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Let b \geq 2 be an integer and S be a finite non-empty set of primes not containing divisors of b. For any non-dense set A \subseteq [0,1) which is invariant under \times b operation, we prove the finiteness of rational numbers in A whose denominators can only be divided by primes in S. A quantitative result on the largest prime divisors of the denominators of rational numbers in A is also obtained. [ABSTRACT FROM AUTHOR]

Let R be a ring and X a complex of R-modules. We give some characterizations of the dimension of X related to special Gorenstein projective precovers. Let R be a ring such that (G P , G P ⊥) forms a cotorsion pair cogenerated by a set, where G P denotes the category consisting of all Gorenstein projective R-modules. Then the dimension related to special Gorenstein projective precovers Gppd (X) is equal to the infimum of the set { n ∈ Z | there exists a complete Gorenstein projective resolution of X T → σ G → X such that σi is bijective for each i ≥ n }. Forthmore, if Gppd (X) is finite and n an integer then Gppd (X) ≤ n if and only if Ext G P i (X , Q) = 0 for any i > n and any projective R-module Q, where Ext G P i (− , −) is the relative cohomology functor. [ABSTRACT FROM AUTHOR]

The system of two inequalities is considered, and an upper bound for the number of its solutions is established. Here , , , , and are given real numbers, and are positive and arbitrarily small, is the distance to the nearest integer, and and are coprime variables from given intervals such that the partial quotients of the continued fraction expansion of belong to a finite alphabet . [ABSTRACT FROM AUTHOR]

The ring $\mathbb Z_{d}$ of d -adic integers has a natural interpretation as the boundary of a rooted d -ary tree $T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin ['Automata finiteness criterion in terms of van der Put series of automata functions', p-Adic Numbers Ultrametric Anal. Appl. 4 (2) (2012), 151–160] showed that $f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p -automatic sequence over a finite subset of $\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers $d\geq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree. [ABSTRACT FROM AUTHOR]

Let K be a mixed characteristic local field whose residue field has cardinality q , and let n be an integer dividing q − 1. In the first part of this document, we construct a K -theoretic enhancement of the n -th power residue symbol K × × K × → μ n. In the second part we construct central extensions of G L m (K) by μ n and we express the n -th power residue symbol in terms of a symbol defined using the extension obtained for m = 1. Our constructions both rely on the study of finite free pointed μ n -sets. [ABSTRACT FROM AUTHOR]

Recently there has been significant progress in classifying integer friezes and \mathrm {SL}_2-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame \mathrm {SL}_2-tilings, positive integer \mathrm {SL}_2-tilings, and tame integer friezes – both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes which generalises the classical construction of Conway and Coxeter for positive integer friezes. [ABSTRACT FROM AUTHOR]

In this article, we prove the integrality of v-adic multiple zeta values (MZVs). For any index s ∈ Nr and finite place v ∈ A:= Fq[θ], Chang and Mishiba introduced the notion of the v-adic MZVs ζA(s)v, which is a function field analogue of Furusho's p-adic MZVs. By estimating the v-adic valuation of ζA(s)v, we show that ζA(s)v is a v-adic integer for almost all v. This result can be viewed as a function field analogue of the integrality of p-adic MZVs, which was proved by Akagi-Hirose-Yasuda and Chatzistamatiou. [ABSTRACT FROM AUTHOR]

