We discuss some of our recent results regarding Generalized Willmore Energies, their critical surfaces, associated gradient flows and real-world applications.

The paper is devoted to exploration of statistical manifolds. In particular we consider exponential family manifolds, its metrics and connections. We proved that a statistical manifold does not admit nontrivial geodesic mapping between the Riemannian connection and any alpha-connection. Besides, a statistical manifold does not admit a Weyl connection which coincides with any alpha-connection. We obtain also that the Fisher information matrix calculated for a mixture family is a Hessian metric.

In this paper, we study on a gradient Einstein-type spacetime. Under what special conditions such spacetimes will be transformed into solitons of special type are among the other topics of this study. Considering a gradient Einstein-type spacetime with a parallel vector field, we establish the subsequent results.

The existence of Dark Matter (DM) is hypothesized as an explanation of the flat rotational curves in galaxies. Modified Newtonian Dynamics (MOND) offers an alternative view. This work introduces the Reparameterization Invariant Scaling Symmetry (RISS) by considering a conformal cosmic scale factor, defined within the Scale Invariant Vacuum (SIV) gauge, that removes the Einstein Cosmological Constant from the corresponding dynamical equations. The paper derives the MOND fundamental relation between the observed gravitational acceleration, the MOND fundamental acceleration, and the Newtonian acceleration of a system. The value obtained for the MOND fundamental acceleration is in the correct order of magnitude as observed in nature.

The dually flat structure is a central notion in information geometry. In this paper, we introduce a Bregman divergence and hence a dually flat structure to a ground-state manifold, the parametric family of the ground states of a parameterized quantum Hamiltonian. Taking an Ising chain as an example, we also discuss an approximation of a quantum Hamiltonian by the projection theorem.

This paper addresses the problem of the existence of global minimal solutions for a class of general Riccati equations (REs) arising in linear quadratic (LQ) optimal control problems for infinite-dimensional linear systems (LSs) with uncountable Markovian jumps (MJs). We establish sufficient conditions for the existence of these solutions in the cases where the REs are defined on ordered spaces of operator-valued measurable functions. This work extends previous results concerning the existence of minimal solutions for REs defined on Banach spaces of sequences of linear operators. Our results can be used to design optimal controls that minimize infinite-horizon quadratic cost functionals associated with infinitedimensional LSs with control and MJs.

In the first part of this article, gradient Einstein-type manifolds are defined. In the second part, properties of a perfect fluid spacetime are given. In the third part, considering gradient Einstein-type spacetimes under the special conditions, some theorems are proved.