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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.

This paper derives a sufficient condition for the existence of a dynamic state-feedback compensator, for a polynomial system, such that a prescribed subset defined by an algebraic inequality is invariant for the resulting closed-loop system. Moreover, we present an algorithm for exactly computing such a compensator. The algorithm consists of finitely-many arithmetic operations of polynomials.

The reaction kinetics of the heavy-light-heavy abstraction reaction F + HCl → HF + Cl on the ground electronic state potential energy surface (PES) is investigated theoretically by a recently developed ring polymer molecular dynamics (RPMD) approach. First, a new PES is developed by the permutation invariant polynomial neural network (PIP-NN) approach based on 30 620 points sampled over a large configuration space from the latest and most accurate Deskevich−Hayes−Takahashi−Skodje−Nesbitt (DHTSN) PES (J. Chem. Phys., 2006, 124, 224303). Excellent fitting performance was achieved with only 521 parameters. The PIP-NN PES is 11 times faster than the DHTSN PES. Besides, the first analytical derivatives with respect to the coordinates of the atoms have been obtained for the PIP-NN PES. The RPMD rate coefficients on the PIP-NN PES are calculated and compared with available theoretical and experimental results. It is found that the experimental rate coefficients are significantly larger than the theoretical results on the DHTSN PES, due to its overestimated reaction barrier. We conclude that a reliable PES for this important heavy-light-heavy reaction is highly desirable.

We study the problem of computing the maximal admissible positively invariant set for discrete time switching affine systems subject to basic semi-algebraic constraints. First, we obtain inner ϵ-approximations of the minimal invariant set. Second, following recent results for switching linear systems (Athanasopoulos and Jungers, 2016), we apply an algebraic lifting on the system and obtain a polyhedral representation of the constraint set. Working on this lifted state space offers two distinct advantages, namely (i) we can verify inclusion of an e-inflation of the minimal invariant set in the constraint set and (ii) under proper assumptions, we can characterize and compute the maximal admissible invariant set, which is also a basic semi-algebraic set. Consequently, we are able to identify and recover admissible invariant sets for switching affine systems even when only non-convex invariant sets exist. The underlying algorithms involve only linear operations and convex hull computations.

Recently, due to the hardware computing enhancement such as quantum computers, the amount of information that can be processed in a short period of time is growing exponentially. The cryptography system proposed by Koblitz and Fellows has a problem that it can not be guaranteed that the problem finding perfect dominating set is NP-complete in specific 3-regular graphs because the number of invariant polynomial can not be generated enough. In this paper, we propose an enhanced method to improve the vulnerability in 3-regular graph by generating plenty of invariant polynomials.

E.M. Sa and R.C. Thompson proved that the invariant factors of a matrix over a commutative principal ideal domain and the invariant factors of its submatrices are related by a set of divisibility inequalities, called the interlacing inequalities for invariant factors. We extend this result to matrices over elementary divisor duo rings.

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.

Assume that a linear space of real polynomials in d variables is given which is translation and dilation invariant. We show that if a sequence in this space converges pointwise to a polynomial, then the limit polynomial belongs to the space, too.

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.

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.

We report a detailed quasi-classical dynamics study on a new full-dimensional multireference spin–orbit-corrected potential energy surface (PES) for the F(2P3/2) + C2H6 → HF + C2H5 reaction. For the PES development, the Robosurfer program package is applied and the MRCI-F12+Q(5,3)/aug-cc-pVDZ energy points are fitted using the monomial symmetrization approach of the permutationally invariant polynomial method. Our simulations provide substantial reaction probabilities and sharply increasing cross sections with an increase in collision energy for this early- and negative-barrier reaction. A direct rebound/stripping mechanism is preferred at low/high collision energies, and the initial translational energy turns out to convert mostly into product recoil, whereas the reaction energy excites the HF vibration. Vibrational and vibrationally resolved rotational state distributions of the HF product obtained from our computations agree well with the single-collision experimental data for the vHF = 1, 2, and 3 states.

