Convex Functions Research Articles

Aspidistra leipingensis (Asparagaceae), a new species from Guangxi, China

Aspidistra leipingensis, a new species of Asparagaceae from Guangxi, southern China, is described and illustrated. The new species is morphologically similar to A. punctatoides, but can be distinguished by its shorter leaf blade and petiole, longer perianth tube, stamens inserted in the lower part of the perianth tube, stigma hemispherical shape, upper surface convex and glabrous. Colour plate and line drawing of the new species are provided.

Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

AbstractThis paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a $$O\left( 1/k^2 \right) $$ O 1 / k 2 convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms.

Symmetric midpoint type inequalities for (α, m)-Convex Functions with Applications

Symmetric midpoint type inequalities for (α, <i>m</i>)-Convex Functions with Applications

On the Redundancy of Hessian Non-Singularity for Linear Convergence Rate of the Newton Method Applied to the Minimization of Convex Functions

A new property of convex functions that makes it possible to achieve the linear rate of convergence of the Newton method during the minimization process is established. Namely, it is proved that, even in the case of singularity of the Hessian at the solution, the Newtonian system is solvable in the vicinity of the minimizer; i.e., the gradient of the objective function belongs to the image of the matrix of second derivatives and, therefore, analogs of the Newton method may be used.

Maximal operators along flat curves with one variable vector field

Abstract We study a maximal average along a family of curves $\{(t,m(x_1)\gamma(t)):t\in [-r,r]\}$ , where $\gamma|_{[0,\infty)}$ is a convex function and m is a measurable function. Under the assumption of the doubling property of $\gamma'$ and $1\leqslant m(x_1)\leqslant 2$ , we prove the $L^p(\mathbb{R}^2)$ boundedness of the maximal average. As a corollary, we obtain the pointwise convergence of the average in r &gt; 0 without any size assumption for a measurable m.

A new helcionelloid mollusc from the Cambrian of Green land and Idaho (Laurentia)

Ressericonella gen. nov., a new genus of helcionelloid mollusc, characterized by a narrow shell with a shallowly convex dorsal surface and flat to shallowly concave lateral areas, is described from the Cambrian of Greenland and Idaho. The type species is Helcionella aequa Resser, 1939 from the Langston Formation (Naomi Peak Limestone Member) of early middle Cambrian age (Miaolingian Series, Wuliuan Stage, Albertella Biozone). Ressericonella pipalukae gen. et sp. nov. occurs in the latest early Cambrian (Cambrian Series 2, Stage 4, Bonnia–Pagetides elegans Biozone) of southern Freuchen Land, North Greenland. Silicified specimens of Ressericonella pipalukae from North Greenland preserve two-layered shell structure in which an inner layer with a transverse fibrous pattern similar to lamello-fibrillar structure is overlain by an outer layer with fine threads radiating from the apex.

Preparation of Golf Ball-like Polymer Particles Bearing Site-Selective Functional Groups.

Golf ball-like particles have emerged as interesting materials in the fields of therapeutics, diagnostics, chemical sensing, etc. because of their unique shapes (with concave and convex moieties) and physical properties. Despite their uniqueness, their surfaces are generally covered with single components lacking functional groups, and a well-established functionalization method for a specific area of these particles, such as their concave or convex surface, is lacking. Thus, we proposed a seeded-dispersion polymerization (SDP)-based method for preparing anionic- and cationic-group-bearing golf ball-like polymer particles mainly on their convex surface. Briefly, we prepared golf ball-like particles with site-selective functional groups by subjecting n-butyl methacrylate (nBMA) to SDP in the presence of polystyrene-seed particles and dodecane using 2,2'-azobis(isobutyronitrile) as an initiator; this yielded a poly(nBMA), PnBMA, concave phase. The shape of the golf ball-like particles was also controlled by adjusting the nBMA amount and solvent composition. Further, the results revealed that the functional groups were introduced onto the convex surface of the golf ball-like particles, and the position of the ionic groups on the particles was confirmed based on metal-nanoparticle adsorption. Thus, we demonstrated the feasibility of preparing convex surfaces of golf ball-like particles to impart such surfaces with anionic or cationic functional groups, thereby expanding the application scope of these heteromorphic particles.

On variants of a load-balancing problem with unit-load jobs

AbstractIn this paper, we reconsider an offline load-balancing problem with unit-time jobs that require one unit from a common resource throughout their execution. In the unit-time case, the jobs have to be assigned to time-slots such that a separable convex function of the load of the resource has to be minimized. Variants of this problem have been studied extensively in the literature under different names. We briefly discuss these problems and give a new implementation for one of them with a better worst-case time complexity than any of the known methods. We also consider the more general preemptive problem in which the execution of the jobs can be interrupted and resumed later. Furthermore, we divide the time horizon into disjoint time intervals, and for each interval, a separable convex cost function is given. The jobs have to be scheduled within their feasible intervals preemptively such that the total cost is minimized, where the cost is determined separately for each interval by the corresponding cost function. We show how to solve this problem in polynomial time by a single minimum-cost-flow computation. For the preemptive problem with one cost function only, we propose a proprietary algorithm for finding a feasible solution which is optimal for any convex cost function. We also present some qualitative computational results.

Sharp inequalities for a class of novel convex functions associated with Gregory polynomials

This paper explores the class CG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{C}_{G}$\\end{document}, consisting of functions g that satisfy a specific subordination relationship with Gregory coefficients in the open unit disk E. By applying certain conditions to related coefficient functionals, we establish sharp estimates for the first five coefficients of these functions. Additionally, we derive bounds for the second and third Hankel determinants of functions in CG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{C}_{G}$\\end{document}, providing further insight into the class’s properties. Our study also investigates the logarithmic coefficients of log(g(t)t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\log \\left ( \\frac{g(t)}{t}\\right ) $\\end{document} and the inverse coefficients of the inverse functions (g−1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(g^{-1})$\\end{document} within the same class.

Efficient Convex Optimization Requires Superlinear Memory

We show that any memory-constrained, first-order algorithm which minimizes d -dimensional, 1-Lipschitz convex functions over the unit ball to 1/poly( d ) accuracy using at most d 1.25 - δ bits of memory must make at least \(\tilde{\Omega }(d^{1 + (4/3)\delta })\) first-order queries (for any constant \(\delta \in [0, 1/4]\) ). Consequently, the performance of such memory-constrained algorithms are at least a polynomial factor worse than the optimal Õ( d ) query bound for this problem obtained by cutting plane methods that use Õ(d 2 ) memory. This resolves one of the open problems in the COLT 2019 open problem publication of Woodworth and Srebro.

Novel inequalities involving exponentially (α, m)-convex function and fractional integrals

In this paper, an identity involving Riemann–Liouville fractional integrals is newly produced. Some new extensions of integral inequalities are proved with the help of Hölder, power-mean integral inequalities and newly established identity. Whereas, the absolute values of first derivatives of real valued functions, which are appeared in these inequalities, are exponentially ( α , m ) -convex. Some special cases are discussed and graphical evidences are presented for validity of proved results.

Accelerated-gradient-based generalized Levenberg–Marquardt method with oracle complexity bound and local quadratic convergence

AbstractMinimizing the sum of a convex function and a composite function appears in various fields. The generalized Levenberg–Marquardt (LM) method, also known as the prox-linear method, has been developed for such optimization problems. The method iteratively solves strongly convex subproblems with a damping term. This study proposes a new generalized LM method for solving the problem with a smooth composite function. The method enjoys three theoretical guarantees: iteration complexity bound, oracle complexity bound, and local convergence under a Hölderian growth condition. The local convergence results include local quadratic convergence under the quadratic growth condition; this is the first to extend the classical result for least-squares problems to a general smooth composite function. In addition, this is the first LM method with both an oracle complexity bound and local quadratic convergence under standard assumptions. These results are achieved by carefully controlling the damping parameter and solving the subproblems by the accelerated proximal gradient method equipped with a particular termination condition. Experimental results show that the proposed method performs well in practice for several instances, including classification with a neural network and nonnegative matrix factorization.

Subclasses of Starlike and Convex Functions Involving Pascal Distribution Series

Subclasses of Starlike and Convex Functions Involving Pascal Distribution Series

Existence and stability of a weakly damped laminated beam with a nonlinear delay

AbstractA weakly damped laminated beam system with nonlinear time delay is studied. The existence and uniqueness are proved by Faedo–Galerkin approach. We prove that the system is stable under some specific conditions on the weight of the delay and the equal wave speeds of propagation. The general energy decay rate is established by using multiplier method and some properties of convex functions. This decay result is obtained without imposing any restrictive growth assumption on the damping term at the origin. In addition, our result improves and develops some existing results in the literature.

