Elasticity of factorizations in R 0 + X R1 + … + X lRl [X]

For an atomic domain R the elasticity ρ(R) is defined by ρ(R) = sup{m/n ¦ u₁ … u _m = v ₁ … v_n where u_i, v_i ∈ R are irreducible}. Let R ₀ ? … ? R _l be an ascending chain of domains which are finitely generated over ? and assume that R _l is integral over R ₀. Let X be an indeterminate. In this paper we characterize all domains D of the form D = R ₀ + XR₁ + … + X^lR_l[X] whose elasticity ρ(D) is finite.     

A New Recursion in the Theory of Macdonald Polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]_m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?_p1 [(H)\tilde]_m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M _μ from S _n to S _n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c _{μ v} occurring in the expansion ?_p1 [(H)\tilde]_m[X; q, t] = ?_{v ? m}c_mv [(H)\tilde]_v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M _μ may be calculated from the recursion F_m (q, t) = ?_{v ? m}c_mv F_v (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c _{μ v} have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F _μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Scaling Limits for Some P.D.E. Systems with Random Initial Conditions

Let X = {X(x, t), x ? R ⁿ , t ? R ₊} be the R ²-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X ₀ = (u ₀, v ₀). We discuss the scaling limit of X under suitable conditions on X ₀. Since the component fields u, v are dependent, even when the initial data u ₀, v ₀ are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u ₀, v ₀) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2 Anh , V.V. , and Leonenko , N.N. 1999 . Non-Gaussian scenarios for the heat equation with singular initial data . Stochastic Process. Appl. 84 : 91 – 114 .[Crossref], [Web of Science ®] , [Google Scholar]] for the single heat equation can be discussed for the two-equation system, in a significant way.     

Some infinite families of large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, B_i) : 1 ≤ i ≤ v ? 2}, where every (X,B_i) is a KTS(v) and all B_i form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qⁿ, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Characterizations of schunck classes of finite groups

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a _i b _j=0 for each coefficient a _i of f and b _j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF ₄, andH _r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH _R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF ₄ over the ringA = ℤ[u ^1/2,u ^-1/2] of Laurent polynomials in an indeterminateu ^1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.     

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.     

PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F ₂ or to the polynomial ring F ₂[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F ₂ or to F ₂[X]. We also construct principal ideal domains R of infinite transcendence degree over F ₂ with the property that 1 is the only unit of R.

     

The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δu+|∇u|=p(|x|)f(u,v), Δv+|∇v|=q(|x|)g(u,v) on RN, N?3, provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.     

A characterization of spherical distributions

It is shown that when the random vector X in Rⁿ has a mean and when the conditional expectation E(u′X|v′X) = 0 for all vectors u, v Rⁿ which satisfy u′v = 0, then the distribution of X is orthogonally invariant. A version of this characterization is also established when X does not have a mean vector.     

Bounds in Polynomial Rings over Artinian Local Rings

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. We prove that several algebraic constructions in the ring R[X] admit uniform bounds on the degrees of their output in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials g _i of degree at most d and if f is a polynomial of degree at most d belonging to I, then f = q ₁ f ₁ + ··· + q _s f _s , for some q _i of degree bounded in terms of d, l and n only. Similarly, the module of syzygies of I is generated by tuples all of whose entries have degree bounded in terms of d, l and n only.     

Bounds in Polynomial Rings over Artinian Local Rings

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. We prove that several algebraic constructions in the ring R[X] admit uniform bounds on the degrees of their output in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials g _i of degree at most d and if f is a polynomial of degree at most d belonging to I, then f = q ₁ f ₁ + ··· + q _s f _s , for some q _i of degree bounded in terms of d, l and n only. Similarly, the module of syzygies of I is generated by tuples all of whose entries have degree bounded in terms of d, l and n only.     

The Bass-Quillen problem on a class of local rings with weak global dimension two

Let (R,m) be a local GCD domain. R is called a U2 ring if there is an element u ∈ m-m2 such that R/(u) is a valuation domain and Ru is a B′ezout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U2 ring, then R and R[x] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over R[X] is free.     

Uniqueness of positive solutions for a class of elliptic systems

In this article, we consider uniqueness of positive radial solutions to the elliptic system Δu+a(|x|)f(u,v)=0, Δv+b(|x|)g(u,v)=0, subject to the Dirichlet boundary condition on the open unit ball in RN (N?2). Our uniqueness results applies to, for instance, f(u,v)=uqvp, g(u,v)=upvq, p,q>0, p+q<1 or more general cases.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.