For subspaces, X and Y , of the space, D, of all derivatives M(X, Y ) denotes the set of all g ∈ D such that fg ∈ Y for all f ∈ X. Subspaces of D are defined depending on a parameter p ∈ [0, ∞]. In Section 6, M(X, D) is determined for each of these subspaces and in Section 7, M(X, Y ) is found for X and Y any of these subspaces. In Section 3, M(X, D) is determined for other spaces of functions on [0, 1] related to continuity and higher order differentiation.
Spaces Oq, q ∈ N, of multipliers of temperate distributions introduced in an earlier paper of the first author are expressed as inductive limits of Hilbert spaces.
We apply the method of quasilinearization to multipoint boundary value problems for ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.
A multiresolution analysis is defined in a class of locally compact abelian groups $G$. It is shown that the spaces of integrable functions $\mathcal L^p(G)$ and the complex Radon measures $M(G)$ admit a simple characterization in terms of this multiresolution analysis.
For the current Western military, the A2/AD (Anti Access/Area Denial) zones establishing and securing is a major challenge. For many this is an overestimated threat of democracy - ''an equal right in civic life as well as in offices'', whereas others perceive the threat absolutely, even uncritically. An actual reaction of the West is, among others, a manufacturing and implementation of F-35 JSF (Joint Strike Fighter). Its activity is linked directly to some of the fundamental scientific programs like the atomic bomb development (The Manhattan Project), B-29 to carry the bombs, ballistic missiles (V-2), landing on the Moon (Project Apollo) or SLAC and CERN accelerators. Economically the F-35 program surpasses all of them; the financial intensity of the program is even higher than the total sum of the above-mentioned projects costs at the current real prices, even after the exponentiation. With the current 2 trillion US dollars, the F-35 is the most expensive technical project in human history…
Clustering is used to organize data for efficient retrieval. A popular technique for clustering is based on k-Means such that the data is partitioned into k clusters. In k-Means clustering a set of n data points in d-dimensional space Rd, an integer k is given and the problem is to determine a set of k-points in Rd called centers, to minimize the mean squared distance from each point to its nearest center. In this method, the number of clusters is predefined and the technique is highly dependent on the initial identification of elements that represent the clusters well. A large area of research in clustering has focused on improving the clustering process such that the clusters are not dependent on the initial identification of cluster representation. In this paper, a modified technique, which grows the clusters without the need to specify the initial cluster representation, has been proposed. Initially a local search single swap heuristic can identify the number of clusters and its centers in the interpolated (bicubic) multispectral image. Then the regular k-Means clustering is implemented using the results of the previous process for the true image data set. The technique achieves an impressive speed up of the clustering process even when the number of clusters is not specified initially and the classification accuracy is improved within a fewer number of iterations.
Multistage stochastic optimization requires the definition and the generation of a discrete stochastic tree that represents the evolution of the uncertain parameters in time and space. The dimension of the tree is the result of a trade-off between the adaptability to the original probability distribution and the computational tractability. Moreover, the discrete approximation of a continuous random variable is not unique. The concept of the best discrete approximation has been widely explored and many enhancements to adjust and fix a stochastic tree in order to represent as well as possible the real distribution have been proposed. Yet, often, the same generation algorithm can produce multiple trees to represent the random variable. Therefore, the recent literature investigates the concept of distance between trees which are candidate to be adopted as stochastic framework for the multistage model optimization. The contribution of this paper is to compute the nested distance between a large set of multistage and multivariate trees and, for a sample of basic financial problems, to empirically show the positive relation between the tree distance and the distance of the corresponding optimal solutions, and between the tree distance and the optimal objective values. Moreover, we compute a lower bound for the Lipschitz constant that bounds the optimal value distance.