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We consider weighted spaces of entire functions obtained as the images of spaces of ultradistributions of minimal type and normal type on the real line under the Fourier–Laplace transform. The divisors of these spaces are studied. Namely, we find conditions on a perturbing sequence under which the sequence of integers perturbed by it will be the zero set of an entire function that is a divisor of one of the above-mentioned spaces. [ABSTRACT FROM AUTHOR]

We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and six sequences were extended. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for further analysis and the database was constructed to be extensible to invariants not considered in this work. [ABSTRACT FROM AUTHOR]

In this work, we define a quadra Fibona-Pell integer sequence $W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}$ for $n\geq4$ with initial values $W_{0}=W_{1}=0$, $W_{2}=1$, $W_{3}=3$, and we derive some algebraic identities on it including its relationship with Fibonacci and Pell numbers. [ABSTRACT FROM AUTHOR]

We prove that five ways to define entry A086377 in the OEIS do lead to the same integer sequence. [ABSTRACT FROM AUTHOR]

Networks have an important role in our daily lives. The effectiveness of the network decreases with the breaking down of some vertices or links. Therefore, a less vulnerable communication network is required for greater stability. Vulnerability is the measure of resistance of the network after failure of communication links. In this article, a graph has been taken for modeling a network and integrity as a measure of vulnerability. The approach is to estimate the integrity or upper bound of integrity of at least one connected graph or network constructed from the given graphic integer sequence. Experiments have been done with random graphs, complex networks and also a comparison between two parameters, namely the vertex connectivity and graph integrity as a measure of the network vulnerability have been carried out by removing vertices randomly from various complex networks. A comparison with the existing method shows that the algorithm proposed in this article provides a much better integrity measurement. [ABSTRACT FROM AUTHOR]

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we will explore alternative varieties of integer multiplication by modifying the product axiom of Dedekind–Peano arithmetic (PA). In addition to studying the elementary properties of the new models of arithmetic that arise, we will see that the truth or falseness of some classical conjectures will be equivalently in the new ones, even though these models have non-commutative and non-associative product operations. To pursue this goal, we will generalize the divisor and prime number concepts in the new models. Additionally, we will explore various general number properties and project them onto each of these new structures. This fact will enable us to demonstrate that indistinguishable properties on PA project different properties within a particular model. Finally, we will generalize the main idea and explain how each integer sequence gives rise to a unique arithmetic structure within the integers. [ABSTRACT FROM AUTHOR]

Sylvester's theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters: t 1 , being even or odd, and t 2 , congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester's theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers. [ABSTRACT FROM AUTHOR]

This paper proposes a novel image encryption algorithm based on an integer form of chaotic mapping and 2-order bit compass diffusion technique. Chaotic mapping has been widely used in image encryption. If the floating-point number generated by chaotic mapping is applied to image encryption algorithm, it will slow encryption and increase the difficulty of hardware implementation. An innovative pseudo-random integer sequence generator is proposed. In chaotic system, the result of one-iteration is used as the shift value of two binary sequences, the original symmetry relationship is changed, and then XOR operation is performed to generate a new binary sequence. Multiple iterations can generate pseudo-random integer sequences. Here integer sequences have been used in scrambling of pixel positions. Meanwhile, this paper demonstrates that there is an inverse operation in the XOR operation of two binary sequences. A new pixel diffusion technique based on bit compass coding is proposed. The key vector of the algorithm comes from the original image and is hidden by image encryption. The efficiency of our proposed method in encrypting a large number of images is evaluated using security analysis and time complexity. The performance evaluation of algorithm includes key space, histogram differential attacks, gray value distribution(GDV),correlation coefficient, PSNR, entropy, and sensitivity. The comparison between the results of coefficient, entropy, PSNR, GDV, and time complexity further proves the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences. [ABSTRACT FROM AUTHOR]

Text The sequence of middle divisors is shown to be unbounded. For a given number n , a n , 0 is the number of divisors of n between n / 2 and 2 n . We explicitly construct a sequence of numbers n ( i ) and a list of divisors in the interesting range, so that the length of the list goes to infinity as i increases. Video For a video summary of this paper, please visit https://youtu.be/drODtRj0gjM . [ABSTRACT FROM AUTHOR]