A tensor v is the sum of at least rank ⁡ (v) elementary tensors. In addition, a 'border rank' is defined: rank _ (w) = r holds if r is the minimum integer such that w is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e. tensors with r = rank _ (w) < rank ⁡ (w) may exist. It is easy to see that in such a case the representation of rank-r tensors v contains diverging elementary tensors as v approaches w. In a first part, we recall results about the uniform strength of the divergence in the case of general nonclosed tensor formats (restricted to finite dimensions). The second part discusses the r-term format for infinite-dimensional tensor spaces. It is shown that the general situation is very similar to the behaviour of finite-dimensional model spaces. The third part contains the main result: it is proved that in the case of rank _ (w) = 2 < rank ⁡ (w) the divergence strength is ≳ ε − 1 / 2 , i.e. if ‖ v − w ‖ < ε and rank ⁡ (v) ≤ 2 , the parameters of v increase at least proportionally to ε − 1 / 2. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, M be a finitely generated R-module and a be an ideal of R. For an arbitrary integer k &#8804:-1, we introduce the concept of k-projective dimension of M denoted by k-pdRM. We show that the finite k-projective dimension of M is at least k-depth(a;R)-k-depth(a;M). As a generalization of the Intersection Theorem, we show that for any finitely generated R-module N, in certain conditions, k-pdRM is nearer upper bound for dimN than pdRM. Finally, if M is k-perfect, dimN ≥ k-gradeM that generalizes the Strong Intersection Theorem. [ABSTRACT FROM AUTHOR]

For any positive integer m , let Z m be the set of residue classes modulo m. For A ⊆ Z m and n ‾ ∈ Z m , let the representation function R A (n ‾) denote the number of solutions of the equation n ‾ = a ‾ + a ′ ‾ with unordered pairs (a ‾ , a ′ ‾) ∈ A × A. Let M be an odd integer with M ≥ 3 and m = 2 M. In this paper, we determine the structure of A , B ⊆ Z m with A ∪ B = Z m and | A ∩ B | = 2 such that R A (n ‾) = R B (n ‾) for all n ‾ ∈ Z m. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) , and let d (x , y) denote the length of a shortest path between nodes x and y in G. For a positive integer k and for distinct x , y ∈ V (G) , let d k (x , y) = min { d (x , y) , k + 1 } and R k { x , y } = { z ∈ V (G) : d k (x , z) ≠ d k (y , z) }. A subset S ⊆ V (G) is a k - truncated resolving set of G if | S ∩ R k { x , y } | ≥ 1 for any pair of distinct x , y ∈ V (G). The k - truncated metric dimension , dim k (G) , of G is the minimum cardinality over all k -truncated resolving sets of G , and the usual metric dimension is recovered when k + 1 is at least the diameter of G. We obtain some general bounds for k -truncated metric dimension. For all k ≥ 1 , we characterize connected graphs G of order n with dim k (G) = n − 2 and dim k (G) = n − 1. For all j , k ≥ 1 , we find the maximum possible order, degree, clique number, and chromatic number of any graph G with dim k (G) = j. We determine dim k (G) when G is a cycle or a path. We also examine the effect of vertex or edge deletion on the truncated metric dimension of graphs, and study various problems related to the truncated metric dimension of trees. [ABSTRACT FROM AUTHOR]

In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of sinc functions. We then give a general formula to compute the integral on the real line of the product of the binomial coefficient and a given function, which, in some cases, turns out to be equal to the series of their values on the integers. Finally, we establish a list of identities obtained by applying these formulas. [ABSTRACT FROM AUTHOR]

Let S be a nonempty commutative semigroup written additively. An element e of S is said to be idempotent if e + e = e. The Erdős-Burgess constant I (S) of the semigroup S is defined as the smallest positive integer ℓ such that any S -valued sequence T of length ℓ contains a nonempty subsequence the sum of whose terms is an idempotent of S. We make a study of I (S) when S is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that I (S) is finite, and we obtain sharp bounds of I (S) in case I (S) is finite, and determine the precise value of I (S) in some cases which unifies some well known results on the precise value of Davenport constant in the setting of commutative semigroups. [ABSTRACT FROM AUTHOR]

For fixed positive integers k and N, there are only finitely many holomorphic eta quotients of weight k for the congruence subgroup \(\Gamma_{0}(N)\). In this article, we obtain a similar finiteness result for holomorphic generalized eta quotients of weight k on \(\Gamma_1(p)\) where k is a positive integer and p is a prime number. We also obtain a criterion for two holomorphic generalised eta quotients which represent the same one. [ABSTRACT FROM AUTHOR]

