Orthonormal sequences in L2(Rd)L^2(R^d) and time frequency localization

We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that there is no orthonormal basis for $L^2(\R)$ for which the time and frequency means as well as the product of dispersions are uniformly bounded. The problem is related to recent results of J. Benedetto, A. Powell, and Ph. Jaming. Our main tool is a time frequency localization inequality for orthonormal sequences in $L^2(\R^d)$. It has various other applications.Comment: 18 page

Deux extensions de Théorèmes de Hamburger (portant sur l'équation fonctionnelle de la fonction dzêta)

We propose two types of extensions to Hamburger's theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate meromorphic functions with functional equation and the tempered distributions with extended S-support condition.Nous proposons deux types d'extensions aux théorèmes de Hamburger sur les séries de Dirichlet avec équation fonctionnelle comme celle de la fonction zêta de Riemann, sous des hypothèses plus faibles. Ceci repose sur le dictionnaire entre les fonctions méromorphes modérées avec cette équation fonctionnelle et les distributions tempérées avec la condition de support S-étendue

A quantum mechanical model of the Riemann zeros

In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.Comment: 42 pages, 12 figure

On the system of the functions x (s) / (s-r)k

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

General covariant xp models and the Riemann zeros

We study a general class of models whose classical Hamiltonians are given by H = U(x) p + V(x)/p, where x and p are the position and momentum of a particle moving in one dimension, and U and V are positive functions. This class includes the Hamiltonians H_I =x (p+1/p) and H_II=(x+ 1/x)(p+ 1/p), which have been recently discussed in connection with the non trivial zeros of the Riemann zeta function. We show that all these models are covariant under general coordinate transformations. This remarkable property becomes explicit in the Lagrangian formulation which describes a relativistic particle moving in a 1+1 dimensional spacetime whose metric is constructed from the functions U and V. General covariance is maintained by quantization and we find that the spectra are closely related to the geometry of the associated spacetimes. In particular, the Hamiltonian H_I corresponds to a flat spacetime, whereas its spectrum approaches the Riemann zeros in average. The latter property also holds for the model H_II, whose underlying spacetime is asymptotically flat. These results suggest the existence of a Hamiltonian whose underlying spacetime encodes the prime numbers, and whose spectrum provides the Riemann zeros.Comment: 34 pages, 3 figure

O-GlcNAcylation Increases ChREBP Protein Content and Transcriptional Activity in the Liver

International audienceOBJECTIVE Carbohydrate-responsive element–binding protein (ChREBP) is a key transcription factor that mediates the effects of glucose on glycolytic and lipogenic genes in the liver. We have previously reported that liver-specific inhibition of ChREBP prevents hepatic steatosis in ob/ob mice by specifically decreasing lipogenic rates in vivo. To better understand the regulation of ChREBP activity in the liver, we investigated the implication of O-linked β-N-acetylglucosamine (O-GlcNAc or O-GlcNAcylation), an important glucose-dependent posttranslational modification playing multiple roles in transcription, protein stabilization, nuclear localization, and signal transduction. RESEARCH DESIGN AND METHODS O-GlcNAcylation is highly dynamic through the action of two enzymes: the O-GlcNAc transferase (OGT), which transfers the monosaccharide to serine/threonine residues on a target protein, and the O-GlcNAcase (OGA), which hydrolyses the sugar. To modulate ChREBPOG in vitro and in vivo, the OGT and OGA enzymes were overexpressed or inhibited via adenoviral approaches in mouse hepatocytes and in the liver of C57BL/6J or obese db/db mice. RESULTS Our study shows that ChREBP interacts with OGT and is subjected to O-GlcNAcylation in liver cells. O-GlcNAcylation stabilizes the ChREBP protein and increases its transcriptional activity toward its target glycolytic (L-PK) and lipogenic genes (ACC, FAS, and SCD1) when combined with an active glucose flux in vivo. Indeed, OGT overexpression significantly increased ChREBPOG in liver nuclear extracts from fed C57BL/6J mice, leading in turn to enhanced lipogenic gene expression and to excessive hepatic triglyceride deposition. In the livers of hyperglycemic obese db/db mice, ChREBPOG levels were elevated compared with controls. Interestingly, reducing ChREBPOG levels via OGA overexpression decreased lipogenic protein content (ACC, FAS), prevented hepatic steatosis, and improved the lipidic profile of OGA-treated db/db mice. CONCLUSIONS Taken together, our results reveal that O-GlcNAcylation represents an important novel regulation of ChREBP activity in the liver under both physiological and pathophysiological conditions

Deux extensions de Théorèmes de Hamburger (portant sur l'équation fonctionnelle de la fonction dzêta)

We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate meromorphic functions with functional equation and the tempered distributions with extended S-support condition.Nous proposons deux types d’extensions aux théorèmes de Hamburger sur les séries de Dirichlet avec équation fonctionnelle comme celle de la fonction zêta de Riemann, sous des hypothèses plus faibles. Ceci repose sur le dictionnaire entre les fonctions méromorphes modérées avec cette équation fonctionnelle et les distributions tempérées avec la condition de support S-étendue

On the system of the functions x (s) / (s-r)k

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