A Note on Gentzen's LJ and NJ Systems Isomorphism.

by De Campos Sanz, Wagner [2007-10-01]

Academic Journal

pages 8

In this paper we are going to examine intuitionistic sequent calculus and its negation rules. We state new negation rules defining, in this way, a new sequent system. It will be used to clarify Gentzen's NJ and LJ systems isomorphism. These new negation rules are a direct reading of new natural deduction negation rules obtained by a slight modification of NJ rules. We also show that the new system is equivalent to LJ and that the Hauptsatz holds for it. [ABSTRACT FROM PUBLISHER]


Editorial.

by Kamareddine, F [2001-06-01]

Academic Journal

pages 3

An introduction is presented in which the editor discusses various reports within the issue on topics including a new type of calculus, a tableau system for dynamic first-order logic (DFOL), and non-standard geometry.


Higher Order β Matching is Undecidable.

by Loader, Ralph [2003-02-01]

Academic Journal

pages 18

We show that the solvability of matching problems in the simply typed λ-calculus, up to β equivalence, is not decidable. This decidability question was raised by Huet [4].Note that there are two variants of the question: that concerning β equivalence (dealt with here), and that concerning βη equivalence.The second of these is perhaps more interesting; unfortunately the work below sheds no light on it, except perhaps to illustrate the subtlety and difficulty of the problem. [ABSTRACT FROM AUTHOR]


Distributional properties of the typical cell of stationary iterated tessellations.

by Maier, Roland, Mayer, Johannes, Schmidt, Volker [2004-06-01]

Academic Journal

pages 16

Distributional properties are considered of the typical cell of stationary iterated tessellations (SIT), which are generated by stationary Poisson-Voronoi tessellations (SPVT) and stationary Poisson line tessellations (SPLT), respectively. Using Neveu’s exchange formula, the typical cell of SIT can be represented by those cells of its component tessellation hitting the typical cell of its initial tessellation. This provides a simulation algorithm without consideration of limits in space. It has been applied in order to estimate the probability densities of geometric characteristics of the typical cell of SIT generated by SPVT and SPLT. In particular, the probability densities of the number of vertices, the perimeter, and the area of the typical cell of such SIT have been determined. [ABSTRACT FROM AUTHOR]


USING SAMPLING AND SIMPLEX DERIVATIVES IN PATTERN SEARCH METHODS.

by Custódio, A. L., Vicente, L. N. [2007-04-01]

Academic Journal

pages 19

In this paper, we introduce ways of making a pattern search more efficient by reusing previous evaluations of the objective function, based on the computation of simplex derivatives (e.g., simplex gradients). At each iteration, one can attempt to compute an accurate simplex gradient by identifying a sampling set of previously evaluated points with good geometrical properties. This can be done using only past successful iterates or by considering all past function evaluations. The simplex gradient can then be used to reorder the evaluations of the objective function associated with the directions used in the poll step or to update the mesh size parameter according to a sufficient decrease criterion, neither of which requires new function evaluations. We present these procedures in detail and apply them to a set of problems from the CUTEr collection. Numerical results show that these procedures can enhance significantly the practical performance of pattern search methods. [ABSTRACT FROM AUTHOR]


A Resolution Calculus for Shortening Proofs.

by Peltier, Nicolas [2005-05-01]

Academic Journal

pages 27

We propose an extended resolution calculus called δm-resolution, aiming at reducing the length of the proofs without increasing too much the branching factor of the procedure. The soundness and refutational completeness of the new calculus is proven. We provide numerous examples showing the possibilities of our calculus and we show that δm-resolution allows to reduce the length of proof by a double exponential factor. We compare our calculus with the quantifier introduction method developed by Baaz, Leitsch and Egly and prove that both techniques are theoretically incomparable in the sense that none of them can polynomially simulate the other. [ABSTRACT FROM PUBLISHER]


Academic Journal

pages 32

Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding α-conversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. We believe that the study of these systems should not develop without being well-tied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names and to do so, we provide the λt-calculus: a λ-calculus a la de Bruijn which can be translated into a λ-calculus with explicit substitutions written with variable names. We present explicitly this translation and use it to obtain preservation of strong normalisation for λt. Moreover, we show several properties of λt, including confluence on closed terms and efficiency to simulate β-reduction. Furthermore, λt is a good example of a calculus written in the λs-style (cf. [19]) that possesses the updating mechanism of the calculi á la λσ (cf. [1, 7, 36]).Key words: lambda calculus, variable names, de Bruijn indices, explicit substitutions [ABSTRACT FROM AUTHOR]


THE TIME SCALE CALCULUS APPROACH TO THE GEODESIC PROBLEM IN 3D DYNAMIC DATA SETS.

by Atmaca, Sibel Paşalı, Akgüller, Ömer [2013-12-01]

Academic Journal

pages 7

Geodesics have a fundamental role in the geometry of curved surfaces, as well as in discrete geometry. We present the time scale analogy of the dynamic data sets parameterized by a tensor product of two times scales. The goal of our study is the find the shortest and straightest path between two points on a point cloud like data sets which also involves continuous data. [ABSTRACT FROM AUTHOR]


