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K.T. Atanassov are Firstly established the Coupled Fibonacci Sequence in 1985. In 1987, The essence of Fibonacci Triple Sequences are examined. Fibonacci Sequence stand out as a kind of super sequence with amazing properties. This is the meteoric expansion in the province of Fibonacci Sequence. Leonardo de Pisa foremost Fibonacci's observation on the growth of the rabbit population as a result in 1202. Triple Fibonacci Sequence are hype in the last years, but Multiplicative Triple Sequence of Recurrence Relations are less known. Extravagant work has been done to course on Fibonacci Triple Sequence in Additive form. In 1995, Multiplicative Coupled Fibonacci Sequence are treated. Our wish of this paper to offer some results of Multiplicative Triple Fibonacci Sequence of fourth order under nine specific schemes. [ABSTRACT FROM AUTHOR]

After proving that numerical sequences such as Fibonacci numbers and prime numbers, can be generated as sequences of equilibrium points of an ideal half-infinite one-dimensional distribution of electric charges, a model of the distribution of primes on the x-axis is proposed, where primes ρ(n) are considered as quantum particles oscillating around the sequence of stationary points r(n) of the Lennard-Jones-like potential of the single-particle Hamiltonian. A particle-counting function πQ(x) is defined over the many-particle system, in the same way as the prime-counting function π(x). Through the application of the Hellmann-Feynman theorem, the existence of a solution n(x) ≈ li(x), coinciding asymptotically with the logarithmic integral function, is proved for the counting function of stationary points, together with the quantum equivalent of the Prime Number Theorem for the particle-counting function πQ(x). The conditions on the sequence of energy eigenvalues E(n) of the system, so that the Riemann hypothesis holds for the function πQ(x), are derived, thus proving the quantum theory results in the same bound of the position of the prime-particle in the n-th quantum state as that derived through analytical methods for the n-th prime. The model suggests new conjectures and statistical procedures which are applied to study the sign of the difference π(x)-li(x) and to explore the region beyond the Skewes number. The data predicted by the theory find confirmation when compared with those known in the literature. A new statistical local test of the Riemann hypothesis, based on the difference π(x)-li(x), is proposed. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence is one of the most fascinating sequences in the field of mathematics. It has been researched on a huge scale in the realm of Number Theory, Applied Mathematics, Computer Science, Biology and Physics. Through this paper, we conduct an indepth study of Tetranacci functions on the real numbers R; i.e., functions φ : R => R such that φ(y + 4) = φ (y + 3) + φ(y + 2) + φ (y + 1) + φ(y),Vy € R. We obtain results for the Tetranacci functions using the concept of f-even and f-odd functions. We also deduce that, if φ is a Tetranacci function, then lim r +n+i ) = αi, where α is a positive real root of y-s-oo tP\y+n) the equation y4 - y3 - y2 - y - 1 = 0 for which α > 1. [ABSTRACT FROM AUTHOR]

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers. [ABSTRACT FROM AUTHOR]

Abstract A pair of sequences of natural numbers is called planar if there exists a simple, bipartite, planar graph for which the given sequences are the degree sequences of its parts. For a pair to be planar, the sums of the sequences have to be equal and Euler's inequality must be satisfied. Pairs that verify these two necessary conditions are called admissible. We prove that a pair of constant sequences is planar if and only if it is admissible (such pairs can be easily listed) and is different from (3 5 | 3 5) and (3 25 | 5 15). [ABSTRACT FROM AUTHOR]

In this paper we discuss the concept of a Z-sequence and use it to define the Fibonacci Z-sequence. Based on Z-sequences one may define analogs of logarithm functions. The special case of the logarithm function associated with the Fibonacci Z-sequence is of interest, since the recursive property of this sequence permits a more detailed study of these functions. They are similar to ordinary logarithm functions which may be based on Z-sequences {an}n∈Z, where a > 1. [ABSTRACT FROM AUTHOR]

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear Σn≡lmodqAf (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+. [ABSTRACT FROM AUTHOR]

