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In this paper, we mainly answer a Lampe's question [3] about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of the group G which is defined by mutations of a cluster algebra. In order to do this, using a rational map φ , we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solution φ (ϵ , ϵ , ϵ , ϵ , ϵ) = (3 , 4 , 4) under the actions of the group G ˜ , and the set S (3 , 4 , 4) is shown to be the orbit of (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of a subgroup of G. Then the criterion is proved as the main conclusion. [ABSTRACT FROM AUTHOR]

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

The main theorem of the paper [2] requires a restriction on the prime p. This statement is being modified based on a certain conjecture on the p -adic zeta functions. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1. [Copyright &y& Elsevier]

Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]

Abstract: We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient . [Copyright &y& Elsevier]

Abstract: Remark on Polickýʼs paper on circular units of a compositum of quadratic number fields is given. [Copyright &y& Elsevier]

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j -invariant is ≫ x log − 1 2 ⁡ (x). In this paper, we assume the abc -conjecture to show the sharp asymptotic ∼ c x log − 1 2 ⁡ (x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve E / Q with short Weierstrass equation y 2 = f (x) for reducible f ∈ Z [ x ] , we show that the number of integers d , | d | ≤ D , for which the quadratic twist d y 2 = f (x) has an integral non-2-torsion point is at most D 2 3 + o (1) , assuming the abc -conjecture. (2) We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O (x 1 − ϵ) for some ϵ > 0. We also exhibit a matching lower bound. If instead of the abc -conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias. [ABSTRACT FROM AUTHOR]

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called edge-ordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edge-order but the same underlying graph may appear with a different edge-order. We prove an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs—we identify the relevant parameter for the Turán number of an edge-ordered graph and call it the order chromatic number. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in discrete geometry. [ABSTRACT FROM AUTHOR]

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied. [ABSTRACT FROM AUTHOR]

Given positive integers n ⩾ 2 k , the Kneser graph K G n , k is a graph whose vertex set is the collection of all k -element subsets of the set { 1 , ... , n } , with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of K G n , k is equal to n − 2 k + 2. In this paper, we study the chromatic number of the random Kneser graph K G n , k (p) , that is, the graph obtained from K G n , k by including each of the edges of K G n , k independently and with probability p. We prove that, for any fixed k ⩾ 3 , χ (K G n , k (1 / 2)) = n − Θ ( log 2 ⁡ n 2 k − 2) , as well as χ (K G n , 2 (1 / 2)) = n − Θ ( log 2 ⁡ n ⋅ log 2 ⁡ log 2 ⁡ n 2). We also prove that, for k ⩾ (1 + ε) log ⁡ log ⁡ n , we have χ (K G n , k (1 / 2)) ⩾ n − 2 k − 10. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. The bound on k in the second result is also tight up to a constant. We also discuss an interesting connection to an extremal problem on embeddability of complexes. [ABSTRACT FROM AUTHOR]

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z -gradings on E of full support. Using methods of elementary number theory, we describe the Z -graded polynomial identities for the so-called 2-induced Z -gradings on E of full support. As a consequence of this fact we provide examples of Z -gradings on E which are PI-equivalent but not Z -isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z -gradings on E and we show that their Z -graded polynomial identities are closely related to the Z 2 -graded polynomial identities of E. [ABSTRACT FROM AUTHOR]

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in R n with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n -dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed. [ABSTRACT FROM AUTHOR]

A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]

In this paper we investigate the factorization behaviour of the binomial polynomials ( x n ) = x (x − 1) ⋯ (x − n + 1) n ! and their powers in the ring of integer-valued polynomials Int (Z). While it is well-known that the binomial polynomials are irreducible elements in Int (Z) , the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int (Z) , that is, ( x n ) m factors uniquely into irreducible elements in Int (Z) for all m ∈ N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n > 10 and n , n − 1 , ..., n − (k − 1) are composite integers, then there exists a prime number p > 2 k that divides one of these integers. [ABSTRACT FROM AUTHOR]

