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Abstract: Let be an integer. The real number is a jump for r if there exists such that for every positive ϵ and every integer , every r-uniform graph with vertices and at least edges contains a subgraph with m vertices and at least edges. A result of Erdős, Stone and Simonovits implies that every is a jump for . For , Erdős asked whether the same is true and showed that every is a jump. Frankl and Rödl gave a negative answer by showing that is not a jump for r if and . Another well-known question of Erdős is whether is a jump for and what is the smallest non-jumping number. In this paper we prove that is not a jump for . We also describe an infinite sequence of non-jumping numbers for . [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and let F be arbitrary field of characteristic not p, containing a primitive pth root of unity ζ. In this paper, we prove a criterion, giving the obstructions to realizability of p-groups as Galois groups over F, having a factor-group of the kind . We apply this to the non-abelian groups of orders and . Where it is possible, we give a description of all Galois extensions realizing these groups. We discuss also automatic realizations and local fields. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give new upper bounds for Euclidean minima of algebraic number fields. In particular, to show that Minkowski''s conjecture holds for the maximal totally real subfields of cyclotomic fields of prime power conductor. [Copyright &y& Elsevier]

Abstract: Gröbner bases as a means of studying ideals in polynomial rings have been generalized to other algebraic structures. The purpose of this paper is to present a general setting for Gröbner bases developed by the author in her habilitation thesis. This thesis was co-refereed by Volker Weispfenning who has contributed to the field of Gröbner bases in various publications. [Copyright &y& Elsevier]

Abstract: Several recently proposed block ciphers such as AES, Camellia, Shark, Square and Hierocrypt use s-boxes that are based on the inversion mapping over . In order to hide the simple algebraic structure in this mapping, an affine transformation over is usually used after the output of the s-box. In some ciphers, an additional affine transformation is used before the input of the s-box as well. In this paper, we study the algebraic properties of a simple approximation in the form , for such s-boxes. The implication of this result on the cryptanalysis of these ciphers remains an open problem. [Copyright &y& Elsevier]

Abstract: For an integer , a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k in , the union of every set of k of these lines has rank . Spikes are very special and important in matroid theory. Wu [On the number of spikes over finite fields, Discrete Math. 265 (2003) 261–296] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number p, a ) representable n-spike is only representable on fields with characteristic p provided that . Moreover, M is uniquely representable over . [Copyright &y& Elsevier]

Abstract: A Riemann surface X is said to be of type if its full automorphism group AutX is cyclic of order n and the quotient surface has genus m. In this paper we determine necessary and sufficient conditions on the integers and γ, where n is odd, for the existence of a Riemann surface of genus g and type admitting a symmetry with γ ovals. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the realizability problem of branched coverings of , namely, given a collection of partitions of a positive integer d, whether there is a branched covering with it as branch data. By reducing to the enumeration of a special type of fat graphs, we deduce a formula in terms of characters of symmetric groups to decide the problem. Computation further shows that the difficulty of the problem only lies in the case when the genus of the covering surface is 0 or 1. [Copyright &y& Elsevier]

Abstract: Given a strictly increasing sequence s of non-negative integers, filtering a word by s consists in deleting the letters such that i is not in the set . By a natural generalization, denote by , where L is a language, the set of all words of L filtered by s. The filtering problem is to characterize the filters s such that, for every regular language L, is regular. In this paper, the filtering problem is solved, and a unified approach is provided to solve similar questions, including the removal problem considered by Seiferas and McNaughton. Our approach relies on a detailed study of various residual notions, notably residually ultimately periodic sequences and residually rational transductions. [Copyright &y& Elsevier]

Abstract: Let Y be a smooth, projective complex curve of genus g ⩾ 1. Let d be an integer ⩾ 3, let e = {e 1, e 2,..., e r } be a partition of d and let | e | = Σ i=1 r − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group S i − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group S d . We prove the irreducibility of these Hurwitz spaces when n − 1 + | e | ⩾ 2d, thus generalizing a result of Graber, Harris and Starr [A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056]. [Copyright &y& Elsevier]

Following the question of Smítal and Štefánková we show that there exist continuous maps of [0,1]n, n=2,3,... possessing an ω-scrambled set of the full n-dimensional Lebesgue measure. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct CCRRS, complete coupling round robin schedules, for n teams each consisting of two pairs. The motivation for these schedules is a problem in scheduling bridge tournaments. We construct for n a positive integer, , with the possible exceptions of . For n odd, we show that a can be constructed using a house with a special property. For n even, a can be constructed from a Howell design, , with a special property called Property P. We use a combination of direct and recursive constructions to construct with Property P. In order to apply our main recursive construction, we need group divisible designs with odd group sizes and odd block sizes. One of our main results is the existence of these group divisible designs. [Copyright &y& Elsevier]

Abstract: Our main aim in this paper is to show that, in Conway''s Angel and Devil game, an Angel of sufficient speed can always escape in three dimensions. We also prove some related results and make some conjectures. [Copyright &y& Elsevier]

