On improperly posed Caucfay problems and their approximate solution

It is well known that, in general, the Cauchy problem for theLaplace equation does not allow a solution and therefore isill-posed in both the Hadamard and the Tikhonov senses. Thepresent work focuses on the question whether the problem hasany meaningful approximate solution for arbitrary boundary conditions.Firstly, it is shown that it is possible to construct an analyticfunction which assumes some prescribed value on part of theboundary of a simply-connected domain. This problem is thenshown to be equivalent to the Cauchy problem under consideration,the solution to which can thus be invariably approximated toany degree of accuracy on the unit circle centred at the originwhen both the potential and the flux are specified as square-integrablefunctions over half the unit circle boundary. The uniquenessof the exact solution to the problem is also established. Theseresults are actually true for any simply-connected domain whichcan be conformally mapped onto the unit circle so that the partof its boundary with prescribed potential and flux correspondsto one-half of the unit circle boundary. Finally, the feasibilityof a boundary element formulation for a generic type of ill-posedboundary value problems is briefly discussed.     

Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

The connection between the functional inequalities

Matrix commutators: their asymptotic metric properties and relation to approximate joint diagonalization

We analyze the properties of the norm of the commutator of two Hermitian matrices, showing that asymptotically it behaves like a metric, and establish its relation to joint approximate diagonalization of matrices, showing that almost-commuting matrices are almost jointly diagonalizable, and vice versa. We show an application of our results in the field of 3D shape analysis.     

On approximate properties of solutions of a nonlinear mixed-type equation

In the paper, we obtain sufficient solvability conditions of a certain boundary-value problem for a nonlinear mixed-type equation and find upper and lower estimates for Kolmogorov widths of the solution set. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 95–107, 2005.     

On properties of Bernstein-type operator polynomials that approximate the Urysohn operator

We generalize properties of Bernstein polynomials to Bernstein-type operator polynomials that approximate the Urysohn operator.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1172–1181, September, 2004.     

On multivariate logarithmic polynomials and their properties

In the paper, the author introduces a new notion “multivariate logarithmic polynomial”, establishes two recurrence relations, an explicit formula, and an identity for multivariate logarithmic polynomials by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, some product inequalities, and logarithmic convexity for multivariate logarithmic polynomials by virtue of some properties of completely monotonic functions.     

On approximate ring homomorphisms

We prove the stability of a generalization of the Bourgin's result on approximate ring homomorphisms.     

On approximate constraint satisfaction

We consider an abstract problem on fulfilling asymptotic constraints. We propose a very general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalizable as finitely additive measures. We study the existence and the structure of the asymptote universal in the range of “asymptotic constraints” not requiring the compactifiability of the space of ordinary solutions.     

On approximate Monetary Unit Sampling

Monetary Unit Sampling (MUS), also known as Dollar-Unit Sampling, is a popular sampling strategy in Auditing, in which all units are to be randomly selected with probabilities proportional to the book value. However, if units sizes have very large variability, no vector of probabilities exists fulfilling the requirement that all probabilities are proportional to the associated book values. In this note we propose a Mathematical Optimization approach to address this issue. An optimization program is posed, structural properties of the optimal solution are analyzed, and an algorithm yielding the optimal solution in time and space linear to the number of population units is given.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

On support properties of functions and their Jacobi transform

The qualitative uncertainty principle proved by Benedicks asserts that f and its Fourier transform \(\hat f\) cannot both concentrated in subsets of finite Lebesgue measure. In this paper we obtain some uncertainty principles concerning sets of finite measure in the Jacobi setting.     

On some properties of Segal algebras and their multipliers

Let A and B be Banach algebras such that A is a right abstract Segal algebra in B. We investigate some properties of A and B and show how these are related to the algebras of right multipliers on A and B.     

On approximate approximations using Gaussian kernels

This paper discusses quasi-interpolation and interpolation withGaussians. Estimates are obtained showing a high-order approximationup to some saturation error negligible in numerical applications.The construction of local high-order quasi-interpolation formulasis given. Supported in part by the International Centre for MathematicalSciences, Edinburgh.     

On an approximate identity of Ramanujan

In his second notebook, Ramanujan says that $$\frac{q}{{x + }}\frac{{q^4 }}{{x + }}\frac{{q^8 }}{{x + }}\frac{{q^{12} }}{{x + }} \cdots = 1 - \frac{{qx}}{{1 + }}\frac{{q^2 }}{{1 - }}\frac{{q^3 x}}{{1 + }}\frac{{q^4 }}{{1 - }} \cdots $$ “nearly” forq andx between 0 and 1. It is shown in what senses this is true. In particular, asq → 1 the difference between the left and right sides is approximately exp ?c(x)/(l-q) wherec(x) is a function expressible in terms of the dilogarithm and which is monotone decreasing with c(0) = π²/4,c(1) = π²/5; thus the difference in question is less than 2· l0^?85 forq = 0·99 and allx between 0 and 1.