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$Arxiv:1404.7630V2 [Math.AT] 5 Nov 2014 Asoffsae,Se[S4 P8,Vr6.Isedof Instead Ver66]$

Arxiv:1404.7630V2 [Math.AT] 5 Nov 2014 Asoffsae,Se[S4 P8,Vr6.Isedof Instead Ver66]

PROPER BASE CHANGE FOR SEPARATED LOCALLY PROPER MAPS OLAF M. SCHNÜRER AND WOLFGANG SOERGEL Abstract. We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental con- structions of sheaf theory, more precisely proper base change, pro- jection formula, and Verdier duality, can be extended from contin- uous maps between locally compact Hausdorff spaces to separated locally proper maps between arbitrary topological spaces. Contents 1. Introduction 1 2. Locally proper maps 3 3. Proper direct image 5 4. Proper base change 7 5. Derived proper direct image 9 6. Projection formula 11 7. Verdier duality 12 8. The case of unbounded derived categories 15 9. Remindersfromtopology 17 10. Reminders from sheaf theory 19 11. Representability and adjoints 21 12. Remarks on derived functors 22 References 23 arXiv:1404.7630v2 [math.AT] 5 Nov 2014 1. Introduction The proper direct image functor f! and its right derived functor Rf! are defined for any continuous map f : Y → X of locally compact Hausdorff spaces, see [KS94, Spa88, Ver66]. Instead of f! and Rf! we will use the notation f(!) and f! for these functors. They are embedded into a whole collection of formulas known as the six-functor-formalism of Grothendieck. Under the assumption that the proper direct image functor f(!) has finite cohomological dimension, Verdier proved that its ! derived functor f! admits a right adjoint f . Olaf Schnürer was supported by the SPP 1388 and the SFB/TR 45 of the DFG. Wolfgang Soergel was supported by the SPP 1388 of the DFG. 1 2 OLAFM.SCHNÜRERANDWOLFGANGSOERGEL In this article we introduce in 2.3 the notion of a locally proper map between topological spaces and show that the above results even hold for arbitrary topological spaces if all maps whose proper direct image functors are involved are locally proper and separated.
$MATH 418 Assignment #7 1. Let R, S, T Be Positive Real Numbers Such That R$

MATH 418 Assignment #7 1. Let R, S, T Be Positive Real Numbers Such That R

MATH 418 Assignment #7 1. Let r, s, t be positive real numbers such that r + s + t = 1. Suppose that f, g, h are nonnegative measurable functions on a measure space (X, S, µ). Prove that Z r s t r s t f g h dµ ≤ kfk1kgk1khk1. X s t Proof . Let u := g s+t h s+t . Then f rgsht = f rus+t. By H¨older’s inequality we have Z Z rZ s+t r s+t r 1/r s+t 1/(s+t) r s+t f u dµ ≤ (f ) dµ (u ) dµ = kfk1kuk1 . X X X For the same reason, we have Z Z s t s t s+t s+t s+t s+t kuk1 = u dµ = g h dµ ≤ kgk1 khk1 . X X Consequently, Z r s t r s+t r s t f g h dµ ≤ kfk1kuk1 ≤ kfk1kgk1khk1. X 2. Let Cc(IR) be the linear space of all continuous functions on IR with compact support. (a) For 1 ≤ p < ∞, prove that Cc(IR) is dense in Lp(IR). Proof . Let X be the closure of Cc(IR) in Lp(IR). Then X is a linear space. We wish to show X = Lp(IR). First, if I = (a, b) with −∞ < a < b < ∞, then χI ∈ X. Indeed, for n ∈ IN, let gn the function on IR given by gn(x) := 1 for x ∈ [a + 1/n, b − 1/n], gn(x) := 0 for x ∈ IR \ (a, b), gn(x) := n(x − a) for a ≤ x ≤ a + 1/n and gn(x) := n(b − x) for b − 1/n ≤ x ≤ b.
(Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008

A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding.
Laurent Schwartz (1915–2002), Volume 50, Number 9

