Showing papers in "Acta Applicandae Mathematicae in 1998"
TL;DR: In this paper, a general theory of moving co-frames is proposed and a practical and easy-to-implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant operators is presented.
Abstract: This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finite-dimensional Lie group actions and infinite Lie pseudo-groups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.
306 citations
TL;DR: In this article, the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions were studied.
Abstract: We study the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions. It is known that if the Dirichlet boundary conditions are prescribed on thewhole boundary and the total influx equals zero, then weak solutions exist globally in time andthey are even unique and smooth in the case of two-dimensional domains. The methods that havebeen used to prove these results fail if non-Dirichlet conditions are applied on a part of theboundary, since there is then no control over the energy flux on this part of the boundary. In thispaper, we prove the existence and the uniqueness of solutions on a (short) time interval. Theproof is performed for Lipschitz domains and a wide class of initial data. The length of the timeinterval on which the solution exists depends only on certain norms of the data.
48 citations
TL;DR: The q-monopole bundle introduced previously is extended to ageneral construction for quantum group bundles with nonuniversaldifferential calculi in this paper, which applies to several other classes of bundles as well, including bicrossproduct quantum groups, thequantum double and combinatorial bundles associated with covers of compactmanifolds.
Abstract: The q-monopole bundle introduced previously is extended to ageneral construction for quantum group bundles with nonuniversaldifferential calculi. We show that the theory applies to several otherclasses of bundles as well, including bicrossproduct quantum groups, thequantum double and combinatorial bundles associated with covers of compactmanifolds.
47 citations
TL;DR: An omission in the outline of the general approach to the inverse problem in Acta Appl. Math. (54 (1998), pp. 233 and 273) is clarified in this paper.
Abstract: An omission in the outline of the general approach to the inverse problem in Acta Appl. Math. (54 (1998), pp. 233–273) is clarified.
42 citations
TL;DR: In this paper, a new minimax inequality and one equivalent geometric form are proved and a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.
Abstract: In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.
38 citations
TL;DR: In this article, a unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard is developed.
Abstract: We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method is essentially algebraic and based on a certain nonlinear extension of similarity (gauge) transformations in the ring of analytic differential operators. The paper provides a systematic and self-consistent review of classical and recent results on Huygens' principle in Minkowski spaces. Most of these results are carried over to more general pseudo-Riemannian spaces with the metric of a plane gravitational wave. A particular attention is given to various connections of Huygens' principle with integrable systems and the soliton theory. We discuss the link to nonlinear KdV-type evolution equations, Darboux–Backlund transformations and the bispectral problem in the sense of Duistermaat, Grunbaum and Wilson.
33 citations
TL;DR: A short review about nonassociative algebraic systems and their physical applications is presented in this article, where the main directions discussed are the octonionic, Lie-admissible, and quasigroup approaches.
Abstract: A short review about nonassociative algebraic systems (mainly nonassociative algebras) and their physical applications is presented. We begin with some motivations, then we give a brief historical overview about the formation and development of the concept of hypercomplex number system and about some earlier applications. The main directions discussed are the octonionic, Lie-admissible, and quasigroup approaches. Also, some problems investigated in Tartu, the octonionic approach, Moufang–Mal'tsev symmetry, and associator quantization are discussed. This review does not pretend to be complete as the accent is placed on ideas and not on the techniques, also the references are quite sporadic (there are many authors and results mentioned in the text without references).
30 citations
TL;DR: In this article, an automatic catalogue of all non-orientable 3-manifolds admitting coloured triangulations with a fixed number of tetrahedra has been created.
Abstract: The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automaticcatalogue of all nonorientable 3-manifolds admitting coloured triangulationswith a fixed number of tetrahedra In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds As a consequence, the following summarising result can be stated:
29 citations
TL;DR: In this paper, it was shown that integral bordism groups can be expressed as extensions of quantum bordisms and these last are extensions of subgroups of usual bordist groups.
