GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

UNIFORM FILTERS AND SEMI-COMPATIBILITY

In [5] García Román, M., Márquez Hernández, M. and Verschoren, A. 1997. Structure Sheaves and Noncommutative Topologies. J. of Algebra, 194: 224–244. [Crossref], [Web of Science ®] , [Google Scholar] we used functors which are compositions of localization functors to construct sheaves over an arbitrary ring R. These functors share some properties with localization, and questions like when is the composition of localizations a localization functor? arise naturally. In this note we answer this question and some related ones using the key concept of semi-compatibility.

     

Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

On sub-polynomial lower error bounds for quadrature of SDEs with bounded smooth coefficients

In recent work of Hairer, Hutzenthaler and Jentzen, [11 Hairer, M., Hutzenthaler, M., and Jentzen, A., 2015. Loss of regularity for Kolmogorov equations. Ann. Probab. 43:468–527.[Crossref], [Web of Science ®] , [Google Scholar]], a stochastic differential equation (SDE) with infinitely differentiable andbounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In this article, we show that this type of bad performance for quadrature of SDEs with infinitely differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely differentiable and bounded by one coefficients such that the first-order derivatives of all diffusion coefficients are bounded by one as well and the first order derivatives of all drift coefficients are uniformly dominated by a single real-valued function and such that the corresponding sequence of mean absolute errors for approximation of the expected value of the first component of the solution at the final time can not converge to zero faster than the given speed.     

Geometric aspects of robust testing for normality and sphericity

Stochastic robustness of control systems under random excitation motivates challenging developments in geometric approach to robustness. The assumption of normality is rarely met when analyzing real data and thus the use of classic parametric methods with violated assumptions can result in the inaccurate computation of p-values, effect sizes, and confidence intervals. Therefore, quite naturally, research on robust testing for normality has become a new trend. Robust testing for normality can have counterintuitive behavior, some of the problems have been introduced in Stehlík et al. [Chemometrics and Intelligent Laboratory Systems 130 (2014 Stehlík, M., St?elec, L., and Thulin, M. 2014. On robust testing for normality in chemometrics. Chemometrics and Intelligent Laboratory Systems 130:98–108.[Crossref], [Web of Science ®] , [Google Scholar]): 98–108]. Here we concentrate on explanation of small-sample effects of normality testing and its robust properties, and embedding these questions into the more general question of testing for sphericity. We give geometric explanations for the critical tests. It turns out that the tests are robust against changes of the density generating function within the class of all continuous spherical sample distributions.     

Stochastic motion under G-framework: II. Kinematics

Continuing the study of stochastic motion that we started [11 Zhang, H., and Kannan, D. 2014. Stochastic motion under G-framework: I. Nelson stochastic derivatives. Stoch Anal Appl 32:1067–1096.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present in this article the kinematics of such a motion. We begin by defining the quadratic derivative of an S2-process, and show that this derivative of the Brownian motion captures the variance uncertainty. We show, under certain vanishing derivatives and independence conditions, the martingale properties of an S1-process. Starting with an S1-process, we derive the equation of motion, an Itô equation corresponding to a G-diffusion process.     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.     

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ? R ² ? J ? Rⁿ (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R ² such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = G_k = (2) × ? × (2).

