Great Mathematician Research Articles

Strong Dominating Sets of Direct Product Graph of Cayley Graphs with Arithmetic Graphs

Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Product of graphs are introduced in Graph Theory very recently and developing rapidly. In this paper, we consider strong product graphs of Cayley graphs with Arithmetic graphs and present strong domination parameter of these graphs.

New Models and Psychoanalytic Theory

Received: June 02, 2014 | Published: June 03, 2014 It was nineteen twenty three when Freud published his second major theory about the mind, the structural theory, a tripartite model that divided the mind into Ego, Id and Superego. During the same decade many leading quantum physicists were publishing some of their bewildering findings. Concurrently the great mathematician/ philosopher, Alfred North Whitehead, began integrating some of these findings into his process philosophy. For psychoanalysis, however, Newtonian physics the accepted model of the time – became normative. Its major tenets, i.e. a strict separation between the observer and the observed (avoiding subjective influence) the replication of findings by other neutral observers and the reliance on strict cause and effect were paramount. Psychoanalysis assumed such models were normative despite its experience of what Freud named the unconscious – an area of psychic experience that shows remarkable parallels with some of the basic findings of quantum mechanics.

Ramanujan’s circle

Ramanujan is one of the greatest mathematicians that India has ever produced. Eminent mathematicians of the world have admitted that Ramanujan was a genius. On the path to achieving fame, Ramanujan received generous support from various persons. In this article, an account of the history of the nurturing of Ramanujan by persons around him is given, along with a brief introduction to each of those generous people from India as well as from abroad.

Some Domination Parameters of Arithmetic Vn Graph

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Nathanson [1] was the pioneer in introducing the concepts of Number Theory, particularly, the „Theory of Congruences‟ in Graph Theory, and paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Inspired by the interplay between Number Theory and Graph Theory several researchers in recent times are carrying out extensive studies on various Arithmetic graphs in which adjacency between vertices is defined through various arithmetic functions. Cayley Graphs are another class of graphs associated with elements of a group. If this group is associated with some Arithmetic function then the Cayley graph becomes an Arithmetic graph and in this paper we study the Efficient domination, Clique domination parameters of Arithmetic Vn Graphs.

‘Een filosofisch geschriftje’: Christiaan Huygens’ gedachten over God in zijn Cosmotheoros en andere geschriften

Although much has been written about Christiaan Huygens (1629–1695), his religious views are almost neglected in historical research. Some scholars have come to the conclusion that God played no part at all in the worldview of this great mathematician. Although Huygens has never written a book exclu- sively devoted to the subject of God, he did leave us some notes in which he explicitly expresses his views about God and the divine. This article focuses on Huygens’ philosophical writings and especially on his last completed work Cosmotheoros (1698), in which he discusses the possibility of extra-terres- trial life. In this book God is an important subject. Huygens’ thoughts show a strong belief in an underlying logic behind the construction of the universe. According to Huygens, the idea that the earth is just a planet among other planets, logically implies that these other planets should have a similar nature as the earth and are inhabited by ‘rational beings’. Eve- rything in the universe is created for a reason, so God would not have created anything for no purpose. Huygens can’t imagine the world being created without a great intelligent deity, but he denies that God has a direct influence on the course of events on earth. Therefore, in his view, the existence of miracles is impossible. This article presents a new perspective on Christiaan Huygens and shows that the concept of God was actually a crucial element in his understanding of the world.

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painleve is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painleve and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painleve transcendents. The contribution of Paul Painleve to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.

Rhazes' description of a case with aortic regurgitation

Abu Bakr Muhammad ibn Zakariya al-Razi (865–925AD), with the Latinized name of Rhazes, was a great Persian physician, mathematician, chemist, philosopher, and astronomer (Fig. 1). He was born in the city of Rayy (near present-day Tehran). Rhazes was educated in medicine in Baghdad (the capital city of today's Iraq) [1]. He is believed to be the first great scientist of the Golden Age of Islamic Medicine (9th–12th centuries AD) [1,2].

