Non-symmetric Forms Research Articles

A special family of non-symmetric semi-classical forms of class one

A form (linear functional) v is called regular if there exists a sequence of polynomials {Sn}n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{S_{n}\\}_{n\\ge 0}$$\\end{document}, deg(Sn)=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg (S_{n})=n$$\\end{document}, which is orthogonal with respect to v. {Sn}n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{S_{n}\\}_{n\\ge 0}$$\\end{document} is fully characterized by the following recurrence relation: Sn+2(x)=(x-βn+1)Sn+1(x)-γn+1Sn(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{n+2}(x)=(x-\\beta _{n+1})S_{n+1}(x)-\\gamma _{n+1}S_{n}(x)$$\\end{document}, n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 0$$\\end{document}, with S0(x)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{0}(x)=1$$\\end{document}, S1(x)=x-β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{1}(x)=x-\\beta _{0}$$\\end{document} and γn+1≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _{n+1}\ e 0$$\\end{document}, n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 0$$\\end{document}. Such a form v is said be semi-classical if there exist polynomials Ψ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi (x)$$\\end{document} and Φ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi (x)$$\\end{document} with deg(Ψ)≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg (\\Psi )\\ge 1$$\\end{document} such that (Φv)′+Ψv=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Phi v)' +\\Psi v=0$$\\end{document}. When v is semi-classical and regular, its corresponding polynomial sequences {Sn}n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{S_{n}\\}_{n\\ge 0}$$\\end{document} are called semi-classical. In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{n}$$\\end{document}, γn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _{n+1}$$\\end{document}, n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 0$$\\end{document} of the semi-classical orthogonal polynomials sequences of class one when βn=tn-1-tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{n}=t_{n-1}-t_{n}$$\\end{document} and γn+1=-tn(c+tn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _{n+1}=-t_{n}(c+t_{n})$$\\end{document} with tn(c+tn)≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_{n}(c+t_{n})\ e 0$$\\end{document}n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 0$$\\end{document}, t-1=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_{-1}=0$$\\end{document} and c∈C-{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c\\in \\mathbb {C}-\\{0\\} $$\\end{document}. There are essentially five canonical cases.

An Averaging Principle for Stochastic Flows and Convergence of Non-Symmetric Dirichlet Forms

We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence of sequences of non-symmetric bilinear forms defined on different spaces; the study of weighted Sobolev spaces on metric graphs or “books”.

Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.

Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure

In this paper we show that the realization in $\mathrm {L}^{p}(X,\nu _{\infty })$ of a nonsymmetric Ornstein-Uhlenbeck operator Lp is sectorial for any $p\in (1,+\infty )$ and we provide an explicit sector of analyticity. Here, $(X,\mu _{\infty },H_{\infty })$ is an abstract Wiener space, i.e., X is a separable Banach space, $\mu _{\infty }$ is a centred nondegenerate Gaussian measure on X and $H_{\infty }$ is the associated Cameron-Martin space. Further, $\nu _{\infty }$ is a weighted Gaussian measure, that is, $\nu _{\infty }=e^{-U}\mu _{\infty }$ where U is a convex function which satisfies some minimal conditions. Our results strongly rely on the theory of nonsymmetric Dirichlet forms and on the divergence form of the realization of L2 in $\mathrm {L}^{2}(X,\nu _{\infty })$ .

First-order least-squares method for the obstacle problem

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide a posteriori bounds that can be used as error indicators in an adaptive algorithm. Numerical studies are presented.

Solvability of nonlocal systems related to peridynamics

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

Non-symmetric forms of non-linear vibrations of flexible cylindrical panels and plates under longitudinal load and additive white noise

Parametric non-linear vibrations of flexible cylindrical panels subjected to additive white noise are studied. The governing Marguerre equations are investigated using the finite difference method (FDM) of the second-order accuracy and the Runge-Kutta method. The considered mechanical structural member is treated as a system of many/infinite number of degrees of freedom (DoF). The dependence of chaotic vibrations on the number of DoFs is investigated. Reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of PDEs (partial differential equations) to ODEs (ordinary differential equations), i.e. the Faedo-Galerkin method in higher approximations and the 4th and 6th order FDM. The Cauchy problem obtained by the FDM is eventually solved using the 4th-order Runge-Kutta methods. The numerical experiment yielded, for a certain set of parameters, the non-symmetric vibration modes/forms with and without white noise. In particular, it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi-periodicity, whereas the previously non-symmetric vibration modes are closer to symmetric ones.

Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents

We analyze the transience, recurrence, and irreducibility properties of general sub-Markovian resolvents of kernels and their duals, with respect to a fixed sub-invariant measure m. We give a unifying characterization of the invariant functions, revealing the fact that an Lp-integrable function is harmonic if and only if it is harmonic with respect to the weak dual resolvent. Our approach is based on potential theoretical techniques for resolvents in weak duality. We prove the equivalence between the m-irreducible recurrence of the resolvent and the extremality of m in the set of all invariant measures, and we apply this result to the extremality of Gibbs states. We also show that our results can be applied to non-symmetric Dirichlet forms, in general and in concrete situations. A second application is the extension of the so called Fukushima ergodic theorem for symmetric Dirichlet forms to the case of sub-Markovian resolvents of kernels.

