Convex Functions Research Articles

Starlike and Convex functions associated with Hypergeometric Matrix functions

Starlike and Convex functions associated with Hypergeometric Matrix functions

Development of a Highly Adaptive Miniature Piezoelectric Robot Inspired by Earthworms.

Miniature resonant piezoelectric robots have the advantages of compact structure, fast response, high speed, and easy control, which have attracted the interest of many scholars in recent years. However, piezoelectric robots usually suffer from the problem of poor adaptability due to the micron-level amplitude at the feet. Inspired by the fact that earthworms have actuation trajectories all around their bodies to move flexibly under the ground, a miniature piezoelectric robot with circumferentially arranged driving feet to improve adaptability is proposed. Notably, a longitudinal-vibration-compound actuation principle with multilegged collaboration is designed to achieve the actuation trajectories around the robot, similar to the earthworms. The structure and operating principle are simulated by the finite element method, and the prototype is fabricated. The robot weighs 22.7g and has dimensions of 35.5×36.5×47mm3. The robot is tethered to an ultrasonic power supply, and the experimental results show that the speed reaches 179.35mms-1 under an exciting signal with a frequency of 58.5kHz and a voltage of 200Vp-p. High adaptability is achieved by the proposed robot, it can move on flat, fold, concave, and convex surfaces, and even in an inclined or rotating tube.

Direct-write printed controllable concave surface for fabricating flexible metal electrode

Structured surface of flexible substrate could generate a positive impact on flexibility of metal electrode due to the surface free energy relaxation, which could make the flexibility of metal electrode meet the performance requirement of flexible electronics. However, the preparation of controllable structured surface of flexible substrate is still a tough and challenge process. In this research work, controllable concave surface of flexible substrate was prepared by direct-write print with a viscoelastic substrate. When the controllable concave surface of flexible substrate was evaporated with a nanoscale metal layer, various metal electrodes could be obtained with a controllable morphology of concave surface. The direct-write printed concave surface efficiently enhanced the flexibility of metal electrodes. When the prepared concave surface generated a continuous connected structure with a 0.1 mm printing space, an excellently enhanced flexibility of metal electrode was achieved with an internal bending flexibility of 93.72%, external bending flexibility of 93.67%, stretchability of 85% and number of stable cyclic bending times over 12,000. Meanwhile, the direct-write printed concave surface generated a reversed convex surface, which could be used for pressure sensing as a flexible pressure sensor. Therefore, this research work provides an efficient method for fabricating flexible electrode, which has important research and application value in the area of high-performance flexible electronics.

Strict convexity of yield surfaces of some weakly-textured materials

Let Sym0 be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function f:Sym0→R is the sum of a texture-independent isotropic part fiso and an anisotropic part which is linear in the relevant texture coefficients. Let c>0 and S≔f−1(c)⊂Sym0 be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ∇2f(S) be positive definite for each S∈S, for a smooth yield surface S to be strictly convex in Sym0. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions f and fiso are smooth; (ii) ∇2fiso(S) is positive definite for each S in Siso≔fiso−1(c)⊂Sym0. We prove that the yield surface S⊂Sym0 of such weakly-textured material is strictly convex if the texture coefficients in f are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.

Effect of nozzle configurations and jet orientations on thermal performance of jet impingement on convex surface

The present study analyze the thermal behaviour of multiple jets impinging on a convex heated surface for two different nozzle geometries (circular and elliptical) and its orientations. Tests are performed with 5 different nozzles at different values of non-dimensional nozzle to surface distance (z/d = 2–10) and Reynolds number (5000–28000). At smaller value of non-dimensional nozzle to plate distance, a distinct pattern of temperature variation is observed that depend on the nozzle shape and orientation, and this pattern diminishes as surface to nozzle distance increases. In the farthest region, elliptical nozzle is found to improve the uniformity (up to 60 %) in Nu variation compared to the circular jets even at the largest surface to nozzle distance. The overall heat transfer is found to increase up to 18 % and the uniformity is found to enhance up to 60 %) for N-2 and N-3 nozzles. An improvement in the thermal performance is observed in the elliptical nozzle in the fountain and impingement zones. The non-uniformity in the Nu behaviour is found to increase with the increase in Reynolds number.

