Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the structure in which the non-zero parallel spinor lives is equivalent to the canonical spin structure associated to the Kähler structure.

     

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

     

A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

In this paper, we derive a new monotonicity formula for the plurisubharmonic functions/positive (1,1) currents on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which, in particular, partially solve a conjecture of Yau.

The methods used in this paper, without the assumption of maximum volume of growth, as observed recently by Chen, Fu, Yin, and Zhu, provide a complete solution to Yau's conjecture.

     

Local Nonautonomous Schrödinger Flows on Kähler Manifolds

In this paper, we prove that the nonautonomous Schrödinger flow from a compact Riemannian manifold into a Kähler manifold admits a local solution     

Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section     

Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics

We consider a compact complex manifold of dimension that admits Kähler metrics and we assume that is a closed complex curve. We denote by the space of classes of Kähler forms that define Kähler metrics of volume 1 on and define by . We show how the Riemann-Hodge bilinear relations imply that any critical point of is the strict global minimum and we give conditions under which there is such a critical point : A positive multiple of is the Poincaré dual of the homology class of . Applying this to the Abel-Jacobi map of a curve into its Jacobian, , we obtain that the Theta metric minimizes the area of within all Kähler metrics of volume 1 on .

     

Upper bound of Kähler angles on the β-symplectic critical surfaces

Let (M,g) be a Kähler surface and \mathrm{\Sigma } be a \beta-symplectic critical surface in M. If L_q\left(\mathrm{\Sigma }\right) is bounded for some q>3, then we give a uniform upper bound for the Kähler angle on \mathrm{\Sigma }. This bound only depends on M,q,\beta and the L_q functional of \mathrm{\Sigma }. For q>4, this estimate is known and we extend the scope of q.     

Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry

Multicomponent evolution equations associated with linear connections on complex manifolds are considered. It is proved that under some general assumptions an equation from this class is integrable by inverse scattering method if the corresponding linear connection is the Levi-Civita connection of an indefinite Kählerian metric of constant holomorphic sectional curvature. This result is based on a certain characterization of the above-mentioned Levi-Civita connections. It is shown that the obtained integrable equations are generalized ferromagnetics, and recurrent formulas for their local conservation laws are given.     

The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds

We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.     

Model selection based on Lorenz and concentration curves,Gini indices and convex order

In order to determine an appropriate amount of premium, statistical goodness-of-fit criteria must be supplemented with actuarial ones when assessing performance of a given candidate pure premium. In this paper, concentration curves and Lorenz curves are shown to provide actuaries with effective tools to evaluate whether a premium is appropriate or to compare two competing alternatives. The idea is to compare the premium income for sub-portfolios gathering low risks (identified as low by means of the premiums under consideration) to the true one, or equivalently, to the actual losses. Numerical illustrations performed on hypothetical data and real ones demonstrate the usefulness of the proposed approach.     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

Fast solvers of integral and pseudodifferential equations on closed curves

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order , (Sobolev norms of periodic functions) in arithmetical operations.

     

Computing rational points on rank 1 elliptic curves via -series and canonical heights

Let be an elliptic curve of rank 1. We describe an algorithm which uses the value of and the theory of canonical heghts to efficiently search for points in and . For rank 1 elliptic curves of moderately large conductor (say on the order of to ) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set contains non-torsion points.

     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Exis tence and st ability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.     

The -method, weak Lefschetz theorems, and the topology of Kähler manifolds

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the -method, avoids moving arguments and gives much stronger results. In particular, it is proved that if and are connected smooth projective varieties of positive dimension and is a holomorphic immersion with ample normal bundle, then the image of in is of finite index. This result is obtained as a consequence of a direct generalization of Nori's theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in () by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold of dimension has a strongly pseudoconvex end and for some positive constant , then, away from , has finite volume.

     

Structure equations and constraint manifolds on Lorentz plane

Calculating the structure equation of a chain is important to represent the position of the end link on the chain. Furthermore, the structure equation helps to determine the constraint manifold of the chain. The constraint manifold satisfies to make geometric interpretations about the form that is obtained. What is more, the constraint forced on the positions of the end link by the rest of the chain is represented by the manifold. In Lorentz space, the structure equations change according to the causal characters of the first link. In this paper, we attain the structure equations of a planar open chain in terms of the causal character of the first link in this space. Later, the constraint manifolds of the chain by using these equations are given. Some geometric comments about these manifolds are explained.     

Exact solution of a class of fractional integro‐differential equations with the weakly singular kernel based on a new fractional reproducing kernel space

In this paper, we construct a new fractional weighted reproducing kernel space, which is the minimum space containing the exact solution. The closed form of the reproducing kernel is obtained. Using this fractional reproducing kernel space, a class of fractional integro‐differential equations with a weakly singular kernel is solved. The error estimation is given. The final numerical experiments demonstrate the correctness of the theory and the effectiveness of the method.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.     

Oscillation result for half‐linear dynamic equations on timescales and its consequences

We study oscillatory properties of half‐linear dynamic equations on timescales. Via the combination of the Riccati technique and an averaging method, we find the domain of oscillation for many equations. The presented main result is not the conversion of a known result from the theory of differential or difference equations, ie, we obtain new results for the timescales (for differential equations) and (for difference equations). Half‐linear equations generalize linear equations (in fact, they coincide with certain one‐dimensional PDEs with p‐Laplacian), but the main result is new also for linear differential and difference equations. The corresponding corollaries and examples are given as well.