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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.

``$\Delta$-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients to close to a coupled cluster (CC) level of accuracy. Here we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation $V_{LL{\rightarrow}CC}=V_{LL}+\Delta{V_{CC-LL}}$, and demonstrated for \ce{CH4}, \ce{H3O+}, and $trans$ and $cis$-$N$-methyl acetamide (NMA), \ce{CH3CONHCH3}. For these molecules, the LL PES, $V_{LL}$, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients, and $\Delta{V_{CC-LL}}$ is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For \ce{CH4} these are new calculations adopting an aug-cc-pVDZ basis, for \ce{H3O+} previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA training is done with 4696 CCSD(T) energies.

Let $R$ be an invariant polynomial ring of a reductive group acting on a vector space, and let $d$ be the minimum integer such that $R$ is generated by those polynomials in $R$ of degree no more than $d$ . To upper bound such $d$ is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups — the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups — the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$ -copies of them are local unitary equivalent for some $r$ .

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

The availability of accurate full-dimensional potential energy surfaces (PESs) is a mandatory condition for efficient computer simulations of molecular systems. Much effort has been devoted to developing reliable PESs with physically sound properties, such as the invariance of the energy with respect to the permutation of chemically identical atoms. In this work, we compare the performance of four neural network (NN)-based approaches with a rigorous permutation symmetry for fitting five typical reaction systems: OH + CO, H + H2S, H + NH3, H + CH4 and OH + CH4. The methods can be grouped into two categories, invariant polynomial based NNs and high-dimensional NN potentials (HD-NNPs). For the invariant polynomial based NNs, three types of polynomials, permutation invariant polynomials (PIPs), non-redundant PIPs (NRPIPs) and fundamental invariants (FIs), are used in the input layer of the NN. In HD-NNPs, the total energy is the sum of atomic contributions, each of which is given by an individual atomic NN with input vectors consisting of sets of atom-centered symmetry functions. Our results show that all methods exhibit a similar level of accuracy for the energies with respect to ab initio data obtained at the explicitly correlated coupled cluster level of theory (CCSD(T)-F12a). The HD-NNP method allows study of systems with larger numbers of atoms, making it more generally applicable than invariant polynomial based approaches, which in turn are computationally more efficient for smaller systems. To illustrate the applicability of the obtained potentials, quasi-classical trajectory calculations have been performed for the OH + CH4 → H2O + CH3 reaction to reveal its complicated mode specificity. peerReviewed

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.

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 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

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

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 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.

Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, ... , Mp : Ip → I the continuous means. We consider the problem of invariance of the graphs of functions ϕ : Jp−1 → I with respect to the mean-type mapping M = (M1, ... , Mp). Applying a result on the existence and uniqueness of an M -invariant mean [7], we prove that if the graph of a continuous function ϕ : Jp−1 → I is M-invariant, then ϕ satisfies a simple functional equation. As a conclusion we obtain a theorem which, in particular, allows to determine all the continuous and decreasing in each variable functions ϕ of the M-invariant graphs. This improves some recent results on invariant curves [8] where the case p = 2 is considered. Soit I un intervalle réel, J un sous-intervalle de I, p ≥ 2 un entier, et M1, ... , Mp : Ip → I les moyennes continues. Nous considérons le problème de l’invariance des graphes des fonctions ϕ : Jp−1 → I par rapport aux applications de type moyenne M = (M1, ... , Mp). En appliquant un résultat d’existence et unicité d’une moyenne M-invariante [7], nous montrons que si le graphe d’une fonction continue ϕ : Jp−1 → I est M-invariant, alors ϕ vérifie une équation fonctionnelle simple. En conclusion, nous obtenons un théorème qui, en particulier, permet de déterminer toutes les fonctions ϕ des graphes M-invariant continues et décroissantes en chaque variable. Ceci améliore les résultats récents sur les courbes invariantes [8] où le cas p = 2 était considéré.

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

Assume the system of differential equations E_4(a, b, c, c'; X, Y ) satisfied by Appell's hypergeometric function F_4(a, b, c, c'; X, Y ) has a finite irreducible monodromy group M_4(a, b, c, c'). The monodromy matrix Γ_ derived from a loop Γ_3 surrounding once the irreducible component C = {(X, Y) | (X − Y)^2 − 2(X + Y) + 1 = 0} of the singular locus of E_4 is a complex reflection. The minimal normal subgroup N_C of M_4 containing Γ_ is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation _2E_1(a, b, c). It is known that N_C is reducible if ε := c + c' - a - b - 1 ∉ Z or if ε ∈ Z and P(G) is a dihedral group. We prove that, if ε ∈ Z, then N_C is the (irreducible) Coxeter group W(D_4), W(F_4)and W(H_4) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.

