Convex Functions Research Articles

Bernstein–Kantorovich operators, approximation and shape preserving properties

Inspired by the construction of Bernstein and Kantorovich operators, we introduce a family of positive linear operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{K}_n$$\\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.

Bounds of some divergence measures using Green’s function and Fink’s identity via Diamond Integrals

Abstract In the present paper, some inequalities regarding Csiszár divergence are generalized for diamond integrals by utilizing Green function and Fink’s identity. By selecting various kinds of convex functions, bound of different entropies and divergence measures are obtained. Furthermore, in seek of applications, bounds of different divergence measures are estimated on diverse fixed time scales. These divergence measures are fundamental in problems pertaining to source and channel coding, large deviations theory, and other sorts of problems.

Printing, Characterizing, and Assessing Transparent 3D Printed Lenses for Optical Imaging

AbstractHigh‐quality lens production has involved subtractive manufacturing methods for centuries. These methods demand specialist equipment and expertise that often render custom high‐grade glass optics inaccessible. A low‐cost, accessible, and reproducible method is developed to manufacture high‐quality three dimensional (3D) printed lenses using consumer‐grade technology. Various planoconvex lenses are produced using a consumer‐grade 3D printer and low‐cost spin coating setup, and printed lenses are compared to commercial glass counterparts. A range of mechanical and optical methods are introduced to determine the surface quality and curvature of 3D printed lenses. Amongst others, high‐resolution interference reflection microscopy methods are used to reconstruct the convex surface of printed lenses and quantify their radius of curvature. The optical throughput and performance of 3D printed lenses are assessed using optical transmissivity measurements and classical beam characterization methods. It is determined that 3D printed lenses have comparable curvature and performance to commercial glass lenses. Finally, the application of 3D printed lenses is demonstrated for brightfield transmission microscopy, resolving sub‐cellular structures over a 2.3 mm field‐of‐view. The high reproducibility and comparable performance of 3D printed lenses present great opportunities for additive manufacturing of bespoke optics for low‐cost rapid prototyping and improved accessibility to high‐quality optics in low‐resource settings.

Numerical Investigation of Air-Steam Condensation in a Vertically Enhanced Tube with Concave and Convex Surfaces

Condensation heat transfer enhancement technology is an important problem for heat exchangers. The air-steam condensation inside a vertically enhanced tube with concave and convex surfaces is analyzed numerically, and the effects of relevant parameters on thermohydraulic performance are discussed. The numerical results show that the drop in steam and temperature between the inlet and outlet is improved by 53.8% and 96.8%, respectively, compared with smooth tubes. In the condensation process, the noncondensable gas accumulates near the wall, which leads to the deterioration of condensation. The structures of concave and convex surfaces disrupt the boundary layer flow and noncondensable gas film and promote flow mixing, leading to enhanced condensation heat transfer. All the numerical results provide guidance values for heat transfer enhancement technology in the applications of condensers.

Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators A and B, both of which are positive and have spectra within the interval m,M, denoted as σA and σB. In addition, let us introduce two monotone continuous functions, namely, g and h, defined on the interval m,M. Let f be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers t and s, we have fts≤ftfs. Under these specified conditions, we establish the following inequality: for all 0≤ν≤1, this outcome highlights the intricate relationship between the numerical range of the expression gνAXh1−ν when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ν. Furthermore, we provide supportive examples to validate these results.

Semiglobal fixed/preassigned-time synchronization of stochastic neural networks with random delay via adaptive control

This paper presents two kinds of novel results about semiglobal synchronization in probability for stochastic systems. To this end, new criteria of semiglobal fixed-time synchronization in probability (SFXTSp) and semiglobal preassigned-time synchronization in probability (SPATSp) are proposed. In contrast to the existing results of semiglobal synchronization in probability, our criteria have wider applicability in SFXTSp and SPATSp due to their flexible forms. Meanwhile, based on the properties of convex functions and Jensen’s inequality, our approachs overcome the issue arising from the non-equivalence between the Lyapunov function and its expectation, making our conclusions more rigorous. In addition, to achieve SFXTSp of stochastic neural networks with random delay (RDSNNs), a unified adaptive controller is developed. By modifying the parameters of the unified controller, RDSNNs reach SPATSp in a predetermined time. Ultimately, numerical simulations are conducted to confirm the practicality and accuracy of the derived outcomes.

