-or-nothing investor is more risk averse toward an approaching goal deadline if well funded, but more risk seeking if not on track with upcoming goals, compared with an investor with flexible goals. This paper was accepted by David Simchi-Levi, finance. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant RGPIN-2020-06290] and Fi-Tek. [ABSTRACT FROM AUTHOR]

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder's and nabla Jensen's inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results. [ABSTRACT FROM AUTHOR]

We study a non-compact version of the carrying simplex for the planar Leslie–Gower and planar Ricker maps when they are written in logarithmic variables. We show that for both of these models there is a convex (unbounded) invariant set $ X_\infty $ X ∞ , and all orbits are attracted to $ X_\infty $ X ∞ . For the Leslie–Gower map, which is injective, the boundary of $ X_\infty $ X ∞ globally attracts all orbits and we identify it with a non-compact carrying simplex. As the Ricker map is not invertible, the boundary of $ X_\infty $ X ∞ may not be invariant. We establish conditions on the parameters of the Ricker map which guarantee that there is a convex non-compact carrying simplex when r, s<1 which maps into a compact carrying simplex in the standard untransformed coordinates. [ABSTRACT FROM AUTHOR]

This paper considers a chief interface between mathematical programming and economics, namely: money-based trade of perfectly divisible and transferable goods. Three important and related features are singled out here: first, convexity enters via acceptable payments, second, convolution of monetary criteria secures Pareto efficiency, and third, competitive equilibrium obtains when agents' subdifferentials intersect. [ABSTRACT FROM AUTHOR]

In the present work, we have established some new fractional integral inequalities for functions whose k th-derivatives are generalized modified h -convex and symmetric about the midpoint involving the Caputo fractional derivatives. Many particular cases are obtained by using the findings. [ABSTRACT FROM AUTHOR]

Abstract The crystalline lens of the human eye possesses a distinct convex shape, a subject that has garnered significant attention in optical research. In our study, we delve into the methodologies for quantifying the convexity of these lenses. Utilizing lens maps as our foundational model, we demonstrate that these mappings exhibit convexity of order β > 0 $\beta >0$ and are starlike of order α > 0 $\alpha >0$ . Furthermore, we establish a direct correlation between these orders and the opening angle of the lens map.

In the present work, we have established some new fractional integral inequalities for functions whose $ k $th-derivatives are generalized modified $ h $-convex and symmetric about the midpoint involving the Caputo fractional derivatives. Many particular cases are obtained by using the findings.

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a convex geometry. In this work, we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled Lamé system only with viscoealstic dampings. By assuming a more general of relaxation functions and by using some properties of convex functions, we establish optimal explicit and general energy decay results to the system. This result improves previous results in the literature. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials C n (q) = 1 [ n + 1 ] q 2 n n q viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions. [ABSTRACT FROM AUTHOR]

This article takes a cognitive approach to natural concepts. The aim is to introduce criteria that are evaluated with respect to how they support the cognitive economy of humans when using concepts in reasoning and communicating with them. I first present the theory of conceptual spaces as a tool for expressing the criteria. Then I introduce the central idea that natural concepts correspond to convex regions of a conceptual space. I argue that this criterion has far-reaching consequences as regards natural concepts. Partly following earlier work, I present some other criteria that further delimit the class of natural concepts. One of these is coherence, which does not seem to have been discussed previously. Finally, I show that convexity and other criteria make it possible to ensure that people mean the same thing when they communicate using concepts. Apart from its philosophical interest, the analysis presented in the article will be relevant for tasks of conceptual engineering in artificial systems that work with concepts. [ABSTRACT FROM AUTHOR]

In this article, we explore and analyze the different variants of Julia set patterns for the complex exponential function W(z) = αezn + βz² + log γt and complex sine function T(z) = sin(zn) + βz² + log γt, where n = 2, α, β ∈ C, γ ∈ C\0, and t ∈ R, t = 1 by employing a viscosity approximation-type iterative method with s-convexity. We utilize a viscosity approximation-type iterative method with s-convexity to derive an escape criterion for visualizing Julia sets. This is achieved by generalizing the existing algorithms, which led to visualization of beautiful fractals as Julia sets. Additionally, we present graphical illustrations of Julia sets to demonstrate their dependence on the iteration parameters. Our study concludes with an analysis of variations in the images and the influence of parameters on the color and appearance of the fractal patterns. Finally, we observe intriguing behaviors of Julia sets with fixed input parameters and varying values of n. [ABSTRACT FROM AUTHOR]