Abstract: Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an application, we relate integral bordism to the spectral term \( E_1^{0,n - 1} \) that represents the space of conservation laws for PDEs. This also gives a general method to associate in a natural way a Hopf algebra to any PDE.
26 citations
TL;DR: In this article, the authors studied the nonlinear nonlocal noncanonical, axiom-preserving isotopies/Q-operator deformations SUQ(2) of the spin-isospin symmetry.
Abstract: In this note, we study the nonlinear-nonlocal-noncanonical, axiom-preserving isotopies/Q-operator deformations SUQ(2) of the SU(2) spin-isospin symmetry We prove the local isomorphism SUQ(2)≈SU(2), construct and classify the isorepresentations of SUQ(2), identify the emerging generalizations of Pauli matrices, and show their lack of unitary equivalence to the conventional representations The theory is applied for the reconstruction of the exact SU(2)-isospin symmetry in nuclear physics with equal p and n masses in isospaces We also prove that Bell's inequality and the von Neumann theorem are inapplicable under isotopies, thus permitting the isotopic completion/Q-operator deformation of quantum mechanics studied in this note which is considerably along the celebrated argument by Einstein, Podolsky and Rosen
24 citations
TL;DR: In this paper, the authors deal with orbit enumeration within the framework of Burnside's Lemma in a special situation when the group possesses two "mutually orthogonal" actions with respect to fixed points of its elements.
Abstract: The paper deals with orbit enumeration within the framework of Burnside’s Lemma in a special situation when the group possesses two ‘mutually orthogonal’ actions with respect to fixed points of its elements. Due to this property, only regular permutations should be taken into account in the enumeration. This considerably facilitates counting combinatorial objects up to isomorphism and obtaining simple closed formulae. Two general approaches reducing the counting of nonisomorphic (unrooted) objects to ‘rooted’ and ‘cycle-rooted’ (and then quotient) objects, respectively, are developed and described. We give here a general description and a survey of methods and results arisen. The idea turns out to be applicable to numerous concrete problems. Most of them can be modelled by tuples of permutations that generate transitive groups and are considered up to conjugacy. Enumerative results are given for counting subgroups of free groups, strong automata, coverings of surfaces, planar maps and plane point configurations. In conclusion, some open questions are posed.
TL;DR: In this paper, a generalization of a Drinfeld-Sokolov scheme of attaching integrable systems of PDEs to affine Kac-Moody algebras is proposed.
Abstract: We propose a generalization of a Drinfeld–Sokolov scheme of attaching integrable systems of PDEs to affine Kac–Moody algebras. With every affine Kac–Moody algebra \(\mathfrak{g} \) and a parabolic subalgebra \($$\) , we associate two hierarchies of PDEs. One, called positive, is a generalization of the KdV hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove a coordinatization theorem which establishes that the number of functions needed to express all PDEs of the the total hierarchy equals the rank of\(\mathfrak{g} \) . The choice of functions, however, is shown to depend in a noncanonical way on \($$\). We employ a version of the Birkhoff decomposition and a ‘2-loop’ formulation which allows us to incorporate geometrically meaningful solutions to those hierarchies. We illustrate our formalism for positive hierarchies with a generalization of the Boussinesq system and for the negative hierarchies with the stationary Bogoyavlenskii equation.
TL;DR: In this article, a unified approach to the Krall-type polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one perturbed by the addition of one or more Dirac deltafunctions is given.
Abstract: We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied.
TL;DR: The use of group actions, double cosets and homomorphisms in the constructive theory of discrete structures is described, as it is found useful from both a theoretical and a practical point of view.
Abstract: In the present paper we describe the use of group actions, double cosets and homomorphisms in the constructive theory of discrete structures, as we found it useful from both a theoretical and a practical point of view. By means of examples we should like to demonstrate that these methods are useful both as unifying principles and as efficient methods for applications.
