Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

Homotopical intersection theory, II: equivariance

This paper is a sequel to (Klein and Williams in Geom Topol 11:939–977, 2007). We develop here an intersection theory for manifolds equipped with an action of a finite group. As in Klein and Williams (2007), our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. We give applications of our theory to embedding problems, equivariant fixed point problems and the study of periodic points of self maps.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

Finite Bolyai–Lobachevskii planes

Finite analogs of the classical Beltrami–Klein model of the Bolyai–Lobachevskii plane arising from ovals, unitals and maximal (k,n)-arcs are of interest in finite geometry. Three new results are obtained which give characterizations of such models equipped with many symmetries.     

On the Quaternion Ball and the Quaternion Projective Space

Since the quaternion ball was used to study the AdS/CFT problems tor spinor fields, it is worthwhile to study further the geometry （in sense of Klein） and analysis on it and on its extended space （in the sense of Behnke-Thullen）, the quaternion projective space.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

Interaction of order and convexity

This is an overview of merging the techniques of vector lattice theory and convex geometry. The text was submitted by the author in English. An expanded version of the talk delivered at the opening of the Russian-German Geometry Meeting dedicated to the 95th anniversary of A. D. Alexandrov, St. Petersburg, June 18–23, 2007.     

Weyl connections and the local sphere theorem for quaternionic contact structures

We apply the theory of Weyl structures for parabolic geometries developed by Čap and Slovák (Math Scand 93(1):53–90, 2003) to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot–Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold, cf. (Geom Appl 28(4):376–394, 2010). In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern–Moser tensor in CR geometry. This tensor was first discovered via different methods by Ivanov and Vasillev (J Math Pures Appl 93:277–307, 2010), and we also get an independent proof of their Local Sphere Theorem. However, as a result of our derivation of this tensor, its fundamental properties—conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure—follow as easy corollaries from the general parabolic theory.     

KZ equation,G-opers,quantum Drinfeld–Sokolov reduction,and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 246–259.     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

Algebraic geometry in first-order logic

In every variety of algebras Θ, we can consider its logic and its algebraic geometry. In previous papers, geometry in equational logic, i.e., equational geometry, has been studied. Here we describe an extension of this theory to first-order logic (FOL). The algebraic sets in this geometry are determined by arbitrary sets of FOL formulas. The principal motivation of such a generalization lies in the area of applications to knowledge science. In this paper, the FOL formulas are considered in the context of algebraic logic. For this purpose, we define special Halmos categories. These categories in algebraic geometry related to FOL play the same role as the category of free algebras Θ⁰ play in equational algebraic geometry. This paper consists of three parts. Section 1 is of introductory character. The first part (Secs. 2–4) contains background on algebraic logic in the given variety of algebras Θ. The second part is devoted to algebraic geometry related to FOL (Secs. 5–7). In the last part (Secs. 8–9), we consider applications of the previous material to knowledge science. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004.     

Sails and Hilbert bases

A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial coneC⊂ ℝⁿ. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone. In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three (e.g., [2]). However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four. CEREMADE, University Paris 9. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 43–49, April–June, 2000. Translated by J.-O. Moussafir     

Hylomorphic solitons and charged Q-balls: Existence and stability

In this paper we present a general theoretical framework from which several known results (a some new ones) on the existence and stability of solitons can be recovered.We give an abstract definition of solitary wave and soliton and we develope an abstract existence theory. This theory provides a powerful tool to study the existence of solitons for the Klein–Gordon equations as well as for gauge theories. Applying this theory, we prove the existence of a continuous family of stable charged Q-balls.     

Discrete convex analysis

A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.     

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

Markov Incremental Constructions

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be “locally” random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley’s theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson–Shor-type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to randomized analysis in computational geometry. This work was supported in part by NSF grants CCR-0306283, CCF-0634958.     

Geometric Variational Crimes: Hilbert Complexes,Finite Element Exterior Calculus,and Problems on Hypersurfaces

A recent paper of Arnold, Falk, and Winther (Bull. Am. Math. Soc. 47:281–354, 2010) showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk (Lecture Notes in Math., vol. 1357, pp. 142–155, 1988) analyzed a class of nodal finite elements for the Laplace–Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis (SIAM J. Numer. Anal. 47:805–827, 2009) to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge–de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk’s and Demlow’s a priori estimates for 2- and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results generalize these earlier estimates dramatically, extending them from nodal finite elements for Laplace–Beltrami to mixed finite elements for the Hodge Laplacian, and from 2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.     

A generalization of an inequality by N. V. Krylov

In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application to an L ^p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242). We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.