On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ℋ_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ℋ ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋ_q,1(B _n )- module D^λ remains irreducible on restriction to ℋ_q(D _n ).     

Cantor spectrum for the almost Mathieu equation

For a dense G_δ of pairs (λ, α) in R², we prove that the operator (Hu)(n) = u(n + 1) + u(n ?1) + λ cos(2παn + θ) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case $\text{α =}\text{p}\text{q}$ of which we mention: (a) If qθ is not an integral multiple of π, then all gaps are open, and (b) If q is even and θ is chosen suitably, then the middle gap is closed for all λ.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

Resonance between automorphic forms and exponential functions

Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum Xn 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).     

Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

Irreducible polynomials over finite fields. I

A function H_n(a₁, a₂, a₃) is found, computing the number of normalized irreducible polynomials of degree n over a finite field F_q, with the first three coefficients a₁, a₂, and a₃ fixed for n = 4. The function is expressed via some sums of characters admitting “good” estimates. In particular, the following theorem is proved: If q = 3m + 1, a ∈ k^*, and N (a) = H₄(0, 0, a), then $$N(a) = \frac{1}{4}(q - 2\operatorname{Re} [(\lambda (a) - \eta ( - 1)\bar \lambda (a/2))J(\lambda ,\lambda )] - \eta ( - 1)),$$ where η is a quadratic character of the field k = F_q, λ is a nontrivial cubic character, and J(λ, λ) is the known Jacobi sum.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

On the completeness of the system of root vectors of the Sturm-Liouville operator with general boundary conditions

We study general boundary value problems with nondegenerate characteristic determinant Δ(λ) for the Sturm-Liouville equation on the interval [0, 1]. Necessary and sufficient conditions for the completeness of root vectors are obtained in terms of the potential. In particular, it is shown that if Δ(λ) ≠ const, q(·) ∈ C ^k [0, 1] for some k ? 0, and q ^(k)(0) ≠ (?1) ^k q ^(k)(1), then the system of root vectors is complete and minimal in L ^p [0, 1] for p ∈ [1,∞).     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Sur un probleme aux valeurs propres non line aire

In this paper, we give some sufficient conditions for which the differential operatorP(λ=P ₀+λP ₁+...+λ ^m?1 P _m?1+λ ^m , depending polynomially on the complex parameter λ, verifies the following statement: there exists λ₀ ∈ ?,u _o=0,u ₀ ∈ ?(? ⁿ ) a Schwartz space of rapidly decreasing functions, such thatP(λ₀)u ₀=0-.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence,uniqueness and stability for solutions of the polynomial-like iterative equation Σni=1λifi(x)=F(x) were obtained in the case of λ 1 = 0.In this paper,we construct C 0 decreasing solutions of the iterative equation in the case that λ 1 can vanish to answer the Leading Coefficient Problem.Moreover,we also give the necessary and sufficiently condition for uniqueness of solutions.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

On the deformation method of study of global asymptotic stability

We consider the one-parameter family of systems $$x' = F(x,\lambda ), x \in \mathbb{R}^n , 0 \leqslant \lambda \leqslant 1,$$ where F: ? ⁿ × [0, 1] → ? ⁿ is a continuous vector field. The solution x(t) = φ(t, y, λ) is uniquely determined by the initial condition x(0) = y = φ(0, y, λ) and can be continued to the whole axis (?∞, +∞) for all λ ∈ [0, 1]. We obtain conditions ensuring the preservation of the property of global asymptotic stability of the stationary solution of such a system as the parameter λ varies.     

On the chromaticity of the 2-degree integral subgraph ofq-trees

A graphG is called to be a 2-degree integral subgraph of aq-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactlyq- 1 triangles. An added-vertexq-treeG with n vertices is obtained by taking two verticesu, v (u, v are not adjacent) in a q-treesT withn -1 vertices such that their intersection of neighborhoods ofu, v forms a complete graphK _q , and adding a new vertexx, new edgesxu, xv, xv ₁,xv ₂, …,xv _{q- 4}, where {v ₁,v ₂,...,v _q?4} ?-K _q . In this paper we prove that a graphG with minimum degree not equal toq -3 and chromatic polynomialP(G;λ) = λ(λ - 1) … (λ -q +2)(λ -q +1)³(λ -q) ^{n- q- 2} withn ≥ q + 2 has and only has 2-degree integral subgraph of q-tree withn vertices and added-vertex q-tree withn vertices.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Asymptotics of the complexity of systems of integer linear forms for additive computations

The computational complexity of integer linear forms is studied. By l ₂(A) we denote the minimal number of the additions and subtractions required for computing the system of p linear forms in q variables x ₁, x ₂, …, x _q that are defined by an integer matrix A of size p × q (repeated use of the results of intermediate computation is permitted). We show that l ₂(A) ? log D(A), where D(A) is the maximum of the absolute values of the minors of A over all minors from order 1 to order min (p, q) (Theorem 1). Moreover, for each sequence of matrices A(n) of size p(n) × q(n) satisfying the condition p + q = o ((log log D(A))^1/2) as n → ∞ the bound l ₂(A) ? log D(A) + o(log D(A)) is valid (Theorem 2). Hence, for all fixed (and even weakly increasing) sizes of matrices that determine a system of integer linear forms, the upper bound on the computational complexity of this system is asymptotically equal to the lower bound.     

A Morita equivalence theorem for Hecke algebra ℋq(Dn) when n is even

 In this paper we prove a Morita equivalence theorem for Hecke algebras of type D _n when n is even, which generalize a similar result obtained by C. Pallikaros ([P, (3.7)]) when n is odd. As a consequence, we construct all the irreducible ℋ _q (D _n )-modules when f _n (q)≠ 0 (see [P, (2.12)] for definition of f _n (q)) and show that ℋ _q (D _n ) is split in this case. Received: 19 February 2001 / Revised version: 26 January 2002     

Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS

Pour toute algèbre enveloppante quantique U_q(𝔤) de Drinfeld–Jimbo et toute famille λ = (λ_ij)1≤i ∈ k^? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien A_λ de U_q(𝔤) et nous montrons que tout objet galoisien de U_q(𝔤) est homotope à un unique objet de la forme A_λ.

For any Drinfeld–Jimbo quantum enveloping algebra U_q(𝔤) and for any family λ = (λ_ij)_{1≤i ∈ k^? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.}