Stanley-Reisner rings and the radicals of lattice ideals

In this article we associate to every lattice ideal I_L,ρ⊂K[x₁,…,x_m] a cone σ and a simplicial complex Δ_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. We assign a simplicial subcomplex Δ_σ(F) of Δ_σ to every polynomial F. If F₁,…,F_s generate I_L,ρ or they generate rad(I_L,ρ) up to radical, then is a spanning subcomplex of Δ_σ. This result provides a lower bound for the minimal number of generators of I_L,ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.     

Some Schroeder-Bernstein type theorems for Banach spaces

We first introduce the notion of (p,q,r)-complemented subspaces in Banach spaces, where p,q,r∈N. Then, given a couple of triples {(p,q,r),(s,t,u)} in N and putting Λ=(q+r−p)(t+u−s)−ru, we prove partially the following conjecture: For every pair of Banach spaces X and Y such that X is (p,q,r)-complemented in Y and Y is (s,t,u)-complemented in X, we have that X is isomorphic Y if and only if one of the following conditions holds:

(a)

Λ≠0, Λ divides p−q and s−t, p=1 or q=1 or s=1 or t=1.

(b)

p=q=s=t=1 and gcd(r,u)=1.

The case {(2,1,1),(2,1,1)} is the well-known Pe?czyński's decomposition method. Our result leads naturally to some generalizations of the Schroeder-Bernstein problem for Banach spaces solved by W.T. Gowers in 1996.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.     

Conjectured statistics for the q,t-Catalan numbers

We introduce the distribution function F_n(q,t) of a pair of statistics on Catalan words and conjecture F_n(q,t) equals Garsia and Haiman's q,t-Catalan sequence C_n(q,t), which they defined as a sum of rational functions. We show that F_n,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that F_n(q,t)=C_n(q,t) when t=1/q. We also show the partial symmetry relation F_n(q,1)=F_n(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word.     

Fourier-Bohr spectrum of sequences related to continued fractions

Let α be a real irrational number and q₁, q₂, q₃,… the sequence of denominators of the convergents of α. Call σ_α(n)) the sum of the digits of the integer n in basis q₁, q₂, q₃,…. The Fourier-Bohr spectrum of the sequence exp(2πiσ_α(n)) and more generally of sequences exp(2πif(n)) is studied, where f is a completely α-additive real sequence. A sufficient condition is derived for the sequence xn + yσ_α(n) to be uniformly distributed modulo one.     

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.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators

For a class of potentialsq which may become as singular asr ^?2 at the origin we show that the eigenfunctionsu of the Friedrichs extensionT _Fof (?Δ+q)?C ₀ ^∞ (R ⁿ /{0}) satisfy the virial theorem 2(?Δu,u)=(u,rq,u). Two proofs of this relation are given. One extends the Fock-Weidmann method of scale transformations, the other makes an argument by Finkelstein rigorous and can be viewed of as a substitute of the “commutator proof” of (*) employed in the physical literature. The virial theorem provides a convenient tool for proving the nonexistence of eigenvalues embedded in the continuum or the total absence of eigenvalues ofT _F.     

On the saturation sequence of the rational normal curve

Let denote the rational normal curve of order d. Its homogeneous defining ideal admits an SL₂-stable filtration J₂⊆J₄⊆…⊆I_C by sub-ideals such that the saturation of each J_2q equals I_C. Hence, one can associate to d a sequence of integers (α₁,α₂,…) which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the α_q, using inter alia the methods of classical invariant theory.     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

On dynamic programming equations for utility indifference pricing under delta constraints

In this paper we study the problem of utility indifference pricing in a constrained financial market, using a utility function defined over the positive real line. We present a convex risk measure −v(•:y) satisfying q(x,F)=x+v(F:u₀(x)), where u₀(x) is the maximal expected utility of a small investor with the initial wealth x, and q(x,F) is a utility indifference buy price for a European contingent claim with a discounted payoff F. We provide a dynamic programming equation associated with the risk measure (−v), and characterize v as a viscosity solution of this equation.     

Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness

We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u^−pv^−q,−Δv=u^−rv^−s,u>0,v>0,x∈Ω,u|_∂Ω=v|_∂Ω=0, where Ω is a bounded domain with smooth boundary in R^N, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318].     

A sufficient condition for F-purity

It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F  -purity. In this note, we prove that if (R,m)$(R,\mathfrak{m})$ is an equidimensional and _S2$S_2$ local ring that admits a canonical ideal I≅_ωR$I\cong {\omega }_R$ such that R/I$R/I$ is F-pure, then R is F-pure. This greatly generalizes one of the main theorems in [2]. We also provide examples to show that not all Cohen–Macaulay F-pure local rings satisfy the above property.     

Characterization of unigraphic and unidigraphic integer-pair sequences

Given a graph (digraph) G with edge (arc) set E(G) = {(u₁}, υ₁), (u₂, υ₂),?,(u_q, υ_q, where q = |E(G)|, we can associate with it an integer-pair sequence S_G = ((a₁, b₁), (a₂, b₂),?, (a_q, b_q)) where a_i, b_i are the degrees (indegrees) of u_i, υ_i respectively. An integer- pair sequence S is said to be graphic (digraphic) if there exists a graph (digraph) G such that _SG = S. In this paper we characterize unigraphic and unidigraphic integer-pair sequences.     

Hamilton cycles in digraphs of unitary matrices

A set S⊆V is called a q⁺-set (q^--set, respectively) if S has at least two vertices and, for every u∈S, there exists v∈S,v≠u such that N⁺(u)∩N⁺(v)≠∅ (N^-(u)∩N^-(v)≠∅, respectively). A digraph D is called s-quadrangular if, for every q⁺-set S, we have |∪{N⁺(u)∩N⁺(v):u≠v,u,v∈S}|?|S| and, for every q^--set S, we have |∪{N^-(u)∩N^-(v):u,v∈S)}?|S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

Boundary representations and rigidity of submodules

This paper considers two problems on the Fock type spaces Fs (0<s?1). Firstly, it is shown that on the space Fs (0<s<1), the identity representation of C^∗(I,T₁,…,Tn) is a boundary representation for the Banach subalgebra B(I,T₁,…,Tn), while on the space F₁, it is not. Secondly, it is shown that all the submodules of F₁ are rigid.