An integer sequence a : N → Z is called a Newton sequence generated by the sequence of integers c : N → Z, if the following Newton identities hold... In particular, f may be the Möbius function, the Euler totient function or the Jordan totient function. [ABSTRACT FROM AUTHOR]

In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set X its cardinal | X | induces the sequential cardinal that associates to each augmented semi-simplicial finite set X an augmented sequence | X | n of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets. [ABSTRACT FROM AUTHOR]

We find the asymptotic behaviour of a sequence considered by Shaun Cooper. [ABSTRACT FROM AUTHOR]

In this brief note, we formulate some modifications of the Pell-Jacobsthal numbers, which belong to a more general class, the Lucas sequences. [ABSTRACT FROM AUTHOR]

In this paper, by utilizing d-dimensional single-particle states, three semiquantum cryptography protocols, i.e., the multi-party semiquantum private comparison (MSQPC) protocol, the multi-party semiquantum multiplication (MSQM) protocol and the multi-party semiquantum summation (MSQS) protocol, can be achieved simultaneously under the assistance of two semi-honest quantum third parties (TPs). Here, the proposed MSQPC scheme is the only protocol which is devoted to judging the size relationship of secret integers from more than two semiquantum participants without a pre-shared key. And the proposed MSQM protocol absorbs the innovative concept of semiquantumness into quantum multiplication for the first time, which can calculate the modulo d multiplication of private inputs from more than two semiquantum users. As for the proposed MSQS protocol, it is the only semiquantum summation protocol which aims to accomplish the modulo d addition of more than three semiquantum users' private integers. Neither quantum entanglement swapping nor unitary operations are necessary in the three proposed protocols. The security analysis verifies in detail that both the external attacks and the internal attacks can be resisted in the three proposed protocols. [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a given integer. For any arbitrary positive integer n, we determine the least integer lk(n) such that every positive integer sequence with terms not exceeding n, sum kn and length at least lk(n) can be separated into k subsequences each with sum n. The methods are elementary. [ABSTRACT FROM AUTHOR]

For every positive integer n and every δ ∈ 0 , 1 , let B (n , δ) denote the probabilistic model in which a random set A ⊆ { 1 , ... , n } is constructed by choosing independently every element of { 1 , ... , n } with probability δ. Moreover, let (u k) k ≥ 0 be an integer sequence satisfying u k = a 1 u k − 1 + a 2 u k − 2 , for every integer k ≥ 2 , where u 0 = 0 , u 1 ≠ 0 , and a 1 , a 2 are fixed nonzero integers; and let α and β , with | α | ≥ | β | , be the two roots of the polynomial X 2 − a 1 X − a 2. Also, assume that α / β is not a root of unity. We prove that, as δ n / log ⁡ n → + ∞ , for every A in B (n , δ) we have log ⁡ lcm (u a : a ∈ A) ∼ δ Li 2 (1 − δ) 1 − δ ⋅ 3 log ⁡ | α / (a 1 2 , a 2) | π 2 ⋅ n 2 with probability 1 − o (1) , where lcm denotes the lowest common multiple, Li 2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ = 1. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ = 1 , and is motivated by an asymptotic formula for lcm (A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui. [ABSTRACT FROM AUTHOR]

This work considers fractional operators (derivatives and integrals) as surfaces  f (x , α)  subject to the function constraints defined by integer operators, which is a mandatory requirement of any fractional operator definition. In this respect, the problem can be seen as the problem of generating a surface constrained at some positive integer values of  α  for fractional derivatives and at some negative integer values for fractional integrals. This paper shows that by using the Theory of Functional Connections, all (past, present, and future) fractional operators can be approximated at a high level of accuracy by smooth surfaces and with no continuity issues. This practical approach provides a simple and unified tool to simulate nonlocal fractional operators that are usually defined by infinite series and/or complicated integrals. [ABSTRACT FROM AUTHOR]