A group G is said to be of finite rank r if every finitely generated subgroup of G can be generated by at most r elements, and r is the least positive integer with a such property. If there is no such r, then the group G is said to be of infinite rank. In the present paper, it is proved that if G is an X -group of infinite rank whose proper subgroups of infinite rank are finite-by-hypercentral (respectively, hypercentral-by-finite), then all proper subgroups of G are finite-by-hypercentral (respectively, hypercentral-by-finite), where X is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations P ´ , P ' and L . [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a partition π = (V 1 , V 2 , ... , V k) of the vertex set V is a transitive partition if V i dominates V j whenever i < j. The transitivity  Tr (G) of a graph G is the maximum order of a transitive partition of V. For any positive integer k , we characterize the smallest tree with transitivity k and obtain an algorithm to determine the transitivity of any tree of finite order. [ABSTRACT FROM AUTHOR]

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

An algorithm presented in this article gives a correct answer to one of the two questions for any (max, +) labeling problem with integer weights: either “What is the best labeling?” or “Is the problem supermodular?”, and this answer is guaranteed to be correct. The algorithm is called self-driven because the user cannot decide which of the two questions will be answered — this decision is up to the algorithm. Also, the algorithm does not need to know the order of labels if the problem is supermodular. The finite execution time of the algorithm is guaranteed for integer weights of vertices and edges and use of subgradient descent. [ABSTRACT FROM AUTHOR]

For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A, B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms A → C → B, then PCSP(A, B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto [Bar19] showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to some finite C is possible, this structure may become arbitrarily large. For every integer n > 1 and every prime p we give A, B of size n with a single relation of arity n p such that PCSP(A, B) reduces via a chain of homomorphisms A → C → B to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime p ≥ 7 we construct A, B of size p − 1 with a single ternary relation such that PCSP(A, B) reduces via A → C → B to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A, B) reduces to a tractable CSP(C) for some finite digraph C, then already A or B has a tractable CSP. This extends results and answers a question of [DSM+21]. [ABSTRACT FROM AUTHOR]

We consider elliptic surfaces E over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or p > 3 and E is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and K 2 unbounded. [ABSTRACT FROM AUTHOR]

Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth cycles containing e. For a vertex v of Γ let {e1,e2,...,ek} be the set of edges incident to v ordered such that n(e1)≤n(e2)≤⋯ ≤n(ek). Then (n(e1),n(e2),...,n(ek)) is called the signature of v. The graph Γ is said to be girth‐regular if all of its vertices have the same signature. Let Γ be a girth‐regular graph with girth g=2d and signature (a1,a2,...,ak). It is known that in this case we have ak≤(k−1)d. In this paper we show that if ak=(k−1)d−ϵ for some nonnegative integer ϵ

Let d be a positive integer and U ⊂ ℤd finite. We study β (U) : = inf A , B ≠ ϕ finite | A + B + U | | A | 1 / 2 | B | 1 / 2 , and other related quantities. We employ tensorization, which is not available for the doubling constant, ∣U + U∣/∣U∣. For instance, we show β (U) = | U | , whenever U is a subset of {0,1}d. Our methods parallel those used for the Prékopa—Leindler inequality, an integral variant of the Brunn—Minkowski inequality. [ABSTRACT FROM AUTHOR]

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers ∑ k = n m n (− 1) k U k , ∑ k = n m n (− 1) k U 2 k and ∑ k = n m n (− 1) k U 2 k − 1 , where m and n are positive integers with m ≥ 2 , n ≥ 2. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers. [ABSTRACT FROM AUTHOR]

This paper aims to study the zero distribution of v -adic multiple zeta values over function fields. We show that the interpolated v -adic MZVs at negative integers only vanish at what we call the "trivial zeros", for degree one finite place over rational function fields. And we conjecture that this result can be generalized to all finite places. [ABSTRACT FROM AUTHOR]

