相关论文: Expanding phenomena over higher dimensional matrix…
Higher-degree polynomial interpolations carried out on uniformly distributed nodes are often plagued by {\it overfitting}, known as Runge's phenomenon. This work investigates Runge's phenomenon and its suppression in various versions of the…
Every commuting set of normal matrices with entries in an AW*-algebra can be simultaneously diagonalized. To establish this, a dimension theory for properly infinite projections in AW*-algebras is developed. As a consequence, passing to…
The goal of this paper is to undertake an in-depth study of the phenomenon behind the Furstenberg--S\'ark\"ozy theorem, which, in its modern form due to Kamae and Mend\`es-France, states that if $E$ is a set of integers with positive…
Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…
We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…
We introduce a new model of random $d$-dimensional simplicial complexes, for $d\geq 2$, whose $(d-1)$-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The…
In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to…
The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices introduced in D. D. Anderson, C. Jayaram, and P. A. Phiri, Baer lattices,…
Consider the special linear group of degree $2$ over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring,…
We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring…
We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent…
Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m\geqslant n+1+p$. We also show the graded ring of sections $R(X, K_X+mA)$ is…
We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an…
A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and…
The work proves that, for three-dimensional upper triangular groups over a field of odd characteristic with an abelian unipotent subgroup, the ring of invariants is polynomial if and only if the unipotent subgroup is generated by…
Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange…
On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the…
In this paper, we study the non trivial idempotents of the $2 \times 2$ matrix ring over the polynomial ring $\mathbb{Z}_{pqr}[x]$ for distinct primes $p, q $ and $r$ greater than $3$. We have classified all the idempotents of this matrix…
In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with…
The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and…