### Book

## Probability and Statistics by Example, Vol. 1

Probability and statistics are as much about intuition and problem solving as they are about theorem proving. Consequently,

students can find it difficult to make a successful transition from lectures, to examinations, to practice because the problems

involved can vary so much in nature. Since the subject is critical in so many applications, from insurance, to telecommunications,

to bioinformatics, the authors have collected more than 200 worked examples to help the students develop a deep

understanding of the subject rather than a superficial knowledge of sophisticated theories. With amusing stories and

historical asides sprinkled thoughout, this enjoyable book will leave students better equipped to solve problems in practice

and under exam conditions.

The complexity of today’s statistical data calls for modern mathematical tools. Many fields of science make use of mathematical statistics and require continuous updating on statistical technologies. Practice makes perfect, since mastering the tools makes them applicable. Our book of exercises and solutions offers a wide range of applications and numerical solutions based on R. In modern mathematical statistics, the purpose is to provide statistics students with a number of basic exercises and also an understanding of how the theory can be applied to real-world problems. The application aspect is also quite important, as most previous exercise books are mostly on theoretical derivations. Also we add some problems from topics often encountered in recent research papers. The book was written for statistics students with one or two years of coursework in mathematical statistics and probability, professors who hold courses in mathematical statistics, and researchers in other fields who would like to do some exercises on math statistics.

The purpose of this paper is the presentation of the ideas and concepts that

form the basis of the concept of mathematical model control some processes

occurring in the Russian market of cereals. The estimated model must have a

stochastic nature, i.e. constitute some random process. Indeed, in a free market

there are objectively random factors that cannot be described by deterministic.

This is the first book on the U.S. presidential election system to analyze the basic principles underlying the design of the existing system and those at the heart of competing proposals for improving the system. The book discusses how the use of some election rules embedded in the U.S. Constitution and in the Presidential Succession Act may cause skewed or weird election outcomes and election stalemates. The book argues that the act may not cover some rare though possible situations which the Twentieth Amendment authorizes Congress to address. Also, the book questions the constitutionality of the National Popular Vote Plan to introduce a direct popular presidential election de facto, without amending the Constitution, and addresses the plan’s “Achilles’ Heel.” In particular, the book shows that the plan may violate the Equal Protection Clause from the Fourteenth Amendment of the Constitution. Numerical examples are provided to show that the counterintuitive claims of the NPV originators and proponents that the plan will encourage presidential candidates to “chase” every vote in every state do not have any grounds. Finally, the book proposes a plan for improving the election system by combining at the national level the “one state, one vote” principle – embedded in the Constitution – and the “one person, one vote” principle. Under this plan no state loses its current Electoral College benefits while all the states gain more attention of presidential candidates.

The controversial question of how J.M. Keyness early philosophical ideas influenced the essence and the method of his economic works, first and foremost *The General Theory of Employment, Interest and Money *is under consideration in a broad historical context. Intellectual sources, basic ideas and concepts of Keyness logic of probability are presented in brief as well as major criticism of Keyness view.

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.