scholarly journals THE DIRICHLET HOPF ALGEBRA OF ARITHMETICS

2007 ◽  
Vol 16 (04) ◽  
pp. 379-438 ◽  
Author(s):  
BERTFRIED FAUSER ◽  
P. D. JARVIS
Keyword(s):  
Hopf Algebra ◽  
Arithmetic Functions ◽  
Algebraic Structures ◽  
Axiom A ◽  
Matrix Mechanics ◽  
Starting Point ◽  
Index Sets ◽  
Coprime Integers ◽  
Free Development ◽  
Quantum Matrix

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

scholarly journals Centrum i peryferia. Obraz Moskwy w powieści Dmitrija Głuchowskiego Tekst

2019 ◽  
pp. 195-208
Author(s):  
Aleksandra Zywert
Keyword(s):  
Russian Literature ◽  
Detective Novel ◽  
Social Diversity ◽  
The Core ◽  
Supportive Role ◽  
Starting Point ◽  
Free Development ◽  
Fundamental Value ◽  
Struggle For Recognition ◽  
City Village

Text is a novel of manners with elements of a thriller, a noir, and a detective novel (but the above-mentioned complementary elements fulfill only a supportive role, because the criminal intrigue exposed at the very beginning is treated marginally amounts to a starting point for deeper considerations of the psychological and sociological nature). Due to the peculiar presentation of the image of the Center (here: Moscow), Text fits into the vision of Moscow as the core of Russian predatory capitalism, exuberant consumerism, glitz, semblance and ruthless struggle for recognition in the ranking of successful people, which is presented in contemporary Russian literature. Its fundamental value is the fact that the realization of the author's idea is mainly due to the confrontation of megalopolis with images of the periphery, which can be regarded as satellite cities of the capital. In his perception of Russia, Glukhovsky is close to Roman Senchin and, similarly to the latter, he believes that the traditionally understood center-periphery, city-village conflict is disappearing, because eventually it turns out that (despite the spatial and social diversity) none of these places (mainly because of conspicuous regressive tendencies) does not give a person a chance for free development and self-realization.

Algebraic combinatorics and QFT

2021 ◽  
pp. 76-94
Author(s):  
Adrian Tanasa
Keyword(s):  
Hopf Algebra ◽  
Hochschild Cohomology ◽  
Algebraic Combinatorics ◽  
Fourth Section ◽  
Analytic Combinatorics ◽  
Multi Scale ◽  
Starting Point ◽  
Feynman Graphs ◽  
Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

The law of large numbers for additive arithmetic functions

1975 ◽  
Vol 78 (1) ◽  
pp. 33-71 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Law Of Large Numbers ◽  
Arithmetic Function ◽  
Sufficient Conditions ◽  
Growth Conditions ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Large Numbers ◽  
Coprime Integers ◽  
Weak Growth ◽  
Image Position

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequenciesWe shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

Algebraic Structures Associated to Orbifold Wreath Products

2010 ◽  
Vol 8 (2) ◽  
pp. 323-338
Author(s):  
Carla Farsi ◽  
Christopher Seaton
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Finite Groups ◽  
Lie Group ◽  
Wreath Products ◽  
Closed Manifold ◽  
Algebraic Structures ◽  
Present Structure ◽  
Smooth Closed Manifold ◽  
Connected Lie Group

AbstractWe present structure theorems in terms of inertial decompositions for the wreath product ring of an orbifold presented as the quotient of a smooth, closed manifold by a compact, connected Lie group acting almost freely. In particular we show that this ring admits λ-ring and Hopf algebra structures both abstractly and directly. This generalizes results known for global quotient orbifolds by finite groups.

scholarly journals Algebraic Structures on Typed Decorated Rooted Trees

2021 ◽  
Author(s):  
Loïc Foissy ◽  
◽  
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Stochastic Pdes ◽  
Algebraic Structures ◽  
Contraction Process

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

High-Power Analogues of the Turán-Kubilius Inequality, and an Application to Number Theory

1980 ◽  
Vol 32 (4) ◽  
pp. 893-907 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Number Theory ◽  
Absolute Constant ◽  
Arithmetic Function ◽  
Standard Form ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Coprime Integers ◽  
Complete Catalogue ◽  
Image Position ◽  
Complex Valued

An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we defineA standard form of the Turán-Kubilius inequality states that(1)holds for some absolute constant c1, uniformly for all complex-valued additive arithmetic functions ƒ (n), and real x ≧ 2. An inequality of this type was first established by Turán [11], [12] subject to some side conditions upon the size of │ƒ(pm)│. For the general inequality we refer to [10].This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [3], [4], and [1].

