Quintasymptotic primes and four results of Schenzel

Let I be an ideal in a Noetherian ring R, let (I)_a be the integral closure of I, and let S be a multiplicatively closed subset of R. Let T₁, T₂, and T₃ be the topologies given by the filtrations {Iⁿ R_S ∩ R | n ≥ 1}, {Iⁿ | n ≥ 1}, and {(Iⁿ)_a | n ≥ 1}. We g results due to Schenzel, characterizing when T₁ is either equivalent or linearly equivalent to either of T₂ or T₃. The characterizations involve the sets of essential primes of I, quintessential primes of I, asymptotic primes of I, and quintasymptotic primes of I.     

Isomorphism Checking of I-graphs

We consider the class of I-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two I-graphs I(n, j, k) and I(n, j ₁, k ₁) are isomorphic if and only if there exists an integer a relatively prime to n such that either {j ₁, k ₁} =? {a j mod n, a k mod n } or {j ₁, k ₁} =? {a j mod n, ? a k mod n }. This result has an application in the enumeration of non-isomorphic I-graphs and unit-distance representations of generalized Petersen graphs.     

On latin squares and the facial structure of related polytopes

By identifying all latin squares of order n with certain n²-element subsets of an n³-element ground set E_n a clutter B_n is obtained, which induces an independence system (E_n, I_n) in a natural way. Starting from Ryser's conditions for the completion of latin rectangles (cf. Mirsky [15]) we present special classes of circuits of (E_n, I_n) and extend Ryser's conditions slightly.Latin squares of order n correspond to the solutions of the planar 3-dimensional assignment problem and, in view of its solution via linear programming techniques, we present some first classes of facet-defining inequalities for P(I_n) resp. P(B_n), the convex hull of all those 0–1 vectors, which correspond to members of I_n resp. B_n.     

The set of irreducible graphs for the projective plane is finite

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K_{3, 3}. For positive integer n, let I_n (P) denote a smallest set of graphs whose maximal valency is n and such that any graph which does not embed in the real projective plane contains a subgraph homeomorphic to a graph in I_n (P) for some n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃ (P), and Glover and Huneke proved that I_n (P) is finite for all n. This note proves that I_n (P) is empty for all but a finite number of n. Hence there is a finite set of graphs for the projective plane analogous to Kuratowski's two graphs for the plane.     

On cubic graphs which are irreducible for nonorientable surfaces

Let ∑?_n denote the nonorientable surface of nonorientable genus n. A graph G is irreducible for ∑?_n if G does not embed in ∑?_n but any proper subgraph does embed. Let I₃∑?_n) denote the set of cubic irreducible graphs for ∑?_n. This note provesTheorem. I₃(∑?_n) is finite for eachn.     

The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra S_n in the title which is an algebra obtained from a polynomial algebra P_n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P_n. The algebra S_n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S_n is 2n; but the weak and the global dimensions of S_n are n. The prime and maximal spectra of S_n are found, and the simple S_n-modules are classified. It is proved that the algebra S_n is central, prime, and catenary. The set I_n of idempotent ideals of S_n is found explicitly. The set I_n is a finite distributive lattice and the number of elements in the set I_n is equal to the Dedekind number d_n.     

Filtrations,rees rings,and ideal transforms

Let I be an ideal, and let f = {K_n|n ≥ 0 } be a filtration of the Noetherian ring R, such that Iⁿ ⊆ K_n for all n ≥ 0. We study when the Rees ring $\text{R}$(f) is either finite or integral over the Rees ring $\text{R}$(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (Iⁿ : J^{m(n) + 1}), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.     

The mathematics of perfect shuffles

There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle O leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group 〈 I, O 〉 generated by the two shuffles, for all n. If 2n is not a power of 2, and if 2n ≠ 12,24, then 〈 I, O 〉 has index 1, 2, or 4 in the Weyl group B_n (the group of all 2ⁿn! signed n × n permutation matrices). If 2n = 2^k, then 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂^k and Z_k. When 2n = 24, 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂¹¹ and M₁₂, the Mathieu group of degree 12. When 2n = 12, 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂⁶ and the group PGL(2,5) of all linear fractional transformations over GF(5).     

Graphs with bounded valency that do not embed in the projective plane

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K₃₃. Let I_n(P) denote the minimal set of graphs whose vertices have miximal valency n such that any graph which does not embed in the real projective plane (or equivalently, does not embed in the Möbius band) contains a subgraph homeomorphic to a graph in I_n(P) for some positive integer n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃(P). This note proves that for each n, I_n(P) is finite.     

Existence of Tchebycheff extensions

Given a Tchebycheff System {y_o ··· y_n} defined on an interval I, it is proved that there exists a function y_n+1, such that the system {y_o ··· y_n, y_n+1} is a Tchebycheff System on I. A function such as y_n+1 is called a Tchebycheff extension of the system {y_i}ⁿ_{(i = 0)}.     

Bitableau bases for Garsia-Haiman modules of hollow type

Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x₁,x₂,…,xn} and Yn={y₁,y₂,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

Nonhomogeneous spectra of numbers

A simple characterization is given of those sequences of integersM_n={a_i}ⁿ_i=1for which there exist real numbers αandβ such thata_i=?iα+β?(1?i?n). In addition, for givenM_n, an open intervalI_nis computed such that α?I_nif and only ifa_i=?iα+β?(1?i?n)for suitableβ=β(α).     

Solution of a system of functional-differential equations arising from an optimization problem

In connection with an optimization problem, all functions ?: Iⁿ → $\text{R}$ with continuous nonzero partial derivatives and satisfying

$\text{??}\text{?x,}{}_{\mathrm{i}}\text{??}\text{?x}{}_{\mathrm{j}}$

for all x_i, x_j ≠ I, i, j = 1,2,…, n (n > 2) are determined (I is an interval of positive real numbers).     

An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions U _n and W _n . Contrary to the discretization scheme enacted by Ablowitz and Ladik, a central difference scheme is applied to the positional derivative term in the matrix Zakharov-Shabat system to arrive at a different discrete linear system. The major effect of the new discretization is that we no longer need the following two conditions in theories based on the Ablowitz-Ladik discretization: (a) invertibility of I _N ?U _n W _n and I _M ?W _n U _n , and (b) I _N ?U _n W _n and I _M ?W _n U _n being nonzero multiples of the respective identity matrices I _N and I _M .     

Stable closed characteristics on partially symmetric convex hypersurfaces

Let n be a positive integer and P=diag(−I_n−κ,I_κ,−I_n−κ,I_κ) for some integer κ∈[0,n]. In this paper, we prove that for any convex compact smooth hypersurface Σ in R²ⁿ with n?2 there always exists at least one closed characteristic on Σ which possesses at least 2n−4κ Floquet multipliers on the unit circle of the complex plane, provided Σ is P-symmetric, i.e., x∈Σ implies Px∈Σ.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and I _Δ its Stanley-Reisner ideal in the polynomial ring S = K[x ₁,..., x _n]. We show that Δ is a matroid (complete intersection) if and only if S/I _Δ ^(m) (S/I _Δ ^(m)) is clean for all m ∈ N and this is equivalent to saying that S/I _Δ ^(m) (S/I _Δ ^(m), respectively) is Cohen-Macaulay for all m ∈ N. By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S/I ^m or S/I ^m is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/I _Δ ⁽²⁾ Δ (S/I _Δ ²) is clean if and only if S/I _Δ ⁽²⁾ (S/I _Δ ², respectively) is Cohen-Macaulay.