Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.     

On degree sequences of undirected,directed, and bidirected graphs

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs:

• –Erdős–Gallai-type results: characterization of net-degree sequences,
• –Havel–Hakimi-type results: complete sets of degree-preserving operations,
• –Extremal degree sequences: characterization of uniquely realizable sequences, and
• –Enumerative aspects: counting formulas for net-degree sequences.

To underline the similarities and differences to their (un)directed counterparts, we briefly survey the undirected setting and we give a thorough account for digraphs with an emphasis on the discrete geometry of degree sequences. In particular, we determine the tight and uniquely realizable degree sequences for directed graphs.     

Linear relations between pattern sequences in a ��q,r��-numeration system

We investigate linear relations between pattern sequences in a ??q,r??-numeration system, and give a basis of the module generated by pattern sequences for words of length not exceeding l. The expressions of pattern sequences using the basis are also studied. Similar results are obtained for the module generated by all pattern sequences.     

The central limit problem for,infinite products of,and Lévy processes of renewal sequences

Summary Necessary and sufficient conditions are obtained for: (i) convergence of row products from a null triangular array of renewal sequences to a particular renewal sequence and (ii) convergence of an infinite product of renewal sequences to a non-trivial limit. These products correspond to intersections of regenerative phenomena of integers. Lévy processes of such regenerative phenomena are constructed.Research partially supported by National Science Foundation Grant MCS 78-01168     

Binary permutation sequences as subsets of Levenshtein codes,spectral null codes,run-length limited codes and constant weight codes

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes. This paper was presented in part at the IEEE Information Theory Workshop, Chengdu, China, October, 2006.     

Almost periodicity,finite automata mappings,and related effectiveness issues

We introduce a class of eventually almost periodic sequences where some suffix is almost periodic (i.e., uniformly recurrent). The class of generalized almost periodic sequences includes the class of eventually almost periodic sequences, and we prove this inclusion to be strict. We also prove that the class of eventually almost periodic sequences is closed under finite automata mappings and finite transductions. Moreover, we obtain an effective form of this result. In conclusion we consider some algorithmic questions related to the almost periodicity.     

Dual Numbers,Weighted Quivers,and Extended Somos and Gale-Robinson Sequences

We investigate a general method that allows one to construct new integer sequences extending existing ones. We apply this method to the classic Somos-4 and Somos-5, and the Gale-Robinson sequences, as well as to more general class of sequences introduced by Fordy and Marsh, and produce a great number of new sequences. The method is based on the notion of “weighted quiver”, a quiver with a \(\mathbb {Z}\)-valued function on the set of vertices that obeys very special rules of mutation.     

Uniform Distribution,Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.     

A short remark on Fibonacci-type sequences,Möbius strips and the ψ-function

A base for linear recursive sequences, such as the sequence of Fibonacci numbers, is defined within the framework of the sum of the digits of a number. Examples of bases of a number of such sequences are then outlined, and a Möbius strip is also used to illustrate the effects diagrammatically.     

On discrete Fourier transform,ambiguity, and Hamming-autocorrelation of pseudorandom sequences

We estimate discrete Fourier transform, ambiguity, and Hamming-auto-correlation of \(m\) -ary sequences in terms of their (periodic) correlation measure of order 4. Roughly speaking, we show that every pseudorandom sequence, that is, any sequence with small correlation measure up to a sufficiently large order, cannot have a large discrete Fourier transform, ambiguity, or Hamming-autocorrelation. Conversely, there are sequences, for example the two-prime generator, with large correlation measure of order 4 but small discrete Fourier transform, ambiguity, autocorrelation, and Hamming-autocorrelation.     

Cayley compositions,partitions, polytopes,and geometric bijections

In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun?s conjecture proved recently by the authors [15].     

LAD models,trees, and an analog of the fundamental theorem of arithmetic

Motivated by applications of Logical Analysis of Data (LAD) in medical contexts, original discrete optimization problems are proposed. When a patient arrives with a presumption of a disease, he is submitted to a sequence of tests. From one patient to another, the tests allowing to detect the disease may vary. A subset of tests whose results detect the disease in a given part of the population is called a pattern, which has its own prevalence in the population.If there is only a limited number of tests that can be done, which ones must be selected in order to maximize the number of spotted patients? Or, if each test has a cost, in which order the tests have to be done, in order to minimize the cost? It is the kind of questions that are investigated in this paper. For various special cases, polynomial algorithms are proposed, especially when the hypergraph whose vertices are the tests and whose edges are the patterns is a tree graph.One of these questions involves a criterion which is not a number but a sequence of numbers. The objective is then to find the best sequence for the lexicographic order. To solve this question, a new product on finite sequences is defined, namely the maximum shuffle product, which maps two sequences to their shuffle that is maximal for the lexicographic order. Surprisingly, this product leads to a theorem similar to the fundamental theorem of arithmetic: every sequence can be written uniquely as the product of prime sequences, with the suitable definition of prime sequences.     

Jacobi-like forms,differential equations,and Hecke operators

We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.     

Bessel sequences,wavelet frames,duals and extensions

From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results.     

Properties of Cordonnier,Perrin and Van der Laan numbers

This paper aims to explore some properties of certain third-order linear sequences which have some properties analogous to the better known second-order sequences of Fibonacci and Lucas. Historical background issues are outlined. These, together with the number and combinatorial theoretical results, provide plenty of pedagogical opportunities for further development.     

Heap games, numeration systems and sequences

We propose and analyze a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences and demonstrate its equivalence with the class of sequences defined for the strategy of our games.     

Pairs,sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.      

GR(4,m)上本原序列最高权位的密码特征

刻划了特征为4的Galois环上本原序列最高权位序列的相关函数、线性度和元素分布等密码特征。     

Complexity,Universal Libraries,dna Sequences

Complexity studies are burgeoning into an ever-increasing number of fields. This paper seeks to abstract from a large number of non-biological complexity areas some exemplar common thread representative of many complexity areas. One such common thread is identified, a thread leading immediately to the notion of Universal Library, as an additional complexity area, containing all possible texts and, correspondingly, encompassing all possible knowledge (amidst, naturally, a welter of partially and wholly senseless texts). This notion is then wedded to an elementary portion of number theory, to indicate where replicas of universal libraries exist and with what certain attributes. Next, a representative example of a biological complexity area—DNA sequences—is introduced. While DNA is often casually termed a library specifying the heritability machinery of individuals and species, this paper briefly explores whether explicit relations exist between DNA sequences and universal libraries, to help test, inter alia, the strength of the initially identified complexity thread to link non-biological and biological complexity areas. This theme of common linking threads has many interesting open research issues.     

Linear Complexity,k-Error Linear Complexity,and the Discrete Fourier Transform

Complexity measures for sequences of elements of a finite field play an important role in cryptology. We focus first on the linear complexity of periodic sequences. By means of the discrete Fourier transform, we determine the number of periodic sequences S with given prime period length N and linear complexity L_N, 0(S)=c as well as the expected value of the linear complexity of N-periodic sequences. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity L_N, k(S) of sequences S with period length N. For some k and c we determine the number of periodic sequences S with given period length N and L_N, k(S)=c. For prime N we establish a lower bound on the expected value of the k-error linear complexity.