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A symbolic analysis of Archimedes's periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci's Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics. [ABSTRACT FROM AUTHOR]

This paper provides another proof of Fermat's theorem. As in the previous work, a geometric approach is used, namely: instead of integers a, b, c, a triangle with side lengths a, b, c is considered. To preserve the completeness of the proof of the theorem in this work, the proof is repeated for the cases of right and obtuse triangles. In this case, the Fermat equation ap+bp = cp has no solutions for any natural number p > 2 and arbitrary numbers a, b, c. When considering the case when the numbers a, b, c are sides of an acute triangle, it is proven that Fermat's equation has no solutions for any natural number p > 2 and non-zero integer numbers a, b, c. Numbers a = k, b = k+m, c = k+n, where k, m, n are natural numbers that satisfy the inequalities n > m, n < k+m, exhaust all possible variants of natural numbers a, b, c, which are the sides of the triangle. In an acute triangle, the following condition is additionally satisfied: k > n -m + √2n(n - m). To study the Fermat equation, an auxiliary function f(k,p) = kp+(k+m)p-(k+n)p, is introduced, which is a polynomial of natural degree p in the variable k. The equation f(k, p) = 0 has a single positive root for any natural p = 2. A recurrent formula connecting the functions f(k,p+1) and f(k,p) has been proven: f(k,p+1) = kf(k, p)-[n(k+n)p-m(k+m)p]. The proof of the main proposition 2 is based on considering all possible relationships between the assumed integer solution of the equation f(k, p+1) = 0 and the number (ñ -m) corresponding to this solution k. The proof was carried out using the mathematical apparatus of number theory, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat's theorem were proved. [ABSTRACT FROM AUTHOR]

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]

The article discusses second Gaussian lecture in 2021 is expected to take place in the upper town hall in Bremen, Germany that will focus on whole numbers and L-functions. Topics discussed include declaration on equal opportunities, diversity and scientific collaboration; new trends in algorithmic randomness and computable analysis; nonlocal conservation laws; mathematical models; and mathematical analysis and quantum systems.

In this brief note we survey a sample of the deep and influential contributions of Jean Bourgain to the field of nonlinear dispersive equations. Bourgain also made many fundamental contributions to other areas of partial differential equations and mathematical physics (as well as to a myriad of other areas in analysis, number theory, combinatorics, theoretical computer science, and more). Quoting the citation of the American Mathematical Society L. P. Steele Prize for Lifetime Achievement awarded to Bourgain in 2018, ''Jean Bourgain is a giant in the field of mathematical analysis, which he has applied broadly and to great effect.'' Jean Bourgain's contributions to mathematics will be remembered forever. Those of us who knew him will also remember his warmth, generosity, and graciousness. [ABSTRACT FROM AUTHOR]

Hausman and Shapiro gave an estimate for the number of practical numbers to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)². In this paper, we generalize Hausman and Shapiro bound by proving the number of t-practical numbers n (n ≤x ), ( t ≥1 ) to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)², and 1≤t≤exp.(( logx ) δ1 ) for any δ1 satisfying 0< δ1 <1-1(1 +√2 β) log2 . We mean by the t-practical number n, the number in which every integer 1≤m≤tn is of the form m= ∑ d/n cd d, 0 ≤ cd≤ t [ABSTRACT FROM AUTHOR]

Abstract We point out a characterization of the Ree curve which involves the number of rational points, the genus, and the shape of two elements of the Weierstrass semigroup at a rational point. [ABSTRACT FROM AUTHOR]

Despite the efforts of many individuals, the disciplines of modern physics and number theory have remained largely divorced, in the sense that the experimentally verified theories of quantum physics and gravity are written in the language of linear algebra and advanced calculus, without reference to several established branches of pure mathematics. This absence raises questions as to whether or not pure mathematics has undiscovered application to physical modeling that could have far reaching implications for human scientific understanding. In this paper, we review physical interpretations of number theoretic concepts developed in the twentieth century, in an attempt to help bridge the divide between pure mathematical truth and testable physical theory. Specifically, the relevance of L-functions and modular forms to the physics of quantum and classical physical systems is addressed, with the objective of motivating general interest among physicists in these mathematical concepts. [ABSTRACT FROM AUTHOR]

