On expansions of (Z,+,0)

A Pisot numeration system [Formula: see text] for [Formula: see text] is one where the sequence of positive integers [Formula: see text], with [Formula: see text], is generated by a recurrence whose polynomial is the minimal polynomial of a Pisot number. The Zeckendorf numeration [Formula: see text] is the simplest such example. We define generalised equations of [Formula: see text]-Mahler type, and we show that if a sequence over a commutative ring is [Formula: see text]-regular, then it is the sequence of coefficients of a series which is a solution of a [Formula: see text]-Mahler equation. Conversely, if the [Formula: see text]-Mahler equation is isolating, then its solutions define [Formula: see text]-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating [Formula: see text]-Mahler equations whose solutions do not define [Formula: see text]-regular sequences. Our proof yields a new construction of weighted automata that generate classical [Formula: see text]-regular sequences. Our results can be generalised to numeration systems generated by recurrences whose characteristic polynomial is the minimal polynomial of a Pisot number.

A set of n homogeneous polynomials in n variables is a regular sequence if the associated polynomial system has only the obvious solution (0,0,...,0) . Denote by p_k(n) the power sum symmetric polynomial in n variables x_1^k+x_2^k+...+x_n^k . The interpretation of the q -analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets A \subset \mathbb N^* of cardinality n such that the set of polynomials p_a(n) with a\in A is a regular sequence. We prove that a necessary condition is that n! divides the product of the degrees of the elements of A . To find an easily verifiable sufficient condition turns out to be surprisingly difficult already for n = 3 . Given positive integers a < b < c with gcd (a,b,c) = 1 , we conjecture that p_a(3), p_b(3), p_c(3) is a regular sequence if and only if abc ≡ 0\ (\mathrm{mod}\ 6) . We provide evidence for the conjecture by proving it in several special instances.

Special rank-one perturbations

A kind of fixed-point problem in the area of discrete tomography is proposed and investigated.Our chief concern in this paper is the case of square windows in the plane.Dealing with the arrays which are bounded, of polynomial growth, and finite-ring-valued, one comes across several interesting phenomena of combinatorial and arithmetic nature.

When asymptotically analysing the summatory function of a $q$-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the "shape" (in particular the growth) of the asymptotic formula. Existing general results for determining the precise behavior (including the Fourier coefficients of the appearing fluctuations) have previously been restricted by a technical condition on these eigenvalues. The aim of this work is to lift these restrictions by providing a insightful proof based on generating functions for the main pseudo Tauberian theorem for all cases simultaneously. (This theorem is the key ingredient for overcoming convergence problems in Mellin--Perron summation in the asymptotic analysis.) One example is discussed in more detail: A precise asymptotic formula for the amount of esthetic numbers in the first~$N$ natural numbers is presented. Prior to this only the asymptotic amount of these numbers with a given digit-length was known.

CHAPTER 4 - Recurrence Relations and Algebraic Equations

In the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems.

Linear dynamical systems over finite rings

This goal of this chapter is to find finitely many canonical representatives of each similarity class of square matrices with entries in a field and correspondingly of each isomorphism class of linear maps from a finite-dimensional vector space to itself.Section 1 frames the problem in more detail. Section 2 develops the theory of determinants over a commutative ring with identity in order to be able to work easily with characteristic polynomials det(?I — A). The discussion is built around the principle of "permanence of identities," which allows for passage from certain identities with integer coefficients to identities with coefficients in the ring in question.Section 3 introduces the minimal polynomial of a square matrix or linear map. The Cayley-Hamilton Theorem establishes that such a matrix satisfies its characteristic equation, and it follows that the minimal polynomial divides the characteristic polynomial. It is proved that a matrix is similar to a diagonal matrix if and only if its minimal polynomial is the product of distinct factors of degree 1. In combination with the fact that two diagonal matrices are similar if and only if their diagonal entries are permutations of one another, this result solves the canonical-form problem for matrices whose minimal polynomial is the product of distinct factors of degree 1.Section 4 introduces general projection operators from a vector space to itself and relates them to vector-space direct-sum decompositions with finitely many summands. The summands of a directsum decomposition are invariant under a linear map if and only if the linear map commutes with each of the projections associated to the direct-sum decomposition.Section 5 concerns the Primary Decomposition Theorem, whose subject is the operation of a linear map L: V ? V with V finite-dimensional. The statement is that if L has minimal polynomial \( P_1 (\lambda )^{l_1 } \cdots P_k (\lambda )^{l_k } \) with the Pj (?) distinct monic prime, then V has a unique direct-sum decomposition in which the respective summands are the kernels of the linear maps \( P_j (L)^{l_j } \), and moreover the minimal polynomial of the restriction of L to the j th summand is \( P_j (\lambda )^{l_j } \).Sections 6–7 concern Jordan canonical form. For the case that the prime factors of the minimal polynomial of a square matrix all have degree 1, the main theorem gives a canonical form under similarity, saying that a given matrix is similar to one in "Jordan form" and that the Jordan form is completely determined up to permutation of the constituent blocks. The theorem applies to all square matrices if the field is algebraically closed, as is the case for C. The theorem is stated and proved in Section 6, and Section 7 shows how to make computations in two different ways.

