Convex Functions Research Articles

Diffractive optical elements on non-planar surfaces for the formation of color 3D images

This paper is dedicated to the development of synthesis methods for diffractive optical elements (DOEs) on non-planar surfaces. To form 3D images, a set of color 2D frames is used, visible to an observer from different observation points. The developed DOEs can be called a stereogram with a 360-deg view when the DOE is illuminated with white light. Methods are developed for computing the phase function of the DOE and are applicable to a wide range of convex surfaces. However, the manufacturing technology used limits this range to conical and cylindrical surfaces. The microrelief of the flat DOE is produced using electron-beam lithography. A video recorded using optical elements produced demonstrates the capabilities of the technologies developed. These DOEs can be used to protect brands against counterfeit.

Filterless Near‐Infrared Narrowband Photodetectors Based on Mixed Metal Perovskite Single Crystals

AbstractIodine‐based Sn‐Pb mixed perovskite single crystals (SCs) are promising candidates for near‐infrared (NIR) narrowband photodetectors due to their low bandgaps. However, they are highly defective when grown in ambient air due to the extremely poor air stability of Sn2+ ions in precursor solutions. It is discovered that the oxidation of Sn2+ to Sn4+ ions not only consumes Sn2+ ions but also increases the precursor solution viscosity. The increased viscosity slows down the solute diffusion and hinders the anisotropic growth of SCs, leading to Sn‐Pb perovskite SCs with curved convex surfaces and unevenly distributed composition. By preventing the solute‐diffusion‐limited growth mode with hypophosphorous acid (H3PO2), high‐quality Sn‐Pb perovskite SCs can be robustly grown in ambient air with low trap density of 6.58 × 109 cm−3 and high carrier mobility‐lifetime product (µτ) of 1.5 × 10−3 cm2 V−1. Accordingly, a series of filterless NIR narrowband photodetectors with gradually tuned detection centers and dual detection modes has been achieved, delivering a maximum external quantum efficiency of 15.2% and detectivity of 1.19 × 1010 Jones.

Descent modulus and applications

The norm of the gradient ‖∇f(x)‖ measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0,∂f(x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f|(x). In this work we propose an axiomatic definition of descent modulus T[f](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

We prove that the image of an isometric embedding into R3 of a two dimensional complete Riemannian manifold (Σ,g) without boundary is a convex surface, provided that, first, both the embedding and the metric g enjoy a C1,α regularity for some α>2/3, and second, the distributional Gaussian curvature of g is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.

A Minimax-Program-Based Approach for Robust Fractional Multi-Objective Optimization

In this paper, by making use of some advanced tools from variational analysis and generalized differentiation, we establish necessary optimality conditions for a class of robust fractional minimax programming problems. Sufficient optimality conditions for the considered problem are also obtained by means of generalized convex functions. Additionally, we formulate a dual problem to the primal one and examine duality relations between them. In our results, by using the obtained results, we obtain necessary and sufficient optimality conditions for a class of robust fractional multi-objective optimization problems.

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Some Classical Inequalities Associated with Generic Identity and Applications

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ.

Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals and (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} derivatives, respectively. Then we investigate some implicit integral inequalities for (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}-convex functions defined on the non-negative part of the real line.

New Quantum Estimates for Midpoint and Trapezoid Type Inequalities Through (α,m)-Convex Functions with Applications

The main goal of current investigation is to present two new q-integral identities for midpoint and trapezoid type inequalities. Then using these identities, we develop several new quantum estimates for midpoint and trapezoid type inequalities via (α, m)-convexity. Some special cases of these new inequalities can be turned into quantum midpoint and quantum trapezoid type inequalities for convex functions, classical midpoint and trapezoid type inequalities for convex functions without having to prove each one separately. Finally, we discuss how the special means can be used to address newly discovered inequalities. 2010 Mathematics Subject Classification. 26D10, 26D15, 26B25.

Legendre-Fenchel duality in m-convexity

We show that m-convex function in n-dimensional can be viewed as the Legendre-Fenchel dual problem. Particularly, we have prove this for certain convex functions.

Fractional Newton‐type integral inequalities by means of various function classes

The authors of the paper present a method to examine some Newton‐type inequalities for various function classes using Riemann‐Liouville fractional integrals. Namely, some fractional Newton‐type inequalities are established by using convex functions. In addition, several fractional Newton‐type inequalities are proved by using bounded functions by fractional integrals. Moreover, we construct some fractional Newton‐type inequalities for Lipschitzian functions. Furthermore, several Newton‐type inequalities are acquired by fractional integrals of bounded variation. Finally, we provide our results by using special cases of obtained theorems and examples.