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function ψ : N → (0 , 1 / 2) with ∑ q ψ (q) = ∞ and any number γ , the following set W (ψ , γ) = { x ∈ [ 0 , 1 ] : | q x − p − γ | < ψ (q) for infinitely many q , p ∈ N } has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if γ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition ψ (q) = O ((q (log ⁡ log ⁡ q) 2) − 1). In particular, this covers the case when γ is not Liouville, for example π , e , ln ⁡ 2 , 2. In general, if γ is irrational, ψ (q) = O (q − 1 (log ⁡ log ⁡ q) − 2) and in addition, ( lim inf Q → ∞ ∑ q = Q Q (log ⁡ Q) 1 / 8 ψ (q)) = ∞ , then W (ψ , γ) has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in F q [ t ]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions d r (f) with an error term that improves the one given by Burr. [ABSTRACT FROM AUTHOR]

Let K be a number field. In the terminology of Nagell a unit ε of K is called exceptional if 1 − ε is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ε 1 + ε 2 + ε 3 = 0 is solvable in units ε 1 , ε 2 , ε 3 of K . Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer k with k ≥ 3 , denoted by ℓ ( K ) , such that the unit equation ε 1 + … + ε k = 0 is solvable in units ε 1 , … , ε k of K . If no such k exists, we set ℓ ( K ) = ∞ . Apart from trivial cases when ℓ ( K ) = ∞ , we give an explicit upper bound for ℓ ( K ) . We obtain several results for ℓ ( K ) in number fields of degree at most 4, cyclotomic fields and general number fields of given degree. We prove various properties of ℓ ( K ) , including its magnitude, parity as well as the cardinality of number fields K with given degree and given odd resp. even value ℓ ( K ) . Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions. [ABSTRACT FROM AUTHOR]

In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations. [ABSTRACT FROM AUTHOR]

While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations. [ABSTRACT FROM AUTHOR]

Let f r (n , v , e) denote the maximum number of edges in an r -uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v + 1 vertices. The study of f r (n , v , e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: n k − o (1) < f r (n , e r − (e − 1) k + 1 , e) = o (n k) holds for all fixed integers r > k ≥ 2 and e ≥ 3 as n → ∞. For r = 3 , e = 3 , k = 2 , the bound n 2 − o (1) < f 3 (n , 6 , 3) = o (n 2) was proved by the celebrated (6,3)-theorem of Ruzsa and Szemerédi. In this paper, we add more evidence for the validity of the conjecture. On one hand, using the hypergraph removal lemma we show that the upper bound part of the conjecture is true for all fixed integers r ≥ k + 1 ≥ e ≥ 3. On the other hand, using tools from additive number theory we present several constructions showing that the lower bound part of the conjecture is true for r ≥ 3 , k = 2 and e = 4 , 5 , 7 , 8. Prior to our results, all known constructions that match the conjectured lower bound satisfy either r = 3 or e = 3. Our constructions are the first ones in the literature that break this barrier. [ABSTRACT FROM AUTHOR]

It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or C ∞) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example. [ABSTRACT FROM AUTHOR]

The Erdős multiplication table problem asks how many distinct integers appear in the N × N multiplication table. The order of magnitude of this quantity was determined by Ford [8]. In this paper we study the number of y -smooth entries of the N × N multiplication table, that is to say entries with no prime factors greater than y. [ABSTRACT FROM AUTHOR]

For positive integers α and β , we define an (α , β) -walk to be any sequence of positive integers satisfying w k + 2 = α w k + 1 + β w k. We say that an (α , β) -walk is n -slow if w s = n with s as large as possible. Slow (1 , 1) -walks have been investigated by several authors. In this paper we consider (α , β) -walks for arbitrary positive α , β. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of n -slow walks for a given n. In addition to this, we study the slowest n -slow walk for a given n amongst all possible α , β. [ABSTRACT FROM AUTHOR]

