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.

An Elementary Proof of Suslin Reciprocity

2005 ◽  
Vol 48 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Matt Kerr
Keyword(s):  
Elementary Proof ◽  
Function Fields ◽  
Algebraic Cycles ◽  
Important Special Case ◽  
Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

Some Spectral Properties of Polar Decompositions

1984 ◽  
Vol 36 (6) ◽  
pp. 973-985
Author(s):  
Bryan E. Cain
Keyword(s):  
Polar Decomposition ◽  
Positive Semidefinite ◽  
Unitary Matrix ◽  
List Type ◽  
Complex Matrix ◽  
Important Special Case ◽  
Definition Of ◽  
Image Position ◽  
Square Complex ◽  
Special Case

The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ1, …, λn be the eigenvalues of A. The questions are:(A) When will |λ1|, …, |λn| be the eigenvalues of P?(B) When will λ1/|λ1|, …, λn/|λn| be the eigenvalues of U?The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U2 and P commute.“Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial

Introduction

2017 ◽  
Author(s):  
Peter Cheyne
Keyword(s):  
Sustained Attention ◽  
Definition Of ◽  
External Powers ◽  
The Mind

This introductory chapter commences with a definition of contemplation as the sustained attention to the ideas of reason, which are not merely concepts in the mind, but real, external powers that constitute and order being and value, and therefore excite reverence or admiration. A contemplative, Coleridgean position is outlined as a defence in the crisis of the humanities, arguing that if Coleridge is right in asserting that ideas ‘in fact constitute … humanity’, then they must be the proper or ultimate studies of the disciplines that comprise the humanities. This focus on contemplation as the access to essential ideas explains why Coleridge progressed from, without ever abandoning, imagination to reason as his thought evolved during his lifetime. A section on ‘Contemplation: How to Get There from Here’, is followed by a descriptive bibliography of Coleridge as discussed by philosophers, intellectual historians, theologians, and philosophically minded literary scholars.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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.

scholarly journals Overall Introduction

2021 ◽  
pp. 1-7
Author(s):  
Wenzhong Shi ◽  
Michael F. Goodchild ◽  
Michael Batty ◽  
Mei-Po Kwan ◽  
Anshu Zhang
Keyword(s):  
Information Technologies ◽  
Interdisciplinary Approach ◽  
Urban Systems ◽  
Data Infrastructure ◽  
Urban Informatics ◽  
New Information ◽  
Wide Range ◽  
Urban Science ◽  
Definition Of

AbstractUrban informatics is an interdisciplinary approach to understanding, managing, and designing the city using systematic theories and methods based on new information technologies. Integrating urban science, geomatics, and informatics, urban informatics is a particularly timely way of fusing many interdisciplinary perspectives in studying city systems. This edited book aims to meet the urgent need for works that systematically introduce the principles and technologies of urban informatics. The book gathers over 40 world-leading research teams from a wide range of disciplines, who provide comprehensive reviews of the state of the art and the latest research achievements in their various areas of urban informatics. The book is organized into six parts, respectively covering the conceptual and theoretical basis of urban informatics, urban systems and applications, urban sensing, urban big data infrastructure, urban computing, and prospects for the future of urban informatics. This introductory chapter provides a definition of urban informatics and an outline of the book’s structure and scope.

Introduction and Overview

2013 ◽  
Author(s):  
Kenneth Prewitt
Keyword(s):  
Eighteenth Century ◽  
Civil Rights ◽  
Classification Scheme ◽  
First Century ◽  
The Political ◽  
Racial Classification ◽  
Civil Rights Era ◽  
Racial Hierarchy ◽  
Definition Of

This introductory chapter discusses how there was a racial classification scheme in America's first census (1790), as there was in the next twenty-two censuses, up until the present. Though the classification was altered in response to the political and intellectual fashions of the day, the underlying definition of America's racial hierarchy never escaped its origins in the eighteenth-century. Even the enormous changing of the racial landscape in the civil rights era failed to challenge a dysfunctional classification, though it did bend it to new purposes. Nor has the demographic upheaval of the present time led to much fresh thinking about how to measure America. The chapter contends that twenty-first-century statistics should not be governed by race thinking that is two and a half centuries out of date.

Drawing Down the Moon: Defining Magic in the Ancient Greco­Roman World

2019 ◽  
pp. 1-34
Author(s):  
Radcliffe G. Edmonds III
Keyword(s):  
Cultural Tradition ◽  
The Moon ◽  
Social Construct ◽  
Perceived Efficacy ◽  
Roman World ◽  
Greco Roman ◽  
The Social ◽  
Definition Of ◽  
Over Time

This introductory chapter provides a definition of magic. One of the most useful adjustments in the recent scholarship on magic has been the turn to considering magic as a dynamic social construct, instead of some particular reality. Magic is not a thing, but a way of talking. Thus, magic is a discourse pertaining to non-normative ritualized activity, in which the deviation from the norm is most often marked in terms of the perceived efficacy of the act, the familiarity of the performance within the cultural tradition, the ends for which the act is performed, or the social location of the performer. Such a discourse always has a history, since such a way of talking about things shifts over time as different people do the talking. When one speaks of “magic,” therefore, one should always explain: “magic for whom?” Any specific piece of evidence from the ancient Greco-Roman world provides an example of magic for that particular person, from one particular perspective. To speak of “magic in the ancient Greco-Roman world” is thus to refer to the whole range of things that various people in those cultures during those times could label as “magic.” The chapter then considers the act of drawing down the moon.

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