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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.

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

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 $\unicode[STIX]{x1D6FD}$.

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.

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

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.

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.

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.

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.

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.

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.

A brief review of works on the modified theory of invariant variational problems, developed by the author, and its applications is presented. The study is based on the Lie–Ovsyannikov infinitesimal formalism and on E. Noether’s fundamental idea of invariant functionals. Theorems on generalized invariance are proved and applications of the developed theory to solving optimal problems of mathematical physics are discussed.

On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the fibers of the boundary fibration. The formula holds for product-type model edge metrics where the degeneration is of conical type in each fiber, but also for general classes of perturbations of the model metrics. The same method produces a Chern- Gauss-Bonnet formula for complete, non-compact manifolds with fibered boundaries in the sense of Mazzeo-Melrose and perturbations thereof, involving the odd Pfaffian of the base of the fibration. We deduce the rationality of the usual Pfaffian form on Riemannian orbifolds, and exhibit obstructions for certain metrics on a fibration to be realized as the model at infinity of a flat metric with conical, edge or fibered boundary singularities., Comment: This second version corrects a statement about the degenerate metric on the blow-up of a submanifold, a few typos and includes new references

The purpose of this paper is to show that it is characteristic of symmetric cones among irreducible homogeneous cones that there exists a non-constant relatively invariant polynomial such that its Laplace transform is the reciprocal of a certain polynomial. To prove our theorem, we need the inductive structure of the basic relative invariants of a homogeneous cone. However, we actually work in a more general setting, and consider the inducing of the basic relative invariants from lower rank cones.

We give a quotient of the ring so that the Miyazawa polynomial is a non-trivial invariant of welded links. Furthermore we show that this is also an invariant under the other forbidden move , and so it is a fused isotopy invariant. Also, we give some quotient ring so that the index polynomial can be an invariant for welded links.

Motivated by the work of Abreu and Freitas [1], we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over .

We introduce a new asymptotic invariant of magnetic fields, namely, the quadratic (and polynomial) helicity. We construct a higher asymptotic invariant of a magnetic field. We also discuss various problems that can be solved by using the magnetic helicity invariant.

In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie–Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional Lie–Bäcklund symmetries to derive invariant subspaces of the two-dimensional operators. As an application, the invariant subspaces for a class of two-dimensional nonlinear quadratic operators are provided. Furthermore, the invariant subspace method in one-dimensional space combined with the Lie symmetry reduction method and the change of variables is used to obtain invariant subspaces of the two-dimensional nonlinear operators.

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

Agraïments: The third author is supported by the grants AGAUR PIV-DGR-2010 and by FCT through CAMGD The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X = (P, Q, R) in R3 whose flow leaves Q invariant. If m1 = degree P, m2 = degree Q and m3 = degree R, we say that m = (m1, m2, m3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that result from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non degenerate conics of the vector field X depending on each quadric Q and of the degrees m1, m2 and m3 of X.

Invariant ergodic measures for generalized Boole-type transformations are studied by using an invariant quasimeasure generating function approach based on special solutions for the Frobenius‐Perron operator. New two-dimensional Boole-type transformations are introduced and their invariant measures and ergodicity properties are analyzed.

We prove that the algebra of polynomial maps Mn(E)→E invariant under simultaneous conjugation is generated by the queer trace polynomials.

The behavior of a dynamical system, in Willems's point of view, is the set of all trajectories of the system. Fuhrmann defines a behavior as a linear, shift invariant, and complete subspace of

In the work we consider a three-dimensional subalgebra embedded in a four- dimensional subalgebra in order to find the set of solutions and to adjoint them with the solutions on subalgebras of higher dimension. Although the aim is not reached yet, we obtain invariant solutions of the rank 1 and partially invariant solutions of the rank 1 and defect 1. We obtain two submodels being invariant and partially invariant, seven solutions depend on arbitrary function and nineteen exact solutions.

There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.

In this paper, we give a characterization of shift-invariant subspaces which are also invariant under additional non-integer translations. Both principal and finitely generated shift-invariant subspaces are studied. Our results improve some known ones.

It is well known that for $a>4$, the dynamical behaviors of the logistic map $f_a(x)=ax(1-x)$ on the maximal invariant compact set are "simple" which could be clearly explained by the theories of hyperbolic dynamics and symbolic dynamics. Is it possible that similar phenomena could be observed in general real polynomial maps? In this paper, we study this problem by investigating the real polynomial map $f_a(x)=ag(x)$, where $a$ is a parameter, and $g$ is a real-coefficient polynomial, which has at least two different real zeros or only one real zero., 21 pages

An invariant of links with two and more components in the thickened torus is constructed; the invariant depends on several variables. The construction uses Kauffman’s formal theory, which is based on Dehn’s representation of knot groups. This invariant is a natural generalization of a polynomial z constructed by Zenkina and Manturov. Some properties of the new invariant are also considered.

We consider an extension of the notion of sub- and super-solutions for an elliptic boundary-value problem and derive the existence of a classical solution in the presence of pairs of well-ordered or non-well-ordered generalized sub- and super-solutions.

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $\sigma$-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the "integer Sierpi\'nski gasket and carpet" tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $\sigma$-finite invariant measure, which is locally finite and unique up to scaling., Comment: 43 pages

We study the amenability property for the group of formal power series in one variable with coefficients in a commutative ring with identity. We show that there exists an invariant mean on the space of uniformly continuous bounded functions on this group. This is equivalent to the fact that every continuous action of on every compact space has an invariant probability measure.

i = j = k = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. We identify C with the subfield R+ Ri ofH. Let V be a right n-dimensional linear space over H. A finite-order element of GL(V ) is said to be a reflection if the fixed-point space of the element is (n− 1)-dimensional. All finite subgroups of GL(V ) generated by reflections are classified in [1]. Aiming at the extension to the quaternion case of the classification of the discrete groups generated by affine Hermitian reflections, which was obtained in [2], we arrive at the problem of describing, up to a homothety (i.e., up to multiplication by a nonzero element of H), the lattices in V that are invariant with respect to the finite subgroups of GL(V ) generated by reflections (by a lattice we mean a free subgroup of rank 4n in the additive group of V ). In this note, such a description is given for n = 1. This description is heavily used in the investigation of the general case, which will be treated elsewhere. Thus, in what follows, we assume that V is H with right multiplication by constants End(V ) is H acting on V by left multiplication, and GL(V ) is the multiplicative group H∗ of the skew field H. All nonidentity elements of finite order inH∗ are reflections. The description of finite subgroups ofH∗ is well known (see, e.g., [3]); it is given by the following proposition (below by 〈M〉 we denote the subgroup ofG generated by some subsetM ⊆ G). Proposition 1. Up to conjugacy, the finite subgroups of H∗ are exhausted by the following pairwise nonconjugate subgroups

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.

Concentration and logarithmic Sobolev inequalities are derived for a class of multidimensional probability distributions, including spherically invariant log-concave measures. Bibliography: 17 titles.