Proof Of Equivalence Research Articles

A Note on Certain Modular Equations about Infinite Products of Ramanujan

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In the same spirit, we also could present elementary and simple proofs of certain Ramanujan's modular equations on infinite products.

Infinitary rewriting: closure operators, equivalences and models

Infinitary Term Rewriting allows to express infinite terms and transfinite reductions that converge to those terms. Underpinning the machinery of infinitary rewriting are closure operators on relations that facilitate the formation of transfinite reductions and transfinite equivalence proofs. The literature on infinitary rewriting has largely neglected to single out such closure operators, leaving them implicit in definitions of (transfinite) rewrite reductions, or equivalence relations. This paper unpicks some of those definitions, extracting the underlying closure principles used, as well as constructing alternative operators that lead to alternative notions of reduction and equivalence. A consequence of this unpicking is an insight into the abstraction level at which these operators can be defined. Some of the material in this paper already appeared in Kahrs (2010). The paper also generalises the notion of equational model for infinitary rewriting. This leads to semantics-based notions of equivalence that tie in with the equivalences constructed from the closure operators.

Leukotriene antagonists in asthma

This recent article published in The New England Journal of Medicine describes a new style of clinical trial, a pragmatic trial. The purpose of the pragmatic design is to take into account the vagaries of clinical practice and simulate ‘real life’. This trial highlights the difficulties in the performance, reporting and interpretation of such trials. The paper includes two parallel, multicentre pragmatic trials that compare leukotriene antagonists and inhaled corticosteroids as first line (n = 306) and add on (n = 352) asthma controller therapy in a primary care setting in the UK. The primary outcome measure was the Mini Asthma Quality of Life Questionnaire. In both trials, there was no difference between groups in the mini AQLQ score [−0.11(CI −0.35 to 0.13) and −0.11 (95% CI, −0.32 to 0.11)] or secondary outcome measures at two months or at two years (see Fig. 1). However, there are many factors producing bias towards equivalence in this study, including enrolment of patients without disease, poor adherence, open label design, use of concomitant therapy, lack of placebo and lack of objective outcome measures. We are concerned that this trial, published in such a high-impact journal, will be misinterpreted by busy clinicians as proof of equivalence between the two therapies in question. Although this paper provides a useful focus for discussion of pragmatic trials, we strongly believe that its results should not influence clinical practice. Time Course of Improvements in ACQ and MiniAQLQ Scores and Peak Expiratory Flow over a Two-Year Period in Patients with Asthma. Panel A shows outcomes over a two-year period for patients receiving a leukotriene-receptor antagonist (LTRa) or an inhaled glucocorticoid as first-line asthma-controller therapy. Panel B shows outcomes over a two-year period for patients receiving an LTRa or a long-acting beta2-agonist (LaBa) as an add-on to an inhaled glucocorticoid. ACQ denotes Asthma Control Questionnaire (on which scores (shown as means ± SD) range from 0 to 6, with higher scores representing worse control and a minimal clinically important difference (MID) of 0.5). Mini Asthma Quality of Life Questionnaire (Minia QLQ) (on which scores (shown as means ± SD) range from 1 to 7, with higher scores representing better quality of life and an MID of 0.5), and PEF peak expiratory flow (shown as medians with I bars representing the interquartile ranges). Reviewers: John Widger and John Massie, Royal Children's Hospital, Melbourne ([email protected])

Comment on “Toward General Type-2 Fuzzy Logic Systems Based on zSlices”

Wagner and Hagras introduced a novel defuzzification formula in their recent paper and showed that it works very well within the framework of their general type-2 fuzzy logic systems based on zSlices ( α-plane representation). This letter aims to point out the hidden connection between the standard centroid defuzzification formula and Wagner and Hagras' new defuzzification formula, which leads to the proof of complete equivalence of the two.

Equivalence checking of static affine programs using widening to handle recurrences

Designers often apply manual or semi-automatic loop and data transformations on array- and loop-intensive programs to improve performance. It is crucial that such transformations preserve the functionality of the program. This article presents an automatic method for constructing equivalence proofs for the class of static affine programs. The equivalence checking is performed on a dependence graph abstraction and uses a new approach based on widening to find the proper induction hypotheses for reasoning about recurrences. Unlike transitive-closure-based approaches, this widening approach can also handle nonuniform recurrences. The implementation is publicly available and is the first of its kind to fully support commutative operations.

Standard SANC modules for NLO QCD radiative corrections to single top-quark production

In this paper we present the results obtained with the newly created standard SANC modules for calculation of the NLO QCD corrections to single top-quark-production processes in s and t channels at the partonic level, as well as to top-decays. The main aim of these results is to prove the correct work of modules. A comprehensive comparison with the results of the CompHEP system is given, where possible. These modules are intended to be used in Monte Carlo generators for single top-quark-production processes at the LHC. As in our recent paper, devoted to the electroweak corrections to these processes, we study the regularization of the top-legs associated infrared divergences with the aid of the complex mass of the top quark. A comparison of QCD corrections with those computed by the conventional method is presented both for top production and for decays. For s-channel production we give an analytic proof of equivalence of the two methods in the limit of low top width.

