Notes on Set Theory, Logic, and Computation
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Proof-Of-Concept Experiments for Quantum Physics in Space
Proof-of-Concept Experiments for Quantum Physics in Space Rainer Kaltenbaek, Markus Aspelmeyer, Thomas Jennewein, Caslav Brukner and Anton Zeilinger Institut f¨ur Experimentalphysik, Universit¨at Wien, Boltzmanngasse 5, A-1090 Wien, Austria Martin Pfennigbauer and Walter R. Leeb Institut f¨ur Nachrichtentechnik und Hochfrequenztechnik, Technische Universit¨at Wien, Gußhaussstraße 25/389, A-1040 Wien, Austria Quantum physics experiments in space using entangled photons and satellites are within reach of current technology. We propose a series of fundamental quantum physics experiments that make advantageous use of the space infrastructure with specific emphasis on the satellite-based distribution of entangled photon pairs. The experiments are feasible already today and will eventually lead to a Bell-experiment over thousands of kilometers, thus demonstrating quantum correlations over distances which cannot be achieved by purely earth-bound experiments. I. INTRODUCTION Space provides a unique ”lab”-environment for quantum entanglement: In the case of massive particles, the weak gravitational interaction enables the expansion of testing fundamental quantum properties to much more massive particles than is possible today [1]. In the case of photons, the space environment allows much larger propagation distances compared to earth-bound free space experiments. This is mainly due to the lack of atmosphere and due to the fact that space links do not encounter the problem of obscured line-of-sight by unwanted objects or due to the curvature of the Earth. Quantum experiments over long distances are usually based on the transmission of photons. Earth-based transmission is limited, however, to some hundred kilometers both for optical fibers [2, 3] and for ground-to-ground free-space links [4]. -
An Introduction to Operad Theory
AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification. -
Nested Complexes and Their Polyhedral Realizations Andrei Zelevinsky
Pure and Applied Mathematics Quarterly Volume 2, Number 3 (Special Issue: In honor of Robert MacPherson, Part 1 of 3 ) 655|671, 2006 Nested Complexes and their Polyhedral Realizations Andrei Zelevinsky To Bob MacPherson on the occasion of his 60th birthday 1. Introduction This note which can be viewed as a complement to [9], presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes and associated fans and polytopes on one side, and cluster complexes and generalized associahedra introduced and studied in [7, 2] on the other side. First a very brief history (more details and references can be found in [10, 9]). Nested complexes appeared in the work by De Concini - Procesi [3] in the context of subspace arrangements. More general complexes associated with arbitrary ¯nite graphs were introduced and studied in [1] and more recently in [10]; these papers also present a construction of associated simple convex polytopes (dubbed graph associahedra in [1], and De Concini - Procesi associahedra in [10]). An even more general setup developed by Feichtner - Yuzvinsky in [6] associates a nested complex and a simplicial fan to an arbitrary ¯nite atomic lattice. In the present note we follow [5] and [9] in adopting an intermediate level of generality, and studying the nested complexes associated to building sets (see De¯nition 2.1 below). Feichtner and Sturmfels in [5] show that the simplicial fan associated to a building set is the normal fan of a simple convex polytope, while Postnikov in [9] gives an elegant construction of this polytope as a Minkowski sum of simplices. -
COMPLEXES of TREES and NESTED SET COMPLEXES 11 K+2 and Congruent to 2 Mod K, and at Least One Internal Edge
COMPLEXES OF TREES AND NESTED SET COMPLEXES EVA MARIA FEICHTNER Abstract. We exhibit an identity of abstract simplicial complexes between the well- studied complex of trees Tn and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another, in the context of nested set complexes more natural, generalization. 1. Introduction In this article we explore the connection between complexes of trees and nested set complexes of specific lattices. Nested set complexes appear as the combinatorial core in De Concini-Procesi wonderful compactifications of arrangement complements [DP1]. They record the incidence struc- ture of natural stratifications and are crucial for descriptions of topological invariants in combinatorial terms. Disregarding their geometric origin, nested set complexes can be defined for any finite meet-semilattice [FK]. Interesting connections between seemingly distant fields have been established when relating the purely oder-theoretic concept of nested sets to various contexts in geometry. See [FY] for a construction linking nested set complexes to toric geometry, and [FS] for an appearance of nested set complexes in tropical geometry. This paper presents yet another setting where nested set complexes appear in a mean- ingful way and, this time, contribute to the toolbox of topological combinatorics and combinatorial representation theory. -
Senior Times!
