Convex Functions Research Articles

Atomic-level analysis of unusual mechanical and failure behaviors in compositionally graded nanowires: A molecular dynamics study

Compositionally graded materials have emerged at the frontier of science involving material science, engineering, physics, and chemistry due to their unusual and tunable material properties derived from spatial variations in compositions. However, achieving a comprehensive understanding of the mechanical and failure behavior of compositionally graded nanostructures, particularly at the atomic level, has remained elusive. In this study, molecular dynamics simulations are used to investigate the tensile mechanical and failure properties of CuNi nanowires with compositional gradients. The nanowires exhibit a dual variation in composition along the longitudinal direction following power-law functions. The findings demonstrate that Young's modulus, ultimate tensile strength, and yielding stress of nanowires decrease with an increase in the gradient index. Remarkably, the mechanical properties exhibit higher sensitivity to changes in the gradient index for nanowires displaying composition distribution in a convex function, while they exhibit lower sensitivity for those with concave functions. Furthermore, the necking and failure processes predominantly occur in regions with highly varied compositions. Moreover, as the gradient index increases, the distance between the necking position and the mid-plane of nanowires decreases. Additionally, this study provides a detailed discussion of the phase transitions that occur during the tension process.

Integral inequalities of Ostrowski type for two kinds of s-logarithmically convex functions

Abstract In the paper, the authors establish several integral inequalities of the Ostrowski type for s-logarithmically convex functions. These integral inequalities modify the conditions and correct errors in two main theorems of the paper [A. O. Akdemir and M. Tunç, Ostrowski type inequalities for s-logarithmically convex functions in the second sense with applications, Georgian Math. J. 22 (2015), no. 1, 1–7].

Hermite-Hadamard type inequalities for $(p,h)$-convex functions on $\mathbb{R}^n$

In this paper, the concept of the $(p,h)$-convex function is introduced, which generalizes the $p$-convex function and the $h$-convex function, and Hermite-Hadamard type inequalities for $(p,h)$-convex functions on $\mathbb{R}^n$ are established. Furthermore, some mappings related to the above inequalities are studied and some known results are generalized.

Approximating Nonlinear Functions With Latent Boundaries in Low-Rank Excitatory-Inhibitory Spiking Networks.

Deep feedforward and recurrent neural networks have become successful functional models of the brain, but they neglect obvious biological details such as spikes and Dale's law. Here we argue that these details are crucial in order to understand how real neural circuits operate. Towards this aim, we put forth a new framework for spike-based computation in low-rank excitatory-inhibitory spiking networks. By considering populations with rank-1 connectivity, we cast each neuron's spiking threshold as a boundary in a low-dimensional input-output space. We then show how the combined thresholds of a population of inhibitory neurons form a stable boundary in this space, and those of a population of excitatory neurons form an unstable boundary. Combining the two boundaries results in a rank-2 excitatory-inhibitory (EI) network with inhibition-stabilized dynamics at the intersection of the two boundaries. The computation of the resulting networks can be understood as the difference of two convex functions and is thereby capable of approximating arbitrary non-linear input-output mappings. We demonstrate several properties of these networks, including noise suppression and amplification, irregular activity and synaptic balance, as well as how they relate to rate network dynamics in the limit that the boundary becomes soft. Finally, while our work focuses on small networks (5-50 neurons), we discuss potential avenues for scaling up to much larger networks. Overall, our work proposes a new perspective on spiking networks that may serve as a starting point for a mechanistic understanding of biological spike-based computation.

Convergence analysis of the maximum principle preserving BDF2 scheme with variable time-steps for the space fractional Allen–Cahn equation

In this work, a fully implicit discrete scheme preserving the maximum principle is analyzed for solving the space fractional Allen–Cahn equation with variable step BDF2 in time. The convergence analysis in L∞ norm is rigorously proved under the same requirement of time-step ratios for the maximum principle by means of the kernels recombination technique. Furthermore, the unique solvability of the proposed scheme is theoretically guaranteed depending on the property of convex functions. The energy decay law with the consideration of Riesz fractional derivative is also proved under sufficient restrictions on time-steps and time-step ratios. Numerical experiments are given to validate the second order accuracy in time, the maximum principle as well as the energy decay law of the proposed numerical scheme. In particular, comparisons of errors and time levels for the adaptive time-stepping strategy and uniform time steps are conducted to demonstrate the efficiency of adaptive time-stepping strategy.