Let k be a field, let \( G \) be a finite group. We describe linear \( G \)-gradings of the polynomial algebra k[x1, ..., xm] such that the unit component is a polynomial k-algebra.

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here., 23 pages, 9 figures

While abstract interpretation is not theoretically restricted to specific kinds of properties, it is, in practice, mainly developed to compute linear over-approximations of reachable sets, aka. the collecting semantics of the program. The verification of user-provided properties is not easily compatible with the usual forward fixpoint computation using numerical abstract domains.

El títol de la versió pre-print de l'article és: Polynomial differential systems in R^3 having an invariant quadric and a Darboux invariant Agraïments: FEDER-UNAB-10-4E-378. The second author is supported by CNPq-Brazil grant 30 8315/2012-0 and by FAPESP grant 12/18413-7. The third author is supported by FAPESP grant 2013/01743-7. All the authors are supported by the Int.Coop. Proj. CAPES/MECD-TQED II and PHB-2009-0025. We give the normal forms of all polynomial differential systems in R^3 which have a nondegenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example, we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric.

Let $X$ be a differentiable manifold endowed with a transitive action $\alpha:A\times X\longrightarrow X$ of a Lie group $A$. Let $K$ be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: {enumerate} equivalence classes of $\alpha$-invariant $K$-connections on $X$, $\alpha$-invariant gauge classes of $K$-connections on $X$, and $\alpha$-invariant isomorphism classes of pairs $(Q,P)$ consisting of a holomorphic $K^\C$-bundle $Q\longrightarrow X$ and a $K$-reduction $P$ of $Q$ (when $X$ has an $\alpha$-invariant complex structure). {enumerate}, Comment: Published version

We study the existence of invariant quadrics for a class of systems of difference equations in R n defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix A and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving A . We show that if A is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric.

For a Lie algebra g and its representation R, the quantum ( g , R) invariant of knots recovers from the Kontsevich invariant through the weight system derived from substitution of g and R into chord diagrams. We expect a similar property for invariants of 3-manifolds; for a Lie group G, the perturbative G invariant of 3-manifolds should recover from the universal perturbative invariant defined in [25] through the weight system derived from substitution of the Lie algebra of G. In this paper we give a rigorous proof of the recovery for G = SO (3).

We study conformally-invariant theories of gravity in six dimensions. In four dimensions, there is a unique such theory that is polynomial in the curvature and its derivatives, namely Weyl-squared, and furthermore all solutions of Einstein gravity are also solutions of the conformal theory. By contrast, in six dimensions there are three independent conformally-invariant polynomial terms one could consider. There is a unique linear combination (up to overall scale) for which Einstein metrics are also solutions, and this specific theory forms the focus of our attention in this paper. We reduce the equations of motion for the most general spherically-symmetric black hole to a single 5th-order differential equation. We obtain the general solution in the form of an infinite series, characterised by 5 independent parameters, and we show how a finite 3-parameter truncation reduces to the already known Schwarzschild-AdS metric and its conformal scaling. We derive general results for the thermodynamics and the first law for the full 5-parameter solutions. We also investigate solutions in extended theories coupled to conformally-invariant matter, and in addition we derive some general results for conserved charges in cubic-curvature theories in arbitrary dimensions., 28 pages. References added

We find the precise growth of some invariant metrics near a point on the boundary of a domain where the Levi form has at least one negative eigenvalue. We also introduce a new invariant pseudometric which is convenient in this context, and give some of its general properties., v5: a lower estimate for the Kobayashi metric is added

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces., Examples of compact quantum group actions on compact Hausdorff spaces added. Comments are welcome

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invari- ant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for han- dling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution{projection strategy.

We study hypercyclicity, Devaney chaos, topological mixing properties and strong mixing in the measure-theoretic sense for operators on topological vector spaces with invariant sets. More precisely, our purpose is to establish links between the fact of satisfying any of our dynamical properties on certain invariant sets, and the corresponding property on the closed linear span of the invariant set, or on the union of the invariant sets. Viceversa, we give conditions on the operator (or C0-semigroup) to ensure that, when restricted to the invariant set, it satisfies certain dynamical property. Particular attention is given to the case of positive operators and semigroups on lattices, and the (invariant) positive cone. We also present examples that illustrate these results., This work is supported in part by MICINN and FEDER, Projects MTM2010-14909 and MTM2013-47093-P, and by GVA, Project PROMETEOII/2013/013. The first author is supported by a Grant from the FPU Program of MEC.

We define a Markov process on the set of countable graphs with spins. Transitions are local substitutions in the graph. It is proved that the scaling macrodimension is an invariant of such dynamics.