Successive upper approximation methods for generalized fractional programs  

The majorization approximation procedure consists in replacing the resolution of a nonlinear optimization problem by solving a sequence of simpler ones, whose objective and constraint functions upper estimate those of the original problem. For generalized fractional programming, i.e., constrained minimization programs whose objective functions are maximums of finite ratios of functions, we propose an adapted scheme that simultaneously upper approximates parametric functions formed by the objective and constraint functions. For directionally convex functions, that is, functions whose directional derivatives are convex with respect to directions, we will establish that every cluster point of the generated sequence satisfies Karush-Kuhn-Tucker type conditions expressed in terms of directional derivatives. The proposed procedure unifies several existing methods and gives rise to new ones. Numerical problems are solved to test the efficiency of our methods, and comparisons with different approaches are given.

The Optimality Conditions for Generalized Uniform η – V – I Invariant Convexity Programming

Generalized uniformly invariant convex programming is an important optimization problem that has numerous practical applications, such as transportation planning, engineering optimization, etc. The study of its optimal conditions can help solve practical problems and improve efficiency. Optimality conditions serve as the foundation of optimization theory by aiding in understanding the nature and characteristics of problems and providing guidance for the design and analysis of optimization algorithms. Therefore, studying the optimality conditions of generalized uniformly invariant convex programming is highly significant for solving practical problems, promoting the development of optimization theory and algorithm design and analysis, as well as fostering interdisciplinary research. In this research, a novel type of generalized uniform η-V-I invexity function is defined through the use of the η - subdifferential. In the case of the new generalized convex functions, a certain of multiobjective programming with this generalized convexity is discussed, and the corresponding sufficient optimality conditions are obtained.

Deformed graphene FET biosensor on textured glass coupled with dielectrophoretic trapping for ultrasensitive detection of GFAP

Numerous efforts have been undertaken to mitigate the Debye screening effect of FET biosensors for achieving higher sensitivity. There are few reports that show sub-femtomolar detection of biomolecules by FET mechanisms but they either suffer from significant background noise or lack robust control. In this aspect, deformed/crumpled graphene has been recently deployed by other researchers for various biomolecule detection like DNA, COVID-19 spike proteins and immunity markers like IL-6 at sub-femtomolar levels. However, the chemical vapor deposition (CVD) approach for graphene fabrication suffers from various surface contamination while the transfer process induces structural defects. In this paper, an alternative fabrication methodology has been proposed where glass substrate has been initially texturized by wet chemical etching through the sacrificial layer of synthesized silver nanoparticles, obtained by annealing of thin silver films leading to solid state dewetting. Graphene has been subsequently deposited by thermal reduction technique from graphene oxide solution. The resulting deformed graphene structure exhibits higher sensor response towards glial fibrillary acidic protein (GFAP) detection with respect to flat graphene owing to the combined effect of reduced Debye screening and higher surface area for receptor immobilization. Additionally, another interesting aspect of the reported work lies in the biomolecule capture by dielectrophoretic (DEP) transport on the crests of the convex surfaces of graphene in a coplanar gated topology structure which has resulted in 10 aM and 28 aM detection limits of GFAP in buffer and undiluted plasma respectively, within 15 min of application of analyte. The detection limit in buffer is almost four decades lower than that documented for GFAP using biosensors which is is expected to pave way for advancing graphene FET based sensors towards ultrasensitive point-of-care diagnosis of GFAP, a biomarker for traumatic brain injury.