We report permutationally invariant polynomial (PIP) fits to energies and gradients for 15-atom tropolone. These include standard, augmented, and fragmented PIP bases. Approximately, 6600 energies and their associated gradients are obtained from direct-dynamics calculations using DFT/B3LYP/6-31+G(d) supplemented by grid calculations spanning an energy range up to roughly 35 000 cm-1. Three fragmentation schemes are investigated with respect to efficiency and fit precision. In addition, several fits are done with reduced weight for gradient data relative to energies. These do result in more precision for the H-transfer barrier height. The properties of the fits such as stationary points, harmonic frequencies, and the barrier to H-atom transfer are reported and compared to direct calculations. A previous 1D model is used to obtain the tunneling splitting for the ground vibrational state and qualitative predictions for excited vibrational states. This model is applied to numerous fits with different barrier heights and then used to extrapolate the H and D atom tunneling splittings to values at the CCSD(T)-F12 barrier. The extrapolated values are 2.3 and 0.14 cm-1, respectively for H and D. These are about a factor of two larger than experiment, but within the expected level of agreement with experiment for the 1D method used and the level of the electronic structure theory.

A six-dimensional (6D) potential energy surface (PES) for the SO(X -HSO(X 3 Σ - )-H2 system is computed using high-level electronic structure theory and fit using a hybrid invariant polynomial method. Full-dimensional quantum close-coupling scattering calculations have been carried out using this potential for rotational and, for the first time, vibrational quenching transitions of SO induced by H2. State-to-state cross sections and rate coefficients of SO are reported for rotational transitions from rotational levels j 1 = 0–10 in the ground vibrational state neglecting fine-structure. Some selected state-to-state rotational rate coefficients are compared with previous theoretical results obtained using a rigid-rotor approximation. For vibrational quenching, state-to-state and total cross sections and rate coefficients were calculated for the transitions in SO( v 1 = 1 , j 1 ) + H2( v 2 = 0 , j 2 ) → SO( v 1 ′ = 0 , j 1 ′ ) + H2( v 2 ′ = 0 , j 2 ′ ) collisions with j 1 = 0–5. Cross sections for collision energies in the range 1 to 3000 cm−1 and rate coefficients in the temperature range of 5–600 K are obtained for both para-H2 ( j 2 = 0) and ortho-H2 ( j 2 = 1) collision partners. The application of the results to astrophysics is discussed.

The representation of high-dimensional potential energy surfaces by way of the many-body expansion and permutationally invariant polynomials has become a well-established tool for improving the resolution and extending the scope of molecular simulations. The high level of accuracy that can be attained by these potential energy functions (PEFs) is due in large part to their specificity: for each term in the many-body expansion, a species-specific training set must be generated at the desired level of theory and a number of fits attempted in order to obtain a robust and reliable PEF. In this work, we attempt to characterize the numerical aspects of the fitting problem, addressing questions which are of simultaneous practical and fundamental importance. These include concrete illustrations of the nonconvexity of the problem, the ill-conditionedness of the linear system to be solved and possible need for regularization, the sensitivity of the solutions to the characteristics of the training set, and limitations of the approach with respect to accuracy and the types of molecules that can be treated. In addition, we introduce a general approach to the generation of training set configurations based on the familiar harmonic approximation and evaluate the possible benefits to the use of quasirandom sequences for sampling configuration space in this context. Using sulfate as a case study, the findings are largely generalizable and expected to ultimately facilitate the efficient development of PIP-based many-body PEFs for general systems via automation.

The mathematical representation of large data sets of electronic energies has seen substantial progress in the past 10 years. The so-called Permutationally Invariant Polynomial (PIP) representation is one established approach. This approach dates from 2003, when a global potential energy surface (PES) for CH5+ was reported using a basis of polynomials that are invariant with respect to the 120 permutations of the five equivalent H atoms. More recently, several approaches from “machine learning” have been applied to fit these large data sets. Gaussian Process (GP) regression is such an approach. Here, we consider the implementation of the (full) GP due to Krems and co-workers, with a modification that renders it permutationally invariant, which we denote by PIP-GP. This modification uses the approach of Guo and co-workers and later extended by Zhang and co-workers, to achieve permutational invariance for neural-network fits. The PIP, GP, and PIP-GP approaches are applied to four case studies for fitting da...