An Integrated Version of Varadhan\u2019s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.

Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and
 non-explosion results

Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first showweak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strictpositivity of the unique continuous version of the density to its invariant measure. This non-symmetric distorted Brownian motion is also proved to be strong Feller. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically hasan unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] that the constructedweak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet formsto prove further properties of the solutions. For example, we obtain new non-explosion criteria for them.We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion relatedto a class of Muckenhoupt weights and corresponding divergence free perturbations.

Detecting rhythms in time series with RAIN.

A fundamental problem in research on biological rhythms is that of detecting and assessing the significance of rhythms in large sets of data. Classic methods based on Fourier theory are often hampered by the complex and unpredictable characteristics of experimental and biological noise. Robust nonparametric methods are available but are limited to specific wave forms. We present RAIN, a robust nonparametric method for the detection of rhythms of prespecified periods in biological data that can detect arbitrary wave forms. When applied to measurements of the circadian transcriptome and proteome of mouse liver, the sets of transcripts and proteins with rhythmic abundances were significantly expanded due to the increased detection power, when we controlled for false discovery. Validation against independent data confirmed the quality of these results. The large expansion of the circadian mouse liver transcriptomes and proteomes reflected the prevalence of nonsymmetric wave forms and led to new conclusions about function. RAIN was implemented as a freely available software package for R/Bioconductor and is presently also available as a web interface.

ChemInform Abstract: Diphosphoinositol Polyphosphates: Polar Stars in Cell Signaling

AbstractReview: [23 refs.

Diphosphoinositol Polyphosphates: Polar Stars in Cell Signaling

The propagation of information inside cells is achieved in nature by small diffusible molecules. Many of these secondary messengers are highly charged species derived from myo -inositol. The diphosphoinositol polyphosphates (X-PP- myo -InsP 5 ) represent a novel group of secondary messengers that regulate various important biological processes. Depending on the phosphorylation pattern, these molecules can occur in symmetric or nonsymmetric forms. In a recent study, we have shown the potential of C 2 symmetric phosphoramidites as reagents for desymmetrizing myo -inositol and for obtaining all four possible nonsymmetric X-PP- myo -InsP 5 stereoisomers in enantiomerically pure form. These results and their potential implications for the design of tools for chemical biology are discussed.

The Dirichlet heat kernel in inner uniform domains: Local results, compact domains and non-symmetric forms

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable real coefficients in inner uniform domains in Rn. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.

Markov Chain Approximations to Nonsymmetric Diffusions with Bounded Coefficients

AbstractWe consider a certain class of nonsymmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to nonsymmetric diffusions. As an application, we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric Markov chains. This extends the results by Stroock and Zheng to the nonsymmetric divergence forms. © 2012 Wiley Periodicals, Inc.

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Itô’s formula in terms of the extended stochastic integrals is obtained.

On Representations of Non-Symmetric Dirichlet Forms

In this paper, we present some structure results on non-symmetric Dirichlet forms. These include, in particular, an analogue of LeJan’s transformation rule for their diffusion parts and a Levy-Khintchine type formula for regular non-symmetric Dirichlet forms on Rd.

Nonsymmetric forms of envelopes of relativistic magnetic stars

Within the framework of a previously proposed relativistic model of a magnetized perfectly conducting material surface without internal stresses, an axisymmetrical equilibriumproblem is considered of the rotation of a stellar envelope, supported by a radial hypersonic stellar wind in a given central gravitational field and dipole magnetic field. In the absence of either the magnetic field or the stellar wind, the equations have an explicit solution; in the general case the solution is obtained numerically. Of special interest are the envelope shapes asymmetric with respect to the equator (relative to the direction of the dipole), which have the greatest stability, in particular, bottle-shaped structures.

Adaptive Variational Multiscale Methods for Multiphase Flow in Porous Media

Porous media are characterized by strongly heterogeneous properties on all scales of observation. This has motivated a sustained research effort in upscaling methods and lately multiscale methods. However, due to the lack of scale separation for realistic problems, static upscaling and multiscale methods may not be suitable in general, and adaptive approaches are needed. In this paper, we outline the application of an adaptive variational multiscale method for two-phase flow in porous media. We give the symmetric and nonsymmetric forms of the mixed variational multiscale method and suggest a framework for adaptively selecting the support of the fine scale Green's operators in space and time. The applicability of our framework is validated through a set of numerical comparisons.

A class of non-symmetric forms on the canonical simplex of $\R^d$

We introduce a class of non-symmetric bilinear forms on the d-dimen\-sional canonical simplex, related with Fleming-Viot type operators. Strong continuity, closedness and results in the spirit of Beurling-Deny criteria are established. Moreover, under suitable assumptions, we prove that the forms satisfy the Log-Sobolev inequality.As a consequence, regularity results for semigroups generated by a class of Fleming-Viot type operators are given.