Sufficiency Conditions for a Class of Convex Functions Connected with Tangent Functions Associated with the Combination of Babalola Operators and Binomial Series

Sufficiency Conditions for a Class of Convex Functions Connected with Tangent Functions Associated with the Combination of Babalola Operators and Binomial Series

Hermite –Hadamard and Hermaite –Hadamard Fejer inequality for new f-divergence measure via conformable fractional integrals

In information geometry, a divergence measure is a type of statistical distance and a binary function which established the separation from one probability distribution to another on a statistical numerous. The basic use of divergence measure is statistical data processing, information storage, decision making etc. The most famous inequality regarding the integral mean of a convex function is HermiteâHadamard's inequality and the weighted version of this is called the Hermite-Hadamard-Fejér inequality. Purpose of this paper is to find Hermaite-Hadamard and Hermite-Hadamard Fejer type inequalities for new f-divergence measure with the help of conformable fractional integrals. Hermaite-Hadamard inequality gives us necessary and sufficient condition for a function must be convex. Here we consider the new f-divergence measure, it has the property of convexity. In this research article we drive some inequalities for t-convex function which gives us the extensions of the previous work for convex and t-convex function and also obtains some fractional midpoint type inequalities. The main purpose of this paper is to establish conformal fractional approximation of Hermaite-Hadamard and Hermite-Hadamard Fejer type inequalities for new f-divergence measure which close the fractional integral and the Riemann-Liouville integrable into single form,also gives us some new results for -Riemann-Liouville integral as special cases of main results. This article gives us most useful link between convexity and symmetry.

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Feasible Online Learning with Gradient Feedback Online gradient descent (OGD) is well-known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of [Formula: see text] for strongly convex cost functions, and (2) in the multiagent setting of strongly monotone games with each agent employing OGD we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of [Formula: see text]. Whereas these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In “Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback,” M. Jordan, T. Lin, and Z. Zhou design a fully adaptive OGD algorithm, AdaOGD, that does not require a priori knowledge of these parameters. In the single-agent setting, the algorithm achieves [Formula: see text] regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs AdaOGD in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of [Formula: see text], again optimal up to log factors. The algorithms are illustrated in a learning version of the classic newsvendor problem, in which, because of lost sales, only (noisy) gradient feedback can be observed. The results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multiretailer settings.

New version of midpoint-type inequalities for co-ordinated convex functions via generalized conformable integrals

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals.

The Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Convex Mapping and a Harmonic Set via Fuzzy Inclusion Relations and Their Applications in Quadrature Theory

The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (F-N-V-Ms), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard (H ⋅ H) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes.

Inclusion and Neighborhood on a Multivalent q-Symmetric Function with Poisson Distribution Operators

In this paper, by using Poisson distribution probability, some characteristics of analytic multivalent q-symmetric starlike and q-symmetric convex functions of order η are examined. Then, by utilizing the Poisson distribution and the concept of the q-analogue Salagean integral operator, the p-valent convergence polynomial was introduced. Furthermore, a number of subclasses of analytic symmetric p-valent functions linked to novel polynomials are also deduced. After that, specific coefficient constraints are determined and symmetric δ,q-neighborhoods for p-valent functions are defined. In relation to symmetric δ,q-neighborhoods of q-symmetric p-valent functions formed by Poisson distributions, this paper presents new inclusion results. In addition, a detailed discussion of certain q-symmetric inequalities of analytic functions with negative coefficients is also provided.