An experimental and numerical study of the influence of the additive manufacturing process in packing properties of particles: the printed shape matters

Investigations into the various properties of granular matter composed of particles with defined shapes have gained increasing attention. Additive manufacturing, with its freedom of shape and rapid prototyping capabilities, has significantly contributed to these studies. However, this technique may introduce defects in the manufactured particles, which can significantly affect the properties of granular materials. The extent of these defects on particles of different shapes is investigated here. Particles of various shapes (cube, octahedron, quatropod, stellated octahedron, tetrahedron, and tetrapod) were manufactured and subsequently imaged using micro-Computed Tomography. The surface roughness, solidity, and convexity of the particles were quantified. Discrete element simulations of granular bed porosity, utilizing both idealized and real particle shapes, were conducted with different surface mesh resolutions and frictional parameters. A clear influence of the manufacturing process on the packing properties of 3D printed particles was identified. This influence is not uniform across all shapes and is directly correlated with the particle convexity. For numerical simulations, a shape-dependent correction of particle density and surface characteristics are imperative for each shape under consideration, despite the fact that the particles were manufactured using the same technique and material.Graphic abstract

The GroupMax Neural Network Approximation of Convex Functions.

We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts which is, for example, a necessary feature to approximate Bellman values when solving linear stochastic optimization problems. The network can be easily adapted to partial convexity. We give an universal approximation theorem in the full convex case and give many numerical results proving its efficiency. The network is competitive with the most efficient convexity-preserving neural networks and can be used to approximate functions in high dimensions.

Impact dynamics of droplets on convex structures: an experimental study with a maximum spreading diameter model for convex surface impacts

Impact dynamics of droplets on convex structures: an experimental study with a maximum spreading diameter model for convex surface impacts

On multiparametrized integral inequalities via generalized α‐convexity on fractal set

This article explores integral inequalities within the framework of local fractional calculus, focusing on the class of generalized ‐convex functions. It introduces a novel extension of the Hermite‐Hadamard inequality and derives numerous fractal inequalities through a novel multiparameterized identity. The primary aim is to generalize existing inequalities, highlighting that previously established results can be obtained by setting specific parameters within the main inequalities. The validity of the derived results is demonstrated through an illustrative example, accompanied by 2D and 3D graphical representations. Lastly, the paper discusses potential practical applications of these findings.

Analyzing the chemical environment of bent bamboo (Phyllostachys iridescens) sections using XRD, XPS, and FTIR analysis

ABSTRACT Bamboo exhibits remarkable flexibility and bent shape stability in thermal-bending. To explore how bamboo responds to bending, this study investigated the chemical properties on the concave, middle and convex walls (marked as CW, MW, VW respectively) of bent bamboo Phyllostachys iridescens though XRD, XPS, and FTIR analyses. Results indicate a reduction in crystallinity from 42% of MW to 39% of CW and 40% of VW. Notably, the peak shifts of CW and VW indicates the neutral layer on CW and VW experience an offset toward the convex surface. Chemical degradation varies on the thermal bent bamboo sections, the O/C ratio increases from 0.26 of MW to 0.36 of CW, accompanied a decrease in the Cox/unox ratio from 0.93 of MW to 0.81 of VW. This indicates that VW undergoes more significant degradation of cellulose and hemicelluloses, while CW experiences a greater loss of extracts. Additionally, FTIR analysis indicates that the ester bond peak at 1731 cm−1 flattens on CW, while it splits into two peaks on VW, with one of the peaks shifting to a higher wavenumber of 1735 cm−1. This suggests a reduction in the intermolecular distance between lignin and hemicelluloses on CW, whereas a higher lignin content and compression stress on VW.

Compensation method of the spiral bevel gear tooth surface error based on eight tooth-surface fitting

Abstract The misalignment error between the gear theoretical coordinate system and actual gear coordinate system is illustrated, as well as its influence on the measurement of tooth surface deviation is discussed. On the analysis of the tooth convex and concave surface error measurement process, the measured points in the theoretical coordinate system are transformed from the actual measured points in the actual coordinate system. Then, an eight tooth-surface fitting method is utilized to calculate the misalignment error and the compensation method is given. Finally, actual measurements of three different spiral bevel gears are analyzed, and the misalignment error are calculated. Using the proposed compensation method, both the eccentricity and tilt are compensated to some extent. And the machining suggestions of each spiral bevel gear are also given according to its compensated tooth surface error distribution.

Liouville type theorems for generalized P-harmonic maps

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature.

New Results on Simpson-Type Inequalities for Co-ordinated Convex Function by Generalized Conformable Integrals

In this paper, firstly, we prove an identity involving generalized conformable integrals. By using this obtained identity, we establish several Simpson-type inequalities for differentiable co-ordinated convex function by means of generalized conformable fractional integrals. More precisely, some Simpson-type inequalities are obtained using the well-known Hölder and Power mean inequalities. Furthermore, we also consider some special cases which can be deduced from the main results.