1. CMB Online first
 Awonusika, Richard; Taheri, Ali

A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt 1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt 1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.
Keywords:Jacobi coefficient, LaplaceBeltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15 

2. CMB 2015 (vol 58 pp. 723)
 Castro, Alfonso; Fischer, Emily

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear LaplaceBeltrami Equations on Spheres
We show that a class of semilinear LaplaceBeltrami equations
on the unit sphere
in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions.
The solutions to
these equations are the solutions to a two point boundary value
problem for a
singular ordinary differential equation. We prove the existence
of such solutions
using energy and phase plane analysis. We derive a
Pohozaevtype
identity
in
order to prove that the energy to an associated initial value
problem tends
to infinity as the energy at the singularity tends to infinity.
The nonlinearity is allowed to grow as fast as $s^{p1}s$ for
$s$ large
with $1 \lt p \lt (n+5)/(n3)$.
Keywords:LaplaceBeltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, subsuper critical nonlinearity Categories:58J05, 35A24 

3. CMB 1999 (vol 42 pp. 169)
 Ding, Hongming

Heat Kernels of Lorentz Cones
We obtain an explicit formula for heat kernels of Lorentz cones, a
family of classical symmetric cones. By this formula, the heat
kernel of a Lorentz cone is expressed by a function of time $t$ and
two eigenvalues of an element in the cone. We obtain also upper and
lower bounds for the heat kernels of Lorentz cones.
Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, LaplaceBeltrami operator, eigenvalues Categories:35K05, 43A85, 35K15, 80A20 
