相关论文: Bounds for the 3x+1 Problem using Difference Inequ…
The declustering problem is to allocate given data on parallel working storage devices in such a manner that typical requests find their data evenly distributed on the devices. Using deep results from discrepancy theory, we improve previous…
One of the first and easy to use techniques for proving run time bounds for evolutionary algorithms is the so-called method of fitness levels by Wegener. It uses a partition of the search space into a sequence of levels which are traversed…
Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid…
In this paper, we show $O(1.415^n)$-time and $O(1.190^n)$-space exact algorithms for 0-1 integer programs where constraints are linear equalities and coefficients are arbitrary real numbers. Our algorithms are quadratically faster than…
We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower…
The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by…
The vast majority of real world classification problems are imbalanced, meaning there are far fewer data from the class of interest (the positive class) than from other classes. We propose two machine learning algorithms to handle highly…
The three distance theorem states that for any given irrational number $\alpha$ and a natural number $n$, when the interval $( 0, 1 )$ is divided into $n+1$ subintervals by integer multiples of $\alpha$, namely, $\{0\}, \{ \alpha \}, \{…
We give a survey of work on the number of vertices of the convex hull of integer points defined by the system of linear inequalities. Also, we present our improvement of some of these.
It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…
We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.
We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory…
We will give new upper bounds for the number of solutions to the inequalities of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form, with integer coefficients, and $h$ is a sufficiently small integer in terms of the…
We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted…
The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha,2\alpha,\ldots,N\alpha$, for any integer $N$ and real number $\alpha$. This statement…
The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $\alpha,2\alpha,\ldots, N\alpha$ take at most three distinct values. Motivated by a question of…
Arithmetic constraints on integer intervals are supported in many constraint programming systems. We study here a number of approaches to implement constraint propagation for these constraints. To describe them we introduce integer interval…
In what follows we improve an inequality related to matrix theory. T. Laffey proved (2013) a weaker form of this inequality [2].
We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier…
By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the…