Related papers: An Elementary Proof of the Restricted Invertibilit…
The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary real $n \times k$ matrix with orthonormal columns a sufficiently "good" $k \times k$ submatrix exists. "Good" in the sense…
An old conjecture of Erd\H{o}s and R\'enyi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x) \in \mathbb{C}[x]$ when its square $g(x)^2$ has a given number of terms. Further conjectures and results arose,…
The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector space, without placing any restrictions on the dimension of the space or on the base field. We define a…
In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer…
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…
We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is…
We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain. Our approach allows…
It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the…
The theorem on the existence of maximal nonnegative invariant subspaces for a special class of dissipative operators in Hilbert space with indefinite inner product is proved in the paper. It is shown in addition that the spectra of the…
We present a simple solution to a question posed by Candes, Romberg and Tao on the uniform uncertainty principle for Bernoulli random matrices. More precisely, we show that a rectangular k*n random subgaussian matrix (with k < n) has the…
It is well-understood that the robustness of mechanical and robotic control systems depends critically on minimizing sensitivity to arbitrary application-specific details whenever possible. For example, if a system is defined and performs…
The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as…
Integer programs (IPs) on constraint matrices with bounded subdeterminants are conjectured to be solvable in polynomial time. We give a strongly polynomial time algorithm to solve IPs where the constraint matrix has bounded subdeterminants…
A linearly constrained framework in $\mathbb{R}^d$ is a bar-joint framework where, in addition, vertices with loops are constrained to lie in given affine subspaces. In the generic case, when each vertex is incident to sufficiently many…
We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous…
In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier…
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can…
The infinite upper triangular Pascal matrix is $T = [\binom{j}{i}]$ for $0\leq i,j$. It is easy to see that any leading principle square submatrix is triangular with determinant $1$, hence invertible. In this paper, we investigate the…
We prove that the space of intertwining operators associated with certain admissible modules over vertex operator algebras is isomorphic to a quotient of the vector space of conformal blocks on a three-pointed rational curve defined by the…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.