Non-degeneracy Property Research Articles

Dynamics for the focusing, energy-critical nonlinear Hartree equation

Abstract By Li, Miao and Zhang (2009) and Miao, Xu and Zhao (2009), the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when E(u 0) < E(W), where W is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy under the assumption of the nondegeneracy of the linearized operator near ground state. Our arguments closely follow those of Duyckaerts and Merle (2008, 2009), Duyckaerts and Roudenko (2010) and Li and Zhang (2009, 2011). The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in L 2d/(d-2)(ℝ d ) is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution. Moreover, the nondegeneracy property of the linearized operator can be shown in the Newtonian's potential case (that is, d = 6).

Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov’s theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a ∈ V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.

Some Generic Properties of Non Degeneracy for Critical Points of Functionals and Applications

We give some generic properties of non degeneracy for critical points of functionals. We apply these results, obtaining some theorems of multiplicity of solutions for the equation $$ \left\{\begin{array}{ll} -\varepsilon^2\Delta_g u + u = |u|^{p-2}u \quad {\rm in}\ M \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad, \\ u \in H_g^1(M) \end{array}\right. $$ where M is a compact Riemannian manifold of dimension n and $${2 < p < \frac{2n}{n\,-\,2}}$$ .

The Ideal Intersection Property for Groupoid Graded Rings

We show that, if a groupoid graded ring has a grading satisfying a certain nondegeneracy property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras with commutative principal component, the principal component is maximal commutative if and only if it has the ideal intersection property.

On nondegenerate triangulation with condensation at the boundary of a domain

We propose new algorithms for refining triangular grids in a domain with smooth boundary with persevering the nondegeneracy property. Bibliography: 7 titles. Illustrations: 1 figure.

On the non-degeneracy property of the longest-edge trisection of triangles

The longest-edge (LE) trisection of a triangle t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite vertex. In this paper we prove that for any given triangle t with smallest interior angle τ > 0 , if the minimum interior angle of the three triangles obtained by the LE-trisection of t into three new triangles is denoted by τ 1 , then τ 1 ⩾ τ / c 1 , where c 1 = π / 3 arctan ( 3 / 5 ) ≈ 3.1403 . Moreover, we show empirical evidence on the non-degeneracy property of the triangular meshes obtained by iterative application of the LE-trisection of triangles. If τ n denotes the minimum angle of the triangles obtained after n iterative applications of the LE-trisection, then τ n > τ / c where c is a positive constant independent of n. An experimental estimate of c ≈ 6.7052025350 is provided.

Nondegeneracy of the random double oscillator

AbstractIt is shown that under mild conditions the double oscillator with random parametric excitation satisfies the nondegeneracy properties that form the basic assumptions for the qualitative theory of stochastic linear systems. Necessary and sufficient conditions for hypoellipticity and weak ellipticity of the associated generator are established as well as the non skew‐symmetrizability of the matrices of the system.

A convergence of the approximated free boundary of regularized functionals via [formula omitted]-convergence

We treat a variational problem with a singular term by the method of approximation via Γ -convergence, which is a constructive way. It is known that the minimizers may have a free boundary in the interior of the domain. The main aim of this paper is to establish a criterion on the location of the free boundary in terms of the approximation. As a corollary, we obtain a non-degeneracy property which means that the convergence of the approximated free boundary is sharp in some sense, and show the corresponding examples in a one-dimensional linear case. In order to derive the non-degeneracy property we need a device to the choice of approximation to the singular term.

Deformation of generic submanifolds in a complex manifold

This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of Thom's transversality theorem, as well as the stratification of real-algebraic sets. Optimal results on the order of nondegeneracy are given.

Spin Calogero models obtained from dynamical r-matrices and geodesic motion

We study classical integrable systems based on the Alekseev–Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, G . We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group G with Lie algebra G , and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle T ∗ G , using the symmetry arising from the adjoint action of G twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework.

Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)) = 0 for n < 0, V(0)=C1, and the contragredient module V' is isomorphic to V as a V-module; (ii) every N-gradable weak V-module is completely reducible; (iii) V is C(2)-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of tau |--> -1/tau, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

Non-degeneracy study of the 8-tetrahedra longest-edge partition

In this paper we show empirical evidence on the non-degeneracy property of the tetrahedral meshes obtained by iterative application of the 8-tetrahedra longest-edge (8T-LE) partition. The 8T-LE partition of an initial tetrahedron t yields an infinite sequence of tetrahedral meshes τ 1 = { t } , τ 2 = { t i 2 } , τ 3 = { t i 3 } , … . We give numerical experiments showing that for a standard shape measure introduced by Liu and Joe ( η), the non-degeneracy convergence to a fixed positive value is guaranteed, that is, for any tetrahedron t i n in τ n , n ⩾ 1 , η ( t i n ) ⩾ c η ( t ) where c is a positive constant independent of i and n. Based on our experiments, estimates of c are provided.