Abstract: For all integers and all natural numbers , let represent the least integer such that for every 2-coloring of the set there exists a monochromatic solution to Let and . In this paper it is shown that whenever , It is also shown that for all values of t, [Copyright &y& Elsevier]

Abstract: The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions. [Copyright &y& Elsevier]

Abstract: The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz''s above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form. [Copyright &y& Elsevier]

Abstract: In this paper, we exhibit a noncongruence subgroup whose space of weight 3 cusp forms admits a basis satisfying the Atkin–Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to by Scholl and also verifies the Atkin–Swinnerton-Dyer congruence conjecture for this space. [Copyright &y& Elsevier]

Abstract: Let n be a positive integer. Let be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by is defined to be the matrix whose -entry is the least common multiple of and . The set S is said to be gcd-closed if for any . For an integer let denote the number of distinct prime factors of m. Define . In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying then the LCM matrix is nonsingular. In this paper, we settle completely Sun''s conjecture. We show the following result: (i). If S is a gcd-closed set satisfying then the LCM matrix is nonsingular. Namely, Sun''s conjecture is true; (ii). For each integer there exists a gcd-closed set S satisfying such that the LCM matrix is singular. [Copyright &y& Elsevier]

Abstract: While parametric solutions of the diophantine equation are known for any integral value of , the complete solution in integers is not known for any value of In this paper, we obtain the complete solution of this equation when [Copyright &y& Elsevier]

Abstract: Let R be a regular noetherian local ring and let V be the ring of a zero-dimensional valuation v of the quotient field of R dominating R. The aim of this paper is to characterize when V can be obtained from R by the sequence of successive quadratic transforms of R along V. We also show some properties of such valuations in terms of the sequence. [Copyright &y& Elsevier]

Let λ be a 2-regular partition of n into two parts and let Dλ denote the corresponding irreducible F2Σn-module. We say that Dλ is of quadratic type if there is a non-degenerate Σn-invariant quadratic form defined on this module. In this paper, we show that Dλ is not of quadratic type precisely when the smaller part of λ is a power of 2, say 2r, where r⩾0, and n≡kmod2r+2, where k is one of the 2r consecutive integers 2r+1+2r-1, …, 2r+2-2. [Copyright &y& Elsevier]

Let K/k be a cyclic totally ramified Kummer extension of degree pn with Galois group G. Let O and o be the rings of integers in K and k, respectively. For n=1, F. Bertrandias and M.-J. Ferton determined the ring EndoG(O) of oG-endomorphisms of O and L.N. Childs constructed the maximal order S in O whose ring of oG-endomorphisms is a Hopf order. A Hopf order whose linear dual is a Larson order in kG is called a dual Larson order. In this paper, the generators of a dual Larson order are given. We determine EndoG(O) and obtain a maximal dual Larson order contained in EndoG(O). We construct a Hopf Galois extension S in O, whose ring H(S) of oG-endomorphisms is a dual Larson order. We affirm an order S′ in O with a dual Larson order H(S′) is a tame H(S′)-extension and show that S is maximal in such orders S′. [Copyright &y& Elsevier]

There have been many results for the system {a−m/2ψ(a−mx−nb)}m,n∈Z to be a frame for L2(R), where ψ∈L2(R), a>1, and b>0. In this paper we study some general conditions for the system ψj,k(x)=sj−1/2ψ(sj−1x−bk), j,k∈Z, to be a frame, where sj>0 and bk are real numbers. First, we give a necessary condition which extremely extends the one obtained by Chui and Shi in 1993. Second, we point out that the necessary condition is also sufficient when ψ is band-limited, moreover, we give sufficient conditions, which exclusively depend on ψ, for {ψj,k} to be a frame for a large class of {sj} and {bk}. [Copyright &y& Elsevier]

In this paper, we describe an algorithm that reduces the computation of the (full) $p$-Selmer group of an elliptic curve $E$ over a number field to standard number field computations such as determining the ($p$-torsion of) the $S$-class group and a basis of the $S$-units modulo $p$th powers for a suitable set $S$ of primes. In particular, we give a result reducing this set $S$ of `bad primes' to a very small set, which in many cases only contains the primes above $p$. As of today, this provides a feasible algorithm for performing a full $3$-descent on an elliptic curve over $\mathbb Q$, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of $E[p]$ is favorable, simplifications are possible and $p$-descents for larger $p$ are accessible even today. To demonstrate how the method works, several worked examples are included. [ABSTRACT FROM AUTHOR]