SEPTEMBER – NOVEMBER 2021 Paid Advertisement Supplement Endless Possibilities NEWS AND ACTIVITIES FOR ORANGE COUNTY’S OLDER ADULTS The New & Improved Senior Times! orangecountync.gov/Aging Paid Advertisement Supplement WELCOME Welcome to the Endless Possibilities News and Activities for Orange County’s Older Adults. We are excited to provide the following information about our many services, programs and opportunities for older adults. 2022-27 Master Aging Plan Table of Contents Community Planning Department on Aging Staff ...................3 From the Director .................................4 Master Aging Plan - What’s most important to you? It’s time to plan for the next five years! From the Editor ......................................5 July 1, 2021 marked the beginning of our planning year for the 2022-27 Master Aging Plan. You'll Want to Know ..............................6 As we go to print with this publication we are out in the community sharing a survey to hear about the issues that you are concerned about for the next five years. Thank you to News.................................................7-12 the many residents that have taken the time to complete the survey. In October we will Volunteer Connect (VC 55+) ......... 13-14 be hosting community engagement events across the county to share with you what we Art Classes .....................................15-16 heard. These drop-in events will provide you with another opportunity to further share Athletic Activities & Lessons...........17-19 your thoughts and to engage with the MAP workgroup leaders. We have seven workgroups Dance, Music & Theatre .................20-21 (Social Participation, Community Supports and Health Services, Transportation, Housing, Educational Opportunities .............22-27 Civic Participation and Employment, Outdoor Spaces, and Communication). -
Binary Integer Programming and Its Use for Envelope Determination
Binary Integer Programming and its Use for Envelope Determination By Vladimir Y. Lunin1,2, Alexandre Urzhumtsev3,† & Alexander Bockmayr2 1 Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 140292 Russia 2 LORIA, UMR 7503, Faculté des Sciences, Université Henri Poincaré, Nancy I, 54506 Vandoeuvre-les-Nancy, France; [email protected] 3 LCM3B, UMR 7036 CNRS, Faculté des Sciences, Université Henri Poincaré, Nancy I, 54506 Vandoeuvre-les-Nancy, France; [email protected] † to whom correspondence must be sent Abstract The density values are linked to the observed magnitudes and unknown phases by a system of non-linear equations. When the object of search is a binary envelope rather than a continuous function of the electron density distribution, these equations can be replaced by a system of linear inequalities with respect to binary unknowns and powerful tools of integer linear programming may be applied to solve the phase problem. This novel approach was tested with calculated and experimental data for a known protein structure. 1. Introduction Binary Integer Programming (BIP in what follows) is an approach to solve a system of linear inequalities in binary unknowns (0 or 1 in what follows). Integer programming has been studied in mathematics, computer science, and operations research for more than 40 years (see for example Johnson et al., 2000 and Bockmayr & Kasper, 1998, for a review). It has been successfully applied to solve a huge number of large-scale combinatorial problems. The general form of an integer linear programming problem is max { cTx | Ax ≤ b, x ∈ Zn } (1.1) with a real matrix A of a dimension m by n, and vectors c ∈ Rn, b ∈ Rm, cTx being the scalar product of the vectors c and x. -
Eric W. Hagedorn Curriculum Vitae
Eric W. Hagedorn Curriculum Vitae Areas of Specialization: Medieval Philosophy Areas of Competence: Metaphysics; Early Modern Philosophy; Philosophy of Religion; Philosophy of Mind; Logic Employment St. Norbert College Associate Professor Philosophy (August 2018-Present) Assistant Professor of Philosophy (January 2014 – August 2018) Teaching Fellow of Philosophy (August 2012 – December 2013) Education University of Notre Dame Ph.D. in Philosophy (January 2013) M.A. in Philosophy (January 2008) Iowa State University B.S. with Honors in Computer Science and Philosophy (May 2004) Journal Articles and Book Chapters “Thomas Aquinas through the 1350s,” in The Cambridge Companion to Medieval Ethics, ed. Thomas Williams (forthcoming). “Ockham’s Scientia Argument for Mental Language,” Oxford Studies in Medieval Philosophy 3 (2015): 145-168. "Is Anyone Else Thinking My Thoughts?: Aquinas's Response to the Too-Many Thinkers Problem," Proceedings of the American Catholic Philosophical Association 84 (2010): 275-86. Book Reviews Review of John Duns Scotus, On Being and Cognition: Ordinatio I.3, John van den Bercken (tr.) in International Journal for the Philosophy of Religion (forthcoming). Review of Claude Panaccio, Mental Language: From Plato to William of Ockham in Notre Dame Philosophical Reviews (August 2017). Review of Jari Kaukua and Tomas Ekenberg (eds.), Subjectivity and Selfhood in Medieval and Early Modern Philosophy in Notre Dame Philosophical Reviews (September 2016). Review of Sander W. de Boer, The Science of the Soul: The Commentary -
Making a Faster Curry with Extensional Types