Abstract We enumerate all the principal congruence link complements in S 3 , thereby answering a question of W. Thurston. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation. [ABSTRACT FROM AUTHOR]

Abstract Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number γ has at most 2 γ minimum dominating sets. Bień gave a counterexample, which allows us to construct forests with domination number γ and 2. 059 8 γ minimum dominating sets. We show that every forest with domination number γ has at most 2. 460 6 γ minimum dominating sets, and that every tree with independence number α has at most 2 α − 1 + 1 maximum independent sets. [ABSTRACT FROM AUTHOR]

Abstract We determine the exact value of the η -constant and the multiwise Davenport constants for finite abelian groups of rank three having the form G ≃ C 2 ⊕ C n 2 ⊕ C n 3 with 2 | n 2 | n 3. Moreover, we determine the Erdős–Ginzburg–Ziv constant of these groups under the assumption that n 2 / 2 has Property D or n 2 = n 3. [ABSTRACT FROM AUTHOR]

Abstract Let S (a , b) denote the normalized Dedekind sum. We study the range of possible values for S (a , b) = k q with gcd ⁡ (k , q) = 1. Girstmair proved local restrictions on k depending on q (mod 12) and whether q is a square and conjectured that these are the only restrictions possible. We verify the conjecture in the cases q even, q a square divisible by 3 or 5, and 2 ≤ q ≤ 200 (the latter by computer), and provide progress towards a general approach. [ABSTRACT FROM AUTHOR]

Abstract We will prove that for every m ≥ 0 there exists an ε = ε (m) > 0 such that if 0 < λ < ε and x is sufficiently large in terms of m , then | { n ≤ x : | [ n , n + λ log ⁡ n ] ∩ P | = m } | ≫ m x log ⁡ x. [ABSTRACT FROM AUTHOR]

Abstract We prove the asymptotic normality of the standardized number of crossings of a centered stationary mixing Gaussian process when both the level and the time horizon go to infinity in such a way that the expected number of crossings also goes to infinity. [ABSTRACT FROM AUTHOR]

Let 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = 1/2 (1 − j) An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞← n An−. Furthermore we will show that μ = 0 if and only if μ − = 0 in this new definition. [ABSTRACT FROM AUTHOR]

Abstract Let G be a bridgeless cubic graph. Oddness (weak oddness) is defined as the minimum number of odd components in a 2-factor (an even factor) of G , denoted as ω (G) (Steffen, 2004) (ω ′ (G) Lukot'ka and Mazák (2016)). Oddness and weak oddness have been referred to as measurements of uncolourability (Fiol et al., 2017, Lukot'ka and Mazák, 2016, Lukot'ka et al., 2015 and, Steffen, 2004), due to the fact that ω (G) = 0 and ω ′ (G) = 0 if and only if G is 3-edge-colourable. Another so-called measurement of uncolourability is resistance, defined as the minimum number of edges that can be removed from G such that the resulting graph is 3-edge-colourable, denoted as r (G) (Steffen, 2004). It is easily shown that ω (G) ≥ ω ′ (G) ≥ r (G). While it has been shown that the difference between any two of these measures can be arbitrarily large, it has been conjectured that ω ′ (G) ≤ 2 r (G) , and that if G is a snark then ω (G) ≤ 2 r (G) (Fiol et al., 2017). In this paper, we disprove the latter by showing that the ratio of oddness to weak oddness can be arbitrarily large. We also offer some insights into the former conjecture by defining what we call resistance reducibility , and hypothesizing that almost all cubic graphs are such resistance reducible. [ABSTRACT FROM AUTHOR]

