Pfaffian Functions Research Articles

Decision problem for a class of univariate Pfaffian functions

We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order 1. We present a new symbolic procedure solving this problem with a computable complexity based on the computation of suitable Sturm sequences. For a general Pfaffian function, we assume the existence of an oracle to determine the sign that a function of the class takes at a real algebraic number. As a by-product, we obtain a new oracle-free effective algorithm solving the same problem for univariate E-polynomials based on techniques that are simpler than the previous ones, and we apply it to solve a similar decision problem in the multivariate setting. Finally, we introduce a notion of Thom encoding for zeros of an E-polynomial and describe an algorithm for their computation.

Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

AbstractThe o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree$D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.

Effective Pila–Wilkie bounds for unrestricted Pfaffian surfaces

We prove effective Pila–Wilkie estimates for the number of rational points of bounded height lying on certain surfaces defined by Pfaffian functions. The class of surfaces to which our result applies includes, for instance, graphs of unrestricted Pfaffian functions defined on the plane.

Pfaffian definitions of Weierstrass elliptic functions

We give explicit definitions of the Weierstrass elliptic functions $$\wp $$ and $$\zeta $$ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function $$\sigma $$ . Our work has immediate applications to Diophantine geometry and we answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves. We also point out further applications, also in connection with Pila–Wilkie type counting problems.

Model-driven regularization approach to straight line program genetic programming

This paper presents a regularization method for program complexity control of linear genetic programming tuned for transcendental elementary functions. Our goal is to improve the performance of evolutionary methods when solving symbolic regression tasks involving Pfaffian functions such as polynomials, analytic algebraic and transcendental operations like sigmoid, inverse trigonometric and radial basis functions. We propose the use of straight line programs as the underlying structure for representing symbolic expressions. Our main result is a sharp upper bound for the Vapnik Chervonenkis dimension of families of straight line programs containing transcendental elementary functions. This bound leads to a penalization criterion for the mean square error based fitness function often used in genetic programming for solving inductive learning problems. Our experiments show that the new fitness function gives very good results when compared with classical statistical regularization methods (such as Akaike and Bayesian Information Criteria) in almost all studied situations, including some benchmark real-world regression problems.

Zero counting for a class of univariate Pfaffian functions

We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order 1. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign determination. In the particular case of E-polynomials, we design an oracle-free effective algorithm solving this task within exponential complexity. In addition, we give an explicit upper bound for the absolute value of the real zeros of an E-polynomial.

Multiplicities of Noetherian Deformations

The Noetherian class is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A conjecture by Khovanskii states that the local geometry of sets defined using Noetherian equations admits effective estimates analogous to the effective global bounds of algebraic geometry. We make a major step in the development of the theory of Noetherian functions by providing an effective upper bound for the local number of isolated solutions of a Noetherian system of equations depending on a parameter $${\varepsilon}$$ , which remains valid even when the system degenerates at $${\varepsilon = 0}$$ . An estimate of this sort has played the key role in the development of the theory of Pfaffian functions, and is expected to lead to similar results in the Noetherian setting. We illustrate this by deducing from our main result an effective form of the Łojasiewicz inequality for Noetherian functions.

Heine Representations and Monotonicity Properties of Determinants and Pfaffians

We establish integral representations of Heine type for certain integrals of determinants. We use these representations to derive monotonicity properties of such determinants. New integral representations of Heine type for biorthogonal functions obtained from the general Gram–Schmidt orthonormalization process are given. We also establish the monotonicity of certain Pfaffians using the de Bruijn formula of Ishikawa and Zeng. The complete or absolute monotonicity of a number of Pfaffian functions is established.

Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity

We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.

Generating the Pfaffian closure with total Pfaffian functions

Given an o-minimal expansionR of the real field, we show that the structure obtained fromR by iterating the operation of adding all total Pfaffian functions overR defines the same sets as the Pfaffian closure ofR. 2010 Mathematics Subject Classification 14P10,03C64 (primary); 58A17 (sec- ondary)

Mildness and the Density of Rational Points on Certain Transcendental Curves

We use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions.

