Jordan Forms and $n$th Order Linear Recurrences

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

The solutions of second order linear recurrences with periodical coefficients are shown to obey Benford’s law.

Perfect powers in second order linear recurrences

Part I: Invited lectures. Domain Decomposition Methods M. Sarkis. Modification and Maintenance of ULV Decompositions J.L. Barlow. Advances in Jacobi Methods Z. Drmac, V. Hari, I. Slapnicar. Modelling of curved rods M. Jurak, J. Tambaca, Z. Tutek. Incompressible Newtonian flow through thin pipes E. Marusic-Paloka. Nonlinear Problems in Dynamics by the Finite Element in Time Method N. Kranjcevic, M. Stegic, N. Vrankovic. First Order Eigenvalue Perturbation Theory and the Newton Diagram J. Moro, F.M. Dopi. Part II: Contributed lectures. Microlocal energy density for hyperbolic systems N. Antonic, M. Lazar. Approximate solutions to some second order linear recurrences K. Balla, V. Horvat. Asymptotic Behaviour of Tension Spline Collocation Matrix I. Beros, M. Marusic. Numerical stability of Krylov subspace iterative methods for solving linear systems N. Bosner. Nonlinear Problems in Dynamics by the Finite Element in Time Method N. Kranjcevic, M. Stegic, N. Vrankovic. On directional bias of the Lp-norm T. Marosevic. A note on slip condition on corrugated boundary E. Marusic-Paloka. Relaxation of some energy functionals related to the formation of microstructure A. Raguz. A Coarse Space for Elasticity M. Sarkis. Numerical Approximations of the Sediment Transport Equations L. Sopta, N. Crnjaric-Zic, S. Vukovic. A model of irregular curved rods J. Tambaca. Existence of the density of states for some alloy type models with single site potentials that change sign I. Veselic. On principal eigenvalue of stationary diffusion problem with nonsymmetric coefficients M. Vrdoljak. Qualitative Analysis of some Solutions of Quasilinear System of Differential Equations B. Vrdoljak, A. Omerspahi . High-Order ENO and WENO Schemes with Flux Gradient and Source Term Balancing S. Vukovi , L. Sopta. Index.

We shall consider two curious divisibility properties due to (1, 5). Our main purpose is to generalize these properties for a general second order linear recursion. We use generating matrix approach for our purposes. By using our results, we derive a new recursive identities for two general second order linear recurrences. We give a more general result extending earlier divisibility properties on the similar subject. 2000 Mathematics Subject Classification: 11B37; 15B36

Sine sequence of second order linear recurrences

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

The topic of permutation polynomials over finite fields is important both in theory and in engineering application. Let p be an odd prime number and q a power of p. A necessary and sufficient condition for a family of polynomials over F q to be permutation polynomials is presented. Moreover, the number of permutation polynomials of this type is obtained. The polynomials discussed in this note are generalizations and improvement of those discussed by Xi-Wang Cao(2009).

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

This paper introduces a novel method for selecting parameters of finite fields formed by 2 × 2 matrices over a finite field of integers modulo a prime p. The method aims to simultaneously determine both the field parameters and primitive elements, thereby optimizing the construction of cryptographic algorithms. The proposed approach leverages the properties of quadratic residues and non-residues, simplifying the process of finding matrix field parameters while maintaining computational efficiency. The method is particularly effective when the prime number p is either a Mersenne prime or (p + 1)/2 is also a prime. This study demonstrates that the resulting matrix fields can be practically computed, offering a high degree of flexibility for cryptographic protocols such as key agreement and secure data transmission. Compared to previous methods, the new method reduces the parameter search space and provides a structured way to identify primitive elements without the need for a separate search procedure. The findings have significant implications for the development of efficient cryptographic systems using matrix-based finite fields.

Deterministic polynomial factoring over finite fields: A uniform approach via [formula omitted]-schemes

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Picard group, where $l$ is a prime number different from the characteristic of the base field.

Efficient parallel factorization and solution of structured and unstructured linear systems

Periodicity properties of 𝑘th order linear recurrences whose characteristic polynomial splits completely over a finite field. I