Control sets for bilinear and affine systems

For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.

Approximation by a new sequence of operators involving Apostol-Genocchi polynomials

The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

Korn and Poincaré-Korn inequalities for functions with a small jump set

In this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ G S B D p , for every $$p>1$$ p > 1 and any dimension $$n\ge 2$$ n ≥ 2 . We show that a displacement in $$GSBD^p$$ G S B D p with a small jump set coincides with a $$W^{1,p}$$ W 1 , p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $$GSBD^p$$ G S B D p .

Fair Allocation of Indivisible Goods: Improvement

We study the problem of fair allocation for indivisible goods. We use the maximin share paradigm introduced by Budish [Budish E (2011) The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. J. Political Econom. 119(6):1061–1103.] as a measure of fairness. Kurokawa et al. [Kurokawa D, Procaccia AD, Wang J (2018) Fair enough: Guaranteeing approximate maximin shares. J. ACM 65(2):8.] were the first to investigate this fundamental problem in the additive setting. They showed that in delicately constructed examples, not everyone can obtain a utility of at least her maximin value. They mitigated this impossibility result with a beautiful observation: no matter how the utility functions are made, we always can allocate the items to the agents to guarantee each agent’s utility is at least 2/3 of her maximin value. They left open whether this bound can be improved. Our main contribution answers this question in the affirmative. We improve their approximation result to a 3/4 factor guarantee.

FFT-Based Numerical Method for Non-Linear Elastic Contact

Based on FFT, a numerical method suitable for elastoplastic and hyperelastic frictionless contact is proposed in this paper, which can be used to solve 2D and 3D contact problems. The non-linear elastic contact problem is transformed to linear elastic contact considering residual deformation (or equivalent residual deformation). Numerical simulations for elastic, elastoplastic and hyperelastic contact between hemisphere and rigid plane are compared with the results of finite element method (FEM) to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the calculation efficiency is improved while ensuring a certain calculation accuracy (pressure error does not exceed 15% while calculation time does not exceed 10 minutes in a 64×64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on load as in linear elastic approximation. Despite a certain error, the simplified numerical method shows a better approximation result than linear elastic contact approximation, which can be used for the fast solution in engineering applications. Finally, taking the sealing application as an example, the contact and leakage rate between 3D complicated rough surfaces are calculated.

Compactness and convergence rates in the combinatorial integral approximation decomposition

The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the $$\hbox {weak}^*$$ weak ∗ topology of $$L^\infty $$ L ∞ if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.

A New Best Approximation Result in (S) Convex Metric Spaces

Consider a self-mapping T defined on the union of p subsets of a metric space, and T is said to be p cyclic if TAi⊆Ai+1 for i=1,…,p with Ap+1=A1. In this article, we introduce the notion of S convex structure, and we acquire a best proximity point for p cyclic contraction in S convex metric spaces.