Let Cn be a finite cyclic group of order n≥2. Every sequence S over Cn can be written in the form S=(n1g),…,(nlg) where g∈Cn and n1,…,nl∈[1,ord(g)], and the index ind(S) of S is defined as the minimum of (n1+⋯+nl)/ord(g) over all g∈Cn with ord(g)=n. Let d>1 and r≥1 be any fixed integers. We prove that, for every sufficiently large integer n divisible by d, there exists a sequence S over Cn of length |S|≥n+n/d+O(n) having no subsequence T of index ind(T)∈[1,r], which has substantially improved the previous results in this direction. [ABSTRACT FROM AUTHOR]

We derive the largest sizes of ( t , m t + 1 ) and ( t , m t − 1 ) -core partitions with distinct parts, and prove that the numbers of such partitions with the largest sizes are at most 2. This work is motivated by Amdeberhan’s conjecture on ( t , t + 1 ) -core partitions with distinct parts and subsequent research on the enumeration, largest sizes and the average sizes of simultaneous core partitions with distinct parts. [ABSTRACT FROM AUTHOR]

An addition chain for n is defined to be a sequence ( a 0 , a 1 , … , a r ) such that a 0 = 1 , a r = n , and, for any 1 ≤ k ≤ r , there exist 0 ≤ i , j < k such that a k = a i + a j ; the number r is called the length of the addition chain. The shortest length among addition chains for n , called the addition chain length of n , is denoted ℓ ( n ) . The number ℓ ( n ) is always at least log 2 ⁡ n ; in this paper we consider the difference δ ℓ ( n ) : = ℓ ( n ) − log 2 ⁡ n , which we call the addition chain defect. First we use this notion to show that for any n , there exists K such that for any k ≥ K , we have ℓ ( 2 k n ) = ℓ ( 2 K n ) + ( k − K ) . The main result is that the set of values of δ ℓ is a well-ordered subset of [ 0 , ∞ ) , with order type ω ω . The results obtained here are analogous to the results for integer complexity obtained in [1] and [3] . We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length. [ABSTRACT FROM AUTHOR]

In this paper, we solve a few Diophantine equations linked to balancing numbers and factorials. The basic problem consists of solving the equation By = x! in positive integers x, y, which has only one nontrivial solution B2 = 6 =3!, as a direct consequence of the theorem of F. Luca [5]. A more difficult problem is to solve By = x2!/x1!, but we were able to handle it under some conditions. Two related problems are also studied. [ABSTRACT FROM AUTHOR]

. In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number x, by W(x) = Σ +∞ n=0 λn cos (2π Nnb x) where A and Nb are two real numbers such that 0 <π< 1, Nb Є N and π Nb > 1, using a sequence a graphs that approximate the studied one. [ABSTRACT FROM AUTHOR]

Let D be a finite and simple digraph with vertex set V ( D ) . For a vertex v ∈ V ( D ) , the degree d ( v ) of v is defined as the minimum value of its out-degree d + ( v ) and its in-degree d − ( v ) . If D is a graph or a digraph with minimum degree δ and edge-connectivity λ , then λ ≤ δ . A graph or a digraph is maximally edge-connected if λ = δ . A graph or a digraph is called super-edge-connected if every minimum edge-cut consists of edges adjacent to or from a vertex of minimum degree. In this note we present degree sequence conditions for maximally edge-connected and super-edge-connected digraphs depending on the clique number of the underlying graph. [ABSTRACT FROM AUTHOR]

Let α , β be real numbers and b ≥ 2 be an integer. Allouche and Shallit showed that the sequence { ⌊ α n + β ⌋ } n ≥ 0 is b -regular if and only if α is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence { ⌊ log b ⁡ ( α n + β ) ⌋ } n ≥ 0 is b -regular if and only if α is rational, where α > 0 , β ≥ 0 . In particular, when α = 2 , β = 0 and b = 2 , we answer a question of Allouche and Shallit that the sequence { ⌊ 1 2 + log 2 ⁡ n ⌋ } n ≥ 1 is not 2-regular, which has previously been proved by Bell, Moshe and Rowland respectively. [ABSTRACT FROM AUTHOR]

