On Whittaker models and the vanishing of Fourier coefficients of cusp forms

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group. This research was partially supported by Grant No. 8400139 from the United States-Israel Bi National Science Foundation (BSF), Jerusalem, Israel.     

Weights of elementary representations in SU(5)-SO(10)-E6 chain

For Dynkin labels of weights in 5 and 10 of SU(5), 10 and 16 of SO(10) and 27 ofE, we find it instructive to (a) explōit the Cartan decom position into a nonsemi simple symmetric subalgebra and coset space, (b) introduce a set of annihilation and creation operators, that may act on an invariant 10>, to represent suitably the shift-action and weight-vectors.     

Un revêtement de l'arbre de GL2 d'un corps local

Let F be a non-archimedean local field with residue class field k. Put G=GL₂(F), =PGL₂(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex , equipped with an action of G × and with a G × -equivariant simplicial projection (for the trivial action of on X). We prove that the cohomology with compact support contains nontrivial representations of G (in particular positive level supercuspidal representations).     

Q-Operator and the Drinfeld Equation

We show that the TQ equation is satisfied by the trace over the quantum space of the product of R-matrices intertwining two representations of the quantum double of the Borel subalgebra of the affine algebra (the standard two-dimensional and the N-dimensional cyclic representations).     

Realization of the Universal sℓ q (2)-Symmetric R-Operator in a Function Space for General and Exceptional Values of the Deformation Parameter

Based on the realization of representations of the algebra s _q (2) in the space of polynomials for general values of the deformation parameter q and on a finite tuple of theta functions, which are a natural generalization of polynomials, we construct the eigenstates and find the related eigenvalues of the universal R-operator for cyclic representations corresponding to q ^N = ±1.     

On a number of poisson matrices in Bang-Bang representations for 3 × 3 embeddable matrices

We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 × 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 × 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 × 3 matrix P with det $\text{P ≥}\text{1}\text{8}$ can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 × 3 stochastic matrix P with det $\text{P ≥}\text{1}\text{8}$ to be embeddable. For an embeddable 3 × 3 matrix P with det $\text{P <}\text{1}\text{8}$ we give a new bound for the number of Poisson matrices in its Bang-Bang representation.     

Boundary Value Problems for Boussinesq Type Systems

We characterise the boundary conditions that yield a linearly well posed problem for the so-called KdV–KdV system and for the classical Boussinesq system. Each of them is a system of two evolution PDEs modelling two-way propagation of water waves. We study these problems with the spatial variable in either the half-line or in a finite interval. The results are obtained by extending a spectral transform approach, recently developed for the analysis of scalar evolution PDEs, to the case of systems of PDEs. The knowledge of the boundary conditions that should be imposed in order for the problem to be linearly well posed can be used to obtain an integral representation of the solution. This knowledge is also necessary in order to conduct numerical simulations for the fully nonlinear systems. Mathematics Subject Classifications (2000) 34A30, 34A34, 35F10.     

Periodic pulse train conformation based on the temporal Radon-Wigner transform

By using the Radon-Wigner transform (RWT), we analyze the temporal selfimaging or Talbot effect for producing well-conformed pulse trains with variable repetition rates and duty-cycles. The relationships linking the selfimaging conditions with the fractional orders of the RWT are first obtained for unchirped pulse trains. Then, we extend the analysis to chirped pulse sequences by deriving the conditions to be fulfilled by an equivalent unchirped pulse train producing the same selfimage irradiances. This result becomes relevant for observing well-defined high order fractional selfimaging, which are of interest due to their repetition rate multiplication. Besides, the effect of the finite extension of the pulse train on the selfimage quality is analyzed and a condition is found for relating the required minimum pulse number with the chirp parameter of the pulses.     

Quantum Deformations of α-Stratified Modules

We construct quantum analogues of a class of generalized Verma modulesinduced from nonsolvable parabolic subalgebras of simple Lie algebras. Weshow that these quantum modules are true deformations of the underlyingclassical modules in the sense that the weight-space decomposition ispreserved.     

New rapidly convergent series representations for

We give three series representations for the values of the Riemann zeta function at positive odd integers. One representation extends Ewell's result for [Amer. Math. Monthly 97 (1990), 219-220] and is considerably simpler than the two generalisations proposed earlier. The second representation is even simpler:

where the coefficients for a fixed are rational in and are explicitly given by the finite sum involving the Bernoulli numbers. The third representation is obtained from the second by the Kummer transformation. We demonstrate the rapid convergence of this series using several examples.