A purpose of critical point theory is the counting of critical points of functions. Principal theorems in the subject state in precise terms that topological complexity of the underlying space is reflected in the existence and nature of critical points of any smooth real-valued function defined on the space. The initial development of critical point theory is peculiarly the work of one man, Mar...

We study principal curvatures of fibers and Heeagaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minimally embedded fiber or Heegaard surface and a lower bound on the injectivity...

We generalize the O( dn 2 )-time (1 + )-approximation algorithm for the smallest enclosing Euclidean ball [2, 10] to point sets in hyperbolic geometry of arbitrary dimension. We guarantee a O ( 1/ 2 ) convergence time by using a closed-form formula to compute the geodesic α-midpoint between any two points. Those results allow us to apply the hyperbolic k-center clustering for statistical locati...

We have extended our method of grouping of Feynman diagrams (GFD theory) to study the transversal and longitudinal Greens functions G ⊥ (k) and G (k) in ϕ 4 model below the critical point (T < T c) in presence of an infinitesimal external field. Our method allows a qualitative analysis not cutting the perturbation series. We have shown that the critical behavior of the Greens functions is consi...

This paper presents a novel approach to continuously and robustly tracking critical (geometrically, perpendicular and/or extremal) distances from a moving plane point p ∈ R to a static parametrized piecewise rational curve γ(s) (s ∈ R). The approach is a combination of local marching, and the detection and computation of global topological change, both based on the differential properties of a ...

This paper is devoted to the mathematical modeling and analysis of a hyperbolic Maxwell quasi-variational inequality (QVI) for Bean-Kim superconductivity model with temperature magnetic field dependence in critical current. Our relies on local (resp. global) boundedness Lipschitz continuity assumptions current respect field). Emerging from Euler time discretization, we analyze corresponding H(c...

In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptotically hyperbolic manifolds, asymptotically complex hyperbolic manifolds, and the Schwarzschild spacetime. ...