Cassius Tocqueville Ionescu Tulcea in occasion of his 90-th Birthday

C. T. Ionescu Tulcea is celebrating this year his 90th Birthday. He is joined by the Romanian Mathematical Diaspora and a great number of colleagues from Romania and United States of America, who are sending him their best wishes for a continuous research activity and long life. Professor Ionescu Tulcea was born in Bucharest, Romania, on October 14, 1923. He received his elementary and secondary education in distinguished schools from Bucharest and in 1942 became a student at his Alma Matter, the University of Bucharest, the Faculty of Sciences, Mathematical Section. He graduated from this school in 1946, obtaining his Diploma in Mathematics. The University of Bucharest counted among its Professors some distinguished mathematicians, like Octav Onicescu (Probability Theory), Miron Nicolescu (Mathematical Analysis), Gheorghe Vranceanu (Geometry), Nicolae Theodorescu (Differential Equations), Simion Stoilow (Complex Variables), all of them former students of great mathematicians from France or Italy. During his four years of study at the University of Bucharest, Ionescu Tulcea has acquired a solid training as a research mathematician, not only in classical mathematics, as represented in Europe before the Second World War, but also in the relatively new branches of Topology or Functional Analysis. Following his graduation in 1946, Ionescu Tulcea has been appointed on a position of Assistant at the Chair of Probability Theory and remained with the University of Bucharest for the next 11 years. He has been promoted from the Assistant position to Associate Professor (1951) and to Full Professor (1957), at the Chair of Mathematical Analysis. As a faculty member, Ionescu Tulcea has developed a teaching activity that qualified him as the main promoter of modern mathematics at the Bucharest University, in the field of Modern Analysis. To be more specific, he gave his lectures using Bourbaki’s books and other up to date treatises, which found their place in the programs only after World War II. Moreover, as a research fellow of the Mathematical Institute of the Romanian Academy (1949-1957), Ionescu Tulcea organized seminars on Operator Theory, substantially contributing to a generation of Romanian specialists in this field, among

The Great Mathematician Project

Introduce mixed-ability classes to a project exploring famous mathematicians and scientists and ignite students' math interest.

해리엇의 기호주의와 방정식론

Thomas Harriot has been introduced in middle school textbooks as a great mathematician who created the sign of inequality. This study is about Harriot's symbolism and the theory of equation. Harriot made symbols of mathematical concepts and operations and used the algebraic visual representation which were combinations of symbols. He also stated solving equations in numbers, canonical, and by reduction. His epoch-making inventions of algebraic equation using notation of operation and letters are similar to recent mathematical representation. This study which reveals Harriot's contribution to general and structural approach of mathematical solution shows many developments of algebra in 16th and 17th centuries from Viete to Harriot and from Harriot to Descartes.

Mathematics and Francoism: the interior exile of the mathematician Ferran Sunyer i Balaguer

The article describes the most salient features of Ferran Sunyer i Balaguer’s scientific career. Self-taught, despite serious physical disabilities Sunyer reached mathematical maturity in the the two decades following Spain’s Civil War (1936–1939), the darkest years of Franco’s dictatorship. A distinguished mathematician by international standards, he was never properly recognized as such in Spain. The article places Sunyer’s career in the context of the scientific institutions during Francoism, discusses his relationships with great mathematicians such as Jacques Hadamard and Szolem Mandelbrojt, and describes his relative marginalization on the basis of the peculiar values that dominated scientific institutions during the Franco’s regime.

Long History of the Monge-Kantorovich Transportation Problem

L eonid Vital’evich Kantorovich (1912–1986) was one of the great mathematicians and economists of the twentieth century. In 2012, the centenary of his birth was marked in St. Petersburg. Short histories were presented describing some of the main parts of his legacy, which continue in importance today: duality in linear programming, the so-called ‘‘MongeKantorovich transportation problem,’’ and the ‘‘Kantorovich metric.’’ Note that 2012 was also the 70th anniversary of the publication of his historic paper on the transport metric. The present article offers a somewhat expanded version ofmy talk on that occasion.