Este artículo se encuentra disponible en la página web de la revista en la siguiente URL: https://www.tandfonline.com/doi/abs/10.1080/10236198.2017.1331889 This is the pre-peer reviewed version of the following article: Jorba, À., Muñoz-Almaraz, FJ. & Tatjer, JC. (2018). On non-smooth pitchfork bifurcations in invertible quasi-periodically forced 1-D maps. Journal of Difference Equations and Applications, vol. 24, n. 4, pp. 588-608, which has been published in final form at https://doi.org/10.1080/10236198.2017.1331889 Este es el pre-print del siguiente artículo: Jorba, À., Muñoz-Almaraz, FJ. & Tatjer, JC. (2018). On non-smooth pitchfork bifurcations in invertible quasi-periodically forced 1-D maps. Journal of Difference Equations and Applications, vol. 24, n. 4, pp. 588-608, que se ha publicado de forma definitiva en https://doi.org/10.1080/10236198.2017.1331889 In this note we revisit an example introduced by T. J ager in which a Strange Nonchaotic Attractor seems to appear during a pitchfork bifurcation of invariant curves in a quasi-periodically forced 1-d map. In this example, it is remarkable that the map is invertible and, hence, the invariant curves are always reducible. In the rst part of the paper we give a numerical description (based on a precise computation of invariant curves and Lyapunov exponents) of the phenomenon. The second part consists in a preliminary study of the phenomenon, in which we prove that an analytic self-symmetric invariant curve is persistent under perturbations. Preprint

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.

In this paper, we study the growth of the codimensions of a finite dimensional -superalgebra over a field of characteristic zero. We prove that has polynomial growth if and only if any finite dimensional -superalgebra satisfying the same -graded identities of has an explicit decomposition into suitable -superalgebras. We also give such a characterization by studying the decomposition of the -cocharacter of . In this case, the main tool is the representation theory of the product of symmetric groups.

In this paper we investigate invariant curves of a planar mapping, which equivalently formulates a second-order difference equation modelled from macroeconomics. We construct all invariant curves in the linear case and prove the existence of invariant curves in the nonlinear case. Furthermore, we discuss the continuous dependence of invariant curves in the nonlinear case.

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.

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.

This article considers the problem of generating polynomial invariants for iterative loops with loop initialization statements and nonsingular linear operators in loop bodies. The set of such invariants forms an ideal in the ring of polynomials in the loop variables. Two algorithms are presented one of which calculates basic invariants for a linear operator in the form of a Jordan cell and the other calculates basic invariants for a diagonalizable linear operator with an irreducible minimal characteristic polynomial. The following theorem on the structure of the basis of the ideal of invariants for such an operator is proved: this basis consists of basic invariants of Jordan cells and basic invariants of the diagonalizable part of the linear operator being considered.

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.

A description of all of the cases of the existence of a linear invariant relation of Kirchhoff's equations is obtained. Existence conditions in a parametric vector form and in an explicit coordinate form are presented. New linear invariant relations and partial solutions are found. The inclusion of well-known linear invariant relations in the general case obtained is indicated. A geometrical interpretation of the conditions for the Chaplygin linear invariant relation to exist is given.

We study the existence and non-existence of periodic orbits and limit cycles for planar polynomial differential systems of degree n having n real invariant straight lines taking into account their multiplicities. The polynomial differential systems with n=1,2,3 are completely characterized.

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 study polynomially nonlinear, continuous Roesser models. First we characterize when a variety V is invariant for an autonomous system. Then we derive conditions to decide whether V is controlled invariant, i.e., whether it can be made invariant through polynomial state feedback.

We study the problem of dictionary learning for signals that can be represented as polynomials or polynomial matrices, such as convolutive signals with time delays or acoustic impulse responses. Recently, we developed a method for polynomial dictionary learning based on the fact that a polynomial matrix can be expressed as a polynomial with matrix coefficients, where the coefficient of the polynomial at each time lag is a scalar matrix. However, a polynomial matrix can be also equally represented as a matrix with polynomial elements. In this paper, we develop an alternative method for learning a polynomial dictionary and a sparse representation method for polynomial signal reconstruction based on this model. The proposed methods can be used directly to operate on the polynomial matrix without having to access its coefficients matrices. We demonstrate the performance of the proposed method for acoustic impulse response modeling., 5 pages, 2 figures

In this paper, we propose an approach to automatically compute invariant clusters for nonlinear semialgebraic hybrid systems. An invariant cluster for an ordinary differential equation (ODE) is a multivariate polynomial invariant g(u, x)=0, parametric in u, which can yield an infinite number of concrete invariants by assigning different values to u so that every trajectory of the system can be overapproximated precisely by the intersection of a group of concrete invariants. For semialgebraic systems, which involve ODEs with multivariate polynomial right-hand sides, given a template multivariate polynomial g(u, x), an invariant cluster can be obtained by first computing the remainder of the Lie derivative of g(u,x) divided by g(u, x) and then solving the system of polynomial equations obtained from the coefficients of the remainder. Based on invariant clusters and sum-of-squares (SOS) programming, we present a new method for the safety verification of hybrid systems. Experiments on nonlinear benchmark systems from biology and control theory show that our approach is efficient.

