q-Partition algebra combinatorics

We study a q-analog Qr(n,q) of the partition algebra Pr(n). The algebra Qr(n,q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space coming from r-iterations of Harish-Chandra restriction and induction. For n?2r, we show that Qr(n,q) has the same semisimple matrix structure as Pr(n). We compute the dimension to be a q-polynomial that specializes as dn,r(1)=nr and dn,r(0)=B(r), the rth Bell number. Our method is to write dn,r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of indexed by n-restricted q-set partitions of {1,…,r} and show that there are dn,r(q) of these.     

Rationality theorems for Hecke operators on GLn

We define n families of Hecke operators for GL_n whose generating series are rational functions of the form q_k(u)⁻¹ where q_k is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially .By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL₃ and GL₄.     

Distance-two labellings of Hamming graphs

Let j≥k≥0 be integers. An ?-L(j,k)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…,?} such that |?(u)−?(v)|≥j if u,v are adjacent and |?(u)−?(v)|≥k if they are distance two apart. Let λ_j,k(G) be the smallest integer ? such that G admits an ?-L(j,k)-labelling. Define to be the smallest ? if G admits an ?-L(j,k)-labelling with ?(V)={0,1,2,…,?} and ∞ otherwise. An ?-cyclic L(j,k)-labelling is a mapping ?:V→Z_? such that |?(u)−?(v)|_?≥j if u,v are adjacent and |?(u)−?(v)|_?≥k if they are distance two apart, where |x|_?=min{x,?−x} for x between 0 and ?. Let σ_j,k(G) be the smallest ?−1 of such a labelling, and define similarly to . We determine λ_2,0, , σ_2,0 and for all Hamming graphs K_q1□K_q2□?□K_qd (d≥2, q₁≥q₂≥?≥q_d≥2) and give optimal labellings, with the only exception being for q≥4. We also prove the following “sandwich theorem”: If q₁ is sufficiently large then for any graph G between K_q1□K_q2 and K_q1□K_q2□?□K_qd, and moreover we give a labelling which is optimal for these eight invariants simultaneously.     

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

Simple compact quantum groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q∈GL(n,C) satisfying , n?2; (b) The quantum automorphism groups Aaut(B,τ) of finite-dimensional C^∗-algebras B endowed with the canonical trace τ when dim(B)?4, including the quantum permutation groups Aaut(Xn) on n points (n?4); (c) The standard deformations Kq of simple compact Lie groups K and their twists , as well as Rieffel's deformation KJ.     

On subgroups of orthogonal groups of isotropic quadratic forms

Let a field K be an algebraic extension of a subfield k of characteristic not 2, n an integer, a non-degenerate isotropic form in n variables over K with coefficients in k. We study subgroups of the orthogonal group O_n(K,Q) that contain the derived subgroup Ω_n(k,Q) of the group O_n(k,Q).     

The evaluation subgroup of a fibre inclusion

Let be a fibration of simply connected CW complexes of finite type with classifying map . We study the evaluation subgroup Gn(E,X;j) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G_∗(E,X;j)⊗Q=(G_∗(X)⊗Q)⊕(π_∗(B)⊗Q) is equivalent to the condition Q(h_?)=0.     

Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.     

Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation

Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(⁻²|⋅|∗²|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

Invariant theory for singular α-determinants

From the irreducible decompositions' point of view, the structure of the cyclic GLn(C)-module generated by the α-determinant degenerates when (1?k?n−1) (see [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gln(C), J. Lie Theory 16 (2006) 393-405]). In this paper, we show that -determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using (GLm,GLn)-duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call (n,k)-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group Snk.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

A note on Browder spectrum of operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the Hilbert space H⊕K of the form . In this note, it is shown that the following results in [Hai-Yan Zhang, Hong-Ke Du, Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl. 323 (2006) 700-707]

     

On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

The number of reachable pairs in a digraph

For a digraph G, let R(G) (respectively, R^(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that u≠v and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range S_n of R(G) and the range of R^(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to S_n. These determine the set S_n for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)^0.57+2, for arbitrary k.     

Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S^∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C⁻¹(I−λ₁A₁−?−λdAd)(λ₁B₁+?+λdBd) such that connecting operator has adjoint U^∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A₁,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A₁,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A₁,…,Ad satisfy an additional stability property.     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).