Evidence supporting clinical decision-making in the specialty of ostomy care regarding prescriptive product use has been sparse. Many clinical decisions in ostomy care have been based on either clinician experience, or from teachings and beliefs that have evolved over many years but are not based in evidence. There is now a rich and varied tapestry of evidence supporting clinicians regarding the use of convexity earlier in the patient's journey, particularly of the more compressible types (soft) convexity products. This article reviews the recent evidence concerning the use of more compressible barriers in ostomy care, a relatively newer addition to the clinician's armamentarium, for managing patients. It provides an analysis of the data that supports using these products, sooner rather than later, in achieving a more secure skin seal, and improving patient outcomes compared with using flat skin barriers. [ABSTRACT FROM AUTHOR]

Driving changes to clinical practice can be a daunting task. However, if meaningful change is to occur, the use of evidence to support the decision for change can be crucial in gaining agreement with stakeholders. This article describes the journey taken at one institution by a clinician following the trail of evidence that has recently been developed regarding the use of convexity with a barrier ring, earlier in the patient journey for creating positive impacts in patient outcomes. [ABSTRACT FROM AUTHOR]

In this article, we build a pair non-differentiable second-order symmetric multiobjective fractional variational programming models with cone constraints, where each objective function component has a support function for a compact convex set. The (C, ρ, θ)-convexity/(C, ρ, θ)-pseudo-convexity/(C, ρ, θ)-quasiconvexity functions are defined and also, constructed concrete numerical examples for existing such type of functions. The duality results are established by using these aforesaid assumptions. [ABSTRACT FROM AUTHOR]

Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on R d that are convex with respect to the Löwner partial order can have a complicated structure and might be very difficult to compute even in simple cases. The aim of this paper is to study subdifferential calculus for such functions and properties of their subdifferentials. We show that many standard results from convex analysis no longer hold true in the matrix-valued case. For example, in this case the subdifferential of the sum is not equal to the sum of subdifferentials, the Clarke subdifferential is not equal to the subdifferential in the sense of convex analysis, etc. Nonetheless, it is possible to provide simple rules for computing nonempty subsets of subdifferentials (in particular, individual subgradients) of convex matrix-valued functions in the general case and to completely describe subdifferentials of such functions defined on the real line. As a by-product of our analysis, we derive some interesting properties of convex matrix-valued functions, e.g. we show that if such function is nonsmooth, then its diagonal elements must be nonsmooth as well. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hadamard-Hadamard type inequalities for tgs-convex functions. Our results are the generalizations of some known results. The new generalized estimate of the midpoints product of two tgs-convex functions is also considered. [ABSTRACT FROM AUTHOR]

We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some broad assumptions, we give the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces, and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce the normalized tail of the Maclaurin power series expansion of the square of the tangent function, find out the logarithmic convexity of the normalized tail in light of the monotonicity rule for the ratio of two series, and expand the logarithm of the normalized tail into a Maclaurin power series with the help of a formula for higher order derivatives of the ratio of two differentiable functions. [ABSTRACT FROM AUTHOR]

Stochastic proximal gradient descent (Prox-SGD) is a standard optimization algorithm for solving stochastic composite optimization problems in machine learning. However, the existing convergence rate analysis of Prox-SGD is mostly focused on convergence in expectation. To this end, we provide an almost sure convergence rate analysis of Prox-SGD in two optimization settings: convexity and strong convexity, showing that any implementation of the stochastic algorithm will converge with probability one. Specifically, we show that the average-iterates almost sure convergence rate of the function values for Prox-SGD is arbitrarily close to the optimal rate $ o(1/\sqrt {T}) $ o (1 / T) in the convex setting. For strongly convex objective functions, we obtain the almost sure convergence rate of the iterative sequence, which is also arbitrarily close to the optimal rate $ o(1/T) $ o (1 / T). Last but not least, we establish the more challenging last-iterate convergence rate of the function values on convex problems, which contrasts with existing results on convergence for a weighted average. [ABSTRACT FROM AUTHOR]

