Symmetric Convex Research Articles

Remarks on Soft Ball Packings in Dimensions 2 and 3

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

Accuracy analysis and improvement methods for revolving parts in counter-rotating electrochemical machining with continuous radial feeding

Counter-rotating electrochemical machining (CRECM) stands out as an efficient and cost-effective method for machining aero-engine casings. To maintain stability in the CRECM process equilibrium state, continuous feeding of the cathode tool is essential. However, this continuous feeding approach with single direction of rotation may result in poor roundness and reduced machining accuracy. This paper aims to analyze the sources and rules of roundness error and asymmetry in CRECM. Based on simulation results, a method involving periodic reverse rotation of electrodes with continuous feeding is proposed for CRECM. This approach aims to mitigate the cumulative machining error, thereby significantly improving machining accuracy. Experimental results validate the efficacy of the proposed method at a rotation speed of 0.2 rpm. The roundness error is markedly reduced from 0.19 mm to 0.04 mm, resulting in a more symmetric convex structure. Additionally, the sidewall taper angle is reduced from 14.5 degrees to 6.4 degrees. This underscores the effectiveness of the proposed method in enhancing machining accuracy.

Ensemble Clustering via Co-Association Matrix Self-Enhancement.

Ensemble clustering integrates a set of base clustering results to generate a stronger one. Existing methods usually rely on a co-association (CA) matrix that measures how many times two samples are grouped into the same cluster according to the base clusterings to achieve ensemble clustering. However, when the constructed CA matrix is of low quality, the performance will degrade. In this article, we propose a simple, yet effective CA matrix self-enhancement framework that can improve the CA matrix to achieve better clustering performance. Specifically, we first extract the high-confidence (HC) information from the base clusterings to form a sparse HC matrix. By propagating the highly reliable information of the HC matrix to the CA matrix and complementing the HC matrix according to the CA matrix simultaneously, the proposed method generates an enhanced CA matrix for better clustering. Technically, the proposed model is formulated as a symmetric constrained convex optimization problem, which is efficiently solved by an alternating iterative algorithm with convergence and global optimum theoretically guaranteed. Extensive experimental comparisons with 12 state-of-the-art methods on ten benchmark datasets substantiate the effectiveness, flexibility, and efficiency of the proposed model in ensemble clustering. The codes and datasets can be downloaded at https://github.com/Siritao/EC-CMS.

Positivity and convexity in incomplete cooperative games

Incomplete cooperative games generalize the classical model of cooperative games by omitting the values of some of the coalitions. This allows for incorporating uncertainty into the model and studying the underlying games and possible payoff distributions based only on the partial information. In this paper, we conduct a systematic investigation of incomplete games, focusing on two important classes: positive and convex games. Regarding positivity, we generalize previous results from a special class of minimal incomplete games to a general setting. We characterize the non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. These results also enable the construction of explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with the case of non-negative, minimal incomplete games. We establish the connection between incomplete games and the problem of completing partial functions and, consequently, provide a characterization of extendability and a full description of the set of symmetric convex extensions. This set serves as an approximation of the set of convex extensions.

Impact of Rh, Ru, and Pd Leads and Contact Topologies on Performance of WSe2 FETs: A First Comparative Ab Initio Study.

2D field-effect transistors (FETs) fabricated with transition metal dichalcogenide (TMD) materials are a potential replacement for the silicon-based CMOS. However, the lack of advancement in p-type contact is also a key factor hindering TMD-based CMOS applications. The less investigated path towards improving electrical characteristics based on contact geometries with low contact resistance (RC) has also been established. Moreover, finding contact metals to reduce the RC is indeed one of the significant challenges in achieving the above goal. Our research provides the first comparative analysis of the three contact configurations for a WSe2 monolayer with different noble metals (Rh, Ru, and Pd) by employing ab initio density functional theory (DFT) and non-equilibrium Green's function (NEGF) methods. From the perspective of the contact topologies, the RC and minimum subthreshold slope (SSMIN) of all the conventional edge contacts are outperformed by the novel non-van der Waals (vdW) sandwich contacts. These non-vdW sandwich contacts reveal that their RC values are below 50 Ω∙μm, attributed to the narrow Schottky barrier widths (SBWs) and low Schottky barrier heights (SBHs). Not only are the RC values dramatically reduced by such novel contacts, but the SSMIN values are lower than 68 mV/dec. The new proposal offers the lowest RC and SSMIN, irrespective of the contact metals. Further considering the metal leads, the WSe2/Rh FETs based on the non-vdW sandwich contacts show a meager RC value of 33 Ω∙μm and an exceptional SSMIN of 63 mV/dec. The two calculated results present the smallest-ever values reported in our study, indicating that the non-vdW sandwich contacts with Rh leads can attain the best-case scenario. In contrast, the symmetric convex edge contacts with Pd leads cause the worst-case degradation, yielding an RC value of 213 Ω∙μm and an SSMIN value of 95 mV/dec. While all the WSe2/Ru FETs exhibit medium performances, the minimal shift in the transfer curves is interestingly advantageous to the circuit operation. Conclusively, the low-RC performances and the desirable SSMIN values are a combination of the contact geometries and metal leads. This innovation, achieved through noble metal leads in conjunction with the novel contact configurations, paves the way for a TMD-based CMOS with ultra-low RC and rapid switching speeds.

