Related papers: Discretized sum-product estimates in matrix algebr…
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…
The theory of generalized matric Massey products has been applied for some time to $A$-modules $M$, $A$ a $k$-algebra. The main application is to compute the local formal moduli $\hat{H}_M$, isomorphic to the local ring of the moduli of…
We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A…
We introduce discrete equational theories where operations are induced by those having discrete arities. We characterize the corresponding monads as monads preserving surjections. Using it, we prove Birkhoff type theorems for categories of…
In this paper we study "discrete polynomial blending," a term used to define a certain discretized version of curve blending whereby one approximates from the "sum of tensor product polynomial spaces" over certain grids. Our strategy is to…
In this paper, the exact values of the structured singular values of some generalized stochastic complex matrices is explicit in term of the constant row (column) sum.
We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Ces\'aro, Euler, Lambert, N\"orlund, and Mittag-Leffler,…
We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation.…
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.
This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary…
We derive new prox-functions on the simplex from additive random utility models of discrete choice. They are convex conjugates of the corresponding surplus functions. In particular, we explicitly derive the convexity parameter of discrete…
In this note, we study a factorization result for graded decomposition maps associated with the specializations of graded algebras. We obtain results previously known only in the ungraded setting.
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…
We prove finite-field analogs of Bourgain's projection theorem in higher dimensions. In particular, for a certain range of parameters we improve on an exceptional set estimate by Chen in all dimensions and codimensions.
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these…
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…
We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of…
In this paper, we present a variant of the Balog-Szemer\'edi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of…
In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find…