The (continuous) finite element approximations of different orders for the computation of the solution to electronic structures were proposed in some papers and the performance of these approaches is becoming appreciable and is now well understood. In this publication, the author proposes to extend this discretization for full-potential electronic structure calculations by combining the refinement of the finite element mesh, where the solution is most singular with the increase of the degree of the polynomial approximations in the regions where the solution is mostly regular. This combination of increase of approximation properties, done in an a priori or a posteriori manner, is well-known to generally produce an optimal exponential type convergence rate with respect to the number of degrees of freedom even when the solution is singular. The analysis performed here sustains this property in the case of Hartree-Fock and Kohn-Sham problems.

This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits. The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.

The authors mainly study the generalized symplectic mean curvature flow in an almost Einstein surface, and prove that this flow has no type-I singularity. In the graph case, the global existence and convergence of the flow at infinity to a minimal surface with metric of the ambient space conformal to the original one are also proved.

The authors construct a solution U _t(x) associated with a vector field on the Wiener space for all initial values except in a 1-slim set and obtain the 1-quasi-sure flow property where the vector field is a sum of a skew-adjoint operator not necessarily bounded and a nonlinear part with low regularity, namely one-fold differentiability. Besides, the equivalence of capacities under the transformations of the Wiener space induced by the solutions is obtained.

Let {X,X _k: k ≥ 1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX > 0. Let τ be a nonnegative integer-valued random variable, independent of {X,X _k: k ≥ 1}. In this paper, the authors obtain the necessary and sufficient conditions for the random sums S_\tau = \sum\limits_{n = 1}^\tau {X_n } to have a consistently varying tail when the random number τ has a heavier tail than the summands, i.e., \frac{{P(X > x)}}{{P(\tau > x)}} \to 0 as x→∞.

Let S be a hyperbolic Riemann surface with a finite area. Let G be the covering group of S acting on the hyperbolic plane H. In this paper, the author studies some algebraic relations in the mapping class group of Ṡ for Ṡ = S\{a point}. The author shows that the only possible relations between products of two Dehn twists and products of mapping classes determined by two parabolic elements of G are the reduced lantern relations. As a consequence, a partial solution to a problem posed by J. D. McCarthy is obtained.

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-manifold with nonnegative sectional curvature, which improves Marenich-Toponogov’s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a closed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.

In this paper, the authors construct a Φ-group for n submodules, which generalizes the classical K-theory and gives more information than the classical ones. This theory is related to the classification theory for indecomposable systems of n subspaces.

The authors introduce and investigate the \mathcal{T}_C -Gorenstein projective, \mathcal{L}_C -Gorenstein injective and \mathcal{H}_C -Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the \mathcal{T}_C -Gorenstein projective or the \mathcal{H}_C -Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the \mathcal{L}_C -Gorenstein injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the \mathcal{T}_C (R)-projective and \mathcal{L}_C (R)-injective dimensions and \mathcal{T}_C (R)-precovers and \mathcal{L}_C (R)-preenvelopes. Finally, the authors study the \mathcal{H}_C -Gorenstein flat modules and extend the Foxby equivalences.

In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Gaussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.

The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Ampère type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.