Glances from Afar: Paul Valéry and Henri Poincaré

The French poet and essayist Paul Valéry was fascinated by Henri Poincaré, who he never met but often observed. His description of Poincaré is an unusual and captivating portrait of the great mathematician.

Editorial

This volume on “Topics in Function Spaces, Differential Operators, Harmonic and Fractal Analysis” is dedicated to our teacher and friend Hans Triebel on the occasion of his seventy-fifth birthday. Hans Triebel was born on February 7, 1936 in Dessau, Germany. He studied mathematics and physics at the Friedrich-Schiller-University of Jena from 1954 to 1959, graduating with a Diploma Degree in mathematics. At the beginning of his academic career he worked in classical complex analysis and received a Ph.D. in Mathematics at the Friedrich-Schiller-University in 1962. Motivated by an interest in both mathematics and physics, he studied Sobolev's famous 1950 book and learned about the theory of distributions as developed by L. Schwartz. This might have been the catalyst for his change of research topic towards partial differential operators and function spaces. As a postdoc he spent one year at the University of Leningrad (St. Petersburg), where he enjoyed the intellectual ferment of the atmosphere created by such great mathematicians as Uraltseva, Ladyzhenskaya, Birman and Solomyak, and attended lectures by Birman on functional analysis, spectral theory and quantum mechanics. Inspired by the Russian School of Mathematics he focused his research on recent developments in linear and nonlinear partial differential operators, spectral theory and functional analysis, rapidly obtaining far-reaching results with a deep impact on further research in this field. In particular, he realized the significance of function spaces and contributed to both theory and applications in a decisive way. He completed his Habilitation Thesis on function spaces and nonlinear analysis in 1966, becoming Full Professor of Analysis at the Friedrich-Schiller-University in 1970. His further studies were also strongly influenced and motivated by personal contacts with S. G. Krein, J. Peetre, as well as by new approaches to the theory of function spaces based on Fourier-analytical techniques due to S. M. Nikol'skij, E. M. Stein and C. Fefferman. The development of the modern theory of function spaces in the last 40 years and its application to various branches in both pure and applied mathematics owes much to his seminal contributions. The bare facts are impressive: he has published more than 200 papers in internationally acknowledged journals, and has written no less than 18 monographs and textbooks; the rate of production of new and interesting results shows no sign of decreasing! Perhaps he is best known by the series of books he has written which present systematic treatments of the theory of function spaces from different points of view, thus revealing its symbiotic relationship with interpolation theory, harmonic analysis, partial differential equations, nonlinear operators, entropy, spectral theory, fractal analysis, wavelet theory, and theoretical numerical analysis. In particular, his books Interpolation Theory, Differential Operators, Function Spaces (finished in 1974 and published in 1978) as well as Theory of Function Spaces (based on earlier lecture notes and published in 1983) are much-quoted standard references and have been translated into Russian. At the textbook level, his Higher Analysis and Analysis and Mathematical Physics are masterpieces, relating mathematical theory to physical applications in an extraordinarily convincing way, and made even more vivid in a series of remarkable lecture courses at Jena. Hans Triebel has supervised nearly 40 Ph.D. students, many of whom have become internationally recognised mathematicians. He is on the editorial boards of various international journals, and in particular was an editor of Mathematische Nachrichten for many years. The reputation of the analysis department at the university of Jena owes much to his pioneering scientific work and the activities of his research group on function spaces. His outstanding scientific achievements were recognised by a National Award of the German Democratic Republic for Science and Technology in 1983 and the award of a Doctor of Science honoris causa by the University of Sussex in 1990. He was elected as (Corresponding) Member of the Academy of Science of the German Democratic Republic in 1978. Since 1993 he has been a Member of the Berlin-Brandenburg Academy of Science (formerly the Prussian Academy of Science). No less remarkable than his mathematical ability are his personal qualities, coupling total integrity, resolution and great warmth with an irreverent sense of humour; stories abound of his preparedness to lecture very early in the morning, sometimes to the surprise of the students! We are glad to have this opportunity to express our deep gratitude to him for sharing with so many of his colleagues his ideas and encyclopaedic knowledge. The present collection of papers is a tribute to his distinguished work and reflects recent developments in the theory of function spaces and related fields by outstanding experts. It is a pleasure to thank all the authors for their contributions.

