The best constant of the discrete Sobolev inequalities on the complete bipartite graph

We have the best constants of three kinds of discrete Sobolev inequalities on the complete bipartite graph with 2N vertices, that is, KN,N. We introduce a discrete Laplacian A on KN,N. A is a 2N ×2N real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is 1 = t(1, 1, … , 1)∈ C2N. A discrete heat kernel, a Green’s matrix and a pseudo Green’s matrix play important roles in giving the best constants

THE BEST CONSTANT OF L^p SOBOLEV INEQUALITY CORRESPONDING TO DIRICHLET-NEUMANN BOUNDARY VALUE PROBLEM

We have obtained the best constant of the following Lp Sobolev inequality sup 0≤y≤1| u(j)(y)| ≤C (∫ 01 | u(M)(x)| p dx)1/p , where u is a function satisfying u(M) ∈ Lp(0, 1), u(2i)(0) = 0 (0 ≤i ≤ [(M − 1)/2]) and u(2i+1)(1) = 0 (0 ≤ i ≤ [(M − 2)/2]), where u(i) is the abbreviation of (d/dx)iu(x). In [9], the best constant of the above inequality was obtained for the case of p = 2 and j = 0. This paper extends the result of [9] under the conditions p > 1 and 0 ≤ j ≤ M −1. The best constant is expressed by Bernoulli polynomials

Positivity and Hierarchical Structure of four Green Functions Corresponding to a Bending Problem of a Beam on a half line

We consider the boundary value problem for fourth order linear ordinary differential equation in a half line (0,∞), which represents bending of a beam on an elastic foundation under a tension. A tension is relatively stronger than a spring constant of elastic foundation. We here treat four self-adjoint boundary conditions, clamped, Dirichlet, Neumann and free edges, at x = 0. We show the positivity and the hierarchical structure of four Green functions

Complete Caudate Lobectomy: Its Definition, Indications, and Surgical Approaches

There are three ways to approach and resect the caudate lobe of the liver, that is; and isolated caudate lobectomy, a combined resection of the liver overlying the caudate lobe, and a transhepatic anterior approach by splitting parenchyma of the liver

Minimally Invasive Spinal Stabilization with Denosumab before Total Spondylectomy for a Collapsing Lower Lumbar Spinal Giant Cell Tumor

A 21-year-old man consulted our hospital for treatment of a spinal giant cell tumor (GCT) of Enneking stage III. Lower lumbar-spine tumors and severe spinal canal stenosis are associated with high risk for surgical mor-bidity. Stability was temporarily secured with a percutaneous pedicle screw fixation in combination with deno-sumab, which shrank the tumor. Total en bloc spondylectomy was then performed 6 months after initiation of denosumab, and the patient was followed for 3 years. There was no local recurrence, and bony fusion was obtained. Minimally invasive surgery and denosumab allowed safer and easier treatment of a collapsing lower lumbar extra-compartmental GCT