Output least squares stability in elliptic systems

In this paper the stability of the solutions of parameter estimation problems in their output least squares formulation is analyzed. The concepts of output least squares stability (OLS stability) is defined and sufficient conditions for this property are proved for abstract elliptic equations. These results are applied to the estimation of the diffusion, convection, and friction coefficient in second-order elliptic equations in ⁿ ,n=2, 3. Results on Tikhonov regularization in a nonlinear setting are also given.Part of this research was carried out while both authors were visitors at the Division of Applied Mathematics at Brown University. The research of F. Colonius was supported in part by the Deutsche Forschungsgemeinschaft. Both authors also acknowledge support from the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under Grant No. S3206.     

Interaction of potassium hexacyanoferrate (II) with histidine

L-histidine substitutes cyano groups of K₄Fe (CN)₆ at pH 7·0 on irradiation with ultraviolet light. The reaction follows first order kinetics with reference to K₄Fe(CN)₆ and zero order with reference to histidine. The kinetic data shows the primary process to be aquation of Fe(CN) ₆ ^4? while the final product is formed through a rapid dark reaction of histidine with Fe(CN)₄ (H₂O) ₂ ^2? to give the product K₂Fe(CN)₂(histidine)₂. The final product has been subjected to chemical and infrared spectral analysis.     

A genetic algorithm with neural network fitness function evaluation for IMRT beam angle optimization

Intensity Modulated Radiotherapy Treatment (IMRT) is a technique used in the treatment of cancer, where the radiation beams are modulated by a multileaf collimator allowing the irradiation of the patient using non-uniform radiation fields from selected angles. Beam angle optimization consists in trying to find the best set of angles that should be used in IMRT planning. The choice of this set of angles is patient and pathology dependent and, in clinical practice, most of the times it is made using a trial and error procedure or simply using equidistantly distributed angles. In this paper we propose a genetic algorithm that aims at calculating good sets of angles in an automated way, given a predetermined number of angles. We consider the discretization of all possible angles in the interval [0 \(^{\circ }\) , 360 \(^{\circ }\) ], and each individual is represented by a chromosome with 360 binary genes. As the calculation of a given individual’s fitness is very expensive in terms of computational time, the genetic algorithm uses a neural network as a surrogate model to calculate the fitness of most of the individuals in the population. To explicitly consider the estimation error that can result from the use of this surrogate model, the fitness of each individual is represented by an interval of values and not by a single crisp value. The genetic algorithm is capable of finding improved solutions, when compared to the usual equidistant solution applied in clinical practice. The genetic algorithm will be described and computational results will be shown.     

Structural adaptation via \mathbb{L}_p -norm oracle inequalities

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary ${\mathbb{L}}_p$ -losses. These results are applied for adaptive estimation in the additive multi-index model.     

Estimation of level set trees using adaptive partitions

We present methods for the estimation of level sets, a level set tree, and a volume function of a multivariate density function. The methods are such that the computation is feasible and estimation is statistically efficient in moderate dimensional cases (\(d\approx 8\)) and for moderate sample sizes (\(n\approx \) 50,000). We apply kernel estimation together with an adaptive partition of the sample space. We illustrate how level set trees can be applied in cluster analysis and in flow cytometry.     

Antipodal covers of distance-transitive generalized polygons

In an earlier paper, the first two authors found all distance-regular antipodal covers of all known primitive distance-transitive graphs of diameter at least 3 with one possible exception. That remaining case is resolved here with the proof that a primitive and distance-transitive collinearity graph of a finite generalized 2d-gon with \(d\ge 3\) has no distance-regular antipodal cover of diameter 2d.     

A Logical Reading of the Nonexistence of Proper Homomorphisms Between Affine Spaces

We provide definitions of and of noncollinearity by positive statements in terms of the ternary predicate of collinearity which are valid in affine n-dimensional geometry. This provides the intrinsic reason for the validity of V. Corbas's theorem stating that surjective maps between affine planes that preserve collinearity are isomorphisms, and of P. Maroscia's higher-dimensional generalization thereof.     

