Zero Object Resolution in Korean

Total Page:16

File Type:pdf, Size:1020Kb

Zero Object Resolution in Korean PACLIC 29 Zero Object Resolution in Korean Arum Park Seunghee Lim Munpyo Hong Dept. of German Linguistics Dept. of German Linguistics Dept. of German Linguistics & Literature, Sungkyunkwan & Literature, Sungkyunkwan & Literature, Sungkyunkwan University / University / University / 25-2, Sungkyunkwan-Ro, 25-2, Sungkyunkwan-Ro, 25-2, Sungkyunkwan-Ro, Jongno-Gu, Jongno-Gu, Jongno-Gu, Seoul, Korea Seoul, Korea Seoul, Korea [email protected] [email protected] [email protected] Zero object is the second most frequent zero Abstract pronoun type in Korean spoken text. Despite of the frequent use of zero objects, most of the previous Korean is one of the well-known „pro-drop‟ works do not deal with the zero objects in Korean. languages. When translating Korean zero object into languages in which objects have to be In this work, we focus on Korean spoken texts, overtly expressed, the resolution of zero object since zero pronouns occur more frequently than in is crucial. This paper proposes a machine a written text. Ryu (2001) showed that a zero learning method to resolve Korean zero object. pronoun rarely appears in written texts when it is We proposed 8 linguistically motivated features compared with spoken texts in Korean. For this for ML (Machine Learning). Our approach has reason, we conduct a study for Korean zero object been implemented with WEKA 3.6.10 and in Korean spoken text and try to find the linguistic evaluated by using 10-fold cross validation clues for the zero object resolution. method. The accuracy of the proposed method In the context of machine translation, the reached 73.37%. resolution of Korean zero objects could be one of the most important issues in order to translate them 1 Introduction into the target language like English and German. One of the reasons that zero objects in Korean is a Korean is one of the so-called pro-drop languages. problem in MT is that the omitted objects in Certain pronouns may be omitted and the omitted Korean have to be translated into overt objects in pronouns are often called zero pronouns. This kind target languages. Unfortunately, the majority of of pronoun also occurs in other languages, such as MT systems do not deal with this problem, because Japanese or Spanish. The omitted pronouns in most of the current commercial MT systems do not Korean can appear in subject and object position, treat the linguistic phenomena that go beyond a whereas in Spanish or Italian, they can appear only sentence level. To illustrate this issue, let's take a in subject position. A zero subject is the most look at the following example (1). frequent type of anaphoric expressions in Korean. Hong (2000) reported that about 57% of the (1) MT results from Korean(a) to German(b) pronouns are a zero subject pronoun in pronoun occurrences in Korean spoken text. (a) Korean A: 여권 i 을 분실했습니다. Corresponding author 439 29th Pacific Asia Conference on Language, Information and Computation pages 439 - 448 Shanghai, China, October 30 - November 1, 2015 Copyright 2015 by Arum Park, Seunghee Lim and Munpyo Hong PACLIC 29 a passport-OBJ lost 2 Related Works (yekwenul punsilhaysssupnita.) Zero pronouns have already been studied in other “I lost a passport.” languages, such as Japanese (e.g. Nakaiwa and Shirai, 1996; Okumura and Tamura, 1996) and Spanish (Park and Hong, 2014; Palomar et al., B: øi 다시 발급받으셔야 합니다. 2001; Ferrández and Peral, 2000). These studies again have to issue are based on the researches about anaphora (tasi palkuppatusyeya hapnita.) resolution. It has been a wide-open research field since 1970 focusing on English. Regardless of “You have to issue a passport again.” languages, similar strategies for anaphora resolution have been applied. Using linguistic (b1) German - Systran translator information is the most representative technique; constraints and preferences methods are A: Verlor den Pass. distinguished in the related works (Baldwin, 1997; B: Fragen wiederholt. Lappin and Leass, 1994; Carbonell and Brown, 1988). (b2) German - Google translator Constraints discard possible antecedents and are considered as absolute criteria. Preferences being A: Ich habe meinen Pass verloren. proposed as heuristic rules tend to be relative. B: Wir müssen neu aufgelegt zu werden. After applying constraints, if there are still unresolved candidate antecedents, preferences set The omitted object is represented by the symbol priorities among candidate antecedents. Nakaiwa ø. In this example, the Korean object is not overtly and Shirai (1996) focus on semantic and pragmatic expressed in the sentence B and it refers to constraints such as cases, modal expressions, „여권‟(yekwen, “Passport”) in sentence A. To verbal semantic attributes and conjunctions in translate the omitted object into German correctly, order to determine the reference of Japanese zero the gender and