Showing papers on "Regular polygon published in 2023"
TL;DR: In this article , a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals.
Abstract: In this manuscript, we aim to establish a connection between the concept of inequalities and the novel Center-Radius order relation. The idea of a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals. Consequently, convexity and nonconvexity contribute to different kinds of inequalities. In spite of this, convex theory turns to Godunova-Levin functions because they are more efficient at determining inequality terms than other convexity classes. Our application of these new definitions has led to many classical and novel special cases that illustrate the key findings of the paper. Using total order relations between two intervals, this study introduces CR-$ (h_1, h_2) $-Goduova-Levin functions. It is clear from their properties and widespread usage that the Center-Radius order relation is an ideal tool for studying inequalities. This paper discusses various inequalities based on the Center-Radius order relation. With the CR-order relation, we can first derive Hermite-Hadamard ($ \mathcal{H.H} $) inequalities and then develop Jensen-type inequality for interval-valued functions ($ \mathcal{IVFS} $) of type $ (h_1, h_2) $-Godunova-Levin function. Furthermore, the study includes examples to support its conclusions.
7 citations
01 Jan 2023
TL;DR: In this paper , a genetic algorithm is used as a search method to find a good part-packing solution for a 3D bin packing problem with non-convex objects having cavities and holes.
Abstract: In this paper we describe a unique three-dimensional bin packing problem with non-convex objects having cavities and holes. Part packing takes place in an environment where parts float as they would in a weightless environment. The application domain is the selective laser sintering rapid prototyping technique. A genetic algorithm is used as a search method to find a good part-packing solution. The fitness evaluation for the genetic algorithm is based on the actual part geometry as described in the Stereolithography (STL) file. Part-intersection detection utilizes several methods common in computational geometry. Initial results are promising, showing that the genetic algorithm is able to find a good solution for such a difficult packing problem.
6 citations
TL;DR: In this article , a new variant of the q-Hermite-Hadamard inequality for convex functions via left and right q-integrals was established, and some new q-midpoint and q-trapezoid type inequalities were established for differentiable functions.
Abstract: Abstract In this paper, we establish a new variant of q-Hermite-Hadamard inequality for convex functions via left and right q-integrals. Moreover, we prove some new q-midpoint and q-trapezoid type inequalities for left and right q-differentiable functions. To illustrate the newly established inequalities, we give some particular examples of convex functions.
6 citations
TL;DR: In this article , the authors considered the well-known fuzzy Hermite-Hadamard (HH) type and associated inequalities, and with the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (𝑈𝒟-convexity), they increase this mileage even further.
Abstract: The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (𝑈𝒟-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower 𝑈𝒟-concave (lower 𝑈𝒟-concave) and upper 𝑈𝒟-convex (concave) fuzzy number valued mappings (ℱ𝒩𝒱ℳs), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued 𝑈𝒟-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.
5 citations
TL;DR: Zhang et al. as discussed by the authors proposed a direct optimization-based dictionary learning algorithm using the minimax concave penalty (MCP) as a sparsity regularizer that can enforce strong sparsity and obtain accurate estimation.
Abstract: Direct-optimization-based dictionary learning has attracted increasing attention for improving computational efficiency. However, the existing direct optimization scheme can only be applied to limited dictionary learning problems, and it remains an open problem to prove that the whole sequence obtained by the algorithm converges to a critical point of the objective function. In this article, we propose a novel direct-optimization-based dictionary learning algorithm using the minimax concave penalty (MCP) as a sparsity regularizer that can enforce strong sparsity and obtain accurate estimation. For solving the corresponding optimization problem, we first decompose the nonconvex MCP into two convex components. Then, we employ the difference of the convex functions algorithm and the nonconvex proximal-splitting algorithm to process the resulting subproblems. Thus, the direct optimization approach can be extended to a broader class of dictionary learning problems, even if the sparsity regularizer is nonconvex. In addition, the convergence guarantee for the proposed algorithm can be theoretically proven. Our numerical simulations demonstrate that the proposed algorithm has good convergence performances in different cases and robust dictionary-recovery capabilities. When applied to sparse approximations, the proposed approach can obtain sparser and less error estimation than the different sparsity regularizers in existing methods. In addition, the proposed algorithm has robustness in image denoising and key-frame extraction.
