Combinatorial Species and Labelled Structures
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K-Quasiderivations
K-QUASIDERIVATIONS CALEB EMMONS, MIKE KREBS, AND ANTHONY SHAHEEN Abstract. A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss sev- eral interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extend- ing a previous result. In particular, we show that any such K- quasiderivation must be linear over R. We then discuss two previ- ously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K- quasiderivations which includes the ordinary derivative as a special case. 1. Introduction In the middle half of the twientieth century|perhaps as a reflection of the mathematical zeitgeist|Lausch, Menger, M¨uller,N¨obauerand others formulated a general axiomatic framework for the concept of the derivative. Their starting point was (usually) a composition ring, by which is meant a commutative ring R with an additional operation ◦ subject to the restrictions (f + g) ◦ h = (f ◦ h) + (g ◦ h), (f · g) ◦ h = (f ◦ h) · (g ◦ h), and (f ◦ g) ◦ h = f ◦ (g ◦ h) for all f; g; h 2 R. (See [1].) In M¨uller'sparlance [9], a K-derivation is a map D from a composition ring to itself such that D satisfies Additivity: D(f + g) = D(f) + D(g) (1) Product Rule: D(f · g) = f · D(g) + g · D(f) (2) Chain Rule D(f ◦ g) = [(D(f)) ◦ g] · D(g) (3) 2000 Mathematics Subject Classification. -
New Combinatorial Computational Methods Arising from Pseudo-Singletons Gilbert Labelle
New combinatorial computational methods arising from pseudo-singletons Gilbert Labelle To cite this version: Gilbert Labelle. New combinatorial computational methods arising from pseudo-singletons. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.247-258. hal-01185187 HAL Id: hal-01185187 https://hal.inria.fr/hal-01185187 Submitted on 19 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2008, Valparaiso-Vi~nadel Mar, Chile DMTCS proc. AJ, 2008, 247{258 New combinatorial computational methods arising from pseudo-singletons Gilbert Labelley Laboratoire de Combinatoire et d'Informatique Math´ematique,Universit´edu Qu´ebec `aMontr´eal,CP 8888, Succ. Centre-ville, Montr´eal(QC) Canada H3C3P8 Abstract. Since singletons are the connected sets, the species X of singletons can be considered as the combinatorial logarithm of the species E(X) of finite sets. In a previous work, we introduced the (rational) species Xb of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that E(X) = exp(Xb) in the context of rational species, where exp(T ) denotes the classical analytical power series for the exponential function in the variable T . -
An Introduction to Combinatorial Species
An Introduction to Combinatorial Species Ira M. Gessel Department of Mathematics Brandeis University Summer School on Algebraic Combinatorics Korea Institute for Advanced Study Seoul, Korea June 14, 2016 The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. If a structure has label set A and we have a bijection f : A B then we can replace each label a A with its image f (b) in!B. 2 1 c 1 c 7! 2 2 a a 7! 3 b 3 7! b More interestingly, it allows us to count unlabeled versions of labeled structures (unlabeled structures). If we have a bijection A A then we also get a bijection from the set of structures with! label set A to itself, so we have an action of the symmetric group on A acting on these structures. The orbits of these structures are the unlabeled structures. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions. -
Can the Arithmetic Derivative Be Defined on a Non-Unique Factorization Domain?