On the theory of KM2O-Langevin equations for non-stationary and degenerate flows

We have developed the theory of KM2O-Langevin equations for stationary and non-degenerate flow in an inner product space. As its generalization and refinement of the results in [14], [15], [16], we shall treat in this paper a general flow in an inner product space without both the stationarity property and the non-degeneracy property. At first, we shall derive the KM2O-Langevin equation describing the time evolution of the flow and prove the fluctuation-dissipation theorem which states that there exists a relation between the fluctuation part and the dissipation part of the above KM2O- Langevin equation. Next we shall prove the characterization theorem of stationarity property, the construction theorem of a flow with any given nonnegative definite matrix function as its two-point covariance matrix function and the extension theorem of a stationary flow without losing stationarity property.

Variational inequalities with lack of ellipticity. Part I: Optimal interior regularity and non-degeneracy of the free boundary

This paper is the first part of a program aimed at studying the regularity of sub-elliptic free boundaries. In the setting of Carnot groups we establish the optimal interior regularity of the solution to the obstacle problem in terms of the Folland-Stein non-isotropic class Γ 1,1 . This result constitutes the sub-elliptic counterpart of the classical C 1,1 regularity for Laplace equation. We also prove non-degeneracy properties of the solution and of its free boundary.

On the adjacencies of triangular meshes based on skeleton-regular partitions

For any 2D triangulation τ, the 1-skeleton mesh of τ is the wireframe mesh defined by the edges of τ, while that for any 3D triangulation τ, the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of τ and the “surface” mesh defined by the triangular faces of τ. A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a refined conforming mesh and refined and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements, faces, edges and vertices of the nth and ( n−1)th meshes. By using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to infinity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D.

MODEL ORIENTATION AND WELL CONDITIONING OF SYSTEM MODELS: SYSTEM AND CONTROL ISSUES

Early stages modelling of processes involves issues of classification of variables into inputs, outputs and internal variables, referred to as Model Orientation Problem (MOP) which may be addressed on state space implicit, or matrix pencil descriptions. Defining orientation is equivalent to producing state space models of the regular or singular type. Studying the conditions, under which such models may be derived, as well as the structural properties of the resulting oriented problems, is one of the issues considered here. Oriented models of the S(A,B,C,D) type have transfer functions which may not be well behaving as far as properties of input, output regularity and nondegeneracy. Defining subsystems by reduction of the number of inputs, outputs such that the resulting transfer function is well behaved as far as preservation of degeneracy, is a problem considered in the context of systems having physical input, output variables, which have to be preserved.

Estimation of the intensity of stationary flat processes

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝ d with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution. Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝ d and (3) with unknown directional distribution. We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case. Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Estimation of the intensity of stationary flat processes

The intensity of a stationary process ofk-dimensional affine subspaces (k-flats) of ℝdwith directional distribution from a given familyRis estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (theR-estimator) using the available information about the directional distribution.Special cases are estimators for the intensity of stationaryk-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝdand (3) with unknown directional distribution.We give sufficient conditions for theR-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poissonk-flat processes with directional distribution inR. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case.Moreover, we consider stationary ergodic flat processes with directional distribution inRand general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Outer inheritance in jordan algebras

In the general structure theory of prime, simple, and division Jordan algebras developed by E.I. Zel’manov and applied in his solution of the Burnside problem, the Jordan classification in characteristic 2 required outer ideals of classical algebras. In this paper we show directly that over any ring of scalars the properties of nondegeneracy, strong primeness, unital simplicity, or divisibility are inherited by any ample outer ideal. This applies in particular to ample subspaces H 0(A,*) of hermitian elements in associative algebras with involution.

A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations

We deal with the initial and boundary value problem for the degenerate parabolic equation u_t = \Delta \beta (u) in the cylinder \Omega \times (0,T) , where \Omega \subset \mathbb R^n is bounded, \beta (0) = \beta’(0) = 0 , and \beta' ≥ 0 (e.g., \beta (u) = u |u|^{m–1} \ (m &gt; 1)) . We study the appearance of the free boundary, and prove under certain hypothesis on \beta that the free boundary has a finite speed of propagation, and is Holder continuous. Further, we estimate the Lebesgue measure of the set where u &gt; 0 is small and obtain the non-degeneracy property |\{ 0 &lt; \beta' (u(x,t)) &lt; \epsilon \} | ≤ c \epsilon^{\frac{1}{2}} .