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In this paper, a new and different neural network, called Zhang Neural Network (ZNN) is appropriated from discrete time-varying matrix problems and applied to the angle parameter-varying matrix field of values (FoV) problem. This problem acts as a test bed for newly discovered convergent 1-step ahead finite difference formulas of high truncation error orders. The ZNN method that uses a look-ahead finite difference scheme of error order 6 gives us 15+ accurate digits of the FoV boundary points in record time when applied to hermitian matrix flows A (t). [ABSTRACT FROM AUTHOR]

Zhang Neural Networks rely on convergent 1-step ahead finite difference formulas of which very few are known. Those which are known have been constructed in ad-hoc ways and suffer from low truncation error orders. This paper develops a constructive method to find convergent look-ahead finite difference schemes of higher truncation error orders. The method consists of seeding the free variables of a linear system comprised of Taylor expansion coefficients followed by a minimization algorithm for the maximal magnitude root of the formula's characteristic polynomial. This helps us find new convergent 1-step ahead finite difference formulas of any truncation error order. Once a polynomial has been found with roots inside the complex unit circle and no repeated roots on it, the associated look-ahead ZNN discretization formula is convergent and can be used for solving any discretized ZNN based model. Our method recreates and validates the few known convergent formulas, all of which have truncation error orders at most 4. It also creates new convergent 1-step ahead difference formulas with truncation error orders 5 through 8. [ABSTRACT FROM AUTHOR]

We revisit an earlier idea of Marcus and Pesce from 1987 for generating fields of values ofbymatricesvia 2 by 2 matrix compressions and their easily constructed elliptical field of values. This approach is used to reduce the cost of finding the FOV boundary curve of a matrixand makes it more accurate at the same time. The new algorithm succeeds by using fewer eigenanalyses and constructing 2 by 2 matrix compression ellipses offor approximatinginstead of only eigenanalyses. An application to verify a counterexample to theZemanek Conjecture for companion matrices of monic polynomials and their normalized derivatives inand, respectively, in dimension 4 is given. [ABSTRACT FROM PUBLISHER]

By a theorem of Frobenius (F.G. Frobenius,Über die mit einer Matrix vertauschbaren Matrizen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1910), pp. 3–15 (also in Gesammelte Abhandlungen, Band 3, Springer 1968. pp. 415–427)), every matrixAn,nover any field 𝔽 is the product of two symmetric ones. Using the algorithm of Huang and Nong (J. Huang and L. Nong,An iterative algorithm for solving finite-dimensional linear operator equations T(x) = f with applications, Linear Algebra Appl. 432 (2010), pp. 1176–1188) for linear systems, we develop an algorithm to compute a symmetric matrixS = ST ∈ 𝔽n,nfor whichSAis symmetric for any given square matrixA ∈ 𝔽n,nwhere 𝔽 = ℝ or ℂ. The algorithm is implemented and tested in MATLAB. [ABSTRACT FROM AUTHOR]

Examines the false mathematical statement familiar to teachers and students. Ways to interpret complex numbers; Use of standard operations for matrix addition and multiplication matrices.

The reaction of phenylphosphine (PhPH2) with ketones showed conversion into a 5-membered heterocyclic ring of the type P2SOC if 2–2.5 equivalents of sulfur (S8) were used. X-Ray data and31P,13C,1H NMR spectra proved the formation of the (Z) diastereomer of 1,3,2,4-oxathiadiphospholane1((PhPS)2SOCMe2) if acetone is used. Mechanistic studies displayed that small changes in the reaction pathway lead toward the formation and characterization of intermediates including 1,3,2,4,5-dithiatriphospholane, 2,4,5-triphenyl-2-sulfide (PhP)3S3and 1,3,2,4-oxathiadiphospholane, and 5,5-dimethyl-2,4-diphenyl-2-sulfide ((PhPS)(PhP)SOCMe2), which undergo reactions with acetone, subsequently yielding the abovementioned heterocyclic ring. [ABSTRACT FROM PUBLISHER]

The reaction of dialkyltinhydrides with dialkylzinc results in a tin metal bond formation. Crystal structures of 1,1,1,2,3,3,3-hepta-phenyl-2-(ethylzincio)tristannane *TMEDA 1, 1,1,2,2,3,3,4,4-octaphenyl-1,4-(ethylzincio)tetrastannane*2TMEDA 2, and triphenyl(phenylzincio)stannane*TMEDA 3 could be obtained. Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file. GRAPHICAL ABSTRACT[image omitted] [ABSTRACT FROM AUTHOR]

A series of novel and previously published silicon dihydrides, R2SiH2(R = benzyl, fluorenyl,p-tolyl,p-n-butyl,p-biphenyl,m-tolyl,o-tolyl, 2,4-xylyl, 2,6-xylyl,p-biphenyl, 1-naphthyl, 9-anthracenyl), were synthesized and characterized using1H,13C, and29Si NMR spectroscopy. Effects of substituent bulk on29Si NMR shifts are compared between reported silicon dihydrides species and novel dihydrides herein. In addition, single crystal X-ray diffraction was employed to elucidate the molecular structure of all solid silicon dihydrides and the at room temperature liquid p-tolyl2SiH2. These compounds display non-covalent intermolecular interactions in the form of edge to face, π−π stacking, and CH3···π interactions. The dependency of substituent type on the nature of interactions present will be highlighted and discussed. [ABSTRACT FROM AUTHOR]

A series of novel and previously published silicon dichlorides, R2SiCl2(R = benzyl, fluorenyl, p-tolyl p-n-butyl, p-biphenyl, m-tolyl, o-tolyl, 2,4-xylyl, 2,6-xylyl, p-biphenyl, 1-naphthyl, 9-anthracenyl), were synthesized and characterized using1H-,13C- and29Si NMR spectroscopy. Effects of substituent bulk on both experimental and calculated29Si NMR shifts are compared between reported silicon dichlorides species and novel dichlorides herein. In addition, single crystal X-ray diffraction was employed to elucidate the molecular structure of all solid diorgano silicon dichlorides. These compounds display non-covalent intermolecular interactions in the form of edge to face, π–π stacking, and CH3···π interactions. Dependency of substituent type on the nature of interactions present will be highlighted and discussed. [ABSTRACT FROM PUBLISHER]

The reaction of 1,1,2,2,3,3,4-hepta-tbutyl-1-(chloromagnesio)tetrastannacyclobutane with 1,2-halogenopropane leading to halogeno substituted four membered tin ring systems is presented. In addition, the crystal structures of the chloro and the bromo substituted tetrastannacyclobutanes will be discussed. Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file. GRAPHICAL ABSTRACT [image omitted] [ABSTRACT FROM AUTHOR]

The synthesis of the cyclic tin-modified silanes 1-6 are described by the reaction of t Bu 2 SnCl 2 with different dichloromono- and -disilanes in the presence of magnesium. Reacting dimethyldichlorosilane with diphenyldichlorosilane and magnesium yielded the 1,1-dimethylhexaphenylcyclotetrasilane 8 in reasonably good yields. [ABSTRACT FROM AUTHOR]

The deoxygenation of phosphine oxides was studied on a series of model reactions using an aralkylsilane, arylsilanes, diarylsilanes, a disiloxane, a polysiloxane, and a disilane under solvent-free, thermal, or microwave-assisted conditions. The deoxygenations were optimized, and the reactivity of the silanes was evaluated. User-friendly accomplishments were elaborated. [ABSTRACT FROM PUBLISHER]