Abstract We extend a method of Olsson and Bessenrodt to determine the number of even partitions that are simultaneously s ̄ -core and t ̄ -core. When p and q are distinct primes, this also determines the number of self-associate characters of S ̃ n that are simultaneously defect 0 for p and q. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

The construction of the complex numbers over the reals has been generalized in many ways leading, amongs others, to the 2-dimensional elliptical complex numbers (= ordinary complex numbers), the parabolic complex numbers and the hyperbolic complex numbers. One can extend these to higher dimensions and also to using an arbitrary ring as the base ring. Here we propose a construction of generalized complex numbers over a near-field and investigate some of the structural properties of this near-ring. [ABSTRACT FROM AUTHOR]

We prove that for most entire functions f in the sense of category, a strong form of the Baker-Gammel-Wills conjecture holds. More precisely, there is an infinite sequence S of positive integers n, such that given any r>0, and multipoint Padé approximants Rn to f with interpolation points in z:z≤r, Rnn∈S converges locally uniformly to f in the plane. The sequence S does not depend on r, or on the interpolation points. For entire functions with smooth rapidly decreasing coefficients, full diagonal sequences of multipoint Padé approximants converge. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of Fibonacci periodicity modulo n, denoting this period by the function ̂F(n). We note that ̂F(n) is an integral multiple of a fundamental frequency ̂f(n), where the ratio ̂F(n)= ̂f(n) is a power of 2 for a collection of observed values of n. It is demonstrated that if a, b are natural numbers with gcd(a, b) = 1, then ̂F(n) = lcmf ̂F(a), ̂F(b)g and thus that ̂F is a non-trivial example of a function which we refer to as radical. From observations it also seems clear that ̂F(ps+1)/̂F(ps) = p for primes p. [ABSTRACT FROM AUTHOR]

Abstract We prove that every triangle-free graph with maximum degree Δ has list chromatic number at most (1 + o (1)) Δ ln ⁡ Δ. This matches the best-known upper bound for graphs of girth at least 5. We also provide a new proof that for any r ≥ 4 every K r -free graph has list-chromatic number at most 200 r Δ ln ⁡ ln ⁡ Δ ln ⁡ Δ. [ABSTRACT FROM AUTHOR]

Abstract This paper proves that the Alon–Tarsi number of any planar graph is at most 5, which solves an open problem posed in Hefetz (2011) [2] and gives an alternate proof of the 5-choosability as well as the 5-paintability of planar graphs. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce new families of the q-Fibonacci and q-Lucas polynomials, which are represented here as the incomplete q-Fibonacci polynomials Fnk (x, s, q) and the incomplete q-Lucas polynomials Lnk (x, s, q), respectively. These polynomials provide the q-analogues of the incomplete Fibonacci and Lucas numbers. We give several properties and generating functions of each of these families q-polynomials. We also point out the fact that the results for the q-analogues which we consider in this article for 0 < q < 1 can easily be translated into the corresponding results for the (p, q)-analogues (with 0 < q < p ≥ 1) by applying some obvious parametric variations, the additional parameter p being redundant. [ABSTRACT FROM AUTHOR]

Abstract. In this work, some properties of supplement elements in lattices are investigated. Some relation between lying above and (weak) supplement elements also studied. Some properties of supplement submodules in modules which given in [8] are generalized to lattices. Let a be a supplement of b in a lattice L. If a=0 has at least one maximal (≠ a) element, then it is possible to define a bijective map between the maximal elements (≠ a) of a/0 and the maximal elements (≠ 1) of 1/b. Let a be a supplement element in a lattice L. If L is amply supplemented, then a/0 is also amply supplemented. If L is weakly supplemented, then a/0 is also weakly supplemented. [ABSTRACT FROM AUTHOR]

For Shilov-type parabolic systems with nonnegative genus and coefficients of bounded smoothness, the properties of differentiability of Green's function with respect to spatial variables are studied. [ABSTRACT FROM AUTHOR]