The linear fractional map f (z) = az)b cz)d on the Riemann sphere with complex coefficients ad - bc ≠ 0 is called Möbius map. If f satisfies ad - bc = 1 and - 2

Let p = (pn) be a sequence of nonnegative numbers and Pn := Σnk=0 pk → ∞ as n → ∞: Let the weighted general control modulo of the oscillatory behavior of integer order m ≥ 1 of a sequence (un) be denoted by (ω(m)n,p(u)). We prove that if the weighted generator sequence of a sequence u = (un) of real numbers is summable to a finite number by the weighted mean method, (σ(1)n,p(ω(m-1) (u))) is increasing, the conditions (λα - 1) lim sup n (1/P[λn] - P[sub n] [λn]Σ k=n+1 (ω(m)k,p(u))q)1/q = o(1), λ → 1+; q > 1, α > - 1 and (1 -λα) lim sup n (1/Pn - P[λn] n Σ k=[λn]+1 (ω(m)k,p(u))q)1/q = o(1), λ → 1-, q > 1, α > -1 are satisfied, and certain conditions on (pn) are hold, then (un) is slowly oscillating. [ABSTRACT FROM AUTHOR]

We provide two ways to show that the R. Thompson group F has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of F on (0, 1), thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group F and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs and gives many implicit examples. We also show that F has a decreasing sequence of finitely generated subgroups F > H1 > H2 > ... such that ∩Hi = {1} and for every i there exist only finitely many subgroups of F containing Hi. [ABSTRACT FROM AUTHOR]

We introduce an up-down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two -component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram , there exists a diagram representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II. [ABSTRACT FROM AUTHOR]

In this paper we prove asymptotic formulas for the behavior of approximation, Gelfand, Kolmogorov and Weyl numbers of embeddings between some weighted sequence spaces. The formulas are applied to the estimates of asymptotic behavior of these s -numbers of Sobolev embeddings between Besov and Triebel–Lizorkin spaces defined on quasi-bounded domains. [ABSTRACT FROM AUTHOR]

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

In this paper, we give a definition of the p -adic Arakawa–Kaneko–Hamahata zeta functions. These zeta functions interpolate Hamahata's poly-Euler polynomials at non-positive integers. We prove the derivative formula, the difference equation and the reflection formula of these zeta functions. Furthermore, we also prove a sums of products identity and a closed form of Hamahata's poly-Euler polynomials in terms of the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group. For a positive integer k , let s ≤ k ( G ) denote the smallest integer l such that each sequence of length l has a non-empty zero-sum subsequence of length at most k . Among other results, we determine s ≤ k ( G ) for all finite abelian groups of rank two. [ABSTRACT FROM AUTHOR]

In 1977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In 2013 Foata’s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 2014. In this paper, we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata’s result. [ABSTRACT FROM AUTHOR]

We show that minimizers of the Heitmann-Radin energy (Heitmann and Radin in J Stat Phys 22(3):281-287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in De Luca and Friesecke (Crystallization in two dimensions and a discrete Gauss-Bonnet Theorem, 2016). [ABSTRACT FROM AUTHOR]

The Erdős–Moser theorem ( EM ) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs ( RT 2 2 ) by providing an alternate proof of RT 2 2 in terms of EM and the ascending descending sequence principle ( ADS ). In this paper, we study the computational weakness of EM and construct a standard model ( ω -model) of simultaneously EM , weak König's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdős–Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner. [ABSTRACT FROM AUTHOR]

