Showing papers on "Algebraic number published in 2011"

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

Abstract: Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications. Higher Order Transformations. Modular Equations and Algebraic Approximations to pi. The Complexity of Algebraic Functions. Algorithms for the Elementary Functions. General Means and Iterations. Some Additional Applications. Other Approaches to the Elementary Functions. Pi. Bibliography. Symbol List. Index.

Mathematical Foundations of Supersymmetry

Abstract: We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups.

Almost Free Modules: Set-theoretic Methods

Abstract: Algebraic Preliminaries. Set Theory. Slender Modules. Almost Free Modules. Pure-Injective Modules. More Set Theory. Almost Free Modules Revisited. Countably-Separable Groups. Quotients of Products of the Integers. Iterated Sums and Products. Topological Methods. The Structure of EXT. The Black Box and Endomorphism Rings. Dual Groups. Open Problems. Bibliography. Index.

Determinantal point processes

Abstract: We present a list of algebraic, combinatorial, and analytic mechanisms that give rise to determinantal point processes.

On the reachability and observability of path and cycle graphs

Abstract: In this paper we investigate the reachability and observability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a path or a cycle. More in detail, we provide necessary and sufficient conditions, based on simple algebraic rules from number theory, to characterize all and only the nodes from which the network system is reachable (respectively observable). Interesting immediate corollaries of our results are: (i) a path graph is reachable (observable) from any single node if and only if the number of nodes of the graph is a power of two, $n=2^i, i\in atural$, and (ii) a cycle is reachable (observable) from any pair of nodes if and only if $n$ is a prime number. For any set of control (observation) nodes, we provide a closed form expression for the (unreachable) unobservable eigenvalues and for the eigenvectors of the (unreachable) unobservable subsystem.

Finite subgroups of algebraic groups

Abstract: Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GLn over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

Random Walks in the Quarter-Plane: Algebraic Methods, Boundary Value Problems and Applications

Abstract: and History.- 1 Probabilistic Background.- 1.1 Markov Chains.- 1.2 Random Walks in a Quarter Plane.- 1.3 Functional Equations for the Invariant Measure.- 2 Foundations of the Analytic Approach.- 2.1 Fundamental Notions and Definitions.- 2.1.1 Covering Manifolds.- 2.1.2 Algebraic Functions.- 2.1.3 Elements of Galois Theory.- 2.1.4 Universal Cover and Uniformization.- 2.1.5 Abelian Differentials and Divisors.- 2.2 Restricting the Equation to an Algebraic Curve.- 2.2.1 First Insight (Algebraic Functions).- 2.2.2 Second Insight (Algebraic Curve).- 2.2.3 Third Insight (Factorization).- 2.2.4 Fourth Insight (Riemann Surfaces).- 2.3 The Algebraic Curve Q(x, y) = 0.- 2.3.1 Branches of the Algebraic Functions on the Unit Circle.- 2.3.2 Branch Points.- 2.4 Galois Automorphisms and the Group of the Random Walk.- 2.4.1 Construction of the Automorphisms ?? and ?? on S.- 2.5 Reduction of the Main Equation to the Riemann Torus.- 3 Analytic Continuation of the Unknown Functions in the Genus 1 Case.- 3.1 Lifting the Fundamental Equation onto the Universal Covering.- 3.1.1 Lifting of the Branch Points.- 3.1.2 Lifting of the Automorphisms on the Universal Covering.- 3.2 Analytic Continuation.- 3.3 More about Uniformization.- 4 The Case of a Finite Group.- 4.1On the Conditions for H to be Finite.- 4.1.1 Explicit Conditions for Groups of Order 4 or 6.- 4.1.2 The General Case.- 4.2 Rational Solutions.- 4.2.1 The Case N(f) ? 1.- 4.2.2 The Case N(f) = 1.- 4.3 Algebraic Solution.- 4.3.1 The Case N(f) = 1.- 4.3.2 The Case N(f) ?.- 4.4 Final Form of the General Solution.- 4.5 The Problem of the Poles and Examples.- 4.5.1 Rational Solutions.- 4.5.1.1 Reversible Random Walks.- 4.5.1.2 Simple Examples of Nonreversible Random Walks.- 4.5.1.3 One Parameter Families.- 4.5.1.4 Two Typical Situations.- 4.5.1.5 Ergodicity Conditions.- 4.5.1.6 Proof of Lemma 4.5.2.- 4.6 An Example of Algebraic Solution by Flatto and Hahn.- 4.7 Two Queues in Tandem.- 5 Solution in the Case of an Arbitrary Group.- 5.1 Informal Reduction to a Riemann-Hilbert-Carleman BVP.- 5.2 Introduction to BVP in the Complex Plane.- 5.2.1 A Bit of History.- 5.2.2 The Sokhotski-Plemelj Formulae.- 5.2.3 The Riemann Boundary Value Problem for a Closed Contour.- 5.2.4 The Riemann BVP for an Open Contour.- 5.2.5 The Riemann-Carleman Problem with a Shift.- 5.3 Further Properties of the Branches Defined by Q(x, y)= 0.- 5.4 Index and Solution of the BVP (5.1.5).- 5.5 Complements.- 5.5.1 Analytic Continuation.- 5.5.2 Computation of w.- 5.5.2.1 An Explicit Form via the Weierstrass ?-Function..- 5.5.2.2 A Differential Equation.- 5.5.2.3 An Integral Equation.- 6 The Genus 0 Case.- 6.1 Properties of the Branches.- 6.2 Case 1: ?01 = ??1,0 = ??1,1 = 0.- 6.3 Case 3: ?11 = ?10 = ?01 = 0.- 6.4 Case 4: ??1,0 = ?0,?1 = ??1,?1= 0.- 6.4.1 Integral Equation.- 6.4.2 Series Representation.- 6.4.3 Uniformization.- 6.4.4 Boundary Value Problem.- 6.5 Case 5: MZ= My= 0.- 7 Miscellanea.- 7.1 About Explicit Solutions.- 7.2 Asymptotics.- 7.2.1 Large Deviations and Stationary Probabilities.- 7.3 Generalized Problems and Analytic Continuation.- 7.4 Outside Probability.- References.