New modular equations in the spirit of Ramanujan

We give proofs for new Ramanujan-type modular equations discovered by M. Somos and establish applications of some of them.

Decorated proofs for computational effects: States

The syntax of an imperative language does not mention explicitly the state, while its denotational semantics has to mention it. In this paper we show that the equational proofs about an imperative language may hide the state, in the same way as the syntax does.

Automatic equivalence proofs for non-deterministic coalgebras

A notion of generalized regular expressions for a large class of systems modeled as coalgebras, and an analog of Kleene’s theorem and Kleene algebra, were recently proposed by a subset of the authors of this paper. Examples of the systems covered include infinite streams, deterministic automata, Mealy machines and labeled transition systems. In this paper, we present a novel algorithm to decide whether two expressions are bisimilar or not. The procedure is implemented in the automatic theorem prover CIRC, by reducing coinduction to an entailment relation between an algebraic specification and an appropriate set of equations. We illustrate the generality of the tool with three examples: infinite streams of real numbers, Mealy machines and labeled transition systems.

Relating Algebraic and Coalgebraic Descriptions of Lenses

Lenses are a heavily studied form of bidirectional transformation, with diverse applications including database view updating, software development and memory management. Previous work has explored lenses category-theoretically, and established that the category of lenses for a fixed 'view' V is, up to isomor- phism, the category of algebras for a particular monad on set=V. It has recently been shown that lenses are the coalgebras for the comonad generated by the carte- sian closure adjunction on set. In this paper, we present an equational proof of the coalgebra correspondence, note that the algebra correspondence extends to arbitrary categories with products and that the coalgebra correspondence extends to arbitrary cartesian closed categories, and show that both correspondences extend to isomor- phisms of categories. The resulting isomorphism between a category of algebras and a category of coalgebras is unexpected, and we analyze its underlying general- ity and the particularity that restricts its applicability. We end with remarks about the utility of the two different treatments of lenses, especially for obtaining further, more realistic, generalizations of the notion of lens.

Proof of the perpetuity equation

A perpetuity is a perpetual annuity. Although there have been a number of different derivations, which we discuss in detail, we present what appears to be the first mathematical proof of the perpetuity equation based on the fundamental properties of the real numbers (Result (2.2.1) of Dieudonne (1960) [14] ).

Equivalence between the mechanical model and energy-transfer theory for the classical decay rates of molecules near a spherical particle

In the classical modeling of decay rates for molecules interacting with a nontrivial environment, it is well known that two alternate approaches exist which include: (1) a mechanical model treating the system as a damped harmonic oscillator driven by the reflected fields from the environment; and (2) a model based on the radiative and nonradiative energy transfers from the excited molecular system to the environment. While the exact equivalence of the two methods is not trivial and has been explicitly demonstrated only for planar geometry, it has been widely taken for granted and applied to other geometries such as in the interaction of the molecule with a spherical particle. Here we provide a rigorous proof of such equivalence for the molecule-sphere problem via a direct calculation of the decay rates adopting each of the two different approaches.

Equivalent Proof and Applicability Evaluation of Two Types of Simplified Bishop’s Methods

In order to consider the influence of water pressure on cohesive soil slope with groundwater, the local and global water pressure methods are proposed. The equivalence and applicability of the two methods based on the simplified Bishop’s method are analyzed. The local water pressure method considers the water pressures on boundaries as components and the global water pressure method considers the water pressures as a resultant force. The physical analysis and numerical verification showed these two methods are completely equivalent. While the global water pressure method is more convenient in hydrostatic condition, the local water pressure method is prior in seepage condition. It is also pointed out that there are significant defects in some regular expressions of simplified Bishop’s method. Because they neglect the moment generated by water pressure on the top boundary of slope or wrongly evaluate of the hydrodynamic failure states with hydrostatic stress parameters.

First-order reasoning for higher-order concurrency

We present a practical first-order theory of a higher-order π-calculus which is both sound and complete with respect to a standard semantic equivalence. The theory is a product of combining and simplifying two of the most prominent theories for HOπ of Sangiorgi et al. and Jeffrey and Rathke [10,21], and a novel approach to scope extrusion. In this way we obtain an elementary labelled transition system where the standard theory of first-order weak bisimulation and its corresponding propositional Hennessy–Milner logic can be applied.The usefulness of our theory is demonstrated by straightforward proofs of equivalences between compact but intricate higher-order processes using witness first-order bisimulations, and proofs of inequivalence using the propositional Hennessy–Milner logic. Finally we show that contextual equivalence in a higher-order setting is a conservative extension of the first-order π-calculus.

