Mathematics

Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's work leads to (...) a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found. (shrink)

The Autism Spectrum Disorder (ASD) represents a set of life-long disorders. In particular, subjects with ASD can display momentary behaviors of acute agitation and aggressiveness called crisis behaviors. These events are problematic for the subject and care providers but little is known about their occurrence, namely, possible relations among intensity, frequency, and duration. A group of ASD subjects (n=33) has been observed for 12 months reporting data on each crisis ( n = 1137 crises). Statistical analysis did not find significant (...) results, while the relation between crisis duration and frequency showed a good fit to a “power law” curve, suggesting the application of Self-Organized Criticality (SOC) model. The SOC is used to describe natural phenomena as earthquakes, bank failures of rivers, mass extinctions, and other systems where a type of “catastrophic events” is necessary to maintain a critical equilibrium. In a sense, subjects at risk of crisis behavior seem to fit the same model as seismic zones at risk of earthquakes. The employment of the same strategies, as those successfully developed for known SOC systems, could lead to important insights for ASD management. Moreover, the SOC model offers possible interpretations of crisis behavior dynamics suggesting that they are unpredictable and, in a sense, necessary. (shrink)

A circumscription of the classical theory of computation building up from the Chomsky hierarchy.

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...) object that changes its shape from a triangle to a circle, and then back to a triangle with every second. (shrink)

Holding students to high standards and assessing, measuring and evaluating their learning with challenging, authentic problems in the midterm and final exams is the goal of the professors who teach core signal processing concepts. However, the heavy reliance of these subjects on mathematics makes it difficult for students to genuinely grasp the concepts and relate to a conceptual framework. Specifically, analyzing the signals and the functionality of systems in Fourier domain; separating the system level analysis from signal level analysis; and (...) understanding how they are related in time domain and frequency domain are among the most challenging concepts. Students’ lower grades observed over past years in the introductory signal processing course exposed a potential disconnect between the actual level of learning and the high expectations set by the professors. In this paper, we present the active learning techniques that we implemented in one of the summer session offerings of this course in our department. The research explored Peer Instruction, pre-class reading quizzes and post-lecture quizzes. In addition to the mid and end of the quarter survey results, the comparison analysis of the grades students achieved in the active learning integrated course in the second summer session and the standard course offered in first summer session is discussed. According to our results, the developed techniques helped students in the active classroom perform significantly better than their peers participating in standard lectures when tested by challenging questions in their exams. (shrink)

The standard model of subatomic particles and the periodic table of the atoms have the common goal to bring order in the bewildering chaos of the constituents of matter. Their success relies on the presence of fundamental symmetries in their core. The purpose of the book is to share the admiration for the power and the beauty of these symmetries. The reader is taken on a journey from the basic geometric symmetry group of a circle to the sublime dynamic symmetries (...) that govern the motions of the particles. The trail follows the lines of parentage linking groups upstream to the unitary symmetry of the eightfold way of quarks, and to the four-dimensional symmetry of the hydrogen atom. Along the way the theory of symmetry groups is gradually introduced with special emphasis on graphical representations. The final challenge is to open up the structure of Mendeleev's table which goes beyond the symmetry of the hydrogen atom. Breaking this symmetry to accommodate the multi-electron atoms requires to leave the common ground of linear algebras and explore the potential of non-linearity. (shrink)

I have been working for a long time about basic laws which direct existence, and some mathematical problems which are waited for a solution. I can count myself lucky, that I could make some important inferences during this time, and I published them in a few papers partially as some propositions. This work aimed to explain and discuss these inferences all together by relating them one another by some extra additions, corrections and explanations being physical phenomena are prior. There are (...) many motivation instruments for exact physical inferences. (shrink)

The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.

Dieses Buch ist ein wichtiges studienbegleitendes Hilfsmittel für alle die Mathematik-Lehrveranstaltungen besuchen. Die Lektüre dieses Buch ermöglicht Ihnen, begriffliche Sicherheit für Mathematik-Vorlesungen und Prüfungen aufzubauen. Die Lektüre jedes Kapitel dieses Buches erlaubt Ihnen, einen Überblick über die Begriffe eines Teilgebiets der Mathematik zu erhalten und diese Begriffe nachhaltig zu erfassen. Wenn Sie als Student einen mathematischen Begriff nicht richtig verstehen oder sich an seine Definition nicht erinnern, können Sie in diesem Buch nachschlagen und erhalten durch paradigmatische Beispiele und Bilder ein (...) fundiertes Verständnis des Begriffs. Arbeiten Sie ein Kapitel dieses Buches in Vorbereitung einer Prüfung durch, so können Sie sich in begrifflicher Hinsicht in der Prüfung sicher fühlen. Insgesamt finden sich in diesem Buch mehr als tausend Definitionen von Begriffen aus vierzehn Teilgebieten der Mathematik. Die Auswahl der Begriffe orientiert sich in jedem Kapitel an den Vorlesungen zum behandelten Thema, die an deutschen Hochschulen gehalten werden. Alle wesentlichen Begriffe, die in Mathematik-Vorlesungen in Bachelorstudiengängen vorkommen und auch alle grundlegenden Begriffe der Mathematik-Vorlesungen in Masterstudiengängen sind in diesem Buch enthalten. Dieses Buch stellt also einen Kanon mathematischer Begriffe vor, der auch für Lehrende von Interesse ist. (shrink)