On the Vapnik-Chervonenkis dimension of computer programs which use transcendental elementary operations

We exhibit upper bounds for the Vapnik-Chervonenkis (VC) dimension of a wide family of concept classes that are defined by algorithms using analytic Pfaffian functions. We give upper bounds on the VC dimension of concept classes in which the membership test for whether an input belongs to a concept in the class can be performed either by a computation tree or by a circuit with sign gates containing Pfaffian functions as operators. These new bounds are polynomial both in the height of the tree and in the depth of the circuit. As consequence we obtain polynomial VC dimension not also for classes of concepts whose membership test can be defined by polynomial time algorithms but also for those defined by well-parallelizable sequential exponential time algorithms.

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii's Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure. Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti numbers.

Model theory of exponentials on Lie algebras

This paper presents an analysis of definitions and decidability for exponential functions on various matrix algebras. The main idea is to show that, generically, the entries of the exponential (or logarithm) of a matrix are Pfaffian functions of the entries of the matrix.

Topology of Definable Hausdorff Limits

Let A ⫅ ℝn+r be a set definable in an o-minimal expansion S of the real field, let A’ ⫅ ℝr be its projection, and assume that the non-empty fibers Aa ⫅ ℝn are compact for all a ∈ A’ and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers Aai, we give an upper-bound for the Betti numbers bk(L) in terms of definable sets explicitly constructed from a fiber Aa. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure SPfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)0 in the special case where Y=∅.

BETTI NUMBERS OF SEMIALGEBRAIC AND SUB-PFAFFIAN SETS

Let X be a subset in [-1, 1](n0) subset of R-n0 defined by the formula X = {x(0) Q(1)x(1) Q(2)x(2) ... Q(v)x(v) ((x(0), x(1), ...,x(v)) is an element of X-v)}, where Q(i) is an element of {There Exists, For All}, Q(i) not equal Q(i+1), x(i) is an element of [-1, 1](ni), and X-v may be either an open or a closed set in being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of X is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving X-v. In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of X-v are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

An effective version of Wilkie's theorem of the complement and some effective o-minimality results

Wilkie (Selecta Math. (N.S.) 5 (1999) 397) proved a “theorem of the complement” which implies that in order to establish the o-minimality of an expansion of R with C ∞ functions it suffices to obtain uniform (in the parameters) bounds on the number of connected components of quantifier free definable sets. He deduced that any expansion of R with a family of Pfaffian functions is o-minimal. We prove an effective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered field R with finitely many C ∞ functions, if there are uniform and computable upper bounds on the number of connected components of quantifier free definable sets, then there are uniform and computable bounds for all definable sets. In such a case the theory of the structure is effectively o-minimal: there is a recursively axiomatized subtheory such that each of its models is o-minimal. This implies the effective o-minimality of any expansion of R with Pfaffian functions. We apply our results to the open problem of the decidability of the theory of the real field with the exponential function. We show that the decidability is implied by a positive answer to the following problem (raised by van den Dries (in: Logic: From Foundations to applications, Oxford Science Publ., Oxford University Press, New York, 1996, p. 137)): given a language L expanding the language of ordered rings, if an L-sentence is true in every L-structure expanding the ordered field of real numbers, then it is true in every o-minimal L-structure expanding any real closed field.

Complexity of gene circuits, Pfaffian functions and the morphogenesis problem

We consider a model of gene circuits. We show that these circuits are capable to generate any spatio–temporal patterns. We give lower bounds on the number of genes required to create a given pattern. To cite this article: S. Vakulenko, D. Grigoriev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Approximation and Complexity II: Iterated Integration

We introduce two classes of real analytic functions W \subset U on an interval. Starting with rational functions to construct functions in W we allow the application of three types of operations: addition, integration, and multiplication by a polynomial with rational coefficients. In a similar way, to construct functions in U we allow integration, addition, and multiplication of functions already constructed in U and multiplication by rational numbers. Thus, U is a subring of the ring of Pfaffian functions [7].

A generalization of Wilkie's theorem of the complement, and an application to Pfaffian closure

Using a modification of Wilkie's recent proof of o-minimality for Pfaffian functions, we give an invariant characterization of o-minimal expansions of $\Bbb R$. We apply this to construct the Pfaffian closure of an arbitrary o-minimal expansion of $\Bbb R$.