Score sets ink-partite tournaments

The set S of distinct scores (outdegrees) of the vertices of ak-partite tournamentT(X ₁, X₂, ···, X_k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set ofn non-negative integers is a score set of somek-partite tournament for everyn ≥k ≥ 2.     

Stability of Numerical Invariants in Free Groups

Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l ₁ and l ₂ be cyclic field extensions of k of degree d with norms n _{l 1/k} and n _{l 2/k} . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n _{l 1/k}  ? n _{l 2/k} of degree d. Let K = k(X ₁,…, X _{d ²} ). We compute polynomials which are similarity factors of a form of the kind N ? (n _{l 2/k}  ? _k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n _{l 1/k}  ? n _{l 2/k} to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n _{l 1/k} .     

Nonsingular Equations over Groups II

Let G be a group, t an element distinct from G, and r(t) = g ₁ t ^{l ₁} …g _k t ^{l _k}  ∈ G* ?t?, where each g _i is an element of G order greater than 2, and the l _i are nonzero integers such that l ₁ + l ₂+…+l _k  ≠ 0 and |l _i | ≠ |l _j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ?G*t | r(t)? is injective. This together with previous results show that the natural map from G is injective for k ≥ 1.     

Min-Wise Independent Families with Respect to any Linear Order

A set of permutations 𝒮 on a finite linearly ordered set Ω is said to be k-min-wise independent, k-MWI for short, if Pr (min (π(X)) = π(x)) = 1/|X| for every X ? Ω such that |X| ≤ k and for every x ∈ X. (Here π(x) and π(X) denote the image of the element x or subset X of Ω under the permutation π, and Pr refers to a probability distribution on 𝒮, which we take to be the uniform distribution.) We are concerned with sets of permutations which are k-MWI families for any linear order. Indeed, we characterize such families in a way that does not involve the underlying order. As an application of this result, and using the Classification of Finite Simple Groups, we deduce a complete classification of the k-MWI families that are groups, for k ≥ 3.     

Rings of Constants of the Form k[f]

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ? k[X]? k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f] in k[X]. A similar construction for fields of rational functions is also given.     

The Primitive Sharp Permutation Groups of Twisted Wreath Type

A permutation group G ≤ Sym(X) on a finite set X is sharp if |G|=∏ _l?L(G)(|X| ? l), where L(G) = {|fix(g)| | 1 ≠ g ? G}. We show that no finite primitive permutation groups of twisted wreath type are sharp.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

A Class of P-polynomial Table Algebras with and without Integer Multiplicities

The homogeneous, monotonic, P-polynomial table algebras with valency k ≥ 2 are classified. It is also determined which of these algebras, when integral, have integer multiplicities. In particular, it is shown that all multiplicities are integers only if k = 2 or the diameter d = 2. Some of these algebras come from distance-regular graphs, and some do not.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

Minimal graded free resolutions for monomial curves in 𝔸4 defined by almost arithmetic sequences

Let m = (m₀, m₁, m₂, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m₀, m₁, m₂, n) = 1, such that m₀ < m₁ < m₂ form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m₀? +m₁? +m₂? +n?. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X₀ = t^m₀, X₁ = t^m₁, X₂ = t^m₂, Y = tⁿ is a graded R-module, where R is the polynomial ring k[X₀, X₁, X₂, Y] with the grading degX_i: = m_i, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].     

Stochastic Dynamics for Passage Times and Diffusion Approximations for Finite Capacity Storage Models with Different Kinds of Barriers

In this article, we discuss a number of storage models of finite capacity with random inputs, random outputs, and linear release policy. They form a class of one-dimensional master equations with separable kernels. For this class of problems, the integral equations for first overflow or first emptiness can be transformed exactly into ordinary differential equations. Analysis is done with separable kernel. For all the stochastic models, two barriers are considered: one at X = 0 and the other at X = k, and the barriers are treated as absorbing or reflecting. The imbedding method is used to derive a third order differential equation. We consider first passage times for overflow without or with emptiness of the dam. We also study the passage times for first emptiness with and without overflows. The expected amount of overflows in a given time is also calculated. Finally, by suitable statistical features, all these models are converted into diffusion process with drift. Closed form solutions are obtained for all the problems in terms of Laplace transform functions. For the diffusion process with drift first passage time density is arrived at by treating X = 0 and X = k as absorbing barriers. One of the barriers as reflecting is also studied.     

Almost perfect powers in consecutive integers (II)

Let k ≥ 4 be an integer. We find all integers of the form by^l where l ≥ 2 and the greatest prime factor of b is at most k (i.e. nearly a perfect power) such that they are also products of k consecutive integers with two terms omitted.     

Die Verteilung der k-vollen und l-freien Zahlen

Zusammenfassung  We consider the distribution ofk-full numbers. But we go into more detail and investigate suchk-full integers which are at the same timel-free. We give asymptotic results for the numberN _k,l (x) ofk- full andl-free integers not exceedingx in cases ofl =k + r with 2 ≤r ≤ 5. Moreover, we consider these cases and the casesk = 2, 3,l ≥k + 2 also under the assumption of Riemann’s Hypothesis.      

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

On the Generalization of Hörmander's Inequality

ABSTRACT

We are concerned with a lower bound with a gain of k/2 + 1 derivatives for the class OP N ^{m, k} (X, Σ) of pesudodifferential operators with characteristics of even multiplicity k ≥ 2. In the case of double characteristics operators (k = 2), we recapture a well-known inequality due to Hörmander.     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

Divisibility Graph for Symmetric and Alternating Groups

Let X be a nonempty set of positive integers and X* = X?{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b ∈ X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S _n or is the alternating group A _n .