number of the antecedent „yekwen‟ pronouns. However, they proposed constraints has to be considered. Since the morphological focusing on zero subjects mainly. Therefore, it is information of „yekwen‟ is „masculine‟ and hard to apply their approach on zero object „singular‟ in German, the omitted object has to be resolution. translated as „ihn‟ considering its case. However, Centering theory (Grosz et al., 1995) is one of the object of the sentence B is not translated in the approaches using heuristic rules. It is claimed German in either MT system. Then, the results that certain entities mentioned in an utterance are would be ungrammatical in German. Therefore, the more central than others, and this property has resolution of Korean zero objects is crucial in MT been applied to determine the antecedent of the systems with Korean as a source language, when anaphor. Walker et al. (1994) applied the centering translating them into languages in which objects model on zero pronoun resolution in Japanese. Roh have to be overtly expressed. and Lee (2003) proposed a generation algorithm of In section 2 we present the related works about zero pronouns using a Cost-based Centering Model anaphora resolution and their limitation. Section 3 which considers the inference cost. It is known that explains zero objects phenomenon in Korean. We the most salient element of the given discourse is suggest the machine learning (ML) method for likely to be realized as a zero pronoun. We take Korean zero object resolution and propose 8 this into account in selecting the features for ML. features for ML method in section 4. In addition, Current anaphora resolution methods rely the effect of using ML is evaluated. Finally, the mainly on constraint and preference heuristics, conclusion is presented in section 5. which employ morpho-syntactic information or shallow semantic analysis. These methods are a deterministic algorithm which always produces the same output in a given particular condition. However, even if the condition is applied, the 440 PACLIC 29 output can be wrong. ML methods which are a Given the hierarchy of the forward looking non-deterministic algorithm have been studied on center ranking, a zero object can be interpreted as anaphora resolution (Connolly et al., 1994; Paul et the topic which is the most salient discourse entity. al., 1999). Since ML learns from data and makes The topic of the sentence contributes to discourse predictions of the most likely candidate on the data, salience and maintains discourse coherence by it can overcome the limitation of the deterministic preferring the CONTINUE transition state. The method. topic of the sentence can be detected easily in Park and Hong (2014) proposed a hybrid Korean using the topic markers „은‟(eun), „는‟(nun) approach to resolve Spanish zero subjects that integrates heuristic rules and ML in the context of and delimiters such as „도‟(to), „만‟(man). Spanish to Korean MT. Since Spanish zero Therefore, it is likely a candidate antecedent is the subjects can be restored from the verb ending, they antecedent of the zero object if the candidate has use morphological flections for verbs. After that, one of the topic markers or delimiters. We can see ML is utilized for some ambiguous cases. Unlike some examples in the following table. this work, our work deals with Korean zero object. Morphological information cannot be utilized for Speaker Korean dialogue Korean because of the difference of the two A 음식 주문 을 languages. For this reason, we use ML method 1 ordering a food-OBJ alone to determine the antecedent of the zero (umsik cumunul objects in spoken Korean. 어떻게 하는 거죠? how can 3 Zero object phenomenon in Korean ettehkey hanun kecyo?) A prominent phenomenon in Korean is the “How can I order a food?” B prevalence of zero pronouns. Unlike English, zero 저 기계 2 에서 메뉴 3 를 pronouns occur very frequently in Korean. In that machine-ADV menu-OBJ Korean, a zero subject is the most frequent type of (ce kikyeyeyse menyulul anaphoric expression. The second most frequent 선택한 후 식권 4 을 type is zero objects, especially when the direct after selecting a meal ticket-OBJ object is omitted. According to Hong (2000), when senthaykhan hu sikkwenul the direct object does not occur in spoken Korean, 뽑으세요. the rate becomes 19.1%. To resolve zero object, centering theory can be buy ppopuseyyo.) utilized. The centering theory endeavors to identify the antecedent of a (zero) pronoun using the idea “You can buy a meal ticket after selecting of the most central entity that a sentence concerns menu from that vending machine.” which tends to be expressed by a (zero) pronoun. A 아침 식사 는 11 시까지만 There are some studies attempting to apply the 5 breakfast-TOP until 11 o‟clock centering theory to anaphora resolution (Choi and (achim siksanun 11sikkaciman Lee, 1999; Hong, 2000; Hong, 2011). The forward looking center rankings for Korean are defined 되는 건가요? differently in the studies.