1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.1.2 47 6 23 11 Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain? Pentti Haukkanen, Mika Mattila, and Jorma K. Merikoski School of Information Sciences FI-33014 University of Tampere Finland [email protected] [email protected] [email protected] Timo Tossavainen School of Applied Educational Science and Teacher Education University of Eastern Finland P.O. Box 86 FI-57101 Savonlinna Finland [email protected] Abstract Given n Z, its arithmetic derivative n′ is defined as follows: (i) 0′ = 1′ =( 1)′ = ∈ − 0. (ii) If n = up1 pk, where u = 1 and p1,...,pk are primes (some of them possibly ··· ± equal), then k k ′ 1 n = n = u p1 pj−1pj+1 pk. X p X ··· ··· j=1 j j=1 An analogous definition can be given in any unique factorization domain. What about the converse? Can the arithmetic derivative be (well-)defined on a non-unique fac- torization domain? In the general case, this remains to be seen, but we answer the question negatively for the integers of certain quadratic fields. We also give a sufficient condition under which the answer is negative. 1 1 The arithmetic derivative Let n Z. Its arithmetic derivative n′ (A003415 in [4]) is defined [1, 6] as follows: ∈ (i) 0′ =1′ =( 1)′ = 0. − (ii) If n = up1 pk, where u = 1 and p1,...,pk P, the set of primes, (some of them possibly equal),··· then ± ∈ k k ′ 1 n = n = u p1 pj−1pj+1 pk. -
Day Convolution for Monoidal Bicategories
Day convolution for monoidal bicategories Alexander S. Corner School of Mathematics and Statistics University of Sheffield A thesis submitted for the degree of Doctor of Philosophy August 2016 i Abstract Ends and coends, as described in [Kel05], can be described as objects which are universal amongst extranatural transformations [EK66b]. We describe a cate- gorification of this idea, extrapseudonatural transformations, in such a way that bicodescent objects are the objects which are universal amongst such transfor- mations. We recast familiar results about coends in this new setting, providing analogous results for bicodescent objects. In particular we prove a Fubini theorem for bicodescent objects. The free cocompletion of a category C is given by its category of presheaves [Cop; Set]. If C is also monoidal then its category of presheaves can be pro- vided with a monoidal structure via the convolution product of Day [Day70]. This monoidal structure describes [Cop; Set] as the free monoidal cocompletion of C. Day's more general statement, in the V-enriched setting, is that if C is a promonoidal V-category then [Cop; V] possesses a monoidal structure via the convolution product. We define promonoidal bicategories and go on to show that if A is a promonoidal bicategory then the bicategory of pseudofunctors Bicat(Aop; Cat) is a monoidal bicategory. ii Acknowledgements First I would like to thank my supervisor Nick Gurski, who has helped guide me from the definition of a category all the way into the wonderful, and often confusing, world of higher category theory. Nick has been a great supervisor and a great friend to have had through all of this who has introduced me to many new things, mathematical and otherwise. -
Dirichlet Product of Derivative Arithmetic with an Arithmetic Function Multiplicative a PREPRINT
DIRICHLET PRODUCT OF DERIVATIVE ARITHMETIC WITH AN ARITHMETIC FUNCTION MULTIPLICATIVE A PREPRINT Es-said En-naoui [email protected] August 21, 2019 ABSTRACT We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibniz rule. The aim of this article is to calculate the Dirichlet product of this map with a function arithmetic multiplicative. 1 Introduction Barbeau [1] defined the arithmetic derivative as the function δ : N → N , defined by the rules : 1. δ(p)=1 for any prime p ∈ P := {2, 3, 5, 7,...,pi,...}. 2. δ(ab)= δ(a)b + aδ(b) for any a,b ∈ N (the Leibnitz rule) . s αi Let n a positive integer , if n = i=1 pi is the prime factorization of n, then the formula for computing the arithmetic derivative of n is (see, e.g., [1, 3])Q giving by : s α α δ(n)= n i = n (1) pi p Xi=1 pXα||n A brief summary on the history of arithmetic derivative and its generalizations to other number sets can be found, e.g., in [4] . arXiv:1908.07345v1 [math.GM] 15 Aug 2019 First of all, to cultivate analytic number theory one must acquire a considerable skill for operating with arithmetic functions. We begin with a few elementary considerations. Definition 1 (arithmetic function). An arithmetic function is a function f : N −→ C with domain of definition the set of natural numbers N and range a subset of the set of complex numbers C. Definition 2 (multiplicative function). A function f is called an multiplicative function if and only if : f(nm)= f(n)f(m) (2) for every pair of coprime integers n,m. -
Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected]
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 1-1-2014 Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: http://repository.upenn.edu/edissertations Part of the Computer Sciences Commons, and the Mathematics Commons Recommended Citation Yorgey, Brent, "Combinatorial Species and Labelled Structures" (2014). Publicly Accessible Penn Dissertations. 1512. http://repository.upenn.edu/edissertations/1512 This paper is posted at ScholarlyCommons. http://repository.upenn.edu/edissertations/1512 For more information, please contact [email protected]. Combinatorial Species and Labelled Structures Abstract The theory of combinatorial species was developed in the 1980s as part of the mathematical subfield of enumerative combinatorics, unifying and putting on a firmer theoretical basis a collection of techniques centered around generating functions. The theory of algebraic data types was developed, around the same time, in functional programming languages such as Hope and Miranda, and is still used today in languages such as Haskell, the ML family, and Scala. Despite their disparate origins, the two theories have striking similarities. In particular, both constitute algebraic frameworks in which to construct structures of interest. Though the similarity has not gone unnoticed, a link between combinatorial species and algebraic data types has never been systematically explored. This dissertation lays the theoretical groundwork for a precise—and, hopefully, useful—bridge bewteen the two theories. One of the key contributions is to port the theory of species from a classical, untyped set theory to a constructive type theory. This porting process is nontrivial, and involves fundamental issues related to equality and finiteness; the recently developed homotopy type theory is put to good use formalizing these issues in a satisfactory way. -
A COMBINATORIAL THEORY of FORMAL SERIES Preface in His
A COMBINATORIAL THEORY OF FORMAL SERIES ANDRÉ JOYAL Translation and commentary by BRENT A. YORGEY This is an unofficial translation of an article that appeared in an Elsevier publication. Elsevier has not endorsed this translation. See https://www.elsevier.com/about/our-business/policies/ open-access-licenses/elsevier-user-license. André Joyal. Une théorie combinatoire des séries formelles. Advances in mathematics, 42 (1):1–82, 1981 Preface In his classic 1981 paper Une théorie combinatoire des séries formelles (A com- binatorial theory of formal series), Joyal introduces the notion of (combinatorial) species, which has had a wide-ranging influence on combinatorics. During the course of researching my PhD thesis on the intersection of combinatorial species and pro- gramming languages, I read a lot of secondary literature on species, but not Joyal’s original paper—it is written in French, which I do not read. I didn’t think this would be a great loss, since I supposed the material in his paper would be well- covered elsewhere, for example in the textbook by Bergeron et al. [1998] (which thankfully has been translated into English). However, I eventually came across some questions to which I could not find answers, only to be told that the answers were already in Joyal’s paper [Trimble, 2014]. Somewhat reluctantly, I found a copy and began trying to read it, whereupon I discovered two surprising things. First, armed with a dictionary and Google Translate, reading mathematical French is not too difficult (even for someone who does not know any French!)— though it certainly helps if one already understands the mathematics. -
Ends and Coends
THIS IS THE (CO)END, MY ONLY (CO)FRIEND FOSCO LOREGIAN† Abstract. The present note is a recollection of the most striking and use- ful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise. Contents Introduction. 1 1. Dinaturality, extranaturality, co/wedges. 3 2. Yoneda reduction, Kan extensions. 13 3. The nerve and realization paradigm. 16 4. Weighted limits 21 5. Profunctors. 27 6. Operads. 33 Appendix A. Promonoidal categories 39 Appendix B. Fourier transforms via coends. 40 References 41 Introduction. The purpose of this survey is to familiarize the reader with the so-called co/end calculus, gathering a series of examples of its application; the author would like to stress clearly, from the very beginning, that the material presented here makes arXiv:1501.02503v2 [math.CT] 9 Feb 2015 no claim of originality: indeed, we put a special care in acknowledging carefully, where possible, each of the many authors whose work was an indispensable source in compiling this note. -
