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.     

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].

     

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 .     

Stratifying Systems via Relative Projective Modules

ABSTRACT

In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]), and also the “Comparison method” given in Platzeck and Reiten (2001 Platzeck , M. I. , Reiten , I. ( 2001 ). Modules of finite projective dimension for standardly stratified algebras . Comm. in Algebra 29 : 973 – 986 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]) and Proposition 2.1 in Ringel (1991 Ringel , C. M. ( 1991 ). The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences . Math. Z. 208 : 209 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]) follows.

     

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.     

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.     

PROJECTIVITY AND FLATNESS OF A MODULE OVER THE SUBRING OF INVARIANTS

Let k be a field and H a Hopf k-algebra with bijective antipode, R an H-module algebra over k and A = R#H the associated smash product. The fixed subring of R under H is denoted by S. Let P be an R#H-module. Thus P is an S-module. The aim of this paper is to study the projectivity of P as a module over S. We get a generalization of some results of J.J. Garcia and Angel Del Rio [4] Garcia, J. J. and Del Rio, A. 1995. On Flatness and Projectivity of a Ring as a Module Over a Fixed Subring. Mathem. Scandin., 76: 179–192.  [Google Scholar] of Ida Doraiswamy [8] Doraiswamy, I. 1982. Projectivity of Modules Over Rings with Suitable Group Action. Comm. Algebra, 10(8): 787–795. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] and of ours [[7] Guédénon, T. 1997. Algèbre Homologique Dans la Catégorie Mod_(R#U(g)). J. Algebra, 197(2): 584–614.  [Google Scholar], section 5].     

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]).     

Grouped Functional Time Series Forecasting: An Application to Age-Specific Mortality Rates

Age-specific mortality rates are often disaggregated by different attributes, such as sex, state, and ethnicity. Forecasting age-specific mortality rates at the national and sub-national levels plays an important role in developing social policy. However, independent forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider reconciling forecasts of age-specific mortality rates, extending the methods of Hyndman et al. in 2011 Hyndman, R. J., Ahmed, R. A., Athanasopoulos, G., and Shang, H. L. (2011), “Optimal Combination Forecasts for Hierarchical Time Series,” Computational Statistics and Data Analysis, 55, 2579–2589.[Crossref], [Web of Science ®] , [Google Scholar] to functional time series, where age is considered as a continuum. The grouped functional time series methods are used to produce point forecasts of mortality rates that are aggregated appropriately across different disaggregation factors. For evaluating forecast uncertainty, we propose a bootstrap method for reconciling interval forecasts. Using the regional age-specific mortality rates in Japan, obtained from the Japanese Mortality Database, we investigate the one- to ten-step-ahead point and interval forecast accuracies between the independent and grouped functional time series forecasting methods. The proposed methods are shown to be useful for reconciling forecasts of age-specific mortality rates at the national and sub-national levels. They also enjoy improved forecast accuracy averaged over different disaggregation factors. Supplementary materials for the article are available online.     

ON RESULTANT CRITERIA AND FORMULAS FOR THE INVERSION OF A POLYNOMIAL MAP

Concerning the inversion of a polynomial map F: K ² ? K ² over an arbitrary field K, it is natural to consider the following questions: (1) Can we find a necessary and sufficient criterion in terms of resultants for F to be invertible with polynomial ((2) resp. rational) inverse such that, this criterion gives an explicit formula to compute the inverse of F in this case? MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar] gave a partial answer to question (1), by giving an explicit expression of the inverse of F, when F is invertible without constant terms. On the other hand, Adjamagbo and van den Essen [3] Adjamagbo, K. and van den Essen, A. 1990. A Resultant Criterion and Formula for the Inversion of a Polynomial Map in Two Variables. J. of Pure and Appl. Algebra., 64: 1–6. North-Holland [Google Scholar] have fully answered question (2) and have furnished a necessary and sufficient criterion which relies on the existence of some constants λ₁, λ₂ in K ^*. We improve this result by giving an explicit relation between λ₁, λ₂ and constants of the Theorem of MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar].

Concerning question (2), Adjamagbo and Boury [2] Adjamagbo, K. and Boury, P. 1992. A Resultant Criterion and Formula for the Inversion of a Rational Map in Two Variables. J. of Pure and Appl. Algebra., 79: 1–13. North-Holland [Google Scholar] give a criterion for rational maps which relies on the existence of two polynomials λ₁, λ₂. We also improve this result, by expliciting the relations between these λ₁, λ₂ and the coefficients of F. This improvement enables us, first to give an explicit proof of the corresponding Theorem of Abhyankhar [1] Abhyankar, S. S. 1990. Algebraic Geometry for Scientists and Engineers. Math. Surveys and Monographs., 5: 267–273.  [Google Scholar], and secondly, to give a counter example where these λ₁, λ₂ are not in K ^*, contrary to claim of Yu [6] Yu, J.-T. 1993. Computing Minimal Polynomials and the Inverse via GCP. Comm. Algebra, 21(No.7): 2279–2294.  [Google Scholar].     

Classical Morita equivalence and linear topologies; applications to quasi-dualities

ABSTRACT

In this note,we answer a question of Hong et al. (2003 Hong , C. Y. , Kim , N. K. , Kwak , T. K. ( 2003 ). On skew Armendariz rings . Comm. Alg. 31 ( 1 ): 103 – 122 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by proving that if α is a monomorphism of a reduced ring R, and R is α-skew Armendariz, then R is α-rigid.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

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]].     

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.     

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]].     

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy [15 Priddy, S. (1970). Koszul resolutions. Trans. Am. Math. Soc. 152:39–60.[Crossref] , [Google Scholar]]. There exist some relations between these algebras and the skew PBW extensions defined in [8 Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Comm. Algebra 39(1):50–75.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In [24 Suárez, H., Reyes, A. (submitted for publications). Koszulity for skew PBW extensions over fields. [Google Scholar]] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

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]).