The Droplet Search Algorithm for Kernel Scheduling

Kernel scheduling is the problem of finding the most efficient implementation for a computational kernel. Identifying this implementation involves experimenting with the parameters of compiler optimizations, such as the size of tiling windows and unrolling factors. This article shows that it is possible to organize these parameters as points in a coordinate space. The function that maps these points to the running time of kernels, in general, will not determine a convex surface. However, this article provides empirical evidence that the origin of this surface (an unoptimized kernel) and its global optimum (the fastest kernel) reside on a convex region. We call this hypothesis the “droplet expectation.” Consequently, a search method based on the Coordinate Descent algorithm tends to find the optimal kernel configuration quickly if the hypothesis holds. This approach—called Droplet Search—has been available in Apache TVM since April of 2023. Experimental results with six large deep learning models on various computing devices (ARM, Intel, AMD, and NVIDIA) indicate that Droplet Search is not only as effective as other AutoTVM search techniques but also 2 to 10 times faster. Moreover, models generated by Droplet Search are competitive with those produced by TVM’s AutoScheduler (Ansor), despite the latter using 4 to 5 times more code transformations than AutoTVM.

Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s=1, λ=α, σ(0)=1, and w=0. In conclusion, the methodology described in this article is expected to stimulate further research in this area.

Characterizations of convex functions by level sets

AbstractIt is shown that a lsc function is convex if and only if the minimal set of any linear perturbation of the function is convex. That fact also yields that the convexity of a function is equivalent to the quasiconvexity of its all linear perturbations.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Computing subgradients of convex relaxations for solutions of parametric ordinary differential equations

A novel subgradient evaluation method is proposed for nonsmooth convex relaxations of parametric solutions of ordinary differential equations (ODEs) arising in global dynamic optimization, assuming that the relaxations always lie strictly within interval bounds during integration. We argue that this assumption is reasonable in practice. These subgradients are computed as the unique solution of an auxiliary parametric affine ODE, analogous to classical forward/tangent sensitivity evaluation methods for smooth dynamic systems. Unlike established subgradient evaluation approaches for nonsmooth dynamic systems, this new method does not require smoothness or transversality assumptions, and is compatible with existing subgradient evaluation methods for closed-form convex functions, as implemented in subgradient evaluation software such as EAGO.jl and MC ++. Moreover, we show that a subgradient for a lower-bounding problem in global dynamic optimization can be directly evaluated using reverse/adjoint sensitivity analysis, which may reduce the overall computational effort for an overarching global optimization method. Numerical examples are presented, based on a proof-of-concept implementation in Julia.

New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature.

Wild solutions to scalar Euler-Lagrange equations

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W 1 , 1 W^{1,1} solutions are necessarily W loc 1 , 2 W^{1,2}_{\operatorname {loc}} , which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl’s classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W 1 , 1 W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W 1 , 1 W^{1,1} solution to be improved to W loc 1 , 2 W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution.

On diagonally structured scheme for nonlinear least squares and data-fitting problems

Recently, structured nonlinear least-squares (NLS) based algorithms gained considerable emphasis from researchers; this attention may result from increasingly applicable areas of these algorithms in different science and engineering domains. In this article, we coined a new efficient structured-based NLS algorithm. We developed a diagonal Hessian-based formulation for solving NLS problems. We derived the quasi-Newton update based on a diagonal matrix scheme subject to a modified structured secant condition. Also, we show that the algorithm search direction satisfies a sufficient descent condition under some standard assumptions. Subsequently, we also prove the global convergence of the algorithm and then eventually show the linear convergence rate for strongly convex functions. Furthermore, we numerically experimented with the proposed algorithm on benchmark test functions available in the literature. Finally, in the scheme, we apply the method to solve some data-fitting problems.

Harmonic conformable refinements of Hermite-Hadamard Mercer inequalities by support line and related applications

ABSTRACT We establish new conformable fractional Hermite-Hadamard (H–H) Mercer type inequalities for harmonically convex functions using the concept of support line. We introduce two new conformable fractional auxiliary equalities in the Mercer sense and apply them to differentiable functions with harmonic convexity. We also use Power-mean, Hölder’s and improved Hölder inequality to derive new Mercer type inequalities via conformable fractional integrals. The accuracy and superiority of the offered technique are clearly depicted through impactful visual illustrations. We also use our technique to derive new estimates for hypergeometric functions and special means of real numbers that are more precise than existing ones. Some applications are provided as well. Our results generalize and extend some existing ones in the literature.