TL;DR: In this paper, a transversally holomorphic foliated homotopy map is shown to have an absolute minimum energy functional (E_T (\varphi ) = \frac{1}{2}\int {_M } \left\| {{\text{d}}_T \varphi } \right\|^2 \mu \) in its foliated class.
Abstract: Any transversally holomorphic foliated map \(\varphi :(M,\mathcal{F}) \to (M\prime ,\mathcal{F}\prime )\) of Kahlerianfoliations with \(\mathcal{F}\) harmonic, is shown to be a transversallyharmonic map and an absolute minimum of the energy functional \(E_T (\varphi ) = \frac{1}{2}\int {_M } \left\| {{\text{d}}_T \varphi } \right\|^2 \mu \) inits foliated homotopy class.
TL;DR: In this article, an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures is presented, where the structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory.
Abstract: We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood within this context, and a number of questions can be most properly addressed (specifically in view of the formulation of QFT on a curved background). In particular, we clarify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory.
TL;DR: In this paper, a general framework is developed to analyze the convergence of linear-programming approximations for Markov control processes in metric spaces, based on aggregation and relaxation of constraints, as well as inner approximation of the decision variables.
Abstract: We develop a general framework to analyze the convergence of linear-programming approximations for Markov control processes in metric spaces. The approximations are based on aggregation and relaxation of constraints, as well as inner approximations of the decision variables. In particular, conditions are given under which the control problem’s optimal value can be approximated by a sequence of finite-dimensional linear programs.
TL;DR: In this article, the authors use Shabat polynomials to introduce a new operation, that of a composition, for combinatorial bicolored plane trees, which may be considered as a generalized symmetry.
Abstract: The connections recently established between combinatorial bicolored plane trees and Shabat polynomials show that the world of plane trees is incredibly rich with different mathematical structures. In this article we use Shabat polynomials to introduce a new operation, that of a composition, for combinatorial bicolored plane trees. The composition may be considered as a generalized symmetry.
TL;DR: In this paper, the authors discuss additional results on this topic and consider questions related to the following problems: • embeddings of varieties (G:P) into the Lie algebra corresponding to Chevalley group G; interpretations of Lie geometries, small Schubert cells, connections between the geometry of G and its associated Weyl geometry in terms of linear algebra, and applications of these problems to calculations performed in Lie geometry and association schemes.
Abstract: Investigations of homogeneous varieties T=(G:P) of all cosets of finite Coxeter or Chevalley groups G by their maximal parabolic subgroups P had been conducted at the Kalužnin seminar at Kiev State University since the 1970’s, as were investigations of their corresponding permutation groups, geometries and association schemes In I A Faradžev et al (eds), Investigations in Algebraic Theory of Combinatorial Objects (Kluwer Acad Publ, 1994), one can find some results on the investigation of noncomplete Galois correspondence between fusion schemes of the orbital scheme for (G,T) and overgroups of (G,T), as well as calculations of the intersectional indices of the Hecke algebra of (G,T) We will discuss additional results on this topic and consider questions related to the following problems: • embeddings of varieties (G:P) into the Lie algebra corresponding to Chevalley group G; • interpretations of Lie geometries, small Schubert cells, connections between the geometry of G and its associated Weyl geometry in terms of linear algebra, and applications of these problems to calculations performed in Lie geometries and association schemes; • constructions of geometric objects arising from Kac–Moody Lie algebras and superalgebras, and applications of these constructions to investigations of graphs of large girth and large size
TL;DR: In this paper, the infinite-dimensional Weyl group is realized in p-adic L2-spaces, and the corresponding L 2-space of padic-valued square integrable functions are constructed.
Abstract: Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L2-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L2-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L2-spaces with respect to a p-adic Gaussian distribution.
TL;DR: In this article, a follow-up to previous recent publications in the field of theoretical economic geography and spatial economics is presented, where earlier results are generalised and simulated in higher dimensions, and given possible undesirable outcomes of the process (which can behave chaotically), application of control methods to it is being studied.
