Duality relation for the Hilbert series of almost symmetric numerical semigroups

We derive the duality relation for the Hilbert series H (d ^m ; z) of the almost symmetric numerical semigroup S (d ^m ) combining it with its dual H (d ^m ; z ⁻¹). We establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F _j , generators d _i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and sufficient conditions for the minimal set d ^m to generate such semigroups.     

Frobenius problem for semigroups \mathsf{S}(d_{1},d_{2},d_{3})

We find the matrix representation of the set Δ(d ³), where d ³=(d ₁,d ₂,d ₃), of integers that are unrepresentable by d ₁,d ₂,d ₃ and develop a diagrammatic procedure for calculating the generating function Φ(d ³;z) for the set Δ(d ³). We find the Frobenius number F(d ³), the genus G(d ³), and the Hilbert series H(d ³;z) of a graded subring for nonsymmetric and symmetric semigroups and enhance the lower bounds of F(d ³) for symmetric and nonsymmetric semigroups.      

Trace formulas for a class of Toeplitz-like operators

LetP _T denote projection onto the space of entire functions of exponential type ≦T which are square summable on the line relative to a measuredΔ and letG denote multiplication by a suitably restricted complex valued function,g. For a reasonably large class of measuresdΔ, which includes Lebesgue measuredγ, it is shown that trace {(P _TGP_T)ⁿ−P_TGⁿP_T} tends boundedly to a limit asT↑∞ and that the limit isindependent of the choice ofdΔ within the permitted class. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special casedΔ=dγ using a different formalism.     

Some combinatorial properties of flag simplicial pseudomanifolds and spheres

A simplicial complex Δ is called flag if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension d−1, then the graph of Δ (i) is (2d−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the d-dimensional cross-polytope. Second, the h-vector of a flag simplicial homology sphere Δ of dimension d−1 is minimized when Δ is the boundary complex of the d-dimensional cross-polytope.     

Point interactions inL p

An interpretation is given to point interactions of the form −Δ+d inL ^p (ℝ ^N ), where Δ is the Laplacian operator andd is a pseudopotential related to the ‘Dirac measure at 0', depending on the dimension. They are described as extensions of −Δ, defined on the space {u∈C ₀ ^∞ (ℝ ^N )|u(0)=0} that are negative generators of analytic semigroups. This is done forN=1,2 and 1<p<∞ and forN=3 and 3/2<p<3.     

Numerical semigroups with many gaps of the same parity

Let g _e (S) (respectively, g _o (S)) be the number of even (respectively, odd) gaps of a numerical semigroup S. In this work we study and characterize the numerical semigroups S that verify 2|g _e (S)−g _o (S)|+1∈S. As a consequence we will see that every numerical semigroup can be represented by means of a numerical semigroup with maximal embedding dimension with all its minimal generators odd. The first author is supported by the project MTM2007-62346 and FEDER funds. The authors want to thank P.A. García-Sánchez and the referee for their comments and suggestions.     

A Five Color Zero-Sum Generalization

Let g_zs(m, 2k) (g_zs(m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , g_zs(m, 2k) by (the integers from 1 to g_zs(m, 2k+1) by ) there exist integers such that 1. there exists j_x such that Δ(x_i) ∈ for each i and ∑_i=1^m Δ(x_i) = 0 mod m (or Δ(x_i)=∞ for each i); 2. there exists j_y such that Δ(y_i) ∈ for each i and ∑_i=1^m Δ(y_i) = 0 mod m (or Δ(y_i)=∞ for each i); and 1. 2(x_m−x₁)≤y_m−x₁. In this note we show g_zs(m, 2)=5m−4 for m≥2, g_zs(m, 3)=7m+−6 for m≥4, g_zs(m, 4)=10m−9 for m≥3, and g_zs(m, 5)=13m−2 for m≥2. Supported by NSF grant DMS 0097317     

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write ${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}For every positive integer n, consider the linear operator U _n on polynomials of degree at most d with integer coefficients defined as follows: if we write \frach(t)(1 - t)^{d + 1}=?_{m 3 0} g(m)  t^m{\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}} , for some polynomial g(m) with rational coefficients, then \fracU_nh(t)(1- t)^d+1 = ?_{m 3 0}g(nm)  t^m{\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}} . We show that there exists a positive integer n _d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n _d , U _n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.     