Recommended publications
  • Complete Objects in Categories
    Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion.
    [Show full text]
  • Research.Pdf (1003.Kb)
    PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS A Dissertation presented to the Faculty of the Graduate School, University of Missouri, Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by KILLIAN MEEHAN Dr. Calin Chindris, Dissertation Supervisor Dr. Jan Segert, Dissertation Supervisor MAY 2018 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS presented by Killian Meehan, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Associate Professor Calin Chindris Associate Professor Jan Segert Assistant Professor David C. Meyer Associate Professor Mihail Popescu ACKNOWLEDGEMENTS To my thesis advisors, Calin Chindris and Jan Segert, for your guidance, humor, and candid conversations. David Meyer, for our emphatic sharing of ideas, as well as the savage question- ing of even the most minute assumptions. Working together has been an absolute blast. Jacob Clark, Brett Collins, Melissa Emory, and Andrei Pavlichenko for conver- sations both professional and ridiculous. My time here was all the more fun with your friendships and our collective absurdity. Adam Koszela and Stephen Herman for proving that the best balm for a tired mind is to spend hours discussing science, fiction, and intersection of the two. My brothers, for always reminding me that imagination and creativity are the loci of a fulfilling life. My parents, for teaching me that the best ideas are never found in an intellec- tual vacuum. My grandparents, for asking me the questions that led me to where I am.
    [Show full text]
  • Separable Commutative Rings in the Stable Module Category of Cyclic Groups
    SEPARABLE COMMUTATIVE RINGS IN THE STABLE MODULE CATEGORY OF CYCLIC GROUPS PAUL BALMER AND JON F. CARLSON Abstract. We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. Contents Introduction1 1. Separable ring-objects4 2. The Kelly radical and the tensor6 3. The case of the group of prime order 14 4. The case of the general cyclic group 16 References 18 Introduction Since 1960 and the work of Auslander and Goldman [AG60], an algebra A over op a commutative ring R is called separable if A is projective as an A ⊗R A -module. This notion turns out to be remarkably important in many other contexts, where the module category C = R- Mod and its tensor ⊗ = ⊗R are replaced by an arbitrary tensor category (C; ⊗). A ring-object A in such a category C is separable if multiplication µ : A⊗A ! A admits a section σ : A ! A⊗A as an A-A-bimodule in C. See details in Section1. Our main result (Theorem 4.1) concerns itself with modular representation theory of finite groups: Main Theorem. Let | be a separably closed field of characteristic p > 0 and let G be a cyclic p-group. Let A be a commutative and separable ring-object in the stable category |G- stmod of finitely generated |G-modules modulo projectives.
    [Show full text]
  • Homological Algebra Lecture 1
    Homological Algebra Lecture 1 Richard Crew Richard Crew Homological Algebra Lecture 1 1 / 21 Additive Categories Categories of modules over a ring have many special features that categories in general do not have. For example the Hom sets are actually abelian groups. Products and coproducts are representable, and one can form kernels and cokernels. The notation of an abelian category axiomatizes this structure. This is useful when one wants to perform module-like constructions on categories that are not module categories, but have all the requisite structure. We approach this concept in stages. A preadditive category is one in which one can add morphisms in a way compatible with the category structure. An additive category is a preadditive category in which finite coproducts are representable and have an \identity object." A preabelian category is an additive category in which kernels and cokernels exist, and finally an abelian category is one in which they behave sensibly. Richard Crew Homological Algebra Lecture 1 2 / 21 Definition A preadditive category is a category C for which each Hom set has an abelian group structure satisfying the following conditions: For all morphisms f : X ! X 0, g : Y ! Y 0 in C the maps 0 0 HomC(X ; Y ) ! HomC(X ; Y ); HomC(X ; Y ) ! HomC(X ; Y ) induced by f and g are homomorphisms. The composition maps HomC(Y ; Z) × HomC(X ; Y ) ! HomC(X ; Z)(g; f ) 7! g ◦ f are bilinear. The group law on the Hom sets will always be written additively, so the last condition means that (f + g) ◦ h = (f ◦ h) + (g ◦ h); f ◦ (g + h) = (f ◦ g) + (f ◦ h): Richard Crew Homological Algebra Lecture 1 3 / 21 We denote by 0 the identity of any Hom set, so the bilinearity of composition implies that f ◦ 0 = 0 ◦ f = 0 for any morphism f in C.