We solve d-dimensional Heesch's problem in the asymptotic sense. Namely, we show that, if d → ∞ , then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space E d ; in other words, given any nonnegative integer n, we can find a dimension d (depending on n) in which there exists a hypersolid whose Heesch number is finite and greater than n. [ABSTRACT FROM AUTHOR]

We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let Γ be a finite simplicial graph with all edges labelled by positive even integers, and let A Γ A_{\Gamma} be its associated Artin group; our new proof implies that if A T A_{T} is poly-free (resp. normally poly-free) for every clique 푇 in Γ, then A Γ A_{\Gamma} is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell–Jones Conjecture for even Artin groups. In particular, we show that if A Γ A_{\Gamma} is an even Artin group such that each clique in Γ either has at most three vertices, has all of its labels at least 6, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by 2), then A Γ A_{\Gamma} satisfies the Farrell–Jones Conjecture. In addition, our methods enable us to obtain results for general Artin groups. [ABSTRACT FROM AUTHOR]

For a finite group G , we associate the quantity β (G) = | L (G) | | G | , where L (G) is the subgroup lattice of G. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of β on the class of p -groups of order p n , where n ≥ 3 is an integer. We show that the set containing the quantities β (G) , where G is a finite (abelian) group, is dense in [ 0 , ∞). Finally, we consider β to be a function on L (G) and we indicate some of its properties, the main result being the classification of finite abelian p -groups G satisfying β (H) ≤ 1 , ∀ H ∈ L (G). [ABSTRACT FROM AUTHOR]

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word λ and the prefixes of a right-infinite word ρ. The main theorem states that the set of minimum lengths of words x and x ′ such that x λ n = ρ n x ′ or λ n x = x ′ ρ n is finite, where n runs over positive integers and λ n and ρ n are respectively the suffix of λ and the prefix of ρ of length n , if and only if λ and ρ are ultimately periodic words of the form λ = u − ∞ v and ρ = w u ∞ for some finite words u , v and w. [ABSTRACT FROM AUTHOR]

In this paper, we characterise the family of finite arc‐transitive bicirculants. We show that every finite arc‐transitive bicirculant is a normal r$r$‐cover of an arc‐transitive graph that lies in one of eight infinite families or is one of seven sporadic arc‐transitive graphs. Moreover, each of these 'basic' graphs is either an arc‐transitive bicirculant or an arc‐transitive circulant, and each graph in the latter case has an arc‐transitive bicirculant normal r$r$‐cover for some integer r$r$. [ABSTRACT FROM AUTHOR]

Given a right-infinite word x over a finite alphabet A, the rank of x is the size of the smallest set S of words over A such that x can be realized as an infinite concatenation of words in S. We show that the property of having rank two is decidable for the class of k-automatic words for each integer k ≥ 2. [ABSTRACT FROM AUTHOR]

A purely inseparable field extension K of a field k of characteristic p 6= 0 is said to be !0-generated over k if K=k is not finitely generated, but L=k is finitely generated for each proper intermediate field L. In 1986, Deveney solved the question posed by R. Gilmer and W. Heinzer, which consists in knowing if the lattice of intermediate fields of an !0-generated field extension K=k is necessarily linearly ordered under inclusion, by constructing an example of an !0-generated field extension where [kp-n \ K: k] = p2n for all positive integer n. This example has proved to be extremely useful in the construction of other examples of !0-generated field extensions (of any finite irrationality degree). In this paper, we characterize the extensions of finite irrationality degree which are !0-generated. In particular, in the case of unbounded irrationality degree, any modular extension of unbounded exponent contains a proper subfield of unbounded exponent over the ground field. Finally, we give a generalization, illustrated by an example, of the !0-generated to include modular purely inseparable extensions of unbounded irrationality degree. [ABSTRACT FROM AUTHOR]

We define positive integer roots of finite matrices in the 2 × 2 case. Where possible, we try to generalise results to finite n × n matrices. [ABSTRACT FROM AUTHOR]