Commutative Gorenstein rings appeared at first in the paper [3] Gorenstein, D. 1952. An Arithmetic Theory of Adjoint Plane Curves. Trans. AMS., 72: 414–436. [Crossref], [Web of Science ®] , [Google Scholar]. Torsion-free modules over commutative Gorenstein domains were investigated in [1] Bass, H. 1963. On the Ubiquity of Gorenstein Rings. Math. Z., 82(1): 8–27. [Crossref] , [Google Scholar]. Noncommutative Gorenstein orders were considered in [2] Drozd, Yu. A., Kirichenko, V. V. and Roiter, A. V. 1967. On Hereditary and Bass Orders. Izv. Akad. Nauk SSSR Ser. Mat., 31: 1415–1436. Math. USSR – Izvestija, 1967, 1, 1357–1375 [Google Scholar] and [10] Roggenkamp, K. W. 1970. Lattices Over Orders II Berlin, Heidelberg, New York: Springer-Verlag. [Crossref] , [Google Scholar]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5] Kirichenko, V. V. 1978. On Quasi-Frobenius Rings and Gorenstein Orders. Trudy Math. Steklov Inst., 148: 168–174. (in Russian) [Google Scholar]. Arbitrary tiled orders were considered in [4] Jategaonkar, V. A. 1974. Global Dimension of Tiled Orders Over a Discrete Valuation Ring. Trans. AMS., 196: 313–330. [Crossref], [Web of Science ®] , [Google Scholar], 11-14 Simson, D. 1992. Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. Vol. 4, Gordon and Breach Science Publishers. Zavadskij, A. G. 1973. The Structure of Orders with Completely Decomposable Representations. Mat. Zametki, 13: 325–335. (in Russian) Zavadskij, A. G. and Kirichenko, V. V. 1976. Torsion-free Modules over Prime Rinqs. Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 57: 100–116. J. Soviet. Math. 1979, 11, 598–612 Zavadskij, A. G. and Kirichenko, V. V. 1977. Semimaximal Rings of Finite Type. Mat. Sbornik, 103(No. 3): 323–345. Math. USSR Sbornik, 1977, 32 (3), 273–291 .     

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

Convergence of the Structure Function of a Multifractal Random Walk in a Mixed Asymptotic Setting

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L ¹ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19 Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 .[Crossref], [Web of Science ®] , [Google Scholar]] and Bacry et al. [1 Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . [Google Scholar]] in the simpler framework of Mandelbrot cascades.     

ON THE NORM OF THE FOURIER-JACOBI PROJECTION

We extend the results of Pollard [7] Pollard, H. 1949. The mean convergence of orthogonal series. III. Duke Math. J., 16: 189–191. [Crossref], [Web of Science ®] , [Google Scholar] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted L_p space.     

Self-Interacting Markov Chains: Some Asymptotics

Abstract

Guided by the self-interaction mechanisms introduced in Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]] and in [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present a more general definition of self-interacting Markov chains (SIMCs) (than in Del Moral and Miclo [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]]). We then establish, for particular self-interaction mechanisms, a stability theorem with error estimation, two central limit theorems, two functional central limit theorems, and the large deviation principle.     

A note on semicentral idempotents

In this note we answer the question raised by Han et al. in [3 Han, J., Lee, Y., Park, S. (2014). Semicentral idempotents in a ring. J. Korean Math. Soc. 51(3):463–472, MR3206399.[Crossref], [Web of Science ®] , [Google Scholar]] whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.     

Scattering Theory Below Energy for a Class of Hartree Type Equations

We prove the global existence and scattering for the Hartree-type equation in H ^s (?³) the low regularity space s < 1. We follow the ideas in Colliander et al. (2004 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2004 ). Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ?³ . Comm. Pure Appl. Math. 57 : 987 – 1014 .[Crossref], [Web of Science ®] , [Google Scholar]) to the Hartree-type nonlinearity, and also develop the theory of the classical multilinear operator modifying the L ^p estimate in Coifman and Meyer (1978 Coifman , R. , Meyer , Y. ( 1978 ). Au delá des opérateurs pseudo-differentiel . Astérisque, Société Mathématique de France 57 . [Google Scholar]).     

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4 Chen , Y. , Giga , Y. , Goto , S. ( 1991 ). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Diff. Geom. 33 : 749 – 786 .[Crossref], [Web of Science ®] , [Google Scholar]] and [12 Evans , L. C. , Spruck , J. ( 1991 ). Motion of level sets by mean curvature. I . J. Diff. Geom. 33 ( 3 ): 635 – 681 . [Google Scholar]]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.     

Standard Koszul standardly stratified algebras

In this paper, we prove that every standard Koszul (not necessarily graded) standardly stratified algebra is also Koszul. This generalizes a similar result of [3 Ágoston, I., Dlab, V., Lukács, E. (2003). Quasi-hereditary extension algebras. Algebras Represent. Theory 6:97–117.[Crossref], [Web of Science ®] , [Google Scholar]] on quasi-hereditary algebras.