On the Banach–Mazur Distance in Small Dimensions

AbstractWe establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in $$\mathbb {R}^3$$ R 3 and $$\mathbb {R}^4$$ R 4 . In dimension three this distance is equal to $$\frac{9}{5}$$ 9 5 , and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies

It is shown that every not-necessarily symmetric convex body K in Rn has an affine image K˜ of K such that the covering numbers of K˜ by growing dilates of the unit Euclidean ball, as well as those of the unit Euclidean ball by growing dilates of K˜, decrease in a regular way. This extends to the non-symmetric case a famous theorem by Pisier, albeit with worse estimates on the rate of decrease of the covering numbers. The affine image K˜ can be chosen to have either barycentre or Santaló point at the origin.In the proof we use Pisier's theorem as a black box, as well as a suggested approach by Klartag and V. Milman. A key new ingredient is Blaschke-Santaló-type inequalities for projections of a body K˜ with Santaló point at the origin, which could be of independent interest. Unlike the application to covering, these (as well as the analogous inequalities for centred convex bodies that were already considered by Klartag and Milman [28]) can be shown to be optimal up to absolute constants.We also present an application to results around the mean norm of isotropic (not-necessarily symmetric) convex bodies.

Sharp bounds for a discrete John’s theorem

AbstractTao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$ . Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$ . We obtain the bound $d^{O(d)} \# C$ , which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem.

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces

Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces

Remark on coisotropic Ekeland–Hofer–Zehnder capacity

In this paper, we give a counter-example to show that coisotropic Ekeland–Hofer–Zehnder capacities do not satisfy superadditivity for hyperplane cuts in higher dimension. Next, we show that the coisotropic Ekeland–Hofer–Zehnder capacity relative to Rn,k is not asymptotic equivalent to any normalized symplectic capacity in general case. But we show on the class of centrally symmetric convex domain in R2n, the coisotropic Ekeland–Hofer–Zehnder capacity is asymptotic equivalent to the Ekeland–Hofer–Zehnder capacity. Finally, we give some other results about the coisotropic Ekeland–Hofer–Zehnder capacity.

Minkowski's successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

Improved log-concavity for rotationally invariant measures of symmetric convex sets

Improved log-concavity for rotationally invariant measures of symmetric convex sets

Distributed bipartite output containment control of heterogeneous multi-agent systems with directed signed graphs

This paper investigates the bipartite output containment control problem by utilising distributed control protocols for heterogeneous multi-agent systems over directed signed graphs. A fully distributed dynamic compensator is proposed to estimate the convex hull and symmetric convex hull spanned by states of leaders. Then, a distributed observer based on neighbour information is presented for estimating the state of each follower. An output feedback control protocol is further given based on the fully distributed dynamic compensator and neighbour-based observer to cope with the bipartite containment control problem of the linear heterogeneous multi-agent systems. Different from the related works, a neighbour-based observer is presented for heterogeneous multi-agent systems, in which the analysis method used in the Luenburger observer cannot be applied directly, because of the heterogeneity. Besides, the dynamics of the agents are allowed to not only have nonidentical coefficient matrices but also have different dimensions. A simulation example is shown to verify the effectiveness of the proposed theoretical results.

Irrationally elliptic closed characteristics on symmetric compact convex hypersurfaces in R^8

Let $\Sigma$ be a $C^3$ compact symmetric convex hypersurface in $\mathbb{R}^{8}$. We prove that when $\Sigma$ carries exactly four geometrically distinct closed characteristics, then there are at least two irrationally elliptic closed characteristics on $\Sigma$.

A note on the reduction of the slicing problem to centrally symmetric convex bodies

In this paper, we obtain the best possible value of the absolute constant C such that for every isotropic convex body K⊆Rn the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied:LK≤CLKn+2(gK). Here LK denotes the isotropic constant of K, gK its covariogram function, which is log-concave, and, for any log-concave function g, Kn+2(g) is a convex body associated to the log-concave function g, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever g satisfies a better type of concavity than the log-concavity, as gK is, indeed 1n-concave.

On the mean width ratio of convex bodies

AbstractGiven two convex bodies in . The mean width ratio of and is defined by , where the infimum is over all linear transformations of for which . For symmetric and containing the origin not necessarily symmetric convex bodies we show that , where is an absolute constant.

Bipartite Containment Control for Delayed Multiagent Systems With Markovian Switching Topologies Under Impulsive Attacks

This paper is concerned with the problem of bipartite containment control design for a class of nonlinear multiagent systems with time-delays in agents’ states under impulsive false-data-injection attacks. The considered multiagent system is subject to Markovian variation in the signed communication topology. The graph among the followers is assumed to be structurally balanced for each Markovian switching mode. A memory distributed control protocol is proposed to achieve bipartite containment within the convex hull of leader agents as well as the symmetric convex hull. The bipartite containment control problem is solved by means of a Markovian switching Lyapunov function and the Razumikhin technique. In addition, the problem of bipartite leader-following consensus is also addressed for delayed nonlinear multiagent systems with one leader. Two examples are provided to show the effectiveness of the proposed control scheme, and comparison results with existing methods are given as well.

A Study on the Modified Form of Riemann-Type Fractional Inequalities via Convex Functions and Related Applications

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals of Riemann-type via convex functions. The mean fractional inequalities for functions with convex absolute value derivatives are discovered. Hermite–Hadamard-type fractional inequalities for a symmetric convex function are explored. These results are achieved using a fresh and innovative methodology for the modified form of generalized fractional integrals. Some applications for the results explored in the paper are briefly reviewed.

On generalized Minkowski arrangements

The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order μ for some 0 < μ < 1 if no member K of the family overlaps the homothetic copy of any other member K′ with ratio μ and with the same center as K′. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order μ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.