Let 71, be a positive integer and let G be a group. We denote by v(G) a certain extension of the non-abelian tensor square G ⨂ G by G x G. Set T⨂(G) = {g ⨂ h | g, h ∈ C}. We prove that if the size of the conjugacy class |xv(G)| ≤ n for every x ∈ T⨂(G), then the second derived subgroup v(G)" is finite with n-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group. [ABSTRACT FROM AUTHOR]

We prove that for every graph H of maximum degree at most 3 and for every positive integer q there is a finite f=f(H,q) such that every Kf‐minor contains a subdivision of H in which every edge is replaced by a path whose length is divisible by q. For the case of cycles we show that for f=O(qlogq) every Kf‐minor contains a cycle of length divisible by q, and observe that this settles a recent problem of Friedman and the second author about cycles in (weakly) expanding graphs. [ABSTRACT FROM AUTHOR]

We give |$L^1$| -norm estimates for exponential sums of a finite sets |$A$| consisting of integers or lattice points. Under the assumption that |$A$| possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee–Pigno–Smith and Konyagin. These theorems improve upon past work of Petridis. [ABSTRACT FROM AUTHOR]

We introduce the notions of τ -exceptional and signed τ -exceptional sequences for any finite dimensional algebra. We prove that for a fixed algebra of rank n , and for any positive integer t ≤ n , there is a bijection between the set of signed τ -exceptional sequences of length t , and (basic) ordered support τ -rigid objects with t indecomposable direct summands. If the algebra is hereditary, our notions coincide with exceptional and signed exceptional sequences. The latter were recently introduced by Igusa and Todorov, who constructed a similar bijection in the hereditary setting. [ABSTRACT FROM AUTHOR]

As a common generalization of Rickart -rings and generalized Baer -rings, we say that a ring with an involution is a generalized Rickart -ring if for all the right annihilator of is generated by a projection for some positive integer depending on . The abelian generalized Rickart -rings are closed under finite direct product. We address the behavior of the generalized Rickart condition with respect to various constructions and extensions, present some families of generalized Rickart -rings, study connections to the related classes of rings, and indicate various examples of generalized Rickart -rings. Also, we provide some large classes of finite and infinite-dimensional Banach -algebras that are generalized Rickart -rings but neither Rickart -rings nor generalized Baer -rings. [ABSTRACT FROM AUTHOR]

A group of matrices with entries in a number field K is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (3 , 3 , 3) , (3 , 3 , ∞) , (3 , 4 , 4) , (3 , 4 , ∞) , (3 , 6 , 6) , (3 , ∞ , ∞) , (4 , 4 , 4) , (4 , 4 , ∞) , (4 , ∞ , ∞) , (6 , 6 , 6) , (6 , 6 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]

For any positive integer m, let Z m be the set of residue classes modulo m. For A ⊆ Z m and n ¯ ∈ Z m , let the representation function R A (n ¯) denote the number of solutions of the equation n ¯ = a ¯ + a ′ ¯ with unordered pairs (a ¯ , a ′ ¯) ∈ A × A . We characterize the partitions of Z 2 p with A ∪ B = Z 2 p and | A ∩ B | = 2 such that R A (n ¯) = R B (n ¯) for all n ¯ ∈ Z 2 p , where p is an odd prime. [ABSTRACT FROM AUTHOR]

Given integers n ≥ m , let Sep (n , m) be the set of separable states on the Hilbert space C n ⊗ C m . It is well-known that for (n , m) = (3 , 2) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set Sep (n , m) has no semidefinite programming description of finite size. As Sep (n , m) is a semialgebraic set this provides a new counterexample to the Helton–Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer's approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions. [ABSTRACT FROM AUTHOR]

We prove that the Maskin monotonicity⁎ condition (proposed by Bergemann et al. (2011)) fully characterizes exact rationalizable implementation in an environment with lotteries and transfers. Different from previous papers, our approach possesses many appealing features simultaneously, e.g., finite mechanisms with no integer game or modulo game are used; no transfers are made in any rationalizable profile; the message space is small; the implementation is robust to information perturbations in the sense of Oury and Tercieux (2012). [ABSTRACT FROM AUTHOR]

