Congruences in modular, Jacobi, Siegel, and mock modular forms with applications

We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have several applications. We describe the Ramanujan congruences in the counting functions for overparitions, overpartition pairs, crank differences, and Andrews' two-coloured generalized Frobenius partitions. We also study Ramanujan congruences in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of Jacobi forms and Siegel modular forms of degree two. Finally, the last chapter contains a completely unrelated result about harmonic weak Maass forms.

Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

The Rankin convolution type Dirichlet series D-F,D-G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D-F,D-G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

Comportamento assintótico dos polinômios ortogonais de Sobolev-Jacobi e Sobolev-Laguerre

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Elementary properties of Modular curves and Modular forms

Lo scopo di questa tesi è introdurre in breve le prime proprietà delle curve modulari e delle forme modulari, per poi mostrarne alcune applicazioni archetipiche. Per farlo, dopo aver richiamato alcune nozioni utili nel primo capitolo, sviluppiamo, nel secondo capitolo, la teoria di base delle curve modulari compatte come superfici di Riemann, calcolandone il genere nel caso dei sottogruppi principali di congruenza. Dunque, nel terzo capitolo, dopo un estesa trattazione dell'esempio delle forme modulari rispetto al gruppo modulare, viene calcolata la dimensione degli spazi delle forme intere e delle forme cuspidali rispetto a un sottogruppo di indice finito del gruppo modulare. Questo capitolo si conclude con tre esempi di applicazione della teoria esposta, tra i quali spiccano la dimostrazione del Grande Teorema di Picard e del Teorema dei quattro quadrati di Jacobi.

Chronic Telogen Effluvium And Female Pattern Hair Loss Are Separate And Distinct Forms Of Alopecia: A Histomorphometric And Immunohistochemical Analysis.

Chronic telogen effluvium (CTE), a poorly understood condition, can be confused with or may be a prodrome to female pattern hair loss (FPHL). The pathogenesis of both is related to follicle cycle shortening and possibly to blood supply changes. To analyze a number of histomorphometric and immunohistochemical findings through vascular endothelial growth factor (VEGF), Ki-67, and CD31 immunostaining in scalp biopsies of 20 patients with CTE, 17 patients with mild FPHL and 9 controls. Ki-67 index and VEGF optical density were analyzed at the follicular outer sheath using ImageJ software. CD31 microvessel density was assessed by a Chalkley grid. Significant follicle miniaturization and higher density of nonanagen follicles were found in FPHL, compared with patients with CTE and controls. Ki-67+ index correlated positively with FPHL histological features. The FPHL group showed the highest VEGF optical density, followed by the CTE and control groups. No differences were found in CD31 microvessel density between the three groups. Histomorphometric results establish CTE as a distinct disorder, separate from FPHL from its outset. Its pathogenic mechanisms are also distinct. These findings support the proposed mechanism of 'immediate telogen release' for CTE, leading to cycle synchronization. For FPHL, accelerated anagen follicular mitotic rates and, thus, higher Ki-67 and VEGF values, would leave less time for differentiation, resulting in hair miniaturization.

Use Of Experimental Design To Optimize A Triple-potential Waveform To Develop A Method For The Determination Of Streptomycin And Dihydrostreptomycin In Pharmaceutical Veterinary Dosage Forms By Hplc-pad.

An HPLC-PAD method using a gold working electrode and a triple-potential waveform was developed for the simultaneous determination of streptomycin and dihydrostreptomycin in veterinary drugs. Glucose was used as the internal standard, and the triple-potential waveform was optimized using a factorial and a central composite design. The optimum potentials were as follows: amperometric detection, E1=-0.15V; cleaning potential, E2=+0.85V; and reactivation of the electrode surface, E3=-0.65V. For the separation of the aminoglycosides and the internal standard of glucose, a CarboPac™ PA1 anion exchange column was used together with a mobile phase consisting of a 0.070 mol L(-1) sodium hydroxide solution in the isocratic elution mode with a flow rate of 0.8 mL min(-1). The method was validated and applied to the determination of streptomycin and dihydrostreptomycin in veterinary formulations (injection, suspension and ointment) without any previous sample pretreatment, except for the ointments, for which a liquid-liquid extraction was required before HPLC-PAD analysis. The method showed adequate selectivity, with an accuracy of 98-107% and a precision of less than 3.9%.

