Showing papers on "Algebraic number published in 2012"

Numerical Solution of Algebraic Riccati Equations

Abstract: This concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic way. Some proofs of theoretical results have been simplified and a unified notation has been adopted. Readers will find a discussion of doubling algorithms, which are effective in solving algebraic Riccati equations, and a detailed description of all classical and advanced algorithms for solving algebraic Riccati equations, along with their MATLAB codes. This will help the reader gain understanding of the computational issues and provide ready-to-use implementation of the different solution techniques. Audience: This book is intended for researchers who work in the design and analysis of algorithms and for practitioners who are solving problems in applications and need to understand the available algorithms and software. It is also intended for students with no expertise in this area who wish to approach this subject from a theoretical or computational point of view. The book can be used in a semester course on algebraic Riccati equations or as a reference in a course on advanced numerical linear algebra and applications.

New Stable Closed Newton-Cotes Trigonometrically Fitted Formulae for Long-Time Integration

Abstract: The closed Newton-Cotes differential methods of high algebraic order for small number of function evaluations are unstable. In this work, we propose a new closed Newton-Cotes trigonometrically fitted differential method of high algebraic order which gives much more efficient results than the well-know ones.

Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem ∗

Abstract: The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymor- phisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near- unanimity terms (proved by McKenzie and Maroti) in an elementary and self-contained way.

A spectral gap theorem in SU$(d)$

Abstract: We establish the spectral gap property for dense subgroups of SU(d) (d=2 ), generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU(2) .

General grey numbers and their operations

Abstract: Purpose – The purpose of this paper is to advance new rules about operations of grey numbers.Design/methodology/approach – The paper first puts forward the definitions of basic element of grey number and general grey number. The operation axiom, operation rules of general grey numbers and a new algebraic system for general grey numbers are built based on the “kernel” and the degree of greyness of grey numbers.Findings – Up to now, the operation of general grey numbers has been transformed to operation of real numbers; thus, the difficult problem for set up operation of general grey numbers has been solved to a certain degree.Practical implications – The method exposed in the paper can be used to integrate information from a different source. The operation of general grey numbers could be extended to the case of grey algebraic equation, grey differential equation and grey matrix which includes general grey numbers. The operation system of general grey numbers also opened a new passageway for research on gr...

LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future

Abstract: 10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions and some new results.

Investigations on Bent and Negabent Functions via the Nega-Hadamard Transform

Abstract: Parker considered a new type of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behavior on combinations of Boolean functions and use this theory to derive several results on negabentness (that is, flat nega-spectrum) of concatenations, and partially symmetric functions. We derive the upper bound ⌈n/2⌉ for the algebraic degree of a negabent function on n variables. Further, a characterization of bent-negabent functions is obtained within a subclass of the Maiorana-McFarland set. We develop a technique to construct bent-negabent Boolean functions by using complete mapping polynomials. Using this technique, we demonstrate that for each l ≥ 2, there exist bent-negabent functions on n = 12l variables with algebraic degree n/4 + 1 = 3l + 1. It is also demonstrated that there exist bent-negabent functions on eight variables with algebraic degrees 2, 3, and 4. Simple proofs of several previously known facts are obtained as immediate consequences of our work.

Spectral curves and the Schroedinger equations for the Eynard-Orantin recursion

Abstract: It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schroedinger equation, and that the characteristic variety of the Schroedinger operator gives the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes. In this paper we present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side. The A-model examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, we show that the Laplace transform of the counting functions satisfies the Eynard-Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schroedinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory.

On the algebraic K-theory of higher categories

Abstract: We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.

How to find groups? (and how to use them in Erdős geometry?)

Abstract: Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ? ?[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be re-written as z = f(x, y), for a polynomial or rational function f ? ?(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on "circle grids"). The main result of our paper concerns low-degree algebraic sets F which contain "too many" points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).

