On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.     

Rationale approximation mit Hilfe pythagoräischer Zahlen

The wealth of Pythagorean number triples is demonstrated afresh by showing that for every rational number $\text{p}\text{q}$, p, q ∈ $\text{N}$ there exist infinitely many Pythagorean number triples (a, b, c) which satisfy

$\text{|}\text{a}\text{b}\text{?}\text{p}\text{q}\text{| ?}\text{1}\text{b}$

where a is odd, b is even and a² + b² = c². The special cases where p = 1 or where q = 1 are considered first as they illustrate the method and yield additional results. Rational approximations are also possible by means of the quotients $\text{c}\text{b}$, provided p > q. The results generalize Pythagorean number triples (a, b, c) where a and b differ by a constant investigated by S. Pignataro.     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

Some big sets of mutually disjoint subgeometries of PG(d, q t )

In PG(d, q ^t ) we construct a set ? of mutually disjoint subgeometries isomorphic to PG(d, q) almost partitioning the point set of PG(d, q ^t ) such that there is a group of collineations of PG(d, q ^t ) operating simultaneously as a Singer cycle on all elements of ?. In PG(t?1,q ^t ) we construct big subsets ? of ? whose elements are far away from each other in the following sense:

u

? If P ₁, P ₂ ∈ ? _k , then no point of P ₁ lies on ak-dimensional subspace of P ₂.

For example, we get a set ofq - 1 subplanes of orderq of PG(2,q ³) such that no point of one subplane lies on a line of another subplane, and such that no three points of three different subplanes are collinear.     

Completely strong path-connected tournaments

Let T = (V, A) be a tournament with p vertices. T is called completely strong path-connected if for each arc (a, b) ∈ A and k (k = 2, 3,…, p), there is a path from b to a of length k (denoted by P_k(a, b)) and a path from a to b of length k (denoted by P′_k(a, b)). In this paper, we prove that T is completely strong path-connected if and only if for each arc (a, b) ∈ A, there exist P₂(a, b), P′₂(a, b) in T, and T satisfies one of the following conditions: (a) T/T₀-type graph, (b) T is 2-connected, (c) for each arc (a, b) ∈ A, there exists a P′_p?1(a, b) in T.     

On the difference between an integer and its m-th power mod n

For any real constants λ ₁, λ ₂ ∈ (0, 1], let $n \geqslant \max \{ [\tfrac{1} {{\lambda _1 }}],[\tfrac{1} {{\lambda _2 }}]\} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ ₁ n] and b ∈ [1, λ ₂ n] satisfy the congruence b ≡ a ^m (mod n). The main purpose of this paper is to study the mean value of (a ? b)^2k for any fixed positive integer k and obtain some sharp asymptotic formulae.     

Accepted Elasticity in Local Arithmetic Congruence Monoids

For certain \({a,b \in \mathbb{N}}\) , an Arithmetic Congruence Monoid M(a, b) is a multiplicatively closed subset of \({\mathbb{N}}\) given by \({\{x\in\mathbb{N}:x \equiv a \pmod{b}\} \cup\{1\}}\) . An irreducible in this monoid is any element that cannot be factored into two elements, each greater than 1. Each monoid element (apart from 1) may be factored into irreducibles in at least one way. The elasticity of a monoid element (apart from 1) is the longest length of a factorization into irreducibles, divided by the shortest length of a factorization into irreducibles. The elasticity of the monoid is the supremum of the elasticities of the monoid elements. A monoid has accepted elasticity if there is some monoid element that has the same elasticity as the monoid. An Arithmetic Congruence Monoid is local if gcd(a, b) is a prime power (apart from 1). It has already been determined whether Arithmetic Congruence Monoids have accepted elasticity in the non-local case; we make make significant progress in the local case, i.e. for many values of a, b.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Polynomials and primitive roots in finite fields

The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds.Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p.Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b¹ modulo p.     

On 2min-sets and 2minmax-sets with respect to certain cycles

Let [n] = {1, 2, …, n}. Suppose we have k linear orderings on [n], say <1, <2, …, <k. Let M ? [n]. Then M has a minimum for each linear ordering <i. So M has at most k minima. A set M ? [n] is called a 2min-set if it has at most two different minima in the linear orderings <1, <2, …, <k. Similarly, a set N ? [n] can have at most k minima and k maxima for any k linear orderings. A set N ? [n] is called a 2minmax-set if there exist a, b ∈ N such that all the elements in N | {a, b} lie in between a and b for every linear ordering <i. In this paper, we shall determine the sizes of 2min-sets and 2minmax-sets for certain k linear orderings.     

An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Raki?, N. ?. Din?i? and D. S. Djordjevi? generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)* = ab, ba² = a, and ab² = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.     

Cubic Thue inequalities with negative discriminant

We give an upper bound for the solutions of the family of cubic Thue inequalities |x³+axy²+by³|?k when a is positive and larger than a certain value depending on b. For the case k=a+|b|+1 and a?360b⁴ we show that these inequalities have only trivial solutions. For the case k=a+|b|+1 and |b|=1,2, we solve these inequalities for all a?1. Our method is based on Padé approximations using Rickert's integrals. We also use a generalization of Legendre's theorem on continued fractions.     

Best methods of interpolation for certain classes of differentiable functions

For the class W^(r)L_q (M;a, b), 1≤q≤∞, we construct the best method of approximation of the functionalf (x), x∈ [a, b], among all the methods using only information about the values off (^k)(x_i) (k=0, 1, ..., r?1; i=1, 2, ..., N).     

Factorisation dans un Ordre Non Maximal d'un Corps Quadratique

Let O_f be an order of index f in a quadratic field. We denote A_f the set of elements of O_f whose norm is relatively prime to f. An element v ∈ A_f is called k-prime if for x, y ∈ A_f, v|xy implies v|x^k or v|y^k where k is the exponent of the group O_f. We prove that the k-th powers of the elements of A_f have a unique representation as a product of elements which are irreductible and k-prime. One criteria for resolution of some diophantine equations is an illustration from it.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

Artinian rings with supersolvable adjoint group

The set of all elements of an associative ring R, not necessarily with a unit element, forms a monoid under the circle operation a ° b = a + b + ab. The group of all invertible elements of this monoid is called the adjoint group of R and is denoted by R ^°. It is proved that an artinian ring R with supersolvable adjoint group R ^° must be Lie supersolvable. An example of a Lie supersolvable ring with non-supersolvable adjoint group is also constructed. Received: 7 December 2007     

Uniqueness of recovering the parameters of sectional operators on simple complex Lie algebras

By a sectional operator on a simple complex Lie algebra g we mean a self-adjoint operator ?: g → g satisfying the identity [?x, a] = [x, b] for some chosen elements a, b ∈ g, a ≠ 0. The problem concerning the uniqueness of recovering the parameters of a given specific operator arises in many areas of geometry. The main result of the paper is as follows: if a and b are not proportional and a is regular and semisimple, then every pair of parameters p, q of the sectional operator is obtained from the pair a, b by multiplying the pair by a nonzero scalar, i.e., the parameters are recovered uniquely in a sense. It follows that the Mishchenko-Fomenko subalgebras for regular semisimple elements of the Poisson-Lie algebra coincide for proportional values of the parameters only.     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.