Stein--Sahi complementary series and their degenerations

The aim of the paper is an introduction to Stein--Sahi complementary series, holomorphic series, and 'unipotent representations'. We also discuss some open problems related to these objects. For the sake of simplicity, we consider only the groups U(n,n).Comment: 40pp, 7fig, revised versio

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial

Applying a method developed by Takamura and Takano for the nonsymmetric Jack polynomial, we present the Rodrigues formula for the nonsymmetric multivariable Hermite polynomial.Comment: 5 pages, LaTe

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

In this paper we analyze Fourier coefficients of automorphic forms on adelic split simply-laced reductive groups $G(\mathbb{A})$. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G(\mathbb{A})$. We prove that any $\eta\in \pi$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whittaker coefficients. A consequence of our results is the non-existence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory

Generalized Second Law of Thermodynamics on the Event Horizon for Interacting Dark Energy

Here we are trying to find the conditions for the validity of the generalized second law of thermodynamics (GSLT) assuming the first law of thermodynamics on the event horizon in both cases when the FRW universe is filled with interacting two fluid system- one in the form of cold dark matter and the other is either holographic dark energy or new age graphic dark energy. Using the recent observational data we have found that GSLT holds both in quintessence era as well as in phantom era for new age graphic model while for holographic dark energy GSLT is valid only in phantom era.Comment: 8 pages, 2 figure

Notes on Stein-Sahi representations and some problems of non L2L^2 harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Comment: 40p

The Burst and Transient Source Experiment (BATSE) Earth Occultation Catalog of Low-Energy Gamma-Ray Sources

The Burst and Transient Source Experiment (BATSE), aboard the Compton Gamma Ray Observatory (CGRO), provided a record of the low-energy gamma-ray sky (20-1000 keV) between 1991 April and 2000 May (9.1y). Using the Earth Occultation Technique to extract flux information, a catalog of sources using data from the BATSE large area detectors has been prepared. The first part of the catalog consists of results from the monitoring of 58 sources, mostly Galactic. For these sources, we have included tables of flux and spectral data, and outburst times for transients. Light curves (or flux histories) have been placed on the world wide web. We then performed a deep-sampling of 179 objects (including the aforementioned 58 objects) combining data from the entire 9.1y BATSE dataset. Source types considered were primarily accreting binaries, but a small number of representative active galaxies, X-ray-emitting stars, and supernova remnants were also included. The deep sample results include definite detections of 83 objects and possible detections of 36 additional objects. The definite detections spanned three classes of sources: accreting black hole and neutron star binaries, active galaxies and supernova remnants. Flux data for the deep sample are presented in four energy bands: 20-40, 40-70, 70-160, and 160-430 keV. The limiting average flux level (9.1 y) for the sample varies from 3.5 to 20 mCrab (5 sigma) between 20 and 430 keV, depending on systematic error, which in turn is primarily dependent on the sky location. To strengthen the credibility of detection of weaker sources (5-25 mCrab), we generated Earth occultation images, searched for periodic behavior using FFT and epoch folding methods, and critically evaluated the energy-dependent emission in the four flux bands.Comment: 64 pages, 17 figures, abstract abridged, Accepted by ApJ

Assessment of acute and sub-acute toxicity of olive pomace in female Wistar rats

Objective: Olive Pomace (OP) is considered to be a rich source of phenolic compounds. Recently many researches showed a broad biological activity of this by-product of the olive oil production process in addition to their emergence as value-added materials with potential applications in the pharmaceutical, food, and nutraceutical industries. The present study is aimed to evaluate in vivo toxicological activities of OP. Materials and Methods: The qualitative phytochemical analysis aims to determine the key phytoconstituents found in OP. For the in vivo study, two types of tests are performed: acute and 28-day repeated oral toxicity studies in Wistar rats for evaluation of hematological, biochemical, and histological parameters. Results: The qualitative phytochemical analysis has revealed the presence of polyphenols, flavonoids, tannins, saponins, quinones, anthraquinones, terpenoids, and compounds reduced in our methanol extract of OP. In acute oral toxicity, no treatment-related death or toxic signs are observed in female rats for 14 days in 200, 2000, 3000, and 5000 mg/kg doses, besides LD50 value is found to be up to 2000 mg/kg bodyweight. As for the Globally Harmonized System of Classification and Labelling of Chemicals. 28-days sub-acute toxicity study is carried in female rats at four dose levels (3.12, 31.25, 125 and 500 mg/kg), no changes in observation related death and toxic signs when compared with control. The hematological and biochemical investigation shows a significant change (p>0.05) in the high-level doses (500 mg/kg). Conclusions: According to the findings of this study, OP extract has the potential to be used to generate new anti-cancer and antioxidant additives for pharmaceutical and food manufacturing. Long-term in vivo toxicological tests should also be conducted to determine a safe dosage of OP extract

Antidiabetic treatment, obesity, and cancer risk in Algerian patients with type 2 diabetes mellitus

OBJECTIVE: Several studies have shown that antidiabetic drugs and obesity can modulate the risk of developing cancer. The objective of this study was to assess the impact of the use of antidiabetic drugs and obesity on the risk of developing cancers in type 2 diabetics. MATERIALS AND METHODS: Data for 1220 patients were collected from the processing of files and a pre-established questionnaire. The anthropobiological parameters and the associated treatment type have been unspecified. RESULTS: Women (OR=17.26; 95% CI=2.88-103.45, p<0.01), overweight individuals (OR=4.81; 95% CI=1.63-14.14, p<0.01) and hypertensive diabetic subjects (OR=3.82; 95% CI=1.39-10.49, p< 0.01) are more exposed to cancers. It is interesting to note that diabetic subjects treated with insulin have a reduced risk of developing cancer (OR=0.22; 95% CI=0.07-0.67, p<0.01). Diabetic subjects treated with metformin have a four and a half times higher risk of developing cancer (OR=4.61; 95% CI=1.48-14.37, p<0.01). CONCLUSIONS: In type 2 diabetic subjects, cancer is significantly linked to overweight, to the presence of essential hypertension in individuals under hypotensive as well as in patients treated with metformin