Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

2994 Results for the subject "Calculus":

AbstractA multiplicative calculus dealing with real valued functions is extended to a multiplicative type complex calculus (MCC) dealing with complex valued functions. Some fundamental theorems and concepts of the classical calculus are interpreted from the view point of the MCC and the analogies between them are given. Also new notations for the MCC are defined. The MCC is distinguished from the classical calculus by calling the classical calculus as the additive type complex calculus (ACC).

Ali Uzer Publication date: 2010/11/01AbstractIn this paper, to compare the situation calculus and event calculus we formulate both as logic programs and prove properties of these by reasoning with their completions augmented with induction. We thus show that the situation calculus and event calculus imply one another. Whereas our derivation of the event calculus from the situation calculus requires the use of induction, our derivation of the situation calculus from the event calculus does not. We also show that in certain concrete applications, such as the missing car example, conclusions that seem to require the use of induction in the situation calculus can be derived without induction in the event calculus. To compare the two calculi, we need to make a number of small modifications to both. As a by-product of these modifications, the resulting calculi can be used to reason about both actual and hypothetical states of affairs, including counterfactual ones. We further show how the core axioms of both calculi can be exte *Read more...*

AbstractThis paper presents an algebraic calculus like the relational calculus for reasoning about sequential phenomena. It provides a common foundation for several proposed models of concurrent or reactive systems. It is clearly differentiated from the relational calculus by absence of a general converse operation. This permits the treatment of temporal logic within the sequential calculus.

Burghard von Karger Publication date: 1995/02/10AbstractIn this chapter, we consider the basic calculus of functions of two or more independent variables, that is, the calculus of multivariate functions, or multivariate calculus. This book has so far focussed on functions of a single independent variable, for example, y(x) or f(z); however, most realistic models of physical and financial systems involve more than one independent variable. In addition, as we saw in the previous chapter with the discussion of exact ODEs, it is often very convenient with work with mathematical tools that require some knowledge of multivariate calculus. We will see an additional example of such tools later in this chapter.

S.J. Garrett Publication date: 2015/01/01AbstractWe describe a simple but expressive calculus of sequential processes, represented as coroutines. We show that this calculus can be used to express a variety of programming language features including procedure calls, labelled jumps, integer references and stacks. We describe the operational properties of the calculus using reduction rules and equational axioms.We describe a notion of categorical model for our calculus, and give a simple example of such a model based on a category of games and strategies. We prove full abstraction results showing that equivalence in the categorical model corresponds to observational equivalence in the calculus, and also to equivalence of evaluation trees, which are infinitary normal forms for the calculus.We show that our categorical model can be used to interpret the untyped λ-calculus and use this fact to extract a sound translation of the latter into our calculus of coroutines.

J. Laird Publication date: 2006/02/07AbstractThree notions of structural inclusion between process terms of the π-calculus are considered, and proven to be decidable and to have axiomatizations that are sound and complete in the multiset semantics Mπ of the π-calculus. All three are strong simulation relations.

Joost Engelfriet Publication date: 2001/05/06AbstractObjective: To critically analyse the formation, composition, ethnic variations and pathogenic potential of subgingival calculus in comparison with supragingival calculus.Data sources: Using CD-ROM and index medicus, scientific papers relating to subgingival calculus or subgingival and supragingival calculus written in the English language since 1960 were considered, with the emphasis on more recent articles.Study selection: Studies were selected for their relevance and contemporary nature re:composition and formation of dental calculus and comparisons of ethnic groups with regard to dental calculus, especially subgingival calculus. Some similar studies were not included.Data extraction: Abstracts of studies were kept brief unless particularly important to the review. Population, methodology, statistics and accurate conclusions were used as important guides to the quality and validity of studies.Data synthesis: Similarities and differences between supragingival and subgingival c *Read more...*

AbstractA new formulation of the π-calculus, where name instantiation is handled explicitly via the introduction of a suitable combinator, is presented. The bisimulation semantics originally developed for the π-calculus are retrieved by giving the description of the corresponding strategies for name instantiation. The explicit handling of name instantiation allows us to reduce the π-calculus transitional semantics to a standard SOS framework. Hence, π-calculus bisimulation models can take fully advantage of the SOS meta-theory developed for ‘static’ process calculi. For instance, complete axiomatic characterizations of π-calculus bisimulation equivalences can be automatically derived by turning SOS rules into equations. This formulation of the π-calculus is very promising for the development of semantic-based automatic verification tools.

Gian-Luigi Ferrari Publication date: 1996/11/10AbstractDental plaque is considered to be a major etiological factor in the development of periodontal disease. Accordingly, the elimination of supra- and sub-gingival plaque and calculus is the cornerstone of periodontal therapy. Dental calculus is mineralized plaque; because it is porous, it can absorb various toxic products that can damage the periodontal tissues. Hence, calculus should be accurately detected and thoroughly removed for adequate periodontal therapy. Many techniques have been used to identify and remove calculus deposits present on the root surface. The purpose of this review was to compile the various methods and their advantages for the detection and removal of calculus.