    [Show full text]
  • Categories of Projective Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 110 (1996) 241-258 Categories of projective spaces Aurelio Carboni *, Marco Grandis Dipartimento di Matematica, Universitri di Genova, Via L.B. Alberti 4, 16132 Geneva, Italy Communicated by by F.W. Lawvere; received 1 December 1994 Abstract Starting with an abelian category d, a natural construction produces a category Pd such that, when d is an abelian category of vector spaces, llp& is the corresponding category of projective spaces. The process of forming the category P& destroys abelianess, but not completely, and the precise measure of what remains of it gives the possibility to reconstruct d out from PzZ, and allows to characterize categories of the form Pd, for an abelian d (projective categories). The characterization is given in terms of the notion of “Puppe exact category” and of an appropriate notion of “weak biproducts”. The proof of the characterization theorem relies on the theory of “additive relations”. 0. Introduction Starting with an abelian category &, a natural construction described in [5] produces a category P& such that, when & is an abelian category of vector spaces, Pd is the corresponding category of projective spaces. This construction is based on the subobject fimctor on d, which in the linear context we call “Grassmannian finctor”. Clearly, categories of the form P&’ for an abelian category ~4 are to be thought of as “categories of projective spaces”, and the natural question arises of their intrinsic characterization.
    [Show full text]
  • Math 256: Linear Algebra Dr
    Math 256: Linear Algebra Dr. Heather Moon contributing-Authors: Dr. Thomas J. Asaki, Dr. Marie Snipes, and Dr. Chris Camfield April 20, 2016 1 Contents 1 Systems of Equations 5 1.1 Review from your basic Algebra class . .5 1.1.1 Substitution . .5 1.1.2 Elimination . .5 1.2 Systems of Equations with more variables . .6 1.3 Using Matrices to solve systems of equations . .8 2 Matrix operations and Matrix Equations 12 2.1 Basic Operations on Matrices . 12 2.1.1 Matrix Addition . 13 2.1.2 Matrix Multiplication . 13 2.2 Matrix Equations and more . 14 2.3 Determinants . 18 3 Radiograpy and Tomography in Linear Algebra Lab #1 24 3.1 Grayscale Images . 24 4 Vector Spaces 26 4.1 Vectors and Vector Spaces . 27 4.2 Subspaces . 32 5 Span 35 5.1 Definition of Span . 36 5.2 Span as a Vector Space . 38 6 Linear Independence/Dependence 41 6.1 Linear Dependence Definitions . 41 6.2 Basis . 43 6.3 Dimension . 44 7 Coordinate Spaces 49 7.1 Coordinates in the Standard Basis . 49 7.2 Coordinates in Other Bases . 50 8 Transmission Radiography and Tomography A Simplified Overview 54 8.1 What is Radiography? . 54 8.1.1 Transmission Radiography . 55 8.1.2 Tomography . 55 8.2 The Incident X-ray Beam . 55 8.3 X-Ray Beam Attenuation . 56 8.4 Radiographic Energy Detection . 57 8.5 The Radiographic Transformation Operator . 58 8.6 Multiple Views and Axial Tomography . 58 8.7 Model Summary . 60 8.8 Model Assumptions .
