相关论文: A tight lower bound for Szemer\'edi's regularity l…
A method for compression of large graphs and non-negative matrices to a block structure is proposed. Szemer\'edi's regularity lemma is used as heuristic motivation of the significance of stochastic block models. Another ingredient of the…
Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions…
This paper records some apparently new results for the partition of integer intervals [1, n] into weakly sum-free subsets. These were produced using a method closely related to that used by Schur in 1917. New lower bounds can be produced in…
Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend…
We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric…
The explicit construction of function fields tower with many rational points relative to the genus in the tower play a key role for the construction of asymptotically good algebraic-geometric codes. In 1997 Garcia, Stichtenoth and Thomas…
We prove an analogue of the Approximation Theorem of L^2-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module…
In this paper, we investigate properties of countable stationary towers. We derive the regularity properties of sets of reals in $L(\mathbf R)$ from some properties of countable stationary towers without explicit use of strong large…
The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…
We prove a variant of the multidimensional polynomial Szemer\'edi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain…
We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find…
We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.
Let $N$ be a large prime and $P, Q \in \mathbb{Z}[x]$ two linearly independent polynomials with $P(0) = Q(0) = 0$. We show that if a subset $A$ of $\mathbb{Z}/N\mathbb{Z}$ lacks a progression of the form $(x, x + P(y), x + Q(y), x + P(y) +…
We continue the study on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We…
This article provides a proof of a generalization of Schur's theorem on the partition regularity of the equation x+y=z, which involves a divisibility condition. This generalization will be utilized to prove the existence of 'small'…
We establish a lower bound for the complexity of multiplying two skew polynomials. The lower bound coincides with the upper bound conjectured by Caruso and Borgne in 2017, up to a log factor. We present algorithms for three special cases,…
The investigation of partitions of integers plays an important role in combinatorics and number theory. Among the many variations, partitions into powers $0<\alpha<1$ were of recent interest. In the present paper we want to extend our…
We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we…
We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound…
We study a class of focusing nonlinear Schroedinger-type equations derived recently by Dumas, Lannes and Szeftel within the mathematical description of high intensity laser beams [7]. These equations incorporate the possibility of a…