Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.

On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

The effect of calculators on the numerical problem-solving skills and attitudes of primary students towards mathematics /

The purpose of this study was to determine the effect that calculators have on the attitudes and numerical problem-solving skills of primary students. The sample used for this research was one of convenience. The sample consisted of two grade 3 classes within the York Region District School Board. The students in the experimental group used calculators for this problem-solving unit. The students in the control group completed the same numerical problem-solving unit without the use of calculators. The pretest-posttest control group design was used for this study. All students involved in this study completed a computational pretest and an attitude pretest. At the end of the study, the students completed a computational posttest. Five students from the experimental group and five students from the control group received their posttests in the form of a taped interview. At the end of the unit, all students completed the attitude scale that they had received before the numerical problem-solving unit once again. Data for qualitative analysis included anecdotal observations, journal entries, and transcribed interviews. The constant comparative method was used to analyze the qualitative data. A t test was also performed on the data to determine whether there were changes in test and attitude scores between the control and experimental group. Overall, the findings of this study support the hypothesis that calculators improve the attitudes of primary students toward mathematics. Also, there is some evidence to suggest that calculators improve the computational skills of grade 3 students.

The use of explicit algorithms and episodic context to teach subtraction to students with learning problems in mathematics /

This research attempted to address the question of the role of explicit algorithms and episodic contexts in the acquisition of computational procedures for regrouping in subtraction. Three groups of students having difficulty learning to subtract with regrouping were taught procedures for doing so through either an explicit algorithm, an episodic content or an examples approach. It was hypothesized that the use of an explicit algorithm represented in a flow chart format would facilitate the acquisition and retention of specific procedural steps relative to the other two conditions. On the other hand, the use of paragraph stories to create episodic content was expected to facilitate the retrieval of algorithms, particularly in a mixed presentation format. The subjects were tested on similar, near, and far transfer questions over a four-day period. Near and far transfer algorithms were also introduced on Day Two. The results suggested that both explicit and episodic context facilitate performance on questions requiring subtraction with regrouping. However, the differential effects of these two approaches on near and far transfer questions were not as easy to identify. Explicit algorithms may facilitate the acquisition of specific procedural steps while at the same time inhibiting the application of such steps to transfer questions. Similarly, the value of episodic context in cuing the retrieval of an algorithm may be limited by the ability of a subject to identify and classify a new question as an exemplar of a particular episodically deflned problem type or category. The implications of these findings in relation to the procedures employed in the teaching of Mathematics to students with learning problems are discussed in detail.

Gender bias in the Trends in Mathematics and Science Study 2003 (TIMMS) for Canadian students /

This study is a secondary data analysis of the Trends in Mathematics and Science Study 2003 (TIMSS) to determine if there is a gender bias, unbalanced number of items suited to the cognitive skill of one gender, and to compare performance by location. Results of the Grade 8, math portion of the test were examined. Items were coded as verbal, spatial, verbal /spatial or neither and as conventional or unconventional. A Kruskal- Wallis was completed for each category, comparing performance of students from Ontario, Quebec, and Singapore. A Factor Analysis was completed to determine if there were item categories with similar characteristics. Gender differences favouring males were found in the verbal conventional category for Canadian students and in the spatial conventional category for students in Quebec. The greatest differences were by location, as students in Singapore outperformed students from Canada in all areas except for the spatial unconventional category. Finally, whether an item is conventional or unconventional is more important than whether the item is verbal or spatial. Results show the importance of fair assessment for the genders in both the classroom and on standardized tests.

Adolescent girls and mathematics: classroom performance, attitudes toward mathematics and perceptions of parental attitudes /

