Noncommutative Dynamical Models with Quantum Symmetries

We define dynamical models on the q-Minkowski space algebra (which is a particular case of the Reflection Equation Algebra) as deformations (quantizations) of dynamical models with rotational symmetries, and we find their integrals. In particular, we introduce a q-analog of the Runge-Lenz vector and a q-analog of the dynamics in space-time with a spherically symmetric metric.     

On $$\alpha $$ α -points of q-analogs of the Fano plane

Designs, Codes and Cryptography - Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of...     

量子环面上一类导子李代数的结构和自同构群

本文研究量子环面上的一类导子李代数,它包含了Virasoro-Like代数及其q类似．首先证明了这 类导子李代数之间的同构一定是分次同构,并进一步给出了代数同构的充要条件及同构映射的具体表达 式,最后确定了该类李代数的自同构群．     

Hermiteq-polynomials andq-oscillators

A new q-analog of the Hermite polynomials is suggested. Its definition is based on the notion of a deformed oscillator and is related to the symmetric (with respect to the change q↔1/q) version of q-analysis. Bibliography: 27 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 81–90. Translated by E. V. Damaskinskii.     

Integral intertwining operators and quantum homogeneous spaces

Integral representations of functions on quantum homogeneous spaces are considered. The Dirichlet problem for the quantum ball is solved and a q-analog of the Cauchy-Szeg formula is derived.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 355–363, December, 1995.     

An involution and dynamics for theq-deformed quantum top

An involution on the phase space of the q-deformed quantum top is constructed which singles out a q-analog of its compact form, and a dynamics compatible with this involution is proposed which imitates the motion of the classical symmetric top. Bibliography:15 titles. To Olga Alexandrovna Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 3–16. Translated by N. B. Lebedinskaya     

q-Shock soliton evolution

By generating function based on Jackson’s q-exponential function and the standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to q-Hermite polynomials with triple recurrence relations similar to [1], our polynomials satisfy multiple term recurrence relations, which are derived by the q-logarithmic function. It allows us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We find solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole–Hopf transformation we obtain a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere, single and multiple q-shock soliton solutions and their time evolution are studied. A novel, self-similarity property of the q-shock solitons is found. Their evolution shows regular character free of any singularities. The results are extended to the linear time dependent q-Schrödinger equation and its nonlinear q-Madelung fluid type representation.     

Four drafts on the representation theory of the group of infinite matrices over a finite field

The history of these drafts is described in the preface written by the first author; the drafts were written in 1997–2000, when the authors studied asymptotic representation theory. Each draft is devoted to one of the main chapters of the representation theory of the group of infinite matrices over a finite field. The list of results includes the definition of the group GLB(q), which is the right analog of the group of matrices over a finite field, or a q-analog of the infinite symmetric group, the latter corresponding to q = 1; a formula for the principal series characters of the group GLB(q) and similar groups; the statistics of Jordan forms and a law of large numbers for the characters; a construction of simplest factor representations. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 5–36.     

On the analytic complexity of discriminants

The paper deals with the notion of analytic complexity introduced by V.K. Beloshapka. We give an algorithm which allows one to check whether a bivariate analytic function belongs to the second class of analytic complexity. We also provide estimates for the analytic complexity of classical discriminants and introduce the notion of analytic complexity of a knot.     

Decompositions of analytic 1-manifolds

In an author’s previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author’s previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action.     

On a global analytic Positivstellensatz

We consider several modified versions of the Positivstellensatz for global analytic functions that involve infinite sums of squares and/or positive semidefinite analytic functions. We obtain a general local-global criterion which localizes the obstruction to have such a global result. This criterion allows us to get completely satisfactory results for analytic curves, normal analytic surfaces and real coherent analytic sets whose connected components are all compact.     

k解析函数的Fourier级数

讨论了复平面上k解析函数的性质,并利用k解析函数的泰勒展开定理研究了k解析函数的Fourier级数,推广了经典的解析函数的Fourier级数理论.     

On Monodromy of Generalized Analytic Functions

We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

On a Generalized Dirichlet Problem for Analytic Family

In this paper, we give the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron in Cn, and get a necessary and sufficient condition for a generalized Dirichlet problem to be solvable for an analytic family on a bounded holomorphic domain. Especially, we derive that this condition is just that the continuous real boundary value is prescribed on and only on the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron.     

Analytic normalization of analytic integrable systems and the embedding flows

This paper provides the normal forms of analytic integrable differential systems and diffeomorphisms via analytic normalizations. Furthermore, we consider the existence of embedding flows of an analytic integrable diffeomorphism.     

Local analytic reduction of families of diffeomorphisms

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1:−1 resonant saddle.