Related papers: Extended Set Difference : Inverse Operation of Min…
We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be…
Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear…
We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections…
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
This paper treats the variation of sets. We attempt to formulate convergence and continuity of set-valued functions in a different way from the theories on sequences of sets and correspondence. In the final section, we also attempt to…
In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on…
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…
Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…
In this paper we study invertible extensions of a symmetric operator in a Hilbert space $H$. All such extensions are characterized by a parameter in the generalized Neumann's formulas. Generalized resolvents, which are generated by the…
Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite…
The extended commutation relations for a generalized uncertainty principle have been based on the assumption of the minimal length in position. Instead of this assumption, we start with a constrained Hamiltonian system described by the…
The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the…
We introduce a new version of arithmetic in all finite types which extends the usual versions with primitive notions of extensionality and extensional equality. This new hybrid version allows us to formulate a strong form of extensionality,…
We generalize McDiarmid's inequality for functions with bounded differences on a high probability set, using an extension argument. Those functions concentrate around their conditional expectations. We further extend the results to…
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…
We establish some weighted $L^2$ inequalities for Fourier extension operators in the setting of orthonormal systems. In the process we develop a direct approach to such inequalities based on generalised Wigner distributions, complementing…
Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented.…
L-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L2-convex sets, is an intriguing object that is closely related to polymatroid intersection.…
Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than $0$. It is shown that unitary…
A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary…