Independence of algebraic monodromy groups in compatible systems

In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

ASSESSMENT OF FALL RISK AMONG ELDERLY IN HOME ENVIRONMENT IN AL-HASA

Introduction and Aim: Fall it is an accident that unintentionally breaks down a person on the floor, the surface of the earth, or any other low level. The goal of this study was to estimate the prevalence of falls among the elderly and investigate the factors that contribute to this study. Method: The study design was quantitative, descriptive that surveyed for elderly risk of fall and their caregivers, using google form. The study was conducted by a two questionnaire type assessment, 8 questions were used to assess each questionnaire, using fall efficacy scale International with four options (not at all concerned, somewhat concerned, fairly concerned, very concerned), and using Modified Barthel index scale with thee options (No, complete dependence, yes, partially assist, and independence). Result: People who are 65 Years old and above were 140 participated in the study. In Barthel index study about (32.9% - 52.1%) was independence, and about (29.3% - 48.6%) were need partial assist, and about (18.6% - 36.4%) were complete dependence. In fall efficacy scale about (6.4% - 42.9%) were not at all concerned, and about (12.9% - 31.4%) were somewhat concerned, and about (12.9% - 29.3%) were fairly concerned, and about (12.9% - 57.1%) were very concerned. Conclusion: The elderly stage is the most stage or period that the human will have many conditions that enhance to fall.

Corrigendum: Around -independence

We correct the proof of the main $\ell$-independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ($p$-adic and $\ell$-adic) cohomology.

THE TOPOLOGICAL PIGEONHOLE PRINCIPLE FOR ORDINALS

Given a cardinal κ and a sequence ${\left( {{\alpha _i}} \right)_{i \in \kappa }}$ of ordinals, we determine the least ordinal β (when one exists) such that the topological partition relation$$\beta \to \left( {top\,{\alpha _i}} \right)_{i \in \kappa }^1$$holds, including an independence result for one class of cases. Here the prefix “top” means that the homogeneous set must be of the correct homeomorphism class rather than the correct order type. The answer is linked to the nontopological pigeonhole principle of Milner and Rado.

A WEIGHT INDEPENDENCE RESULT FOR QUATERNIONIC HECKE ALGEBRAS

Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.

Countable Fréchet Boolean groups: An independence result

It is relatively consistent with ZFC that every countable FUfin space of weight ℵ1 is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].