scholarly journals Boundary Values of the Resolvent of a Self-Adjoint Operator: Higher order Estimates

1996 ◽  
pp. 9-52 ◽  
Author(s):  
Anne Boutet De Monvel ◽  
Vladimir Georgescu
Keyword(s):  
Adjoint Operator ◽  
Higher Order ◽  
Boundary Values

Pointwise lower bounds on the heat kernels of higher order elliptic operators

1999 ◽  
Vol 125 (1) ◽  
pp. 105-111 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Space Dimension ◽  
Adjoint Operator ◽  
Integral Kernel ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order Elliptic Operators

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

scholarly journals Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

2019 ◽  
Vol 69 (4) ◽  
pp. 1627-1678 ◽  
Author(s):  
Ariel Barton ◽  
Steve Hofmann ◽  
Svitlana Mayboroda
Keyword(s):  
Elliptic Equations ◽  
Neumann Boundary ◽  
Higher Order ◽  
Boundary Values

scholarly journals Spectral Bounds Using Higher-Order Numerical Ranges

2005 ◽  
Vol 8 ◽  
pp. 17-45 ◽  
Author(s):  
E. B. Davies
Keyword(s):  
Anderson Model ◽  
Adjoint Operator ◽  
Higher Order ◽  
Basic Properties ◽  
Two Space Dimensions ◽  
Spectral Bounds

AbstractThis paper describes how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what are referred to here as ‘its higher-order numerical ranges’. Proofs of some of their basic properties are given, as well as an explanation of how to compute them. Finally, they are used to obtain new spectral insights into the non-self-adjoint Anderson model in one and two space dimensions.

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.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

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