On Carlitz’s trilinear generating functions

In this paper, Carlitz’s trilinear generating functions for Rogers–Szegö and Hahn polynomials are obtained by using Carlitz’s q-operator. In addition, an equivalent proof of Srivastava–Jain’s mixed trilinear generating function is gained. Moreover, a Carlitz’s trilinear generating function is deduced. At last, an application to the multiple Carlitz’s generating functions for Hahn polynomials is given.

Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side

We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally 𝒞0, α. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.

An evolution strategy in structural optimization problems for plates and shells

For multilayered plated and shell structures the formulation of the optimization problem is strongly dependant on the definition of the design variables. Therefore, the first part of the work is devoted to the definition of design variables and the forms of objective functions. Those design variables define stacking sequences of structures have discrete fiber orientations 0°, ±45°, 90° and a finite number of key points that are required in the evaluation of the curve Γ characterizing an external boundary of the structure or a structural shape understood in the sense of a structural geometry representing a shell/plate mid-surface or thickness distribution of structures. For the curve definition we have adopted one dimensional B-splines. Each curve is formed by an assembly of subsegments passing through certain key points. The positions of key points are randomly generated so that in the generation process it is possible to fulfill the required set of equality or inequality constraints. It is necessary to emphasize that the proposed method is very general and can be applicable to a very broad class of optimization problems. The generality of the approach is confirmed by the proof of the direct equivalence and mapping between discrete fiber orientations and continuous angle ply orientations. The evolution strategy is proposed herein as the optimization algorithm. Similarly as classical ones (e.g. ACO, SS, PS or ISM) it combines all features and advantages of evolution algorithms. It is worth to note that in the evolution strategy the number of children produced in one generation is not limited and it is not necessary to conduct mutation operations as in genetic algorithms. It simplifies significantly the effectiveness of numerical procedures. Then, two numerical examples have been solved to demonstrate the effectiveness of the proposed formulations and the optimization algorithm. They deal with thickness and stacking sequence optimization problems for circular cylindrical shells subjected to various dynamic and static constraints, respectively.

2-Dimensional Directed Type Theory

Recent work on higher-dimensional type theory has explored connections between Martin-Löf type theory, higher-dimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSetA B to be the isomorphisms between A and B. The crucial observation is that all of the familiar type and term constructors can be equipped with a functorial action that describes how they preserve such proofs. The key benefit of higher-dimensional type theory is that programmers and mathematicians may work up to isomorphism and higher equivalence, such as equivalence of categories.In this paper, we consider a further generalization of higher-dimensional type theory, which associates each type with a directed notion of transformation between its elements. Directed type theory accounts for phenomena not expressible in symmetric higher-dimensional type theory, such as a universe set of sets and functions, and a type Ctx used in functorial abstract syntax. Our formulation requires two main ingredients: First, the types themselves must be reinterpreted to take account of variance; for example, a Π type is contravariant in its domain, but covariant in its range. Second, whereas in symmetric type theory proofs of equivalence can be internalized using the Martin-Löf identity type, in directed type theory the two-dimensional structure must be made explicit at the judgemental level. We describe a 2-dimensional directed type theory, or 2DTT, which is validated by an interpretation into the strict 2-category Cat of categories, functors, and natural transformations. We also discuss applications of 2DTT for programming with abstract syntax, generalizing the functorial approach to syntax to the dependently typed and mixed-variance case.

유연한 수학적 사고에 의한 개념의 동치성 비교 - 사례 연구 -

The flexible mathematical thinking - the ability to generate and connect various representations of concepts - is useful in understanding mathematical structure and variation in problem solving. In particular, the flexible mathematical thinking with the inventive mathematical thinking, the original mathematical problem solving ability and the mathematical invention is a core concept, which must be emphasized in all branches of mathematical education. In this paper, the author considered a case of flexible mathematical thinking with an inventive problem solving ability shown by his student via real analysis courses. The case is on the proofs of the equivalences of three different definitions on the concept of limit superior shown in three different real analysis books. Proving the equivalences of the three definitions, the student tried to keep the flexible mathematical thinking steadily.

The Equivalence of Tree Adjoining Grammars and Monadic Linear Context-free Tree Grammars

The equivalence of leaf languages of tree adjoining grammars and monadic linear context-free grammars was shown about a decade ago. This paper presents a proof of the strong equivalence of these grammar formalisms. Non-strict tree adjoining grammars and monadic linear context-free grammars define the same class of tree languages. We also present a logical characterisation of this tree language class showing that a tree language is a member of this class iff it is the two-dimensional yield of an MSO-definable three-dimensional tree language.