Poisson Hopf algebra related to a twisted quantum group

A Poisson algebra ?[G] considered as a Poisson version of the twisted quantized coordinate ring ?_q,p[G], constructed by Hodges et al. [11 Hodges, T. J., Levasseur, T., Toro, M. (1997). Algebraic structure of multi-parameter quantum groups. Adv. Math. 126:52–92.[Crossref], [Web of Science ®] , [Google Scholar]], is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of ?[G] are characterized. Further it is shown that ?[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ?[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ?[G] onto the Poisson primitive ideal space.     

A Generalized Renewal Equation for Perturbed Compound Poisson Processes with Two-Sided Jumps

Abstract

In this article, we study the discounted penalty at ruin in a perturbed compound Poisson model with two-sided jumps. We show that it satisfies a renewal equation under suitable conditions and consider an application of this renewal equation to study some perpetual American options. In particular, our renewal equation gives a generalization of the renewal equation in Gerber and Landry [2 Gerber , H.U. , and Landry , B. 1998 . On the discounted penalty at ruin in a jump-diffusion and the perpetual put option . Insurance: Mathematics and Economics 22 : 263 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]] where only downward jumps are allowed.     

A ring with FT-rank two

We construct a ring which admits a 2-generated, faithful torsion module but lacks a cyclic faithful torsion module. This answers a question by Oman and Schwiebert [1 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules. Commun. Algebra 40(6):2184–2198.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules II. J. Algebra Appl. 11(3):1250054 (12 p.).[Crossref], [Web of Science ®] , [Google Scholar]].     

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]).     

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries

In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 .[Crossref], [Web of Science ®] , [Google Scholar]] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L ^n/2. In the process, we derive L ^p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Breadth and characteristic sequence of nilpotent Lie algebras

The notion of breadth of a nilpotent Lie algebra was introduced and used to approach problems of classification up to isomorphism in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]]. In the present paper, we study this invariant in terms of characteristic sequence, another invariant, introduced by Goze and Ancochea in [1 Ancochea-Bermúdez, J. M., Goze, M. (1986). Sur la classification des algèbres de Lie nilpotentes de dimension 7. C. R. Acad. Sci. Paris 302:611–613. [Google Scholar]]. This permits to complete the determination of Lie algebras of breadth 2 studied in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]] and to begin the work for Lie algebras with breadth greater than 2.     

Parabolic NTA domains in ℝ2

We show that each connected component of the boundary of a parabolic NTA domain in ?² is given by a graph. We then apply this observation to classify blowup solutions in ?² to a free boundary problem for caloric measure first considered by Hofmann, Lewis and Nyström [16 Hofmann, S., Lewis, J.L., Nyström, K. (2004). Caloric measure in parabolic flat domains. Duke Math. J. 122(2):281–346. MR 2053754 (2005e:35092)[Crossref], [Web of Science ®] , [Google Scholar]].     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.     

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.     

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.     

Polynomial extensions of SCN rings need not be SCN

In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2 Kwak, T. K., Lee, M. J., Lee, Y. (2014). On sums of coe?cients of products of polynomials. Comm. Algebra 42(9):4033–4046.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]].     

Discrete Morse theory and a reformulation of the K(π,1)-conjecture

A recent theorem of Dobrinskaya [20 Dobrinskaya, N.È. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] states that the K(π,1)-conjecture holds for an Artin group G if and only if the canonical map BM→BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of [13 Charney, R., Meier, J., Whittlesey, K. (2004). Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105:171–188.[Crossref], [Web of Science ®] , [Google Scholar]], and a small chain complex for computing its monoid homology, similar to the one of [44 Squier, C. C. (1994). The homological algebra of Artin groups. Math. Scand. 75(1):5–43.[Crossref], [Web of Science ®] , [Google Scholar]].     

Gorenstein coresolving categories

Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8 Enochs, E. E., Jenda, O. M. G. (1995). Gorenstein injective and projective modules. Math. Z. 220:611–633.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein FP-injective modules [20 Mao, L. X., Ding, N. Q. (2008). Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7:491–506.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein AC-injective modules [3 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44:2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢?_𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢?_𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.     

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].     

Global Stability and Bounds for Coarsening Rates Within the Mean-Field Theory for Domain Coarsening

We investigate the long-time behavior of solutions to the classical mean-field model for coarsening by Lifshitz–Slyozov and Wagner (LSW). In the original work (Lifshitz and Slyozov, 1961 Lifshitz , I. M. , Slyozov , V. V. ( 1961 ). The kinetics of precipitation from supersaturated solid solutions . J. Phys. Chem. Solids 19 : 35 – 50 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Wagner 1961 Wagner , C. ( 1961 ). Theorie der Alterung von Niederschlägen durch Umlösen . Z. Elektrochemie 65 : 581 – 594 . [CSA]  [Google Scholar]) convergence of solutions to a uniquely determined self-similar solution was predicted. However, it is by now well known (Giron et al., 1998 Giron , B. , Meerson , B. , Sasorov , V. P. ( 1998 ). Weak selection and stability of localized distributions in Ostwald ripening . Phys. Rev. E 58 : 4213 – 4216 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Niethammer and Pego 1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the long-time behavior of solutions depends sensitively on the initial data. In Niethammer and Pego (1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) a necessary criterion for convergence to any self-similar solution which behaves like a finite power at the end of its (compact) support is given. It says that the data have to be regularly varying at the end of their support with the same power. This criterion is also shown to be sufficient if the power is sufficiently small and for data which are close to self-similar.

In this article we extend the local stability result to the whole range of self-similar solutions with compact support. Our first main result establishes global stability of self-similar solutions with not too large power. The proof relies on a global contraction argument for the spreading of characteristics. In addition, we also establish upper and lower bounds for the coarsening rates of the system for a suitable class of initial data whose variation is bounded at the end of the support but not necessarily regular.     

Density Estimates for Compact Surfaces with Total Boundary Curvature Less Than 4π

In this paper we obtain density estimates for compact surfaces immersed in ? ⁿ with total boundary curvature less than 4π and with sufficiently small L ^p norm of the mean curvature, p ≥ 2. In fact we show that these estimates hold for compact branched immersions. Our results generalize the main results in [2 Ekholm , T. , White , B. , Wienholtz , D. ( 2002 ). Embeddedness of minimal surfaces with total curvature at most 4π . Ann. Math. 155 : 209 – 234 .[Crossref], [Web of Science ®] , [Google Scholar]]. We then apply our estimates to discuss the geometry and topology of such surfaces.     

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.     

Almost Exponential Decay Near Maxwellian

ABSTRACT

By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 Guo , Y. ( 2002 ). The Landau equation in a periodic box . Comm. Math. Phys. 231 ( 3 ): 391 – 434 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2003a Guo , Y. ( 2003a ). Classical solutions to the Boltzmann equation for molecules with an angular cutoff . Arch. Ration. Mech. Anal. 169 ( 4 ): 305 – 353 . [CSA] [CROSSREF]  [Google Scholar] b Guo , Y. ( 2003b ). The Vlasov-Maxwell-Boltzmann system near maxwellians . Invent. Math. 153 ( 3 ): 593 – 630 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and Strain and Guo (2004 Strain , R. M. , Guo , Y. ( 2004 ). Stability of the relativistic Maxwellian in a collisional plasma . Comm. Math. Phys. 251 ( 2 ): 263 – 320 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results (Desvillettes and Villani, 2005 Desvillettes , L. , Villani , C. ( 2005 ). On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation . Invent. Math. 159 ( 2 ): 245 – 316 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.     

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]).