In this paper we introduce new densities on the set of primes of a number field. If K/K0 is a Galois extension of number fields, we associate to any element x ∈ ... a density δK/K0,x on the primes of K. In particular, the density associated to x = 1 is the usual Dirichlet density on K. We also give two applications of these densities (for x ≠ 1): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg. [ABSTRACT FROM AUTHOR]

Green and Lindqvist proved that for any 2‐colouring of N, there are infinitely many monochromatic solutions to x+y=z2. In fact, they showed the existence of a monochromatic solution in every interval [N,cN8] with large enough N. In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval [N,104N4] with large enough N. A 2‐colouring of [N,(1/27)N4] avoiding monochromatic solutions to x+y=z2 is also presented, which shows that in 104N4 only the constant factor can be reduced. [ABSTRACT FROM AUTHOR]

We prove the statement in the title and exhibit examples of quotients of arbitrary nilpotency class. This answers a question by Holt. [ABSTRACT FROM AUTHOR]

In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ2, the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number 0 < ε < 1 2 , we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ2] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution. [ABSTRACT FROM AUTHOR]

We study compositions (ordered partitions) of n. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is 'grounded'). Via generating functions and asymptotic analysis, we consider compositions of n whose Durfee squares are of size less than s × s. This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n. [ABSTRACT FROM AUTHOR]

Abstract We obtain a simple two-sided inequality for the ratio L ν (x) / L ν − 1 (x) in terms of the ratio I ν (x) / I ν − 1 (x) , where L ν (x) is the modified Struve function of the first kind and I ν (x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for I ν (x) / I ν − 1 (x) to immediately deduce bounds for L ν (x) / L ν − 1 (x). We note some consequences and obtain further bounds for L ν (x) / L ν − 1 (x) by adapting techniques used to bound the ratio I ν (x) / I ν − 1 (x). We apply these results to obtain new bounds for the condition numbers x L ν ′ (x) / L ν (x) , the ratio L ν (x) / L ν (y) and the modified Struve function L ν (x) itself. Amongst other results, we obtain two-sided inequalities for x L ν ′ (x) / L ν (x) and L ν (x) / L ν (y) that are given in terms of x I ν ′ (x) / I ν (x) and I ν (x) / I ν (y) , respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature. [ABSTRACT FROM AUTHOR]

Let E(n)kk denote the minimax (i.e., best supremum norm) error in approximation of xn on [0, 1] by rational functions of type (k, k) with k < n. We show that in an appropriate limit E(n)kk ~ 2Hk+1/2 independently of n, where H ≈ 1/9.28903 is Halphen's constant. This is the same formula as for minimax approximation of ex on (-∞, 0]. [ABSTRACT FROM AUTHOR]

Let {Xn, n ⩾ 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {dni, 1 ⩽ i ⩽ n, n ⩾ 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the Lp convergence and the complete convergence. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers for AANA random variables is obtained. [ABSTRACT FROM AUTHOR]

The Hanf number for a set S of sentences in Lω1,ω (or some other logic) is the least infinite cardinal κ such that for all φ∈S, if φ has models in all infinite cardinalities less than κ, then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is ℶω1CK. The same argument proves that ℶω1CK is the Hanf number for Scott sentences of hyperarithmetical structures. [ABSTRACT FROM AUTHOR]

Abstract We provide an upper bound on the number of n 2 × n 2 Sudoku squares, and explain intuitively why there is reason to believe that the bound is tight up to a multiplicative factor of a much smaller order of magnitude. A similar bound is established for Sudoku squares with rectangular regions. [ABSTRACT FROM AUTHOR]

Abstract In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for p p ¯ o (n) , the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer k , p p ¯ o (n) is almost always divisible by 2 k. In this paper, we prove several congruences for p p ¯ o (n) modulo higher powers of 2 in an elementary way. [ABSTRACT FROM AUTHOR]

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial. [ABSTRACT FROM AUTHOR]