Control-space properties of cooperative games

If two or more players agree to cooperate while playing a game, they help one another to minimize their respective costs as long as it is not to their individual disadvantages. This leads at once to the concept of undominated solutions to a game. Anundominated orPareto-optimal solution has the property that, compared to any other solution, at least one playerdoes worse or alldo the same if they use a solution other than the Pareto-optimal one.Closely related to the concept of a Pareto-optimal solution is theabsolutely cooperative solution. Such a solution has the property that, compared to any other permissible solution,every playerdoes no better if a solution other than the absolutely cooperative one is employed.This paper deals with control-space properties of Pareto-optimal and absolutely cooperative solutions for both static, continuous games and differential games. Conditions are given for cases in which solutions to the Pareto-optimal and absolutely cooperative games lie in the interior or on the boundary of the control set.The solution of a Pareto-optimal or absolutely cooperative game is related to the solution of a minimization problem with avector cost criterion. The question of whether or not a problem with a vector cost criterion can be reduced to a family of minimization problems with ascalar cost criterion is also discussed.An example is given to illustrate the theory.This research was supported in part by NASA Grant No. NGR-03-002-011 and ONR Contract No. N00014-69-A-0200-1020.     

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.     

Normal relatively convex subgroups of solvable orderable groups

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed. To the memory of N. Ya. Medvedev Supported by RFBR (project No. 03-01-00320). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009.     

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998. The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical Society Fund.     

On the Marcinkiewicz theorem for the binary Perron integral

A criterion for Perron integrability of the derivative of a function is stated in terms of the variation of the function with respect to a differential base. The criterion is used to construct an example showing that the Marcinkiewicz theorem, which asserts that ordinary Perron integrability follows from the existence of at least one continuous Perron minorant and at least one continuous Perron majorant, cannot be generalized to the binary Perron integral.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 267–277, February, 1996.This work was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00417.     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

Some properties of autostable models

Established are (1) a nonuniform criterion for the stability of models in terms of enumeration reducibility of constructivizations; (2) a criterion for the autostability of certain particular classes of models close to algebraic number fields; (3) a uniform autostability of each 1-constructive model that is autostable. Supported by RFFR grant No. 096-01-01525 and by ISF grant NQ 6000. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 685–698, November–December, 1996.     

Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon‐Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K‐property, the criterion is boosted by a zero‐one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon‐Wolff rank‐one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.© 2015 Wiley Periodicals, Inc.     

Solution of an equiweighted minimax location problem on a hemisphere

A particular continuous single facility minimax location problem on the surface of a hemisphere is discussed. We assume that all the demand points are equiweighted. An algorithm, based on spherical trigonometry, for finding the minimax point is presented. The minimax point thus obtained is unique and the algorithm is O(n ²) in the worst case.     

Fields of representatives of commutative local rings and maximal scalar fields of finite-dimensional algebras

The structure of scalar fields for a directly indecomposable finite-dimensional algebra treated as a ring is studied. Scalar fields are assumed similar if their action on a ring is identical modulo an annihilator. The criterion for a class of maximal scalar fields to be unique under a similitude relation is established. Supported by RFFR grant No. 96-01-01675, and by RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 667–686, November–December, 1998.     

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C _P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000.     

Separability of normalizable superintuitionistic propositional logics

The problem of separability of superintuitionistic propositional logics that are extensions of the intuitionistic propositional logic is studied. A criterion of separability of normal superintuitionistic propositional logics, as well as results concerning the completeness of their subcalculi is obtained. This criterion makes it possible to determine whether a normalizable superintuitionistic propositional logic is separable. By means of these results, the mistakes discovered by the author in the proofs of certain statements by McKay and Hosoi are corrected.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 606–615, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-00944.     

The evolution of transformation media in spherical trigonometry in 17th- and 18th-century China,and its relation to “Western learning”

Problems of spherical trigonometry in 17th- and 18th-century China were often reduced to problems in plane trigonometry and then solved by means of the proportionality of corresponding sides of similar right triangles. Nevertheless, in the literature on the history of Chinese mathematics, there is not much discussion on the transformation and reduction of spherical problems to the plane, and how the techniques utilized for such transformations evolved over time. In this article, I investigate the evolution of the transformation media involved. I will show that in the trigonometric treatises by Mei Wending (1633–1721) and Dai Zhen (1724–1777), the authors’ views on Western learning shaped their choices of transformation media, and conversely their choices of transformation media offered support to their views on trigonometry in the debate of Chinese versus Western methods. Based on my analysis, I also propose a reassessment of Dai’s treatise of trigonometry, which was controversial ever since its publication in the 18th century.     

The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups

The statement “no non-Abelian simple group can be obtained from a non-simple one by adding one generator and one relator” first is equivalent to the Kervaire-Laudenbach conjecture, and second, becomes true under the additional assumption that an initial non-simple group is either finite or torsion free.Supported by RFBR grant No. 02-01-00170.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 399–437, July–August, 2005.     

Uniformn-analytic polynomial approximations of functions on rectifiable contours in ℂ

We study approximations of functions byn-analytic polynomials in the uniform norm on closed rectifiable Jordan curves in the complex plane. It is shown that, in contrast to the case of uniform approximations by complex polynomials, there are no topological criteria for the existence of such approximations. We obtain a criterion for the existence ofn-analytic polynomial approximations in terms of analytic properties of these curves. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 603–609, April, 1996. The author is extremely grateful to A. G. Vitushkin and P. V. Paramonov for the statement of the problem and their attention to the work. The work was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00225.     

On the acceptance of trigonometry in wasan: Evidence from a text of Aida Yasuaki

The Dutch introduced trigonometry to Japan in the middle of the 17th century, but the use of trigonometry was rarely seen until the 18th century, and its use was limited to practical purposes such as surveying and astronomy. It was rarely used to solve geometry problems currently called wasan, the reason for which is not yet fully explained. In this paper, I summarize a book by Aida Yasuaki (1747–1817) in which he strongly criticizes the use of trigonometry for wasan, and try to find the origins of his antipathy.     

The swap conjecture of tennant and turner

We argue against the conjecture which says that any two finite generating sets for G of the same cardinality are swap equivalent. The latter means that one is changed to another by a finite sequence of generating sets such that all the neighboring sets differ only in a single entry. Namely, it is proved that a free metabelian group of rank 3 has non swap equivalent bases.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 448–463, July-August, 1995.Supported by the Russian Foundation for Fundamental Research.     

Probability, geometry, and irreversibility in quantum mechanics

The main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established.     

On problematic aspects in learning trigonometry

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.

The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.