Interleukin-22 forms dimers that are recognized by two interleukin-22R1 receptor chains

Interleukin-22 (IL-22) is a class 2 cytokine whose primary structure is similar to that of interleukin 10 (IL-10) and interferon-gamma (IFN-gamma). IL-22 induction during acute phase immune response indicates its involvement in mechanisms of inflammation. Structurally different from IL-10 and a number of other members of IL-10 family, which form intertwined inseparable V-shaped dimers of two identical polypeptide chains, a single polypeptide chain of IL-22 folds on itself in a relatively globular structure. Here we present evidence, based on native gel electrophoresis, glutaraldehyde cross-linking, dynamic light scattering, and small angle x-ray scattering experiments, that human IL-22 forms dimers and tetramers in solution under protein concentrations assessable by these experiments. Unexpectedly, low-resolution molecular shape of IL-22 dimers is strikingly similar to that of IL-10 and other intertwined cytokine dimeric forms. Furthermore, we determine an ab initio molecular shape of the IL-22/IL-22R1 complex which reveals the V-shaped IL-22 dimer interacting with two cognate IL-22R1 molecules. Based on this collective evidence, we argue that dimerization might be a common mechanism of all class 2 cytokines for the molecular recognition with their respective membrane receptor. We also speculate that the IL-22 tetramer formation could represent a way to store the cytokine in nonactive form at high concentrations that could be readily converted into functionally active monomers and dimers upon interaction with the cognate cellular receptors.

Synthesis of Cyclic Guanidine Intermediates of Anatoxin-a(s) in Both Racemic and Enantiomerically Pure Forms

The alkyl chain of anatoxin-a(s) (cyclic guanidines), which can be used as an intermediate in the total synthesis of anatoxin-a(s), was synthesized in both racemic and enantiomerically pure forms. These enantiomerically pure cyclic compounds can be used as chiral inductors in some reactions. The two racemic routes disclosed herein have the advantages of high overall yield and mild reaction conditions. Both routes proceed through an intermediate 2,3-diaminoacid - an important synthetic scaffold - with good yields. Furthermore, the N,N-dimethyl-2(tosylimino)imidazolidine-4-carboxamide might be obtained from 2-(tosylimino)imidazolidine-4-carboxylic acid followed by selective reduction of the carbonyl functionality. All synthesized compounds were analyzed by mass spectrometry and (1)H NMR and (13)C NMR spectroscopy.

Monocyte-derived dendritic cells from patients with severe forms of chromoblastomycosis induce CD4(+) T cell activation in vitro

Dendritic cells (DCs) have been described as initiators and modulators of the immune response. Recently we have shown a predominant production of interleukin-10 cytokine, low levels of interferon-gamma and inefficient T cell proliferation in patients with severe forms of chromoblastomycosis. Chromoblastomycosis starts with subcutaneous inoculation of Fonsecaea pedrosoi into tissue where DCs are the first line of defence against this microorganism. In the present study, the interaction of F. pedrosoi and DCs obtained from patients with chromoblastomycosis was investigated. Our results showed that DCs from patients exhibited an increased expression of human leucocyte antigen D-related (HLA-DR) and co-stimulatory molecules. In the presence of conidia, the expression of HLA-DR and CD86 was up-regulated by DCs from patients and controls. Finally, we demonstrate the reversal of antigen-specific anergy and a T helper type 1 response mediated by DCs incubated with F. pedrosoi conidea.