    [Show full text]
  • Generalized 2-Vector Spaces and General Linear 2-Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 212 (2008) 2069–2091 www.elsevier.com/locate/jpaa Generalized 2-vector spaces and general linear 2-groups Josep Elgueta∗ Dept. Matematica` Aplicada II, Universitat Politecnica` de Catalunya, Spain Received 28 February 2007; received in revised form 28 October 2007 Available online 29 February 2008 Communicated by I. Moerdijk Abstract A notion of generalized 2-vector space is introduced which includes Kapranov and Voevodsky 2-vector spaces. Various kinds of such objects are considered and examples are given. The corresponding general linear 2-groups are explicitly computed in a special case which includes Kapranov and Voevodsky 2-vector spaces. c 2008 Elsevier B.V. All rights reserved. MSC: 18E05; 18D35; 18D20 1. Introduction In the last decade and a half, the need to introduce a categorical analog of the notion of vector space over a field K , usually called a 2-vector space over K , has become more and more obvious. This need was made explicit, for example, when looking for the right framework to study the generalizations of the Yang–Baxter equation to higher dimensions. It was indeed in this setting that the notion of (finite-dimensional) 2-vector space over a field K was first introduced by Kapranov and Voevodsky [14]. The main point in Kapranov and Voevodsky’s definition was to take the category VectK of finite-dimensional vector spaces over K as the analog of the field K and to define a (finite-dimensional) n ≥ 2-vector space over K as a “VectK -module category” equivalent in the appropriate sense to VectK for some n 0 (see [14] for details).
    [Show full text]
  • Arxiv:1409.5934V1 [Math.CT] 21 Sep 2014 I field a Fix O-Esrajnto.Teui for Unit the Adjunction
    REFLEXIVITY AND DUALIZABILITY IN CATEGORIFIED LINEAR ALGEBRA MARTIN BRANDENBURG, ALEXANDRU CHIRVASITU, AND THEO JOHNSON-FREYD Abstract. The “linear dual” of a cocomplete linear category C is the category of all cocontinuous linear functors C → Vect. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a correspond- ing copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category Qcoh(X) of quasi-coherent sheaves on a stack X is not dualizable if X is the classifying stack of a semisimple algebraic group in positive characteristic or if X is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if X is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset. 1. Introduction Fix a field K. For cocomplete K-linear categories C and D, let Hom(C, D)= Homc,K(C, D) denote the category of cocontinuous K-linear functors and natural transformations from C to D, and let C⊠D = C⊠c,KD denote the universal cocomplete K-linear category receiving a functor C×D →C⊠D which is cocontinuous and K-linear in each variable (while holding the other variable fixed). If C and D are both locally presentable (see Definition 2.1), then Hom(C, D) and C ⊠ D both exist as locally small categories.
    [Show full text]
  • Notes on Category Theory
    Notes on Category Theory Mariusz Wodzicki November 29, 2016 1 Preliminaries 1.1 Monomorphisms and epimorphisms 1.1.1 A morphism m : d0 ! e is said to be a monomorphism if, for any parallel pair of arrows a / 0 d / d ,(1) b equality m ◦ a = m ◦ b implies a = b. 1.1.2 Dually, a morphism e : c ! d is said to be an epimorphism if, for any parallel pair (1), a ◦ e = b ◦ e implies a = b. 1.1.3 Arrow notation Monomorphisms are often represented by arrows with a tail while epimorphisms are represented by arrows with a double arrowhead. 1.1.4 Split monomorphisms Exercise 1 Given a morphism a, if there exists a morphism a0 such that a0 ◦ a = id (2) then a is a monomorphism. Such monomorphisms are said to be split and any a0 satisfying identity (2) is said to be a left inverse of a. 3 1.1.5 Further properties of monomorphisms and epimorphisms Exercise 2 Show that, if l ◦ m is a monomorphism, then m is a monomorphism. And, if l ◦ m is an epimorphism, then l is an epimorphism. Exercise 3 Show that an isomorphism is both a monomorphism and an epimor- phism. Exercise 4 Suppose that in the diagram with two triangles, denoted A and B, ••u [^ [ [ B a [ b (3) A [ u u ••u the outer square commutes. Show that, if a is a monomorphism and the A triangle commutes, then also the B triangle commutes. Dually, if b is an epimorphism and the B triangle commutes, then the A triangle commutes.