The main result of this paper is the proof of the following partition property of the family of all two-element sets of the first n positive integers.There is a real constant C>0 such that for every partition of the pairs of the set [n]={1,2,…,n} into two parts, there exists a homogeneous set H⊆[n] (i.e., all pairs of H are contained in one of the two partition classes) with min H⩾2 such that∑lower limit h∈H 1/logh⩾Cloglogloglogn/logloglogloglogn⩾C loglogloglogn/logloglogloglogn.This answers positively a conjecture of Erdo¨s (see “On the combinatorial problems which I would most like to see solved”, Combinatorica 1 (1981) 25). [Copyright &y& Elsevier]

In this paper we prove that if (r,12)⩽3, then the set of positive odd integers k such that kr−2n has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to kr2n+1 is also true. [Copyright &y& Elsevier]

The aim of this paper is to give a characterisation of the determinants and signatures of integral ideal lattices over a given algebraic number field. This is then used to obtain an existence criterion for automorphisms of given characteristic polynomial. In particular, we give a different proof of a result of B. Gross and C. McMullen [in preparation]. [Copyright &y& Elsevier]

Let A and B be two finite subsets of a field F. In this paper, we provide a non-trivial lower bound for ∣{a+b:a∈A, b∈B, and P(a,b)≠0}∣ where P(x,y)∈F[x,y]. [Copyright &y& Elsevier]

The first major study of p-groups of maximal class was made by Blackburn in 1958. He showed that an important invariant of these groups is the ‘degree of commutativity.’ Recently (1995) Ferna´ndez-Alcober proved a best possible inequality for the degree of commutativity in terms of the order of the group. Recent computations for primes up to 43 show that sharper results can be obtained when an additional invariant is considered. A series of conjectures about this for all primes have been recorded in [A. Vera-Lo´pez et al., preprint]. In this paper, we prove two of these conjectures. [Copyright &y& Elsevier]

The trade spectrum of a graph G is essentially the set of all integers t for which there is a graph H whose edges can be partitioned into t copies of G in two entirely different ways. In this paper we determine the trade spectrum of complete partite graphs, in all but a few cases. [Copyright &y& Elsevier]

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a2+b, the “length” of b being an even number. [Copyright &y& Elsevier]

We prove that any orderOof any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-orderingO, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Gro¨bner basis given a finite generating set for an ideal. It is shown that our theory of Gro¨bner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Gro¨bner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper. [Copyright &y& Elsevier]

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory. [Copyright &y& Elsevier]

This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups. The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups. [ABSTRACT FROM AUTHOR]

David R. Richman proved that for $n \geq k \geq 2$ every integral $n \times n$ matrix is a sum of seven $k$-th powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every $n \times n$ matrix $ (n \geq k \geq2)$ over an order of an algebraic number field to be a sum of (seven) $k$-th powers. [ABSTRACT FROM AUTHOR]

This paper presents a series of simple but realistic examples in which common algebraic indexing conventions are not so convenient. In particular, it analyzes the difficulties caused by the conventions that each model component must have a fixed number of indices and that the order of the indices is significant to their meaning. To deal with these difficulties compensating extensions to algebraic notation are proposed. The proposed notation is compared to existing notation in terms of the human abilities to understand, maintain and verify model descriptions. [ABSTRACT FROM AUTHOR]

A generalization of P. A. Minlos, V. V. Sazonov's theorem is proved in a case of bounded evenly positive definite function given in Hilbert space. The integral representation is obtained for a family of bounded commutative self-adjoint operators which are connected by algebraic relationship. [ABSTRACT FROM AUTHOR]

Abstract: A graph is called supereulerian if has a spanning Eulerian subgraph. Let be the maximum number of independent edges in the graph . In this paper, we show that if is a 2-edge-connected simple graph and , then is supereulerian if and only if is not for some odd number . [Copyright &y& Elsevier]

Abstract: For a given set A of nonnegative integers the representation functions , are defined as the number of solutions of the equation with , , respectively. In this paper we give a simple proof to two results by Sándor. [Copyright &y& Elsevier]

Abstract: Let E be a CM elliptic curve defined over an algebraic number field F. In general E will not be modular over F. In this paper, we determine extensions of F, contained in suitable division fields of E, over which E is modular. Under some weak assumptions on E, we construct a minimal subfield of division fields over which E is modular. [Copyright &y& Elsevier]

Kramer and Magliveras constructed simple 5-(24,8,λ) designs for λ⩽9. Betten et al. constructed simple 5-(24,8,λ) designs for λ=16,17,…,484. In this paper, we construct 15 mutually disjoint Steiner systems S(5,8,24) and thus we construct simple 5-(24,8,λ) design for λ=10,11,12,13,14 and 15. As a consequence there exists a simple 5-(24,8,λ) design for any positive integer λ⩽λmax=969. [Copyright &y& Elsevier]

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ' R = T ' for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral. [ABSTRACT FROM AUTHOR]

Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects. Emergent multifractality is the object of both fundamental and technology-oriented research. Here, the authors demonstrate and characterize multifractality in the optical resonances of aperiodic arrays of nanoparticles designed from fundamental structures of algebraic number theory. [ABSTRACT FROM AUTHOR]

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.