Showing papers on "Field (mathematics) published in 2009"

F-thresholds of hypersurfaces

Abstract: ′) for every λ ′ 0 that is F-finite, i.e. such that the Frobenius morphism F : R − → R is finite. If a is an ideal in R and if λ is a nonnegative real number, then the corresponding test ideal is denoted by τ(a � ). In this context we say that λ is an F-jumping exponent (or an F-threshold) if τ(a � ) 6 τ(a � ′ ) for every λ ′ < λ. The following is our main result about F-jumping exponents in positive characteristic. Theorem 1.1. If R is an F-finite regular ring, and if a = (f) is a principal ideal, then the F-jumping exponents of a are rational, and they form a discrete set. The discreteness and the rationality of F-jumping numbers has been proved in [BMS] for every ideal when the ring R is essentially of finite type over an F-finite field. We mention also that for R = k[[x, y]], with k a finite field, the above result has been proved in [Ha] using a completely different approach. We stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic. Even

Algebraic characterization of X-states in quantum information

Abstract: A class of two-qubit states called X-states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information. Maximally entangled Bell states and 'Werner' states are subsets of them. Apart from being so named because their density matrix looks like the letter X, there is not as yet any characterization of them. The su(2) × su(2) × u(1) subalgebra of the full su(4) algebra of two qubits is pointed out as the underlying invariance of this class of states. X-states are a seven-parameter family associated with this subalgebra of seven operators. This recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties. At the same time, it points to other groups of seven-parameter states that, while not at first sight appearing similar, are also invariant under the same subalgebra. And it opens the way to analyzing invariant states of other subalgebras in bipartite systems.

Graphs and Ideals generated by some 2-minors

Abstract: Let G be a finite graph on [n] = {1,2,3,...,n}, X a 2 times n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this paper, we study about ideals I_G of S generated by 2-minors [i,j] of X which correspond to edges {i,j} of G. In particular, we construct a Groebner basis of I_G as a set of paths of G and compute a primary decomposition.

Realization of Symmetry in the ERG Approach to Quantum Field Theory

Abstract: We review the use of the exact renormalization group for realization of symmetry in renormalizable field theories. The review consists of three parts. In part I (sects. 2,3,4), we start with the perturbative construction of a renormalizable field theory as a solution of the exact renormalization group (ERG) differential equation. We show how to characterize renormalizability by an appropriate asymptotic behavior of the solution for a large momentum cutoff. Renormalized parameters are introduced to control the asymptotic behavior. In part II (sects. 5--9), we introduce two formalisms to incorporate symmetry: one by imposing the Ward-Takahashi identity, and another by imposing the generalized Ward-Takahashi identity via sources that generate symmetry transformations. We apply the two formalisms to concrete models such as QED, YM theories, and the Wess-Zumino model in four dimensions, and the O(N) non-linear sigma model in two dimensions. We end this part with calculations of the abelian axial and chiral anomalies. In part III (sects. 10,11), we overview the Batalin-Vilkovisky formalism adapted to the Wilson action of a bare theory with a UV cutoff. We provide a few appendices to give details and extensions that can be omitted for the understanding of the main text. The last appendix is a quick summary for the reader's convenience.

A minkowski-style bound for the orders of the finite subgroups of the cremona group of rank 2 over an arbitrary field

Abstract: Let Cr(k) = Autk(X; Y ) be the Cremona group of rank 2 over a eld k. We give a sharp multiplicative bound M(k) for the orders of the

The E11 origin of all maximal supergravities. The hierarchy of field-strengths

Abstract: Starting from E11 and the space-time translations we construct an algebra that promotes the global E11 symmetries to local ones, and consider all its possible massive deformations. The Jacobi identities imply that such deformations are uniquely determined by a single tensor that belongs to the same representation of the internal symmetry group as the D−1 forms specified by E11. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible gauged maximal supergravity theory in any dimension. All the possible deformed algebras are in one to one correspondence with all the possible massive maximal supergravity theories. The hierarchy of fields inherent in the E11 formulation plays an important role in the derivation. The tensor that determines the deformation can be identified with the embedding tensor used previously to parameterise gauged supergravities. Thus we provide a very efficient, simple and unified derivation of all the field strengths and gauge transformations of all maximal gauged supergravities from E11. The dynamics arises as a set of first order duality relations among these field strengths.

