Sharp Bounds on the Number of Resonances for Conformally Compact Manifolds with Constant Negative Curvature Near Infinity

Abstract

The purpose of this article is to prove a sharp bound on the number of resonances for the Laplacian on conformally compact manifolds with constant negative curvature near infinity, thus improving the polynomial bound of Guillopé and Zworki (Guillopé, L., Zworski, M. ([1995b] Guillopé, L. and Zworski, M. 1995b. Upper bounds on the number of resonances for noncompact Riemann surfaces. J. Funct. Anal., 129: 364–389. [Crossref], [Web of Science ®] , [Google Scholar]). Polynomial bound on the number of resonances for some complete spaces of constant negative curvature near infinity. Asympt. Anal. 11:1–22).     

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT

In his recent work, [1] Simson, D. 2000. An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II. J. Algebra, 227: 670–705. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Simson, D. 2001. On Small Right Pure Semisimple Rings and the Structure of their Auslander-Reiten Quiver. Communic. in Algebra, 29 in press[Web of Science ®] , [Google Scholar], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.     

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]).     

2 × 2 Permanental Ideals of Hypermatrices

We analyze the structure of ideals generated by some classes of 2 × 2 permanents of hypermatrices, generalizing [9 Laubenbacher , R. C. , Swanson , I. ( 2000 ). Permanental ideals . J. Symbolic Comput. 30 : 195 – 205 .[Crossref], [Web of Science ®] , [Google Scholar]] on 2 × 2 permanental ideals of generic matrices. We compare the obtained structure to that of the corresponding determinantal ideals in [11 Swanson , I. , Taylor , A. ( 2013 ). Minimal primes of ideals arising from conditional independence statements . J. Algebra 392 : 299 – 314 .[Crossref], [Web of Science ®] , [Google Scholar]]: as expected, the permanental ideals have many more (minimal) components. In the last two sections, we examine a few related classes of permanental ideals.     

Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004 van Oystaeyen , F. , Zhang , P. ( 2004 ). Quiver Hopf algebras . J. Algebra 280 ( 2 ): 577 – 589 . [CSA] [CROSSREF]  [Google Scholar]). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000 Andruskiewitsch , N. , Schneider , H.-J . ( 2000 ). Finite quantum groups and Cartan matrices . Adv. Math. 154 : 1 – 45 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.     

Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ? ^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996 Namah , G. ( 1996 ). Asymptotic solution of a Hamilton–Jacobi equation . Asymptotic Anal. 12 ( 4 ): 355 – 370 . [CSA] [Web of Science ®] , [Google Scholar]), Namah and Roquejoffre (1999 Namah , G. , Roquejoffre , J.-M . ( 1999 ). Remarks on the long-time behavior of the solutions of Hamilton–Jacobi equations . Comm. PDE 24 ( 5–6 ): 883 – 893 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Roquejoffre (1998 Roquejoffre , J.-M . ( 1998 ). Comportement asymptotique des solutions d’équations de Hamilton–Jacobi monodimensionnelles . C. R. Acad. Sci. Paris Sér. I Math. 326 ( 2 ): 185 – 189 . [CSA] [Crossref] , [Google Scholar]), Fathi (1998 Fathi , A. ( 1998 ). Sur la convergence du semi-groupe de Lax–Oleinik semigroup . C. R. Acad. Sci. Paris Sér. I Math. 327 ( 3 ): 267 – 270 . [CSA] [Crossref] , [Google Scholar]), Barles and Souganidis (2000 Barles , G. , Souganidis , P. E. ( 2000 ). On the large time behavior of solutions of Hamilton–Jacobi equaitons . SIAM J. Math. Anal. 31 ( 4 ): 925 – 939 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Barles , G. , Souganidis , P. E. ( 2001 ). Space-time periodic solutions and long-time behavior of solutions to quasi-periodic parabolic equations . SIAM J. Math. Anal. 32 ( 6 ): 1311 – 1323 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.     

Khasminskii-Type Theorems for Stochastic Differential Delay Equations

Abstract

The classical Khasminskii theorem (see [6 Khasminskii , R. Z. 1980 . Stochastic Stability of Differential Equations . Alphen : Sijtjoff and Noordhoff (translation of the Russian edition, Moscow: Nauka 1969) .[Crossref] , [Google Scholar]]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13 Mao , X. 2002 . A note on the LaSalle-type theorems for stochastic differential delay equations . J. Math. Anal. Appl. 268 : 125 – 142 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied.     

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ? × ? ^d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17 Kenig , C.E. , Merle , F. ( 2008 ). Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation . Acta Math. 201 : 147 – 212 .[Crossref], [Web of Science ®] , [Google Scholar]] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26 Planchon , F. ( 2003 ). On uniqueness for semilinear wave equations . Math. Z. 244 : 587 – 599 .[Web of Science ®] , [Google Scholar]].     