We consider the problem of computing a linear recurrence relation (or equivalently a Linear Feedback Shift Register) of minimum order for a finite sequence over a field, with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is non-zero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we will call it bidirectional. If the field is finite, a sequence is periodic if and only if it admits a bidirectional linear recurrence relation. For solving the above problem we propose an algorithm similar to the Berlekamp-Massey algorithm and prove its correctness. We also describe the set of all solutions to this problem and prove some properties of the minimal polynomials of initial segments of the sequence.

A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

In this paper, we studied symmetries of string scattering amplitudes in the high energy limits of both the fixed angle or Gross regime (GR) and the fixed momentum transfer or Regge regime (RR). We calculated high energy string scattering amplitudes (SSA) at arbitrary mass levels for both regimes. We discovered the infinite linear relations among fixed angle string amplitudes and the ifinite recurrence relations among Regge string amplitudes. The linear relations we obtained in the GR corrected the saddle point calculations by Gross, Gross and Mende. In addition, for the high energy closed string scatterings, our results differ from theirs by an oscillating prefactor which was crucial to recover the KLT relation valid for all energies. We showed that all the high energy string amplitudes can be solved using the linear or recurrence relations, so that all the string amplitudes can be expressed in terms of a single string amplitude. We further found that, at each mass level, the ratios among the fixed angle amplitudes can be extracted from the Regge string scattering amplitudes. Finally, we reviewed the recent developments on the discovery of infinite number of recurrence relations valid for all energies among Lauricella SSA. The symmetries or relations among SSA at various limits obtained previously can be exactly reproduced. It leads us to argue that the known S L ( K + 3 , C ) dynamical symmetry of the Lauricella function may be crucial to probe spacetime symmetry of string theory.

This dissertation is a study on Generating Functions and the Linear Recurrence Relations and is divided in three chapters in the following manner:On the first chapter we present Ordinary Generating Functions and Exponential Generating Functions along with their definitions, properties and a few theorems with their respective demonstrations.Still on this chapter we explore the concepts of Partitions and some Partition Generating Functions.On the second chapter we explore the concept of Recurrence Relations, specifically the Linear ones.We show a few methods for the resolution of Recurrence Relations, highlighting the resolution based on Generating Functions and also an interesting method using matrix diagonalization.The methods are very well detailed and are easy to comprehend.Chapter 3 is made of four applications on Recurrence Relations and Generating Functions.(n,k)-block fountain is a method of piling up coins on a plane obeying a few criteria.Steiner's pizza consists of dividing the pizza and obtaining the greatest number of slices with n cuts, triangulation of a convex n-agon is a problem of diving polygons in triangles and lastly one of the most famous sequences in Mathematics, the Fibonacci Sequence, where we show the sequence recurrence relation and its respective generating function.

Polynomials and matrices have played a very important role in the development of different branches of mathematics. Indeed, several mathematicians have introduced classical polynomials very useful for the scientific community such as the Lagrange’s interpolation polynomials, the Chebyshev’s polynomials and the Bernstein’s polynomials [1,2]. Also, there is a strong link between polynomials and matrices through the notions of the determinant, the characteristic polynomial and the minimal polynomial. Similarly, we will introduce in this article two polynomials which we will call twin polynomials as well as a matrix called kernels matrix. Finally, we will present some applications such as the resolution of recurrent sequences with second member and the establishment of several sum formulas.