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For every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree.

International audience; Assuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action.

The study of quadratic polynomial differential systems on the plane have been shown a tough challenge, there exist hundreds of papers about them which are dated for over a century and until now there exist several topics to be studied and concluded. For instance, the complete characterization of phase portraits of quadratic systems remains unknown and the complete topological classification of such systems has been a complex work. It is well known that the greatest difficult of working with quadratic systems is the quantity of parameters. A (generic) quadratic system is defined by 12 parameters, however by using affine transformations and time rescaling one can reduce this number by five, but yet this is a very large number, once the corresponding bifurcation diagram is a fivedimensional euclidean space. So, it is convenient to use some tools (as the Invariant Theory) in order to study families of quadratic systems with specific properties (for instance, according to the structural stability or possessing classes of invariant algebraic curves) with the purpose of reducing even more (when it is possible) this quantity of parameters. The main goal of this thesis is to contribute to the classification of the quadratic systems on the plane. More precisely, we present the complete study (modulo islands) of the bifurcation diagram of two families of quadratic systems possessing specific properties on their singularities, we do the complete topological classification (modulo limit cycles) of all the phase portraits of two sets of quadratic systems of codimension two and we perform the classification of quadratic differential systems with invariant ellipses according to their configurations of invariant ellipses and invariant lines. It is worth mentioning that these three works represent three different approaches to the study of quadratic systems and each one of them uses different techniques, which all together are useful towards the final goal of classifying phase portraits. O estudo dos sistemas diferenciais polinomiais quadráticos no plano tem se demonstrado desafiador, existem centenas de artigos datados de mais de um século sobre esse tema e ainda existem muitos tópicos para serem estudados e concluídos. Por exemplo, a caracterização completa dos retratos de fase de sistemas quadráticos permanece desconhecida e a classificação topológica completa de tais sistemas tem sido um trabalho complexo. É bem sabido que a principal dificuldade de se trabalhar com os sistemas quadráticos é a quantidade de parâmetros. Um sistema quadrático (genérico) é definido por 12 parâmetros, entretanto, usando transformações afins e reescala temporal podese reduzir este número para cinco, mas ainda são muitos parâmetros, uma vez que o correspondente diagrama de bifurcação é um espaço euclideano de dimensão cinco. Desta forma, fazse conveniente utilizar algumas ferramentas (a Teoria dos Invariantes, por exemplo) de modo a estudar famílias de sistemas quadráticos com propriedades específicas (por exemplo, de acordo com a estabilidade estrutural ou possuindo classes de curvas algébricas invariantes) para reduzir ainda mais (quando possível) essa quantidade de parâmetros. Nesta tese objetivamos contribuir com a classificação dos sistemas quadráticos no plano. Mais precisamente, apresentamos o estudo completo (módulo ilhas) do diagrama de bifurcação de duas famílias de sistemas quadráticos com propriedades específicas em suas singularidades. Fazemos a classificação topológica completa de todos os retratos de fases (módulo ciclos limites) de dois conjuntos de sistemas quadráticos de codimensão dois e fazemos a classificação de todos os sistemas quadráticos que possuem elipses invariantes de acordo com a chamada configuração de elipses invariantes e retas invariantes. Vale a pena ressaltar que esses trabalhos representam três abordagens distintas para o estudo dos sistemas quadráticos, e cada um deles utiliza técnicas diferentes, que em conjunto são úteis para o objetivo final de classificar retratos de fases.

Zeta functions for linear codes were defined by Iwan Duursma in 1999. They were generalized to the case of some invariant polynomials by the preset author. One of the most important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis., Comment: 9 pages

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.

Agraïments: The first author has been supported by AGAUR FI-DGR 2010. The third author has been supported by AGAUR PIV-DGR-2010, FCT grant PTDC/MAT/117106/2010 and through CAMG SD. We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian non-degenerate centers of linear plus cubic homogeneous planar polynomial vector fields.

We provide a constructive method to obtain polynomial vector fields in ℝ3 having infinitely many limit cycles starting from polynomial vector fields in ℝ2 with a period annulus. We present two examples of polynomial vector fields in ℝ3 having infinitely many limit cycles, one of them of degree 2 and the other one of degree 12. The main tools of our method are the Melnikov integral and the Hamiltonian structure.

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix. We consider the corresponding Diophantine equation and prove an analog of a famous result due to Conway and Coxeter. We also observe that Chebyshev polynomials of the first kind arise as Pfaffians., Comment: 8 pages

We consider harmonic polynomials of real variables $x,y,z$ that are eigenfunctions of the rotations about the axis $z$. They have the form $(x\pm yi)^{n}p(x,y,z)$, where $p$ is a rotation invariant polynomial. Let ${\mathfrak R}_{m}$ be the family of the polynomials $p$ of degree $m$ which are reducible over the rationals. We describe ${\mathfrak R}_{m}$ for $m\leq5$ and prove that ${\mathfrak R}_{6}$ and ${\mathfrak R}_{7}$ are finite.