The purpose of this study was to determine the extent to which gender differences exist in student attitudes toward mathematics and in their performance in mathematics at the Grade Seven and Eight level. The study also questioned how parents influence the attitudes of this grade level of male and female students toward mathematics. Historically, the literature has demonstrated gender differences in the attitudes of students toward mathematics, and in parental support for classroom performance in mathematics. This study was an attempt to examine these differences at one senior public school in the Peel Board of Education. One hundred three Grade Seven and Eight students at a middle school in the Peel Board of Education volunteered to take part in a survey that examined their attitudes toward mathematics, their perceptions of their parents' attitudes toward mathematics and support for good performance in the mathematics classroom, parental expectations for education and future career choices. Gender differences related to performance levels in the mathematics classroom were examined using Pearson contingency analyses. Items from the survey that showed significant differences involved confidence in mathematics and confidence in writing mathematics tests, as well as a belief in the ability to work on mathematics problems. Male students in both the high and low performance groups demonstrated higher levels of confidence than the females in those groups. Female students, however, indicated interest in careers that would require training and knowledge of higher mathematics. Some of the reasons given to explain the gender differences in confidence levels included socialization pressures on females, peer acceptance, and attribution of success. Perceived parental support showed no significant differences across gender groups or performance levels. Possible explanations dealt with the family structure of the participants in the study. Studies that, in the past, have demonstrated gender differences in confidence levels were supported by this study, and discussed in detail. Studies that reported on differences in parental support for student performance, based on the gender of the parent, were not confirmed by this study, and reasons for this were also discussed. The implications for the classroom include: 1) build on the female students' strengths that will allow them to enjoy their experiences in mathematics; 2) stop using the boys as a comparison group; and 3) make students more aware of the need to continue studying mathematics to ensure a wider choice of future careers.

An analysis of the photoelectron and Rydberg states of formaldehyde /|nby Carol R. Lessard. -- 260 St. Catharines [Ont.] : Dept. of Chemistry, Brock University,

Although it is generally accepted that Rydberg orbitals are very large and diffuse, and that electron promotion to a Rydberg orbital is not too different from ionization of the molecule, analysis of the two types of transitions proves otherwise. The photoelectron spectrum of the 2B2 (n) ion has very little vibrational structure attached to the origin band; on the other hand, several of the Rydberg transitions which involve the promotion of the n(bZ) electron exhibit a great deal of vibrational activity. In particular, the members of the n=3 Rydberg\ series interact with and perturb each other through pseudo-Jahn-Teller vibronic coupling. The vacuum ultraviolet spectrum contains a number of features which are difficult to explain, and two unusually sharp bands can only be identified as representing some form of electron promotion in formaldehyde.

An examination of grade three mathematics textbooks for problem solving strategies

Three grade three mathematics textbooks were selected arbitrarily (every other) from a total of six currently used in the schools of Ontario. These textbooks were examined through content analysis in order to determine the extent (i. e., the frequency of occurrence) to which problem solving strategies appear in the problems and exercises of grade three mathematics textbooks, and how well they carry through the Ministry's educational goals set out in The Formative Years. Based on Polya's heuristic model, a checklist was developed by the researcher. The checklist had two main categories, textbook problems and process problems and a finer classification according to the difficulty level of a textbook problem; also six commonly used problem solving strategies for the analysis of a process problem. Topics to be analyzed were selected from the subject guideline The Formative Years, and the same topics were selected from each textbook. Frequencies of analyzed problems and exercises were compiled and tabulated textbook by textbook and topic by topic. In making comparisons, simple frequency count and percentage were used in the absence of any known criteria available for judging highor low frequency. Each textbook was coded by three coders trained to use the checklist. The results of analysis showed that while there were large numbers of exercises in each textbook, not very many were framed as problems according to Polya' s model and that process problems form a small fraction of the number of analyzed problems and exercises. There was no pattern observed as to the systematic placement of problems in the textbooks.

A study of thermal decompositions of an allylic hydroperoxide /|nby Thomas A. McCarrick. -- 260 St. Catharines [Ont.] : Dept. of Chemistry, Brock University,

The thermal decomposition of 2,3-di~ethy l - J-hydr operox y- 1 - butene , p r epared f rol") singl e t oxygen, has been studied i n three solvents over the tempe r a ture r ange from 1500e to l o00e and t!1e i 111 t ial ~oncentrfttl nn r Ange from O. 01 M to 0.2 M. Analys i s of the kine tic data ind ica te s i nduced homolysis as the n ost probRble mode of d e composition, g iving rise to a 3/2 f S order dependence upon hy d.roperoxide concent :r8.tl on . Experimental activation e nergies for the decomposition were f ound to be between 29.5 kcsl./raole and 30.0 k cal./mole .• \,iith log A factors between 11 . 3 and 12.3. Product studies were conducted in R variety of solvents a s well as in the pr esence of a variety of free r adical initiators . Investigation of the kinetic ch a in length indicated a chain length of about fifty. A degenerat i ve chain branching mechanism 1s proposed which predicts the multi t ude of products which Rre observed e xperimentally as well as giving activation energies and log A factors si~il a r to those found experimentally .