Toward a History of Mathematics Focused on Procedures.

by Błaszczyk, Piotr, Kanovei, Vladimir, Katz, Karin, Katz, Mikhail, Kutateladze, Semen, Sherry, David [2017-12-01]

Academic Journal

pages 21

Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson's framework is more helpful in understanding their procedures than a Weierstrassian framework. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 8

EU lawmakers have long refused the cultivation of "Genetically Modified Organisms". An example of this struggle is the revision of the accepted level of contaminants in maize: rather than admitting that Bt maize is safer than "non-GMO" varieties, and therefore European farmers should be allowed not only to import it, but also to produce it, politicians have raised the threshold of the poisonous fumonisins that may be legally present in food and feed. This decision is an example of a "Schumpeterian" approach to policy, where public choices are not inspired by a science-based mindset, but are substantially dictated by a calculus of consent; economic/commercial protectionism has also been considered as a motivation. While scholars must continue to explain that every policy decision should have a basis in sound science, no way out of the "GMO" imbroglio seems to be foreseeable, as long as politicians stick to the Schumpeterian iron law. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 1

The article presents a discussion on the application of a nonstandard explicit integration to solve Green and Ampt infiltration equation. The discussion had earlier proposed a quick nonstandard algorithm, to which the author of the current extension of the discussion adds his own points. The points added take into consideration, issues like dependence on time step size and the percentage of error.


A compact kernel for the calculus of inductive constructions.

by Asperti, A., Ricciotti, W., Coen, C. Sacerdoti, Tassi, E. [2009-02-01]

Academic Journal

pages 74

The paper describes the new kernel for the Calculus of Inductive Constructions (CIC) implemented inside the Matita Interactive Theorem Prover. The design of the newkernel has been completely revisited since the first release, resulting in a remarkably compact implementation of about 2300 lines of OCaml code. The work is meant for people interested in implementation aspects of Interactive Provers, and is not self contained. In particular, it requires good acquaintance with Type Theory and functional programming languages. [ABSTRACT FROM AUTHOR]


The Design and Formalization of Mezzo, a Permission-Based Programming Language.

by BALABONSKI, THIBAUT, POTTIER, FRANÇOIS, PROTZENKO, JONATHAN [2016-08-01]

Academic Journal

pages 94

The programming language Mezzo is equipped with a rich type system that controls aliasing and access to mutable memory. We give a comprehensive tutorial overview of the language. Then we present a modular formalization of Mezzo's core type system, in the form of a concurrent λ-calculus, which we successively extend with references, locks, and adoption and abandon, a novel mechanism that marries Mezzo's static ownership discipline with dynamic ownership tests. We prove that well-typed programs do not go wrong and are data-race free. Our definitions and proofs are machine checked. [ABSTRACT FROM AUTHOR]


POLY-GENOCCHI POLYNOMIALS WITH UMBRAL CALCULUS VIEWPOINT.

by TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, JONGKYUM KWON [2019-01-01]

Academic Journal

pages 18

In this paper, we would like to exploit umbral calculus in order to derive explicit expressions, some properties, recurrence relations and identities for poly-Genocchi polynomials. [ABSTRACT FROM AUTHOR]


UMBRAL CALCULUS APPROACH TO DEGENERATE POLY-GENOCCHI POLYNOMIALS.

by TAEKYUN KIM, DAE SAN KIM, LEE CHAE JANG, GWAN-WOO JANG [2019-01-01]

Academic Journal

pages 17

In this paper, we apply umbral calculus techniques in order to derive explicit expressions, some properties, recurrence relations and identities for degenerate poly-Genocchi polynomials. Furthermore, we derive several explicit expressions of degenerate poly-Genocchi polynomials as linear combinations of some of the well-known families of special polynomials. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 24

We investigate the complexity of the standard translation of lambda calculus into combinatory logic. The main result shows that the asymptotic growth rate of the size of a trans- lated term is Θ(n3) in worst-case, where n denotes the size of the lambda term. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 14

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 24

I argue that European schools of thought on memory and memorization were critical in enabling growth of the scientific method. After giving a historical overview of the development of the memory arts from ancient Greece through 17th century Europe, I describe how the Baconian viewpoint on the scientific method was fundamentally part of a culture and a broader dialogue that conceived of memorization as a foundational methodology for structuring knowledge and for developing symbolic means for representing scientific concepts. The principal figures of this intense and rapidly evolving intellectual milieu included some of the leading thinkers traditionally associated with the scientific revolution; among others, Francis Bacon, Renes Descartes, and Gottfried Leibniz. I close by examining the acceleration of mathematical thought in light of the art of memory and its role in 17th century philosophy, and in particular, Leibniz's project to develop a universal calculus. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 4