Laurent Schwartz (1915–2002) François Treves, Gilles Pisier, and Marc Yor classics scholar and that of a mathematician. He had won the Concours Général in Latin; the Con- Biographical cours Général was and still is the most prestigious nationwide competition in France for high school- Sketch ers. Meanwhile he had become fascinated by the beauty of geometry, and, in the end, with the en- François Treves couragement of one of his professors in classics and of his uncle, the pediatrician Robert Debré, and Laurent Schwartz died in Paris on the 4th of July despite the rather unhelpful attitude of Hadamard, 2002. He was born in Paris on March 5, 1915. His dismayed that the sixteen-year-old Laurent was father, Anselme Schwartz, had been born in 1872 not acquainted with the Riemann zeta function, he in a small Alsatian town soon after the annexation of Alsatia by Germany. A fervent patriot, Anselme tried for admission to the science classes of the Schwartz had emigrated to France at the age of four- École Normale Supérieure (ENS), the most selective teen (before speaking French). In Paris he man- and most scholarly oriented of the “Grandes aged to carry out successful studies in medicine and Écoles”. He underwent the rather grueling two-year was to become a prominent surgeon in France in training (“hypotaupe”, followed by “taupe”, the the years between the two world wars. In 1907, tunnelling “submole” and “mole” years, so to speak) having just become the first Jewish surgeon ever preparatory to entrance to the ENS, where he was officially employed in a Paris hospital, Anselme admitted in 1934.
Mathematics Support Class Can Succeed and Other Projects To

Mathematics Support Class Can Succeed and Other Projects to Assist Success Georgia Department of Education Divisions for Special Education Services and Supports 1870 Twin Towers East Atlanta, Georgia 30334 Overall Content • Foundations for Success (Math Panel Report) • Effective Instruction • Mathematical Support Class • Resources • Technology Georgia Department of Education Kathy Cox, State Superintendent of Schools FOUNDATIONS FOR SUCCESS National Mathematics Advisory Panel Final Report, March 2008 Math Panel Report Two Major Themes Curricular Content Learning Processes Instructional Practices Georgia Department of Education Kathy Cox, State Superintendent of Schools Two Major Themes First Things First Learning as We Go Along • Positive results can be • In some areas, adequate achieved in a research does not exist. reasonable time at • The community will learn accessible cost by more later on the basis of addressing clearly carefully evaluated important things now practice and research. • A consistent, wise, • We should follow a community-wide effort disciplined model of will be required. continuous improvement. Georgia Department of Education Kathy Cox, State Superintendent of Schools Curricular Content Streamline the Mathematics Curriculum in Grades PreK-8: • Focus on the Critical Foundations for Algebra – Fluency with Whole Numbers – Fluency with Fractions – Particular Aspects of Geometry and Measurement • Follow a Coherent Progression, with Emphasis on Mastery of Key Topics Avoid Any Approach that Continually Revisits Topics without Closure Georgia Department of Education Kathy Cox, State Superintendent of Schools Learning Processes • Scientific Knowledge on Learning and Cognition Needs to be Applied to the classroom to Improve Student Achievement: – To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills.
Stone-Weierstrass Theorems for the Strict Topology

STONE-WEIERSTRASS THEOREMS FOR THE STRICT TOPOLOGY CHRISTOPHER TODD 1. Let X be a locally compact Hausdorff space, E a (real) locally convex, complete, linear topological space, and (C*(X, E), ß) the locally convex linear space of all bounded continuous functions on X to E topologized with the strict topology ß. When E is the real num- bers we denote C*(X, E) by C*(X) as usual. When E is not the real numbers, C*(X, E) is not in general an algebra, but it is a module under multiplication by functions in C*(X). This paper considers a Stone-Weierstrass theorem for (C*(X), ß), a generalization of the Stone-Weierstrass theorem for (C*(X, E), ß), and some of the immediate consequences of these theorems. In the second case (when E is arbitrary) we replace the question of when a subalgebra generated by a subset S of C*(X) is strictly dense in C*(X) by the corresponding question for a submodule generated by a subset S of the C*(X)-module C*(X, E). In what follows the sym- bols Co(X, E) and Coo(X, E) will denote the subspaces of C*(X, E) consisting respectively of the set of all functions on I to £ which vanish at infinity, and the set of all functions on X to £ with compact support. Recall that the strict topology is defined as follows [2 ] : Definition. The strict topology (ß) is the locally convex topology defined on C*(X, E) by the seminorms 11/11*,,= Sup | <¡>(x)f(x)|, xçX where v ranges over the indexed topology on E and </>ranges over Co(X).
Combinatorics of Square-Tiled Surfaces and Geometry of Moduli Spaces