We construct a scalar polynomial curvature invariant that transforms covariantly under a conformal transformation from any spherically symmetric metric. This invariant has the additional property that it vanishes on the event horizon of any black hole that is conformal to a static spherical metric., 5 pages, minor corrections to two equations and references

This is a survey of our research (conducted over the last few years) on invariant control systems, the associated optimal control problems, and the associated Hamilton–Poisson systems. The focus is on equivalence and classification. State space and detached feedback equivalence of control systems are characterized in simple algebraic terms; several classes of systems (in three dimensions, on the Heisenberg groups, and on the six-dimensional orthogonal group) are classified. Equivalence of cost-extended systems is shown to imply equivalence of the associated Hamilton–Poisson systems. Cost-extended systems of a certain kind are reinterpreted as invariant sub-Riemannian structures. A classification of quadratic Hamilton–Poisson systems in three dimensions is presented. As an illustrative example, the stability and integration of a typical system is investigated.

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.

Part 1: MAKE Topology; International audience; Topological data analysis is a new approach to processing digital data, focusing on the fact that topological properties are quite important for efficient data comparison. In particular, persistent topology and homology are relevant mathematical tools in TDA, and their study is attracting more and more researchers. As a matter of fact, in many applications data can be represented by continuous real-valued functions defined on a topological space X, and persistent homology can be efficiently used to compare these data by describing the homological changes of the sub-level sets of those functions. However, persistent homology is invariant under the action of the group $$\mathrm {Homeo}(X)$$ of all self-homeomorphisms of X, while in many cases an invariance with respect to a proper subgroup G of $$\mathrm {Homeo}(X)$$ is preferable. Interestingly, it has been recently proved that this restricted invariance can be obtained by applying G-invariant non-expansive operators to the considered functions. As a consequence, in order to proceed along this line of research we need methods to build G-invariant non-expansive operators. According to this perspective, in this paper we prove some new results about the algebra of GINOs.

Dimirovski,Georgi M. (Dogus Author) -- Conference full title: 20th IFAC World Congress, Toulouse, France ; 9 July 2017 thorugh 14 July 2017 We present a simple algorithm for computation of H2-optimal tracking and disturbance rejection controller of discrete-time systems possessing invariant zeros on the unit circle, based on polynomial spectral factorization. We prove that the column degrees of the associated para-hermitian polynomial matrix to be factorized are equal to the plant controllability indices. A numerical/computer simulation example is given.

In Chap. 6 we discussed the existence of labelings by utilizing the ⊗ h -product of digraphs, which could be expressed algebraically as a generalization of voltage assignments, a classical technique used in topological graph theory. In this chapter, we introduce an algebraic method: Combinatorial Nullstellensatz.

Chvatal-Gomory (CG) cuts captures useful and efficient linear programs that the bounded degree Lasserre/Sum-of-Squares (\({\textsc {sos}}\)) hierarchy fails to capture. We present an augmented version of the \({\textsc {sos}}\) hierarchy for 0/1 integer problems that implies the Bienstock-Zuckerberg hierarchy by using high degree polynomials (when expressed in the standard monomial basis). It follows that for a class of polytopes (e.g. set covering and packing problems), the \({\textsc {sos}}\) approach can optimize, up to an arbitrarily small error, over the polytope resulting from any constant rounds of CG cuts in polynomial time.

In this chapter, we focus on rotation invariant valuations. We give an overview of the results available in the literature, concerning characterization of such valuations. In particular, we discuss the characterization theorem, derived in Alesker (Ann Math 149:977–1005, 1999), for continuous rotation invariant polynomial valuations on \(\mathcal{K}^{n}\). Next, rotational Crofton formulae are presented. Using the new kinematic formulae for trace-free tensor valuations presented in Chap. 4, it is possible to extend the rotational Crofton formulae for tensor valuations, available in the literature. Principal rotational formulae for tensor valuations are also discussed. These formulae can be derived using locally defined tensor valuations, as introduced in Chap. 2 A number of open questions in rotational integral geometry are presented.

Let $\pi\colon P\to M$ be a principal bundle and $p$ an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map $\chi^{k} : H_{2r-k-1}(M)\times H_{k}(\mathcal{F}/\mathcal{G})\to \mathbb{R}/\mathbb{Z}$, for $k, Comment: 22 pages

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.