Zero-divisor graphs in commutative rings

Abstract: This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography.

Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

Abstract: For large scale problems, an effective approach for solving the algebraic Lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation. Although Krylov subspaces have been used for a long time, only more recent developments have shown that rational Krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit (ADI) recurrence. In this paper we develop a convergence analysis of the rational Krylov subspace method (RKSM) based on the Kronecker product formulation and on potential theory. Moreover, we propose new enlightening relations between this approach and the ADI method. Our results provide solid theoretical ground for recent numerical evidence of the superiority of RKSM over ADI when the involved parameters cannot be computed optimally, as is the case in many practical application problems.

The r-Bell numbers

Abstract: The notion of generalized Bell numbers was appeared in several works but there is no a systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers which in any cases generalize the similar properties of the Bell numbers. Most of these properties seems to be new. It turns out that in a paper of Whitehead these numbers appeared in a very dierent context. In addition, we introduce the so-called

Highly Nonlinear Boolean Functions with Optimal Algebraic Immunity and Good Behavior Against Fast Algebraic Attacks.

Abstract: In this paper, we present a new combinatorial conjecture about binary strings. Based on the new conjecture, two classes of Boolean functions of 2k variables with optimal algebraic immunity are proposed, where k ≥ 2. The first class contains unbalanced functions having high algebraic degree and nonlinearity. The functions in the second one are balanced and have maximal algebraic degree and high nonlinearity. It is checked that, at least for small numbers of variables, both classes of functions have a good behavior against fast algebraic attacks. Compared with the known Boolean functions resisting algebraic attacks and fast algebraic attacks, the two classes of functions possess the highest lower bounds on nonlinearity. These bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Moreover, these values are very good and much better than for the previously found functions having all the necessary features for being used in the filter model of pseudo-random generators.

Impulsive Consensus for Complex Dynamical Networks with Nonidentical Nodes and Coupling Time-Delays

Abstract: This paper investigates the problem of global consensus between a complex dynamical network (CDN) and a known goal signal by designing an impulsive consensus control scheme. The dynamical network is complex with respect to the uncertainties, nonidentical nodes, and coupling time-delays. The goal signal can be a measurable vector function or a solution of a dynamical system. By utilizing the Lyapunov function and Lyapunov-Krasovskii functional methods, robust global exponential stability criteria are derived for the error system, under which global exponential impulsive consensus is achieved for the CDN. These criteria are expressed in terms of linear matrix inequalities (LMIs) and algebraic inequalities. Thus, the impulsive controller can be easily designed by solving the derived inequalities. Meanwhile, the estimations of the consensus rate for global exponential consensus are also obtained. Two examples with numerical simulations are worked out for illustration.

Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

Abstract: We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant -24n+1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.

Varieties fibered by good minimal models

Abstract: Let f : X → Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map with relative dimension no more than three, then X has a good minimal model.

Symbol-pair codes: Algebraic constructions and asymptotic bounds

Abstract: For the recently proposed model of symbol-pair channels, we advance the pair-error coding theory with algebraic cyclic-code constructions and asymptotic bounds on code rates. Cyclic codes for pair-errors are constructed by a careful use of duals of known tools from cyclic-code theory. Asymptotic lower bounds on code rates show that codes for pair-errors provably exist for rates strictly higher than codes for the Hamming metric.

Global Lie-Tresse theorem

Abstract: We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

Linear Time-Varying Systems: Algebraic-Analytic Approach

Abstract: Part I Mathematical tools.- Part II Algebraic Theory of Linear Systems.- Part III Applications.- Part IV Complements.

Algebraic Design Theory

Abstract: Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.

Algebraic model structures

Abstract: We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove "algebraic" analogs of classical results. Using a modified version of Quillen's small object ar- gument, we show that every cofibrantly generated model structure in the usual sense underlies a cofibrantly generated algebraic model structure. We show how to pass a cofibrantly generated algebraic model structure across an adjunction, and we characterize the algebraic Quillen adjunc- tion that results. We prove that pointwise algebraic weak factorization systems on diagram categories are cofibrantly generated if the original ones are, and we give an algebraic generalization of the projective model structure. Finally, we prove that certain fundamental comparison maps present in any cofibrantly generated model category are cofibrations when the cofibrations are monomorphisms, a conclusion that does not seem to be provable in the classical, nonalgebraic, theory.