Abstract: The present study is a follow-up to previous recent publications in the field of theoretical economic geography and spatial economics. Earlier results are generalised and simulated in higher dimensions (in terms of variables and topological dimensions), and given possible undesirable outcomes of the process (which can behave chaotically), application of control methods to it is being studied.
TL;DR: In this paper, the k-orbit reconstruction problem is solved for transitive Abelian and Hamiltonian groups: all k-orbits of Abelian groups are reconstructible if k>3 and the same is true for Hamiltonians if k >4.
Abstract: Let G be a permutation group on a set Ω. Then G acts in the natural way on the collection Ω{k} of all k-element subsets. Orbits under this action are called k-orbits. A problem similar to the Edge-Reconstruction Conjecture in graph theory can be posed for k-orbits of a general group G. Here the k-orbit reconstruction problem is solved for transitive Abelian and Hamiltonian groups: all k-orbits of Abelian groups are reconstructible if k>3 and the same is true for Hamiltonian groups if k>4.
TL;DR: A large variety of Lp(p 0) form very general Opial type inequalities are presented in this paper, which are based on a generalization of Taylor's formula for generalized fractional derivatives.
Abstract: A large variety of Lp(p 0) form very general Opial type inequalities arepresented engaging different order generalized fractional derivatives of a function. These arebased on a generalization of Taylor’s formula for generalized fractional derivatives. In the finalresults of this work, a monotonicity property of the involved function/highest-order generalizedfractional derivative is used.
TL;DR: In this paper, it was shown that the geometry of light-like hypersurfaces of the de Sitter space Sn+11 is directly connected with the hypersurface geometry of the conformal space Cn.
Abstract: It is proved that the geometry of lightlike hypersurfaces of the de Sitter space Sn+11 is directly connected with the geometry of hypersurfaces of the conformal space Cn. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.
TL;DR: In this paper, the authors apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation and find approximate solutions.
Abstract: In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible unitary representations of the motion group and discuss the corresponding Fourier transform of functions on the motion group. The main reasons why exact solutions of this convolution equation do not exist in many cases are discussed. Techniques for regularization of the problem and numerical methods for finding approximate solutions are presented. Examples are considered and approximate solutions are found.
TL;DR: In this paper, two continuation methods are described based on essential maps and 0-epi maps, where the first one is based on the essential map and the second one is on the 0epi map.
Abstract: Two continuation methods are described in this paper. The first is based on essential mapswhereas the second is based on 0-epi maps.
TL;DR: In this article, continuous Moufang transformations are introduced and discussed, and commutation relations for infinitesimal moufang transformation relations are established, and the resulting Lie algebra has quite impressive structure equations well known from the theory of alternative algebras.
Abstract: Continuous Moufang transformations are introduced and discussed. Commutation relations for infinitesimal Moufang transformations are established. The resulting Lie algebra has quite impressive structure equations, well known from the theory of alternative algebras.
TL;DR: In this paper, some classes of initial boundary value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having nonsmooth coefficients.
Abstract: In this work, some classes of initial boundary-value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having, in general, nonsmooth coefficients. A new type of Riemann function concept is given for these problems, which is more natural than the classical Riemann-type function concept, and an integral form of the solutions of nonhomogeneous problems can be found more naturally using this concept.
TL;DR: In this article, local conditions of effectivity and normality for Backlund transformations are introduced, having implications for the solution generating properties, for the pKdV, the sine-Gordon, and the Tzitzeica equation.
Abstract: For Backlund transformations, treated as relations in the categoryof diffieties, local conditions of effectivity and normality are introduced,having implications for the solution generating properties. We check themfor the pKdV, the sine-Gordon, and the Tzitzeica equation.
TL;DR: In this article, the problem of estimating a mixture of probability measures in an abstract setting is considered and twelve examples are worked out in order to show the applicability of the theory.
Abstract: We consider the problem of estimating a mixture of probability measures in an abstract setting Twelve examples are worked out, in order to show the applicability of the theory