Contravariantly finite subcategories closed under predecessors

Let A be an Artin algebra and mod A be the category of finitely generated right A-modules. We prove that an additive full subcategory C of mod A closed under predecessors is contravariantly finite if and only if its right Ext-orthogonal is covariantly finite, or if and only if the Ext-injectives in C define a cotilting module (over the support algebra of C) or, equivalently, if and only if C is the support of the representable functors given by the Ext-injectives. (C) 2009 Elsevier Inc. All rights reserved.

Hierarchies of circuit classes that are closed under complement

We examine three hierarchies of circuit classes and show they are closed under complementation. (1) The class of languages recognized by a family of polynomial size skew circuits with width O(w), are closed under complement. (2) The class of languages recognized by family of polynomial size circuits with width O(w) and polynomial tree-size, are closed under complement. (3) The class of languages recognized by a family of polynomial size, O(log(n)) depth, bounded AND fan-in with OR fan-in f (f⩾log(n)) circuits are closed under complement. These improve upon the results of (i) Immerman (1988) and Szelepcsenyi (1988), who show that 𝒩L𝒪𝒢 is closed under complementation, and (ii) Borodin et al. (1989), who show that L𝒪𝒢𝒞ℱL is closed under complement

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.

A decidable temporal logic with repeating values

Various logical formalisms with the freeze quantifier have been recently considered to model computer systems even though this is a powerful mechanism that often leads to undecidability. In this paper, we study a linear-time temporal logic with past-time operators such that the freeze operator is only used to express that some value from an infinite set is repeated in the future or in the past. Such a restriction has been inspired by a recent work on spatio-temporal logics. We show decidability of finitary and infinitary satisfiability by reduction into the verification of temporal properties in Petri nets. This is a surprising result since the logic is closed under negation, contains future-time and past-time temporal operators and can express the nonce property and its negation. These ingredients are known to lead to undecidability with a more liberal use of the freeze quantifier.

Temporal Logics of Repeating Values

Various logical formalisms with the freeze quantifier have been recently considered to model computer systems even though this is a powerful mechanism that often leads to undecidability. In this article, we study a linear-time temporal logic with past-time operators such that the freeze operator is only used to express that some value from an infinite set is repeated in the future or in the past. Such a restriction has been inspired by a recent work on spatio-temporal logics that suggests such a restricted use of the freeze operator. We show decidability of finitary and infinitary satisfiability by reduction into the verification of temporal properties in Petri nets by proposing a symbolic representation of models. This is a quite surprising result in view of the expressive power of the logic since the logic is closed under negation, contains future-time and past-time temporal operators and can express the nonce property and its negation. These ingredients are known to lead to undecidability with a more liberal use of the freeze quantifier. The article also contains developments about the relationships between temporal logics with the freeze operator and counter automata as well as reductions into first-order logics over data words.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Matrices whose hermitian part is positive definite

We are concerned with the class ∏_n of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite.

Various connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diagonal entries, then there is a positive diagonal F such that FA ϵ ∏_n.

Powers are investigated and it is found that the only matrices A for which A^m ϵ ∏_n for all integers m are the Hermitian elements of ∏_n. Products and sums are considered and criteria are developed for AB to be in ∏_n.

Since ∏_n n is closed under inversion, relations between H(A)^-1 and H(A^-1) are studied and a dichotomy observed between the real and complex cases. In the real case more can be said and the initial result is that for A ϵ ∏_n, the difference H(adjA) - adjH(A) ≥ 0 always and is ˃ 0 if and only if S(A) = A-A*/2 has more than one pair of conjugate non-zero characteristic roots. This is refined to characterize real c for which cH(A^-1) - H(A)^-1 is positive definite.

The cramped (characteristic roots on an arc of less than 180°) unitary matrices are linked to ∏_n and characterized in several ways via products of the form A ^-1A*.