Geometry of maximum likelihood estimation in Gaussian graphical models

Abstract: Author(s): Uhler, Caroline | Advisor(s): Sturmfels, Bernd | Abstract: Algebraic statistics exploits the use of algebraic techniques to develop new paradigms and algorithms for data analysis. The development of computational algebra software provides a powerful tool to analyze statistical models. In Part I of this thesis, we use methods from computational algebra and algebraic geometry to study Gaussian graphical models. Algebraic methods have proven to be useful for statistical theory and applications alike. We describe a particular application to computational biology in Part II.Part I of this thesis investigates geometric aspects of maximum likelihood estimation in Gaussian graphical models. More generally, we study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. In Chapter 2, we examine these problems at the interface of statistics and optimization from the perspective of convex algebraic geometry and characterize the cone of all sufficient statistics for which the maximum likelihood estimator (MLE) exists. In Chapter 3, we develop an algebraic elimination criterion, which allows us to find exact lower bounds on the number of observations needed to ensure that the MLE exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also present the first instance of a graph for which the MLE exists with probability one even when the number of observations equals the treewidth. Computational algebra software can be used to study graphs with a limited number of vertices and edges. In Chapter 4, we study the problem of existence of the MLE from an asymptotic point of view by fixing a class of graphs and letting the number of vertices grow to infinity. We prove that for very large cycles already two observations are sufficient for the existence of the MLE with probability one. Part II of this thesis describes an application of algebraic statistics to association studies. Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on determining associations between single loci and a specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. We develop a method for finding interacting genes (i.e. epistasis) using Markov bases. We test our method on simulated data and compare it to a two-stage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a genome-wide dog data set and identify epistasis associated with canine hair length.

Algebraic trace functions over the primes

Abstract: We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann Hypothesis over finite fields, we prove general estimates with power-saving for such sums. We then derive various concrete applications.

Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems

Abstract: SUMMARY We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin (DG) discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections, and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system associated with this space is constructed algebraically using a Galerkin approach with the natural embedding as the prolongation operator. This embedding operator needs to be provided, which means that the approach is not fully algebraic. For the construction of the linear systems on the subsequent coarser levels, non-smoothed aggregation AMG techniques are used. In a series of numerical experiments, we establish for the first time the efficiency and robustness of an AMG method for various symmetric and non-symmetric interior penalty DG methods (including the higher-order cases) on problems with complicated, high contrast jumps in the coefficients. The solver is robust with respect to an increase in the polynomial degree of the DG approximation space (at least up to degree 6), computationally efficient, and affected only mildly by the coefficient jumps and by the mesh size h (i.e., number of iterations = O(log h−1)). Copyright © 2012 John Wiley & Sons, Ltd.

Essential Convexity and Complexity of Semi-Algebraic Constraints

Abstract: Let \Gamma be a structure with a finite relational signature and a first-order definition in (R;*,+) with parameters from R, that is, a relational structure over the real numbers where all relations are semi-algebraic sets. In this article, we study the computational complexity of constraint satisfaction problem (CSP) for \Gamma: the problem to decide whether a given primitive positive sentence is true in \Gamma. We focus on those structures \Gamma that contain the relations \leq, {(x,y,z) | x+y=z} and {1}. Hence, all CSPs studied in this article are at least as expressive as the feasibility problem for linear programs. The central concept in our investigation is essential convexity: a relation S is essentially convex if for all a,b\inS, there are only finitely many points on the line segment between a and b that are not in S. If \Gamma contains a relation S that is not essentially convex and this is witnessed by rational points a,b, then we show that the CSP for \Gamma is NP-hard. Furthermore, we characterize essentially convex relations in logical terms. This different view may open up new ways for identifying tractable classes of semi-algebraic CSPs. For instance, we show that if \Gamma is a first-order expansion of (R;*,+), then the CSP for \Gamma can be solved in polynomial time if and only if all relations in \Gamma are essentially convex (unless P=NP).

Essential convexity and complexity of semi-algebraic constraints

Abstract: Let \Gamma be a structure with a finite relational signature and a first-order definition in (R;*,+) with parameters from R, that is, a relational structure over the real numbers where all relation ...