Exploring the impact of a hands-on intervention on grade 7 girls and their feelings about self, mathematics, science, technology and related careers

Forty-five 12- and 13-year-old females attending Grade 7 in North York, Ontario were randomly selected from a group of 100 females who had volunteered to participate in a oneday hands-on workshop called It's Your Choice at Seneca College. The goals of this intervention were to broaden the career horizons of these students and to help them realize the need to continue mathematics and science through high school in order to keep occupational options unlimited. The young women were given a pre- and post-attitude survey to provide background information. In the month following participation in the workshop the students were interviewed in small groups (S students per group) to discover their perceptions of the impact of the workshop. The interviews revealed that participants felt that after the workshop their feelings of self-confidence increased, specifically with respect to working with their hands. Participants felt more aware of the usefulness and importance of the study of mathematics, science and technology, They also felt that It's Your Choice increased their interest in careers in these domains and helped them to see that these careers are viable choices for females. The interviews also revealed that many of the participants felt that in this society their roles and their choices were influenced and probably limited by the fact that they are female.

Reduced local energy calculations on X¹Sigmaâ ½gHâ and 1¹S He /|nGerald F. Thomas. -- 260 St. Catharines [Ont.] : Dept. of Chemistry, Brock University,

The one-electron reduced local energy function, t ~ , is introduced and has the property < tL)=(~>. It is suggested that the accuracy of SL reflects the local accuracy of an approximate wavefunction. We establish that <~~>~ <~2,> and present a bound formula, E~ , which is such that where Ew is Weinstein's lower bound formula to the ground state. The nature of the bound is not guaranteed but for sufficiently accurate wavefunctions it will yield a lower bound. ,-+ 1'S I I Applications to X LW Hz. and ne are presented.

Some aspects of the chemistry of Reissert compounds / |nDurga Prasad Aysola. -- 260 St. Catharines [Ont. : s. n.],

1-(0- and m-Ohlorobenzoyl)isoquinolines have been synthesized by two routes involving Reissert compounds. One route involves condensation of 2-benzoyl-l,2-dihydroisoquinaldonitrile with the appropriate chlorobenzaldehyde and the second involves rearrangement of the appropriate Z-(chlorobenzoyl)-l,Z-dihydroisoquinaldonitrile under basic conditions. The action of potassamide in anhydrous liquid ammonia on both ketones gave unexpectedly N-(l-isoquinolyl)benzamide (67) as the major product and the use of dibenzo-18-crown-6-ether 98% substantially improved the yd..e.ld in the case of l-chloroketone. This amide (67) exhibits unusual hydrogen bonding. 1-(o-chlorobenzoyl)-6,7-dimethoxyisoquinoline (79) was prepared in very s,amll quantities by the route involving condensation of 2-benzoyll, Z-dihydro-6,7-dimethoxyisoquinaldonitrile with o-chlorobenzaldehyde. The poor yields are due to the instability of the anion of 2-benzoyl1, Z-dihydro-6,7-dimethoxyisoquinaldonitrile. Attempted preparation of the ketone (79) by rearrangement of 2-(o-chlorobenzoyl)-l,2-dihydro6,7- dimethoxyisoquinaldonitrile under basic conditions yielded the start~ng material (Reissert compound) and 6,7-dimethoxyisoquinoline. The action of potassamide in anhydrous liquid ammonia on l-(o-bromo-4,5-dimethoxybenzoyl)isoquinoline (85), which was prepared by the route involving the condensation of 2-benzoyl-l,4-dihydroisoquinaldonitrile with o-bromo-4,5-dimethoxybenzaldehyde, gave two products, which have not yet been identified. The ketone (85) and its precursors are interest~ng in that their 20 eV and 70 eV mass spectra do not show molecular ions.