Classical inequalities for Hermitian positive definite matrices are studied in ∏_n and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of ∏_n in which the precise statement of Hadamardis inequality holds is isolated while in another large subclass its reverse is shown to hold. In the second Hadamard's inequality is weakened in such a way that it holds throughout ∏_n. Both approaches contain the original Hadamard inequality as a special case.

O trabalho político do profissional de saúde mental em um processo de desinstitucionalização: um estudo sobre integralidade e reforma psiquiátrica

O trabalho pretende apresentar uma cartografia das atividades desenvolvidas pelos profissionais de saúde mental que prestam assistência aos moradores dos serviços residenciais terapêuticos do município de Carmo, no Estado do Rio de Janeiro. Estas moradias constituem uma etapa do processo de desinstitucionalização de um hospital psiquiátrico estadual do tipo colônia agrícola, fundado na década de 40 do século passado, onde estavam internados cerca de 280 pacientes. A partir de 2003, com a extinção do hospital coordenada pelo gestor estadual e a municipalização dos recursos, uma pequena parcela dos internos retornou ao seio familiar, mas a maioria (cerca de 160) foi alocada em moradias assistidas, espalhadas pelas áreas urbana e rural do município. Tomando a tarefa de reinserção social como o viés político da Reforma Psiquiátrica Brasileira, o trabalho cotidiano da equipe multiprofissional é focalizado. Neste plano relacional o texto busca a conceitualização de Política e de um trabalho político, através das idéias de Arendt, Gramsci, Deleuze e Guattari, entre outros; já a ergologia possibilita uma metodologia para a abordagem do trabalho humano. Com base na pesquisa de campo, a cartografia revela como a tarefa política é realizada, nas atividades dos técnicos, quer dizer, na micropolítica dos encontros dos profissionais uns com os outros, com a sociedade civil e com os moradores, onde os valores da Reforma Psiquiátrica são negociados. No fim, trata-se de uma etapa em um processo, ainda a meio caminho entre a gestão estadual e a gestão municipal, onde a proposta mantém-se sustentada pela vontade política do gestor e, na via da hegemonia, deve ser trabalhada entre os profissionais e na sociedade civil. O panorama é heterogêneo, e a dinâmica revela a diversidade de entendimentos e interesses. No cenário do trabalho cotidiano, conceitos como autonomia e cidadania se atualizam em atividades que caracterizam a vida nas cidades e se desdobram em torno de certos temas, como o uso do dinheiro, ou a apropriação do espaço. A experiência dos técnicos envolvidos mais diretamente com os moradores, como os cuidadores, produz uma técnica de escuta e mobilização, que não admite cartilhas nem regras pré-estabelecidas ou imutáveis. Este trabalho conjunto, formador de redes e sustentado na interação, é indicador de integralidade na execução da proposta da Reforma Psiquiátrica.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

An Eigenfunction Approach for Volatility Modeling.

In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp. infinitesimal generator) operator associated to the state variable in discrete (resp. continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials respectively. The eigenfunction approach has at least six advantages: i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and, hence, simple for forecasting and inference purposes; (iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation.

A study on semigroup action on fuzzy subsets, inverse fuzzy automata and related topics