Microwave spectrum of trichloroacetonitrile

The microwave spectra of different isotopic species of trichloroacetonitrile—CCl ₃ ³⁵ CN, CCl ₂ ³⁵ Cl³⁷CN, CCl³⁵Cl ₂ ³⁷ CN and CCl ₃ ³⁷ CN—in the region 20,000–40,100 Mc./sec. have been studied both at room temperature and at low temperature. From the measurements of B₀ values of the symmetric tops, assuming the C-C and C-N bond distances, the other two parameters are determined. They are: C-Cl=1·771 A, C-C-Cl=108° 52′.     

Consistent estimation of a convex density at the origin

Motivated by Hampel’s birds migration problem, Groeneboom, Jongbloed, and Wellner [7] established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, g ₀, at a fixed point t ₀ > 0. However, estimation of the distribution function of the birds’ resting times involves estimation of g′₀ at 0, a boundary point at which the estimators are not consistent.In this paper, we focus on the Least Squares estimator, \(\tilde g_n \). Our goal is to show that consistent estimators of both g ₀(0) and g′₀(0) can be based solely on \(\tilde g_n \). Following the idea of Kulikov and Lopuhaä [14] in monotone estimation, we show that it suffices to take \(\tilde g_n \)(n ^?α ) and \(\tilde g'_n \)(n ^?α ), with α ∈ (0, 1/3). We establish their joint asymptotic distributions and show that α = 1/5 should be taken as it yields the fastest rates of convergence.     

On one transformation of parameter-spaces of real interpolation method

In connection with estimation of interpolation orbits and coorbits we introduce a new transformation acting in the class of all parameter-space of real interpolation K-method. We “calculate” the result of the transformation of classical parameters. It is revealed that the transformation of weighted L₁-spaces leads to Orlicz spaces.     

Accessibility to a Road Network: Definitions and Applications

In 1953 A. Shimbel proposed definitions of integral and relative accessibility in a network. When this definition of integral accessibility is applied to a detailed route network, it does not give an exact and coherent measure of the accessibility to the territory. Instead of considering the number of edges as a basic measure to calculate the relative accessibility, we use the length of, or the travelling time of, the edges. Moreover, attempts were made to improve the definition in various other ways:

1. a)
   by limiting the reference nodes to a hexagonal grid covering the territory regularly;
    
2. b)
   by limiting the reference nodes to the principal towns or cities of economical and social significance;
    
3. c)
   by weighting the relative accessibilities in function of the distance in order to obtain local measures.
    

The accessibility is obtained from the shortest path between nodes of the network. The algorithm of DIJKSTRA was used to obtain all these shortest paths. Computer experiences will be given for the Belgian route network, built up of more than 900 nodes. Automatically plotted geographical maps will be shown for several applications (route, network, relative and integral accessibility, weighted accessibility, location of public facilities, location of distribution centres, evalution of added new routes,...,).     

Two-dimensional fin with convective base condition

When solving a two-dimensional model of an isolated fin, researchers have mainly concentrated on either a constant or a periodic fin base temperature. It is possible to obtain a numerical solution by a convective boundary condition on the fin base. However, in an analytical solution, one cannot calculate an arbitrary constant because of the convective boundary condition of the separation of variables. Therefore a heat balance is applied here to resolve this difficulty. In addition, a modified solution is presented which does not involve any additional mathematics with respect to the classical approach of solving a one-dimensional model. For different values of the Biot number $B_{22}$, a comparison of one- and two-dimensional solutions is given. Relative errors of the heat flow rates predicted by the classical and modified one-dimensional solutions, and the respective exact two-dimensional solution with respect to an, are computed. It is found that, for large values of $B_{22}$ (say 50.0) modified solution, by using a convective condition at the fin base gives significant accuracy improvements in comparison to the classical one-dimensional technique.     

Numerical distribution functions for seasonal unit root tests with OLS and GLS detrending

This paper implements the approach introduced by MacKinnon (J Bus Econ Stat 12:167–176, 1994, J Appl Econom 11:601–618, 1996) to estimate the response surface of the test statistics of seasonal unit root tests with OLS and GLS detrending for quarterly and monthly time series. The Gauss code that is available in the supplementary material of the paper produces p values for five test statistics depending on the sample size, deterministic terms and frequency of the data. A comparison with previous studies is undertaken, and an empirical example using airport passenger arrivals to a tourist destination is carried out. Quantile function coefficients are reported for simple computation of critical values for tests at 1, 5 and 10 % significance levels.