In diesem Buch finden Sie Perlen der Mathematik aus 2500 Jahren, beginnend mit Pythagoras und Euklid über Euler und Gauß bis hin zu Poincaré und Erdös. Sie erhalten einen Überblick über schöne und zentrale mathematische Sätze aus neun unterschiedlichen Gebieten und einen Einblick in große elementare Vermutungen. -/- Die Vielfalt an schönen Resultaten bietet eine einzigartige mathematisch-allgemeinbildende Lektüre auf akademischem Niveau. -/- Die Beweise in diesem Buch sind möglichst einfach und kurz gehalten und vermitteln Ihnen wesentliche Ansätze, Ideen und Strategien (...) ohne große Vorkenntnisse vorauszusetzen. Die verwendeten Begriffe werden zumeist im Text eingeführt und zu grundlegenden Begriffen steht Ihnen zusätzlich ein Anhang zur Verfügung. -/- Als Student der Mathematik oder Naturwissenschaften können Sie das Buch verwenden, um Ihre Perspektive zu erweitern und Ihre mathematische Bildung zu vertiefen. Hochschullehrer können jedes Kapitel des Buches zur Ausgestaltung eines Proseminars heranziehen. Wenn Sie einfach nur an Mathematik interessiert sind, und die Analysis und Lineare Algebra ein wenig kennen, wird Sie dieses Buch in das Reich der reinen Mathematik entführen. -/- Die vorliegende zweite Auflage ist vollständig durchgesehen und um etliche Themen wie zum Beispiel den Satz von Hall und Kettenbrüche ergänzt. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...) question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)

In the paper it is demonstrated that Bell's theorem is unproveable.

The present Yearbook (which is the fourth in the series) is subtitled Trends & Cycles. Already ancient historians (see, e.g., the second Chapter of Book VI of Polybius' Histories) described rather well the cyclical component of historical dynamics, whereas new interesting analyses of such dynamics also appeared in the Medieval and Early Modern periods (see, e.g., Ibn Khaldūn 1958 [1377], or Machiavelli 1996 [1531] 1). This is not surprising as the cyclical dynamics was dominant in the agrarian social systems. With (...) modernization, the trend dynamics became much more pronounced and these are trends to which the students of modern societies pay more attention. Note that the term trend – as regards its contents and application – is tightly connected with a formal mathematical analysis. Trends may be described by various equations – linear, exponential, power-law, etc. On the other hand, the cliodynamic research has demonstrated that the cyclical historical dynamics can be also modeled mathematically in a rather effective way (see, e.g., Usher 1989; Chu and Lee 1994; Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Turchin and Nefedov 2009; Nefedov 2004; Korotayev and Komarova 2004; Korotayev, Malkov, and Khaltourina 2006; Korotayev and Khaltourina 2006; Korotayev 2007; Grinin 2007), whereas the trend and cycle components of historical dynamics turn out to be of equal importance. (shrink)

The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical (...) methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink)

After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden (...) ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. (shrink)

From the exponential function of Euler’s equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures.

In this paper, I recreate a paradox from my earlier work (“Paradox with just self-reference”) albeit entirely within the language of arithmetic. (In lieu of self-referential expressions, Gödel numbering is used to generate the paradox.) This can suggest that Robinson arithmetic and its extensions are unsound; however, I claim instead that the metalanguage may be to blame, owing to arithmetization of syntax. If so, then the moral would be to restrict arithmetization in the metalanguage somehow, rather than distrust the arithmetical (...) theory in the object language. (shrink)