In the present paper an approach to some questions in the theory of invariant (quasi-invariant) measures is discussed. It is useful in certain situations, where given topological groups or topological vector spaces are equipped with various nonzero σ -finite left invariant (left quasi-invariant) measures. Keywords: Invariant measure, Quasi-invariant measure, Extensions of measures, Surjective homomorphism

In this paper we consider the question of under which conditions multi-valued dynamical systems admit invariant measures. We give results on the existence of invariant measures with full support on orbit spaces of multi-valued dynamical systems. We use these measures on the orbit space to induce measures on the original dynamical system. We focus on the question of when a non-atomic invariant measure on the orbit space induces an atomic invariant measure on the multi-valued dynamical system. This phenomenon is an indicator of complicated multi-periodic behaviour.

Let k be a field of positive characteristic p. We introduce the notion of a pseudo-derivation of a k-algebra A, and give a one-to-one correspondence between the set of all pseudo-derivations of A and the set of all p-unipotent automorphisms of A. We classify p-unipotent triangular automorphisms of a polynomial ring k[x, y, z] in three variables over k up to conjugation of automorphisms of k[x, y, z]. We prove that if a p-cyclic group ℤ/pℤ acts triangularly on the polynomial ring k[x, y, z], the modular invariant ring k[x, y, z] ℤ/pℤ is a hypersurface ring.

A complete invariant defined for (closed, connected, orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself is reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In previous papers by the first author, a characteristic lattice point invariant and a characteristic rational invariant defined for the 3-manifolds were constructed which also produced a smooth real function with the definition interval ( − 1 , 1 ) as a characteristic invariant defined for the 3-manifolds. In this paper, a complex number-valued characteristic invariant for the 3-manifolds whose norm is smaller than or equal to one half is introduced by using an embedding of a set of lattice points called the ADelta set, distinct from the PDelta set, into the set of complex numbers. The distributive situation for the invariants of the 3-manifolds of lengths up to 10 is plotted in the complex number plane with radius smaller than or equal to one half. By using this complex number-valued characteristic invariant, a holomorphic function with the unit open disk as the definition domain which is called the characteristic quantity function is constructed as a characteristic invariant defined for the 3-manifolds.

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

We consider the random $\unicode[STIX]{x1D6FD}$-transformation $K_{\unicode[STIX]{x1D6FD}}$ introduced by Dajani and Kraaikamp [Random $\unicode[STIX]{x1D6FD}$-expansions. Ergod. Th. & Dynam. Sys.23 (2003), 461–479], which is defined on $\{0,1\}^{\mathbb{N}}\times [0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We give an explicit formula for the density function of a unique $K_{\unicode[STIX]{x1D6FD}}$-invariant probability measure absolutely continuous with respect to the product measure $m_{p}\otimes \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$, where $m_{p}$ is the $(1-p,p)$-Bernoulli measure on $\{0,1\}^{\mathbb{N}}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ is the normalized Lebesgue measure on $[0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters $p$ and $\unicode[STIX]{x1D6FD}$.

We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of $Inc(\mathbb{N})$ and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gr\"obner basis consisting of finitely many orbits. As our second main result we prove that the Krull dimension and multiplicity of ideals in an invariant filtration grow eventually linearly and exponentially, respectively, and we determine the terms that dominate this growth., Comment: 30 pages; the title has been modified

In this paper, we find a lower bound for the order of the group 〈 θ + α 〉 ⊂ F ‾ q ⁎ , where α ∈ F q , θ is a generic root of the polynomial F A , r ( X ) = b X q r + 1 − a X q r + d X − c ∈ F q [ X ] and a d − b c ≠ 0 .

We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.

Consider a nonlinear input-affine control system x(t) = f(x(t)) + g(x(t))u(t), y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of =). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x = (f + gα)(x). If it is possible to achieve this goal with a polynomial output feedback law u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods.

We study invariant sets for switched systems. We focus on a class of third-order switched systems. In this class of switched systems we solve the problem of finding an invariant set for a switching law. For each switched system we provide the invariant set and the switching law, in addition, the invariant set is a polyhedral cone and the switching law is switching on the boundary. To accomplish this we reduce the problem to a simplified system. We provide a procedure calculating the invariant set. The method is based on calculating an invariant set for a simplified system and then transformating the invariant set to the original system. We illustrate this method with an example.