The paper studies recoverability of missing values for sequences in a pathwise setting without probabilistic assumptions. This setting is oriented on a situation where the underlying sequence is considered as a sole sequence rather than a member of an ensemble with known statistical properties. Sufficient conditions of recoverability are obtained; it is shown that sequences are recoverable if there is a certain degree of degeneracy of the Z-transforms. We found that, in some cases, this degree can be measured as the number of the derivatives of Z-transform vanishing at a point. For processes with non-degenerate Z-transform, an optimal recovering based on the projection on a set of recoverable sequences is suggested. Some robustness of the solution with respect to noise contamination and truncation is established. [ABSTRACT FROM AUTHOR]

In this paper, by the exponential complete Bell polynomials, we establish two general asymptotic expansions for the Barnes G -function, which contain only even power terms or odd power terms. We also derive the recurrences for the coefficient sequence or the parameter sequences in these two general expansions, and obtain some special asymptotic expansions by specifying the parameters. [ABSTRACT FROM AUTHOR]

Let B t ( n ) be the number of set partitions of a set of size t into at most n parts and let B t ′ ( n ) be the number of set partitions of { 1 , … , t } into at most n parts such that no part contains both 1 and t or both i and i + 1 for any i ∈ { 1 , … , t − 1 } . We give two new combinatorial interpretations of the numbers B t ( n ) and B t ′ ( n ) using sequences of random-to-top shuffles, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phatarfod on the eigenvalues of the random-to-top shuffle. We also prove analogous results for random-to-top shuffles that may flip certain cards. The proofs use the Solomon descent algebras of Types A, B and D. We give generating functions and asymptotic results for all the combinatorial quantities studied in this paper. [ABSTRACT FROM AUTHOR]

In [VP] , V.V. Volkov and F.V. Petrov consider the problem of existence of the so-called n -universal sets (related to simultaneous p -orderings of Bhargava) in the ring of Gaussian integers. A related problem concerning Newton sequences was considered by D. Adam and P.-J. Cahen in [AC] . We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov–Petrov on minimal cardinality of n -universal sets. Along the way, we discover a link with Euler–Kronecker constants and prove a lower bound on Euler–Kronecker constants which is of the same order of magnitude as the one obtained by Ihara. [ABSTRACT FROM AUTHOR]

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin-Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz's inhomogeneous (1958) version of Khintchine's Theorem (1924). For example, given any real sequence , we build a divergent series of non-negative reals such that for any , almost no real number is inhomogeneously -approximable with inhomogeneous parameter . Furthermore, given any second sequence not intersecting the rational span of , and assuming a dynamical version of Erdős' Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously -approximable with any inhomogeneous parameter . Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin-Schaeffer Conjecture and Duffin-Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by reduced fractions. [ABSTRACT FROM AUTHOR]

Let λ = (λn)n≥1be a nondecreasing sequence of positive numbers tending to infinity such that λ1 = 1 and λn+1 ≥ λn + 1 for all n, and let In = [n -- λn + 1; n] for n = 1; 2;,.... Then for any given nonzero sequence µ, we define by Δ+(µ) the operator that generalizes the operator of the First difference and is defined by Δ+(µ)xk = µk(xk -- xk+1). In this article, for any given integer r ≤ 1, we deal with the λ+r(µ)-statistical convergence that generalizes in a certain sense the well-known λE-rstatistical convergence, main results consist in determining sets of sequences χ and χ' of the form sξ0 satisfying χ ⊂ [V; λ]0(Δ+r(µ)) ⊂ χ' and sets κ and κ' of the form sξ satisfying χ ⊂ [V, λ]∞(Δ+r(µ)) ≥ κ'. This study is justified since the infinite matrix associated with the operator Δ+r(µ) cannot be explicitly calculated for all r. [ABSTRACT FROM AUTHOR]

The Jacobsthal sequence has been generalized in many ways, some by preserving the initial conditions, others by preserving the recurrence relation. In this paper, we define a new generalization {Mk,n}∞ n = 1,with initial conditions Mk,o = 2Mk,1= m+k, which is generated by the recurrence relation Mk,n+1 = kMk,n + 2Mk,n-1 for n = 1, where k, m are integer numbers. Some new properties concerning this new generalization have been described using matrix algebra. [ABSTRACT FROM AUTHOR]

