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## Publications

Publications (89)

Motivation
Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filte...

The ideal of the arc scheme of a double point or, equivalently, the differential ideal generated by the ideal of a double point is a primary ideal in an infinite-dimensional polynomial ring supported at the origin. This ideal has a rich combinatorial structure connecting it to singularity theory, partition identities, representation theory, and dif...

Quadratization refers to a transformation of an arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that facilitate model analysis, simulation, and control and offer a convenient parameterization for data-driven approaches...

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given mod...

The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and sufficient condition) for the infinite-dimensionality of its space V L of solutions belonging to R is the presen...

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and...

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model's formal analysis and mechanistic und...

Dynamical systems are commonly used to represent real-world processes. Model reduction techniques are among the core tools for studying dynamical systems models, they allow to reduce the study of a model to a simpler one. In this poster, we present an algorithm for computing exact nonlinear reductions, that is, a set of new rational function macro-...

Detailed dynamical systems models used in life sciences may include dozens or even hundreds of state variables. Models of large dimension are not only harder from the numerical perspective (e.g., for parameter estimation or simulation), but it is also becoming challenging to derive mechanistic insights from such models. Exact model reduction is a w...

Structural global parameter identifiability indicates whether one can determine a parameter's value from given inputs and outputs in the absence of noise. If a given model has parameters for which there may be infinitely many values, such parameters are called non-identifiable. We present a procedure for accelerating a global identifiability query...

Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore,...

Symbolic computation for systems of differential equations is often computationally expensive. Many practical differential models have a form of polynomial or rational ODE system with specified outputs. A basic symbolic approach to analyze these models is to compute and then symbolically process the polynomial system obtained by sufficiently many L...

Detailed dynamical systems models used in life sciences may include dozens or even hundreds of state variables. Models of large dimension are not only harder from the numerical perspective (e.g., for parameter estimation or simulation), but it is also becoming challenging to derive mechanistic insights from such models. Exact model reduction is a w...

Identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from the model equations. The IO-equation method is a method for verifying identifiability. It is important because the additional insights it provides can be used to analyze and improve models, but its theoretical grounds and applicability...

The equation $x^m = 0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x, x', x^{(2)}, \ldots]$ by all differential consequences of $x^m = 0$. This infinite-dimensional algebra admits a natural filtration by finite dimensional algebras corresponding to the truncations of arcs. We...

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that...

Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a nat...

Structural identifiability properties of models of ordinary differential equations help one assess if the parameter's value can be recovered from experimental data. This theoretical property can be queried without the need for data collection and is determined with help of differential algebraic tools. We present a web-based Structural Identifiabil...

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the siz...

Parameter identifiability describes whether, for a given differential model, one can determine parameter values from model equations. Knowing global or local identifiability properties allows construction of better practical experiments to identify parameters from experimental data. In this work, we present a web-based software tool that allows to...

Kinetic models of biochemical systems used in the modern literature often contain hundreds or even thousands of variables. While these models are convenient for detailed simulations, their size is often an obstacle to deriving mechanistic insights. One way to address this issue is to perform an exact model reduction by finding a self-consistent low...

Ritt’s theorem of zeroes and Siedenberg’s embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riqu...

Parameter identifiability describes whether, for a given differential model, one can determine parameter values from model equations. Knowing global or local identifiability properties allows construction of better practical experiments to identify parameters from experimental data. In this work, we present a web-based software tool that allows to...

Finite Elements are a common method for solving differential equations via discretization. Under suitable hypotheses, the solution \(\mathbf {u}=\mathbf {u}(t,\vec x)\) of a well-posed initial/boundary-value problem for a linear evolutionary system of PDEs is approximated up to absolute error \(1/2^n\) by repeatedly (exponentially often in n) multi...

Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. Quadratization problem is, given a system of ODEs with polynomial right-hand side, transform the system to a system with quadratic right-hand side by introducing new va...

Motivation
Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a low...

Kinetic models of biochemical systems used in the modern literature often contain hundreds or even thousands of variables. While these models are convenient for detailed simulations, their size is often an obstacle to deriving mechanistic insights. One way to address this issue is to perform an exact model reduction by finding a self-consistent low...

Quadratization problem is, given a system of ODEs with polynomial right-hand side, transform the system to a system with quadratic right-hand side by introducing new variables. Such transformations have been used, for example, as a preprocessing step by model order reduction methods and for transforming chemical reaction networks. We present an alg...

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input–output equations and, in particular, characteristic sets of differential ideals. The precis...

Motivation:
Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a l...

Elimination of unknowns in systems of equations, starting with Gaussian elimination, is a problem of general interest. The problem of finding an a priori upper bound for the number of differentiations in elimination of unknowns in a system of differential-algebraic equations (DAEs) is an important challenge, going back to Ritt (1932). The first cha...

Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable pro...

Transformation of a polynomial ODE system to a special quadratic form has been successfully used recently as a preprocessing step for model order reduction methods. However, to the best of our knowledge, there has been no practical algorithm for performing this step automatically with any optimality guarantees.
We present an algorithm that, given a...

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input-output equations and, in particular, characteristic sets of differential ideals. The precis...

We establish effective elimination theorems for ordinary differential-difference equations. Specifically, we find a computable function B ( r , s ) B(r,s) of the natural number parameters r r and s s so that for any system of algebraic ordinary differential-difference equations in the variables x = x 1 , … , x q \mathbfit {x} = x_1, \ldots , x_q an...

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the un...

Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensio...

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the siz...

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that...

We present an algorithm which for any given ideal $I\subseteq\mathbb{K} [x,y]$ finds all elements of $I$ that have the form $f(x) - g(y)$, i.e., all elements in which no monomial is a multiple of $xy$.

Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investi...

Many important real-world processes are modeled using systems of ordinary differential equations (ODEs) involving unknown parameters. The values of these parameters are usually inferred from experimental data. However, due to the structure of the model, there might be multiple parameter values that yield the same observed behavior even in the case...

Structural identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. The method of input-output equations is one method for verifying structural identifiability. This method stands out in its importance because the additional insights it pr...

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension F⊂E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{am...

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (e.g., standard difference schemes) and difference equations in functions on words. On the universal...

Biological processes are often modeled by ordinary differential equations with unknown parameters. The unknown parameters are usually estimated from experimental data. In some cases, due to the structure of the model, this estimation problem does not have a unique solution even in the case of continuous noise-free data. It is therefore desirable to...

Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. This paper contains the r...

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension $F \subset E$ of such fields of zero characteristic such that $\bullet$ $E$ is generated over $F$ by finitely many elements using th...

We establish effective elimination theorems for differential-difference equations. Specifically, we find a computable function $B(r,s)$ of the natural number parameters $r$ and $s$ so that for any system of algebraic differential-difference equations in the variables $\mathbf{x} = x_1, \ldots, x_q$ and $\mathbf{y} = y_1, \ldots, y_r$ each of which...

Biological processes are often modeled by ordinary differential equations with unknown parameters. The unknown parameters are usually estimated from experimental data. In some cases, due to the structure of the model, this estimation problem does not have a unique solution even in the case of continuous noise-free data. It is therefore desirable to...

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group G ⊂ GL n ( C ) G \subset \operatorname {GL}_n(C) can be arb...

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist - sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, - powerful randomized algorithms for computing triangular decompositions using Hensel...

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel...

Many real-world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide the theory, an algorithm, and software for deciding...

We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations,...

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree cu...

We present a new upper bound for the orders of derivatives in the Rosenfeld–Gröbner algorithm under weighted rankings. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomp...

We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the low-density power series expansion requires heavy and nontrivial computations. Despite of the exponential computational complexity, we compute almost three times more terms t...

In the present paper, we describe an algebraic point of view on the notion of the solution of a system of algebraic differential equations. We apply Capelli’s rank theorem to prime and simple differential algebras.

It is well-known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree cu...

For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded into the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfies the standard Lie...

In the present paper, we give a full description of the jet schemes of the polynomial ideal $\left( x_1\ldots x_n \right) \in k[x_1, \ldots, x_n]$ over a field of zero characteristic. We use this description to answer questions about products and intersections of ideals emerged recently in algorithmic studies of algebraic differential equations.

The effective differential Nullstellensatz is a fundamental result in the computational theory of algebraic differential equations. It allows one to reduce problems about differential equations to problems about polynomial equations. In particular, it provides an algorithm for checking consistency of a system of algebraic differential equations and...

In her PhD thesis, Szanto proposed an algorithm for determining an unmixed representation for a given radical polynomial ideal. Szanto estimated an upper bound for the degrees of the polynomials occurring in the computation. The degrees of these polynomials in their leading variables was bounded effectively. But for non-leading variables, Szanto ga...

We show that nonfree modules of Gorenstein dimension zero over a graph algebra exist if and only if the graph is a tree. A classification of such modules is given. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

Показывается, что несвободные модули нулевой горенштейновой размерности над алгеброй графа существуют в том и только том случае, если граф - дерево. Дается классификация таких модулей. Библиография: 19 названий.

Throughout the paper all fields are assumed to be of characteristic zero and “differential” means “ordinary differential”.

We compute an upper bound for the orders of derivatives in the Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used t...

In this paper, we prove a differential analog of the Noether normalization
lemma.
It states, that in a finitely generated integral differential $k$-algebra
($char$ $k = 0$) of differential transcendence degree~$d$ there are
differentially independent elements $b_1, \ldots, b_d$ such that, for every
prime differential ideal $\mathfrak{p} \subset A =...

Let $E$ be a differential field finitely generated over its subfield $k
\subset E$. We prove that if $trdeg_k E < \infty$ and $E$ contains a
nonconstant, then $E$ is generated as a differential field by a single element.
This result is a refinement of the Kolchin's theorem.

We construct a monomorphism from the differential algebra k{x}/[x
m
] to a Grassmann algebra endowed with a structure of differential algebra. Using this monomorphism, we prove the primality of k{x}/[x
m
] and its algebra of differential polynomials, solve one of so-called Ritt problems related to this algebra, and give a new proof of the integrali...

We consider some kind of Hopf algebra assigned to any finite-dimensional Lie algebra. This algebra was pointed out by Hochschild. We prove several statements on its embeddings into an algebra of formal power series. In particular, we obtain similar results for Lie algebras. More precisely, a Lie algebra can be embedded into a Lie algebra of special...

We construct the monomorphism from the differential algebra $k\{x\} / [x^m]$
to a Grassmann algebra endowed with a structure of differential algebra. Using
this monomorphism we prove primality of the $k\{x\} / [x^m]$ and its algebra of
differential polynomials, solve one of Ritt problems and give a new proof of
integrality of the ideal $[x^m]$.

An isomorphism between the algebra of formal power series and the dual space to the universal enveloping Lie algebra is constructed. This isomorphism maps the maximal locally finite-dimensional submodule to the preimage of the algebra of rational functions under the Borel transform.

An associative multilinear polynomial depending on 16 variables and being skew-symmetric with respect to 12 of them is presented.
This polynomial provides us with a mapping recovering the algebra of regular functions of an irreducible affine variety from
any smooth involutive distribution of dimension 2.