The Algebraic Structure of Physical Quantities

P ublic support for U.S. federal expenditures for basic research gained momentum with the science and technology breakthroughs that contributed to the Allies' victory in World War II. After World War II, the Korean and Cold Wars concentrated research appropriations in the Department of Defense (DOD). DOD research expenditures were massive from 1950 to 1990. They included not only direct investments in weapons and intelligence systems but support of the underlying science, centered in physical sciences and engineering. What is less well remembered today is that DOD's basic research investments were broadly based, ranging far beyond the physical sciences and engineering into the life and social sciences. Through its service research offices and the Advanced Research Projects agencies, DOD also fostered interdisciplinary research in promising new areas, such as computation, and developed new modes for the performance of research, as in university materials research laboratories. The large investment in the physical sciences also contributed indirectly to medical science and health care, giving rise to many technologies used today. The life sciences now account for more than 50 percent of U.S. federal investment in basic research. Biomedical research funding has followed a pattern of steady significant growth over four decades, and the National Institutes of Health (NIH) have slowly come to dominate that funding. Today's strong federal support for the life sciences is warranted, because biomedical research is on the cusp of a revolution in preventative medicine and treatment. Nevertheless, today's overall research budget is increasingly out of balance. Federal funding of many fields in the physical sciences and engineering is down substantially since 1993 (9 to 36% in real terms in fields such as chemistry, physics, and electrical and chemical engineering). This loss, if continued, will imperil advances in these disciplines and endanger the continued flow of valuable discoveries and technologies that have been important to biomedical research and health care. National science and technology (S&T) policy over the past four decades has largely been led by physical scientists who first gained national experience in World War II. They crafted policies for broad investment in basic research and infrastructure, including the life sciences. As we enter the 21st century, biological scientists must assume broader leadership responsibilities in S&T policy, and they must speak out about the importance of support for all disciplines, including the physical sciences and engineering. NIH's Harold Varmus and the National Science Foundation's Rita Colwell recognize the imbalance in the current federal research portfolio and have begun advocating increased investment in all areas of research. Their leadership is to be commended, but it is not enough. Government, organizations, institutions, and industry can and should do more to bolster all basic research. NIH must set the example by much more broadly supporting innovative interdisciplinary research embracing all science, just as DOD did when it was the prime funder of R&D. Some of this has already begun in NIH's cross-institute bioengineering initiative and in prospective increased NIH support of information technology for biocomputation. Similar initiatives should be launched in other areas of the physical sciences and in the social and behavioral sciences. Funding for these initiatives should become a much larger percentage of NIH's overall expenditures. The life science professional societies must speak broadly for basic research and not just argue their own disciplinary cases. Disease advocacy groups must also articulate the case for the physical sciences in their work with the public and before Congress. University leaders and corporate executives must make the case for investment in all research disciplines. Most of all, biologists in the laboratory—working scientists and their students—need to appreciate that their research rests on the legacy not only of the life sciences but also the physical and other sciences. Today's life scientists and the next generation they are training must be the national leaders of the future who will increasingly guide all of U.S. basic research policy in the 21st century.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Bounded subgroups as a von Neumann radical of an Abelian group