Repeated Measurement Designs, with two treatments,n(experimental) units andpperiods are examined, the two treatments are denotedAandB. The model with independent observations within and between treatment sequences is used. Optimal designs are derived for: (i) the difference of direct treatment effects and the difference of residual effects, (ii) the difference of direct treatment effects, and (iii) the difference of residual effects. We prove that for three periods whennis odd the optimal design in the three cases (i), (ii), and (iii) is determined by taking the sequences BAA and ABB in numbers differing by one. Ifnis even, the optimal design in cases (i), (ii), and (iii) is again the same, by taking the sequences ABB and BAA in equal numbers. In case (i), forneven or odd, in the optimal design there is no correlation between the two estimated parameters. Forneven, case (i) was solved by Cheng and Wu in 1980. The above imply that with two treatments in practice are preferable to use three periods instead of two. [ABSTRACT FROM PUBLISHER]

In a recent work, Luca and Stănică examined quotients of the form φ ( C m ) φ ( C n ) , where φ is Euler's totient function and C 0 , C 1 , C 2 … is the sequence of the Catalan numbers. They observed that the number 4 (and analogously 1 4 ) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients φ ( C n + 1 ) φ ( C n ) , but also that the value 4 k has similar “special” properties with respect to the quotients φ ( C n + k ) φ ( C n ) , and in particular we show that Dickson's conjecture implies that, for each k , the number 4 k appears infinitely often as a value of the quotients φ ( C n + k ) φ ( C n ) . [ABSTRACT FROM AUTHOR]

In this note, we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method. [ABSTRACT FROM AUTHOR]

In this paper, we study the log-behavior of a new sequence $\{S_{n}\} _{n=0}^{\infty}$ , which was defined by Z-W Sun. We find that the sequence is log-convex by using the interlacing method. Additionally, we consider ratio log-behavior of $\{S_{n}\}_{n=0}^{\infty}$ and find the sequences $\{S_{n+1}/S_{n}\}_{n=0}^{\infty}$ and $\{\sqrt[n]{S_{n}}\} _{n=1}^{\infty}$ are log-concave. Our results give an affirmative answer to a conjecture of Z-W Sun on the ratio monotonicity of this new sequence. [ABSTRACT FROM AUTHOR]

We consider synthesis problems for a discrete event model of a real time distributed computational system. The model reflects information exchange events between software units of the system and is intended to construct special test sequences that detect faults in these exchanges. [ABSTRACT FROM AUTHOR]

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The m -eternal domination number, γ m ∞ ( G ) , of a graph G is the minimum number of guards needed to defend G against any such sequence. We characterize the class of trees of order n with maximum possible m -eternal domination number, which is ⌈ n 2 ⌉ . [ABSTRACT FROM AUTHOR]

Student learning in the middle grades is afforded and constrained by basic numerical operations characterized in the number sequences framework (Steffe, 2010a), originally developed with younger children. I investigated its use with four sixth-grade students through a constructivist teaching experiment. I found that one student, Adam, was able to strategically operate on composite units, which is a strong indication that the he had constructed an explicitly nested number sequence (ENS). However, he was not able to reversibly disembed subsequences or construct iterable units of 1, both principal operations of an ENS. I conclude that Adam was constrained to the construction of the less sophisticated tacitly nested number sequence (TNS), and that his advanced behavior is a natural outgrowth of his increased experience operating with a TNS. I suggest a refinement of the number sequences framework to support research and teaching at the middle grades. [ABSTRACT FROM AUTHOR]

Text Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals 1 2 , 1 3 , 2 3 , 1 4 , 2 4 , 3 4 , 1 5 , ⋯ into an infinite continued fraction expansion, then this new number is normal with respect to the continued fraction expansion. We show a variety of new constructions of continued fraction normal numbers, including one generated by the subsequence of rationals with prime numerators and denominators: 2 3 , 2 5 , 3 5 , 2 7 , 3 7 , 5 7 , ⋯ . Video For a video summary of this paper, please visit https://youtu.be/L7uyAQ7hS74 . [ABSTRACT FROM AUTHOR]

