scholarly journals HEAT KERNEL BOUNDS, POINCARÉ SERIES, AND L2 SPECTRUM FOR LOCALLY SYMMETRIC SPACES

2008 ◽  
Vol 78 (1) ◽  
pp. 73-86 ◽  
Author(s):  
ANDREAS WEBER
Keyword(s):  
Heat Kernel ◽  
Symmetric Spaces ◽  
Discrete Group ◽  
Nonpositive Curvature ◽  
Upper Bounds ◽  
Poincaré Series ◽  
Locally Symmetric Spaces ◽  
Poincare Series ◽  
Gaussian Bounds ◽  
Kernel Bounds

AbstractWe derive upper Gaussian bounds for the heat kernel on complete, noncompact locally symmetric spaces M=Γ∖X with nonpositive curvature. Our bounds contain the Poincaré series of the discrete group Γ and therefore we also provide upper bounds for this series.

scholarly journals Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

1999 ◽  
Vol 42 (2) ◽  
pp. 155-161 ◽  
Author(s):  
H. E. A. Campbell ◽  
A. V. Geramita ◽  
I. P. Hughes ◽  
R. J. Shank ◽  
D. L. Wehlau
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
Invariant Theory ◽  
Upper Bounds ◽  
Poincaré Series ◽  
Faithful Representation ◽  
Gorenstein Ring ◽  
Poincare Series ◽  
Trivial Group

AbstractThis paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

scholarly journals Sharp constants in higher-order heat kernel bounds

2000 ◽  
Vol 61 (2) ◽  
pp. 189-200 ◽  
Author(s):  
Nick Dungey
Keyword(s):  
Heat Kernel ◽  
Laplace Operator ◽  
Adjoint Operator ◽  
Higher Order ◽  
Sharp Constants ◽  
Polynomial Type ◽  
Gaussian Bounds ◽  
Precise Constant ◽  
Kernel Bounds ◽  
The Laplace Operator

We consider a space X of polynomial type and a self-adjoint operator on L2(X) which is assumed to have a heat kernel satisfying second-order Gaussian bounds. We prove that any power of the operator has a heat kernel satisfying Gaussian bounds with a precise constant in the Gaussian. This constant was previously identified by Barbatis and Davies in the case of powers of the Laplace operator on RN. In this case we prove slightly sharper bounds and show that the above-mentioned constant is optimal.

scholarly journals Upper bounds for geodesic periods over rank one locally symmetric spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1065-1077 ◽  
Author(s):  
Jan Frahm ◽  
Feng Su
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Locally Symmetric Spaces ◽  
Ambient Manifold ◽  
Rank One ◽  
General Rank ◽  
Laplace Eigenvalues ◽  
Periods Of Automorphic Forms

AbstractWe prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

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