4 citations
TL;DR: In this paper , two new subclasses of bi-close-to-convex functions associated with Janowski functions were introduced and studied. And the Faber polynomial expansion method was used to determine the general coefficient bounds for the functions belonging to these classes.
Abstract: Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the general coefficient bounds for the functions belonging to these classes. It also finds initial coefficients of bi-close-to-convex and bi-quasi-convex functions by using Janowski functions. Some known consequences of the main results are also highlighted.
4 citations
TL;DR: In this article , the authors gave sharp bounds for the Hankel determinant H2(3) for the coefficients of functions in the class of convex functions related to the three-leaf-like domain.
4 citations
TL;DR: In this article , the authors established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes.
Abstract: It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.
4 citations
TL;DR: In this article , the authors generalize this result for self-joinings of convex cocompact groups and show that the Hausdorff dimension of the limit set of a self-joining subgroup is equal to the critical exponent of the whole subgroup.
Abstract: The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $\Gamma<\text{SO}^\circ (n,1)$, $n\ge 2$, the Hausdorff dimension of the limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $\Delta$ be a finitely generated group and $\rho_i:\Delta\to \text{SO}^\circ(n_i,1)$ be a convex cocompact faithful representation of $\Delta$ for $1\le i\le k$. Associated to $\rho=(\rho_1, \cdots, \rho_k)$, we consider the following self-joining subgroup of $\prod_{i=1}^k \text{SO}(n_i,1)$: $$\Gamma=\left(\prod_{i=1}^k\rho_i\right)(\Delta)=\{(\rho_1(g), \cdots, \rho_k(g)):g\in \Delta\} .$$ (1). Denoting by $\Lambda\subset \prod_{i=1}^k \mathbb{S}^{n_i-1}$ the limit set of $\Gamma$, we first prove that $$\text{dim}_H \Lambda=\max_{1\le i\le k} \delta_{\rho_i}$$ where $\delta_{\rho_i}$ is the critical exponent of the subgroup $\rho_{i}(\Delta)$. (2). Denoting by $\Lambda_u\subset \Lambda$ the $u$-directional limit set for each $u=(u_1, \cdots, u_k)$ in the interior of the limit cone of $\Gamma$, we obtain that for $k\le 3$, $$ \frac{\psi_\Gamma(u)}{\max_i u_i }\le \text{dim}_H \Lambda_u \le \frac{\psi_\Gamma(u)}{\min_i u_i }$$ where $\psi_\Gamma:\mathbb{R}^k\to \mathbb{R}\cup\{-\infty\}$ is the growth indicator function of $\Gamma$.
4 citations
TL;DR: In this paper , a detailed performance analysis for the kernel-based regularized pairwise learning model associated with a strongly convex loss is presented, and the robustness for the model is analyzed by applying an improved convex analysis method.
Abstract: This paper presents a detailed performance analysis for the kernel-based regularized pairwise learning model associated with a strongly convex loss. The robustness for the model is analyzed by applying an improved convex analysis method. The results show that the regularized pairwise learning model has better qualitatively robustness according to the probability measure. Some new comparison inequalities are provided, with which the convergence rates are derived. In particular an explicit learning rate is obtained in case that the loss is the least square loss.
4 citations
TL;DR: The concept of nondegeneracy was introduced by Cruz et al. as discussed by the authors , who showed that taking closures or interiors of open or closed realizations does not change the code that is realized, and gave the first general criteria for precluding a code from being closed-convex.