COMBINATORICS OF SQUARE-TILED SURFACES AND GEOMETRY OF MODULI SPACES VINCENT DELECROIX, QUENTIN GENDRON, AND CARLOS MATHEUS Abstract. This text corresponds to the lecture notes of a minicourse delivered from August 16 to 20, 2021 at the IMPA{ICTP online summer school \Aritmética,Grupos y Análisis(AGRA) IV". In particular, we discuss the same topics from our minicourse, namely, the basic theory of origamis and its connections to the calculation of Masur{Veech volumes of moduli spaces of translation surfaces. Contents 1. Square-tiled surfaces, Veech groups and arithmetic Teichmüllercurves 2 1.1. Basic definitions and some examples 2 1.2. Action of SL(2; Z) and Veech groups 4 1.3. Arithmetic Teichmüllercurves 5 1.4. Exercises 5 2. Some properties of Veech groups 7 2.1. Characteristic origamis 7 2.2. Origamis with prescribed Veech groups 8 2.3. Non-congruence Veech groups within H(2) 8 2.4. Exercises 10 3. Translation surfaces, moduli spaces and Masur{Veech volumes 11 3.1. Translation surfaces and GL(2; R)-action 11 3.2. Strata of translation surfaces 11 3.3. Masur-Veech volume and enumeration of square-tiled surfaces 12 3.4. Reduced versus non-reduced origamis 12 3.5. The Masur-Veech volume of H(2) 13 3.6. Further readings 15 4. Enumeration of square-tiled surfaces via quasi-modular forms 17 4.1. Frobenius formula and the generating function of square-tiled surfaces 17 4.2. Bloch-Okounkov and Kerov-Olshanski theorems 17 4.3. Computations 17 5. Classification of SL(2; Z)-orbits of origamis 18 Date: August 17, 2021.
$Learning Support Competencies-Math$

Learning Support Competencies-Math

Mathematics Learning Support Curriculum The mathematics curriculum sub-committee was tasked with developing a curriculum designed to prepare students to be successful in their first college level mathematics course. To satisfy federal guidelines for financial aid, alignment of curriculum with the Department of Education high school mathematics curriculum standards established a floor for the mathematics learning support curriculum. The mathematics sub- committee was directed to use the ACT college readiness standards and alignment with a mathematics ACT benchmark sub score of 22 as guidelines to set a ceiling for curriculum designated for learning support credit. Based on the statewide curriculum survey, comparisons’ with ACT readiness standards as well as the committee members’ teaching experience, the mathematics curriculum sub-committee recommends the following: Primary Recommendations: 1) Mathematics curriculum should be defined and organized into competencies points for assessment, placement, and transferability. 2) In order to prepare students to enter into non-algebra intensive college-level math courses, the committee recommends that students demonstrate mastery of five competencies points prior to enrollment in any college level math course. 3) Additional curriculum is necessary to prepare students for algebra intensive courses and should be provided at the college level. The students will o Develop study skills for success . Understand students’ learning styles. Use the textbook, software and note taking to assist the learning process . Read and follow instructions. o Communicate mathematically (with appropriate vocabulary) . Articulate the process of finding and interpreting the meaning of solutions. Use symbols, diagrams, graphs and words to reason logically and form appropriate implications. o Be problem solvers . Experiment with problem solving strategies.
The Top Mathematics Award

Fields told me and which I later verified in Sweden, namely, that Nobel hated the mathematician Mittag- Leffler and that mathematics would not be one of the do- mains in which the Nobel prizes would The Top Mathematics be available." Award Whatever the reason, Nobel had lit- tle esteem for mathematics. He was Florin Diacuy a practical man who ignored basic re- search. He never understood its impor- tance and long term consequences. But Fields did, and he meant to do his best John Charles Fields to promote it. Fields was born in Hamilton, Ontario in 1863. At the age of 21, he graduated from the University of Toronto Fields Medal with a B.A. in mathematics. Three years later, he fin- ished his Ph.D. at Johns Hopkins University and was then There is no Nobel Prize for mathematics. Its top award, appointed professor at Allegheny College in Pennsylvania, the Fields Medal, bears the name of a Canadian. where he taught from 1889 to 1892. But soon his dream In 1896, the Swedish inventor Al- of pursuing research faded away. North America was not fred Nobel died rich and famous. His ready to fund novel ideas in science. Then, an opportunity will provided for the establishment of to leave for Europe arose. a prize fund. Starting in 1901 the For the next 10 years, Fields studied in Paris and Berlin annual interest was awarded yearly with some of the best mathematicians of his time. Af- for the most important contributions ter feeling accomplished, he returned home|his country to physics, chemistry, physiology or needed him.
Unify Manifold System Hot Runner Installation Manual

