Rational Vector Fields Research Articles

Moment evolution equations for rational random dynamical systems: an increment decomposition method

Abstract Statistical moments are commonly used tools for exploring the ensemble behavior in gene regulation and population dynamics, where the rational vector fields are particularly ubiquitous, but how to efficiently derive the corresponding moment evolution equations was not much involved. Traditional derivation methods rely on fractional reduction and Itô formula, but it may become extremely complicated if the vector field is described by multivariate fractional polynomials. To resolve this issue, we present a novel incremental decomposition method, by which the rational vector field is divided into two parts: (proper) fractional polynomials and non-fractional polynomials. For the non-fractional polynomial part, we deduce the variation rate of a statistical moment by the Itô formula, but for the fractional polynomial part we acquire the corresponding variation rate by a relation analogous to that between the moment generating function and the distinct statistical moments. As application of the novel technique, the resultant infinite-dimensional moment systems associated with two typical examples are truncated with the schemes of derivative matching closure and the Gaussian moment closure. By comparing the lower-order statistical moments obtained from the closed moment systems with the counterparts obtained from direct simulation, the correctness of the proposed technique is verified. The present study is significant in facilitating the development of moment dynamics towards more complex systems.

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

On the separatrix graph of a rational vector field on the Riemann sphere

We consider the rational flow ξR(z)=R(z)(d/dz) where R is given by the quotient of two polynomials without common factors on the Riemann sphere. The separatrix graph ΓR is the boundary between trajectories with different properties. We characterize the properties of a planar directed graph to be the separatrix graph of a rational vector field on the Riemann sphere.

Remarks on Rational Vector Fields on mathbb {C}mathbb {P}^{1}

In this paper, we introduce geometric tools to study the families of rational vector fields of a given degree over mathbb {C}mathbb {P}^{1}. To a generic vector field of such a parametric family, we associate several geometric objects: a periodgon, a star domain, and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields. They are used to describe the bifurcations inside the families. We specialize to the case of rational vector fields of degree 4.

Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal $$\mathbb {C}\times \mathbb {S}^1$$ -bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic $$\mathbb {C}$$ -actions $$\mathscr {A}$$ on the space of polynomials of degree n. For each orbit $$\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}$$ of $$\mathscr {A}$$ , we study the dynamical problem of the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$ such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit $$\{ s \cdot P(z) = 0 \}$$ . Regarding the above $$\mathbb {C}$$ -action coming from the $$\mathbb {C}\times \mathbb {S}^1$$ -bundle structure, we prove the existence of a complex rational vector field $$\mathbb {X}(z)$$ on $$\mathbb {C}$$ , which describes the geometric change of the n-root configuration in the unitary disk $$\mathbb {D}$$ of a $$\mathbb {C}$$ -orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in $$\mathbb {C}\backslash \overline{\mathbb {D}}$$ , by constructing a principal $$\mathbb {C}^* \times \mathbb {S}^1$$ -bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.

Region of attraction analysis for nonlinear vehicle lateral dynamics using sum-of-squares programming

ABSTRACTIn this study we will estimate the region of attraction (RoA) of the lateral dynamics of a nonlinear single-track vehicle model. The tyre forces are approximated using rational functions that are shown to capture the nonlinearities of tyre curves significantly better than polynomial functions. An existing sum-of-squares (SOS) programming algorithm for estimating regions of attraction is extended to accommodate the use of rational vector fields. This algorithm is then used to find an estimate of the RoA of the vehicle lateral dynamics. The influence of vehicle parameters and driving conditions on the stability region are studied. It is shown that SOS programming techniques can be used to approximate the stability region without resorting to numerical integration. The RoA estimate from the SOS algorithm is compared to the existing results in the literature. The proposed method is shown to obtain significantly better RoA estimates.

The projective translation equation and rational plane flows. II. Corrections and additions

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments and additions are needed to claim the validity of the classification results. With these new results in a disposition, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension $2$. Previously we found such flows symmetric with respect to a linear involution $i(x,y)=(y,x)$. Here we find all rational flows symmetric with respect to a non-linear $1$-homogeneous involution $i(x,y)=(\frac{y^2}{x},y)$. We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.

Automorphisms of smooth canonically polarized surfaces in positive characteristic

This paper investigates the geometry of a smooth canonically polarized surface X defined over an algebraically closed field of characteristic p>0 in the case when the automorphism scheme of X is not smooth. In particular, it is shown that a smooth canonically polarized surface X with 1≤KX2≤2 and non-smooth automorphism scheme tends to be uniruled and simply connected and is the purely inseparable quotient of a ruled or rational surface by a rational vector field. Moreover, restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth.

Application of Gaussian moment method to a gene autoregulation model of rational vector field

We take a lambda expression autoregulation model driven by multiplicative and additive noises as example to extend the Gaussian moment method from nonlinear stochastic systems of polynomial vector field to noisy biochemical systems of rational polynomial vector field. As a direct application of the extended method, we also disclose the phenomenon of stochastic resonance. It is found that the transcription rate can inhibit the stochastic resonant effect, but the degradation rate may enhance the phenomenon. These observations should be helpful in understanding the functional role of noise in gene autoregulation.