Abstract: Previous work on convexity of neural codes has produced codes that are open-convex but not closed-convex—or vice-versa. However, why a code is one but not the other, and how to detect such discrepancies are open questions. We tackle these questions in two ways. First, we investigate the concept of nondegeneracy introduced by Cruz et al. We extend their results to show that nondegeneracy precisely captures the situation when taking closures or interiors of open or closed realizations, respectively, does not change the code that is realized. Second, we give the first general criteria for precluding a code from being closed-convex (without ruling out open-convexity), unifying ad-hoc geometric arguments in prior works. One criterion is built on a phenomenon we call a rigid structure, while the other can be stated algebraically, in terms of the neural ideal of the code. These results complement existing criteria having the opposite purpose: precluding open-convexity but not closed-convexity. Finally, we show that a family of codes shown by Jeffs to be not open-convex is in fact closed-convex and realizable in dimension three.
TL;DR: In this article , a novel radar-communication spectrum sharing framework for transmission design is presented, which maximizes the radar signal-to-interference-plus-noise ratio under the constraints concerning the transmit energy, the radar similarity with a radar reference waveform, and the communication system rate achievable.
Abstract: In this work, we present a novel radar-communication spectrum sharing framework for transmission design. The design degrees of freedom (DoFs) under control are the radar transmit waveforms and the communication system code-book. The formulation of the spectrum sharing problem is stated as the maximization of the radar signal-to-interference-plus-noise ratio (SINR) under the constraints concerning the transmit energy, the radar similarity with a radar reference waveform, the communication system rate achievable. Compared with the traditional alternating convex transformation approach for the resulting non-convex problem, a novel algorithm is proposed based on the derived local design for radar waveform (LDFRW) to reduce the computational complexity. The numerical results show the merits of the proposed design.
TL;DR: In this paper , a single machine common due-window (denoted by CONW) assignment scheduling problem with position-dependent weights and resource allocations was studied under just-in-time production, and it was shown that the weighted sum of scheduling cost and resource consumption cost minimization is polynomially solvable.
Abstract: Under just-in-time production, this paper studies a single machine common due-window (denoted by CONW) assignment scheduling problem with position-dependent weights and resource allocations. A job’s actual processing time can be determined by the resource assigned to the job. A resource allocation model is divided into linear and convex resource allocations. Under the linear and convex resource allocation models, our goal is to find an optimal due-window location, job sequence and resource allocation. We prove that the weighted sum of scheduling cost (including general earliness–tardiness penalties with positional-dependent weights) and resource consumption cost minimization is polynomially solvable. In addition, under the convex resource allocation, we show that scheduling (resp. resource consumption) cost minimization is solvable in polynomial time subject to the resource consumption (resp. scheduling) cost being bounded.
01 Apr 2023
TL;DR: In this article , the authors used Density Functional Tight Binding (DFTB) with the Clusters Approach to Statistical Mechanics (CASM) software package for the first time.
Abstract: Defects in materials significantly alter their electronic and structural properties, which affect the performance of electronic devices, structural alloys, and functional materials. However, calculating all the possible defects in complex materials with conventional Density Functional Theory (DFT) can be computationally prohibitive. To enhance the efficiency of these calculations, we interfaced Density Functional Tight Binding (DFTB) with the Clusters Approach to Statistical Mechanics (CASM) software package for the first time. Using SiC and ZnO as representative examples, we show that DFTB gives accurate results and can be used as an efficient computational approach for calculating and pre-screening formation energies/convex hulls. Our DFTB+CASM implementation allows for an efficient exploration (up to an order of magnitude faster than DFT) of formation energies and convex hulls, which researchers can use to probe other complex systems.
TL;DR: The ellipsotope representation as discussed by the authors combines the advantages of ellipsoids and zonotopes while ensuring convex collision checking, and has been shown to be closed under affine maps, Minkowski sums, and intersections.
