Complexity of solution of linear systems in rings of differential operators

Suppose given a k₁×k₂ system of linear equations over the Weyl algebraA _n = F[X₁,...X₁,D₄,...,D_n] or over the algebra of differential operatorsK _n = F[X₁,...X₁,D₄,...,D_n], where the degree of each coefficient of the system is less than d. It is proved that if the system is solvable overA _n, orK _n, respectively, then it has a solution of degree at most (k, d)²⁰⁽ⁿ⁾.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 47–59, 1991.     

The List-Chromatic Number of Infinite Graphs Defined on Euclidean Spaces

If D is a countable set of positive reals, 2≤n<ω, let X _n (D) be the graph with the points of R ⁿ as vertices where two vertices are joined iff their distance is in D. We determine the list-chromatic number of X _n (D) as much as possible.     

Semistar-Krull and valuative dimension of integral domains

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type ⋆[X] on the polynomial ring D[X], such that, if n := ⋆-dim(D), then n+1 ≤ ⋆[X]-dim(D[X]) ≤ 2n+1. We also establish that if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain, then ⋆[X]-dim(D[X]) = ⋆- dim(D)+1. Moreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dim _v (D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dim _v (D) = n if and only if ⋆[X ₁, . . . , X _n ]-dim(D[X ₁, . . . , X _n ]) = 2n, and that if ⋆-dim _v (D) < ∞ then ⋆[X]-dim _v (D[X]) = ⋆-dim _v (D) + 1. In general ⋆-dim(D) ≤ ⋆-dim _v (D) and equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domains D such that ⋆-dim(D) < ∞ and ⋆-dim(D) = ⋆-dim _v (D). As an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆′)-linked overring T of D, is a ⋆′-Jaffard domain, where ⋆′ is a stable semistar operation of finite type on T. As a consequence of this result we obtain that a Krull domain D, must be a w _D -Jaffard domain.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

Central limit theorems in D[0, 1]

Summary Let X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {X _n} of independent copies of X, let S _n=X₁++X_n. Conditions which are either necessary or sufficient for the weak convergence of n ^–1/2(S _n–ES_n) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

Crossed products and blocks with normal defect groups

Abstract

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with d ⁿ  = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that x ⁿ D ∩ y ⁿ D is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ? D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XD _S [X] is an AGCD domain if and only if D and D _S [X] are AGCD domains and S is an almost splitting set.     

Secant planes of projective curves embedded by nonspecial line bundles

For a nonspecial line bundle ? on a smooth curve X we consider a presentation ??𝒦_X?D+E which is minimal with respect to deg E. If ? is very ample, then this minimality means that any n-points of φ_?(X) with n≤deg E?1 are in general position while φ_?(E) spans a (deg E?2)-plane. In this work, we investigate conditions on D and E for ??𝒦_X?D+E to be minimal. We also observe s-secant (s?k?1)-planes which are minimal with respect to the secant degree s for a given k. We apply minimal presentations to problems about the exactness of Green-Lazarsfeld’s conjecture on property (N_p).     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Quasi-reversibility of a discrete-time queue and related models

We consider a discrete-time queueing system and its application to related models. The model is defined byX _n+1=X_n+A_n-D_n+1 with discrete states, whereX _n is the queue-length at the nth time epoch,A _n is the number of arrivals at the start of the nth slot andD _n+1 is the number of outputs at the end of the nth slot. In this model, the arrival process {A _n} is described as a sequence of independently and identically distributed random variables. The departureD _n+1 depends only on the system sizeX _n+A_n at the beginning of the time slot.We study the reversibility for the model. The departure discipline in which the system has quasi-reversibility is determined. Models with special arrival processes were studied by Walrand [8] and sawa [7]. In this paper, we generalize their results. Moreover, we consider discrete-time queueing networks with some reversible nodes. We then obtain the product-form solution for these networks.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e _n ) _n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L _diag(X) of diagonal operators with respect to (e _n ) _n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $ \mathfrak{X} $ \mathfrak{X} _D with a Schauder basis (e _n ) _n∈ℕ such that $ \mathfrak{X} $ \mathfrak{X} *_D is isometric to L _diag($ \mathfrak{X} $ \mathfrak{X} _D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L _diag($ \mathfrak{X} $ \mathfrak{X} _D) is of the form T = λI + K, where K is a compact operator.     

On the rigidity of sphericalt-designs that are orbits of finite reflection groups

The concept of rigid sphericalt-designs was introduced by Bannai. He conjectured that there is a functionf(t, d) such that ifX is a sphericalt design in thed-dimensional Euclidean space so that |X|>f(t, d), theX is non-rigid. Furthermore, he asked to find examples of rigid but not tight sperical designs. In the present article we shall investigate the case whenX is an orbit of a finite reflection group and prove thatX is rigid iff tight for the groupsA _n ,B _n ,C _n ,D _n ,E ₆,E ₇,F ₄,I ₃.Research was partially supported by Hungarian National Research fund Grant No. 4267.     

Semigroups with I-Quasi Length and Syntactic Semigroups

A module M is said to be extending (𝒢-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X ∩ D is essential in both X and D). It is known that for a nonsingular module the concepts of 𝒢-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article, we obtain a characterization of the right 𝒢-extending generalized triangular matrix rings. This result and its corollaries improve and generalize the existing results on right extending generalized triangular matrix rings. It is well known that the ring of n-by-n triangular matrices over a right selfinjective ring is not, in general, right extending. One application of our characterization shows that such rings are right 𝒢-extending. Connections to Operator Theory and a characterization of the class of right extending right SI-rings are also obtained. Examples are given to illustrate and delimit the theory.     

Termination of (many) 4-dimensional log flips

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K _X +D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K _X +D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.     

The distinguishing number of the direct product and wreath product action

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D _G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀_Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S _m × S _n on [m] × [n].     

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 .