An explicit, coupled, single-step method for the numerical solution of initial value problems for systems of ordinary differential equations is presented. The method was designed to be general purpose in nature but to be especially efficient when dealing with stiff systems of differential equations. it is, in general, second order except for the case of a linear system with constant coefficients and linear forcing terms; in that case, the method is third order. It has been implemented and put to routine usage in biological applications—where stiffness frequently appears—with favorable results. When compared to a standard fourth order Runge-Kutta implementation, computation time required by this method has ranged from comparable for certain nonstiff problems to better than two orders of magnitude faster for some highly stiff systems. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 4

The solution of the nonlinear differential equation Y" = F(x, Y, Y') with two-point boundary conditions is opproximated by a quintic or cubic spline function y(x). The method is well suited to nonuniform mesh size and dynamic mesh size allocation. For uniform mesh size h, the error in the quintic spline y(x) is O(h4), with typical error one-third that from Numerov's method. Requiring the differential equation to be satisfied at the mesh points results in a set of difference equations, which are block tridiagonal and so are easily solved by relaxation or other standard methods. [ABSTRACT FROM AUTHOR]


PERFORMANCE ANALYSIS OF INTRODUCTORY STATISTICS COURSE DUE TO DIFFERENT MATHEMATICAL BACKGROUND.

by Choudhury, Askar H., Radhakrishnan, Ramaswamy [2009-06-01]

Conference

pages 1

This paper addresses the issue of students' different mathematical background that differentiates students' performance in statistics course. Students can choose one of several mathematics based prerequisite to gain necessary background knowledge for the Statistics course. Statistics is one of the required courses for business and economics majors. Among several possible prerequisite courses we considered two different calculus courses (Applied Calculus and Calculus-I) as background knowledge for statistics course to compare. Therefore, the objective of this paper is to observe the significance and magnitude of differential effect of two different calculus courses on statistics course performance. Statistical tests provided evidence that differential effect exist due to different calculus background knowledge. Specifically, we have found that students who took the Calculus-I received higher average grade in Statistics course than students who took Applied Calculus. Thus, students with added traditional calculus orientation do have greater statistical proficiency. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 16

We present an approach to the numerical integration of ordinary differential equations based on the algebraic theory of Butcher (Math. Comp. 26, 79–106, 1972) and the [InlineMediaObject not available: see fulltext.]-series theory of Hairer and Wanner (Computing 13, 1–15, 1974). We clarify the differences of these two approaches by equating the elementary weight functions and showing the differences of the composition rules. By interpreting the elementary weight function as a mapping from input values to output values and introducing some special mappings, we are able to derive the order conditions of several types of integration methods in a straight-forward way. The simplicity of the derivation is illustrated by linear multistep methods that use the second derivative as an input value, Runge-Kutta type methods that use the second as well as first derivatives, and general two-step Runge-Kutta methods. We derive new high stage-order methods in each example. In particular, we found a symmetric and stiffly-accurate method of order eight in the second example. [ABSTRACT FROM AUTHOR]


Sorting realization of well-ordered sets based on π-calculus.

by Hao Bu, Rong Zhu, Shihong Chen, Xiaoqiong Tan [2017-09-01]

Academic Journal

pages 17

π-calculus is one of the most effective means for the modeling of the concurrent mobile system at the present stage. The paper starts with a brief introduction of π-calculus, which concerns mutual simulation, the fundamentals of mobile computing, and the concept of interaction with some basic grammars and regulations of π-calculus while the system mobility is discussed with special focuses on the relation between the referent and the mobility, and the relation between the granular size and mobility. Then the analysis goes on to discuss the sorting problems that frequently occur in modeling concurrent systems. Through the discussion of the underlying sorting algorithms based on π-calculus, five processes are established by applying such artless thoughts as "looking for the seats in the cinema" or "changing keys among a group" to do quick concurrent locating and instant sorting, so that the sorting problems of specific elements in the well-ordered sets could be solved on the basis of π-calculus. In this way, the sorting results will eventually be displayed in the specified process and conveniently help the readings of other processes. [ABSTRACT FROM AUTHOR]


Academic Journal

pages 18

Using data from the first National study on high school preparation for college calculus success, the Factors Influencing College Success in Mathematics (FICSMath) project, this article connects student high school instructional experiences to college calculus performance. The findings reported here reveal that students were better prepared for college calculus success by high school instructional experiences that emphasized mathematical definitions, vocabulary, reasoning, functions, and hands-on activities. These findings serve to inform high school mathematics teachers about promising instructional practices. They can also inform teacher education programs about how to better prepare secondary mathematics educators to discuss conceptual understanding on the widely used Educative Teacher Performance Assessment (edTPA). [ABSTRACT FROM AUTHOR]


Miraculous Mathematics: André Bazin's Film Theory.

by Vacche, Angela Dalle [2016-05-01]

Academic Journal

pages 25

The article focuses on assessment of film critic André Bazin's theory for films. Topics discussed include adoption of Henri Bergson's philosophy for mathematical analysis by Bazin with calculus method for computation; criticism of the "Miracle in Milan" film under supervision of director Vittorio De Sica by Bazin with analysis of its special effects; and examination of neorealism in the film "The Bicycle Thief" by Bazin.