Explicit methods for Hilbert modular forms

We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.

Verifying Harder's Conjecture for Classical and Siegel Modular Forms

A conjecture by Harder shows a surprising congruence between the coefficients of “classical” modular forms and the Hecke eigenvalues of corresponding Siegel modular forms, contigent upon “large primes” dividing the critical values of the given classical modular form. Harder’s Conjecture has already been verified for one-dimensional spaces of classical and Siegel modular forms (along with some two-dimensional cases), and for primes p 37. We verify the conjecture for higher-dimensional spaces, and up to a comparable prime p.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

INFERENCE FOR EIGENVALUES AND EIGENVECTORS OF GAUSSIAN SYMMETRIC MATRICES

This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic background radiation data, where the observations are, respectively, 3 x 3 and 2 x 2 symmetric positive definite matrices. The parameter sets involved in the inference problems for eigenvalues and eigenvectors are subsets of Euclidean space that are either affine subspaces, embedded submanifolds that are invariant under orthogonal transformations or polyhedral convex cones. We show that for a class of sets that includes the ones considered in this paper, the MLEs of the mean parameter do not depend on the covariance parameters if and only if the covariance structure is orthogonally invariant. Closed-form expressions for the MLEs and the associated LLRs are derived for this covariance structure.

Eigenvalues of Casimir operators for gl(m/infinity)

A full set of Casimir operators for the Lie superalgebra gl(m/infinity) is constructed and shown to be well defined in the category O-FS generated by the highest-weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from gl(m/infinity) are also determined.

Transfer matrix eigenvalues of the anisotropic multiparametric U model

A multiparametric extension of the anisotropic U model is discussed which maintains integrability. The R-matrix solving the Yang-Baxter equation is obtained through a twisting construction applied to the underlying U-q(sl (2/1)) superalgebraic structure which introduces the additional free parameters that arise in the model. Three forms of Bethe ansatz solution for the transfer matrix eigenvalues are given which we show to be equivalent.

Eigenvalues of Casimir invariants for unitary irreps of Uq(gl(m vertical bar n))

A fully explicit formula for the eigenvalues of Casimir invariants for U-q(gl(m/n)) is given which applies to all unitary irreps. This is achieved by making some interesting observations on atypicality indices for irreps occurring in the tensor product of unitary irreps of the same type. These results have applications in the determination of link polynomials arising from unitary irreps of U-q(gl(m/n)).

On simple bounds for eigenvalues of symmetric tridiagonal matrices

How much can be said about the location of the eigenvalues of a symmetric tridiagonal matrix just by looking at its diagonal entries? We use classical results on the eigenvalues of symmetric matrices to show that the diagonal entries are bounds for some of the eigenvalues regardless of the size of the off-diagonal entries. Numerical examples are given to illustrate that our arithmetic-free technique delivers useful information on the location of the eigenvalues.

Ensaios sôbre lepra experimental: inoculações de três amostras de bacilos ácido-álcool resistentes (amostras "Chaves II", "Emilia" e "Hecke") isoladas de leprosos, em trinta doentes da Colônia Mirueira

Com o fim de produzir o fenômeno de KOCH, os A. A. inocularam em 30 leprosos da Colônia Mirueira (Recife), de várias idades e formas clínicas, emulsões vivas de três culturas de bacilos ácido-álcool resistentes isolados de leprosos pro um dêles (S.A.). As doses inoculadas foram de 0,2 cc., por via intradérmica, em cada doente, das amostras "CII", "E" e "H" e mais da Leprolina S.A. (antígeno morto). No 10º dia da inoculação verificou-se que 24 dos 30 pacientes tiveram reação geral intensa; 2, reação moderada e 4, nenhuma reação geral. 16 dos 30 tiveram reação leprótica, sendo 10 em casos ativos (lepromatosos) e 6 em inativos, e 17 dos 30 tiveram adenopatias inguinais. O inóculo "CII" produziu escaras de 1 x 1 e 2 x 2 cm. de diâmetro, com destruição total da pele, nos 30 pacientes (o total dêles); o inóculo "E" produziu escaras de igual intensidade em 29, o inóculo "H", escaras muitos mais benignas em 23, e a Leprolina em 10, naturalmente por ação concomitante de um dos outros três inóculos. No 10º dia foram semeadas em meio de LOEWENSTEIN secreções das escaras de sete dos 30 doentes, num total de 20 tubos, dos quais 19 produziram retroculuras, a amioria contaminada por fungos ou por bactérias cianófilas. De um doente foi obtido retrocultura cromogênica da escara produzida na intradérmoreação pela "Leprolina S. A.", macro e microscòpricamente indiferencável das amostras "CII" e "E". Aliás, pela extensiva experimentação feita com estas duas amostras, estamos nos inclinando por considerá-las como idênticas. No 18º dia da inoculação foram feitos 30 esfregaços de secreções de lesões experimentais de 13 doentes, com 15 resultados positivos (50%), apesar do exame tardio. As morfologias macro e microscópica das retroculturas obtidas em Recife confirmam os caracteres descritos nas culturas originais. Dêste rápido ensaio se conclui que a maioria dos pacientes apresentou o fenômeno de KOCH parcial ou integral, com as clássicas reações gerais, focais e locais. A falta de recursos de laboratório na Colônia não permitiu melhor aproveitamento de tão precioso material experimental, e por isso êste trabalho apresenta várias lacunas.

Coefficients of Drinfeld modular forms and Hecke operators

"Vegeu el resum a l'inici del document del fitxer adjunt."

Van Hecke [Alfons, coureur cycliste belge] : [photographie de presse] / [Agence Rol]

Référence bibliographique : Rol, 58739

Effect of nuclear motion on the energy eigenvalues in muonic atoms

I have investigated the effect of the nuclear motion on the energy eigenvalues in muonic atoms. In addition to the usually used reduced-mass correction, I have calculated the relativistic influences including the magnetic and retardation interaction between the nucleus and the muon for the inner orbitals of muonic atoms.

On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.