Strong underrelaxation in Kaczmarz's method for inconsistent systems

Summary We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.     

Coloring update methods

For linear update methods (such as SOR), a coloring method is introduced for which the multicolor iteration matrix has the same spectrum as the original iteration matrix. It applies to general linear systems, not necessarily arising from PDEs. When the iteration matrices are nonsingular, it is shown that they are similar to each other.     

Three-body effective interactions in the f72 region

The three-body effective forces in a pure $\text{(0}\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{)}{}^{\mathrm{<mtext>n</mtext>}}$ model space are calculated to second order in perturbation theory. The full effect of the $\text{(0f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{1}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{)}{}^3$ configuration on the effective interaction in the $\text{(0}\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{)}{}^3$ model space leading to three-body forces is calculated to all orders in G. The resulting three-body force matrix elements come out much weaker than demanded by phenomenological fits. Their main effect is to push the spectra as a whole up in energy, improving the fit of the calculated ground state binding energies with experiment. The biggest three-body-force contribution appears toward the end of the $\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ shell where its main object is to compensate for Pauli violations arising from the two-body diagrams.     

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.     

Laser-microprobe elemental determinations with an optical multichannel detection system

The results obtained when a laser microprobe is adapted to imaging detectors for multiwavelength detection are described. Detectors evaluated are a silicon-intensified target vidicon and a second-generation photodiode array. Data are presented to illustrate how the combined system can be used to monitor both surface and depth profiles of elemental content of a variety of sample types including a ruby rod and ceramic material with blemishes, electrical capacitors, integrated circuits, and surface-coated electrical conductors. It is also shown how the gating capability of the intensified vidicon can be used to monitor time-dependent changes in the several nanosecond range during the laser microprobe excitation process. Detection limits obtained with both detectors are in the range 2–500 ppm depending on the element, the wavelength used, the matrix, and other variables. The uncertainty associated with the measurement step can be improved by a factor of 2–3 by using ratios of spectral lines. Principal limitations of the laser microprobe method are the nonlinear response of intensity vs. concentration and the resulting need for reference materials with matrices similar to samples for calibration purposes.     

Convergence and perturbation resilience of dynamic string-averaging projection methods

We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of string-averaging projection (SAP) methods for the CFP, analyzing their convergence and their perturbation resilience. In the past, SAP methods were formulated with a single predetermined set of strings and a single predetermined set of weights. Here we extend the scope of the family of SAP methods to allow iteration-index-dependent variable strings and weights and term such methods dynamic string-averaging projection (DSAP) methods. The bounded perturbation resilience of DSAP methods is relevant and important for their possible use in the framework of the recently developed superiorization heuristic methodology for constrained minimization problems.     

Super-Resolved Raman Spectroscopy

In this paper we present a super-resolved Raman spectroscopy configuration where, due to the proposed resolution improvement, more chemical species can be simultaneously detected. The idea includes usage of the basic property of Raman in which shifting the excitation wavelength will shift the absolute wavelength of the Raman spectral lines. Thus, encoding the spectra prior to analysis by the spectrometer allows obtaining spectral super-resolved sensing via time multiplexing super-resolution concepts applied over the spectral domain.     

Poly(3,4-ethylenedioxyselenophene)

The first highly conductive polyselenophene, namely, poly(3,4-ethylenedioxyselenophene) (PEDOS), was synthesized by taking advantage of a novel method for efficiently contracting the selenophene ring. PEDOS shows a relatively low band gap (1.4 eV), very high stability in the oxidized state, and a well-defined spectroelectrochemistry.     

The physics of categorization

In the context of cognition, categorization is the process through which several elements (i.e., words) are grouped into a single set which by naming becomes an abstraction of its elements. For example, tiger, kitty, and max can be categorized as Cats. In this article, we aim to show how the physical, biological and cognitive dimensions are related in the process of categorization or abstraction through the physics of computation. Drawing on Landauer's principle, we show that the price paid in terms of entropy is higher when grouping elements of low ranking (high probability) than when grouping elements of high ranking (low probability). Therefore, the logic of the cognitive process of abstraction is explained through constraints imposed by memory on the computation of categories. © 2015 Wiley Periodicals, Inc. Complexity 21: 269–274, 2016     

Restrictions on forcings that change cofinalities

In this paper we investigate some properties of forcing which can be considered “nice” in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some “damage” to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of “nice” properties, when changing cofinalities to be uncountable.