On elliptic and parabolic systems with x‐dependent multivalued graphs

We consider elliptic and parabolic systems with multivalued x‐dependent graphs. The existence of solutions for elliptic equation was established in (Ann. Inst. H. Poincare Anal. Non Linéaire 1990; 7 (3):123–160; Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2004; 7 (1):23–59). We extend this result to the elliptic and parabolic systems, in particular to the systems describing a flow of non‐Newtonian incompressible fluids. Contrary to these two papers we follow the spirit of the compactness method of J. L. Lions for variational‐type operators, however, expanded on the framework of measure‐valued solutions. The main concept consists in applying the relation between x‐dependent maximal monotone graphs and 1‐Lipschitz Carathéodory functions to introduce the generalized Young measures. The method was announced in the short note (C. R. Math. Acad. Sci. Paris 2005; 340 (7):489–492). Copyright © 2006 John Wiley & Sons, Ltd.     

Construction of the functional model of B. S.-Nagy-C. Foias

Methodological paper: quick and simple construction of the Nagy—Foias functional model, using only elementary information about Hilbert spaces and operators on them. A formula is derived for a characteristic function. The account is based on an idea of B. S. Pavlov (Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 123–148 (1975)) on symmetrization of the notation for a model space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 16–23, 1977.     

Ergodicity of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with H¹‐initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the L²‐norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. Copyright © 2017 John Wiley & Sons, Ltd.     

Transitivity and Dense Periodicity for Graph Maps

In 1984, Blokh proved [A. M. Blockh, On transitive mappings of one-dimensional branched manifolds, Differential-Difference Equations and Problems of Mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 131, pp. 3–9, 1984] that any topologically transitive continuous map from a graph into itself which has periodic points has a dense set of periodic points and has positive topological entropy (in this proof a crucial role is played by the specification property, which implies these two statements). Also, he characterized the topologically transitive continuous graph maps without periodic points. Unfortunately, this clever paper is only available in Russian (except for a translation to English of the statements of the theorems without proofs—see [A. M. Blockh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat. Nauk, 42(5(257)) (1987), pp. 209–210]).     

On Nonexistence of any λ0-invariand positive harmonic function,a counter example to stroock' conjecture

Consider a sub-Markovian semigroup such that λ₀, the border number between recrrence and transience, equals zero. In 1982, D. W. Stroock conjectured that under general hypotheses on the semi-group the corresponding process always admits an invariant measure.

In this paper we present an example of a second order elliptic operator P with a generalized principal eigenvalue λ₀ which equals zero such that the parabolic equation does not admit any positive invariant P—harmonic function and also any invariant measure. This gives a counter example to Stroock's conjecture for diffusion processes. We also present an example of a complete Riemannian manifold M which does not admit any positive invariant harmonic function while λ₀, the bottom of the spectrum of M, is zero. This gives a partial answer to a question of Stroock and Sullivan.     

Arithmetic of pall orders and representation of integers by some ternary quadratic forms

The discrete ergodic method (Yu. V. Linnik, Izv. Akad. Nauk SSSR, Ser. Mat.,4, 363–402 (1940); A. V. Malyshev, Tr. Mat. Inst. Akad. Nauk SSSR,65) is applied to the study of properties of integral points on the ellipsoids $$\sum\nolimits_{g,m} : g(x) = m,x = (x_1 ,x_2 ,x_3 ), g(x) = \bar f(Cx),$$ where \(\bar f\) is the adjoint of one of the 39 quadratic forms of Pall (Trans. Am. Math. Soc,59, 280–332 (1946);C is an integral matrix,¦detC¦?1. We construct a flow of integral points on the genus surface of the ellipsoidsg(x)=m. The ergodicity of this flow and a mixing theorem are proved. We obtain an asymptotic formula for the number of representations ofm belonging to a given domain on the ellipsoid and lying in a given residue class.     

Some inequalities for algebraic polynomials with the Laguerre weight

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L²-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e^-xx^α (α>-1).     

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.     

Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential

In this article, we give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy's inequality with respect to an invariant measure μ, we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein–Uhlenbeck operator perturbed by an inverse-square potential in L ²(? ^N ,?μ). In the case of the classical Ornstein–Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.     

Applications of autoreproducing kernel grammian moduli to S (U,H)-valued stationary random functions

It is shown that the analytical characterizations of q-variate interpolable and minimal stationary processes obtained by H. Salehi (Ark. Mat., 7 (1967), 305–311; Ark. Mat., 8 (1968), 1–6; J. Math. Anal. Appl., 25 (1969), 653–662), and later by A. Weron (Studia Math., 49 (1974), 165–183), can be easily extended to Hilbert space valued stationary processes when using the two grammian moduli that respectively autoreproduce their correlation kernel and their spectral measure. Furthermore, for these processes, a Wold-Cramér concordance theorem is obtained that generalizes an earlier result established by H. Salehi and J. K. Scheidt (J. Multivar. Anal., 2 (1972), 307–331) and by A. Makagon and A. Weron (J. Multivar. Anal., 6 (1976), 123–137).     

The size of edge chromatic critical graphs with maximum degree 6

In 1968, Vizing [Uaspekhi Mat Nauk 23 (1968) 117–134; Russian Math Surveys 23 (1968), 125–142] conjectured that for any edge chromatic critical graph with maximum degree , . This conjecture has been verified for . In this article, by applying the discharging method, we prove the conjecture for . © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 149–171, 2009     

Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms.     

Pharpening of a feller theorem on the law of iterated logarithms

In the paper, a sharpened version of a theorem of W. Feller on the law of iterated logarithms is proved. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 117–123. Translated by the author.     

Invariant Measures for a Stochastic Kuramoto–Sivashinsky Equation

Abstract

For the one-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are presented.     

Sur l'existence de solutions d'équations différentielles stochastiques progréssives rétrogrades couplées

Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others.

Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded.

The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation.

The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation.

The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient.

At the end of the paper we give some extensions of our result.     

New characterizations of the Takagi function via functional equations

We provide two new characterizations of the Takagi function as the unique bounded solution of some systems of two functional equations. The results are independent of those obtained by Kairies (Wy? Szko? Ped Krakow Rocznik Nauk Dydakt Prace Mat 196:73–82, 1998), Kairies (Aequ Math 53:207–241, 1997), Kairies (Aequ Math 58:183–191, 1999) and Kairies et al. (Rad Mat 4:361–374, 1989; Errata, Rad Mat 5:179–180, 1989).     

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.     

Cyclic even cycle systems of the complete graph

In this article, it is proved that for each even integer m?4 and each admissible value n with n>2m, there exists a cyclic m‐cycle system of K_n, which almost resolves the existence problem for cyclic m‐cycle systems of K_n with m even. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:23–39, 2012     

Curves of minimal action over metric spaces

Given a metric space X, we consider a class of action functionals, generalizing those considered in Brancolini et al. (J Eur Math Soc 8:415–434, 2006) and Ambrosio and Santambrogio (Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei Mat Appl 18: 23–37, 2007), which measure the cost of joining two given points x ₀ and x ₁, by means of an absolutely continuous curve. In the case X is given by a space of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus on the possibility to split the mass in its moving part and its part that (in some sense) has already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.