Complements on adic spaces

2020 ◽  
pp. 35-40
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Cartier Divisor ◽  
General Criterion ◽  
Good Theory ◽  
Closed Immersion ◽  
Coherent Sheaves ◽  
Uniformity Condition

This chapter analyzes a collection of complements in the theory of adic spaces. These complements include adic morphisms, analytic adic spaces, and Cartier divisors. It turns out that there is a very general criterion for sheafyness. In general, uniformity does not guarantee sheafyness, but a strengthening of the uniformity condition does. Moreover, sheafyness, without any extra assumptions, implies other good properties. Ultimately, it is not immediately clear how to get a good theory of coherent sheaves on adic spaces. The chapter then considers Cartier divisors on adic spaces. The term closed Cartier divisor is meant to evoke a closed immersion of adic spaces.

Hawking a Good Theory

1987 ◽  
Vol 32 (10) ◽  
pp. 854-856
Author(s):  
Stephen R. Briggs
Keyword(s):  
Good Theory

Materials Issues and Characterization of Low-k Dielectric Materials

MRS Bulletin ◽  
1997 ◽  
Vol 22 (10) ◽  
pp. 49-54 ◽  
Author(s):  
E. Todd Ryan ◽  
Andrew J. McKerrow ◽  
Jihperng Leu ◽  
Paul S. Ho
Keyword(s):  
Process Integration ◽  
Dielectric Materials ◽  
Propagation Delay ◽  
Thermomechanical Properties ◽  
General Criterion ◽  
Crosstalk Noise ◽  
Total Delay ◽  
Interlayer Dielectrics ◽  
Performance Improvements ◽  
And Performance

Continuing improvement in device density and performance has significantly affected the dimensions and complexity of the wiring structure for on-chip interconnects. These enhancements have led to a reduction in the wiring pitch and an increase in the number of wiring levels to fulfill demands for density and performance improvements. As device dimensions shrink to less than 0.25 μm, the propagation delay, crosstalk noise, and power dissipation due to resistance-capacitance (RC) coupling become significant. Accordingly the interconnect delay now constitutes a major fraction of the total delay limiting the overall chip performance. Equally important is the processing complexity due to an increase in the number of wiring levels. This inevitably drives cost up by lowering the manufacturing yield due to an increase in defects and processing complexity.To address these problems, new materials for use as metal lines and interlayer dielectrics (ILDs) and alternative architectures have surfaced to replace the current Al(Cu)/SiO2 interconnect technology. These alternative architectures will require the introduction of low-dielectric-constant k materials as the interlayer dielectrics and/or low-resistivity conductors such as copper. The electrical and thermomechanical properties of SiO2 are ideal for ILD applications, and a change to material with different properties has important process-integration implications. To facilitate the choice of an alternative ILD, it is necessary to establish general criterion for evaluating thin-film properties of candidate low-k materials, which can be later correlated with process-integration problems.

Private and Public Law

2020 ◽  
pp. 574-592
Author(s):  
Thomas W. Merrill
Keyword(s):  
Law And Economics ◽  
Public Law ◽  
Private Law ◽  
General Criterion ◽  
The Public ◽  
Private Ordering ◽  
Public Rights ◽  
Private And Public ◽  
The Relationship ◽  
Derivatives Of

This chapter explores the relationship between private and public law. In civil law countries, the public-private distinction serves as an organizing principle of the entire legal system. In common law jurisdictions, the distinction is at best an implicit design principle and is used primarily as an informal device for categorizing different fields of law. Even if not explicitly recognized as an organizing principle, however, it is plausible that private and public law perform distinct functions. Private law supplies the tools that make private ordering possible—the discretionary decisions that individuals make in structuring their lives. Public law is concerned with providing public goods—broadly defined—that cannot be adequately supplied by private ordering. In the twentieth and twenty-first centuries, various schools of thought derived from utilitarianism have assimilated both private and public rights to the same general criterion of aggregate welfare analysis. This has left judges with no clear conception of the distinction between private and public law. Another problematic feature of modern legal thought is a curious inversion in which scholars who focus on fields of private law have turned increasingly to law and economics, one of the derivatives of utilitarianism, whereas scholars who concern themselves with public law are increasingly drawn to new versions of natural rights thinking, in the form of universal human rights.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

scholarly journals An Overview of Fatigue Testing Systems for Metals under Uniaxial and Multiaxial Random Loadings

Metals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 447
Author(s):  
Julian M. E. Marques ◽  
Denis Benasciutti ◽  
Adam Niesłony ◽  
Janko Slavič
Keyword(s):  
Fatigue Testing ◽  
Local State ◽  
Specimen Geometry ◽  
Present Review ◽  
General Criterion ◽  
Classification Criterion ◽  
Test Type ◽  
State Of Stress ◽  
The Relationship

This paper presents an overview of fatigue testing systems in high-cycle regime for metals subjected to uniaxial and multiaxial random loadings. The different testing systems are critically discussed, highlighting advantages and possible limitations. By identifying relevant features, the testing systems are classified in terms of type of machine (servo-hydraulic or shaker tables), specimen geometry and applied constraints, number of load or acceleration inputs needed to perform the test, type of loading acting on the specimen and resulting state of stress. Specimens with plate, cylindrical and more elaborated geometry are also considered as a further classification criterion. This review also discusses the relationship between the applied input and the resulting local state of stress in the specimen. Since a general criterion to classify fatigue testing systems for random loadings seems not to exist, the present review—by emphasizing analogies and differences among various layouts—may provide the reader with a guideline to classify future equipment.

scholarly journals A general criterion for avoiding infinite unfolding during partial deduction

1992 ◽  
Vol 11 (1) ◽  
pp. 47-79 ◽  
Author(s):  
Maurice Bruynooghe ◽  
Danny De Schreye ◽  
Bern Martens
Keyword(s):  
General Criterion ◽  
Partial Deduction

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

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