Infinitesimal Invariance Research Articles

Infinitesimal invariance of completely Random Measures for 2D Euler Equations

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.

Finite element approximation of invariant manifolds by the parameterization method

We combine the parameterization method for invariant manifolds with the finite element method for elliptic PDEs, to obtain a new computational framework for high order approximation of invariant manifolds attached to unstable equilibrium solutions of nonlinear parabolic PDEs. The parameterization method provides an infinitesimal invariance equation for the invariant manifold, which we solve via a power series ansatz. A power matching argument leads to a recursive systems of linear elliptic PDEs—the so called homological equations—whose solutions are the power series coefficients of the parameterization. The homological equations are solved recursively to any desired order N using finite element approximation. The end result is an N-th order polynomial approximation of a chart map of the manifold, with coefficients in an appropriate finite element space. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two dimensional polygonal domains (not necessary simply connected), for equilibrium solutions with Morse indices one and two. We implement a-posteriori error indicators which provide numerical evidence in support of the claim that the manifolds are computed accurately.

The Steinhaus-Weil Property: I. Subcontinuity and Amenability

The Steinhaus-Weil theorem that concerns us here is the simple, or classical, `interior-points' property -- that in a Polish topological group a non-negligible set $B$ has the identity as an interior point of$\ B^{-1}B.$ There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a \textit{Haar} reference measure $\eta $. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. This the first of four companion papers (we refer to the others as II [BinO11], III, [BinO12], and IV, [BinO13], below). Here (Propositions 1.1-1.7 and Theorems 11.-1.4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of \textit{selective} reference measures $\sigma $, drawing on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In II, we turn to a converse of the Steinhaus-Weil theorem, the Simmons-Mospan theorem, and related results. In III, we discuss Weil topologies, linking the topological group-theoretic and measure-theoretic aspects. We close in IV with some other interior-point results related to the Steinhaus-Weil theorem.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space

We consider a locally uniformly strictly elliptic second order partial differential operator in Rd, d≥2, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt process is known to solve a corresponding stochastic differential equation that is pathwise unique. In this situation, we study the relation of invariance, infinitesimal invariance, recurrence, transience, conservativeness and Lr-uniqueness, and present sufficient conditions for non-existence of finite infinitesimally invariant measures as well as finite invariant measures. Our main result is that recurrence implies uniqueness of infinitesimally invariant measures, as well as existence and uniqueness of invariant measures, both in subclasses of locally finite measures. We can hence make in particular use of various explicit analytic criteria for recurrence that have been previously developed in the context of (generalized) Dirichlet forms and present diverse examples and counterexamples for uniqueness of infinitesimally invariant, as well as invariant measures and an example where L1-uniqueness fails for one infinitesimally invariant measure but holds for another and pathwise uniqueness holds. Furthermore, we illustrate how our results can be applied to related work and vice versa.

Massey Products and Fujita decompositions on fibrations of curves

Let $$f:S\rightarrow B$$ be a fibration of curves and let $$f_*\omega _{S/B}={{\mathcal {U}}}\oplus {{\mathcal {A}}}$$ be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $${{\mathcal {U}}}$$. The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole $${{\mathcal {U}}}$$ of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.

Lie Algebroid Invariants for Subgeometry

We investigate the infinitesimal invariants of an immersed submanifold $\Sigma $ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $\Sigma $. The invariants are derived from the logarithmic derivative of the immersion of $\Sigma $ into $M$, a complete invariant introduced in the companion article, 'A characterization of smooth maps into a homogeneous space'. Applications of the Lie algebroid approach to subgeometry include a new interpretation of Cartan's method of moving frames and a novel proof of the fundamental theorem of hypersurfaces in Euclidean, elliptic and hyperbolic geometry.

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in a nonholonomic system of Chetaev’s type

For a nonholonomic system of Chetaev’s type, the conformal invariance and the conserved quantity of Mei symmetry for Appell equations are investigated. First, under the infinitesimal one-parameter transformations of group and the infinitesimal generator vectors, Mei symmetry and conformal invariance of differential equations of motion for the system are defined, and the determining equation of Mei symmetry and conformal invariance for the system are given. Then, by means of the structure equation to which the gauge function is satisfied, the Mei-conserved quantity corresponding to the system is derived. Finally, an example is given to illustrate the application of the result.

A geometric method for observability and accessibility of discrete impulsive nonlinear systems

This article is concerned with the observability and accessibility of discrete impulsive nonlinear systems. A geometric method based on the differential geometric analysis and Lie group investigation is proposed. The infinitesimal invariance principle in Lie group theory is extended to the case of discrete impulsive nonlinear systems. By characterising the infinitesimal principle in terms of the sequences of codistribution and distribution, explicit criteria for the local observability and local accessibility of the system are derived, respectively. Additionally, two examples are provided to show that the criteria are convenient to check.

Geometric approach for observability and accessibility of discrete‐time non‐linear switched impulsive systems

This study is concerned with observability and accessibility of discrete-time non-linear switched impulsive systems, for which the problem is more complicated than that for general discrete systems and has novel features. A new method from the combination of differential geometric theory and Lie group analysis is developed. Specifically, under the geometric framework of the system, infinitesimal invariance for the observability and accessibility is investigated, respectively. Based on the invariance investigation, explicit criteria for local observability and local accessibility are derived. Numerical examples are discussed to show the effectiveness of the proposed methods, and the features of observability and accessibility for discrete switched impulsive systems.

Asymptotic Stability of N-Solitary Waves of the FPU Lattices

We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Backlund transformation and use the result to analyze the linearized FPU equation.

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.

The gauge theory of dislocations: conservation and balance laws

We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J , L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.

The symplectic ideal and a double centraliser theorem

We interpret a result of S. Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on \bigoplus_{r=0}^s\otimes^rV, where V is the natural module for the symplectic group. This result was obtained in characteristic zero by H. Weyl. Furthermore we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G]^g of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain.

Involutions of reductive Lie algebras in positive characteristic

Let G be a reductive group over a field k of characteristic ≠2, let g = Lie ( G ) , let θ be an involutive automorphism of G and let g = k ⊕ p be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G θ on p is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic. Among other results, we prove that the variety N of nilpotent elements of p has a dense open orbit, and that the same is true for every fibre of the quotient map p → p ▪ G θ . However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of N , extending a result of Sekiguchi for k = C . Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants k [ p ] k .

Generalized Lie bialgebras and Jacobi structures on Lie groups

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras and we prove that they can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structure. We also propose a method generalizing the Yang-Baxter equation method to obtain generalized Lie bialgebras. Finally, we classify the compact generalized Lie bialgebras.

Jacobi groupoids and generalized Lie bialgebroids

Jacobi groupoids are introduced as a generalization of Poisson and contact groupoids and it is proved that generalized Lie bialgebroids are the infinitesimal invariants of Jacobi groupoids. Several examples are discussed.

Algebraic cycles and infinitesimal invariants on Jacobian varieties

Algebraic cycles and infinitesimal invariants on Jacobian varieties

L1-Uniqueness of regularized 2D-Euler and stochastic Navier–Stokes equations

A regularization of the 2D-Euler equation with periodic boundary conditions is introduced, having the same infinitesimal invariants as the Euler equation. A flow of measure-preserving transformations is constructed on L 1-spaces induced by the Gaussian measure with covariance given by the inverse of the enstrophy and it is shown that this flow is the only measure-preserving flow inducing a strongly continuous semigroup on the corresponding L 1-space. We also prove similar uniqueness results for a corresponding class of regularized stochastic 2D-Navier–Stokes equations.