Related papers: Webster Sequences, Apportionment Problems, and Jus…
We present a deterministic algorithm which, given a graph G with n vertices and an integer 1<m < n, computes in n^{O(ln m)} time the sum of weights w(S) over all m-subsets S of the set of vertices of G, where w(S)=exp{gamma t m +O(1/m)}…
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The…
This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of…
In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the…
We propose an $\widetilde{O}(n + 1/\eps)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time…
Given $iid$ observations from an unknown absolute continuous distribution defined on some domain $\Omega$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function.…
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $\alpha_c(\Delta)$ and provide (i) for $\alpha <…
We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $\rho\in(0,1)$. The rate of passage of particles to the right…
We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $\beta(A)=\frac{1}{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty$. Steinerberger [1]…
The restricted partition function $p_N(n)$ counts the partitions of the integer $n$ into at most $N$ parts. In the nineteenth century Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and,…
Relatively little is known about the arithmetic properties of Gamma-function derivatives evaluated at arbitrary points $q\in\mathbb{Q}\setminus\mathbb{Z}_{\leq0}$. In recent work, we showed that the sequence…
We consider the SUBSET SUM problem and its important variants in this paper. In the SUBSET SUM problem, a (multi-)set $X$ of $n$ positive numbers and a target number $t$ are given, and the task is to find a subset of $X$ with the maximal…
A subset $A$ of an abelian group $G$ is sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_i$ for $1 \leq i \leq k$,…
We consider the problem of partitioning $n$ integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen…
The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by…
We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…
For integers $n$ and $k$, the density Hales-Jewett number $c_{n,k}$ is defined as the maximal size of a subset of $[k]^n$ that contains no combinatorial line. We show that for $k \ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to…
We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: For any $\delta > 0$ there is some $N_0 \geq 1$ such that whenever $A \subseteq [k]^N$ with $N \geq N_0$ and $|A|\geq \delta k^N$, $A$…
At low densities, with decreasing temperatures, in symmetric nuclear matter alpha-particles are formed, which eventually give raise to a quantum condensate with four-nucleon alpha-like correlations (quartetting). Starting with a model of…
In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a…