Unify Manifold System Hot Runner Installation Manual Original Instructions v 2.2 — March 2021 Unify Manifold System Issue: v 2.2 — March 2021 Document No.: 7593574 This product manual is intended to provide information for safe operation and/or maintenance. Husky reserves the right to make changes to products in an effort to continually improve the product features and/or performance. These changes may result in different and/or additional safety measures that are communicated to customers through bulletins as changes occur. This document contains information which is the exclusive property of Husky Injection Molding Systems Limited. Except for any rights expressly granted by contract, no further publication or commercial use may be made of this document, in whole or in part, without the prior written permission of Husky Injection Molding Systems Limited. Notwithstanding the foregoing, Husky Injection Molding Systems Limited grants permission to its customers to reproduce this document for limited internal use only. Husky® product or service names or logos referenced in these materials are trademarks of Husky Injection Molding Systems Ltd. and may be used by certain of its affiliated companies under license. All third-party trademarks are the property of the respective third-party and may be protected by applicable copyright, trademark or other intellectual property laws and treaties. Each such third- party expressly reserves all rights into such intellectual property. ©2016 – 2021 Husky Injection Molding Systems Ltd. All rights reserved. ii Hot Runner Installation Manual v 2.2 — March 2021 General Information Telephone Support Numbers North America Toll free 1-800-465-HUSKY (4875) Europe EC (most countries) 008000 800 4300 Direct and Non-EC + (352) 52115-4300 Asia Toll Free 800-820-1667 Direct: +86-21-3849-4520 Latin America Brazil +55-11-4589-7200 Mexico +52-5550891160 option 5 For on-site service, contact your nearest Husky Regional Service and Sales office.
Tempered Distributions and the Fourier Transform

CHAPTER 1 Tempered distributions and the Fourier transform Microlocal analysis is a geometric theory of distributions, or a theory of geomet- ric distributions. Rather than study general distributions { which are like general continuous functions but worse { we consider more specific types of distributions which actually arise in the study of differential and integral equations. Distri- butions are usually defined by duality, starting from very \good" test functions; correspondingly a general distribution is everywhere \bad". The conormal dis- tributions we shall study implicitly for a long time, and eventually explicitly, are usually good, but like (other) people have a few interesting faults, i.e. singulari- ties. These singularities are our principal target of study. Nevertheless we need the general framework of distribution theory to work in, so I will start with a brief in- troduction. This is designed either to remind you of what you already know or else to send you off to work it out. (As noted above, I suggest Friedlander's little book [4] - there is also a newer edition with Joshi as coauthor) as a good introduction to distributions.) Volume 1 of Hörmander'streatise [8] has all that you would need; it is a good general reference. Proofs of some of the main theorems are outlined in the problems at the end of the chapter. 1.1. Schwartz test functions To fix matters at the beginning we shall work in the space of tempered distribu- tions. These are defined by duality from the space of Schwartz functions, also called the space of test functions of rapid decrease.
Distributions: How to Think About the Dirac Delta Function

1 Distributions on R Definition 1. Let X be a topological space, let f : X ! C, then define the support of f to be the set supp(f) = fx 2 X : f(x) 6= 0g: Definition 2. Let ΩRn, then define D(Ω) ½ C1(Ω) to be the space of infinitely differentiable (partials of all orders) functions f : Rn ! R which have compact support. We can see that D(Ω) is a real vector space, an algebra under pointwise multiplication, and an ideal in C1(Ω). We can always thing of D(Ω) ½ D(Rn), and for brevity I will call D(R) = D. n n Definition 3 (Convergence in D(R )). Let f'ig be a sequence in D(R ) n and let ' 2 D(R ), then we say that f'ig ! ' if n 1. supp('i) ½ K for some compact K ½ R for all i 2 N. p p n 2. fD 'ig ! D ' uniformly for each p 2 N . This topology defined by some semi-norm? n n Theorem 4. The space D(R ) is dense in C0(R ), the space of continuous real function of compact support with topology given by uniform conver- n n gence, i.e. for any f 2 C0(R ) with supp(f) ½ U for some open U ½ R and for any " > 0, there exists a ' 2 D(Rn) with supp(') ½ U, and such that for all x 2 Rn jf(x) ¡ '(x)j < ": Definition 5. A distribution T is a continuous linear map T : D(Rn) ! C, i.e. 1. T (®' + ¯Ã) = ®T (') + ¯T (Ã) for all ®; ¯ 2 C, '; Ã 2 D.