Regular Almost Interconnection of Multidimensional Behaviors

An E-group is a group in which each element commutes with any of its endomorphic images. Hence E-groups provide a generalization of abelian groups. It is easy to show that the endomorphism near-ring (the group generated additively by the endomorphisms) of an E-group is a ring, just as the endomorphism near-ring of an abelian group is a ring. In this paper, we establish the fact that, like abelian groups, E-groups have no proper semidirect sum decompositions (i.e. a semidirect sum decomposition of an E-group must be a direct sum decomposition).

In this paper, we study the decompositions of an amply supplemented module M into a direct sum of cotype submodules. For this purpose, we introduce some conatural classes, such as the class of all q-molecular modules, the class of all q-bottomless modules, the class of all q-discrete modules and the class of all q-continuous modules, which are defined for every ring. We also find conditions on the structure of the cotype submodules of an amply supplemented module M, in order for M to have certain direct sum decompositions.

This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.

The structure of a complex Clifford algebra is studied by direct sum decompositions into eigenspaces of specific linear operators.

The possibility of obtaining certain decompositions with finite dimensional intersection, for a given two-dimensional behavior, are investigated and related to the factor skew-primeness property of suitable matrix pairs involved in the behavior description. The analysis carried out represents a fist step toward a complete generalization of the preliminary results about direct sum decompositions presented by the authors.

In the paper, the notions of characteristic set and, in particular, of characteristic cone of a two-dimensional (2-D) behavior are introduced. Autonomous behaviors are (linear shift-invariant) complete 2-D behaviors endowed with nontrivial characteristic sets. For this class of behaviors, a characterization of all characteristic cones, based on the supports of the greatest common divisors (g.c.d.'s) of the maximal order minors of any matrix involved in the behavior description, is given. Stability property of an autonomous behavior, with respect to any of its characteristic cones, is defined first for finite-dimensional behaviors and then for autonomous behaviors which are kernels of nonsingular square matrices. For both classes, stability is related to the algebraic varieties of the Laurent polynomial matrices appearing in the behavior representations. Finally, upon explicitly proving that any autonomous behavior can be expressed as the sum of a finite-dimensional behavior and of a square autonomous one, stability of general 2-D autonomous behaviors is stated and characterized.

The asymptotic stability of stable and time-autonomous discrete multidimensional behaviors

Direct-Sum Decompositions over Local Rings

The compatible Grassmannian

We have shown that a particular class of non-local free field theory has conformal symmetry in arbitrary dimensions. Using the local field theory counterpart of this class, we have found the Noether currents and Ward identities of the translation, rotation and scale symmetries. The operator product expansion of the energy-momentum tensor with quasi-primary fields is also investigated.

On the dynamic cumulative residual entropy

The recent introduction of a specific electronic readout chip, designed for the processing of the same signal pulses from an x-ray sensor in one integrating and one counting channel in each pixel E. Kraft et al., [IEEE Trans. Nucl. Sci. 54, 383-390 (2007)]; E. Kraft, "Counting and integrating microelectronics development for direct conversion X-ray imaging," Ph.D. thesis, 2007; J. Fink, "Characterization of the imaging performance of the simultaneously counting and integrating X-ray detector CIX," Ph.D. thesis, 2010, enables an alternative to conventional dual-energy imaging methods. As shown in a previous paper, the proposed alternative benefits from positively correlated noise in the raw data, inherent to the counting-integrating readout (CIX), which results in a reduction of basis image noise after dual-energy decomposition. In the present paper, the authors compare the new approach to dual-energy imaging to two conventional dual-energy techniques, the dual-kV technique and the dual-crystal technique and to a photon-counting technique based on two energy windows. The study is based on x-ray computed tomography (CT) simulations of an anthropomorphic head phantom, where the signal-to-noise ratios and the contrast-to-noise ratios in basis material images and quasi-monochromatic images at 60 keV are compared. Moreover, a simple pictorial illustration of the relevance of input noise correlations in the noise-propagation process during the dual-energy basis material decomposition are presented, as well as measurement data for the correlation obtained as a function of the x-ray flux rate. Under the idealized assumptions of the absence of scatter and detector imperfections like K-escape, crosstalk and detector noise, our comparison shows that among the dual-energy techniques investigated, the CIX concept together with the dual-kV technique performs best, confirming the mitigating effect of correlated measurement data on the dual-energy basis material decomposition. The novel concept of simultaneous counting of photons and integrating the x-ray energy flux has a large potential for dual-energy applications in both projection and tomographic x-ray imaging. Future work will have to focus on the sensitivity of the CIX dual-energy concept to scattered radiation, detector imperfections and high x-ray fluxes.

A new stochastic algorithm inspired on genetic algorithms to estimate signals with finite rate of innovation from noisy samples

We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

This work is a continuation of the paper [1]. Recently, in an attempt to give further meaning to several local theories for one dimensional singular integral operators, the authors defined a general concept of local factorization of matrix functions on a contour F in the complex plane. This notion of local factorization is a natural analogue of the well known global (Wiener-Hopf) factorization. It was established in [1] that the existence of a local factorization at each point of the contour is equivalent to the existence of a global factorization. In the present paper we obtain the equivalence of the existence of a local factorization and various notions of local invertibility of singular integral operators. This equivalence is demonstrated in the framework of scalar coefficients on the space L2(F), where F is a Liapounov contour. The paper consists of three sections. In Sect. 1 it is shown how a local factorization relative to Lp at a point on the contour can be pulled back to a global factorization. Section 2 discusses the equivalence of various notions of local invertibility of one dimensional singular integral operators. In Sect. 3 the equivalence of local factorization relative to L 2 and local invertibility is obtained. An earlier version of this work was submitted in the autumn of 1976 for publication in a special issue dedicated to I.N. Vekua. The paper was prepared in response to an official invitation from Moscow to the second author. During the past two years we did not receive any information. Recently we were told by our western colleagues that our paper did not appear in this issue (we image the reasons were non-mathematical ones). The authors were sorry to hear that Professor Vekua died recently. We would like to dedicate this work to the late Professor Vekua who had a broad influence on the development in the theory of one dimensional singular integral operators. This is an updated version of the 1976 manuscript. It takes into account subsequent developments of the local theory. See, [9] and [10].