Algebraic and abelian solutions to the projective translation equation

Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) $(1-z)\phi(\mathbf{x})=\phi(\phi(\mathbf{x}z)(1-z)/z)$, $\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are parametrized by abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR) the orbits of these flows can be transformed into algebraic curves $x^{A}(x-y)^{B}y^{C}\equiv\mathrm{const.}$ (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines $y\equiv\mathrm{const.}$ (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.

The projective translation equation and unramified 2-dimensional flows with rational vector fields

Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1−z)ϕ(x) = ϕ(ϕ(xz)(1−z)/z); here ϕ(x) = (u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in \({\mathbb{C}^{2}\setminus\{\text{union of curves}\}}\)) projective flows whose vector field is still rational. If an orbit of the flow is given by a homogeneous rational function of degree N, then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, unramified non-singular flows are of 6 types: (1) the identity flow; (2) one flow for each non-negative integer N—these flows are rational of level N; (3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; (4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; (5) the level 4 flow expressable in terms of lemniscatic elliptic functions; (6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya–Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.

Some properties of the k-dimensional Lyness' map

This paper is devoted to study some properties of the k-dimensional Lyness' map F(x1, …, xk) = (x2, …, xk, (a + ∑ki=2xi)/x1). Our main result presents a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ⩽ 5, to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F○F which allows us to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period.

CONTROL SYNTHESIS OF SYSTEMS WITH UNCERTAIN PARAMETERS BY CONVEX OPTIMIZATION

The last few years witnessed an increasing interest in the problem of control synthesis of nonlinear systems. A recently derived stability criterion for nonlinear systems-which has a remarkable convexity property- and the development of numerical methods for verification of positivity allows the computation-via semidefinite programming- of stabilizing controllers for the case of systems with polynomial or rational vector fields. Using the theory of semialgebraic sets these computational tools are extended in this paper for the case of polynomial or rational systems with uncertainty parameters.

On simultaneous uniformization and local nonuniformizability

We prove existence of a one-dimensional holomorphic foliation (with isolated irremovable singularities) tangent to a rational vector field on appropriate affine algebraic surface of dimension 2 such that the family of leaves intersecting arbitrary given cross-section does not admit a uniformization holomorphic in the parameter by a family of simply connected domains in C . We show that such a foliation can be chosen transversally affine, having a Liouvillian first integral, with dense and hyperbolic leaves and an attracting cycle. This extends the author's result [4] giving a negative answer to Ilyashenko's simultaneous uniformization conjecture and answers negatively to the local version of this conjecture recently proposed by Shcherbakov. To cite this article: A. Glutsyuk, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 489–494.

Model completion of Lie differential fields

We define a Lie differential field as a field A of characteristic 0 with an action, as derivations on A , of some given Lie algebra L . We assume that L is a finite-dimensional vector space over some sub-field F⊆ A given in advance. As an example take the field of rational functions on a smooth algebraic variety, with L={ rational vector fields } . For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique (up to isomorphism). We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for DCF 0 . We also show that this model completion is ω-stable.

Complete vector fields on (ℂ*)ⁿ

We prove necessary and sufficient conditions for a rational vector field on ( C ∗ ) n (\mathbb {C}^*)^n to be complete.

Finitely Generated Algebras Associated with Rational Vector Fields

Let k be an algebraically closed field of characteristic zero and R a polynomial ring in n variables over k. A normal k-subalgebra S of R which is finitely generated over k is called cofinite if R is integral over S. It is one of the central problems in affine algebraic geometry to consider the structures of cofinite normal k-subalgebras of R. If n s 2, it is known Ž w x. 2 cf. 2 that Spec S is isomorphic to the quotient of the affine plane A k Ž . 2 modulo a finite subgroup G of GL 2, k acting linearly on A . In the case k n G 3, nothing essential is known. In the present article, we propose one concrete method to construct a k-subalgebra of R. w x For R s k x , . . . , x , we consider a rational vector field 1 n

A note on wild fibers of elliptic surfaces

is equal to d (resp. less than d). In characteristic 0, there does not exist a wild fiber by the cohomological flatness. In positive characteristic, however, the existence of wild fibers makes the situation complicated. The notion of wild fiber was introduced in Bombieri and Mumford [l], and Raynaud [S] examined the structure of wild fiber in detail. In this note, we consider elliptic surfaces obtained as quotients of the product of a curve and a supersingular elliptic curve by rational vector fields in positive characteristic. We calculate numerical invariants of wild fibers of such elliptic surfaces (cf. Theorem 3.5). Moreover, we give a characterization of such elliptic surfaces over the projective line P’ (cf. Theorem 4.2). To calculate numerical invariants, Raynaud’s results on wild fibers play an important role (cf. [5]). For the case of the product of a curve and an ordinary elliptic curve, we already treated this in [3]. 1. Preliminaries In this section, we recall some basic facts on elliptic surfaces and Raynaud’s theory on wild fibers. For details, see Bombieri and Mumford [l] and Raynaud [S].

Uniformization of the leaves of a rational vector field

Uniformization of the leaves of a rational vector field

Realization of semisimple lie algebras with polynomial and rational vector fields

(1993). Realization of semisimple lie algebras with polynomial and rational vector fields. Communications in Algebra: Vol. 21, No. 11, pp. 4065-4081.