A novel gear mechanism named high-order multi-segment deformed eccentric non-circular gear was proposed for achieving the unification of non-circular gears with typical-form pitch curve and non-circular gears with free-form pitch curve. The transmission mechanism of the high-order multi-segment deformed eccentric non-circular gear was analyzed, and the unified mathematical expression of eccentric gears was established. The non-circular gears with free-form pitch curve could be constructed based on the proposed high-order multi-segment deformed eccentric non-circular gear by changing the parameters. Moreover, the transmission characteristics were discussed, such as the transmission ratio relationship, convexity distinguished conditions, curvature radius of pitch curve. The visual design and simulation software and generation software of tooth profile for non-circular gears are compiled based on MATLAB, and was verified with the example. This novelty gear was applied to the drive mechanism of the metering pump. The mathematical model of non-circular gear-crank slider mechanism is established and the 3D model and virtual prototype experiment of the mechanism are accomplished. The application showed that the high-order multi-segment deformed eccentric non-circular gears were feasible in practice. [ABSTRACT FROM AUTHOR]

We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of I (P + t) can be extended to polynomials in variables t ∈ R d within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions. [ABSTRACT FROM AUTHOR]

We consider functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups. [ABSTRACT FROM AUTHOR]

The elastic energy of a bending-resistant interface depends on both its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham--Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal L²-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

We investigate the geometric properties for a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. We extend earlier results of Epstein, Hiai, Carlen and Lieb. [ABSTRACT FROM AUTHOR]

In this manuscript, we explore some new stunning fractals of Julia sets by developing the escape criteria for novel type of complex function p(z) = a exp[d sin(zn)] bz + c, where n,|d|≥ 2 and a,b,c,d Ⲉ C and furnish some graphical illustrations of the generated amazing fractals, utilizing the Jungck fourstep iteration scheme equipped with s-convexity as well as four-step iterative method. Moreover, we conclude this work by examining variation in images and the impact of parameters on the deviation of dynamics, color, and appearance of fractals. At some fixed input parameters, we observe the engrossing behavior of Julia sets for different n via the considered algorithms. [ABSTRACT FROM AUTHOR]

In this study, a specific identity was derived for functions that possess two continuous derivatives. Through the utilization of this identity and Riemann-Liouville fractional integrals, several fractional Milne-type inequalities were established for functions whose second derivatives inside the absolute value are convex. Additionally, an example and a graphical representation are included to clarify the core findings of our research. [ABSTRACT FROM AUTHOR]

Applying a parametric approach, in this paper we studied a new class of multidimensional extremization models with data uncertainty. Concretely, first, we derived the robust conditions of necessary optimality. Thereafter, we established robust sufficient optimality conditions by employing the various forms of convexity of the considered functionals. In addition, we formulated an illustrative example to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying classical multiquadric function. Based on the translates of a fractal multiquadric function defined on a grid, we propose two fractal quasi-interpolants L C α f and L D α f to approximate smooth and irregular functions. We study the convergence of L C α f and L D α f to f using uniform error estimates. We investigate the linear polynomial reproducing property, convexity/concavity and monotonicity features of these quasi-interpolation operators. The advantages of fractal quasi-interpolants over the classical quasi-interpolants are demonstrated by various examples. [ABSTRACT FROM AUTHOR]

In this note, we will present an intermediary result for giving a simple proof of a problem presented at the American Putnam Competition in 1998. [ABSTRACT FROM AUTHOR]

In this study, a specific identity was derived for functions that possess two continuous derivatives. Through the utilization of this identity and Riemann-Liouville fractional integrals, several fractional Milne-type inequalities were established for functions whose second derivatives inside the absolute value are convex. Additionally, an example and a graphical representation are included to clarify the core findings of our research.