165 research outputs found
On boole's formula for factorials
Journal ArticleWe present a simple new proof and a new generalization of Boole's formula n! ∑nσj=1 (-1)n-j (n j)jn (n ∈ N)
On an Inequality of H. Minc and L. Sathre
AbstractWe prove that if we let n be a positive integer, then we have for all positive real numbers r, [formula]. Both bounds are best possible
Monotonicity and logarithmic convexity relating to the volume of the unit ball
Let stand for the volume of the unit ball in for
. In the present paper, we prove that the sequence
is logarithmically convex and that the sequence
is strictly
decreasing for . In addition, some monotonic and concave properties of
several functions relating to are extended and generalized.Comment: 12 page
Finitely Many Dirac-Delta Interactions on Riemannian Manifolds
This work is intended as an attempt to study the non-perturbative
renormalization of bound state problem of finitely many Dirac-delta
interactions on Riemannian manifolds, S^2, H^2 and H^3. We formulate the
problem in terms of a finite dimensional matrix, called the characteristic
matrix. The bound state energies can be found from the characteristic equation.
The characteristic matrix can be found after a regularization and
renormalization by using a sharp cut-off in the eigenvalue spectrum of the
Laplacian, as it is done in the flat space, or using the heat kernel method.
These two approaches are equivalent in the case of compact manifolds. The heat
kernel method has a general advantage to find lower bounds on the spectrum even
for compact manifolds as shown in the case of S^2. The heat kernels for H^2 and
H^3 are known explicitly, thus we can calculate the characteristic matrix.
Using the result, we give lower bound estimates of the discrete spectrum.Comment: To be published in JM
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