In this note, we estimate the upper bound of volume closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss equality case. particular, Boundary Conjecture holds when is achieved. Our theorem can be applied to Riemannian manifolds non-smooth boundary, which generalizes Heintze and Karcher's classical comparison theorem. main tool gradient flo...

The discontinuous slip-line form of upper bound plasticity analysis is considered using an equilibrium forces approach. It demonstrated that the underlying basis approach can be written in terms stress functions provide a continuum state interpretation solution. An alternative proof theorem, applicable to both associative and non-associative materials, presented. broader nature strict condition...

In the light of the Sturm-Liouville theorem, the Levinson theorem for the Schrödinger equation with both local and non-local cylindrically symmetric potentials is studied. It is proved that the two-dimensional Levinson theorem holds for the case with both local and non-local cylindrically symmetric cutoff potentials, which is not necessarily separable. In addition, the problems related to the p...

The probelm of generalizing Gleason’s theorem to the non separable case arose in correspondence with Paul Chernoff. I am very grateful to him for suggesting this charming problem to me. Let H be a Hilbert space. The coefficient field K of H can be either the reals or the complexes. We let P(H) denote the collection of all closed subspaces of H. A Gleason measure on H is a map μ : P(H) → [0, 1] ...

Prolog systems currently work with the restriction that negative subgoals not contain free variables, although some authors have suggested that this restriction can be lifted by running the theorem prover to exhaustion on the subgoal being considered. We suggest that this approach is likely not to be computationally viable in practice, and describe and formalize a new approach to dealing with q...

An r-perfect code of a graph G = (V, E) is a set C ⊆ V such that the r-balls centered at vertices of C form a partition of V . It is proved that the direct product of Cm and Cn (r ≥ 1, m,n ≥ 2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r + 1)2 + r2 and that the direct product of Cm, Cn, and C` (r ≥ 1, m,n, ` ≥ 2r +1) contains an r-perfect code if and only if m...

The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the kth integral of a function. The tightness of this upper bound is then analyzed for the c...