Related papers: Counting Polyominoes in a Rectangle b x h
In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting…
The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method.
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
In contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding…
Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…
We construct automata over a binary alphabet with $2n$ states, $n\geq 2$, whose states freely generate a free group of rank $2n$. Combined with previous work, this shows that a free group of every finite rank can be generated by finite…
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…
In this work, we will explore some polygons that individually are capable of filling the plane in an aperiodic way. These polygons were recently discovered by some researchers and constitute a great discovery for Mathematics. We will…
We study rectangles inscribed in lines in the plane by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other…
Recursive maps of high order of convergence $m$ (say $m=2^{10}$ or $m=2^{20}$) induce certain monotone step functions from which one can filter relevant information needed to globally separate and compute the real roots of a function on a…
The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…
The number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors. For the polynomial selection stage of the algorithm, Montgomery proposed a method of generating polynomials…
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…
In this paper, we study a particular family of polyhypercubes in dimension $d\geq 3$, the directed plateau polyhypercubes, according to to the width and a new parameter the lateral area. We give an explicit formula and we also propose an…
The automated generation of exercises may substantially reduce the time educators devote to manual exercise design. A major obstacle to the integration of such automation into teaching practice, however, lies in the ability to control the…
Generation of pseudo random sequences by cellular automata, as well as by hybrid cellular automata is surveyed. An application to the fast evaluation and FPGA implementation of some classes of boolean functions is sketched out.
In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an…