Lefschetz elements of artinian gorenstein algebras and hessians of homogeneous polynomials

Abstract: We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

Algebraic Geometry in Coding Theory and Cryptography

Abstract: This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available.Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books

On the physical part of the factorized correlation functions of the XXZ chain

Abstract: It was recently shown by Jimbo et al (2008 arXiv:0811.0439) that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with a disorder parameter α and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions ρ and ω. Here we approach the description of the same correlation functions and, in particular, of the function ω from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field α. We explicitly factorize the integrals for m = 2. Based on this, we present an alternative description of the function ω in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of ω describe the same function. The definition in the work of Jimbo et al (2008 arXiv:0811.0439) was crucial for the proof of the factorization. The definition given here together with the known description of ρ in terms of the solutions of nonlinear integral equations is useful for performing, e.g., the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite α.

Existence and properties of solutions for neural field equations

Abstract: The first goal of this work is to study the solvability of the neural field equation (known as ‘Amari equation’) which is an integro-differential equation in m+ 1 dimensions. In particular, we show the existence of global solutions for smooth activation functions f with values in [0, 1] and L1 kernels w via the Banach fixpoint theorem. We note that this setting is much more general than in most related studies, e.g. Ermentrout and McLeod (Proceedings of the Royal Society of Edinburgh 1993; 123A:461–478). For a Heaviside-type activation function f, we show that the approach above fails. However, with slightly more regularity on the kernel function w (we use Holder continuity with respect to the argument x) we can employ compactness arguments, integral equation techniques and the results for smooth nonlinearity functions to obtain a global existence result in a weaker space. Finally, general estimates on the speed and durability of waves are derived. We show that compactly supported waves with directed kernels (i.e. w(x, y)⩽0 for x⩽y ) decay exponentially after a finite time and that the field has a well-defined finite speed. Copyright © 2009 John Wiley & Sons, Ltd.

Blowup Algebras of Square-Free Monomial Ideals and Some Links to Combinatorial Optimization Problems

Abstract: Let I=(x^{v_1},...,x^{v_q} be a square-free monomial ideal of a polynomial ring K[x_1,...,x_n] over an arbitrary field K and let A be the incidence matrix with column vectors {v_1},...,{v_q}. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedrons and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornu\'ejols using an algebraic approach.

Colored Group Field Theory

Abstract: Group field theories are higher dimensional generalizations of matrix models. Their Feynman graphs are fat and in addition to vertices, edges and faces, they also contain higher dimensional cells, called bubbles. In this paper, we propose a new, fermionic Group Field Theory, posessing a color symmetry, and take the first steps in a systematic study of the topological properties of its graphs. Unlike its bosonic counterpart, the bubbles of the Feynman graphs of this theory are well defined and readily identified. We prove that this graphs are combinatorial cellular complexes. We define and study the cellular homology of this graphs. Furthermore we define a homotopy transformation appropriate to this graphs. Finally, the amplitude of the Feynman graphs is shown to be related to the fundamental group of the cellular complex.

Holonomic -modules and positive characteristic

Abstract: We discuss a hypothetical correspondence between holonomic -modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic.

Algebraic Geometry and Kinematics

Abstract: In this overview paper we show how problems in computational kinematics can be translated into the language of algebraic geometry and subsequently solved using techniques developed in this field. The idea to transform kinematic features into the language of algebraic geometry is old and goes back to Study. Recent advances in algebraic geometry and symbolic computation gave the motivation to resume these ideas and make them successful in the solution of kinematic problems. It is not the aim of the paper to provide detailed solutions, but basic accounts to the used tools and examples where these techniques were applied within the last years. We start with Study’s kinematic mapping and show how kinematic entities can be transformed into algebraic varieties. The transformations in the image space that preserve the kinematic features are introduced. The main topic are the definition of constraint varieties and their application to the solution of direct and inverse kinematics of serial and parallel robots. We provide a definition of the degree of freedom of a mechanical system that takes into account the geometry of the device and discuss singularities and global pathological behavior of selected mechanisms. In a short paragraph we show how the developed methods are applied to the synthesis of mechanical devices.

A closed galois field cryptographic system

Abstract: A cryptographic system (CS) comprised of generators ( 502 ), ( 504 ), ( 510 ), an encryption device (ED), and a decryption device (DD). The generator ( 502 ) generates a data sequence (DS) including payload data. The generator ( 504 ) generates an encryption sequence (ES) including random numbers. The ED ( 506 ) is configured to perform a CGFC arithmetic process. As such, the ED is comprised of a mapping device (MD) and an encryptor. The MD is configured to map the DS and ES from Galois field GF[p k ] to Galois extension field GF[p k+1 ]. The encryptor is configured to generate an encrypted data sequence (EDS) by combining the DS and ES utilizing a Galois field multiplication operation in Galois extension field GF[p k+1 ]. The generator ( 510 ) is configured to generate a decryption sequence (DS). The DD ( 508 ) is configured to generate a decrypted data sequence by performing an inverse of the CGFC arithmetic process utilizing the EDS and DS.

Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra

Abstract: We present an orthogonal basis of gauge invariant operators constructed from some complex matrices for the free matrix field, where operators are expressed with the help of Brauer algebra. This is a generalisation of our previous work for a signle complex matrix. We also discuss the matrix quantum mechanics relevant to N=4 SYM on S^{3} times R. A commuting set of conserved operators whose eigenstates are given by the orthogonal basis is shown by using enhanced symmetries at zero coupling.

Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

Abstract: We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain \({\mathbb{R}^n}\) with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun’s extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun’s theorem and Whitney’s theorem. In fact this result holds not only in Euclidean \({\mathbb{R}^n}\) but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi–Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.

Discreteness and rationality of $F$-jumping numbers on singular varieties

Abstract: We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.

Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

Abstract: Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed.

Shuffle Relations for Function Field Multizeta Values

Abstract: Despite the failure of naive analogs of the sum shuffle or the integral shuffle relations, we prove the existence of ‘shuffle’ relations for the multizeta values (for a general A, with a rational place at infinity) introduced [T04] by the author in the function field context. This makes the Fp-span of the multizeta values into an algebra. We effectively determine and prove all the Fp-coefficients identities (but not the Fp(t)-coefficients identities). 0. Introduction Multizeta values introduced and studied originally by Euler have been pursued recently again with renewed interest because of their emergence in studies in mathematics and mathematical physics connecting diverse viewpoints. See eg. introduction to [T09b] and references there. This paper is sequel to [T09b]. The author defined and studied two types of multizeta [T04, Sec 5.10] for function fields, one complex valued (generalizing the Artin-Weil zeta function) and the other with values in Laurent series ring over finite fields (generalizing the Carlitz zeta values). (For general background on function field arithmetic, we refer to [G96, T04]. ) For the Fq[t] case, the first type was completely evaluated in [T04] (see [M06] for more detailed study in the higher genus case). For the second type, the failure of sum and integral shuffle identities was noted, but different combinatorially involved identities were established or conjectured in [T04, T09b] as well as in the Masters thesis work [L09, L?] of Jose Alejandro Lara Rodriguez done at the University of Arizona. Also, period interpretation for these multizeta values was given in [AT09] in terms of explicit iterated extensions of the Carlitz-Tate t-motives. In contrast to the classical division between the convergent versus the divergent (normalized) values, all the values are convergent in our case. In place of the sum or the integral shuffle relations, we have different kinds of relations: the shuffle type relations with Fp-coefficients and the relations with Fp(t)-coefficients. (Classically, of course, there is no such distinction, the rational number field being the prime field in that case). In this paper, we show the existence of shuffle type relations proving that the product of multizeta values can also be expressed as a sum of some multizeta values, so that the Fp-span of all multizeta values is an algebra. While [T09b, L09, L?] conjectured and proved many such interesting relations (in the special case A = Fq[t]), which are combinatorially quite involved to describe unlike the classical case, here we prove the existence directly (for general A, defined below) rather than proving those conjectures. Date: October 16, 2009. The author was supported by NSA grant H98230-08-1-0049.

Dp-minimality: basic facts and examples

Abstract: We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

Zeta Functions and Motives

Abstract: We study properties of rigid K-linear ⊗-categories A, where K is a field of characteristic 0. When A is semi-simple, we introduce a notion of multiplicities for an object of A: they are rational integers in important cases including that of pure numerical motives over a field. This yields an alternative proof of the rationality and functional equation of the zeta function of an endomorphism, and a simple proof that the number of rational points modulo q of a smooth projective variety over Fq only depends on its “birational motive”. The multiplicities of motives of abelian type over a finite field are equal to ±1. We also study motivic zeta functions, and an abstracted version of the Tate conjecture over finite fields.

Jumping numbers on algebraic surfaces with rational singularities

Abstract: In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

H-vectors of simplicial complexes with Serre's conditions

Abstract: We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of $\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced homology groups of $\Delta$ over a field $K$. The main result of this paper is that if $\Delta$ satisfies Serre's condition ($S_r$) then (i) $h_k(\Delta)$ is non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(\Delta)$ is non-negative.