    [Show full text]
  • Basics of Monoidal Categories
    4 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of think­ ing about category theory (which will be especially useful throughout these notes) is that category theory is a refinement (or “categorifica­ tion”) of ordinary algebra. In other words, there exists a dictionary between these two subjects, such that usual algebraic structures are recovered from the corresponding categorical structures by passing to the set of isomorphism classes of objects. For example, the notion of a (small) category is a categorification of the notion of a set. Similarly, abelian categories are a categorification 1 of abelian groups (which justifies the terminology). This dictionary goes surprisingly far, and many important construc­ tions below will come from an attempt to enter into it a categorical “translation” of an algebraic notion. In particular, the notion of a monoidal category is the categorification of the notion of a monoid. Recall that a monoid may be defined as a set C with an associative multiplication operation (x; y) ! x · y (i.e., a semigroup), with an 2 element 1 such that 1 = 1 and the maps 1·; ·1 : C ! C are bijections. It is easy to show that in a semigroup, the last condition is equivalent to the usual unit axiom 1 · x = x · 1 = x. As usual in category theory, to categorify the definition of a monoid, we should replace the equalities in the definition of a monoid (namely, 2 the associativity equation (xy)z = x(yz) and the equation 1 = 1) by isomorphisms satisfying some consistency properties, and the word “bijection” by the word “equivalence” (of categories).
    [Show full text]
  • Linear Algebra Jim Hefferon
    Answers to Exercises Linear Algebra Jim Hefferon ¡ ¢ 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯1 2¯ ¯3 1¯ ¡ ¢ 1 x · 1 3 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯x · 1 2¯ ¯ ¯ x · 3 1 ¡ ¢ 6 8 ¡ ¢ 2 1 ¯ ¯ ¯ ¯ ¯6 2¯ ¯8 1¯ Notation R real numbers N natural numbers: {0, 1, 2,...} ¯ C complex numbers {... ¯ ...} set of . such that . h...i sequence; like a set but order matters V, W, U vector spaces ~v, ~w vectors ~0, ~0V zero vector, zero vector of V B, D bases n En = h~e1, . , ~eni standard basis for R β,~ ~δ basis vectors RepB(~v) matrix representing the vector Pn set of n-th degree polynomials Mn×m set of n×m matrices [S] span of the set S M ⊕ N direct sum of subspaces V =∼ W isomorphic spaces h, g homomorphisms, linear maps H, G matrices t, s transformations; maps from a space to itself T,S square matrices RepB,D(h) matrix representing the map h hi,j matrix entry from row i, column j |T | determinant of the matrix T R(h), N (h) rangespace and nullspace of the map h R∞(h), N∞(h) generalized rangespace and nullspace Lower case Greek alphabet name character name character name character alpha α iota ι rho ρ beta β kappa κ sigma σ gamma γ lambda λ tau τ delta δ mu µ upsilon υ epsilon ² nu ν phi φ zeta ζ xi ξ chi χ eta η omicron o psi ψ theta θ pi π omega ω Cover. This is Cramer’s Rule for the system x1 + 2x2 = 6, 3x1 + x2 = 8.
    [Show full text]
  • Reflecting Algebraically Compact Functors
    Reflecting Algebraically Compact Functors Vladimir Zamdzhiev Université de Lorraine, CNRS, Inria, LORIA, F 54000 Nancy, France A compact T-algebra is an initial T-algebra whose inverse is a final T -coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes involving mixed-variance functors, such as function space. The construction of compact algebras is usually done in categories with a zero object where some form of a limit-colimit coincidence exists. In this paper we consider a more abstract approach and show how one can construct compact algebras in categories which have neither a zero object, nor a (standard) limit-colimit coincidence by reflecting the compact algebras from categories which have both. In doing so, we provide a constructive description of a large class of algebraically compact functors (satisfying a compositionality principle) and show our methods compare quite favorably to other approaches from the literature. 1 Introduction Inductive datatypes for programming languages can be used to represent important data structures such as lists, trees, natural numbers and many others. When providing a denotational interpretation for such languages, type expressions correspond to functors and one has to be able to construct their initial alge- bras in order to model inductive datatypes [9]. If the admissible datatype expressions allow only pairing and sum types, then the functors induced by these expressions are all polynomial functors, i.e., functors constructed using only coproducts and (tensor) product connectives, and the required initial algebra may usually be constructed using Adámek’s celebrated theorem [2].
    [Show full text]