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This paper investigates the behavior of sets and functions at infinity by introducing new concepts, namely directional normal cones at infinity for unbounded sets, along with limiting and singular subdifferentials at infinity in the…
The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive…
The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained.…
Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint…
This short note solves the following problem: Given a map of normal toric varieties corresponding to a coherent subdivision of a cone, find an ideal such that the given map is the blowup of that ideal.
We propose a convex optimization formulation with the nuclear norm and $\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the…
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…
We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in R^4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point…
We investigate the structure of $\omega$-limit (resp. $\alpha$-limit) sets for a monotone map $f$ on a regular curve $X$. %Let $X$ be a regular curve and let $f: X\longrightarrowX$ be a monotone map. We show that for any $x\in X$ (resp. for…
A closed convex conic subset $\mathcal{S}$ of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of $\mathcal{S}$ is closely related to the…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
The minimum rank problem is to determine for a graph $G$ the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of $G$. Here $G$ is taken to be a circulant graph,…
The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in…
A large number matrix optimization problems are described by orthogonally invariant norms. This paper is devoted to the study of variational analysis of the orthogonally invariant norm cone of symmetric matrices. For a general orthogonally…
We consider Lie groups equipped with left-invariant subbundles of their tangent bundles and norms on them. On these sub-Finsler structures, we study the normal curves in the sense of control theory. We revisit the Pontryagin Maximum…
For regularized distributions we establish stability of the characterization of the normal law in Cramer's theorem with respect to the total variation norm and the entropic distance. As part of the argument, Sapogov-type theorems are…
We consider the classical minimum and maximum cut problems: find a partition of vertices of a graph into two disjoint subsets that minimize or maximize the sum of the weights of edges with endpoints in different subsets. It is known that if…
A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general…
Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} =…
The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is…