A procedure to compute prime filtration

Let K be a field, S = K[x ₁, … x _n ] be a polynomial ring in n variables over K and I ⊂ S be an ideal. We give a procedure to compute a prime filtration of S/I. We proceed as in the classical case by constructing an ascending chain of ideals of S starting from I and ending at S. The procedure of this paper is developed and has been implemented in the computer algebra system Singular.     

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x ₁,…,x _n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I _n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I _n,d )=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I _n,d )≤⌊(n−d)/(d+1)⌋+d.     

Two-sided bounds uniform in the real argument and the index for modified Bessel functions

Bounds uniform in the real argument and the index for the functionsa _ν (x)=xI′ _ν (x)/I′ _ν (x) andb _ν (x)=xK′ _ν (x)/K _ν (x), as well as for the modified Bessel functionsI _ν(x) andK _ν(x), are established in the quadrantx>0, ν≥0, except for some neighborhoods of the pointx=0, ν=0. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 681–692, May, 1999.     

Almost continuous orbit equivalence for non-singular homeomorphisms

Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III ₁ or that they are both of type III _λ, 0 < λ < 1 and, in the III _λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G _δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T| _X′ and S| _Y′, that is ϕ{T ⁱ x} = {S ⁱ ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ ⁻¹ Sϕ, we have T _x = S′ ^n(x) x and S′x = T ^m(x) x where n and m are continuous on X′.     

Reverse lexicographic and lexicographic shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ_lex—an operation that transforms a monomial ideal of S = K[x_i: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence } are distinct. The limit ideal is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I. Research partially supported by NSF grants DMS 0070571 and DMS 0100141.     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.     

A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Let G be a graph and S ⊂ V(G). We denote by α(S) the maximum number of pairwise nonadjacent vertices in S. For x, y ∈ V(G), the local connectivity κ(x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define . In this paper, we show that if κ(S) ≥ 3 and for every independent set {x ₁, x ₂, x ₃, x ₄} ⊂ S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian.     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

On the Regularity of Operations of Ideals

Let I be a homogeneous ideal of a polynomial ring K[x₁,…, x_n] over a field K, and denote the Castelnuovo–Mumford regularity of I by reg(I). When I is a monomial complete intersection, it is proved that reg(I^m) ≤ mreg(I) holds for any m ≥ 1. When n = 3, for any homogeneous ideals I and J of K[x₁, x₂, x₃], one has that reg(I ? J), reg(IJ) and reg(I ∩ J) are all upper bounded by reg(I) +reg(J), while reg(I + J) ≤reg(I) +reg(J) ?1.     

Dye’s theorem in the almost continuous category

Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G _δ subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures.     

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal I ? S = K [x₁, …, x_n ]. This allows us to compute the depth of S /I in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of S /I holds provided it holds whenever S /I is Cohen–Macaulay. We also discuss a conjecture of Soleyman Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simplicial complexes with rigid depth

We extend a result of Minh and Trung (Adv. Math. 226:1285–1306, 2011) to get criteria for depth ${I = \rm {depth}\sqrt{I}}$ , where I is an unmixed monomial ideal of the polynomial ring S?=?K[x ₁, . . . , x _n ]. As an application we characterize all the pure simplicial complexes Δ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal ${I\subset S}$ with ${\sqrt{I}=I_\Delta}$ one has depth(I)?=?depth(I _Δ).     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

Regularity of ideals and their radicals

In this paper we compare the regularityreg I of a homogeneous idealI⊂K[x ₁,...,x _n] with that of its radical. We prove that regI ≥ reg ifR/I is a BuchsbaumR-module or ifI is a monomial ideal. We also prove the same result when defines a non-singular curve inP ³ under some additional hypotheses     

A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.     

On Monomial Reduction and Polynomial Expressions with Respect to Binomial Ideals

Let I ? K[x ₁,…, x _n ] be an ideal and G be the reduced Gröbner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x ₁,…, x _n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods

For a given polyhedron K(?)M,the notation RM(K)denotes a regular neigh- borhood of K in M.The authors study the following problem:find all pairs(m,k) such that if K is a compact k-polyhedron and M a PL m-manifold,then R_M(f(K))≌R_M(g(K))for each two homotopic PL embeddings f,g:K→M.It is proved that R_S~(k 2)(S~k)(?)S~k×D~2 for each k(?)2 and some PL sphere S~k(?)S~(k 2)(even for any PL sphere S~k(?)S~(K 2)having an isolated non-locally flat point with the singularity S~(k-1)(?) S~(k 1)such thatπ_1(S~(k 1)-S~(k-1))(?)Z).     

Characterization of polynomials and divided difference

For distinct points x₁,x₂,…,x_n in ℛ (the reals), letϕ[x₁, x₂,…,x_n] denote the divided difference ofϕ. In this paper, we determine the general solutionϕ,g: ℛ → ℛ of the functional equationϕ[x₁,x₂,…,x_n] =g(x₁,+ x₂ + … + x_n) for distinct x₁,x₂,…, x_n in ℛ without any regularity assumptions on the unknown functions.     

Geometries and amalgams of J1

ABSTRACT

Let S = 𝕂[x ₁, …, x _n ] be a polynomial ring over a field 𝕂 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕂 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.