Numerous studies show that humans tend to be more cooperative than expected given the assumption that they are rational maximizers of personal gain. As a result, theoreticians have proposed elaborated formal representations of human decision-making, in which utility functions including “altruistic” or “moral” preferences replace the purely self-oriented "Homo economicus" function. Here we review mathematical approaches that provide insights into the mathematical stability of alternative ways of representing human decision-making in social contexts. Candidate utility functions may be evaluated with help (...) of game theory, classical modeling of social evolution that focuses on behavioral strategies, and modeling of social evolution that focuses directly on utility functions. We present the advantages of the latter form of investigation and discuss one surprisingly precise result: “Homo economicus” as well as “altruistic” utility functions are less stable than a function containing a preference for the golden rule or for the common welfare that is only expressed in social contexts composed of individuals with similar preferences. We discuss the contribution of mathematical models to our understanding of human other-oriented behavior, with a focus on the classical debate over psychological altruism. We conclude that human can be psychologically altruistic, but that psychological altruism evolved because it was generally expressed towards individuals that contributed to the actor’s fitness, such as own children, romantic partners and long term reciprocators. (shrink)

A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.

It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist non-standard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheim-Skolem theorem, most theorists agree that the number theoretic version (...) does not have skeptical consequences about the reference of “natural number” analogous to the ‘relativity’ Skolem claimed pertains to notions such as “uncountable” and “cardinal.” In this paper I argue that recent proposals by Shapiro, Lavine, McGee and Field which aim to distinguish the number and set theoretic indeterminacy arguments by locating extra-mathematical constraints on the interpretation of our number theoretic vocabulary are inadequate. I then suggest that if we consider the manner in which the natural numbers figure in our computational practices – e.g. in characterizing how we compute sums and products or determine prime factorizations – it follows that our application of computational terms such as “computable function” and “tractable computational problem” indirectly uniquely determines their structure. (shrink)

We introduce the category of mereotopology Mtop as an alternative category to that of topology Top, stating ontological consequences throughout. We consider entities such as boundaries utilizing Brentano’s thesis and holes utilizing homotopy theory with a rigorous proof of Hausdorff Spaces satisfying [GEM]TC axioms. Lastly, we mention further areas of study in this category.

The purpose of this study was to determine whether the application of learning brain-friendly through the whole brain teaching a positive effect on the character of creative students, to study the response of the students, and to determine whether the students' response to the application of learning brain-friendly through the whole brain teaching positively correlated with the character of creative students in mathematics. The research method used is quantitative. The instruments used are student questionnaire responses related to the application of (...) brain-friendly learning through the whole brain teaching and observation sheet on student creativity in the learning of mathematics after the implementation of this method. Results of research with correlation analysis show that the greater significance of the alpha value (5%), which means accepting and rejecting H0 Ha which means students' response to the application of brain-friendly learning through the whole brain teaching is not positively correlated with the creative character of students in the learning of mathematics. The average score of the students' response to this methodology very well categorized in the amount of 85%. The results of the observation of the creative character of the students after the implementation of this method, the average score of 68% were categorized quite creative. (shrink)

Mathematical thinking skills are very important in mathematics, both to learn math or as learning goals. Thinking skills can be seen from the description given answers in solving mathematical problems faced. Mathematical thinking skills can be seen from the types, levels, and process. Proportionally questions given to students at universities in Indonesia (semester I, III, V, and VII). These questions are a matter of description that belong to the higher-level thinking. Students choose 5 of 8 given problem. Qualitatively, the answers (...) were analyzed by descriptive to see the tendency to think mathematically used in completing the test. The results show that students tend to choose the issues relating to the calculation. They are more use cases, examples and not an example, to evaluate the conjecture and prove to belong to the numeric argumentation. Used mathematical thinking students are very personal (intelligence, interest, and experience), and the situation (problems encountered). Thus, the level of half of the students are not guaranteed and shows the level of mathematical thinking. (shrink)

Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF.

This came into existence when one day (10 -12 years back), all of a sudden I draw a line and divided the line into same segment and draw one more line above the previous line and mark the points on the both lines and magically I got a triangle (while attach all the points with each other), with at point 1 no of points 1 at 2 no of points 3 and at 3 no of points 5. Now an idea (...) born into my mind while I should not expand it more and I got Amazing Result which is in front of All Mathematics Lovers. (shrink)

Motivation -/- God is great, whatever (Air, Water, Earth, Fire & Space) we have it is just because of a Power, which governs the whole universe and more than this Power, it has created human beings, and what we have got or did not get yet, trying to get. The Method, The Principle, The Knowledge or more by which we get or prove in front of World People, basically it is called Science, who GET it, are called Scientists, Mathematicians, Inventors (...) etc i.e. we all breath we all know but in ancient time we don’t know what we breath as the time passes away one man’s curiosity bring out the name of gas which we breathe i.e. Oxygen. There are lot examples or more yet to come. -/- In this regard my little contribution in the field of Mathematics, it starts from my early childhood I used to play with numbers. After a lot of times devoted on numbers one day while squaring 11, 12, 13 so on I found that there is an amazing symmetry of 2 & it’s great. (shrink)