This corrigendum replaces the family g u (x) in Theorem 1 of the paper "Continued fractions for rational torsion" published in the Journal of Number Theory, vol. 189 (2018) 115–130. [ABSTRACT FROM AUTHOR]

In 1973, Erdős et al. introduced the anti-Ramsey number for a graph G in K n , which is defined to be the maximum number of colors in an edge-coloring of K n which does not contain any rainbow G . This is always regarded as one of rainbow generalizations of the classic Ramsey theory. Since then the anti-Ramsey numbers for several special graph classes in complete graphs have been determined. Also, the researchers generalized the host graph for the anti-Ramsey number from the complete graph to general graphs, including bipartite graphs, complete split graphs, planar graphs, and so on. In this paper, we study the anti-Ramsey number of matchings in the complete split graph. Since the complete split graph contains the complete graph as a subclass, the results in this paper cover the previous results about the anti-Ramsey number of matchings in the complete graph. [ABSTRACT FROM AUTHOR]

Text By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence , and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences { G n } that have the same linear recurrence relations as the N th order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G n . We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns . [ABSTRACT FROM AUTHOR]

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers. [ABSTRACT FROM AUTHOR]

Abstract upper densities are monotone and subadditive functions from the power set of positive integers into the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. At the open problem session of the Workshop "Densities and their application", held at St. Étienne in July 2013, G. Grekos asked a question whether there is a "nice" abstract upper density, whose the family of null sets is precisely a given ideal of subsets of N , where "nice" would mean the properties of the familiar densities consider in number theory. M. Di Nasso, R. Jin (2018) [3] showed that the answer is positive for the summable ideals (for instance, the family of finite sets and the family of sequences whose series of reciprocals converge) when "nice" density means translation invariant and rich density (i.e. density which is onto the unit interval). In this paper we extend their result to all ideals with the Baire property. [ABSTRACT FROM AUTHOR]

Truth Discovery is an important learning problem arising in data analytics related fields. It concerns about finding the most trustworthy information from a dataset acquired from a number of unreliable sources. The problem has been extensively studied and a number of techniques have already been proposed. However, all of them are of heuristic nature and do not have any quality guarantee. In this paper, we formulate the problem as a high dimensional geometric optimization problem, called Entropy based Geometric Variance. Relying on a number of novel geometric techniques, we further discover new insights to this problem. We show, for the first time, that the truth discovery problem can be solved with guaranteed quality of solution. Particularly, it is possible to achieve a (1 + ϵ) -approximation within nearly linear time under some reasonable assumptions. We expect that our algorithm will be useful for other data related applications. [ABSTRACT FROM AUTHOR]

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K ×. We consider Kummer extensions of G of the form K (ζ 2 m , G 2 n ) / K (ζ 2 m ) , where n ⩽ m. In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K (ζ 4). We prove how properties of G over K explicitly determine the divisibility parameters over K (ζ 4). This result yields a clear computational advantage, since no field extension is required. [ABSTRACT FROM AUTHOR]

In this paper, we consider the system x ˙ = y (1 + x) 2 − ϵ P (x , y) , y ˙ = − x (1 + x) 2 + ϵ Q (x , y) where P (x , y) and Q (x , y) are arbitrary quadratic polynomials. We study the maximum number of limit cycles bifurcating from the periodic orbits by using the Melnikov function of any order. We prove that the upper bound for the number of limit cycles is 3 and reached. [ABSTRACT FROM AUTHOR]