Let Sn,k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Waiters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn,k is connected is at least ε, then the probability that Sn,k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s (n) with s (n) = o(log n), the probability Sn,k(n) + ⌊C's log log n⌋ is s-connected (i.e. remains connected after the deletion of any s -- 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Waiters (2009). [ABSTRACT FROM AUTHOR]

Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1, . . ., mn, . . . , and on a sequence of complex numbers z1 , . . . , zn , . . . of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived. [ABSTRACT FROM AUTHOR]

We study the conjugation of overpartitions and give the generating function for the number of self-conjugate overpartitions of an integer. Following the recent introduction of over q-binomial coefficients, we obtain the over q-analogue of the Chu-Vandermonde identity. Consequently a new generating function for the number of overpartitions is proved. We also give a new over q-analogue of the Chu-Vandermonde identity. [ABSTRACT FROM AUTHOR]

Many sequences of p -adic integers project modulo p α to p -automatic sequences for every α ≥ 0 . Examples include algebraic sequences of integers, which satisfy this property for every prime p , and some cocycle sequences, which we show satisfy this property for a fixed p . For such a sequence, we construct a profinite automaton that projects modulo p α to the automaton generating the projected sequence. In general, the profinite automaton has infinitely many states. Additionally, we consider the closure of the orbit, under the shift map, of the p -adic integer sequence, defining a shift dynamical system. We describe how this shift is a letter-to-letter coding of a shift generated by a constant-length substitution defined on an uncountable alphabet, and we establish some dynamical properties of these shifts. [ABSTRACT FROM AUTHOR]

An integer sequence is said to be 3-free if no three elements form an arithmetic progression. A Stanley sequence { a n } is a 3-free sequence constructed by the greedy algorithm. Namely, given initial terms a 0 < a 1 < ⋯ < a k , each subsequent term a n > a n − 1 is chosen to be the smallest such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth: Type 1 sequences satisfy a n = Θ ( n log 2 3 ) and appear well-structured, while Type 2 sequences satisfy a n = Θ ( n 2 / log n ) and appear disorderly. In this paper, we define the notion of regularity , which is based on local structure and implies Type 1 asymptotic growth. We conjecture that the reverse implication holds. We construct many classes of regular Stanley sequences, which include all known Type 1 sequences as special cases. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate the surprising fact that certain Stanley sequences possess proper subsets that are also Stanley sequences. [ABSTRACT FROM AUTHOR]

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function x x , establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function x x and for related functions and integer sequences. [ABSTRACT FROM AUTHOR]

The partition method for a power series expansion is a method that utilizes standard integer partitions to evaluate the coefficients in power series expansions and generating functions. Here it is shown how an existing code based on the method can be adapted to deal with partitions homologous to standard integer partitions. Consequently, with the aid of further processing in Mathematica this work presents polynomial expressions for both kinds of Stirling numbers in two cases where: (1) the secondary variable is fixed and (2) it becomes a variable. Interestingly, the second case requires the results produced in the first case for the Stirling numbers of the first kind. In the second case the highest power in the primary variable is found to be dependent upon the secondary variable and the coefficients become polynomials in terms of this variable, whereas in the first case the coefficients are rational. The results represent a major advance on already published results of both kinds of the Stirling numbers due to the introduction of partitions into the analysis. Finally, new results for the related Worpitzky numbers and Stirling polynomials are also presented. [ABSTRACT FROM AUTHOR]

For a sequence f = (f1; f2, ....) of nonzero integers, define Δ(f) to be the numerical triangle that lists all the generalized binomial coefficients { n k } f = fnfn1 ... fnk+1 fk fk1 ... f1: Sequence f is called binomid if all entries of Δ(f) are integers. For I = (1; 2; 3;: : :), Δ(I) is Pascal's Triangle and I is binomid. Surprisingly, every row and column of Pascal's Triangle is also binomid. For any f, the rows and columns of (f) generate their own triangles and all those triangles t together to form the "Binomid Pyramid" BP(f). Sequence f is binomid at every level if all entries of BP(f) are integers. We prove that several familiar sequences are binomid at every level. For instance, every sequence L satisfying a linear recurrence of order 2 has that property provided L(0) = 0. The sequences I, the Fibonacci numbers, and (2n - 1)n≥1 provide examples. [ABSTRACT FROM AUTHOR]

Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra A. In this setting, we prove that the indecomposable uniserial A -modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the n + 3 -gon associated with the cluster-tilted algebra of type A n to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type A n . In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules. [ABSTRACT FROM AUTHOR]

Let α = { A n } n ≥ 1 be a sequence of left-open and right-closed intervals which partition ( 0 , 1 ] . The α -Lüroth transformation L α is defined as an infinite piecewise linear map which maps A n linearly onto ( 0 , 1 ] for every n ≥ 1 . Then every point x ∈ ( 0 , 1 ] is attached with a finite or infinite integer sequence { ℓ n } n ≥ 1 by looking at the coding of its trajectory. In this note, we consider the size of the recurrent set in such a system. More precisely, let x 0 ∈ ( 0 , 1 ] with an infinite α -Lüroth expansion, and { t n } n ≥ 1 an arbitrary non-decreasing sequence of natural numbers. The recurrence set of α -Lüroth transformation L α is defined as F ( x 0 ) = { x ∈ [ 0 , 1 ] : L α n ( x ) ∈ A ( ℓ 1 , ℓ 2 , ⋯ , ℓ t n ) ( x 0 ) , F ( x 0 ) = { infinitely often n ∈ N } , where A ( ℓ 1 , ℓ 2 , ⋯ , ℓ t n ) ( x 0 ) denotes the t n -th cylinder containing x 0 in α -Lüroth expansion. The Hausdorff dimension of F ( x 0 ) is obtained. [ABSTRACT FROM AUTHOR]

We give a different presentation of a recent bijection due to Chapuy and Dolega for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier-Di Francesco-Guitter-like generalization of the Cori-Vauquelin-Schaeffer bijection in the context of general nonorientable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and this allows us to recover a famous asymptotic enumeration formula found by Gao. [ABSTRACT FROM AUTHOR]

This paper is devoted to the characterization of spectrum candidates with a new tree structure to be the spectra of a spectral self-similar measure μ N , D generated by the finite integer digit set D and the compression ratio N − 1 . The tree structure is introduced with the language of symbolic space and widens the field of spectrum candidates. The spectrum candidate considered by Łaba and Wang is a set with a special tree structure. After showing a new criterion for the spectrum candidate with a tree structure to be a spectrum of μ N , D , three sufficient and necessary conditions for the spectrum candidate with a tree structure to be a spectrum of μ N , D were obtained. This result extends the conclusion of Łaba and Wang. As an application, an example of spectrum candidate Λ (N , B) with the tree structure associated with a self-similar measure is given. By our results, we obtain that Λ (N , B) is a spectrum of the self-similar measure. However, neither the method of Łaba and Wang nor that of Strichartz is applicable to the set Λ (N , B) . [ABSTRACT FROM AUTHOR]

Let { a i j } ( 1 ≤ i , j < ∞ ) be i.i.d. real-valued random variables with zero mean and unit variance and let an integer sequence ( N m ) m = 1 ∞ satisfy m / N m ⟶ z for some z ∈ ( 0 , 1 ) . For each m ∈ N denote by A m the N m × m random matrix ( a i j ) ( 1 ≤ i ≤ N m , 1 ≤ j ≤ m ) and let s m ( A m ) be its smallest singular value. We prove that the sequence ( N m − 1 / 2 s m ( A m ) ) m = 1 ∞ converges to 1 − z almost surely. Our result does not require boundedness of any moments of a i j 's higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold. [ABSTRACT FROM AUTHOR]

Abstract: We present a linear-space data structure that maintains a dynamic set of n points with coordinates of real numbers in the plane to support orthogonal range counting in worst-case time, and insertions and deletions in amortized time. This provides faster support for updates than previous results with the same bounds on space cost and query time. We also consider the same problem for points on a grid, and design the first succinct data structure for a dynamic integer sequence, S, to support range counting queries defined as follows: Given a range, , of indices and a range, , of values, count the number of entries of that store integers from . [Copyright &y& Elsevier]

Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers involved in the different processes are also given. [ABSTRACT FROM AUTHOR]

Let T(G; X, Y) be the Tutte polynomial for graphs. We study the sequence ta,b(n) = T(Kn; a, b) where a, b are integers, and show that for every μ ∈ ℕ the sequence ta,b(n) is ultimately periodic modulo μ provided a ≠ 1 mod μ and b ≠ 1 mod μ. This result is related to a conjecture by A. Mani and R. Stones from 2016. The theorem is a consequence of a more general theorem which holds for a wide class of graph polynomials definable in Monadic Second Order Logic. This gives also similar results for the various substitution instances of the bivariate matching polynomial and the trivariate edge elimination polynomial ξ(G; X, Y, Z) introduced by I. Averbouch, B. Godlin and the second author in 2008. All our results depend on the Specker-Blatter Theorem from 1981, which studies modular recurrence relations of combinatorial sequences which count the number of labeled graphs. [ABSTRACT FROM AUTHOR]

Abstract: In the study of Zeilbergerʼs conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let denote the rising factorial, and let denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from to defined by for some , then for any Schur function , the value is positive. In this paper, we provide an affirmative answer to Lassalleʼs conjecture by using the Laguerre–Pólya–Schur theory of multiplier sequences. [Copyright &y& Elsevier]

Abstract: For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in the case of the Erdős–Gallai criterion (for simple undirected graphs) and the Gale–Ryser criterion (for bipartite graphs), it has been shown that the number of inequalities that must be checked can be reduced significantly. We show that similar reductions hold for the corresponding criteria for many other types of graphs, including bipartite -multigraphs, bipartite graphs with structural edges, directed graphs, -multigraphs, and tournaments. We also prove a reduction for imbalance sequences. [Copyright &y& Elsevier]

Abstract: Let , , , , be a simple continued fraction determined by an infinite integer sequence . We are interested in finding an effective irrationality measure as explicit as possible for the irrational number τ. In particular, our interest is focused on sequences with an upper bound at most , where and . In addition to our main target, arithmetic of continued fractions, we shall pay special attention to studying the nature of the inverse function of . [Copyright &y& Elsevier]

Abstract: Denote by the length of a shortest integer sequence on a circle containing all permutations of the set as subsequences. Hansraj Gupta conjectured in that . In this paper, we confirm the conjecture for the case where is even and show that if is odd. [Copyright &y& Elsevier]

Let F and G be integer polynomials where F has degree at least 2. Define the sequence (an) by an = F(an-1) for all n ≥ 1and a0 = 0. Let BF,G,k be the set of all positive integers nsuch that k | gcd(G(n), an) and if p | gcd(G(n), an) for some p, then p | k. Let AF,G,k be the subset of BF,G,k such that AF,G,k = {n ≥ 1 : gcd(G(n), an) =k}. In this article, we prove that the asymptotic density of AF,G,k and BF,G,k exists for a class of (F,G) and also compute the explicit density of AF,G,k and BF,G,k for G(x) = x. [ABSTRACT FROM AUTHOR]

For every nonconstant monic polynomial g ∈ Z [ X ] , let M (g) be the set of positive integers m for which there exist an integer linear recurrence (s n) n ≥ 0 having characteristic polynomial g and a positive integer M such that (s n) n ≥ 0 has exactly m distinct residues modulo M. Dubickas and Novikas proved that M (X 2 - X - 1) = N . We study M (g) in the case in which g is divisible by a monic quadratic polynomial f ∈ Z [ X ] with roots α , β such that α β = ± 1 and α / β is not a root of unity. We show that this problem is related to the existence of special primitive divisors of certain Lehmer sequences, and we deduce some consequences on M (g) . In particular, for α β = - 1 , we prove that m ∈ M (g) for every integer m ≥ 7 with m ≠ 10 and 4 ∤ m . [ABSTRACT FROM AUTHOR]

