Do we create mathematics or do we gradually discover theories which exist somewhere independently of us?
This is a preface to the paper A. Borel, "Mathematics: Art and Science" reprinted in EMS Newsletter, No. 103, March 2017.
This book series features volumes composed of select contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.
This article consider The project of the scientific and educational Center for integration of multimedia technologies in science, education and culture, as space-technological environment for the implementation of innovative scientific and educational projects of the 21st century, which should become the support for the master's programs, especially interdisciplinary; at the intersection of science, art and information technologies, and implementation of innovative scientific and commercial projects, which are to become a master's thesis.
A neat split-up of methods into qualitative and quantitative ‘boxes’ works with just a bunch of elementary and time-tested research devices. It would not easily apply such a division to multiplying cases of new designs for productive investigations. Often they are compound research capacities. A new and trendy ‘box’ termed mixed methods is ready at hand. However, compound structures are not just amalgamations. Their effectiveness rests on structural propensities and not on amassing of their initial components. Furthermore, steadily multiplying new research principles rest on neither quantity nor quality but on something transcending the quality – quantity dichotomy, e.g. Qualitative Comparative Analysis (QCA) or Lijphartian analysis of patterns of democratic rule. Methodological domains can diverge or converge. They can dispose of all ‘deceitful’ aberrations and shrink to a single ‘authentic’ set of algorithms (methodological monism or in its radical display methodological rigorism). They can also entangle alternative research capacities (methodological liberalism or pluralism). The authors would explicate their methodological stance as ‘democratic’. This is more than just a pompous political analogy. Modern democracy converges all sorts of rule to make them good enough for accountable and inclusive governance. Likewise, advanced methods of our age merge any kinds of exploratory faculties to make them good enough for valid and comprehensive investigation. Just as modern democratic practices and conventions have been emerging only recently, current multidisciplinary and transdisciplinary methods still evolve as trial aptitudes for making research far-reaching and reliable enough. Both modern democracy and transdisciplinary are more of a promise rather than long established paragons. The authors perceive the entire methodological realm as shaped into three overlapping but still very distinct major methodological domains – mathematical, morphological and semiotic organons of learning and research. They coalesce around fundamental and abiding principles. Their mundane and transient apparitions are grand methodological approaches and paradigms not say nothing about claims and technical devices of specific schools of thought and research. Mathematical organon integrates a relatively comprehensive domain. Morphological and semiotic ones only crudely amalgamate assortments of areas, branches and endeavors of research that are still at variance with each other. The task is to overcome residual discrepancies and to advance integrating principles of general or ‘pure’ semiotics (Morris) and morphology. The principles of organons derive from our basic sensoria and other primary cognitive abilities. Some originate in our sense of order, measure and quantity to produce mathematical organon. Others commence with our perception of forms, shapes and configurations to yield a would-be morphological organon. Further ones amplify our faculty to re-create and discover meanings in our intercourse with the world and each other to commence a budding semiotic organon. Immanuel Kant, Charles Sanders Pierce and other great minds provide guidelines for trichotomous structure of organons. It is tempting to proclaim analogy between the trichotomous structure of organons and current vague distinction of quantitative, qualitative and ‘mixed’ clusters of methods. One has to explore the analogy. Correlations between configurational comparative studies and morphology or between qualitative studies and semiotics are still problematic. Furthermore, it would be premature to expect a quick integration of entire domains of morphology or semiotics. It is pragmatic to work for integration of selected focal core areas. Possible options are reshaping of neo-institutional paradigms into morphological ones, integration of biological and linguistic morphologies as well as further advancement of biosemiotics and biopolitics.
Although research collaboration has been studied extensively, we still lack understanding regarding the factors stimulating researchers to collaborate with different kinds of research partners including members of the same research center or group, researchers from the same organization, researchers from other academic and non-academic organizations as well as international partners. Here, we provide an explanation of the emergence of diverse collaborative ties. The theoretical framework used for understanding research collaboration couples scientific and technical human capital embodied in the individual with the social organization and cognitive characteristics of the research field. We analyze survey data collected from Slovenian scientists in four scientific disciplines: mathematics; physics; biotechnology; and sociology. The results show that while individual characteristics and resources are among the strongest predictors of collaboration, very different mechanisms underlie collaboration with different kinds of partners. International collaboration is particularly important for the researchers in small national science systems. Collaboration with colleagues from various domestic organizations presents a vehicle for resource mobilization. Within organizations collaboration reflects the elaborated division of labor in the laboratories and high level of competition between different research groups. These results hold practical implications for policymakers interested in promoting quality research.
The goal in putting together this unique compilation was to present the current status of the solutions to some of the most essential open problems in pure and applied mathematics. Emphasis is also given to problems in interdisciplinary research for which mathematics plays a key role. This volume comprises highly selected contributions by some of the most eminent mathematicians in the international mathematical community on longstanding problems in very active domains of mathematical research. A joint preface by the two volume editors is followed by a personal farewell to John F. Nash, Jr. written by Michael Th. Rassias. An introduction by Mikhail Gromov highlights some of Nash’s legendary mathematical achievements.
The treatment in this book includes open problems in the following fields: algebraic geometry, number theory, analysis, discrete mathematics, PDEs, differential geometry, topology, K-theory, game theory, fluid mechanics, dynamical systems and ergodic theory, cryptography, theoretical computer science, and more. Extensive discussions surrounding the progress made for each problem are designed to reach a wide community of readers, from graduate students and established research mathematicians to physicists, computer scientists, economists, and research scientists who are looking to develop essential and modern new methods and theories to solve a variety of open problems.
Full papers (articles) of 2nd Stochastic Modeling Techniques and Data Analysis (SMTDA-2012) International Conference are represented in the proceedings. This conference took place from 5 June by 8 June 2012 in Chania, Crete, Greece.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.