Exponentiable Functors Between Quantaloid-Enriched Categories

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor’s exponentiability follows from the works of Niefield (J. Pure Appl. Algebra 23:147–167, 1982) and Dyckhoff and Tholen (J. Pure Appl. Algebra 49:103–116, 1987).      

Sections of a differential spectrum and factorization-free computations

We construct sections of a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of multiplicative systems generated by one differential polynomial and call it D-localization. Owing to this technique one can construct sections of a differential spectrum of a differential ring without computation of diffspec . We compare our construction with Kovacic’s structure sheaf and with the results obtained by Keigher [J. Pure Appl. Algebra, 27, 163–172 (1983)]. We show how to compute sections of factor-rings of rings of differential polynomials. All computations in this paper are factorization-free. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 133–144, 2003.     

Generalising Conduché’s Theorem

In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009) we proved a form of Conduché’s theorem for LSymcat-categories, where L was a meet-semilattice monoid. The original theorem was proved in Conduché (CR Acad Sci Paris 275:A891–A894, 1972) for ordinary categories. We showed also that the “lifting factorisation condition” used to prove the theorem is strictly related to the notion of state for processes whose semantics is modeled by LSymcat-categories. In this note we resume the content of Kasangian and Labella (J Pure Appl Algebra, 2009) in order to generalise the theorem to other situations, mainly arising from computer science. We will consider PSymcat-categories, where P is slightly more general than a meet-semilattice monoid, in which the lifting factorisation condition for a PSymcat-functor still implies the existence of a right adjoint to its corresponding inverse image functor.     

Prüfer-Like Conditions in Subring Retracts and Applications

In this article, we consider five possible extensions of the Prüfer domain notion to the case of commutative rings with zero divisors. We investigate the transfer of these Prüfer-like properties between a commutative ring and its subring retract. Our results generate new families of examples of rings subject to a given Prüfer-like conditions.     

The free loop space equivariant cohomology algebra of some formal spaces

Let \mathbbK{\mathbb{K}} be a field of characteristic p > 0 and S ¹ the unit circle. We construct a model for the negative cylic homology of a commutative cochain algebra with two stages Sullivan minimal model. Using the notion of shc-formality introduced in Bitjong and Thomas (Topology 41:85–106), the main result of Bitjong and El Haouari (Math Ann 338:347–354) and techniques of Vigué-Poirrier (J Pure Appl Algebra 91:347–354) we compute the S ¹-equivariant cohomology algebras of the free loop spaces of the infinite complex projective space \mathbbCP(￥){\mathbb{CP}(\infty)} and the odd spheres S ^2q+1.     

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.     

A Little More on Coz-Unique Frames

Coz-unique frames were defined and characterized by Banaschewski and Gilmour (J Pure Appl Algebra 157:1–22, 2001). In this note we give further characterizations of these frames along the lines of characterizations of absolutely z-embedded spaces obtained by Blair and Hager (Math Z 136:41–52, 1974) on the one hand, and by Hager and Johnson (Canad J Math 20:389–393, 1968) on the other. We also extend to frames certain characterizations of z-embedded spaces; namely, we give a characterization of coz-onto frame homomorphisms in terms of normal covers.      

Point-Free Version of Kakutani Duality

There are many results proved using the Axiom of Choice. Using point-free topology, we can prove some of these results without using this axiom. B. Banaschewski in [Pointfree Topology and the Spectra of f-rings, Ordered algebraic structures (Curacoa, 1995), Kluwer, Dordrecht, 123–148], studying the spectra of f-rings, describes the point-free version of the classical Gelfand duality without using the Axiom of Choice In this paper, referring to [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of ℓ-Modules, To appear in J. Pure Appl. Algebra], we describe a point-free version of the classical Kakutani duality. For this, using one of the spectra given in [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we find an adjunction between the category of compact completely regular frames with frame maps and the category of Archimedean bounded Riesz spaces with continuous Riesz maps.     

Vaught’s conjecture for modules over a commutative Prüfer ring

We prove that Vaught’s conjecture is true for modules over a commutative Prüfer ring. It is shown that a positive solution to Vaught’s conjecture for modules over 1-dimensional Noetherian domains would imply the same for modules over finitely presented algebras. This article was written during the visit of the second author to the University of Manchester supported by EPSRC grant GR/L68827. She would like to thank the University for hospitality. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 419–435, July–August, 1999.     

Measures on circle coarse-grained systems of sets

We show that a (non-negative) measure on a circle coarse-grained system of sets can be extended, as a (non-negative) measure, over the collection of all subsets of the circle. This result contributes to quantum logic probability (de Lucia in Colloq Math 80(1):147–154, 1999; Gudder in Quantum Probability, Academic Press, San Diego, 1988; Gudder in SIAM Rev 26(1):71–89, 1984; Harding in Int J Theor Phys 43(10):2149–2168, 2004; Navara and Pták in J Pure Appl Algebra 60:105–111, 1989; Pták in Proc Am Math Soc 126(7):2039–2046, 1998, etc.) and completes the analysis of coarse-grained measures carried on in De Simone and Pták (Bull Pol Acad Sci Math 54(1):1–11, 2006; Czechoslov Math J 57(132) n.2:737–746, 2007), Gudder and Marchand (Bull Pol Acad Sci Math 28(11–12):557–564, 1980) and Ovchinnikov (Construct Theory Funct Funct Anal 8:95–98, 1992).     