Arxiv:2011.12106V1 [Math.AT] 24 Nov 2020 Over the Graded Ring E∗ := Π∗(E)
FLAT REPLACEMENTS OF HOMOLOGY THEORIES DANIEL SCHÄPPI Abstract. To a homology theory one can associate an additive site and a new homological functor with values in the category of additive sheaves on that site. If this category of sheaves can be shown to be equivalent to a category of comodules of a Hopf algebroid, then we obtain a new homology theory by composing with the underlying module functor. This new homology theory is always flat and we call it a flat replacement of the original theory. For example, Pstrągowski has shown that complex cobordism is a flat replacement of singular homology. In this article we study the basic properties of the sites associated to homology theories and we prove an existence theorem for flat replacements. Contents 1. Introduction 1 Acknowledgements 3 2. Grothendieck tensor categories associated to homology theories 3 2.1. Coefficient categories 3 2.2. The additive site of a homology theory 12 2.3. Flat replacements 17 3. Locally free objects 24 3.1. Preliminaries 24 3.2. Recollections about Adams algebras 25 3.3. Recognizing locally free objects 27 4. Examples of homology theories with flat replacements 32 4.1. Detecting duals 32 4.2. Existence of flat replacements 36 4.3. Locally free objects in module categories 41 References 44 1. Introduction Let SH denote the stable homotopy category. If E is a commutative ring object in SH, then E∗X := π∗(E^X) defines a homological functor with values in modules arXiv:2011.12106v1 [math.AT] 24 Nov 2020 over the graded ring E∗ := π∗(E). -
Notions of Computation As Monoids (Extended Version)
Notions of Computation as Monoids (extended version) EXEQUIEL RIVAS MAURO JASKELIOFF Centro Internacional Franco Argentino de Ciencias de la Informaci´ony de Sistemas CONICET, Argentina FCEIA, Universidad Nacional de Rosario, Argentina September 20, 2017 Abstract There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as instances of a unifying abstract concept: monoids in monoidal categories. We demonstrate that even when working at this high level of generality one can obtain useful results. In particular, we give conditions under which one can obtain free monoids and Cayley representations at the level of monoidal categories, and we show that their concretisation results in useful constructions for monads, applicative functors, and arrows. Moreover, by taking advantage of the uniform presentation of the three notions of computation, we introduce a principled approach to the analysis of the relation between them. 1 Introduction When constructing a semantic model of a system or when structuring computer code, there are several notions of computation that one might consider. Monads [37, 38] are the most popular notion, but other notions, such as arrows [22] and, more recently, applicative functors [35] have been gaining widespread acceptance. Each of these notions of computation has particular characteristics that makes them more suitable for some tasks than for others. Nevertheless, there is much to be gained from unifying all three different notions under a single conceptual framework. In this article we show how all three of these notions of computation can be cast as monoids in monoidal categories. -
Analytic Combinatorics for Computing Seeding Probabilities
algorithms Article Analytic Combinatorics for Computing Seeding Probabilities Guillaume J. Filion 1,2 ID 1 Center for Genomic Regulation (CRG), The Barcelona Institute of Science and Technology, Dr. Aiguader 88, Barcelona 08003, Spain; guillaume.fi[email protected] or guillaume.fi[email protected]; Tel.: +34-933-160-142 2 University Pompeu Fabra, Dr. Aiguader 80, Barcelona 08003, Spain Received: 12 November 2017; Accepted: 8 January 2018; Published: 10 January 2018 Abstract: Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here, we present an approach to estimate such quantities based on the theory of analytic combinatorics. The strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. The generating function can also be used to set up a simple recurrence to compute the probabilities with greater precision. We use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors, and we show that the estimates are accurate in practical situations. More generally, this work shows novel strategies based on analytic combinatorics to compute probabilities of interest in bioinformatics.