On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.Comment: 16 pages; no figures; replaces math.SP/0403179; completely re-writte

Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians

Importance of Verification and Validation of Data Sources in Attaining Information Superiority

Information superiority has been defined as a state that is achieved when a competitive advantage is derived from the ability to exploit a superior information position. To achieve such a superior information position enterprises and nations, alike, must not only collect and record correct, accurate, timely and useful information but also ensure that information recorded is not lost to competitors due to lack of comprehensive security and leaks. Further, enterprises that aim to attain information superiority must also ensure mechanisms of validating and verifying information to reduce the chances of mis-information. Although, research has been carried out into ways to increase the security of information and detect leaks, not enough focus has been given to the key elements of information, namely data and context. This paper outlines the importance of data in contributing to information superiority and highlights the lack of data centric approach in attaining information superiority. The paper also discusses the importance of verification and back tracking of information to ascertain the data, its source and context in validating information for its correctness, validity and accuracy. A brief list of consequences of information leaks is also provided in the document to emphasize the importance of information security in the context of data collected. Further, this paper examines the McCumber model, which outlines the various states and elements of information, to accommodate a data centric, quantitative approach. Outlining simple protocols for verification of data in the information superiority context, this paper also highlights a few steps that can be taken to verify the sources of data

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian (Payne-Polya-Weinberger, Hile-Protter, etc.), as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.Comment: 21 pages; revised version: minor misprints correcte

Inverse Steklov spectral problem for curvilinear polygons

This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than π, we prove that the asymptotics of Steklov eigenvalues obtained in [LPPS19] determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard–Weierstrass factorisation theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots

Isospectral domains with mixed boundary conditions

We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M. Kac.Comment: 9 figures. Statement of Theorem 5.1 correcte

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations

Trace Identities and Universal Estimates for Eigenvalues of Linear Pencils

We describe the method of constructing the spectral trace identities and the estimates of eigenvalue gaps for the linear self-adjoint operator pencils A B