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Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds…
A \itbf{cover} of a string $x = x[1..n]$ is a proper substring $u$ of $x$ such that $x$ can be constructed from possibly overlapping instances of $u$. A recent paper \cite{FIKPPST13} relaxes this definition --- an \itbf{enhanced cover} $u$…
Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…
A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with…
We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for…
Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary…
The problem of variable-rate lossless data compression is considered, for codes with and without prefix constraints. Sharp bounds are derived for the best achievable compression rate of memoryless sources, when the excess-rate probability…
An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…
More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum…
Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite…
Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every n-dimensional representation of R is…
A positive integer $n$ is said to be $k$-layered if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we start the systematic study of these class of numbers. In particular, we state some algorithms to find some…
Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm…
We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the upper bound for the permanent with a…
Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…
New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…
The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with…
Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a…
In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \setminus P$ doesn't remains a basis. The existence of an essential subset for a basis…
Numerical analysis has no satisfactory method for the more realistic optimization models. However, with constraint programming one can compute a cover for the solution set to arbitrarily close approximation. Because the use of constraint…