Standing wave bed forms at Plage de la Grande Roche, le Verdelet, Pléneuf-Val-André, Cote d’Armor (France)

Coastal Photograph by Hubert Chanson This photograph of standing wave bed forms was taken at very low tide. The tidal range was 10 m. The bed forms were located on the island of Le Verdelet, in a channel between Le Grande Jaune and Le Verdelet. It is likely that these standing wave bed forms were formed during transcritical shallow water flows at the end of ebb tide. The author’s watch is in the foreground for scale. (Coastal Photograph by Hubert Chanson, Division of Civil Engineering, the University of Queensland, Brisbane, Queensland 4072, Australia.)

Cost-benefit analysis of alternative forms of hedgerow intercropping in the Philippine uplands

Considerable resources have been expended promoting hedgerow intercropping with shrub legumes to farmers in the Philippine uplands. Despite the resources committed to research and extension, persistent adoption by farmers has been limited to low cost versions of the technology including natural vegetation and grass strips. In this paper, cost-benefit analysis is used to compare the economic returns from traditional open-field maize farming with returns from intercropping maize between leguminous shrub hedgerows, natural vegetation strips and grass strips. An erosion/productivity model, Soil Changes Under Agroforestry, was used to predict the effect of erosion on maize yields. Key informant surveys with experienced maize farmers were used to derive production budgets for the alternative farming methods. The economic incentives revealed by the cost-benefit analysis help to explain the adoption of maize farming methods in the Philippine uplands. Open-field farming without hedgerows has been by far the most popular method of maize production, often with two or more fields cropped in rotation. There is little persistent adoption of hedgerow intercropping with shrub legumes because sustained maize yields are not realised rapidly enough to compensate farmers for establishment and maintenance costs. Natural vegetation and grass strips are more attractive to farmers because of lower establishment costs, and provide intermediate steps to adoption. Rural finance, commodity pricing and agrarian reform policies influence the incentives for maize farmers in the Philippine uplands to adopt and maintain hedgerow intercropping.

Familial forms broaden the horizons for primary aldosteronism

The identification of familial forms of primary aldosteronism (PAL) has led to its detection in relatives of affected patients not suspected previously of having PAL. Many ave normokalemic and some ave even normotensive. This broadens the spectrum of PAL, permitting the study of its evolution and of intervention with specific therapy when hypertension develops. The genetic basis of one form involves steroid biosynthetic enzymes and the other form predisposes to hyperplasia and benign neoplasia.

Structures of free and complexed forms of Escherichia coli xanthine-guanine phosphoribosyltransferase

Structures of free, substrate-bound and product-bound forms of Escherichia coli xanthine-guanine phosphoribosyltransferase (XGPRT) have been determined by X-ray crystallography. These are compared with the previously determined structure of magnesium and sulphate-bound XPRT. The structure of free XGPRT at 2.25 Angstrom resolution confirms the flexibility of residues in and around a mobile loop identified in other PRTases and shows that the cis-peptide conformation of Arg37 at the active site is maintained in the absence of bound ligands. The structures of XGPRT complexed with the purine base substrates guanine or xanthine in combination with cPRib-PP, an analog of the second substrate PRib-PP, have been solved to 2.0 Angstrom resolution. In these two structures the disordered phosphate-binding loop of uncomplexed XGPRT becomes ordered through interactions with the 5'-phosphate group of cPRib-PP. The cyclopentane ring of cPRib-PP has the C3 exo pucker conformation, stabilised by the cPRib-PP-bound Mg2+. The purine base specificity of XGPRT appears to be due to water-mediated interactions between the 2-exocyclic groups of guanine or xanthine and side-chains of Glu136 and Asp140, as well as the main-chain oxygen atom of Ile135. Asp92, together with Lys115, could help stabilise the N7-protonated tautomer of the incoming base and could act as a general base to remove the proton from N7 .when the nucleotide product is formed. The 2.6 Angstrom resolution structure of XGPRT complexed with product GMP is similar to the substrate-bound complexes. However, the ribose ring of GMP is rotated by similar to 24 degrees compared with the equivalent ring in cPRib-PP. This rotation results in the loss of all interactions between the ribosyl group and the enzyme in the product complex. (C) 1998 Academic Press.