On some new simple perfect squared squares

AbstractTransformation techniques due to Federico and Willcocks are applied to obtain the first known simple perfect squared square of order 24 from Duijvestijn's computer results for lower order. Similarly, twelve new simple squared squares of order 25 are added to the existing eight found by Federico and Wilson. Our emphasis is on four simple perfect squared squares of order 25 and common reduced side 540. They have 12 elements in common but have pairwise identical elements differently arranged

Compound Perfect Squared Squares of the Order Twenties

P. J. Federico used the term low-order for perfect squared squares with at most 28 squares in their dissection. In 2010 low-order compound perfect squared squares (CPSSs) were completely enumerated. Up to symmetries of the square and its squared subrectangles there are 208 low-order CPSSs in orders 24 to 28. In 2012 the CPSSs of order 29 were completely enumerated, giving a total of 620 CPSSs up to order 29.Comment: 44 pages, 10 figures. For associated pdf illustrations of enumerated compound perfect squared squares up to order 29, see http://squaring.net/downloads/downloads.html#cps

Approximating a similarity matrix by a latent class model: A reappraisal of additive fuzzy clustering

Let Q be a given n×n square symmetric matrix of nonnegative elements between 0 and 1, similarities. Fuzzy clustering results in fuzzy assignment of individuals to K clusters. In additive fuzzy clustering, the n×K fuzzy memberships matrix P is found by least-squares approximation of the off-diagonal elements of Q by inner products of rows of P. By contrast, kernelized fuzzy c-means is not least-squares and requires an additional fuzziness parameter. The aim is to popularize additive fuzzy clustering by interpreting it as a latent class model, whereby the elements of Q are modeled as the probability that two individuals share the same class on the basis of the assignment probability matrix P. Two new algorithms are provided, a brute force genetic algorithm (differential evolution) and an iterative row-wise quadratic programming algorithm of which the latter is the more effective. Simulations showed that (1) the method usually has a unique solution, except in special cases, (2) both algorithms reached this solution from random restarts and (3) the number of clusters can be well estimated by AIC. Additive fuzzy clustering is computationally efficient and combines attractive features of both the vector model and the cluster mode

Interatomic forces, phonons, the Foreman-Lomer Theorem and the Blackman Sum Rule

Foreman and Lomer proposed in 1957 a method of estimating the harmonic forces between parallel planes of atoms of primitive cubic crystals by Fourier transforming the squared frequencies of phonons propagating along principal directions. A generalized form of this theorem is derived in this paper and it is shown that it is more appropriate to apply the method to certain combinations of the phonon dispersion relations rather than to individual dispersion relations themselves. Further, it is also shown how the method may be extended to the non-primitive hexagonal close packed and diamond lattices. Explicit, exact and general relations in terms of atomic force constants are found for deviations from the Blackman sum rule which itself is shown to be derived from the generalized Foreman-Lomer theorem.Comment: 13 pages pd

Feature Extraction in Signal Regression: A Boosting Technique for Functional Data Regression

Main objectives of feature extraction in signal regression are the improvement of accuracy of prediction on future data and identification of relevant parts of the signal. A feature extraction procedure is proposed that uses boosting techniques to select the relevant parts of the signal. The proposed blockwise boosting procedure simultaneously selects intervals in the signal’s domain and estimates the effect on the response. The blocks that are defined explicitly use the underlying metric of the signal. It is demonstrated in simulation studies and for real-world data that the proposed approach competes well with procedures like PLS, P-spline signal regression and functional data regression. The paper is a preprint of an article published in the Journal of Computational and Graphical Statistics. Please use the journal version for citation

Computing apparatus Patent

Describing circuit for obtaining sum of squares of number

Macroeconomic structure and policy in Zimbabwe, analysis and empirical model : 1965-1988

The authors develop and apply a macroeconomic general equilibrium model for Zimbabwe. The country faces the challenge of engaging in a program of fiscal stabilization and structural reform to address its current fiscal imbalance, high unemployment, and low growth prospects. The authors discuss macroeconomic changes over the last two decades, provide a model of the macroeconomic structure, and estimate aggregate equations for the main goods and asset markets. The macroeconomic framework they model integrates three features of the country's macroeconomy: (a) the noninflationary and almost exclusively domestic financing of the public sector deficit, which has been similar in gross terms to the private sector surplus; (b) sustained negative or low real interest rates, together with no apparent sign of excess demand in credit markets; and (c) the fact that sustained, high growth has never materialized after the dramatic economic declines of the late 1970s that resulted from economic sanctions and civil war.Economic Theory&Research,Environmental Economics&Policies,Economic Stabilization,Macroeconomic Management,Financial Intermediation