Making a Faster Curry with Extensional Types Paul Downen Simon Peyton Jones Zachary Sullivan Microsoft Research Zena M. Ariola Cambridge, UK University of Oregon [email protected] Eugene, Oregon, USA [email protected] [email protected] [email protected] Abstract 1 Introduction Curried functions apparently take one argument at a time, Consider these two function definitions: which is slow. So optimizing compilers for higher-order lan- guages invariably have some mechanism for working around f1 = λx: let z = h x x in λy:e y z currying by passing several arguments at once, as many as f = λx:λy: let z = h x x in e y z the function can handle, which is known as its arity. But 2 such mechanisms are often ad-hoc, and do not work at all in higher-order functions. We show how extensional, call- It is highly desirable for an optimizing compiler to η ex- by-name functions have the correct behavior for directly pand f1 into f2. The function f1 takes only a single argu- expressing the arity of curried functions. And these exten- ment before returning a heap-allocated function closure; sional functions can stand side-by-side with functions native then that closure must subsequently be called by passing the to practical programming languages, which do not use call- second argument. In contrast, f2 can take both arguments by-name evaluation. Integrating call-by-name with other at once, without constructing an intermediate closure, and evaluation strategies in the same intermediate language ex- this can make a huge difference to run-time performance in presses the arity of a function in its type and gives a princi- practice [Marlow and Peyton Jones 2004]. -
Self-Similarity in the Foundations
Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . -
3.2 Supplement
Introduction to Functional Analysis Chapter 3. Major Banach Space Theorems 3.2. Baire Category Theorem—Proofs of Theorems June 12, 2021 () Introduction to Functional Analysis June 12, 2021 1 / 6 Table of contents 1 Proposition 3.1. The Nested Set Theorem 2 Proposition 3.2. Baire’s Theorem 3 Corollary 3.3. Dual Form of Baire’s Theorem () Introduction to Functional Analysis June 12, 2021 2 / 6 Next, suppose both x, y ∈ Fn for all n ∈ N. Then kx − yk ≤ diam(Fn) for all n ∈ N and since diam(Fn) → 0, then kx − yk = 0, or x = y, establishing uniqueness. Proposition 3.1. The Nested Set Theorem Proposition 3.1. The Nested Set Theorem Proposition 3.1. The Nested Set Theorem. Given a sequence F1 ⊇ F2 ⊇ F3 ⊇ · · · of closed nonempty sets in a Banach space such that diam(Fn) → 0, there is a unique point that is in Fn for all n ∈ N. Proof. Choose some xn ∈ Fn for each n ∈ N. Since diam(Fn) → 0, for all ε > 0 there exists N ∈ N such that if n ≥ N then diam(Fn) < ε. Since the Fn are nested, then for all m, n ≥ N, we have kxn − xmk < ε, and so (xn) is a Cauchy sequence. Since we are in a Banach space, there is x such that (xn) → x. Now for n ∈ N, the sequence (xn, xn+1, xn+2,...) ⊆ Fn is convergent to x and since Fn is closed then Fn = F n. So x ∈ Fn by Theorem 2.2.A(iii). That is x ∈ Fn for all n ∈ N. -
Complexes of Trees and Nested Set Complexes
Pacific Journal of Mathematics COMPLEXES OF TREES AND NESTED SET COMPLEXES EVA MARIA FEICHTNER Volume 227 No. 2 October 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 227, No. 2, 2006 COMPLEXES OF TREES AND NESTED SET COMPLEXES EVA MARIA FEICHTNER We exhibit an identity of abstract simplicial complexes between the well- studied complex of trees Tn and the reduced minimal nested set complex of the partition lattice. We conclude that the order complex of the partition lattice can be obtained from the complex of trees by a sequence of stellar subdivisions. We provide an explicit cohomology basis for the complex of trees that emerges naturally from this context. Motivated by these results, we review the generalization of complexes of trees to complexes of k-trees by Hanlon, and we propose yet another generalization, more natural in the context of nested set complexes. 1. Introduction In this article we explore the connection between complexes of trees and nested set complexes of specific lattices. Nested set complexes appear as the combinatorial core in De Concini and Pro- cesi’s [1995] wonderful compactifications of arrangement complements. They record the incidence structure of natural stratifications and are crucial for descrip- tions of topological invariants in combinatorial terms. Disregarding their geometric origin, nested set complexes can be defined for any finite meet-semilattice [Feicht- ner and Kozlov 2004]. Interesting connections between seemingly distant fields have been established when relating the purely order-theoretic concept of nested sets to various contexts in geometry. See [Feichtner and Yuzvinsky 2004] for a construction linking nested set complexes to toric geometry, and [Feichtner and Sturmfels 2005] for an appearance of nested set complexes in tropical geometry. -
Solving Multiclass Learning Problems Via Error-Correcting Output Codes
Journal of Articial Intelligence Research Submitted published Solving Multiclass Learning Problems via ErrorCorrecting Output Co des Thomas G Dietterich tgdcsorstedu Department of Computer Science Dearborn Hal l Oregon State University Corval lis OR USA Ghulum Bakiri ebisaccuobbh Department of Computer Science University of Bahrain Isa Town Bahrain Abstract Multiclass learning problems involve nding a denition for an unknown function f x whose range is a discrete set containing k values ie k classes The denition is acquired by studying collections of training examples of the form hx f x i Existing ap i i proaches to multiclass learning problems include direct application of multiclass algorithms such as the decisiontree algorithms C and CART application of binary concept learning algorithms to learn individual binary functions for each of the k classes and application of binary concept learning algorithms with distributed output representations This pap er compares these three approaches to a new technique in which errorcorrecting co des are employed as a distributed output representation We show that these output representa tions improve the generalization p erformance of b oth C and backpropagation on a wide range of multiclass learning tasks We also demonstrate that this approach is robust with resp ect to changes in the size of the training sample the assignment of distributed represen tations to particular classes and the application of overtting avoidance techniques such as decisiontree pruning Finally we show thatlike