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1. CMB 2011 (vol 54 pp. 506)
On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order |
On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the Sturm-Liouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the Sturm-Liouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the Sturm-Liouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.
Keywords:turning point, singularity, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvalues Categories:34B05, 34Lxx, 47E05 |
2. CMB 2006 (vol 49 pp. 560)
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain non-Kummer, singular
K3 surface
are dense. We will also compute the entire N\'eron--Severi group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 |
3. CMB 1999 (vol 42 pp. 169)
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, Laplace-Beltrami operator, eigenvalues Categories:35K05, 43A85, 35K15, 80A20 |