A matrix approach for generalizing two curious divisibility properties

Let $p$ be a prime number with $p\neq 2$. We consider sequences generated by $n$th order linear recurrence relations over the finite field $Z_p$. In the first part of this paper we generalize some of the ideas in [6] to $n$th order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in $Z_p$ of multiplicity $n$. In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.

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

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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.

Sine sequence of second order linear recurrences

A block parallel partitioning method for evaluating general linearmth order recurrence equations is presented and applied to solve the eigenvalues of symmetric tridiagonal matrices. The algorithm is based on partitioning, in a way that ensures load balance during computation. This method is applicable to both shared memory and distributed memory MIMD systems. The algorithm achieves a speedup ofO(p) on a parallel computer withp-fold parallelism, which is linear and is greater than the existing results, and the data communication between processors is less than that required for other methods. For solving symmetric tridiagonal eigenvalue problems, our method is faster than the best previous results. The results were tested and evaluated on an MIMD machine, and were within 79% to 96% of the predicted performance for the second order linear recurrence problem.

Polynomial values in linear recurrences, II

We establish a connection between the hypergeometric solutions of a first order linear recurrence systems and the determinant of the system matrix. This enables us to find hypergeometric solutions for systems in a way similar to the scalar case. Our result works in the in the single basic and in the multibasic case.

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

Lucas Functions is a special form of second order linear recurrence relation. This function has been used in the LUC Cryptosystems. The encryption process of this system is the computations of V(e), while the decryption process is done by the computations of V(d). The V(e) and V(d) are both Lucas Function. The performances of computations of LUC are influence by the size of e and d. It is also depends on the size of message, M and two primes p and q. In the case of e, d, M, p and q are in a large number, we are sure that the existing algorithm would suffers a huge computations time and spaces. In this paper, we are presenting a new and efficient computations algorithm for LUC Cryptosystems. We have found that the binary sequence used in a new algorithm is shorter than a special sequence used in an existing algorithm. Once we get a generated binary sequence, we shall use this sequence to perform the LUC computations. We are examining the efficiency of this new algorithm by comparing the computation time with an existing algorithm.

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

In this paper, we investigate a public key cryptosystem which is derived from a third order linear recurrence relation and is analogous to the RSA and LUC cryptosystems. The explicit formulation involves a generalisation of the rule for composition of powers and of the calculus of the Euler totient function which underlie the algebra of the RSA cryptosystem. The security of all these systems appears to be comparable and to depend on the intractability of factorization but the systems do not seem to be mathematically equivalent.

In this paper, we investigate the generalized Guglielmo sequences and we deal with, in detail, four special cases, namely, triangular, triangular-Lucas, oblong and pentagonal sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Let G be a closed subgroup of the general linear group. Let BFI be the classifying space for G-foliated microbundles of rank q. (The G-foliation is not assumed to be integrable.) The homotopy fiber Fit of the classifying map v: BI -3 BG is shown to be (q 1)-connected. For the orthogonal group, this implies FRIY is (q 1)-connected. The indecomposable classes in H*(RWq) therefore are mapped to linearly independent classes in H*(FRIF); the indecomposable variable classes are mapped to independently variable classes. Related results on the homotopy groups r* (FRq) also follow. 1. The main theorem. Let BR fq be the Haefliger classifying space for Riemannian foliations, BO(q) the classifying space for O(q)-bundles and t: BR j'q BO(q) the map classifying the normal bundle of the universal R7q-structure on BR j'q [3]. Let FR Iq be the homotopy theoretic fiber of t. H*() will denote singular cohomology with real coefficients. In this note we show THEOREM 1.1. FR F'q is (q 1)-connected. This implies there is a section of v' over the q-skeleton of BO(q), so z*: Hq(BO(q)) -Hq(BR Iq) is injective. On the other hand, the vanishing Theorem of J. Pastemack [9] implies p*: HqI l(BO(q)) -> H 1(BR Iq) is the zero map. Theorem 1.1 is therefore the best result possible for q = 4k + 3. For other q, it would be interesting to know whether FR rq has higher connectivity. Theorem 1.1 is a special case of a more general result. Let G C Gl(q, R) be a closed subgroup. A foliation on a manifold M is said to be a G-foliation [1], [7] if there is given (i) a model manifold B of dimension q with a G-structure on TB, (ii) an open covering { U, } of M and local submersions fra: Ua -B defining the foliation such that the transition functions Ya are local G-morphisms of B. A G-foliation is integrable if it is modeled on Rq with the flat G-structure. A classifying space for G-foliations is constructed as follows: Let DL(G, Rq) denote the total space of the sheaf of local C -sections of the bundle Rq x Gl(q, R)/ G -> Rq. This is a (non-Hausdorff) C '-manifold, with a canonical Gstructure. Let 9G be the pseudogroup of all local, C ', G-diffeomorphisms of Received by the editors October 18, 1979 and, in revised form, February 22, 1980. AMS (MOS) subject classifications (1970). Primary 57D30, 57D20; Secondary 55F40, 55E15.