The modal logic of Reverse Mathematics

The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results

Mathematics Abilities of Physics Students: Implication for the Application and Analysis of Sound Waves

The study investigated the effect of mathematics abilities of students on their performance in Sound Waves concept in physics in Ikwerre Local Government Area of Rivers State, Nigeria. A quasi-experimental pretest posttest design comprising of three experimental and one control group was used, each group was taught with a different Instructional method. A purposively selected sample of fifty- five (55) physics students of Senior Secondary 2 (SS2) class was involved in the study. Two instruments- Mathematics Ability Test (MAT) and Physics Performance Test on Sound Waves (PPTSW) with reliability coefficients of 0.97 and 0.85 respectively were used. The performances of the students were considered at the levels of application and analysis of Sound waves. Data collected was analysed using Mean scores and Percentages for the research questions, while 4x3 Multivariate Analysis of Covariance was used to test the hypotheses. Analysis of results showed that there was a significant difference in the effect of mathematics abilities of students on their performances in Sound waves. There was also a significant difference in the effect instructional methods on the performance of the students in Sound waves. The Post hoc analyses showed that the significant difference in the mathematics abilities was credited to students with high mathematics ability while Guided-Discovery method accounted for the significant difference found in instructional methods. The Implications of the findings were discussed and relevant recommendations made thereafter. Keywords: Mathematics Abilities, Sound Waves, Physics, Instructional method

A new process foundation for the applied topos

The world is in turmoil for want of sound reasoning. Economics and the environment are but two of many areas of human endeavour badly betrayed through a failed combination of physical and information science and the rule of law. Logic is the fabric of pure mathematics as the foundation of applied mathematics on which all science is based from the physical through biological and medical to the social sciences. However the symbolic logic of today seems of scarce more use than the syllogisms of Aristotle as observed by Francis Bacon nearly 400 years ago: The logic now in use serves rather to fix and give stability to the errors which have their foundation in commonly received notions than to help the search after truth. So it does more harm than good [Novum Organon Aphorism XII, 1620]

MATHEMUSIC – Numbers and Notes A Mathematical Approach To Musical Frequencies

Mathematics and Music, the most sharply contrasted fields of scientific activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind [8]. Music theorists sometimes use mathematics to understand music. Mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical". In today’s technology, without mathematics it is difficult to imagine anything feasible. In this paper we have discussed the relation between music and mathematics. How piano keys are interrelated with mathematics, frequencies are correlated and discussed.  Frequencies of musical instrument (piano) are analyzed using regression and geometric progression. Comparisons   between both the methods are done in this paper. This paper will also be helpful for music seekers and mathematician to understand easily and practicing of musical instruments. Keywords: Musical notes, Regression analysis, Geometric progression

Training mathematical skills in school children: Some preliminary results.

The present study is a further development of an earlier research on school failure. The aim is to develop sound, simple, effective evaluation and training procedures for children with difficulties to learn mathematics. A review of research on mathematical knowledge reveals some confusion in using terms and a lack of empirical studies on intervention strategies. Nevertheless, some progress has been done in defining the necessary skills for children to solve mathematical problems. In the current study, two cases of primary school children (ISCED 1) with difficulties to learn mathematics are presented. They were trained to acquire a set of pre-mathematical or mathematical skills. The core of the intervention procedures is Behavior Skills Training (BST), a highly effective technique for teaching individuals with different disabilities a wide variety of skills. Evaluation of the training is carried out by comparing the percentage of attained objectives before (pre-test) and after training (post-test). Future developments of this research are explaine

Satisfiability Modulo Transcendental Functions via Incremental Linearization

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted in the abstract space, which is described in terms of the combined theory of linear arithmetic on the rationals with uninterpreted functions, and are incrementally axiomatized by means of upper- and lower-bounding piecewise-linear functions. Suitable numerical techniques are used to ensure that the abstractions of the transcendental functions are sound even in presence of irrationals. Our experimental evaluation on benchmarks from verification and mathematics demonstrates the potential of our approach, showing that it compares favorably with delta-satisfiability /interval propagation and methods based on theorem proving

Groups of order 8 and 16

This document is rather a course about groups than a research paper. However, it can be of interest for many master students in mathematics which are devoted to the p-group classification theory. This paper is inspirated of the course of David Clausen from the university of Puget Sound (USA) on the classification of groups of order 16. His publication was made on GNU Licence. However, his publication contained some typo errors which make difficult the understanding of his method. Moreover, his proof, although interesting, is not the simplest. In my document, while keeping the development that he proposed, I greatly simplified his proof of classification of groups of order 16.Comment: 18 page

The Math of Music: An Algorithm of Allegros

Math and music, two things that to most people, seem entirely different. However, they are more alike in countless ways than they are different. As early as the Greek philosophers, there have been brilliant people who have dedicated their time to creating a link between math and music. It is in this attitude, that I attempt to bridge the relationship between math and music.At its core, music is created entirely of mathematics. The humming of strings and the strike of a piano key may indeed carry out sound, but it also carries a plethora of mathematical equations and theorems to support it. Whether it be a relation of Hertz, or even a derivative of a famous mathematical equation like Pythagorean’s Theorem, this paper promotes the ideal that mathematics can be used to describe every type of music.Whether you prefer rock to classical, rap to pop, all music is broken down into notes and chords. Sound is created using electronic equipment that has been perfected over centuries. Instruments have been refined to make the most brilliant sound that their materials allow, all of this and more is in part due to the brilliant relationship that music and mathematics share

A bifurcated circular waveguide problem

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The definitive publisher-authenticated version A D Rawlins. A bifurcated circular waveguide problem. J.I.M.A. 54 (1995) 59-81. Oxford University press is available online at: http://imamat.oxfordjournals.org/cgi/reprint/54/1/59.pdfA rigorous and exact solution is obtained for the problem of the radiation of sound from a semi-infinite rigid duct inserted axially into a larger acoustically lined tube of infinite length. The solution to this problem is obtained by the Wiener-Hopf technique. The transmission and reflection coefficients, when the fundamental mode propagates in the semi-infinite tube, are obtained. The present results could be of use for exhaust design, and as a possible instrument for impedance measurement