Results 1  10
of
45
An Extension Of Zeilberger's Fast Algorithm To General Holonomic Functions
 DISCRETE MATH
, 2000
"... We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy. ..."
Abstract

Cited by 65 (5 self)
 Add to MetaCart
We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Factoring and decomposing a class of linear functional systems, in "Linear Algebra Appl
"... Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators ..."
Abstract

Cited by 28 (21 self)
 Add to MetaCart
Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, overdetermined, underdetermined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, timedelay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M ′, where M (resp., M ′ ) is a module intrinsically associated with the linear functional system R y = 0 (resp., R ′ z = 0). These morphisms define applications sending solutions of the system R ′ z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a noninjective endomorphism of the module M is equivalent to the existence of a nontrivial factorization R = R2 R1 of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system R y = 0 is equivalent to a system R ′ z = 0, where R ′ is a blocktriangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system R y = 0 to the integration of two independent systems R1 y1 = 0 and R2 y2 = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system R y = 0, i.e., they allow us to compute an equivalent system R ′ z = 0, where R ′ is a blockdiagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a Maple package Morphisms based on the library OreModules.
Rings with Auslander Dualizing Complexes
, 1998
"... A ring with an Auslander dualizing complex is a generalization of an AuslanderGorenstein ring. We show that many results which hold for AuslanderGorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show th ..."
Abstract

Cited by 24 (15 self)
 Add to MetaCart
A ring with an Auslander dualizing complex is a generalization of an AuslanderGorenstein ring. We show that many results which hold for AuslanderGorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras. We also prove some results of a categorical nature, most notably the functoriality
The colored Jones function is qholonomic
, 2005
"... A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3–space, we prove from first principles that the colored Jones function is a multisum of a q–properhypergeometric function, and thus it is q–holonomic. We demonstrate our results by computer calculations.
On the characteristic and deformation varieties of a knot
 in: Proceedings of the CassonFest
, 2004
"... Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the ndimensional irreducible representation of sl2. It was recently shown by TTQ Le and the author that the colored Jones function of ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the ndimensional irreducible representation of sl2. It was recently shown by TTQ Le and the author that the colored Jones function of a knot is qholonomic, i.e., that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1dimensional variety in 2. We then compare it with the character variety of SL2 ( ) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the socalled noncommutative Apolynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter. Contents
Effective scalar products of Dfinite symmetric series
 Journal of Combinatorial Theory Series A, 112:1
"... Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We ext ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to kregular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective DFinite Symmetric Functions”.)
Weyl Closure of a Linear Differential Operator
 J. SYMBOLIC COMPUT
, 2000
"... We study the Weyl closure Cl(L ) = K(x)h@iL " D for an operator L of the first Weyl ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
We study the Weyl closure Cl(L ) = K(x)h@iL " D for an operator L of the first Weyl
Quantum Cohomology via Dmodules
, 2003
"... Quantum cohomology first arose in physics, and its (mathematically conjectural) properties were supported by physical intuition. A rigorous mathematical definition came later, based on deep properties of certain moduli spaces. Both approaches are therefore dependent on rather specialized foundations ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Quantum cohomology first arose in physics, and its (mathematically conjectural) properties were supported by physical intuition. A rigorous mathematical definition came later, based on deep properties of certain moduli spaces. Both approaches are therefore dependent on rather specialized foundations. We shall develop another point of view on quantum cohomology, closer in spirit to differential geometry. The main ingredient in our approach is a flat connection, considered as a holonomic Dmodule (or maximally overdetermined system of p.d.e.). This object itself is not new: Givental’s “quantum cohomology Dmodule ” is already well known ([Gi1]), and the associated flat connection appears in Dubrovin’s theory of Frobenius manifolds ([Du]). But, in the existing literature, the Dmodule plays a subservient role, being a consequence of the construction of the GromovWitten invariants and the quantum cohomology algebra. For us, the Dmodule will be the main object of interest. We define a quantization of a (commutative) algebra A to be a (noncommutative) Dmodule M h which satisfies certain properties. The quantum cohomology Dmodule is a particular kind of quantization, which arises in the following way. For a Kähler manifold
Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems
 Adv. Math
, 2000
"... Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinar ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasiexactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.