arXiv subjects

Werner M. Seiler

Publications and source records attributed to Werner M. Seiler.

At least 19 recordsLinked to original sources

arXiv2026

Mathematical Modeling of Early Embryonic Cell Cycles of Drosophila melanogaster

In the early stages of development, Drosophila melanogaster embryos possess very fast and well-coordinated cell cycles. In the cell cycle, CDK activity is essentially regulated by binding CDK and CycB to form an active complex and by phosphorylating CDK via CDC25 and dephosphorylating it via Wee1. We develop a mathematical model for the embryonic cell cycle which is biochemically sound and which can be rigorously analysed after a model reduction. We show that there exists a region in the parameter space where the model describes oscillations. We then focus on the role of two parameters: the CycB synthesis and the activation coefficient of APC. Our main biological hypothesis is that the first one is responsible for the period lengthening over the first 14 cycles which can be experimentally observed and this hypothesis is supported by numerical simulations of our model: if the CycB synthesis is made time-dependent with a prescribed dynamics, then our simulations show qualitatively a very similar behavior to experimental data reported in the literature.

q-bio.QM↗

arXiv2026

Structural Identifiability and Discrete Symmetries

We discuss the use of symmetries for analysing the structural identifiability and observability of control systems. Special emphasis is put on the role of discrete symmetries, in contrast to the more commonly studied continuous or Lie symmetries. We argue that discrete symmetries are the origin of parameters which are structurally locally identifiable, but not globally. We exploit this fact to present a methodology for structural identifiability analysis that detects such parameters and characterizes the symmetries in which they are involved. We demonstrate the use of our methodology by applying it to four case studies.

math.DS↗

arXiv2026

Sympatric speciation by symmetry-breaking: The three-clade case

In this paper we expand the concept of biological speciation by symmetry breaking of Golubitsky and Stewart to the case of three clades in which N populations following the same dynamical laws can separate. The underlying differential equation is based on a fifth order polynomial of a trait variable with first or second order coupling. We present some general strategies to find all possible steady states and their stabilities. Numerical data are given for a specific system. We show the locations of three-clade distributions in dependence on the coupling and an environmental parameter. The results show a decrease of the number of stable states with higher coupling and a higher probability of ending in a three-clade state for larger N. Limits and potentials of the approach if zero roots for the trait variable occur are discussed.

q-bio.PE↗

arXiv2024

Cohen-Macaulay, Gorenstein and complete intersection conditions by marked bases

Using techniques coming from the theory of marked bases, we develop new computational methods for detection and construction of Cohen-Macaulay, Gorenstein and complete intersection homogeneous polynomial ideals. Thanks to the functorial properties of marked bases, an elementary and effective proof of the openness of arithmetically Cohen-Macaulay, arithmetically Gorenstein and strict complete intersection loci in a Hilbert scheme follows, for a non-constant Hilbert polynomial.

math.AC↗

arXiv2024

Machine Learning Conservation Laws of Dynamical systems

Conservation laws are of great theoretical and practical interest. We describe a novel approach to machine learning conservation laws of finite-dimensional dynamical systems using trajectory data. It is the first such approach based on kernel methods instead of neural networks which leads to lower computational costs and requires a lower amount of training data. We propose the use of an "indeterminate" form of kernel ridge regression where the labels still have to be found by additional conditions. We use here a simple approach minimising the length of the coefficient vector to discover a single conservation law.

physics.comp-ph↗

arXiv2024

On the minimal free resolution of the Rees algebra of tri-generated bivariate monomial ideals

Let $I$ be a monomial ideal in two variables generated by three monomials and let $\mathcal{R}(I)$ be its Rees ideal. We describe an algorithm to compute the minimal generating set of $\mathcal{R}(I)$. Based on the data obtained by this algorithm, we build a graph that encodes the minimal free resolution of $\mathcal{R}(I)$. We explicitly describe the modules and differentials on the minimal free resolution of $\mathcal{R}(I)$.

math.AC↗

arXiv2024

Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases

We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points.

math.AC↗

arXiv2023

On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations

We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.

math.NA↗

arXiv2021

Singularities of Algebraic Differential Equations

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notoriously difficult notion ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

math.AC↗

arXiv2020

Singular Initial Value Problems for Scalar Quasi-Linear Ordinary Differential Equations

We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form $g(x)u''=f(x,u,u')$ with an initial condition imposed at a simple zero of $g$. This generalises results by Liang and also makes them more transparent via our geometric approach.

math.DS↗

arXiv2020

No Chaos in Dixon's System

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

math.DS↗

arXiv2020

On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.

math.DS↗

arXiv2020

A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equations

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be characterised as points where the behaviour of a certain linear system of equations changes. These points can be discovered using a specifically adapted parametric generalisation of Gaussian elimination combined with heuristic simplification techniques and real quantifier elimination methods. We demonstrate the relevance and applicability of our approach with computational experiments using a prototypical implementation in Reduce.

math.LO↗

arXiv2018

A Pommaret Bases Approach to the Degree of a Polynomial Ideal

In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bezout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.

math.AG↗

arXiv2018

Computation of Pommaret Bases Using Syzygies

We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method. We conclude the paper by proposing an involutive variant of the signature based algorithm of Gao et al. to compute simultaneously a Grobner basis for a given ideal and for the syzygy module of the input basis. All the presented algorithms have been implemented in Maple and their performance is evaluated via a set of benchmark ideals.

math.AG↗

arXiv2018

Computing Quot schemes via marked bases over quasi-stable modules

Let $ \Bbbk$ be a field of arbitrary characteristic, $A$ a Noetherian $ \Bbbk$-algebra and consider the polynomial ring $A[\mathbf x]=A[x_0,\dots,x_n]$. We consider homogeneous submodules of $A[\mathbf x]^m$ having a special set of generators: a marked basis over a quasi-stable module. Such a marked basis inherits several good properties of a Gröbner basis, including a Noetherian reduction relation. The set of submodules of $A[\mathbf x]^m$ having a marked basis over a given quasi-stable module has an affine scheme structure that we are able to exhibit. Furthermore, the syzygies of a module generated by such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in $\oplus^{m'}_{i=1} A[\mathbf x](-d_i)$). We apply the construction of marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we can explicitely construct (up to the action of a general linear group) an open cover of the corresponding Quot functor made up of open functors represented by affine schemes. This gives a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points.

math.AC↗

arXiv2017

Improved Computation of Involutive Bases

In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Moller et al., we present a more efficient variant of Gerdt's algorithm (than the algorithm presented by Gerdt-Hashemi-M.Alizadeh) to compute minimal involutive bases. Further, by using the involutive version of Hilbert driven technique, along with the new variant of Gerdt's algorithm, we modify the algorithm, given by Seiler, to compute a linear change of coordinates for a given homogeneous ideal so that the new ideal (after performing this change) possesses a finite Pommaret basis. All the proposed algorithms have been implemented in Maple and their efficiency is discussed via a set of benchmark polynomials.

cs.SC↗

arXiv2017

Dimension-Dependent Upper Bounds for Grobner Bases

We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Grobner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions.

cs.SC↗