The data structure at the core of large-scale search engines is the inverted index, which is essentially a collection of sorted integer sequences called inverted lists. Because of the many documents indexed by such engines and stringent performance requirements imposed by the heavy load of queries, the inverted index stores billions of integers that must be searched efficiently. In this scenario, index compression is essential because it leads to a better exploitation of the computer memory hierarchy for faster query processing and, at the same time, allows reducing the number of storage machines. The aim of this article is twofold: first, surveying the encoding algorithms suitable for inverted index compression and, second, characterizing the performance of the inverted index through experimentation. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence, whose first terms are {0, 1, 1, 2, 3, 5, . . .}, is generated using the recursive formula Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 1. This sequence is one of the most famous integer sequences because of its fascinating mathematical properties and connections with other fields such as biology, art, and music. Closely related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence, whose first terms are {2, 1, 3, 4, 7, 11, . . .}, is generated using the recursive formula Ln+2 = Ln+1 + Ln with L0 = 2 and L1 = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F² 3n+4 + F² 3n+2 is even for all n - N0. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. [ABSTRACT FROM AUTHOR]

• Laguerre transform performs integer based calculation and therefore the operations are faster. • The binary factorization of the transform matrix enables efficient hardware implementation. • The computational complexity of the Laguerre transform is O (n × log(n)). • Proposed method supports widely used image formats such as BMP, PPM, PGM, and TIFF. • Variable payload with considerable stego-image's quality distortion is achieved. • StegExpose tool ensures that the robustness of the stego-images is very high. Steganography, an approach used to conceal information into the digital media, generally works in two domains: spatial and transform. Though spatial domain methods are simpler, but transform domain methods are good at identifying the features which make the end system more secure. In this work, a novel Steganographic scheme based on an integer sequence named Laguerre transform (LT) is proposed. The Cover image is decomposed into non-overlapping m-pixel groups and then each such pixel group is transformed by applying LT. Variable length bits from the secret information are fabricated into the transformed components. A post-embedding adjustment is applied over these components to minimize the distortion. By applying Inverse LT (ILT), the m-pixel groups are re-computed from the resulting adjusted components. Experimental results reveal that disparity between cover and stego-pixels increases as m increases. Proposed scheme offers better stego image and higher payload compared to some state-of-the-art techniques. Code of this method is publicly available here. Image, graphical abstract [ABSTRACT FROM AUTHOR]

The ubiquitous Variable-Byte encoding is one of the fastest compressed representation for integer sequences. However, its compression ratio is usually not competitive with other more sophisticated encoders, especially when the integers to be compressed are small which is the typical case for inverted indexes. This paper shows that the compression ratio of Variable-Byte can be improved by $2\times$ 2 × by adopting a partitioned representation of the inverted lists. This makes Variable-Byte surprisingly competitive in space with the best bit-aligned encoders, hence disproving the folklore belief that Variable-Byte is space-inefficient for inverted index compression. Despite the significant space savings, we show that our optimization almost comes for free, given that: we introduce an optimal partitioning algorithm that does not affect indexing time because of its linear-time complexity; we show that the query processing speed of Variable-Byte is preserved, with an extensive experimental analysis and comparison with several other state-of-the-art encoders. [ABSTRACT FROM AUTHOR]

Highly dispersive S‐boxes are desirable in cryptosystems as nonlinear confusion sub‐layers for resisting modern attacks. For a near optimal cryptosystem resistant to modern cryptanalysis, a highly nonlinear and low differential probability (DP) value is required. We propose a method based on a piecewise linear chaotic map (PWLCM) with optimization conditions. Thus, the linear propagation of information in a cryptosystem appearing as a high DP during differential cryptanalysis of an S‐box is minimized. While mapping from the chaotic trajectory to integer domain, a randomness test is performed that justifies the nonlinear behavior of the highly dispersive and nonlinear chaotic S‐box. The proposed scheme is vetted using well‐established cryptographic performance criteria. The proposed S‐box meets the cryptographic performance criteria and further minimizes the differential propagation justified by the low DP value. The suitability of the proposed S‐box is also tested using an image encryption algorithm. Results show that the proposed S‐box as a confusion component entails a high level of security and improves resistance against all known attacks. [ABSTRACT FROM AUTHOR]