Amid the debate over infinitesimals, Bolzano introduced the alternative notion of variable quantities ω in his 1816 work on the binomial theorem. It has often been assumed that his 1817 definition of continuity using them is practically the modern one. This paper explores Bolzano's early mathematical works and diaries to gain insight into the subtleties in his definition of ω and some of his mathematical procedures. We show that those quantities are not clearly 'proto-Weierstrassian' and argue that Bolzano was in the process of refinement and increasing abstraction of the idea of quantity that eventually led to the development of a theory of real numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study Iwasawa theory for p = 2. First of all, we show that the classical Iwasawa main conjecture holds true even for p = 2 over a totally real field k assuming μ = 0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k , we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k = Q and K / Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal. [ABSTRACT FROM AUTHOR]

We study the topos of sets equipped with an action of the monoid of regular 2 × 2 matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient GL 2 (Z ˆ) \ M 2 (A f) / GL 2 (Q) , so they classify the groups Z 2 ⊆ A ⊆ Q 2 up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of Connes and Consani on the Arithmetic Site. As an application to number theory, we show that classifying extensions of Q by Z up to isomorphism relates to Goormaghtigh conjecture. For a video summary of this paper, please visit https://youtu.be/5cA1MOG26ng. [ABSTRACT FROM AUTHOR]

In this paper we study the number of twisted conjugacy classes (the Reidemeister number) for automorphisms of crystallographic groups. We present two main algorithms for crystallographic groups whose holonomy group has finite normaliser in GL n (Z). The first algorithm calculates whether a group has the R ∞ -property; the second calculates the Reidemeister spectrum. We apply these algorithms to crystallographic groups up to dimension 6. [ABSTRACT FROM AUTHOR]

Let k be an even integer and S k be the space of cusp forms of weight k on SL 2 (Z). Let S = ⊕ k ∈ 2 Z S k. For f , g ∈ S , we let R (f , g) be the set of ratios of the Fourier coefficients of f and g defined by R (f , g) : = { x ∈ P 1 (C) | x = [ a f (p) : a g (p) ] for some prime p } , where a f (n) (resp. a g (n)) denotes the n th Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R (f , g) is finite, then f = c g for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL 2 (Z). We apply it to study the number of representations of a positive integer by a quadratic form. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q. In this paper, for q even, we give a simple way to construct small degree polynomials over F q with no roots in the set μ q + 1 = { a ∈ F q 2 | a q + 1 = 1 } of the (q + 1) -th roots of unity. Further, we use these small degree polynomials over F q to construct and classify certain new families of permutation quadrinomials over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the number of F q n -rational points on the affine curve X d , a , b given by the equation y d = a x Tr (x) + b , where Tr denote the trace function from F q n to F q and d is a positive integer. In particular, we present bounds for the number of F q -rational points on X d , a , b and, for the cases where d satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case d = 2. As a consequence of our results, we compute the number of elements α in F q n such that α and Tr (α) are quadratic residues in F q n . [ABSTRACT FROM AUTHOR]

Knowledge of source characteristics is important for the calculation of sound pressure level and insertion loss of a silencer in sourced ducts. Measurement is usually the only feasible approach for the determination of source characteristics. The present paper is concerned with the measurement methods based on the Helmholtz-Thévenin equivalent of ducted one-port plane wave sources. Existing methods are classified as crisp and over-determined methods. In the crisp methods, the measured data are just sufficient for unique characterization of the source. This calls for two loads if their phase relative to the source is known, otherwise three loads are required. Over-determined methods use more number of loads than the crisp methods and are motivated for possible minimization of variations due to measurement errors. The point estimates for the source parameters are, however, still subject to some uncertainty, but estimation of confidence intervals is not feasible because the loads do not constitute a probability sample. The present paper proposes an approach which can yield the source characteristics in intervals from auto-power spectral density measurements with only two loads. The method is based on a novel Apollonian circle formulation. It is called the fuzzy two-load method, because uncertainty inherent to measurements is modelled by fuzzification of a characteristic parameter of the Apollonian circle of two loads. Fuzzy number transformations leading to the source pressure strength, sound pressure level and insertion loss interval predictions are discussed in depth. The paper includes an application showing the working features of the fuzzy two-load method. [ABSTRACT FROM AUTHOR]

Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center x ˙ = − y + 16 3 x 2 − 4 3 y 2 , y ˙ = x + 8 3 x y by the averaging method of first order, when it is perturbed inside a class of discontinuous quadratic polynomial differential systems. The Chebyshev criterion is used to show that this maximum number is 5 and can be realizable. In some sense, the result and that in paper [6] also answer the questions left in the paper [9] . [ABSTRACT FROM AUTHOR]

Text In this paper, we study the distribution of the a -values of the Selberg zeta-function attached to finite volume Riemann surfaces. We obtain a -value free regions, Weyl's law, and investigate the horizontal distribution of a -values. Video For a video summary of this paper, please visit https://youtu.be/cREe7owxYNo . [ABSTRACT FROM AUTHOR]

In this note, we study properties of certain parabolas which were recently defined by Hutchinson and Zaharescu in their paper on parabolas infiltrating the Ford circles. We examine this connection between Ford circles and these parabolas. In particular, we characterize the intersection points of these parabolas in terms of tangent Ford circles. Further, we provide a finer analysis of area under the arcs of these parabolas with end points joining the centers of two neighboring Ford circles, than discussed in their paper. [ABSTRACT FROM AUTHOR]

This paper discusses two related questions. First, given a G -invariant character θ of a normal subgroup N of a solvable group, what can we say if the number of characters of G above θ is in some sense as small as possible? Isaacs and Navarro [5] have shown that under certain assumptions about primes dividing the order of the group, one can show that G / N must have a very particular structure. Here we show that these assumptions can be weakened to obtain results about all solvable groups. We also discuss a related question about blocks. For a prime p and a p -block B of G , we let k ( B ) denote the number of ordinary characters in B . It is relatively easy to show that k ( B ) is bounded below by k ( G , D ) , which is the number of conjugacy classes of G that intersect the defect group D of B . In this paper we ask what can be said if equality is achieved. We show that for p -solvable groups, if k ( B ) = k ( G , D ) , then B is nilpotent and thus k ( B ) = | Irr ( D ) | . In addition, we show that this result holds for many blocks of arbitrary finite groups, including all blocks of the symmetric groups. We also extend a result on fully ramified coprime actions in [5] . [ABSTRACT FROM AUTHOR]

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a 1 x 1 + ⋯ + a k x k ≡ b ( mod n ) , with gcd ⁡ ( x i , n ) = t i ( 1 ≤ i ≤ k ), where a 1 , t 1 , … , a k , t k , b , n ( n ≥ 1 ) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of a i = t i = 1 ( 1 ≤ i ≤ k ) . Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [ Duke Math. J. 39 (1972) 521–527] gave a nice explicit formula for the number of such solutions when ( a 1 , … , a k ) = t i = 1 ( 1 ≤ i ≤ k ) . The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions. [ABSTRACT FROM AUTHOR]

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoï, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each. [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 Based on classical results of Rees and on multivariate Hilbert polynomials, we define new mixed multiplicities of two arbitrary ideals in a local ring (A , m) and use them to express the local degrees of all varieties appearing in the Gaffney–Gassler construction of Segre cycles. We prove that the classical mixed multiplicities of m and an arbitrary ideal I , which are a special case of the new ones, are equal to the generalized Samuel multiplicities of an ideal in the Rees algebra R I (A). This equality is used to improve a result of Jeffries, Montaño and Varbaro on the degree of the fiber cone of an ideal. We conclude the paper with formulas (and their inverses) which express the degrees of Segre classes of subschemes of arbitrary projective varieties by generalized Samuel multiplicities or by classical mixed multiplicities. Using the mixed multiplicities of balanced rational normal scrolls, which have been computed by Hoang and Lam, we find the mixed multiplicities of all rational normal scrolls as well as their Segre classes and their generalized Samuel multiplicities. [ABSTRACT FROM AUTHOR]