Why do the fundamental forces of nature (i.e., the forces that appear to cause all the movements and interactions in the universe) manifest in the way, shape, and form that they do? This is one of the greatest ontological questions that science can investigate. In this article, we are going to consider this crucial question (and relevant issues) via a new axiomatic mathematical formalism. -/- In Part I (pp. 1-10) of this article we provide a general overview (and analysis) of (...) the basic properties of a number of current notable discontinuous approaches to fundamental physics. -/- In Part II (the main part, pp.11-99, Ref. [37]) of this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Sec.2), “it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms”; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to fundamental laws of nature is as follows. First based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) is derived. Then by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field’ invariant mass. In addition, it is shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1+2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, corresponding to the Dirac and Rarita–Schwinger equations. -/- As a particular consequence, it is shown that a certain massive formalism of general relativity – with a definite form of torsion field appeared originally as the generator of gravitational field’s invariant mass – is obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. It has been also proved that Lagrangian densities specified for the originally derived new massive forms of the Maxwell, Yang-Mills and Dirac field equations, are also gauge invariant, where the invariant mass of each field is generated solely by the corresponding torsion field. In addition, in agreement with recent astronomical data, a new form of massive boson is identified (corresponding to U(1) gauge group) with invariant mass: m_γ ≈ 4.90571×10^-50 kg, generated by a coupled torsion field of the background space-time geometry. -/- Moreover, based on the definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the T-symmetry of these equations). It is shown that CPT is the only combined form of C, P, and T symmetries of interacting fields. In addition, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, it has been shown that the metric of background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the "SU(2)_L x U(2)_R" symmetry group for spin-1/2 fermion fields (representing “1+3” generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the "SU(2)_L x U(2)_R" and "SU(3)" gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including four new charge-less right-handed spin-1/2 fermions (including two leptons and two quarks), a spin-3/2 fermion, and also three new spin-1 (massive) bosons are predicted uniquely by this axiomatic approach. As a particular result, based on the definite formulation of the derived Maxwell (and Yang-Mills) field equations, it has been also concluded that magnetic monopoles could not exist in nature. -/- Comments: 99 Pages. A summary of a submitted and accepted research project (On "Foundations of Physics"), Ramin Zahedi, Japan, 2012 - 2015. -/- Invited and Presented Research Article at: The 4th International Conference on New Frontiers in Physics ICNFP2015, Europe (CERN), (https://indico.cern.ch/e/icnfp2015); The 2016 SIAM International Conference on Mathematical Aspects of Materials Science, Philadelphia, USA, (https://www.siam.org/meetings/ms16/); The 17th International Conference on Quantum Foundations: Quantum and Beyond, International Centre for Mathematical Modeling in Physics, Engineering and Cognitive Sciences ( ICMM), Linnaeus University, Sweden, 2016, (https://lnu.se/en/qb/); The 2016 International Conference on Algebraic Geometry and Mathematical Physics (AGMP 2016), University of Tromsø, Norway, (https://site.uit.no/); International Conference on Noncommutative Geometry, Quantum Symmetries and Quantum Gravity (II), XXXVII Max Born International Symposium, Wroclaw University, Poland, 2016, (http://ift.uni.wroc.pl/~mborn37/); 2016 GRavitational-wave Astronomy International Conference (Meeting) in Paris, The Institute d’Astrophysique de Paris (IAP), The University of Paris VI - Sorbonne University, CNRS, LERU, EUA, France, 2016, (Supported by the European Union's Seventh Framework Program: FP7/PEOPLE-2011-CIG); The 22nd Internnational Australian Institute of Physics Congress (AIP), University of Queensland, Brisbane, Australia, 2016, (http://appc-aip2016.org.au/); The 21st International Conference on General Relativity and Gravitation, Columbia University, New York, USA, 2016, (http://www.gr21.org/). -/- Acknowledgements: Special thanks are extended to Prof. and Academician Vitaly L. Ginzburg (Russia), Prof. and Academician Dmitry V. Shirkov (Russia), Prof. Leonid A . Shelepin (Russia), Prof. Vladimir Ya. Fainberg (Russia), Prof. Wolfgang Rindler (USA), Prof. Roman W. Jackiw (USA), Prof. Roger Penrose (UK), Prof. Steven Weinberg (USA), Prof. Ezra T. Newman (USA), Prof. Graham Jameson (UK), Prof. Sergey A. Reshetnjak (Russia), Prof. Sir Michael Atiyah (UK) (who, in particular, kindly encouraged me to continue this work as a new unorthodox (primary) mathematical approach to fundamental physics), and many others for their support and valuable guidance during my studies and research. -/- External URLs of the Article (Including updated versions): https://inspirehep.net/record/1387680/, https://indico.cern.ch/event/344173/session/22/contribution/422/attachments/1140145/1646101/R.a.Zahe di--OnDiscretePhysics-Jan.2015-signed.pdf, https://cds.cern.ch/record/1980381/, https://dumas.ccsd.cnrs.fr/TDS-MACS/hal-01476703 , http://eprints.lib.hokudai.ac.jp/dspace/handle/2115/59279/. DOI: https://dx.doi.org/10.17605/OSF.IO/PQGA9 . (shrink)