Reflected BSDE and Locally Periodic Homogenization of Semilinear PDEs with Nonlinear Neumann Boundary Condition

We study the homogenization of semilinear partial differential equations (PDEs) with nonlinear Neumann boundary condition, locally periodic coefficients, and highly oscillating drift and nonlinear term. Our method is entirely probabilistic, as in a periodic case by Ouknine and Pardoux [14 Ouknine , Y. , and Pardoux , É. 2002 . Homogenization of PDEs with non linear boundary condition, Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progresses of Probability, 52, Birkhäuser, Basel , pp. 229 – 242 . [Google Scholar]] and builds on our earlier work [5 Diakhaby , A. , and Ouknine , Y. 2006 . Locally periodic homogenization of reflected diffusion . Journal of Applied Mathematics and Stochastic Analysis . [Google Scholar]], which gives us the locally periodic counterpart of Theorem 2.2 in Tanaka [21 Tanaka , H. 1984 . Homogenization of diffusion processes with boundary conditions . Stochastic Analysis and Applications 7 : 411 – 437 . Advanced Probability and Related Topics 7, Dekker, New York . [Google Scholar]].     

Inverse Boundary Value Problem for Maxwell Equations with Local Data

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov [4 Isakov , V. ( 2007 ). On uniqueness in the inverse conductivity problem with local data . Inverse Probl. Imaging 1 : 95 – 105 .[Crossref], [Web of Science ®] , [Google Scholar]] for the Schrödinger equation to Maxwell equations.     

The Level Set Method for Systems of PDEs

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996 Evans , L. C. ( 1996 ). A geometric interpretation of the heat equation with multivalued initial data . SIAM J. Math. Anal. 27 ( 4 ): 932 – 958 .[Crossref], [Web of Science ®] , [Google Scholar]) for the heat equation and by Giga and Sato (2001 Giga , Y. , Sato , M.-H. ( 2001 ). A level set approach to semicontinuous viscosity solution for Cauchy problems . Comm. Partial Differential Equations 26 ( 5–6 ): 813 – 839 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) for Hamilton–Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each nonzero sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction–diffusion equations.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

In this paper we pursue the work initiated in [6 Bahuaud , E. ( 2009 ). Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics . Pacific J. Math. 239 : 231 – 249 .[Crossref], [Web of Science ®] , [Google Scholar], 7 Bahuaud , E. , Gicquaud , R. ( 2011 ). Conformal compactification of asymptotically locally hyperbolic metrics . J. Geom. Anal. 21 : 1085 – 1118 .[Crossref], [Web of Science ®] , [Google Scholar]]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to ?1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.     

Elementary matrix reduction over J-stable rings

A commutative ring R is J-stable provided that R∕aR has stable range 1 for all a?J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3 Gillman, L., Henriksen, M. (1956). Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82:362–365.[Crossref] , [Google Scholar], Theorem 8], [4 Larsen, M., Lewis, W., Shores, T. (1974). Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187:231–248.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 4.1], [7 McGovern, W. W. (2008). Bézout rings with almost stable range 1. J. Pure Appl. Algebra 212:340–348.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.7], [8 Moore, M. E. (1975). A strongly complement property of Dedekind domain. Czechoslovak Math. J. 25(100):282–283. [Google Scholar], Theorem], [9 Moore, M., Steger, A. (1971). Some results on completability in commutative rings. Pacific J. Math. 37:453–460.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.1], [14 Zabavsky, B. V. (1996). Generalized adequate rings. Ukrainian Math. J. 48:614–617.[Crossref] , [Google Scholar], Theorem 1] and [18 Zabavsky, B. V., Komarnyts’kyi, M. Y. (2010). Cohn-type theorem for adequacy and elementary divisor rings. J. Math. Sci. 167:107–111.[Crossref] , [Google Scholar], Theorem 7].     

On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993 Helminck , A. G. , Wang , S. P. ( 1993 ). On rationality properties of involutions of reductive groups . Adv. Math. 99 ( 1 ): 26 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.     

Sharp estimates for solutions of mean field equations with collapsing singularity

The pioneering work of Brezis-Merle [7 Brezis, H., Merle, F. (1991). Uniform estimates and blow-up behavior for solutions of ?Δu = V(x)e^u in two dimensions. Commun. Partial Differential Equation 16:1223–1254.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Li-Shafrir [27 Li, Y.Y., Shafrir, I. (1994). Blow-up analysis for solutions of ?Δu = V(x)e^u in dimension two. Indiana Univ. Math. J. 43:1255–1270.[Crossref], [Web of Science ®] , [Google Scholar]], Li [26 Li, Y.Y. (1999). Harnack inequality: the method of moving planes. Commun. Math. Phys. 200:421–444.[Crossref], [Web of Science ®] , [Google Scholar]], and Bartolucci-Tarantello [3 Bartolucci, D., Tarantello, G. (2002). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229:3–47.[Crossref], [Web of Science ®] , [Google Scholar]] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30 Lin, C.S., Tarantello, G. (2016). When “blow-up” does not imply “concentration”: A detour from Brezis-Merle’s result. C. R. Math. Acad. Sci. Paris 354:493–498.[Crossref], [Web of Science ®] , [Google Scholar]] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).