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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.

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.

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

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.

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.

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.

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.

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.

The odd-even invariant for graphs is the graphic version of the odd-even invariant for oriented matroids. Here, simple properties of this invariant are verified, and for certain graphs, including chordal graphs and complete bipartite graphs, its value is determined. The odd-even chromatic polynomial is introduced, its coefficients are briefly studied, and it is shown that the absolute value of this polynomial at -1 equals the odd-even invariant, in analogy with the usual chromatic polynomial and the number of acyclic orientations.

In [3] Bozek introduced a class of solvable Lie groups M2n+1. Calvaruso, Kowalski and Marinosci in [9] have studied homogeneous geodesics on these homogeneous spaces with an arbitrary odd dimension. In [1] we have studied some other geometrical properties on these spaces with dimension five. Our aim in this paper is to extend those geometrical properties for an arbitrary odd dimension in both Riemannian and Lorentzian cases. In fact we first obtain all of the descriptions of their homogeneous Lorentzian and Riemannian structures and their types. Then we calculate the energy of an arbitrary left-invariant vector field X on these spaces and in the Lorentzian case we prove that no left-invariant unit time-like vector fields on these spaces are critical points for the space-like energy. There is also a proof of non-existence of invariant contact structures and left-invariant Ricci solitons on these homogeneous spaces.

We search for invariant solutions of the conformal Killing–Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We classify the 3-dimensional Lie groups with left invariant metrics carrying invariant conformal Killing–Yano 2-forms. publishedVersion Fil: Andrada, Adrián Marcelo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Andrada, Adrián Marcelo. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Andrada, Adrián Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Barberis, María Laura. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Barberis, María Laura. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Barberis, María Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Dotti, Isabel Graciela. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Dotti, Isabel Graciela. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Dotti, Isabel Graciela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Matemática Pura

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 describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess four distinct infinite singularities and eight invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system with four infinite distinct singularities whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer.

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess eight invariant straight lines, including the line at infinity and including their multiplicities and in addition they possess three distinct infinite singularities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of affine invariant polynomials. The invariant polynomials allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.

The rigorous proofs are given: (1) for the existence of the unbounded invariant curves, containing the fixed point – source (μ + 1; 1), of the maps from the one-parameter family Fμ(x,y) = (xy, (x − μ)2), μ ∈ [0, 2]; (2) for the birth of the closed invariant curve from the elliptic fixed point (μ − 1; 1) for μ = 3 / 2. Numerical results are presented for the main steps of the evolution of this invariant curve, when μ changes in the interval (3 / 2, 2).

We prove a new non-splitting result for the cohomology of the Milnor fiber, reminiscent of the classical result proved independently by Lazzeri, Gabrielov, and Le in 1973-74.

We classify all arrangements of ten lines in the real projective plane that are invariant under a polynomial differential equation of degree 4.

All the cases of the existence of a linear invariant relation of the Poincare–Zhukovskii equations with symmetric Hamiltonian matrices are found. A similar geometrical meaning of the existence conditions is indicated for well-known special cases. Precessional motions with a linear invariant relation are identified.

We give a general formula of the quantum sl2-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra g=sl2 in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum 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.

A particular result towards a positive solution of a problem by V.I. Arnol’d about a higher analog of the ergodic asymptotic invariant of magnetic fields is presented.

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X = { x 1 , … , x n } to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n = 1 , for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials K 〈 X 〉 , we give all identities which are invariant and universal invariant by gametization.

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.

For piecewise smooth systems we describe mechanisms to obtain a similar reduction to a lower dimensional system as has been achieved for smooth systems via the center manifold approach. It turns out that for nonsmooth systems there are invariant quantities as well which can be used for a bifurcation analysis but the form of the quantities is more complicated. The approximation by piecewise linear systems (PWLS) provides a useful concept. In the case of PWLS, the invariant sets are given as invariant cones. For nonlinear perturbations of PWLS the invariant sets are deformations of those cones. The generation of invariant manifolds and a bifurcation analysis establishing periodic orbits are demonstrated; also an example for which multiple cones exist is provided.

Let Γ be a finite group of unitary matrices on ℂ n . For z ∈ ℂ n , the Hermitian symmetric polynomial has been used to construct group-invariant CR-mappings from the sphere to hyperquadrics. We show that the ensuing mappings always contain a complete system of fundamental invariants. This result gives a new algorithm for computing systems of fundamental invariants in invariant rings. A special instance for certain cyclic representations illuminates work by Loehr, Warrington and Wilf who made no mention of invariant theory.