This thesis comprises five chapters including the introductory chapter. This includes a brief introduction and basic definitions of fuzzy set theory and its applications, semigroup action on sets, finite semigroup theory, its application in automata theory along with references which are used in this thesis. In the second chapter we defined an S-fuzzy subset of X with the extension of the notion of semigroup action of S on X to semigroup action of S on to a fuzzy subset of X using Zadeh's maximal extension principal and proved some results based on this. We also defined an S-fuzzy morphism between two S-fuzzy subsets of X and they together form a category S FSETX. Some general properties and special objects in this category are studied and finally proved that S SET and S FSET are categorically equivalent. Further we tried to generalize this concept to the action of a fuzzy semigroup on fuzzy subsets. As an application, using the above idea, we convert a _nite state automaton to a finite fuzzy state automaton. A classical automata determine whether a word is accepted by the automaton where as a _nite fuzzy state automaton determine the degree of acceptance of the word by the automaton. 1.5. Summary of the Thesis 17 In the third chapter we de_ne regular and inverse fuzzy automata, its construction, and prove that the corresponding transition monoids are regular and inverse monoids respectively. The languages accepted by an inverse fuzzy automata is an inverse fuzzy language and we give a characterization of an inverse fuzzy language. We study some of its algebraic properties and prove that the collection IFL on an alphabet does not form a variety since it is not closed under inverse homomorphic images. We also prove some results based on the fact that a semigroup is inverse if and only if idempotents commute and every L-class or R-class contains a unique idempotent. Fourth chapter includes a study of the structure of the automorphism group of a deterministic faithful inverse fuzzy automaton and prove that it is equal to a subgroup of the inverse monoid of all one-one partial fuzzy transformations on the state set. In the fifth chapter we define min-weighted and max-weighted power automata study some of its algebraic properties and prove that a fuzzy automaton and the fuzzy power automata associated with it have the same transition monoids. The thesis ends with a conclusion of the work done and the scope of further study.

Systems of Parallel Communicating Restarting Automata

In der vorliegenden Dissertation werden Systeme von parallel arbeitenden und miteinander kommunizierenden Restart-Automaten (engl.: systems of parallel communicating restarting automata; abgekürzt PCRA-Systeme) vorgestellt und untersucht. Dabei werden zwei bekannte Konzepte aus den Bereichen Formale Sprachen und Automatentheorie miteinander vescrknüpft: das Modell der Restart-Automaten und die sogenannten PC-Systeme (systems of parallel communicating components). Ein PCRA-System besteht aus endlich vielen Restart-Automaten, welche einerseits parallel und unabhängig voneinander lokale Berechnungen durchführen und andererseits miteinander kommunizieren dürfen. Die Kommunikation erfolgt dabei durch ein festgelegtes Kommunikationsprotokoll, das mithilfe von speziellen Kommunikationszuständen realisiert wird. Ein wesentliches Merkmal hinsichtlich der Kommunikationsstruktur in Systemen von miteinander kooperierenden Komponenten ist, ob die Kommunikation zentralisiert oder nichtzentralisiert erfolgt. Während in einer nichtzentralisierten Kommunikationsstruktur jede Komponente mit jeder anderen Komponente kommunizieren darf, findet jegliche Kommunikation innerhalb einer zentralisierten Kommunikationsstruktur ausschließlich mit einer ausgewählten Master-Komponente statt. Eines der wichtigsten Resultate dieser Arbeit zeigt, dass zentralisierte Systeme und nichtzentralisierte Systeme die gleiche Berechnungsstärke besitzen (das ist im Allgemeinen bei PC-Systemen nicht so). Darüber hinaus bewirkt auch die Verwendung von Multicast- oder Broadcast-Kommunikationsansätzen neben Punkt-zu-Punkt-Kommunikationen keine Erhöhung der Berechnungsstärke. Desweiteren wird die Ausdrucksstärke von PCRA-Systemen untersucht und mit der von PC-Systemen von endlichen Automaten und mit der von Mehrkopfautomaten verglichen. PC-Systeme von endlichen Automaten besitzen bekanntermaßen die gleiche Ausdrucksstärke wie Einwegmehrkopfautomaten und bilden eine untere Schranke für die Ausdrucksstärke von PCRA-Systemen mit Einwegkomponenten. Tatsächlich sind PCRA-Systeme auch dann stärker als PC-Systeme von endlichen Automaten, wenn die Komponenten für sich genommen die gleiche Ausdrucksstärke besitzen, also die regulären Sprachen charakterisieren. Für PCRA-Systeme mit Zweiwegekomponenten werden als untere Schranke die Sprachklassen der Zweiwegemehrkopfautomaten im deterministischen und im nichtdeterministischen Fall gezeigt, welche wiederum den bekannten Komplexitätsklassen L (deterministisch logarithmischer Platz) und NL (nichtdeterministisch logarithmischer Platz) entsprechen. Als obere Schranke wird die Klasse der kontextsensitiven Sprachen gezeigt. Außerdem werden Erweiterungen von Restart-Automaten betrachtet (nonforgetting-Eigenschaft, shrinking-Eigenschaft), welche bei einzelnen Komponenten eine Erhöhung der Berechnungsstärke bewirken, in Systemen jedoch deren Stärke nicht erhöhen. Die von PCRA-Systemen charakterisierten Sprachklassen sind unter diversen Sprachoperationen abgeschlossen und einige Sprachklassen sind sogar abstrakte Sprachfamilien (sogenannte AFL's). Abschließend werden für PCRA-Systeme spezifische Probleme auf ihre Entscheidbarkeit hin untersucht. Es wird gezeigt, dass Leerheit, Universalität, Inklusion, Gleichheit und Endlichkeit bereits für Systeme mit zwei Restart-Automaten des schwächsten Typs nicht semientscheidbar sind. Für das Wortproblem wird gezeigt, dass es im deterministischen Fall in quadratischer Zeit und im nichtdeterministischen Fall in exponentieller Zeit entscheidbar ist.