Thermo-hydro-mechanical modelling of the heterogeneous subsidence and swelling in the desiccation cracked clayey strata

Soil desiccation cracking as a consequence of severe environmental changes alters soil deformation mechanisms significantly. Therefore, this study aims to explore the effect of crack characteristics and environmental conditions on the heterogeneous deformation of desiccation-cracked soils using thermo-hydro-mechanical analyses. The model framework consists of balance equations, thermal, hydraulic, and mechanical constitutive equations, while the model scenarios were determined based on statistical analyses. The meteorological record of Qom city was used for three years, from 2015 to 2017, to capture long-term behaviour under wetting-drying cycles. Findings revealed that cracks extend the deformation range, potentially up to six times, with variation based on crack dimensions and spacing. Notably, narrower cracks experienced more pronounced deformation than wider ones. The cracked soil with a crack depth of 2.5 m showed 1.5 times higher swelling and subsidence than crack depth of 1 m. Furthermore, the wider cracks indicated a lower rate of increase in their dimensions compared to the initial state during drying. The investigation also highlights the mechanisms of soil surface shape due to swelling and shrinkage, resulting in concave and convex surfaces, respectively. The results provide new perspectives on the behaviour of fine-grained deposits in arid to semi-arid climates with deep groundwater levels.

Generalization of some integral inequalities in multiplicative calculus with their computational analysis

UDC 517.9 We focus on generalizing some multiplicative integral inequalities for twice differentiable functions. First, we derive a multiplicative integral identity for multiplicatively twice differentiable functions. Then, with the help of the integral identity, we prove a family of integral inequalities, such as Simpson, Hermite–Hadamard, midpoint, trapezoid, and Bullen types inequalities for multiplicatively convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities to prove the validity of the results for multiplicatively convex functions. The generalized forms obtained in our research offer valuable tools for researchers in various fields.

Bounds in Normed Spaces Using Convex Functions

Bounds in Normed Spaces Using Convex Functions

Vapor Nucleation on Hybrid-Wetting Nanoconvex Surfaces: The Competition between Intrinsic Wettability and Topography.

Hybrid-wetting surfaces with hydrophilic spots reduced from the micrometer to nanometer scale have been confirmed to enhance vapor nucleation while simultaneously minimizing droplet pinning. Given that surface topography also plays a critical role in influencing nucleation characteristics, the effect of competition between intrinsic wettability and topography on nucleation remains unclear when both surface topography and hydrophilic regions approach the critical nucleation size. This work investigated vapor nucleation on two types of hybrid-wetting nanoconvex surfaces. On random hybrid-wetting convex surfaces, the most negative potential energy sites were located at the sides of the convex structures, leading vapor to preferentially nucleate at these locations, consistent with observations on homogeneous surfaces. Despite similar average potential energy values across the surface, wettability variations in hydrophilic and hydrophobic atoms significantly alter the surface energy distribution. As the wettability difference between hydrophilic and hydrophobic atoms increases, stronger hydrophilic atoms generate relatively higher local energy regions, promoting vapor rapid nucleation. The edge effect still exists at a hydrophilic atom ratio of 10%, and competition among hydrophilic spots impedes vapor nucleation and growth. However, when the ratio increases to 40%, the increased surface average potential energy promotes the probability of vapor contacting the surface, leading to rapid vapor nucleation on the sides of the convex structures. In addition, surface potential energy analysis and the Monte Carlo method revealed that nucleation locations on nanoconvex surfaces are governed by the competition between intrinsic wettability and topography. When the magnitude of the potential energy generated by the hydrophilic atoms exceeds that from the topography, stronger solid-liquid interactions at the top of the convex structure increase the likelihood of vapor contacting the surface, resulting in nucleation at the top. Conversely, when the magnitude of the potential energy generated by hydrophilic atoms is lower than that from topography, nucleation preferentially still occurs on the sides.

On q-Hermite–Hadamard–Mercer and midpoint-Mercer inequalities for general convex functions with their computational analysis

This paper establishes some new inequalities of Hermite–Hadamard–Mercer type for [Formula: see text]-convex functions in the framework of [Formula: see text]-calculus and classical calculus. Some new [Formula: see text]-midpoint-Mercer type inequalities for the [Formula: see text]-differentiable [Formula: see text]-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and [Formula: see text]-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for [Formula: see text]-convex functions in [Formula: see text]-calculus.

An infinite family of chiral, non-slice, non-alternating hyperbolic knots with vanishing Upsilon invariants

For a knot in the 3-sphere, its Upsilon invariant is a continuous, piecewise linear function defined on the interval [0,2]. It is a concordance invariant, so vanishes for slice knots. Since it only changes the sign for the mirror image with reversed orientation, it also vanishes for amphicheiral knots. For alternating knots, more generally, quasi-alternating knots, the Upsilon invariant is determined only by the signature. In particular, it vanishes if such a knot has zero signature. On the other hand, it is known that the Upsilon invariant is a non-zero convex function for L-space knots. In this paper, we construct an infinite family of hyperbolic knots, each of which has zero Upsilon invariant, but is chiral, non-slice, non-alternating. To confirm that the Upsilon invariant vanishes, we calculate the full knot Floer complex.