Related papers: On particular diameter bounds for integral point s…
For better learning, large datasets are often split into small batches and fed sequentially to the predictive model. In this paper, we study such batch decompositions from a probabilistic perspective. We assume that data points (possibly…
In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and…
We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.
We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.
Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the…
We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…
We give uniform upper bounds for the number of integral points of bounded height on affine hypersurfaces, which generalise earlier results of Browning,Heath-Brown and the author.
The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that…
We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…
A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the…
We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. We also define the corresponding concept of $n$-dimensional difference…
We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the…
A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,...,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of…
In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
We consider the set multi-cover problem in geometric settings. Given a set of points P and a collection of geometric shapes (or sets) F, we wish to find a minimum cardinality subset of F such that each point p in P is covered by (contained…
We prove finite-sample concentration and anti-concentration bounds for dimension estimation using Gaussian kernel sums. Our bounds provide explicit dependence on sample size, bandwidth, and local geometric and distributional parameters,…