Some aspects of the chemistry of 1, 3, 4 - Thiadiazolium salts and related compounds

The work described in this thesis has been divided into seven sections. The first section involves the preparation of N'-acyl-N'-arylN- benzothiohydrazides by the acylation of N'-aryl-N-benzothiohydrazides and is followed by a brief discussion of their possible conformation in solution. The second section deals with the preparation of 1,3,4-thiadiazolium salts by the action of perchloric acid/acetic anhydride on N'-acylN'- aryl-N-benzothiohydrazides and also by the reaction of N'-arylN- benzothiohydrazides with nitriles in an acidic medium. The preparation of 2-methylthio-I,3,4-thiadiazolium methosulfate by methylating the corresponding thione is also described. The third section deals with the reaction of 2-phenyl- and 2-methyl-I,3,4-thiadiazolium salts with alcohols in the presence of base. The stability and spectra of these compounds are discussed. Treatment of the 2-methyl-I,3,4-thiadiazolium salt with base was found to give rise to a dimeric anhydrobase and evidence supporting its structure is given. The anhydrobase could be trapped by a variety of acylating and thioacylating agents before dimerization occurred. In the fourth section, the reaction of N'-acyl-N'-aryl-N-benzothiohydrazides with a variety of acid anhydrides is described. These compounds were found to be identical with those obtained by acylating the anhydrobase. The mass spectral fragmentation of these compounds is described and the anomolous product obtained upon thiobenzoylation of 3-methyl-l-phenyl-pyrazal-5-one is also discussed. The fifth section deals with thioacyl derivatives of the anhydrobase which were prepared by the action of phosphorus pentasulfide upon the oxygen analogues and also obtained as the major product of the reaction of thioacetic acid with compounds related to N'-aryl-N-benzothiohydrazides. The mass spectra and p.m.r. spectra of these compounds are discussed. In the sixth section, the reaction of the 2-methylthio-l,3,4- thiadiazolium salt with active methylene compounds to give acyl and diacyl derivatives of the anhydrobase is described. Some aspects of these compounds are discussed. The seventh section describes the synthesis of ncyanine~' type dyes incorporating the l,3,4-thiadiazole ring and their spectra are briefly discussed.

The chemistry of heterocyclic methine bases and of their acyl and thioacyl derivatives

The work described in this thesis has been dtvided into six sections . The first section involves the reaction of 3,5-diphenyl-2-methyl-l,3,4-oxadiazolium perchlorate with acetic and benzoic anhydrides. The second section deals with the preparation and reactions of 1,3,4-thia diazolium salts. Some monomeric 1,3,4-thiadiazoline methine bases have also been prepared by reacting 1,3,4-thia d iaz ol ium s al t s with concen trated ammonium hydroxide solution. Variable temperature p.m.r. of 2-(3-acetylacetonylidene)-3,5-diphenyl-A4 -1,3,4-thiadiazoline has also been described. The third section deals with prepar a tion and reactions of some compounds in benzoxazole series. The fourth section deals with the prep a ration and reactions of N-alkyl-2-methylbenzothi azolium salts with base , a nd with some a cetylating and thioacetylating agents. Treatment of 2,3-dimethylbenzothiazolium iodide and of 3-ethyl-2-methylbenzothia zolium iodide with base wa s found to give the corresponding dimeric methine b a ses and evidence supporting their structure is also given. Thiol acetic acid was found to exchange 0 for S in its reactions with 2-acetonylidene-3-methylbenzothiazoline and 2-acetophenonylidene-3-methylbenzothi a zoline. (ii) In th e fifth section, the r eactions of 2,3-dimethylbenzselenazolium iodide with a variety of ac e tylating and thioacetylating agents has been described. The treatment of 2,3-dimethylbenzselenazolium iodide with base was found to give rise to a dimeric methine base and evidence supporting its structure is also given. The reactions of this dimeric methine b a se with benzoic anhydride and phenylisothiocyanate have also been described. The sixth section deals with the preparation and reactions of l-alkyl-2-methylquinolinium salts. Treatment of 1,2-dimethylquinolinium iodide and l-ethyl-2-methylquinolinium iodide was found to give the corresponding monomeric methine bases and evidence supporting their structure is also given. The E-type geometry of the olefinic bond in 2-acetonylidene-l-methylquinoline has been established on the basis of an N.O.E. experiment.