In this paper we give the normal form of all polynomial differential systems in $$\mathbb {R}^3$$ having a weighted homogeneous surface $$f=0$$ as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when $$f=0$$ is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Nino atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface.

The method of localization of invariant compact sets was proposed to study for asymptotic stability the equilibrium points of an autonomous system of differential equations. This approach relies on the necessary and sufficient conditions for asymptotic stability formulated in terms of positive invariant sets and invariant compact sets, and enables one to study the equilibrium points for asymptotic stability in the cases where it is impossible to use the first approximation or the method of Lyapunov functions. The possibilities of the method were illustrated by examples.

Here we give abstract formulations of some generalized versions of the classical Vitali theorem on Lebesgue nonmeasurable sets which are due to Kharazishvili and Solecki. Keywords: Transformation group, Invariant (quasi-invariant) measure, Invariant set, k-additive measurable structure, k+-saturation, Ulam (k,k+)-matrix, k-independent (strictly k-independent) class, Almost invariant set, Small system

Let G be a Sylow p -subgroup of the unitary groups GU(3,q2)GU(3,q2), GU(4,q2)GU(4,q2), the symplectic group Sp(4,q)Sp(4,q) and, for q odd, the orthogonal group O+(4,q)O+(4,q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an interesting and complicated invariant. The third rank of successive quotients in the lower central series of the fundamental group was called the global invariant of the arrangement by Falk. Falk gave a general formula to compute the global invariant, and asked for a combinatorial interpretation of the global invariant. Schenck and Suciu proved that the global invariant of a graphic arrangement is double of the number of cliques with three or four vertices in the graph with which the arrangement associated. This solved Falk’s problem in the case of graphic arrangements. While in the case of signed graphic arrangements, we obtained a similar combinatorial formula. In this paper, we give a direct and simple proof for this combinatorial formula.

A criterion is suggested for defining such properties of the right-hand sides in the Lienard polynomial system that guarantee its first absolute invariant turning to a constant.

For an ideal I on ω, we introduce the notions of I -invariant and bi- I -invariant injections from ω to ω. We study injections that are invariant with respect to selected classes of ideals. We show some applications to ideal convergence.

Let $G$ be a finite cover of a closed connected transpose-stable subgroup of $GL(n,\bR)$ with complexified Lie algebra $\frg$. Let $K$ be a maximal compact subgroup of $G$, and assume that $G$ and $K$ have equal rank. We prove a translation principle for the Dirac index of virtual $(\frg,K)$-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subalgebra of $\frg$. This ``index polynomial'' generates an irreducible representation of the Weyl group contained in the coherent continuation representation. We show that the index polynomial is the exact analogue on the compact Cartan subgroup of King's character polynomial. The character polynomial was defined in \cite{K1} on the maximally split Cartan subgroup, and it was shown to be equal to the Goldie rank polynomial up to a scalar multiple. In the case of representations of Gelfand-Kirillov dimension at most half the dimension of $G/K$, we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occuring in the associated cycle of the corresponding coherent family., 28 pages

International audience; In this article, we conjecture that the Links–Gould invariant of links, which we know is a generalization of the Alexander–Conway polynomial, shares some of its classical features. In particular, it seems to give a lower bound for the genus of links and to provide a criterion for fibredness of knots. We give some evidence for these two assumptions.

We review the familiar method of reducing a symmetric ordinary differential equation via invariants of the symmetry group. Working exclusively with polynomial invariants is problematic: Generator systems of the polynomial invariant algebra, as well as generator systems for the ideal of their relations, may be prohibitively large, which makes reduction unfeasible. In the present paper we propose an alternative approach which starts from a characterization of common invariant sets of all vector fields with a given symmetry group, and uses suitably chosen localizations. We prove that there exists a reduction to an algebraic variety of codimension at most two in its ambient space. Some examples illustrate the approach.

Let F be a finite field of characteristic p and K a field which contains a primitive pth root of unity and char K ≠ p. Suppose that a classical group G acts on the F-vector space V. Then it can induce the actions on the vector space V V and on the group algebra K[V V], respectively. In this paper we determine the structure of G-invariant ideals of the group algebra K[V V], and establish the relationship between the invariant ideals of K[V] and the vector invariant ideals of K[V V], if G is a unitary group or orthogonal group. Combining the results obtained by Nan and Zeng (2013), we solve the problem of vector invariant ideals for all classical groups over finite fields.

By using the invariants of nonconjugate subgroups of the Poincare group P(1,4) [conjugation is considered with respect to the group P(1,4)], we propose ansatzes that reduce some linear and nonlinear five-dimensional d’Alembert equations to ordinary differential equations. On the basis of the solutions of the reduced equations, we construct the invariant solutions of these five-dimensional d’Alembert equations.

Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years. The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on Information Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.

It is known that a spectral gap condition is required for the existence and smoothness of invariant manifolds and invariant foliations near a fixed point. However, any planar mapping with a unipotent linear part, having a center manifold on the whole phase plane, does not have such a gap. In this paper, we give the existence and smoothness of invariant manifolds, which are invariant submanifolds of the center manifold, for the aforementioned planar mapping. Corresponding to the invariant submanifolds, we further prove the existence and smoothness of invariant foliations on the center manifold.

The symmetric group \(\mathsf {S}_n\) and the partition algebra \(\mathsf {P}_k(n)\) centralize one another in their actions on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the n-dimensional permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\). The duality afforded by the commuting actions determines an algebra homomorphism \(\varPhi _{k,n}: \mathsf {P}_k(n) \rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) from the partition algebra to the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is a surjection for all \(k, n \in \mathbb {Z}_{\ge 1}\), and an isomorphism when \(n \ge 2k\). We present results that can be derived from the duality between \(\mathsf {S}_n\) and \(\mathsf {P}_k(n)\), for example, (i) expressions for the multiplicities of the irreducible \(\mathsf {S}_n\)-summands of \(\mathsf {M}_n^{\otimes k}\), (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra \(\mathsf {P}_k(n)\). When \(2k >n\), the map \(\varPhi _{k,n}\) has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of \(\varPhi _{k,n}\) in terms of the orbit basis of \(\mathsf {P}_k(n)\) and explain how the surjection \(\varPhi _{k,n}\) can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible types of invariant solutions is given in the case n = 2 for the equation possessing an infinitely dimensional symmetry algebra. The results obtained are generalized for the hyperbolic and other fourth-order parabolic equations of thin film and nonlinear dispersion type.

For each invariant polynomial Φ, we construct a global CR invariant via the renormalized characteristic form of the Cheng–Yau metric on a strictly pseudoconvex domain. When the degree of Φ is 0, the invariant agrees with the total Q ′ -curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant I Φ ′ which integrates to the corresponding CR invariant. These are generalizations of the I ′ -curvature on CR five-manifolds, introduced by Case–Gover.

In this paper, a full symmetry group classification for isothermal multiphase drift flux model is presented. All invariant functions are developed for the Lie algebra, which play a vital role in construction of optimal systems. Further, with the help of one dimensional optimal classification group, invariant solutions are obtained which describe the asymptotic behavior of general solution.

It is shown that, for any nonzero σ -finite translation invariant (translation quasi-invariant) measure μ on the real line R , the cardinality of the family of all translation invariant (translation quasi-invariant) measures on R extending μ is greater than or equal to 2 ω 1 , where ω 1 denotes the first uncountable cardinal number. Some related results are also considered.

Let G be a finite group with a faithful representation V over a field F of odd characteristic p, where the order of G is divisible by p. In this article, it is proved that the generalized invariant module of G coincides with the (polynomial) invariant ring of the representation under a necessary and sufficient group theoretical condition. Moreover, the structure of the generalized invariant module for any finite group is determined, providing a tool for computing this module. (C) 2016 Elsevier B.V. All rights reserved.

The D -invariant polynomial subspaces play a crucial role in ideal interpolation. In this paper, we analyze the structure of a second-degree D -invariant polynomial subspace P 2 . As an application for ideal interpolation, we solve the discrete approximation problem for ? z P 2 ( D ) under certain conditions, i.e., we compute pairwise distinct points, such that the limiting space of the evaluation functionals at these points is the given space ? z P 2 ( D ) , as the evaluation sites all coalesce at one site z .

We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Ces𝑎ro sense as time increases unboundedly. Here the Ces𝑎ro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the `standard' Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov-Krylov theorem using generalized functions and the Hahn-Banach theorem. The properties of signed invariant measures are also discussed.

Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.

We present a theory of abstract interpretations in the framework of invariant sets by trans- lating the notions of lattices and Galois connections into t his framework, and presenting their proper- ties in terms of finitely supported objects. We introduce the notions of invariant correctness relation and invariant representation function, emphasize an equivalence between them, and establish the re- lationship between these notions and invariant Galois connections. Finally, we provide some widen- ing and narrowing techniques in order to approximate the least fixed points of finitely supported transition functions.

We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the nonmodular case. By diagonalization, such a group action can be described by integer matrices of orders and exponents. We make use of integer linear algebra to compute a minimal generating set of invariants along with the substitution needed to rewrite any invariant in terms of this generating set. In addition, we show how to construct a minimal generating set that consists only of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems whose solution set is invariant by a finite abelian group action. Finally, we also provide an algorithm to find such symmetries given a polynomial system.

This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.