Abstract: Ellipsoids are a common representation for reachability analysis, because they can be transformed efficiently under affine maps, and they allow conservative approximation of Minkowski sums, which let one incorporate uncertainty and linearization error in a dynamical system by expanding the size of the reachable set. Zonotopes, a type of symmetric, convex polytope, are similarly frequently used, because they allow efficient numerical implementations of affine maps and exact Minkowski sums. Both of these representations also enable efficient, convex collision detection for fault detection or formal verification tasks, wherein one checks if the reachable set of a system collides (i.e., intersects) with an unsafe set. However, both representations often result in conservative representations for reachable sets of arbitrary systems, and neither is closed under intersection. Recently, representations such as constrained zonotopes and constrained polynomial zonotopes have been shown to overcome some of these conservativeness challenges, and are closed under intersection. However, constrained zonotopes can not represent shapes with smooth boundaries such as ellipsoids, and constrained polynomial zonotopes can require solving a non-convex program for collision checking or fault detection. This paper introduces ellipsotopes, a set representation that is closed under affine maps, Minkowski sums, and intersections. Ellipsotopes combine the advantages of ellipsoids and zonotopes while ensuring convex collision checking. The utility of this representation is demonstrated on several examples.
TL;DR: In this article , the authors established the notion and properties of angle rigidity for 3D multi-point frameworks with angle constraints, and then designed direction-only control laws to stabilize angle rigid formations of mobile agents in 3D.
Abstract: This paper establishes the notion and properties of angle rigidity for 3D multi-point frameworks with angle constraints, and then designs direction-only control laws to stabilize angle rigid formations of mobile agents in 3D. Angles are defined using the interior angles of triangles within the given framework, which are independent of the choice of coordinate frames and can be conveniently measured using monocular cameras and direction-finding arrays. We show that 3D angle rigidity is a local property, which is in contrast to the 3D bearing rigidity as has been proved to be a global property in the literature. We demonstrate that such angle rigid and globally angle rigid frameworks can be constructed through adding repeatedly new points to the original small angle rigid framework with carefully chosen angle constraints. We also investigate how to merge two 3D angle rigid frameworks by connecting three points of one angle rigid framework simultaneously to the other. When angle constraints are given only in the surface of a framework, angle rigidity of convex polyhedra is studied, in which the cases of triangular face and triangulated face are considered, respectively. The proposed 3D angle rigidity theory is then utilized to design decentralized formation control strategies using local direction measurements for teams of mobile agents. Simulation examples are provided to validate the convergence of the formations.
TL;DR: In this article , the best approximation results for 1-set contractive set-valued mappings in locally p-convex or p-vector spaces are established. But the results are not applicable to non-self mappings with either inward or outward set conditions under various situations.
Abstract: Abstract The goal of this paper is to develop some new and useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex or p-vector spaces for In particular, we first develop general fixed point theorems for both set-valued and single-valued condensing mappings which provide answers to the Schauder conjecture in the affirmative way under the setting of (locally p-convex) p-vector spaces, then the best approximation results for upper semi-continuous and 1-set contractive set-valued are established, which are used as tools to establish some new fixed points for non-self set-valued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis.
TL;DR: In this paper , the second Hankel determinant of strongly starlike and strongly convex functions has been shown to have sharp bounds for strongly star-like and convex function.
Abstract: Abstract Sharp bounds are given for the second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions.
TL;DR: In this paper , a new class of convex functions associated with strong η-convexity is proposed and the Hermite-Hadamard inequality is derived for this family of functions.
Abstract: It is the purpose of this paper to propose a novel class of convex functions associated with strong η-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong η-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction.
TL;DR: In this paper , it was shown that the Gardner-Zvavitch conjecture is true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures.
Abstract: We prove that the (B) conjecture and the Gardner–Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance, to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.
TL;DR: The sharp bound for the third Hankel determinant for the coefficients of the inverse function of convex functions is obtained in this paper, thus answering a recent conjecture concerning invariance of coefficient functionals for convex function.
Abstract:
The sharp bound for the third Hankel determinant for the coefficients of the inverse function of convex functions is obtained, thus answering a recent conjecture concerning invariance of coefficient functionals for convex functions.
TL;DR: In this paper , the authors investigated common (slack) due-date assignment single-machine scheduling with controllable processing times within a group technology environment and showed that the problem is polynomially solvable in O(℘3) time, where ℘ is the number of jobs.
