Convex Functions Research Articles

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.

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.

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.

REMARKS ON INEQUALITIES WITH PARAMETER BY CONFORMABLE FRACTIONAL INTEGRALS

We prove that an equation holds for differentiable convex functions, and this result has been derived using conformable integrals. With the help of this equality, several parameterized inequalities are established by using the conformable fractional integrals. Namely, we show that our main inequalities reduce to Ostrowski-, Hermite–Hadamard-, Simpson-, and Bullen-type inequalities which are proved in earlier published papers. More precisely, some inequalities are acquired by taking advantage of the convexity, the Hölder, and the power mean inequalities. Finally, examples are given to illustrate the investigated results.

On Jensen’s inequality involving divided differences of convex functions

On Jensen’s inequality involving divided differences of convex functions

The Impact of Metal Ions on MXene Membranes: Critical Role of Titanium Vacancies.

Two-dimensional transition metal carbides and nitrides (MXenes) and MXene-based membranes hold promise for applications including water purification and seawater desalination; however, their environmental behavior and fate in these matrices remain unknown. In this study, we systematically assessed the reaction efficiencies of Ti3C2Tx at varying important environmental conditions. Our experiments revealed that copper and iron ions accelerated the oxidation rate of Ti3C2Tx 55.4 and 33.4 times, respectively. TiO2 and amorphous carbon were identified as the primary solid products. Based on in situ water-phase atomic force microscopy, atomic high-angle annular dark-field scanning transmission electron microscopy, and theoretical results, we postulate that metal ions enhance Ti3C2Tx oxidation by spontaneously migrating and anchoring at Ti vacancies, which then become active sites for this reaction. This process increases the adsorption of H2O and oxygen, making the Ti vacancy-rich surface convex area the most vulnerable site to attack. The findings in this study provide useful information for a comprehensive understanding of the interaction between MXene structural defects and metal ions as well as for the design and modification of MXene membranes resistant to metal ion impact.

Hydrodynamic Performance of Half Pipe Breakwaters Experimentally and Numerically

The main objective of constructing barriers is to safeguard harbours and shores from waves and sea currents. This paper aims at presenting a study on innovative and non-traditional alternatives to breakwaters, experimentally and numerically, to assess the hydrodynamic efficiency of the suggested models in order to choose the most appropriate one. Two models of semi-submersible-type breakwaters have been studied, in the form of a cross-section of a half pipe with an internal diameter and thickness of 30.00 and 1.00 cm, respectively. Model (a) is a circular half-pipe with an upward concave, whereas model (b) has a sideways concave towards the water. Several scenarios for the breakwaters suggested in this study have been simulated using FLOW 3D. The results indicate that the transmission coefficient (Tc), provided by model (b), is 3-7% lower than model (a), while the reflection coefficient (Rc) values for model (b) are 4-8% higher than model (a). The results show that model (b) has a 10% lower (Tc) at d/h = 0.80 compared to d/h = 0.60, while (Rc) is 8% higher at d/h = 0.80 than at d/h = 0.60 for the same model. The suggested breakwater reduces inclined waves' energy more quickly than perpendicular waves. Regarding the hydrodynamic performance, the proposed breakwater in model (b) is shown to be more efficient and effective than the breakwater in model (a), so it is recommended to use it due to its effectiveness in providing protection for coastal wave zones and benefiting from the energy of waves to generate electricity.

Unified Generalizations of Hardy-Type Inequalities Through the Nabla Framework on Time Scales

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results.

A duality and free boundary approach to adverse selection

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.

Ulam’s Method for Computing Stationary Densities of Invariant Measures for Piecewise Convex Maps with Countably Infinite Number of Branches

Let [Formula: see text] be a piecewise convex map with countably infinite number of branches. In [Góra et al., 2022], the existence of Absolutely Continuous Invariant Measure (ACIM) [Formula: see text] for [Formula: see text] and the exactness of the system [Formula: see text] have been proven. In this paper, we develop an Ulam method for approximation of [Formula: see text], the density of ACIM [Formula: see text]. We construct a sequence [Formula: see text] of maps [Formula: see text] s.t. [Formula: see text] has a finite number of branches and the sequence [Formula: see text] converges to [Formula: see text] almost uniformly. Using supremum norms and Lasota–Yorke-type inequalities, we prove the existence of ACIMs [Formula: see text] for [Formula: see text] with the densities [Formula: see text]. For a fixed [Formula: see text], we apply Ulam’s method with [Formula: see text] subintervals to [Formula: see text] and compute approximations [Formula: see text] of [Formula: see text]. We prove that [Formula: see text] as [Formula: see text] both a.e. and in [Formula: see text]. We provide examples of piecewise convex maps [Formula: see text] with countably infinite number of branches and their approximations by [Formula: see text]’s with finite number of branches. For the increasing values of parameter [Formula: see text] we calculate the errors [Formula: see text].