Clique Dominating Sets of Euler Totient Cayley Graphs

Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Using the number theoretic function Euler totient function we have defined an Euler totient Cayley graph and in this paper we study the Clique domination parameters of Euler totient Cayley graphs.

Alhazen’s hyperbolic billiard problem

Given two points inside a circle in the hyperbolic plane, we study the problem of finding an isosceles triangle inscribed in the circle so that the two points belong to distinct congruent sides. By means of a reduction to the corresponding result in Euclidean geometry, we prove that this problem cannot generally be solved with hyperbolic ruler and compass. In his treatise on optics, written in Arabic, the scientist and mathematician Abu Ali al-H . asan ibn al-Haytham (965‐1039) posed the problem of finding the light path between a source and an observer by way of a fixed spherical mirror, and gave a geometric solution for it. The problem may have been formulated much earlier, by the great Greek mathematicians of the Hellenistic era, but no surviving testimony confirms this. Thus it is fit that it carries al-H . asan ibn al-Haytham’s name, which was rendered as Alhazen in the Latin translation of his book — a document that played an important role in the development of modern science. Alhazen recognized that the problem is essentially two-dimensional — the path must lie in a plane determined by the center of the sphere, the source and the observer. His solution is long, in part because he is actually studying a more general problem; see [Sabra 1982] for details. It is not a ruler-and-compass construction, as it requires an auxiliary hyperbola; in fact, apart from special cases, the problem turns out not to be solvable with ruler and compass alone, though it seems this was only proved some 50 years ago ([Elkin 1965]; see also [Riede 1989; Neumann 1998]). In this paper, we study the hyperbolic version of Alhazen’s problem and relate it to its classical Euclidean counterpart. We use the following formulation of the problem: Given a circle (in the Euclidean or the hyperbolic plane) and two points A and B inside it, construct an inscribed, isosceles triangle with A on one equal leg and B on the other. The isosceles condition is equivalent to the condition that the two legs meet at equal angles the diameter of the circle that goes through their common vertex, so this is Alhazen’s problem all right. (One can also imagine a round billiard table

Srinivas Ramanujan - The Genius

This year is the 125th birth anniversary of Srinivas Ramanujan. The Government of India has declared it, as the year of Mathematics; and his birth date 22nd December has been declared as the National Mathematics Day. On this occasion, this is a tribute to this great mathematician.

Srinivas Ramanujan – The Genius

This year is the 125th birth anniversary of Srinivas Ramanujan. The Government of India has declared it, as the year of Mathematics; and his birth date 22nd December has been declared as the National Mathematics Day. On this occasion, this is a tribute to this great mathematician. Srinivas Ramanujan—the man who knew infinity; the man who loved numbers; a natural mathematical genius.

Matching Dominating Sets of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Graph Theory is one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Science & Technology. It is applied in diverse areas such as social sciences, linguistics, physical sciences, communication engineering etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Every branch of mathematics employs some notion of a product that enables the combination or decomposition of its elemental structures. Product of graphs are introduced in Graph Theory very recently and developing rapidly. In this paper, we consider direct product graphs of Cayley graphs with Arithmetic graphs and present Matching dominating set of these graphs.

Some Domination Parameters of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Nathanson [1] paved the way for the emergence of a new class of graphs, namely Arithmetic Graphs by introducing the concepts of Number Theory, particularly, the Theory of Congruences in Graph Theory. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph. Inspired by the interplay between Number Theory and Graph Theory several researchers in recent times are carrying out extensive studies on various Arithmetic graphs in which adjacency between vertices is defined through various arithmetic functions.