Homology of the Fermat Tower and Universal Measures for Jacobi Sums

2016 ◽  
Vol 59 (3) ◽  
pp. 624-640
Author(s):  
Noriyuki Otsubo
Keyword(s):  
Power Series ◽  
Group Ring ◽  
Simple Proof ◽  
Homology Group ◽  
Interpolation Property ◽  
Cyclic Module ◽  
Precise Description ◽  
Fermat Curve ◽  
Jacobi Sums ◽  
Definition Of

AbstractWe give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson’s adelic beta functions, in a manner similar to Ihara’s definition of ℓ-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

scholarly journals ABELIAN COVERS OF SURFACES AND THE HOMOLOGY OF THE LEVEL L MAPPING CLASS GROUP

2011 ◽  
Vol 03 (03) ◽  
pp. 265-306 ◽  
Author(s):  
ANDREW PUTMAN
Keyword(s):  
Group Ring ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Homology Group ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Rational Group ◽  
Rational Homology ◽  
Abelian Covers

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian ℤ/L-cover of the surface. If the surface has one marked point, then the answer is ℚτ(L), where τ(L) is the number of positive divisors of L. If the surface instead has one boundary component, then the answer is ℚ. We also perform the same calculation for the level L subgroup of the mapping class group. Set HL = H1(Σg; ℤ/L). If the surface has one marked point, then the answer is ℚ[HL], the rational group ring of HL. If the surface instead has one boundary component, then the answer is ℚ.

scholarly journals Homology group on manifolds and their foldings

2010 ◽  
Vol 41 (1) ◽  
pp. 31-38 ◽  
Author(s):  
M. Abu-Saleem
Keyword(s):  
Homology Group ◽  
Homology Groups ◽  
Definition Of

In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.

Compound Invariants and Mixed F-, DF-Power Spaces

1998 ◽  
Vol 50 (6) ◽  
pp. 1138-1162 ◽  
Author(s):  
P. A. Chalov ◽  
T. Terzioğlu ◽  
V. P. Zahariuta
Keyword(s):  
Power Series ◽  
Tensor Products ◽  
Main Tool ◽  
Proper Definition ◽  
Definition Of ◽  
Image Position ◽  
Isomorphic Classification

AbstractThe problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed F-, DF-power series spaces, i.e. the spaces of the following kind where ai (p, q) = exp((p - λiq)ai), p,q ∈ ℕ, and λ = (λi)i∈ℕ, a = (ai)i∈ℕ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of F- and DF-types, respectively. The mrectangle characteristic of the space G(λ a) is defined as the number of members of the sequence (ïiÒ ai)i2N which are contained in the union of m rectangles Pk = (δk, εk] ✗ (τk, tk], k = 1, 2 , . . . , m. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

Effectively nowhere simple sets

1984 ◽  
Vol 49 (1) ◽  
pp. 129-136 ◽  
Author(s):  
D. Miller ◽  
J. B. Remmel
Keyword(s):  
Simple Proof ◽  
Recursive Function ◽  
Finite Sets ◽  
Definition Of ◽  
Simple Sets

An r.e. set A is nowhere simple if for every r.e. set We such that We − A is infinite, there is an infinite r.e. set W such that W ⊆ We − A. The definition of nowhere simple sets is due to R. Shore in [4]. In [4], Shore studied various properties of nowhere simple sets and showed that they could be used to give an elegant and simple proof of the fact that every nontrivial class of r.e. sets C closed under recursive isomorphisms is an automorphism base for , the lattice of r.e. sets modulo finite sets, (that is, an automorphism α of is completely determined by its action on C; see Theorem 8 of [4]). Shore also defined the notion of effectively nowhere simple sets.Definition. An r.e. set A is effectively nowhere simple if there is a recursive function f such that for every i, Wf(i) ⊆ Wi − A and Wf(i) is infinite iff Wi − A is infinite. f is called a witness function for A.Other than to produce examples of effectively nowhere simple sets and nowhere simple sets that are not effectively nowhere simple, Shore did not concern himself with the properties of effectively nowhere simple sets since he felt that effectively nowhere simple sets were unlikely to be lattice invariant in either E, the lattice of r.e. sets, or in .

scholarly journals POWER SERIES EXPANSIONS OF MODULAR FORMS AND THEIR INTERPOLATION PROPERTIES