The Hadwiger Theorem on Convex Functions, I

AbstractA complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb{R}}^{n}$ is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

On Riemann-Liouville integrals and Caputo Fractional derivatives via strongly modified (p, h)-convex functions.

The paper introduces a new class of convexity named strongly modified (p, h)-convex functions and establishes various properties of these functions, providing a comprehensive understanding of their behavior and characteristics. Additionally, the paper investigates Schur inequality and Hermite-Hadamard (H-H) inequalities for this new class of convexity. Also, H-H inequalities are proved within context of Riemann-Liouville integrals and Caputo Fractional derivatives. The efficiency and feasibility of Schur inequality and H-H inequalities are supported by incorporating multiple illustrations, that demonstrate the applicability of strongly modified (p, h)-convex functions. The results contribute to the field of mathematical analysis and provide valuable insights into the properties and applications of strongly modified (p, h)-convex functions.

On the Fekete-Szegö Inequalities of the Generalized Mittag-Leffler Function Associated with a Lambert Series

This study aims to investigate the Fekete-Szegö problem for the linear operator generated by the convolution (the Hadamard product) involving one of the generalized forms of the Mittag-Leffler function and the well-known Lambert series. The findings will mainly apply on some subclasses of starlike and convex functions.

Extension of Fejér's inequality to the class of sub-biharmonic functions

Abstract Fejér’s integral inequality is a weighted version of the Hermite-Hadamard inequality that holds for the class of convex functions. To derive his inequality, Fejér [Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390] assumed that the weight function is symmetric w.r.t. the midpoint of the interval. In this study, without assuming any symmetry condition on the weight function, Fejér’s inequality is extended to the class of sub-biharmonic functions, namely, the set of functions f ∈ C 4 ( I ) f\in {C}^{4}\left({\mathbb{I}}) satisfying f ″ ″ ≤ 0 {f}^{^{\prime\prime} ^{\prime\prime} }\le 0 , where I {\mathbb{I}} is an interval of R {\mathbb{R}} . In the special case when the weight function is symmetric w.r.t. the midpoint of some interval, and the function f f is convex and sub-biharmonic, an interesting refinement of Fejér’s inequality is deduced. Moreover, this inequality is extended to a new class of functions defined on the plane, that we call the set of sub-biharmonic functions on the coordinates. Assuming in addition that the function is convex on the coordinates, a refinement of an inequality due to Dragomir [On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), 775–788] is obtained.

Integral-Type Representations for the Subdifferential of Suprema

AbstractAn integral-like representation is provided for the $$\varepsilon $$ ε -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ f t ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.

Methane Adsorption and Transport in Tortuous Slit-like Nanochannels: A Molecular Simulation Study.

Tortuosity is a crucial characteristic of porous materials, such as the shale matrix where shale gas is stored. The presence of tortuous nanochannels significantly affects the adsorption and transport of nanoflows. In this research, we use molecular dynamics simulation (MD) to study the adsorption and transport properties of shale gas (methane) in a curved slit-like nanochannel constructed from bent graphene sheets. Our findings reveal that the curvature of the tortuous channel influences methane adsorption: convex surfaces exhibit stronger adsorption, while concave surfaces exhibit weaker adsorption; the discrepancy is amplified by the nanoflow. Additionally, nanoflow velocity is heterogeneously distributed within the curved channel, with higher tangential flow velocities observed near the entrance and the outer surface. We also identify a "bouncing effect", where the nanoflow not only moves tangentially along the channel but also bounces between the inner and outer walls. Furthermore, methane in narrower channels exhibits higher tangent flow velocity and higher bouncing frequency but smaller flux, whereas larger curvature results in shorter channel length and smaller tortuosity, but the transport tangent velocity and flux are both reduced. The findings of this study can help in the better understanding of shale gas nanoflow properties in tortuous media and provide insights for simulating more general nonstraight nanoflows.