On a Class of Generalized Lacunary Difference Sequence Spaces Defined by Orlicz Functions

In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞using m~(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞and the Cesàro summable and strongly Cesàro summable sequences.     

A probabilistic approach to the first Dirichlet eigenvalue on non-compact Riemannian manifold

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+∇h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for ∇h=0. Moreover, the first Dirichlet eigenvalue for elliptic operators onR ^d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for ∇h≠0. Research supported in part by NFSC and the State Education Commission of China     

Existence of <Emphasis Type="Italic">g</Emphasis><Superscript>1</Superscript><Subscript><Emphasis Type="Italic">g</Emphasis>–2</Subscript>’s on a double covering of a hyperelliptic curve

Abstract Let X be a non–hyperelliptic curve of genus g which is a double covering of a hyperelliptic curve C of genus h. In this paper, we prove that, if h≥ 3 and g≥ 4h+5, then X admits a complete, base point free g¹_g–2. Moreover, if h=3, this result holds under the mild condition g≥ 4h+3=15. Keywords: Double covering of hyperelliptic curves, Pencil of degree g–2 Mathematics Subject Classification (2000:) 14H30, 14H45     

Subalgebras of the Stanley—Reisner Ring

In [2], Billera proved that the R -algebra of continuous piecewise polynomial functions (C ⁰ splines) on a d -dimensional simplicial complex Δ embedded in R ^d is a quotient of the Stanley—Reisner ring A _Δ of Δ. We derive a criterion to determine which elements of the Stanley—Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C ^r _k (Δ) is upper semicontinuous in the Zariski topology. Using the criterion, we give an algorithm for obtaining the defining equations of the set of vertex locations where the dimension jumps. Received June 2, 1997, and in revised form December 22, 1997, and March 24, 1998.     

On the general additive divisor problem

We obtain a new upper bound for the sum Σ _h≤H Δ _k (N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ _k (N, h) is the (expected) error term in the asymptotic formula for Σ _N<n≤2N d _k (n)d _k (n + h), and d _k (n) is the divisor function generated by ζ(s) ^k . When k = 3, the result improves, for H ≥ N ^1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3.     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions

In this paper we define the sequence space ℓ _M (Δ _υ ^m , p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties. We prove some inclusion relations involving ℓ _M (Δ _υ ^m , p, q, s). spaces     

Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems

We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P ₁ or P ₂ finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h ^m+1 + h ² + Δt ^q ), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form O(\frach^m+1Dt+h²+Dt^q){O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})} ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h ^m  + h ² + Δt ^q ); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h ². Numerical experiments support these theoretical results.     

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e ^h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \triangle_m{\triangle_{\mu}} has a positive lower bound λ₁(M) > 0 and the m(m > n)-dimensional Bakry-émery curvature is bounded from below by -\fracm-1m-2l₁(M){-\frac{m-1}{m-2}\lambda_1(M)}, then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.     

Maximal Attractors for the <Emphasis Type="Italic">m</Emphasis>-Dimensional Cahn-Hilliard System

In this paper we discuss maximal attractors of the m-dimensional Cahn-Hilliard System in the product spaces (L^2(Ω))^m and (H^2(Ω))^m in terms of D.Henry‘s general theory and from the viewpoint of compactness and absorptivity of semigroups as R. Temam did. After giving the existence and uniqueness of global solutions, we technically restrict our discussion to some subspaces, give estimates with a new graph norm, and obtain the existence of maximal attractors and some properties of them.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Shellability and higher Cohen-Macaulay connectivity of generalized cluster complexes

Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex Δ^m(Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that Δ^m(Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part Δ₊^m(Φ) of Δ^m(Φ). An explicit homotopy equivalence is given between Δ₊^m(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.