On the Neumann function and the method of images in spherical and ellipsoidal geometry

The invention of an image system for a boundary value problem adds to a significant understanding of the structure of the problem, both at the mathematical and at the physical level. In this paper, the interior and exterior Neumann functions for the Laplacian in the cases of spherical and ellipsoidal domains are represented in terms of images. Besides isolated images, the presence of the normal derivative in the Neumann condition demands an additional continuous distribution of images, which in the spherical cases, can be restricted to a one‐dimensional manifold, whereas for the ellipsoid, both a one‐dimensional and a two‐dimensional distribution of images is needed. Copyright © 2012 John Wiley & Sons, Ltd.     

On a vector-valued integral in the “noncommutative” statistical decision theory

A generalization of a vector-valued integral arising in the noncommutative (quantum) statistical decision theory is considered.     

“Unintended effects”: A theorem for complex systems

Unintended effects are well known to economists and sociologists and their consequences may be devastating. The main objective of this article is to formulate a mathematical theorem, based on Gödel's famous incompleteness theorem, in which it is shown, that from the moment deontical modalities (prohibition, obligation, permission, and faculty) are introduced into the social system, responses are allowed by the system that are not produced, however, prohibited responses or unintended effects may occur. © 2014 Wiley Periodicals, Inc. Complexity 21: 342–354, 2015     

On the “largeur d'arborescence”

Let la(G) be the invariant introduced by Colin de Verdière [J. Comb. Theory, Ser. B., 74:121–146, 1998], which is defined as the smallest integer n≥0 such that G is isomorphic to a minor of K_n×T, where K_n is a complete graph on n vertices and where T is an arbitrary tree. In this paper, we give an alternative definition of la(G), which is more in terms of the tree‐width of a graph. We give the collection of minimal forbidden minors for the class of graphs G with la(G)≤k, for k=2, 3. We show how this work on la(G) can be used to get a forbidden minor characterization of the graphs with (G)≤3. Here, (G) is another graph parameter introduced in the above cited paper. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 24–52, 2002     

Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions

An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

     

On the equivalence of viscosity and distribution solutions of second-order PDEs with Neumann boundary conditions

We apply a probabilistic approach to prove that the viscosity solutions and the distribution ones to the Neumann problem of second order elliptic and parabolic equations are equivalent.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

A note on “A Continuous Approach to Nonlinear Integer Programming”

Ge and Huang (1989) proposed an approach to transform nonlinear integer programming problems into nonlinear global optimization problems, which are then solved by the filled function transformation method. The approach has recently attracted much attention. This note indicates that the formulae to determine a penalty parameter in two fundamental theorems are incorrect, and presents the corrected formulae and revised theorems.     

Optimization in “self‐modeling” complex adaptive systems

When a dynamical system with multiple point attractors is released from an arbitrary initial condition, it will relax into a configuration that locally resolves the constraints or opposing forces between interdependent state variables. However, when there are many conflicting interdependencies between variables, finding a configuration that globally optimizes these constraints by this method is unlikely or may take many attempts. Here, we show that a simple distributed mechanism can incrementally alter a dynamical system such that it finds lower energy configurations, more reliably and more quickly. Specifically, when Hebbian learning is applied to the connections of a simple dynamical system undergoing repeated relaxation, the system will develop an associative memory that amplifies a subset of its own attractor states. This modifies the dynamics of the system such that its ability to find configurations that minimize total system energy, and globally resolve conflicts between interdependent variables, is enhanced. Moreover, we show that the system is not merely “recalling” low energy states that have been previously visited but “predicting” their location by generalizing over local attractor states that have already been visited. This “self‐modeling” framework, i.e., a system that augments its behavior with an associative memory of its own attractors, helps us better understand the conditions under which a simple locally mediated mechanism of self‐organization can promote significantly enhanced global resolution of conflicts between the components of a complex adaptive system. We illustrate this process in random and modular network constraint problems equivalent to graph coloring and distributed task allocation problems. © 2010 Wiley Periodicals, Inc. Complexity 16: 17–26, 2011     

On the invertibility of “rectangular” bi-infinite matrices and applications in time–frequency analysis

Finite dimensional matrices with more columns than rows have no left inverses while those with more rows than columns have no right inverses. We give generalizations of these simple facts to bi–infinite matrices. Our results are then used to obtain density results for p-frames of time–frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn–Nirenberg symbols.     

Green’s function,temperature in a convectively cooled sphere with arbitrarily located spherical heat sources

Steady state heat conduction in a convectively cooled sphere having arbitrarily located spherical heat sources inside is treated with the method of Green’s function accompanied by a coordinate transform. Green’s function of the heat diffusion operator for a finite sphere with Robin boundary condition is obtained by spherical harmonics expansion. Verification of the analytical solution is exemplified in some generic cases related to the pebbles of South-African PBMR as of year 2000 with 268 MW thermal power. Analytical results for different sectors of the sphere (pebble) are compared with the results of computational fluid dynamics code FLUENT^™. This work is motivated through a modest effort to assess the stochastic effects of distribution and volumetric effects of fuel kernels within the pebbles of future-promising pebble bed reactors.     

On the Neumann problem for the nonstationary Stokes system in angles and cones

The authors consider the Neumann problem for the nonstationary Stokes system in a two-dimensional angle or a three-dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.     

Eratosthenes on the “measurement” of the earth

In this paper it is argued that Eratosthenes's measurement of the earth depended on estimated distances and ratios as well as approximation procedures, and that precise observations were not involved. His method is reconstructed here from a number of ancient texts, and it is concluded that Cleomedes, or his source, misunderstood and misrepresented what Eratosthenes did.     

A finite difference method for the wide‐angle “parabolic” equation in a waveguide with downsloping bottom

We consider the third‐order wide‐angle “parabolic” equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range‐dependent bathymetry. It is known that the initial‐boundary‐value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape, if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this article, we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well‐posed problem, in fact making it L² ‐conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank–Nicolson‐type finite difference scheme, which is proved to be unconditionally stable and second‐order accurate and simulates accurately realistic underwater acoustic problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013