Deepa G. Kamath Publication date: 2014/01/01AbstractThis article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of the first-year calculus, (b) construction of connections and relationships between and among concepts and skills, (c) the ability to use the ideas of the first-year calculus, and (d) a deep sense of the context and purpose of the calculus. Each end goal for student understanding is explored in detail and the potential for using the framework as an organizational tool is discussed.

Kimberly S. Sofronas Publication date: 2011/06/01AbstractWe introduce explicit fusions of names. An explicit fusion is a process that exists concurrently with the rest of the system and enables two names to be used interchangeably. Explicit fusions provide a small-step account of reaction in process calculi such as the pi calculus and the fusion calculus. In this respect they are similar to the explicit substitutions of Abadi, Cardelli and Curien, which do the same for the lambda calculus. In this paper, we give a technical foundation for explicit fusions. We present the pi-F calculus, a simple process calculus with explicit fusions, and define a strong bisimulation congruence. We study the embeddings of the fusion calculus and the pi calculus. The former is fully abstract with respect to bisimulation.

Lucian Wischik Publication date: 2005/08/31Publisher SummaryThis chapter discusses the concept of differential calculus. Calculus, probably the single most valuable intellectual achievement of man for solving problems of the physical world, consists essentially of two complementary processes, namely, differentiation and integration. Mathematics involving the former is called differential calculus while that depending upon the latter is known as integral calculus. Equations involving derivatives are called differential equations, the order of such an equation being that of the highest derivative in it.

A.F. HORADAM Publication date: 1968/01/01AbstractWe present a case of a 21 year old male who presented with symptomatic right upper ureteric calculus measuring 5 cm × 1.5 cm fulfilling the criteria to be named as giant ureteric calculus. Laparoscopic right ureterolithotomy was performed and the giant ureteric calculus was retrieved.

Prasad V. Magdum Publication date: 2015/09/01AbstractIn Hoare and He's unifying theories of programming, the alphabetised relational calculus is used to describe and relate different programming paradigms, including functional, imperative, logic, and parallel programming. In this paper, we give a formal semantics of the alphabetised relational calculus, and use our definition to create a deep embedding of the calculus in Z. This allows us to use one of the standard theorem provers for Z, in order to provide mechanised support for reasoning about programs in the unifying theory.

Gift Nuka Publication date: 2004/05/17AbstractLinear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation of focused proofs, a complete subclass of linear sequent proofs known to have a much stronger structure than the standard sequent calculus for linear logic. Despite the classical setting, the interpretation relates proofs to a refined linear λ-calculus, and we investigate its properties and relation to other calculi, such as the usual λ-calculus, the λμ-calculus, and their variants based on sequent calculi.

Taus Brock-Nannestad Publication date: 2015/12/21AbstractWe introduce a complex calculus of variations, analogous to the classical calculus of variations, but about functions from Cn into C. This calculus is based on the minimum of a complex function. The complex minimum allows us to obtain explicit solutions for complex Hamilton-Jacobi equations, and in particular to generalize the Hopf-Lax formula.

M. Gondran Publication date: 2001/08/01AbstractWe develop a ϰ-symmetry calculus for the d = 2 and d = 3, N = 2 massive superparticles, which enables us to construct higher order ϰ-invariant actions. The method relies on a reformulation of these models as supersymmetric sigma models that are invariant under local worldline superconformal transformations. We show that the ϰ-symmetry is embedded in the superconformal symmetry so that a calculus for the ϰ-symmetry is equivalent to a tensor calculus for the latter. We develop such a calculus without the introduction of a worldline supergravity multiplet.

Jerome P. Gauntlett Publication date: 1991/11/28AbstractSeveral symbolic notions of bisimilarity have been defined for the spi calculus and the applied pi calculus. In this paper, we treat a spi calculus with a general constructor-destructor message algebra, and define a symbolic bisimilarity that is both sound and complete with respect to its concrete counterpart.

Johannes Borgström Publication date: 2009/08/07AbstractAn interpretation of Abadi and Cardelli's first-order Imperative ς-calculus into a typed π-calculus is presented. The interpretation validates the subtyping relation and the typing judgments of the ς-calculus, and is computationally adequate. The proof of computational adequacy makes use of (a π-calculus version) of ready simulation, and of a factorization of the interpretation into a functional part and a very simple imperative part. The interpretation can be extended to accommodate various type features. The interpretation can be used to compare and contrast the Imperative and the Functional ς-calculus, and to prove properties about them, within a unified framework.

Josva Kleist Publication date: 2002/09/01AbstractThe lambda calculus with constructors decomposes the pattern matching à la ML into some atomic rules. Some of them do not match with the usual computational intuitions (in particular with typing intuitions). However it is possible to define an abstract notion of model for the untyped calculus, that has a trivial syntactic instance.Nevertheless, the question of devising a non-syntactic model for this calculus was still unresolved. In this paper we answer this question in the untyped setting, by going back to the first motivation of the lambda-calculus with constructors: the simulation of an abstract machine with two independent stacks. This provides immediately a CPS translation into the usual lambda calculus. At the semantic level, it appears that this translation transforms any continuation model of the untyped lambda calculus into a model of the lambda calculus with constructors. In particular, any Scott domain can be turned into such a model.

Barbara Petit Publication date: 2012/09/24