Applicability of Convex and Concave Shaped Surface Texturing to Improve Damping Performance of Mechanical Tensioner for Timing Chain of Internal Combustion Engine

Applicability of Convex and Concave Shaped Surface Texturing to Improve Damping Performance of Mechanical Tensioner for Timing Chain of Internal Combustion Engine

Elucidating vaccine efficacy using a correlate of protection, demographics, and logistic regression

BackgroundVaccine efficacy (VE) assessed in a randomized controlled clinical trial can be affected by demographic, clinical, and other subject-specific characteristics evaluated as baseline covariates. Understanding the effect of covariates on efficacy is key to decisions by vaccine developers and public health authorities.MethodsThis work evaluates the impact of including correlate of protection (CoP) data in logistic regression on its performance in identifying statistically and clinically significant covariates in settings typical for a vaccine phase 3 trial. The proposed approach uses CoP data and covariate data as predictors of clinical outcome (diseased versus non-diseased) and is compared to logistic regression (without CoP data) to relate vaccination status and covariate data to clinical outcome.ResultsClinical trial simulations, in which the true relationship between CoP data and clinical outcome probability is a sigmoid function, show that use of CoP data increases the positive predictive value for detection of a covariate effect. If the true relationship is characterized by a decreasing convex function, use of CoP data does not substantially change positive or negative predictive value. In either scenario, vaccine efficacy is estimated more precisely (i.e., confidence intervals are narrower) in covariate-defined subgroups if CoP data are used, implying that using CoP data increases the ability to determine clinical significance of baseline covariate effects on efficacy.ConclusionsThis study proposes and evaluates a novel approach for assessing baseline demographic covariates potentially affecting VE. Results show that the proposed approach can sensitively and specifically identify potentially important covariates and provides a method for evaluating their likely clinical significance in terms of predicted impact on vaccine efficacy. It shows further that inclusion of CoP data can enable more precise VE estimation, thus enhancing study power and/or efficiency and providing even better information to support health policy and development decisions.

Parametrized multiplicative integral inequalities

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers’ comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus.

Preferential water vapor condensation on a corrugated surface: A molecular dynamics study

In this study, we investigate the nucleation and subsequent growth of water clusters on corrugated surfaces using molecular dynamics simulations, with a specific focus on scenarios where the nanostructure sizes are comparable to the critical nucleus size. The primary objective is to reveal the coupled effects of different surface morphologies, i.e., plane, convex, and concave, on water vapor condensation. Regardless of the nanostructures, the condensation process can be divided into two stages, the nucleation stage and the growth stage. It is shown that in the nucleation stage, the concave areas are more conducive to nucleation compared to plane areas, aligning with classical nucleation theory. The total number of condensed molecules is generally proportion to the nucleation ability, while the average size of condensation clusters is affected by the surface nanostructures, with the combination of concave and plane surfaces exhibiting the strongest capability for both water vapor nucleation and cluster growth. Compared to the plane regions, the convex regions not only inhibit nucleation but also block collision and coalescence between clusters. Additionally, increasing the characteristic width of concave and convex structures leads to condensation behaviors approaching those on the plane surface. This indicates that the regulation of the condensation process is feasible by nanostructures within prescribed structural size limits.

Counterexamples in rotundity of norms in Banach spaces

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense.In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C∞-smooth norm in c0 whose dual norm is not strictly convex.

An inexact variable metric variant of incremental aggregated Forward-Backward method for the large-scale composite optimization problem

ABSTRACT This paper focuses on the large-scale composite optimization problem, which is the sum of a finite number of smooth (possibly nonconvex) functions and a convex (possibly nonsmooth) function. A common method to solve this problem is proximal incremental aggregated gradient (PIAG) method (N. D. Vanli, M. Gurbuzbalaban and A. Ozdaglar, SIAM Journal on Optimization, 2018, 28(2): 1282–1300), which exploits the gradient information from previous iterations to approximate the gradient at the current point. However, as many first-order algorithms, it may suffer from slow convergence. To address this issue, this paper proposes an inexact variable metric variant of incremental aggregated Forward-Backward (iVMFB-IAG) method. This methods incorporates a variable metric strategy to accelerate the PIAG method and introduces an inexact criterion to enhance its practicality. Under the Kurdyka-Łojasiewicz (KL) property and the uniformly bounded positive definiteness of the scaling matrix, the sequence generated by the iVMFB-IAG method converges globally to the critical point of the problem. Additionally, the convergence rate is established under the KL framework. Moreover, under certain restrictive conditions, the method exhibits local superlinear convergence. Through the numerical experiments on image reconstruction applications, we demonstrate the effectiveness of the iVMFB-IAG method by restoring two distinct images.

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].

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