The aim of this paper is to present the technique of the fixed point partition with respect to an operator and a fixed point structure, to study the data dependence of the fixed points, Ostrowski property and well posedness of the fixed point problem.

The functions satisfying the mean value property for an n-dimensional cube are determined explicitly. This problem is related to invariant theory for a finite reflection group, especially to a system of invariant differential equations. Solving this problem is reduced to showing that a certain set of invariant polynomials forms an invariant basis. After establishing a certain summation formula over Young diagrams, the latter problem is settled by considering a recursion formula involving Bernoulli numbers. Keywords: polyhedral harmonics; cube; reflection groups; invariant theory; invariant differential equations; generating functions; partitions; Young diagrams; Bernoulli numbers., Comment: 18 pages, 3 figures

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

We show that a finite type invariant of order at most 4 for genus 2 handlebody-knots is a constant map. For this purpose, we give a concrete basis for the vector space of all finite type invariants of order at most 4 for spatial theta-curves, which makes a correction to Koikeʼs result. Each invariant of the basis is derived from the HOMFLYPT polynomial of the associated 3-component link of a spatial theta-curve.

Let h be an orientation reversing planar homeomorphism and X be an invariant plane separating continuum. We show that there is a natural linear order on the invariant components of R2∖X that resemble the one found in connected unions of circles invariant under the reflection r(x,y)=(−x,y). The main result relates to the Nielsen fixed point theory and work of Krystyna Kuperberg on fixed points of planar homeomorphisms in invariant continua.

We study the notion of linear invariance on the unit ball of a J B ⁎ -triple X , and we obtain some connection between the norm-order of a linear invariant family and the starlikeness of order 1/2. Also, we give some result concerning the radius of univalence of some linear invariant families. Finally, if the dimension of X is finite and if the norm-order of a linear invariant family is finite, then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm-order. In particular, our results are valid for the classical Cartan domains and the unit polydisc.

The main purpose of this paper is to give a survey of algorithms in invariant theory, with emphasis on nonreductive groups and on recent developments. But the article has some novel elements: it contains a new algorithm for computing invariant rings, which works under the condition that the invariant field is the field of fractions of the invariant ring. We also prove that if n is the dimension of the invariant ring, then there exists a separating set of invariants whose size is at most 2n + 1.

Let X, Y be cell complexes, let U be an Abelian group, and let f:[X,Y]{yields}U be a homotopy invariant. By definition, the invariant f has order at most r if the characteristic function of the rth Cartesian power of the graph of a continuous map a:X{yields}Y determines the value f([a]) Z-linearly. It is proved that, in the stable case (that is, when dimX

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 define an invariant for two component spatial graphs. Although the definition of the invariant is alike a linking number, it is different from the absolute value of a linking number. We show that the invariant is not a finite type invariant.

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.

In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m > 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invariant disc or an invariant Cantor set with positive (n + 2)-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its linearized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invariant tori carrying quasi-periodic solutions.

The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

We study invariant subspaces in the context of the work of Katavolos and Power [9] and [10] when one of the semigroups considered is replaced by a discrete one. As a consequence, a rather striking connection is given with the study of the lattice of invariant subspaces of composition operators induced by automorphisms of the unit disc acting on the classical Hardy space. As a particular instance, our study concerns the lattice of invariant subspaces of those composition operators induced by hyperbolic automorphisms, and therefore with the Invariant Subspace Problem.

We present a topological method for the detection of normally hyperbolic type invariant sets for maps. The invariant set covers a sub-manifold without a boundary in $\mathbb{R}^k$. For the method to hold we only need to assume that the movement of the system transversal to the manifold has directions of topological expansion and contraction. The movement in the direction of the manifold can be arbitrary. The result is based on the method of covering relations and local Brouwer degree theory.

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃i=1nTi(K) for some contractions T1, …, Tn of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.

We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen-Cahn equation.

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all points at infinity (in the Poincare compactification) as singularities. We prove that all such systems have invariant affine lines of total multiplicity 3, give all their configurations of invariant lines and show that all these systems are integrable via the method of Darboux having cubic polynomials as inverse integrating factors. After constructing the topologically distinct phase portraits in this class we give invariant necessary and sufficient conditions in terms of the 12 coefficients of the systems for the realization of each one of them and give representatives of the orbits under the action of the affine group and time rescaling. We construct the moduli space of this class for this action and give the corresponding bifurcation diagram.

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction., Comment: 23 pages, results of math.QA/0510382 are included