Here we have two textbooks from the US market directed at two distinctive groups of students. The Physical Sciences is for non-science majors whereas Physics for Scientists and Engineers, as its title implies, is for science and engineering majors. Although many similar topics are covered, their styles are appropriately very different. The Physical Sciences is intended to integrate physics, chemistry and earth science. However, most of the content consists of material which we associate with physics or its borders. It contains very little mathematics. Not only is calculus excluded but the mathematical equations are kept at a very simple level, usually simple equalities and substitution of numbers. Despite the limited mathematics, the authors have included a wide range of advanced physics topics. A chapter on energy is followed by one on thermodynamics, which proceeds as far as entropy. There is a chapter on quantum mechanics where the only equation is that for Heisenberg's uncertainty principle, and there is another chapter on relativity, which includes time dilation and the equation for apparent mass of moving objects but little further mathematics. The broad scope of the book can be seen from a selection of other chapters. Physics ones include electricity and magnetism, waves and electromagnetic radiation, the atom and fundamental particles. More general chapters are introductory ones on the language of science and the development of basic classical mechanics via the ordered universe. Later ones concern chemical bonding and reactions, properties of matter, earth science, atmospheric science, geology, the stars and cosmology. It is unusual for a textbook to develop quite so many advanced concepts with such little mathematics. The text is clear and the words and illustrations are more than adequate to provide the reader with a good qualitative knowledge. In contrast, Physics for Scientists and Engineers follows a specific syllabus followed in the USA, a much tighter syllabus than the A-level and university syllabuses we see in Britain. It is written by an experienced physicist and author from California State University and is very similar in content and level to many other texts on the market. When the texts are used in the UK, it is usually for foundation year and traditional first-year university physics students, although the scope of the contents goes beyond. This is because they use calculus, but do not use higher level mathematics. Experience shows that the subtle differences in the texts only show up as one teaches from them, and even then there is wide variation of preference by both teachers and students. Physics for Scientists and Engineers covers all the usual traditional topics of physics. Thus there are extensive sections on mechanics and gravitation, thermodynamics, wave motion, electrostatics and electricity and magnetism, optics and relativity. Unusually, modern physics has been covered in a supplementary book which repeats the chapters involving relativity. It goes on to provide a quantum approach to light and the atom, the wave-like properties of matter including application of the Schrödinger equation, further quantum systems including the nucleus, short sections on semiconductors, superconductors and superfluids, and finally a chapter on fundamental particles. A huge range of examples are included, both worked out and for the student to attempt. Achieving a spread in general understanding of the physical sciences needs books such as The Physical Sciences. However, it is not adequate for the intending physicist or engineer. So there remains the fundamental question as to whether the non-specialist will spend the time such coverage would require. But the book provides an excellent general overview of traditional and modern physics topics including technological applications. The alternative specialist approach is provided by Physics for Scientists and Engineers but it is difficult to identify any particularly distinctive feature to the book compared with others available. The quality of production of both books is excellent with large numbers of coloured pictures. As is usual with American texts, instructor's manuals and other back-up material are available for each.

Minimal sumsets in finite solvable groups

Professor Michael J. Pikal: Scientist, family man, educator, colleague and friend

Enumeration theorems in infinite Abelian groups

We formulate U(1)bda Chern-Simons theory, which is also called BF theory, on a lattice, adapting a method proposed by Kantor and Susskind [1] for the groups R and ZN. Our method applies to any finite or infinite abelian group. We study the discrete symmetries and use the model to provide a rigorous treatment of the composite fermion theory of the fractional quantum Hall effect (FQHE) [2][3][4], with no ambiguities relating to intersecting Wilson/'t Hooft lines. We derive Jain's fractions, and one can also calculate corrections to the mean field solution within this framework. We also generalize the formalism to higher form gauge models in arbitrary dimension, and suggest a possible non-Abelian extension.

Let A be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual A∗∗. We refer to such an algebra as a WeSeBai algebra. In the present paper, we examine the Arens regularity properties of closed ideals of algebras in this class and observe that, although WeSeBai algebras themselves are always strongly Arens irregular, a variety of Arens regularity properties can be found among their closed ideals. After characterizing Arens regular ideals and strongly Arens irregular ideals, we focus on the main examples of WeSeBai algebras: the convolution group algebras L1(G), G compact, and the Fourier algebras A(Γ), Γ discrete and amenable. We find examples of Arens regular ideals in L1(G) and A(Γ), reflexive and nonreflexive, and examples of strongly Arens irregular ideals that are not in the WeSeBai class. For this, we introduce a new class of Riesz sets that are not Λ(p), for any p>1, and show how to obtain them in a broad family of noncommutative groups. As a consequence of our approach, we prove that every infinite Abelian group contains a Rosenthal set that is not Λ(p), for any p>0.

The P-DNP Problem for Infinite Abelian Groups

Research powers innovation and technoscientific advance, but it is due for a rethink, one consistent with its deeply holistic nature, requiring deeply human nurturing. Research is a deeply human endeavor that must be nurtured to achieve its full potential. As with tending a garden, care must be taken to organize, plant, feed, and weed—and the manner in which this nurturing is done must be consistent with the nature of what is being nurtured. In The Genesis of Technoscientific Revolutions , Venkatesh Narayanamurti and Jeffrey Tsao propose a new and holistic system, a rethinking of the nature and nurturing of research. They share lessons from their vast research experience in the physical sciences and engineering, as well as from perspectives drawn from the history and philosophy of science and technology, research policy and management, and the evolutionary biological, complexity, physical, and economic sciences. Narayanamurti and Tsao argue that research is a recursive, reciprocal process at many levels: between science and technology; between questions and answer finding; and between the consolidation and challenging of conventional wisdom. These fundamental aspects of the nature of research should be reflected in how it is nurtured. To that end, Narayanamurti and Tsao propose aligning organization, funding, and governance with research; embracing a culture of holistic technoscientific exploration; and instructing people with care and accountability.

