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This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods. [ABSTRACT FROM AUTHOR]

For terminal value problems of fractional differential equations of order α ∈ (0 , 1) that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method. [ABSTRACT FROM AUTHOR]

For a linear matrix function f in X ∈ R m × n we consider inhomogeneous linear matrix equations f (X) = E for E ≠ 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution. [ABSTRACT FROM AUTHOR]

For general complex or real 1-parameter matrix flows A(t)n, n and for static matrices A ∈ ℂ n , n alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Aℓ(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Aℓ) ⋅ C with each diagonal block Ak(t) or Ak square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed 'eigenvectors' of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting 'divide and conquer' methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms. [ABSTRACT FROM AUTHOR]

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula. [ABSTRACT FROM AUTHOR]

A Correction to this paper has been published: https://doi.org/10.1007/s11075-021-01216-4 [ABSTRACT FROM AUTHOR]

Alkali metals can react explosively with water and it is textbook knowledge that this vigorous behaviour results from heat release, steam formation and ignition of the hydrogen gas that is produced. Here we suggest that the initial process enabling the alkali metal explosion in water is, however, of a completely different nature. High-speed camera imaging of liquid drops of a sodium/potassium alloy in water reveals submillisecond formation of metal spikes that protrude from the surface of the drop. Molecular dynamics simulations demonstrate that on immersion in water there is an almost immediate release of electrons from the metal surface. The system thus quickly reaches the Rayleigh instability limit, which leads to a 'coulomb explosion' of the alkali metal drop. Consequently, a new metal surface in contact with water is formed, which explains why the reaction does not become self-quenched by its products, but can rather lead to explosive behaviour. [ABSTRACT FROM AUTHOR]

It is generally assumed that the hydrated electron occupies a quasi-spherical cavity surrounded by only a few water molecules in its equilibrated state. However, in the very moment of its generation, before water has had time to respond to the extra charge, it is expected to be significantly larger in size. According to a particle-in-a-box picture, the frequency of its absorption spectrum is a sensitive measure of the initial size of the electronic wavefunction. Here, using transient terahertz spectroscopy, we show that the excess electron initially absorbs in the far-infrared at a frequency for which accompanying ab initio molecular dynamics simulations estimate an initial delocalization length of ≈40 Å. The electron subsequently shrinks due to solvation and thereby leaves the terahertz observation window very quickly, within ≈200 fs. [ABSTRACT FROM AUTHOR]

Focuses on the author's response to comments made by mathematician Frank Uhlig on how to teach a first course on linear algebra published in the 50th volume of the journal "Educational Studies in Mathematics."

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of `mathematical proof'. This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary (DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early `salient point' proofs lead to an intuitive understanding of `math proof'. Once the basic concept of `proof' is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a `math proof' becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning. [ABSTRACT FROM AUTHOR]

Hydrido substituted stannasilanes of the type or ( Z = H, Me, Ph; R, R′ = alkyl, Ph) are accessible by reaction of either alkali metal stannides ( MSn( Z) R2; M = Li, Na) with halogen substituted silanes ( ; X = F, Cl) or chlorostannanes ( R2SnCl2, Ph3SnCl) and fluorosilanes in the presence of magnesium. Stannasilanes with halogen substituents at the silicon as well as the tin atom are formed by treatment of the hydrido substituted stannasilanes with CHCl3 or CCl4. The hydrido substituted stannasilanes decompose in contact with air to distannanes and siloxanes or to the linear ( tBu2Sn(–O– tBu2Si–OH)2) and cyclic ((– tBu2Sn–O– iPr2Si–O–)2) stannasiloxanes. [ABSTRACT FROM AUTHOR]

We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on $D$ –orthogonal reduction techniques where $D= \mbox{ diag}(\pm1)$ is computed from the given matrix $A$ . It operates on a similarity reduction of a real matrix $A_{nn}$ to general tridiagonal form $T$ and computes all of $A$ 's eigenvalues in $\frac{4}{3} n^3 + O(n^2)$ operations, where the $O(n^2)$ part of the operations is possibly performed over ${\Bbb C}$ , instead of the 7–8 $n^3$ real flops required by the $QR$ eigenvalue algorithm. Potential breakdo wn of the $DQR$ algorithm can occur in the reduction to tridiagonal form and in the $D$ –orthogonal $DQR$ reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions $n \leq 300$ and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the $DQR$ steps. We prove an implicit $DQ$ theorem which allows implicit shifts, give a convergence proof for the $DQR$ algorithm and show that $DQR$ is conditionally stable for general balanced tridiagonal matrices $T_{\rm b}$ . [ABSTRACT FROM AUTHOR]

We show that the coefficients of a polynomial can be rather illconditioned functions of its given roots. In certain cases, however, the coefficients are well conditioned, e.g. if the roots lie symmetrically in a closed angular sector for an angle less than π centered at the origin of the complex planeℂ. We introduce and study condition numbers B for the sensitivity of the polynomial coefficient a to changes in the root x and develop an algorithm to compute the condition numbers and to decide whether a given set of roots will generate the polynomial coefficients in an ill way. This algorithm is an essential preliminary for certain numerical pole placement algorithms in linear control theory. [ABSTRACT FROM AUTHOR]