On fuzzy ideals and fuzzy filters of fuzzy lattices

In the literature there are several proposals of fuzzi cation of lattices and ideals concepts. Chon in (Korean J. Math 17 (2009), No. 4, 361-374), using the notion of fuzzy order relation de ned by Zadeh, introduced a new notion of fuzzy lattice and studied the level sets of fuzzy lattices, but did not de ne a notion of fuzzy ideals for this type of fuzzy lattice. In this thesis, using the fuzzy lattices de ned by Chon, we de ne fuzzy homomorphism between fuzzy lattices, the operations of product, collapsed sum, lifting, opposite, interval and intuitionistic on bounded fuzzy lattices. They are conceived as extensions of their analogous operations on the classical theory by using this de nition of fuzzy lattices and introduce new results from these operators. In addition, we de ne ideals and lters of fuzzy lattices and concepts in the same way as in their characterization in terms of level and support sets. One of the results found here is the connection among ideals, supports and level sets. The reader will also nd the de nition of some kinds of ideals and lters as well as some results with respect to the intersection among their families. Moreover, we introduce a new notion of fuzzy ideals and fuzzy lters for fuzzy lattices de ned by Chon. We de ne types of fuzzy ideals and fuzzy lters that generalize usual types of ideals and lters of lattices, such as principal ideals, proper ideals, prime ideals and maximal ideals. The main idea is verifying that analogous properties in the classical theory on lattices are maintained in this new theory of fuzzy ideals. We also de ne, a fuzzy homomorphism h from fuzzy lattices L and M and prove some results involving fuzzy homomorphism and fuzzy ideals as if h is a fuzzy monomorphism and the fuzzy image of a fuzzy set ~h(I) is a fuzzy ideal, then I is a fuzzy ideal. Similarly, we prove for proper, prime and maximal fuzzy ideals. Finally, we prove that h is a fuzzy homomorphism from fuzzy lattices L into M if the inverse image of all principal fuzzy ideals of M is a fuzzy ideal of L. Lastly, we introduce the notion of -ideals and - lters of fuzzy lattices and characterize it by using its support and its level set. Moreover, we prove some similar properties in the classical theory of - ideals and - lters, such as, the class of -ideals and - lters are closed under intersection. We also de ne fuzzy -ideals of fuzzy lattices, some properties analogous to the classical theory are also proved and characterize a fuzzy -ideal on operation of product between bounded fuzzy lattices L and M and prove some results.

Black hole partition function using the hybrid formalism of superstrings

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Algebraic geometric codes over rings

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings.