The invariant polynomials characterizing U(n) tensor operators p, q,…, q, 0,…, 0 having maximal null space

We make use of the “path sum” function to prove that the family of stretched operator functions characterized by the operator irrep labels p,q,…,q, 0,…, 0 satisfy a pair of general difference equations. This family of functions is a generalization of Milne's p,q,…,q, 0, functions for U(n) and Biedenharn and Louck's p,q, 0 functions for U(3). The fact that this family of stretched operator functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations and symmetry properties we give an explicit formula for the polynomials characterized by the operator irrep labels p, 1, 0,…, 0.     

An umbral calculus for polynomials characterizing U(n) tensor operators

After the change of variables Δ_i = γ_i ? δ_i and x_{i,i + 1} = δ_i ? δ_{i + 1} we show that the invariant polynomials _μG⁽ⁿ⁾_q(, Δ_i, ; , x_i,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions S_λ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ₁ ,…, γ_n} and {δ₁ ,…, δ_n}. We obtain a similar result for the yet more general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁ ,…, γ_n; δ₁ ,…, δ_m). Making use of properties of skew Schur functions $\text{S}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\rho </mtext>}}$ and S_λ(γ ? δ) we put together an umbral calculus for ^m_μG⁽ⁿ⁾_q(γ; δ). That is, working entirely with polynomials, we uniquely determine ^m_μG⁽ⁿ⁾_q(γ; δ) from ^m_μG⁽ⁿ⁾_{q ? 1}(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of ^m_μG⁽ⁿ⁾_q(γ; δ) as a linear combination of S_λ(γ ? δ).) As an application we deduce “conjugation” symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.     

Length functions and Demazure operators for G(e,1,n), II

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function ?(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W.In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space H_W in terms of Demazure operators, and show that H_W is isomorphic to the coinvariant algebra S_W, which enables us to define a homogeneous basis on S_W parametrized by w?W.     

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU（2,2）/S（U（2） × U（2））

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.     

Quasiconvex functions, SO(n) and two elastic wells

We use W^1,∞ approximations of minimizing sequences to study the growth of some quasiconvex functions near their zero sets. We show that for SO(n), the quasiconvexification of the distance function dist²(·, SO(n)) can be bounded below by the distance function itself. In certain cases of the incompatible two elastic well structure, we establish a similar result. We also prove that for small Lipschitz perturbations of SO(n) and of the two well structure, the Young measure limits of gradients supported on these perturbed sets are Dirac masses.     

Lax Operator for the Quantised Orthosymplectic Superalgebra Uq[osp(m|n)]

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U _q [osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U _q [osp(m|n)] module. The case W = V is studied as an example. Presented by A. Verschoren.     

Archimedean L -factors on GL ( n ) × GL ( n ) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, π _v × π _v ^′ ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π _v and _v ^′ . In this paper we show that, if ν is archimedean, and π _v and _v ^′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.     

AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m + n log n) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(m log log n + n log n), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the log log nfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(m log log n + n log n) algorithm for finding a maximum-weight clique—which matches the best known algorithm.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if m_a = m_K*d_a*m_K{\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaK\subseteq SU(n)} , for K = SO(n), then m_a^k{\mu _{a}^{k}} is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that m_a^n-1{\mu _{a}^{n-1}} is singular to the Haar measure.     

Markov inequality for polynomials of degree n with m distinct zeros

Let be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatfor every . This is far away from what we expect. We conjecture that the Markov factor 32·8^mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for on a finite interval when mc log n.     

Newton functions generating symmetric fields and irreducibility of Schur polynomials

We prove (Theorem 1.1) that if e₀>>e_r>0 are coprime integers, then the Newton functions , i=0,…,r, generate over the field of symmetric rational functions in X₁,…,X_r. This generalizes a previous result of us for r=2. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Theorem 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Theorem 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Theorem 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1.     

Upper bounds for A(n,4) and A(n,6) derived from Delsarte''s linear programming bound

An appropriate version of the linear programming bound of Delsarte for binary codes is used to find explicit upper bounds for A(n, d), with d ε {4,6}. These bounds are expected to be at least as good as the linear programming bound of Delsarte itself. It is re-established that the Preparata codes are optimal.     

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

Relative ( pn , p , pn , n )-difference sets with GCD ( p , n )=1

Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)=1 through a group ring approach. We then give a construction of (p(p+1),p,p(p+1),p+1) relative difference sets with p a Mersenne prime. Supported by National Natural Science Foundation of China (10331030).     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.     

Cartographic line simplification and polygon CSG formulæ in O(n log n) time

A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. By using a data structure that maintains convex hulls of polygonal lines under splits, both were known to have O(n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log^* n) algorithm for maintaining convex hulls under splits at extreme points. It opens the question of whether there are practical, linear-time solutions to these problems.     

Singular discrete (n, p) boundary value problems

This paper discusses a (n, p) singular discrete boundary value problem. Existence of a nonnegative solution is established.