Enlargements of schemes

In this article we use our previous constructions (L. Brünjes, C. Serpé, Theory Appl. Categ. 14:357–398, 2005) to lay down some foundations for the application of A. Robinson’s nonstandard methods to modern algebraic geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.     

An FIO calculus for marine seismic imaging, II: Sobolev estimates

We establish sharp L ²-Sobolev estimates for classes of pseudodifferential operators with singular symbols [Guillemin and Uhlmann (Duke Math J 48:251–267, 1981), Melrose and Uhlmann (Commun Pure Appl Math 32:483–519, 1979)] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in \mathbb R²{\mathbb R^2} with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics [Felea (Comm PDE 30:1717–1740, 2005), Felea and Greenleaf (Comm PDE 33:45–77, 2008), Nolan (SIAM J Appl Math 61:659–672, 2000)].     

m-Fold Hypergeometric Solutions of Linear Recurrence Equations Revisited

We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (Appl Algebra Eng Commun Comput 17:83–115, 2005; J. Pure Appl Algebra 139:109–131, 1998) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek (Discrete Math 180:3–22, 1998; J Symb Comput 14(2–3):243–264, 1992) for recurrence and q-recurrence equations and the m-fold algorithm from Petkovšek and Salvy (ISSAC 1993 Proceedings, pp 27–33, 1993) for recurrence equations. We will refer to the classical algorithms as van Hoeij or Petkovšek respectively. To formulate our ideas, we first need to introduce an adapted version of an m-fold Newton polygon and its characteristic polynomials for the classical case and q-case, and to prove the important properties in this case. Using the data from the Newton polygon, we are able to present efficient m-fold versions of the van Hoeij and Petkovšek algorithms for the classical shift case and for the q-case, respectively. Furthermore, we show how one can use the Newton polygon and our characteristic polynomials to conclude for which m ? \mathbbN{m\in \mathbb{N}} there might be an m-fold hypergeometric solution at all. Again by using the information obtained from the Newton polygon, the presentation of the q-Petkovšek algorithm can be simplified and streamlined. Finally, we give timings for the ‘classical’ q-Petkovšek, our q-van Hoeij and our modified q-Petkovšek algorithm on some classes of problems and we present a Maple implementation of the m-fold algorithms for the q-case.     

Prüfer ⋆–multiplication domains and ⋆–coherence

Abstract The purpose of this paper is to deepen the study of the Prüfer ⋆–mul-tiplication domains, where ⋆ is a semistar operation. For this reason, we introduce the ⋆–domains, as a natural extension of the v-domains. We investigate their close relation with the Prüfer ⋆-multiplication domains. In particular, we obtain a characterization of Prüfer ⋆-multiplication domains in terms of ⋆–domains satisfying a variety of coherent-like conditions. We extend to the semistar setting the notion of -domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the –domains, the Prüfer ⋆-multiplication domains coincide with the ⋆-domains. Keywords: Star and semistar operation, Prüfer (⋆-multiplication) domain, -domain, Localizing system, Coherent domain, Divisorial and invertible ideal Mathematics Subject Classification (2000): 13F05, 13G05, 13E99     

The Chow Rings of Generalized Grassmannians

Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol. 56 (part 1), pp. 1–26, AMS, Providence, 1994) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, 2005) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, 2006), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces).     

On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations

In this paper we improve the regularity in time of the gradient of the pressure field arising in Brenier’s variational weak solutions (Comm Pure Appl Math 52:411–452, 1999) to incompressible Euler equations. This improvement is necessary to obtain that the pressure field is not only a measure, but a function in . In turn, this is a fundamental ingredient in the analysis made by Ambrosio and Figalli (2007, preprint) of the necessary and sufficient optimality conditions for the variational problem by Brenier (J Am Mat Soc 2:225–255, 1989; Comm Pure Appl Math 52:411–452, 1999).     

An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors

An algorithm for computing a Gr?bner basis of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, is presented; such rings include \mathbb Z_n\mathbb {Z}_n and \mathbb Z_n[i]\mathbb {Z}_n[i], where n is not a prime number. The algorithm is patterned after (1) Buchberger’s algorithm for computing a Gr?bner basis of a polynomial ideal whose coefficients are from a field and (2) its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger’s algorithm when a polynomial ideal is over a field and as Kandri-Rody–Kapur’s algorithm when a polynomial ideal is over a Euclidean domain. The proposed algorithm and the related technical development are quite different from a general framework of reduction rings proposed by Buchberger in 1984 and generalized later by Stifter to handle reduction rings with zero divisors. These different approaches are contrasted along with the obvious approach where for instance, in the case of \mathbb Z_n{\mathbb {Z}}_n, the algorithm for polynomial ideals over \mathbb Z{\mathbb {Z}} could be used by augmenting the original ideal presented by polynomials over \mathbb Z_n{\mathbb {Z}}_n with n (similarly, in the case of \mathbb Z_n[i]{\mathbb {Z}}_n[i], the original ideal is augmented with n and i² + 1).     

Directions for the Long Exact Cohomology Sequence in Moore Categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced by Bourn (Cahiers Topologie Géom Différentielle Catég XL:297–316, 1999; J Pure Appl Algebra 168:133–146, 2002). The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good context for non-abelian cohomology, in particular for the Baer Extension Theory.      

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.