Quantum simpson like type inequalities for q-differentiable convex functions

Quantum simpson like type inequalities for q-differentiable convex functions

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

Altered Foreign Body Response at the Posterior Surface Compared to the Anterior Surface of Human Silicone Breast Implants.

Soft implantable devices are crucial to optimizing form and function for many patients. However, periprosthetic capsule fibrosis is one of the major challenges limiting the use of implants. Currently, little is understood about how spatial and temporal factors influence capsule physiology and how the local capsule environment affects the implant structure. In this work, we analyzed breast implant capsule specimens with staining, immunohistochemistry, and real-time polymerase chain reaction to investigate spatiotemporal differences in inflammation and fibrosis. We demonstrated that in comparison to the anterior capsule against the convex surface of breast implants, the posterior capsule against the flat surface of the breast implant displays several features of a dysregulated foreign body reaction including increased capsule thickness, abnormal extracellular remodeling, and infiltration of macrophages. Furthermore, the expression of pro-inflammatory cytokines increased in the posterior capsule across the lifespan of the device, but not in the anterior capsule. We also analyzed the surface oxidation of breast explant samples with XPS analysis. No significant differences in surface oxidation were identified either spatially or temporally. Collectively, our results support spatiotemporal heterogeneity in inflammation and fibrosis within the breast implant capsule. These findings presented here provide a more detailed picture of the complexity of the foreign body reaction surrounding implants destined for human use and could lead to key research avenues and clinical applications to treat periprosthetic fibrosis and improve device longevity.

Extension of convex functions from a hyperplane to a half-space

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} has an extension to a convex function on the half-space Rn×[0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n\ imes [0,\\infty )$$\\end{document} which is finite and smooth on the open half-space Rn×(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n\ imes (0,\\infty )$$\\end{document}. The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy density ψ(Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (Dy)$$\\end{document} is best expressed when ψ(A)→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (A)\\rightarrow \\infty $$\\end{document} as detA→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\det A\\rightarrow 0+$$\\end{document}.

Strong geometric derivatives

For a real valued function f, defined on an open interval I, and an arbitrary real number h, we consider the lower and upper limits of2n(f(y+h2n)−f(y)) whenever n tends to infinity and y tends to a fixed element x of I. We consider two families of functions determined by the properties of these limits. The first interesting property is when these lower and upper limits are finite and equal to each other for every real number h and every x in I. The second notable family is determined by the property that both limits are finite and increasing (in a specific sense) with respect to x for every positive number h. These properties are motivated by the families of continuously differentiable functions and convex functions, respectively. However, restrictions of additive mappings belong to these classes as well. Our decomposition theorems establish that these motivating examples generate the whole classes. Namely, every function belonging to the first family can be represented as the sum of a continuously differentiable function and an additive one, while every function taken from the second family turns out to be the sum of a convex function and an additive one. We apply our results in order to give a local and approximate characterization of affine functions and Wright-convex functions, respectively.

Further Fractional Hadamard Integral Inequalities Utilizing Extended Convex Functions

This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions.

General decay of solutions for a viscoelastic porous system with Kelvin-Voigt damping

This study investigates the asymptotic stability of a viscoelastic porous system with Kelvin-Voigt damping. The system under consideration involves coupled equations describing the displacement of the solid elastic material, the volume fraction, and a viscoelastic or memory term. Previous studies have shown that introducing control to one equation of these systems can lead to uniform stability for weak solutions when the wave speeds are equal. However, when the wave propagation speeds differ, the decay rate of classical solutions becomes slower. In this work, we improve previous findings by considering the viscoelastic porous elasticity system with Kelvin-Voigt damping applied to the volume fraction equation. By employing the multiplier method in conjunction with the characteristics of convex functions, we establish a general stability estimate independent of the wave speeds. Our results provide valuable insights into the stability properties of viscoelastic porous systems. This study contributes to the existing literature on the subject and has implications for the analysis and control of such systems in various engineering applications.