This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...) have a natural talent for it, you will find it very dense or impossible. Being somewhere in the middle I skimmed thru most of it and slowed down when it got interesting. If you have only a little time I would suggest the last chapter ´The Abyss` about Georg Cantor and transfinite arithmetic. -/- At points they wax philosophical and ask the perennial question: is math is out there in the world or in here in our heads. Why not ask this about art or music or literature or computer programs or philosophy itself? In a very general way math must come from the same place that words and ideas and images come from---our brain evolved to make them and they must in many ways (every way?) reflect the structure of our brains, which reside in our DNA, which was shaped by natural selection, which was shaped by the geology of the earth and the structure of our universe, which comes from particle physics which comes from the laws of nature which are just there. -/- I have written extensively on the nature of math and language and mind and how they are all one in my many other reviews so please see them if these topics interest you. Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX -/- . (shrink)

Based on our research regarding the relationship between physics and mathematics in HPS, and recently on Geneva Edition of Newton's Philosophiae Naturalis Principia Mathematica (1739–42) by Thomas Le Seur (1703–70) and François Jacquier (1711–88), in this paper we present some aspects of such Edition: a combination of editorial features and scientific aims. The proof of Proposition XLIII is presented and commented as a case study.

In the framework of p-adic analysis (the simplest version of analysis on trees in which hierarchic structures are presented through ultrametric distance) applied to formalize psychic phenomena, we would like to propose some possible first hypotheses about the origins of human consciousness centered on the basic notion of time symmetry breaking as meant according to quantum field theory of infinite systems. Starting with Freud's psychophysical (hydraulic) model of unconscious and conscious flows of psychic energy based on the three-orders mental representation, (...) the emotional order, the thing representation order, and the word representation order, we use the p-adic (treelike) mental spaces to model transition from unconsciousness to preconsciousness and then to consciousness. Here we explore theory of hysteresis dynamics: conscious states are generated as the result of integrating of unconscious memories. One of the main mathematical consequences of our model is that trees representing unconscious and conscious mental states have to have dierent structures of branching and distinct procedures of clustering. The psychophysical model of Freud in combination with the p-adic mathematical representation gives us a possibility to apply (for a moment just formally) the theory of spontaneous symmetry breaking of infinite dimensional field theory, to mental processes and, in particular, to make the first step towards modeling of interrelation between the physical time (at the level of the emotional order) and psychic time at the levels of the thing and word representations. Finally, we also discuss some related topological aspects of the human unconscious, following Jacques Lacan's psychoanalytic ideas. (shrink)

We consider the target-reaching problem in plane scenes for a point robot which has a tactile sensor and can locate the target ray. It might have a compass, too, but it is not able to perceive the coordinates of its position nor to measure distances. The complexity of an algorithm is measured by the number of straight moves until reaching the target, as a function of the number of vertices of the scene. It is shown how the target point can (...) be reached by exhaustive search without using a compass, with the complexity exp). Using a compass, there is a target-reaching algorithm, based on rotation counting, with the complexity O. The decision problem, to recognize if the target cannot be reached because it belongs to an obstacle, cannot be solved by our type of robot. If the behaviour of a robot without compass is periodic in a homogeneous environment, it cannot solve the target-reaching problem. (shrink)

‘‘What Is Algebra?-Why This Book?’’ This is the amazing prelude to Taming the Unknown by Victor J. Katz, emeritus professor of mathematics at the University of the District of Columbia and Karen Hunger Parshall, professor of history of mathematics at the University of Virginia. This is an excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians. [continue...].

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early development of (...) algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

A guide to the first steps into the world of Geometry, Trigonometry and their lines-of-reasoning widely used through the high school and first years of college, in the exact-sciences context.