An Overview of the Proof

2019 ◽  
pp. 3-22
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Motivic Cohomology ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Definition Of ◽  
Special Case

This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Large Scale ◽  
Residue Theorem ◽  
Geometric Constructions ◽  
Complete Proof ◽  
Motivic Cohomology ◽  
Symmetric Powers ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Many Sources ◽  
First Time

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

scholarly journals The Structure of n Harmonic Points and Generalization of Desargues’ Theorems

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1018
Author(s):  
Xhevdet Thaqi ◽  
Ekrem Aljimi
Keyword(s):  
Fourth Harmonic ◽  
New Findings ◽  
Rank 2 ◽  
Definition Of ◽  
Special Case

: In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective complete n-points, which are used to implement the definition of the generalization of harmonic points. We present new findings regarding the uniquely determined and constructed sets of H-points and their structure. The well-known fourth harmonic points represent the special case (n = 4) of the sets of H-points of rank 2, which is indicated by P42.

Computer Simulations

2019 ◽  
pp. 9-20
Author(s):  
Paul Humphreys
Keyword(s):  
Computer Simulation ◽  
Numerical Methods ◽  
Computer Simulations ◽  
Scientific Progress ◽  
Computational Science ◽  
Working Definition ◽  
Mathematical Techniques ◽  
Definition Of ◽  
Special Case ◽  
Numerical Treatments

The need to solve analytically intractable models has led to the rise of a new kind of science, computational science, of which computer simulations are a special case. It is noted that the development of novel mathematical techniques often drives scientific progress and that even relatively simple models require numerical treatments. A working definition of a computer simulation is given and the relation of simulations to numerical methods is explored. Examples where computational methods are unavoidable are provided. Some epistemological consequences for philosophy of science are suggested and the need to take into account what is possible in practice is emphasized.

Introduction

2020 ◽  
pp. 1-6
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Number Fields ◽  
Important Special Case ◽  
Langlands Correspondence ◽  
Key Innovation ◽  
Perfectoid Spaces ◽  
Definition Of ◽  
Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

Notes on general topology

1965 ◽  
Vol 61 (4) ◽  
pp. 877-878 ◽  
Author(s):  
A. J. Ward
Keyword(s):  
General Topology ◽  
Standard Definition ◽  
Definition Of ◽  
Image Position ◽  
Special Case ◽  
Close Parallelism

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there existsuch that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

Game genesis of justice in the teachings of Huizinga

2020 ◽  
Vol 7 (1) ◽  
pp. 59-63
Author(s):  
Yury A. Tsvetkov
Keyword(s):  
Human Culture ◽  
Religious Cult ◽  
Points Of View ◽  
The Law ◽  
Johan Huizinga ◽  
Definition Of ◽  
Special Case

The article presents the concept of the game origin of justice, developed by the Dutch historian and philosopher Johan Huizinga, in the context of the general teaching about human culture as a game. From the work of the historian, the game signs are distinguished, and the definition of its concept is formulated. The highlighted game signs correlate with the justice signs. The interpretation of some proto-legal phenomena and statements about their gaming origin are compared with the points of view of other legal historians, namely, J. Davi and V. Ehrenberg. This paper presents the author's interpretation in relation to contemporary developments in the law. An explanation is given for why the theory about the game origin of justice has not received support and development in the lawyers work. The identification of justice with a religious cult is carried out through similar gaming practices. The paper concludes by stating that there are direct genetic links among the game, justice, and religious worship. It is hypothesized that the theory about the game origin of justice can be considered a special case of a higher-level theory about the origin of state and law from the game.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

On unrolled Hopf algebras

2018 ◽  
Vol 27 (10) ◽  
pp. 1850053
Author(s):  
Nicolás Andruskiewitsch ◽  
Christoph Schweigert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Nichols Algebra ◽  
Small Quantum ◽  
Definition Of ◽  
Special Case

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra [Formula: see text] of a Yetter–Drinfeld module [Formula: see text] on which a Lie algebra [Formula: see text] acts by biderivations. As a special case, we find unrolled versions of the small quantum group.

scholarly journals Economic Nature of the Phenomenon “Reebie” in Russian Student Community

2019 ◽  
pp. 1-9
Author(s):  
Marina V. Ryzhkova ◽  
Darja V. Alimova
Keyword(s):  
Special Form ◽  
Educational Process ◽  
Entrepreneurial Activity ◽  
Student Community ◽  
Depth Interview ◽  
Definition Of ◽  
Economic Nature ◽  
Economic Goods ◽  
The Relationship ◽  
Special Case

The article is devoted to the problems of behavioral economics in terms of formation of the attitude and perception of goods with zero price in their special form – “freebie” (or in Russian – “haljava”) as a special form of free (or almost free) good. The study showed the relationship between economic and non-economic goods. The definition of “freebie” is given as a situation of receiving a good in which an individual (recipient) bears zero or insignificant (inconspicuous) economic or physical costs with a perceived high assessment of the usefulness of the good. Three situations were considered: the recipient of a good is a consumer, an employee and a special case of employee – a student obtaining grades in the educational process. Market surpluses in these situations were analyzed in terms of “freebie” and “pure freebie”. An in-depth interview was conducted among students which revealed that 95% of respondents drew a parallel between “freebie” and luck but “freebie” can be prepared to. An interesting finding is the fact that parents’ money is perceived starting from the third year as a “freebie”, while in younger courses it perceived as the help of parents. When studying the issue of morality in a situation “freebie”, it turned out that if a “freebie” does not harm anyone, then this phenomenon is allowed and, moreover, is compared with entrepreneurial activity. Such phenomena as “free money” and “freeloader” as a stable “free” strategy were also analyzed. The latter was negatively judged by respondents. In conclusion, questions are raised for further research of the phenomenon

scholarly journals DEVELOPMENT OF A CLASS OF SYSTEM AND RETURN SYSTEM MATRIXES PROVIDING INCREASE IN NOISE IMMUNITY OF SPECTRALLY EFFECTIVE MODULATION SCHEMES ON BASIS OF ORTHOGONAL CODING

2018 ◽  
pp. 75-79 ◽  
Author(s):  
A. V. Rabin ◽  
S. V. Michurin ◽  
V. A. Lipatnikov
Keyword(s):  
Signal To Noise Ratio ◽  
Inverse System ◽  
System Matrix ◽  
Synthesis Algorithm ◽  
Convolutional Coding ◽  
Integer Coefficients ◽  
Orthogonal Codes ◽  
Definition Of ◽  
Effective Use ◽  
Special Case

In work it is proposed in the digital systems of messages transmission for noise immunity's increase with the fixed code rate to use an additional coding called by the authors orthogonal. The way of a definition of orthogonal codes is presented, the synthesis algorithm of system and inverse system matrices of orthogonal codes is developed, and the main parameters of some matrices constructed by the offered algorithm are specified. Orthogonal coding as a special case of convolutional coding is defined by matrices, which elements are polynomials in the delay variable with integer coefficients. Code words are given by multiplication of an information polynomial by a system matrix, and decoding is performed by multiplication by an inverse system matrix. Basic ratios for orthogonal coding are given in article, and properties of system and inverse matrices are specified. Parameters of system and inverse system matrices assure additional gain in signal-to-noise ratio. This gain is got as a result of a more effective use of energy of transmitted signals. For transmission of one symbol energy of several symbols is accumulated.

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