Combinatorial Conditions Forcing Commutativity of an Infinite Group

The main goal of this Focus Point volume is to explore new advances on theoretical and computational techniques for uncertainty quantification (UQ) of modeling and simulation in relevant problems appearing in physics sciences. The papers submitted to this volume were received by invitation. As a result we have collected a number of high-quality contributions dealing with a rich and update picture of UQ analysis that includes its interplay with laws in physics problems, mainly, formulated via differential equations. The following text highlights some of the primary aspects of the topics covered as well as the contributions of the published papers. The consideration of randomness in the analysis of mathematical models for analyzing relevant problems related to health is currently a main research topic. The paper "The role of noise in the tumor dynamics under chemotherapy treatment" proposes a two-dimensional stochastic nonlinear differential system, taking into account the impact of chemotherapy on both the tumor and the healthy cells. The randomness is driven by two uncorrelated Gaussian noises with different levels of intensity. Authors analyze this interesting problem combining a number of techniques that include bifurcation diagrams, phase portraits, time series and probability density plots. The contribution is aimed at offering a trustworthy explanation of tumor growth under medical treatment. The Focus Point volume continues with the study of a second model also belonging to Medicine, specifically, to Epidemiology. In "Random resampling numerical simulations applied to a SEIR compartmental model," authors revisit the SEIR model addressed to study the dynamics of COVID-19 in Spain during its first pandemic-wave in 2020. Authors concentrate on estimating model parameters taking into account data uncertainties by means of bootstrap sampling. The validation of the stochastic analysis is performed by different goodness-of-fit measures. They show that the point estimates obtained by the bootstrap samples improve the ones of the original data. The use of fractional and stochastic calculus has emerged, mainly, during the last decades with important applications in Physics. In "Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise," authors combine the Caputo fractional derivative with the fractional Laplace operator to analyze the heat equation with noise driven by a colored noise. The paper establishes Hölder continuity of mild random field solutions of the aforementioned PDE by, first, taking advantage of key properties associated to the space-time fractional Green function. In "Asymptotic controllability of nonlinear Fokker-Planck equations," the author studies a problem related with stabilization of the control for a Fokker-Plank PDE. This equation often appears in kinetic theory of statistical mechanics and, contains, as a particular case, the classical Einstein-Smoluchowski equation, which describes the time evolution of the probability density of Brownian particles in a field of forms with a given potential. In the paper, one identifies a class of final states (reachable states) for which the problem has solution. The results can have interesting applications in control theory and physics. The stochastic of multiphysics systems is often a challenging problem because different physical scales and physical effects must be properly combined to accurately conduct the corresponding analysis. When the study is done in the practical setting, it usually involves the rigorous treatment of large sets of data. In the paper "Random stick-slip oscillations in a multiphysics system" (Fig. 12 ), one combines stick-slip oscillations, electromechanical coupling and uncertainties to characterize and quantify the stochastic response of a multiphysics system in terms of two physical quantities, duration and position, which are dependent random variables that must be jointly studied.

This edited volume Solar Cells is a collection of reviewed and relevant research chapters offering a comprehensive overview of recent developments in the field of renewable energy. The book comprises single chapters authored by various researchers and is edited by a group of experts active in the physical sciences, engineering, and technology research areas. All chapters are complete in themselves but united under a common research study topic. This publication aims at providing a thorough overview of the latest research efforts by international authors on physical sciences, engineering, and technology, and opens new possible research paths for further novel developments.

U.S. Federal investments in basic research transformed life and commerce in the 20th century. They sent us to the Moon and beyond, revolutionized communications, helped to feed the planet, reinvented work processes, and drove the remarkable economic growth of the post-1950s era in the United States. These advances and more grew out of the convergence between engineering and the early 20th-century discoveries in the physical sciences. The United States can anticipate comparable world-changing innovations in the 21st century if we adapt our education and research funding strategies to capitalize on new opportunities emerging at the convergence of the life sciences with the physical sciences and engineering.