A collective neurodynamic approach to distributed resource allocation with event-triggered communication

To solve a distributed optimal resource allocation problem, a collective neurodynamic approach based on recurrent neural networks (RNNs) is proposed in this paper. Multiple RNNs cooperatively solve a global constrained optimization problem in which the objective function is a total of local non-smooth convex functions and is subject to local convex sets and a global equality constraint. Different from the projection dynamics to deal with local convex sets in the existing work, an internal dynamics with projection output is designed in the algorithm to relax the Slater’s condition satisfied by the optimal solution. To overcome continuous-time communication in a group of RNNs, an aperiodic communication scheme, called the event-triggered scheme, is presented to alleviate communication burden. It is analyzed that the convergence of the designed collective neurodynamic approach based on the event-triggered communication does not rely on global information. Furthermore, it is proved the freeness of the Zeno behavior in the event-triggered scheme. Two examples are presented to illustrate the obtained results

Probability rate optimization of positive Markov jump linear systems via DC programming

AbstractWe investigate the stabilization problem of positive Markov jump linear systems by optimizing their transition probability rates. By using the convex property of posynomials and the standard mathematical programming that deals with the difference in convex functions, we show that transition probability rate synthesis problems can be solved via difference‐of‐convex (DC) programming. A numerical example is used to illustrate the effectiveness of our results.

On stability and H∞ control synthesis of sampled‐data systems: A multiple convex function approximation approach

AbstractThis article deals with the stability and control synthesis for a class of linear sampled‐data systems. First, a novel integral inequality with the cubic term of integral interval length instead of reciprocal one of integral interval length is constructed by non‐orthogonal polynomials. This can make full use of slack matrix variables and additional information about sawtooth structural sampling pattern. Then, on the basis of the constructed integral inequality, a cubic‐term‐dependent discontinuous exponent Lyapunov–Krasovskii Functional (LKF) is developed containing additional information about system states to design an sampled‐data state feedback controller. To better estimate the upper bound of the derivative of LKF, a kind of multiple convex function approximation approach is developed, which can be used to deal with the two‐variable polynomial negative definite issue by dividing the variable interval into multiple subintervals. By the constructed integral inequality and multiple convex function approximation approach, sufficient conditions with less conservatism are deduced for the feasibility of sampled‐data state feedback controller. Moreover, by the inner convex approximation solution technique and proposed iterative algorithm, the bilinear matrix inequalities can be transformed into linear matrix ones, which can be easily tackled, to obtain the desired sampled‐data controller gain. Finally, by comparing with existing results, the advantages and superiority of our proposed approaches can be confirmed via two numerical examples.

Global convergence analysis of Caputo fractional Whittaker method with real world applications

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

A Gradient Complexity Analysis for Minimizing the Sum of Strongly Convex Functions with Varying Condition Numbers

A Gradient Complexity Analysis for Minimizing the Sum of Strongly Convex Functions with Varying Condition Numbers

A comprehensive study on Milne-type inequalities with tempered fractional integrals

AbstractIn the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.

The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values −∞˜ and +∞˜ at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional g-derivative for the extended fuzzy-valued convex function.

A PDMS-encapsulated cylindrical non-closed-packed photonic crystals composite with Bragg-enhanced Fresnel reflectance for optical gain and spectral selection

According to the Fresnel theory, the reflectivity intensity of spherical and cylindrical convex surfaces decreases from their edge to center, and it is noteworthy and interesting for optical gain to study the enhancement of center reflectance. In this paper, a polydimethylsiloxane (PDMS) − encapsulated cylindrical non-closed-packed photonic crystals (NPCs) composite with Bragg-enhanced Fresnel reflectance was designed for spectral selectivity and optical gain. Theoretically and experimentally, the periodically ordered structure of NPCs achieved high-reflection of light in photonic bandgap and high-transmission in other bands, which enhanced Fresnel reflectivity of the convex center to specific bands. Furtherly, the cylindrical NPCs hydrogel with stretchability was applied for the dynamic tuning of optical signals. The reflection peak of the PDMS-encapsulated cylindrical NPCs composite blue-shifted from 608 nm to 413 nm with 50 % tensile strain and achieved a rapid transition of structural color from orange to blue-violet in 60 cycles. The new kind of photonic crystals composite for optical gain and spectral selection broke through the limitations of traditional Fresnel curved mirrors with the lowest central reflectivity and inability to perform spectral selectivity, and have great significance and application prospects in fields of signal transmission, optical measurement, and instrument design.