Abstract: This paper investigates common (slack) due-date assignment single-machine scheduling with controllable processing times within a group technology environment. Under linear and convex resource allocation functions, the cost function minimizes scheduling (including the weighted sum of earliness, tardiness, and due-date assignment, where the weights are position-dependent) and resource-allocation costs. Given some optimal properties of the problem, if the size of jobs in each group is identical, the optimal group sequence can be obtained via an assignment problem. We then illustrate that the problem is polynomially solvable in O(℘3) time, where ℘ is the number of jobs.
20 Jan 2023
TL;DR: In this article , the authors rigorously investigate closed Minkowski/Finsler billiard trajectories on convex bodies and develop an algorithm for computing length-minimizing closed closed billiard trajectory in the plane.
Abstract: Abstract We rigorously investigate closed Minkowski/Finsler billiard trajectories on $n$-dimensional convex bodies. We outline the central properties in comparison and differentiation from the Euclidean special case and establish two main results for length-minimizing closed Minkowski/Finsler billiard trajectories: one is a regularity result, the other is of geometric nature. Building on these results, we develop an algorithm for computing length-minimizing closed Minkowski/Finsler billiard trajectories in the plane. Mathematics Subject Classification (2010) MSC 37C83
TL;DR: In this paper , the authors prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions.
Abstract: We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis–Székelyhidi, in particular, we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the [Formula: see text]-convex hull of the constitutive set. An important application of our scheme has been exhibited in recent work by Gallenmüller–Wiedemann.
TL;DR: In this article , the sharp bounds of the second Hankel determinant of logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk were investigated.
Abstract: In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.
19 Jun 2023
TL;DR: Adaptive online switching (AOS) as mentioned in this paper is a deterministic algorithm that is (1 + δ)-competitive if predictions are perfect, while maintaining a uniformly bounded competitive ratio of 2~O (1/(α δ)) even when predictions are adversarial.
Abstract: We examine the problem of smoothed online optimization, where a decision maker must sequentially choose points in a normed vector space to minimize the sum of per-round, non-convex hitting costs and the costs of switching decisions between rounds. The decision maker has access to a black-box oracle, such as a machine learning model, that provides untrusted and potentially inaccurate predictions of the optimal decision in each round. The goal of the decision maker is to exploit the predictions if they are accurate, while guaranteeing performance that is not much worse than the hindsight optimal sequence of decisions, even when predictions are inaccurate. We impose the standard assumption that hitting costs are globally α-polyhedral. We propose a novel algorithm, Adaptive Online Switching (AOS), and prove that, for a large set of feasible δ > 0, it is (1+δ)-competitive if predictions are perfect, while also maintaining a uniformly bounded competitive ratio of 2~O (1/(α δ)) even when predictions are adversarial. Further, we prove that this trade-off is necessary and nearly optimal in the sense that any deterministic algorithm which is (1+δ)-competitive if predictions are perfect must be at least 2~Ω (1/(α δ)) -competitive when predictions are inaccurate. In fact, we observe a unique threshold-type behavior in this trade-off: if δ is not in the set of feasible options, then no algorithm is simultaneously (1 + δ)-competitive if predictions are perfect and ζ-competitive when predictions are inaccurate for any ζ < ∞. Furthermore, we discuss that memory is crucial in AOS by proving that any algorithm that does not use memory cannot benefit from predictions. We complement our theoretical results by a numerical study on a microgrid application.
TL;DR: In this paper , the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces was investigated.
Abstract: In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.
TL;DR: In this article , a scalable stochastic optimization model and a Markov chain-based scenario generation method are proposed to benefit from an active distribution network's (ADN) flexibility, where the optimization variables are the dispatch plan, such as the active and reactive power of battery energy storage (BES) and photovoltaic (PV) systems, as well as the flexibilities given to the transmission network at the point of common coupling (PCC).
TL;DR: In this article , the authors investigated several integral inequalities held simultaneously for q and h-integrals in implicit form for symmetric functions using certain types of convex functions on the non-negative part of the real line.
Abstract: This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.