2011 ◽  
Vol 07 (02) ◽  
pp. 529-577 ◽  
Author(s):  
ANDREA MORI
Keyword(s):  
Power Series ◽  
Modular Form ◽  
Modular Forms ◽  
Power Series Expansion ◽  
Base Change ◽  
Square Roots ◽  
Type K ◽  
Power Series Expansions ◽  
Expansion Coefficients ◽  
Definition Of

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

A simple proof of the chaoticity of shift map under a new definition of chaos

2011 ◽  
Vol 27 (4) ◽  
pp. 332-339
Author(s):  
Indranil Bhaumik ◽  
Binayak S. Choudhury ◽  
Basudeb Mukhopadhyay
Keyword(s):  
Simple Proof ◽  
Shift Map ◽  
Definition Of

scholarly journals Design of a Process and Role Model for Internal Crowdsourcing

2020 ◽  
pp. 79-101
Author(s):  
Hannah Ulbrich ◽  
Marco Wedel
Keyword(s):  
Role Model ◽  
Successful Implementation ◽  
Practical Application ◽  
Precise Description ◽  
Energy Provider ◽  
Definition Of ◽  
Theorie Und Praxis ◽  
Agile Model ◽  
A Company ◽  
And Task

AbstractThe successful implementation of internal crowdsourcing (IC) in a company requires a precise description and definition of the personnel responsibilities for the various process levels and process components within each process phase of IC. As part of the research project ‘ICU—Internal Crowdsourcing in Companies’, we have developed a new role model for internal crowdsourcing based on a practical application of IC in the company GASAG AG, an energy provider located in Berlin, Germany. The aim of this article is to present the main features of this role model (Some aspects of this article will also be published in German. Please be referred to Daum, M., Wedel, M., Zinke-Wehlmann, C., Ulbrich, H. (ed.) (2020): Gestaltung vernetzt-flexibler Arbeit. Beiträge aus Theorie und Praxis für die digitale Arbeitswelt. Berlin: Springer Vieweg). It is based on the roles of the agile model of Scrum, because partial aspects of the internal crowdsourcing process and certain process steering tasks have similarities with the procedure and task descriptions of Scrum. Scrum, as a mature and practice-proven set of rules with role descriptions, rules, events and artefacts, provides helpful implications for the design of an internal crowdsourcing role model as we will prove in further detail.

scholarly journals A Note on Stirling's Theorem

1933 ◽  
Vol 28 ◽  
pp. xii-xiii
Author(s):  
J. R. Wilton
Keyword(s):  
Simple Proof ◽  
Definition Of ◽  
Image Position

Let Γ(1 + x) = √(2πx)xxe–x φ(x);This result is Stirling's theorem. A simple proof is given in § 1.87 of Titchmarsh's Theory of Functions (Oxford Univ. Press, 1932).Rather more than Stirling's theorem can be proved by a method which assumes nothing but the definition of the Γ-function, and Γ (½) = √π, from which it follows that

Role of psychopathology in elucidating the underlying neural mechanisms

2016 ◽  
Vol 33 (S1) ◽  
pp. S65-S65
Author(s):  
F. Oyebode
Keyword(s):  
Psychiatric Disorders ◽  
Subjective Experience ◽  
Right Hemisphere ◽  
Neural Mechanisms ◽  
Subjective Experiences ◽  
Precise Description ◽  
Competing Interest ◽  
Definition Of ◽  
Misidentification Syndromes

IntroductionPsychopathology is the systematic study of abnormal subjective experience and behaviour and it aims to give precise description, categorisation and definition of abnormal subjective experiences.AimI aim to demonstrate that the most appropriate approach to elucidating the biological origins of psychiatric disorders is firstly to identify elementary abnormal phenomena and then to relate these to their underlying neural mechanisms. I will exemplify this by drawing attention to studies of Delusional Misidentification Syndromes (DSM).ResultsI will show that there are impairments in face recognition memory in individuals with DSM without impairments in the recognition of emotion and that there are abnormalities of right hemisphere function and of the autonomic recognition pathways that determine sense of familiarity.ConclusionsBasic psychopathological phenomena are more likely to throw light on the basic neural mechanisms that are important in psychiatric disorders than studying disease level categories.Disclosure of interestThe author has not supplied his declaration of competing interest.

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