Weil and Grothendieck approaches to adelic points

Abstract: In [We, Ch. 1], Weil defines a process of “adelization” of algebraic varieties over global fields. There is an alternative procedure, due to Grothendieck, using adelic points. One aim of this (largely) expository note is to prove that for schemes of finite type over global fields (i.e., without affineness hypotheses), and also for separated algebraic spaces of finite type over such fields, Weil’s adelization process naturally coincides (as a set) with the set of adelic points in the sense of Grothendieck (and that in the affine case the topologies defined by these two viewpoints coincide; Grothendieck’s approach doesn’t provide a topology beyond the affine case). The other aim is to prove in general that topologies obtained by Weil’s method satisfy good functorial properties, including expected behavior with respect to finite flat Weil restriction of scalars. The affine case suffices for most applications, but the non-affine case is useful (e.g., adelic points of G/P for connected reductive groups G and parabolic subgroups P ). We also discuss topologizing X(k) for possibly non-separated algebraic spaces X over locally compact fields k; motivation for this is given in Example 5.5. Although everything we prove (except perhaps for the case of algebraic spaces) is “well known” folklore, and [Oes, I, §3] provides an excellent summary in the affine case, some aspects are not so easy to extract from the available literature. Moreover, (i) some references that discuss the matter in the non-affine case have errors in the description of the topology on adelic points, and (ii) much of what we prove is needed in my paper [Con], or in arithmetic arguments in [CGP]. In effect, these notes can be viewed as an expanded version of [Oes, I, §3], and I hope they will provide a useful general reference on the topic of adelic points of algebro-geometric objects (varieties, schemes, algebraic spaces) over global fields. In §2 we carry out Grothendieck’s method in the affine case over any topological ring R, characterizing the topology on sets of R-points by means of several axioms. The generalization to arbitrary schemes of finite type via a method of Weil is developed in §3. We explore properties of these topologies in §4, especially for adelic points and behavior with respect to Weil restriction of scalars. Finally, in §5 everything is generalized to the case of algebraic spaces. Notation. We write AF to denote the adele ring of a global field F , and likewise A n F denotes Euclidean n-space over AF . There is no risk of confusion with the common use of such notation to denote affine n-space over SpecF since we avoid ever using this latter meaning for the notation.

Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field

Abstract: Over an arbitrary field of characteristic ≠ 2, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.

Tailoring Three-Point Functions and Integrability IV. Theta-morphism

Abstract: We compute structure constants in N=4 SYM at one loop using Integrability. This requires having full control over the two loop eigenvectors of the dilatation operator for operators of arbitrary size. To achieve this, we develop an algebraic description called the Theta-morphism. In this approach we introduce impurities at each spin chain site, act with particular differential operators on the standard algebraic Bethe ansatz vectors and generate in this way higher loop eigenvectors. The final results for the structure constants take a surprisingly simple form. For some quantities we conjecture all loop generalizations. These are based on the tree level and one loop patterns together and also on some higher loop experiments involving simple operators.

The absolute arithmetic continuum and the unification of all numbers great and small

Abstract: In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including −ω, ω/2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Godel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-a-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.

Solving polynomial systems over finite fields: improved analysis of the hybrid approach

Abstract: The Polynomial System Solving (PoSSo) problem is a fundamental NP-Hard problem in computer algebra. Among others, PoSSo have applications in area such as coding theory and cryptology. Typically, the security of multivariate public-key schemes (MPKC) such as the UOV cryptosystem of Kipnis, Shamir and Patarin is directly related to the hardness of PoSSo over finite fields. The goal of this paper is to further understand the influence of finite fields on the hardness of PoSSo. To this end, we consider the so-called hybrid approach. This is a polynomial system solving method dedicated to finite fields proposed by Bettale, Faugere and Perret (Journal of Mathematical Cryptography, 2009). The idea is to combine exhaustive search with Grobner bases. The efficiency of the hybrid approach is related to the choice of a trade-off between the two methods. We propose here an improved complexity analysis dedicated to quadratic systems. Whilst the principle of the hybrid approach is simple, its careful analysis leads to rather surprising and somehow unexpected results. We prove that the optimal trade-off (i.e. number of variables to be fixed) allowing to minimize the complexity is achieved by fixing a number of variables proportional to the number of variables of the system considered, denoted n. Under some natural algebraic assumption, we show that the asymptotic complexity of the hybrid approach is 2(3.31-3.62 log2(q)-1)n, where q is the size of the field (under the condition in particular that log(q) L n). This is to date, the best complexity for solving PoSSo over finite fields (when q > 2). We have been able to quantify the gain provided by the hybrid approach compared to a direct Grobner basis method. For quadratic systems, we show (assuming a natural algebraic assumption) that this gain is exponential in the number of variables. Asymptotically, the gain is 21.49n when both n and q grow to infinity and log(q) L n.

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

Abstract: This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.

Finiteness theorems for algebraic groups over function fields

Abstract: We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

Comparison between analytic and algebraic constructions of toroidal compactifications of pel-type shimura varieties

Abstract: Using explicit identications between algebraic and analytic theta functions, we compare the algebraic constructions of toroidal compactications by Faltings{Chai and the author, with the analytic constructions of toroidal compactications following Ash{Mumford{Rapport{Tai. As one of the appli- cations, we obtain the corresponding comparison for Fourier{Jacobi expansions of holomorphic automorphic forms.

An algebraic approach to the radius of comparison

Abstract: The radius of comparison is an invariant for unital C∗-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C∗-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C∗-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C∗-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C∗-algebra D without bounded 2-quasitraces or unital quotients has the property that Mn(D) is stable.