On the Jump Number of Lexicographic Sums of Ordered Sets

Let Q be the lexicographic sum of finite ordered sets Q _x over a finite ordered set P. For some P we can give a formula for the jump number of Q in terms of the jump numbers of Q _x and P, that is, , where s(X) denotes the jump number of an ordered set X. We first show that where w(X) denotes the width of an ordered set X. Consequently, if P is a Dilworth ordered set, that is, s(P) = w(P)–1, then the formula holds. We also show that it holds again if P is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical.     

On higher heine-stieltjes polynomials

Take a linear ordinary differential operator $\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}}$\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}} with polynomial coefficients and set r = max _i=1,…,k(deg Q _i (z) − i). If d(z) satisfies the conditions: (i) r ≥ 0 and (ii) deg Q _k (z) = k + r, we call it a non-degenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [13] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.     

Uniqueness of a solution to the problem of atmosphere dynamics

We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm

Some essential properties ofQ p (∂Δ)-spaces

For p∈(?∞, ∞) letQ _p (?Δ) be the space of all complex-valued functions f on the unit circle ?Δ satisfying $\mathop {\sup }\limits_{I \subset \partial \Delta } \left| I \right|^{ - p} \int_I {\int_I {\frac{{\left| {f(z) - f(w)} \right|^2 }}{{\left| {z - w} \right|^{2 - p} }}\left| {dz} \right|\left| {dw} \right|< \infty } } $ , where the supremum is taken over all subarcs I ? ?Δ with the arclength |I|. In this paper, we consider some essential properties ofQ _p (?Δ). We first show that if p>1, thenQ _p (?Δ)=BMO(?Δ), the space of complex-valued functions with bounded mean oscillation on ?Δ. Second, we prove that a function belongs toQ _p (?Δ) if and only if it is Möbius bounded in the Sobolev spaceL _p ² (?Δ). Finally, a characterization ofQ _p (?Δ) is given via wavelets.     

Enveloping algebras of Slodowy slices and Goldie rank

Let U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra \mathfrakg = \textLie(G) \mathfrak{g} = {\text{Lie}}(G) and let I be a primitive ideal of the enveloping algebra U( \mathfrakg ) U\left( \mathfrak{g} \right) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that I = \textAn\textn_{U( \mathfrakg )}( Q_e ?_{U( \mathfrakg,e )}V ) I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) for some finite dimensional irreducible U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) -module V, where Q _e stands for the generalised Gelfand–Graev \mathfrakg \mathfrak{g} -module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient U( \mathfrakg )

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

In this paper we prove a nonvanishing theorem for central values of L-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TiZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell–Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic \mathbb Q{{\mathbb Q}} -curves A(D) of B. Gross [Gr].     

Second-order conditions on the overflow traffic function from the Erlang-B system: a unified analysis

This paper presents a unified treatment of the mathematical properties of the second-order derivatives of the overflow traffic function from an Erlang loss system, assuming the number of circuits to be a nonnegative real number. It is shown that the overflow traffic function [^(A)]( a,x ) \widehat A\left( {a,x} \right) is strictly convex with respect to x (number of circuits) for x ≥ 0, taking the offered traffic, a, as a positive real parameter. It is also shown that [^(A)]( a,x ) \widehat A\left( {a,x} \right) is a strictly convex function with respect to a, for all (a, x) ∈ ℝ⁺ × ℝ⁺. Following a similar process, it is shown that [^(A)]( a,x ) \widehat A\left( {a,x} \right) is a strict submodular function in this domain and that the improvement function introduced by K. O. Moe [11] is strictly increasing in a. Finally, based on some particular cases and numerous numerical results, there is a conjecture that the function [^(A)]( a,x ) \widehat A\left( {a,x} \right) is strictly jointly convex in areas of low blocking where the standard offered traffic is less than −1.     

(<Emphasis Type="Italic">o</Emphasis>)-Topology in *-algebras of locally measurable operators

We consider the topology t( M ) t\left( \mathcal{M} \right) of convergence locally in measure in the *-algebra LS( M ) LS\left( \mathcal{M} \right) of all locally measurable operators affiliated to the von Neumann algebra M \mathcal{M} . We prove that t( M ) t\left( \mathcal{M} \right) coincides with the (o)-topology in LS_h( M ) = { T ? LS( M ):T* = T } L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} if and only if the algebra M \mathcal{M} is σ-finite and is of finite type. We also establish relations between t( M ) t\left( \mathcal{M} \right) and various topologies generated by a faithful normal semifinite trace on M \mathcal{M} .     

Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings

We say that is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) -module U, the dual module U* is a simple left (right) -module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.     

The Number of Linear Extensions of the Boolean Lattice

Let L(Q ^t ) denote the number of linear extensions of the t-dimensional Boolean lattice Q ^t . We use the entropy method of Kahn to show that

A Volumetric Penrose Inequality for Conformally Flat Manifolds

We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to \mathbbRⁿ\ W, n 3 3{\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}, and so that their boundary is a minimal hypersurface. (Here, W ì \mathbbRⁿ{\Omega\subset \mathbb{R}^{n}} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by \frac12(V/b_n)^(n-2)/n{\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}, where V is the Euclidean volume of Ω and β _n is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost.     

On pairs of commuting nilpotent matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe in terms of λ if has at most two parts.     

Slices for biparabolics of index 1

Let \mathfraka \mathfrak{a} be an algebraic Lie subalgebra of a simple Lie algebra \mathfrakg \mathfrak{g} with index \mathfraka \mathfrak{a}  ≤ rank \mathfrakg \mathfrak{g} . Let Y( \mathfraka ) Y\left( \mathfrak{a} \right) denote the algebra of \mathfraka \mathfrak{a} invariant polynomial functions on \mathfraka^* {\mathfrak{a}^*} . An algebraic slice for \mathfraka \mathfrak{a} is an affine subspace η + V with h ? \mathfraka^* \eta \in {\mathfrak{a}^*} and V ì \mathfraka^* V \subset {\mathfrak{a}^*} subspace of dimension index \mathfraka \mathfrak{a} such that restriction of function induces an isomorphism of Y( \mathfraka ) Y\left( \mathfrak{a} \right) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which h ? \mathfraka h \in \mathfrak{a} is ad-semisimple, η is a regular element of \mathfraka^* {\mathfrak{a}^*} which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to ( \textad  \mathfraka )h \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta in \mathfraka^* {\mathfrak{a}^*} . The classical case is for \mathfrakg \mathfrak{g} semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \mathfrakg \mathfrak{g} . Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \mathfraka \mathfrak{a} be a truncated biparabolic of index one. (This only arises if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \mathfraka \mathfrak{a} is the restriction of a particularly carefully chosen regular nilpotent element of \mathfrakg \mathfrak{g} . A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.     

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

Carleman estimates for degenerate parabolic operators with applications to null controllability

We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Combinatorics of character formulas for the lie superalgebra \mathfrak{g}\mathfrak{l}\left( {m,n} \right)

Let \mathfrakg \mathfrak{g} be the Lie superalgebra \mathfrakg\mathfrakl( m,n ) \mathfrak{g}\mathfrak{l}\left( {m,n} \right) . Algorithms for computing the composition factors and multiplicities of Kac modules for \mathfrakg \mathfrak{g} were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph G \mathcal{G} defined using these diagrams. Each vertex of G \mathcal{G} corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If E \mathcal{E} is the subgraph of G \mathcal{G} obtained by deleting all edges of positive weight, then E \mathcal{E} is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

A vanishing theorem for differential operators in positive characteristic

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A _n ) or a smooth quadric (in this case p is supposed to be odd) then Hⁱ( X,End( \sfF_*O_X ) ) = 0 {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \sfF_*O_X {\sf{F}_*}{\mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.     

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE _M,L ^* [1]. We recall that two sequencesa _n,b _nare asymptotically equal, and we writea _n ≃b _n,if and only if lim _n→∞(a _n/b _n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM _k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26.