Zur Geschichte der Dienstrechtspolitik im Innenministerium

2021 ◽  
Keyword(s):  
Civil Service ◽  
Future Goals ◽  
Starting Point ◽  
The Future ◽  
History Of ◽  
Free Development ◽  
Service Policy ◽  
Administrative History

The ruptures and continuities that the German civil service policy has experienced are impressively illuminated in the contributions by Andreas Wirsching and Frieder Günther on the history of today's Department D "Civil Service" in the Federal Ministry of the Interior, for Building and Community. The starting point for this account of administrative history is the tradition of the civil service. Since 1949, there has been a continuous, but by no means conflict free development. The range of tasks in the department has been subject to considerable fluctuations over the decades. Ansgar Hollah takes a look at the future goals of service law policy against this background of administrative history. With contributions by Dr. Frieder Günther, Ansgar Hollah and Prof. Dr. Andreas Wirsching.

Crystal structure images of large gold clusters and growing crystals by HRTEM

1984 ◽  
Vol 42 ◽  
pp. 428-429
Author(s):  
L.R. Wallenberg ◽  
J.-O. Bovin ◽  
G. Schmid
Keyword(s):  
Crystal Structure ◽  
Nucleation And Growth ◽  
Close Packing ◽  
Gold Clusters ◽  
Molecular Weight Determination ◽  
Growth Mechanisms ◽  
Starting Point ◽  
Different Types ◽  
Nucleation And Growth Mechanisms ◽  
Points Of View

Metallic clusters are interesting from various points of view, e.g. as a mean of spreading expensive catalysts on a support, or following heterogeneous and homogeneous catalytic events. It is also possible to study nucleation and growth mechanisms for crystals with the cluster as known starting point.Gold-clusters containing 55 atoms were manufactured by reducing (C6H5)3PAuCl with B2H6 in benzene. The chemical composition was found to be Au9.2[P(C6H5)3]2Cl. Molecular-weight determination by means of an ultracentrifuge gave the formula Au55[P(C6H5)3]Cl6 A model was proposed from Mössbauer spectra by Schmid et al. with cubic close-packing of the 55 gold atoms in a cubeoctahedron as shown in Fig 1. The cluster is almost completely isolated from the surroundings by the twelve triphenylphosphane groups situated in each corner, and the chlorine atoms on the centre of the 3x3 square surfaces. This gives four groups of gold atoms, depending on the different types of surrounding.

Two isoprenylated flavonoids from Dorstenia psilurus activate AMPK, stimulate glucose uptake, inhibit glucose production and lower glycemia

2019 ◽  
Vol 476 (24) ◽  
pp. 3687-3704 ◽  
Author(s):  
Aphrodite T. Choumessi ◽  
Manuel Johanns ◽  
Claire Beaufay ◽  
Marie-France Herent ◽  
Vincent Stroobant ◽  
...  
Keyword(s):  
Glucose Uptake ◽  
Human Cancer ◽  
Glucose Production ◽  
Liver Mitochondria ◽  
New Drugs ◽  
Structural Isomer ◽  
Rat Liver Mitochondria ◽  
Ionization Mass ◽  
Ampk Activation ◽  
Starting Point

Root extracts of a Cameroon medicinal plant, Dorstenia psilurus, were purified by screening for AMP-activated protein kinase (AMPK) activation in incubated mouse embryo fibroblasts (MEFs). Two isoprenylated flavones that activated AMPK were isolated. Compound 1 was identified as artelasticin by high-resolution electrospray ionization mass spectrometry and 2D-NMR while its structural isomer, compound 2, was isolated for the first time and differed only by the position of one double bond on one isoprenyl substituent. Treatment of MEFs with purified compound 1 or compound 2 led to rapid and robust AMPK activation at low micromolar concentrations and increased the intracellular AMP:ATP ratio. In oxygen consumption experiments on isolated rat liver mitochondria, compound 1 and compound 2 inhibited complex II of the electron transport chain and in freeze–thawed mitochondria succinate dehydrogenase was inhibited. In incubated rat skeletal muscles, both compounds activated AMPK and stimulated glucose uptake. Moreover, these effects were lost in muscles pre-incubated with AMPK inhibitor SBI-0206965, suggesting AMPK dependency. Incubation of mouse hepatocytes with compound 1 or compound 2 led to AMPK activation, but glucose production was decreased in hepatocytes from both wild-type and AMPKβ1−/− mice, suggesting that this effect was not AMPK-dependent. However, when administered intraperitoneally to high-fat diet-induced insulin-resistant mice, compound 1 and compound 2 had blood glucose-lowering effects. In addition, compound 1 and compound 2 reduced the viability of several human cancer cells in culture. The flavonoids we have identified could be a starting point for the development of new drugs to treat type 2 diabetes.

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