Fix a ∈ Z , a ∉ { 0 , ± 1 }. A simple argument shows that for each ϵ > 0 , and almost all (asymptotically 100% of) primes p , the multiplicative order of a modulo p exceeds p 1 2 − ϵ. It is an open problem to show the same result with 1 2 replaced by any larger constant. We show that if a , b are multiplicatively independent, then for almost all primes p , one of a , b , a b , a 2 b , a b 2 has order exceeding p 1 2 + 1 30 . The same method allows one to produce, for each ϵ > 0 , explicit finite sets A with the property that for almost all primes p , some element of A has order exceeding p 1 − ϵ. Similar results hold for orders modulo general integers n rather than primes p. [ABSTRACT FROM AUTHOR]

Let Γ (G) denote the Coprime Graph of a finite group G. In this paper we study the Laplacian eigenvalues of the Coprime Graph of the finite cyclic group ℤ n and the Dihedral group D n where n > 2. We find the characteristic polynomial of Γ (ℤ n) for any n > 2 and determine the eigenvalues of Γ (ℤ n) for n = p m , p q where p , q are primes and m is a positive integer. We characterize the values of n for which algebraic and vertex connectivity of Γ (ℤ n) are equal. We also discuss about the largest and the second largest eigenvalue of Γ (ℤ n). Finally, the spectra of Γ (D n) has been determined for n = p m , p q where p , q are primes and m is a positive integer. [ABSTRACT FROM AUTHOR]

By using the work of Frantzikinakis and Wierdl, we can see that for all d ∈ N , α ∈ (d , d + 1) , and integers k ≥ d + 2 and r ≥ 1 , there exist infinitely many n ∈ N such that the sequence (⌊ (n + r j) α ⌋) j = 0 k − 1 is represented as ⌊ (n + r j) α ⌋ = p (j) , j = 0 , 1 , ... , k − 1 , by using some polynomial p (x) ∈ Q [ x ] of degree at most d. In particular, the above sequence is an arithmetic progression when d = 1. In this paper, we show the asymptotic density of such numbers n as above. When d = 1 , the asymptotic density is equal to 1 / (k − 1). Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields. [ABSTRACT FROM AUTHOR]

Let G be a permutation group of a set Ω and k be a positive integer. The k-closure of G is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(Ω) which has the same orbits as has G under the componentwise action on the set Ωk. It is proved that the k-closure of a finite nilpotent group coincides with the direct product of k-closures of all of its Sylow subgroups. [ABSTRACT FROM AUTHOR]

A P-recursive sequence is a sequence which satisfies a linear recurrence relation with polynomial coefficients. For a P-recursive sequence { a n } n ≥ 0 and its power { a n m } n ≥ 0 , we give a method to construct a polynomial X (n) so that the sum of ∑ k = 0 n − 1 X (k) a k m equals a finite sum of the products of a polynomial and a n − n 1 a n − n 2 ⋯ a n − n k , where n 1 , ... , n k are integers. When these summands have a common factor f (n) , we will derive that f (n) is a factor of the sum ∑ k = 0 n − 1 X (k) a k m . This method can be used to derive some congruences. We give some examples to exhibit the applications of the method. [ABSTRACT FROM AUTHOR]

A numerical scheme with the piecewise linear interpolation polynomial method was established to solve a class of nonlinear fractional ordinary differential equations including the Hadamard finite part integral. In the time direction, the fractional derivative was approximated with the piecewise linear interpolation polynomial method, and the integer order time derivative was discretized by means of the 2nd-order backward difference scheme. Through detailed proof, the error estimates with an accuracy of 0(τmin{１＋α, １＋β} )were obtained. The comparison between the numerical results and the theoretical solution shows the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1,... , k − 1}2 satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural restrictions. Some of these formulae are obtained as consequences of finite Fourier series expansions of the greatest common divisor function, whereas some are shown to be closely connected with the notion of unitary divisors. [ABSTRACT FROM AUTHOR]