The decimal expansions of the numbers 1/n (such as ? = 0:3333:::, 1/7 = 0:142857:::) are most often viewed as tools for approximating quantities to a desired degree of accuracy. The aim of this exposition is to show how these modest expressions in fact deserve have much more to offer, particularly in the case when the expansions are infinitely long. First we discuss how simply asking about the period (that is, the length of the repeating sequence of digits) of the decimal expansion of 1/n naturally leads to more sophisticated ideas from elementary number theory, as well as to unsolved mathematical problems. Then we describe a surprising theorem of K. Girstmair showing that the digits of the decimal expansion of 1/p, for certain primes p, secretly contain deep facts that have long delighted algebraic number theorists. [ABSTRACT FROM AUTHOR]

Text A set A of positive integers is called d -complete if every sufficiently large integer is the sum of distinct terms taken from A such that no one divides the other. In 1996, Erdős and Lewin proved that the sequence { 2 α 5 β p γ } is d -complete for every prime p with 6 < p < 20 and commented that they cannot prove that the sequence { 2 α 5 β 23 γ } is d -complete. In this paper, we prove that, for any integer p > 6 with ( p , 10 ) = 1 , if there exists a number c = c ( p ) such that every integer n with c < n < 25 p c is the sum of distinct terms taken from { 2 α 5 β p γ } such that no one divides the other, then the sequence { 2 α 5 β p γ } is d -complete. In particular, we prove that the sequence { 2 α 5 β p γ } is d -complete for p ∈ { 9 , 21 , 23 , 27 , 29 , 31 } . Video For a video summary of this paper, please visit https://youtu.be/lq7XleRNLf8 . [ABSTRACT FROM AUTHOR]

The secure-domination number ϒs(G) of a finite simple graph G is the smallest size of a set of vertices in G which is both secure and dominating in G. It is elementary that... ≥ ½ . A cur- rently unsolved problem: Find the largest number g ∈ (½, 1] such that there is a sequence (Hn) of distinct connected graphs such that limn→∞... = g. In early work on this problem the full balanced binary trees Gn (which has 2n leafs, each a distance n from the root vertex) were of interest. They are no longer, but the unexpected dif- ficulty of determining s(Gn) has made this determination of interest in itself. In this paper we show that lim infn→∞... ≤ 8/15 andgive a construction of a secure-dominating set Sn ? V (Gn) such that limn→∞... = 11/20 . However, we find that Sn is not a minimum secure-dominating set in Gn for n ≥ 6. [ABSTRACT FROM AUTHOR]

Let L(s, F) be the L-function associated with a Hecke-Maass form F for SL(3, Z), with AF(n, 1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N ≥ 2 and any α ∈ R, there exists an effective constant c > 0 such that ∑n≤N ∧(n)AF(n, 1)e(nα) ≪ N exp(-c √ log N), where ∧(n) is the von Mangoldt function, and the implied constant depends only on F. [ABSTRACT FROM AUTHOR]

Testing of reactive systems is challenging because long input sequences are often needed to drive them into a state to test a desired feature. This is particularly problematic in on-target testing, where a system is tested in its real-life application environment and the amount of time required for resetting is high. This article presents an approach to discovering a test case chain-a single software execution that covers a group of test goals and minimizes overall test execution time. Our technique targets the scenario in which test goals for the requirements are given as safety properties. We give conditions for the existence and minimality of a single test case chain and minimize the number of test case chains if a single test case chain is infeasible. We report experimental results with our ChainCover tool for C code generated from Simulink models and compare it to state-of-the-art test suite generators. [ABSTRACT FROM AUTHOR]

For an triangular arrayof real numbers and a sequenceof random variables extended negatively dependent stochastically dominated by a random variableXhaving finite second absolute moment, conditions are given to ensureas. [ABSTRACT FROM AUTHOR]