On the poles of regular differentials of singular curves

We describe the pole behaviour of the regular differentials of projective algebraic curves in terms of discrete invariants of the singular points.     

Categorical Abstract Algebraic Logic: Algebraic Semantics for (\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf{\pi }$\end{document})‐Institutions

Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π‐institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N‐rule based extension of an N‐rule based π‐institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok‐Rebagliato‐style characterization of those π‐institutions with a mono‐unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π‐institution to possess an algebraic semantics is provided.     

The matrix sign function and computations in systems

The matrix sign function has several interesting properties which form the basis of new solution algorithms for problems which occur frequently in systems and control theory applications. Presented in this paper are new algorithms, based on the matrix sign function, for the solution of algebraic matrix Riccati equations, Lyapunov equations, coupled Riccati equations, spectral factorization, matrix square roots, pole assignment, and the algebraic eigenvalue-eigenvector problem. Examples of the application of each algorithm are also presented.     

On many‐sorted algebraic closure operators

A theorem of Birkhoff‐Frink asserts that every algebraic closure operator on an ordinary set arises, from some algebraic structure on the set, as the corresponding generated subalgebra operator. However, for many‐sorted sets, i.e., indexed families of sets, such a theorem is not longer true without qualification. We characterize the corresponding many‐sorted closure operators as precisely the uniform algebraic operators. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pole Assignment in Hybrid Differential-Difference Systems

We present a general approach to the pole assignment problem for linear stationary hybrid differential-difference systems as a coefficient control problem for their characteristic equations. Various scales (classes) of linear feedback controllers are considered. Special attention is paid to the solvability of the pole assignment problem for such systems in the scale of general differential-difference controllers and in a general scale that, along with differential-difference controllers, contains integral controllers whose kernels are compactly supported functions. A general scheme for constructing such controllers is proposed based on the algebraic properties of the shift operator, the Paley–Wiener theorem on compactly supported functions, and the methods of interpolation theory in the class of entire functions of exponential type. Examples and counterexamples illustrating the results are given.     

Peripheral spectrum of positive quasi-compact operators onC 0(X)

LetX be a locally compact space, andT, a quasi-compact positive operator onC ₀(X), with positive spectral radius,r. Then the peripheral spectrum ofT is a finite set of poles containingr, and the residue of the resolvent ofT at each peripheral pole is of finite rank. Using the concept of closed absorbing set, we develop an iterative process that gives the order,p, ofr, some special bases of the algebraic eigenspaces ker(T-r) ^p and ker(T ^*-r) ^p , and finally the dimension of the algebraic eigenspace associated to each peripheral pole.     

Is There Something to be Gained from Guessing? Middle School Students' Use of Systematic Guess and Check

This article will share results from research that investigated how sixth‐, seventh‐, and eighth‐grade students who had not been exposed to formal algebraic methods approached word problems of an algebraic nature. Student use of systematic guess and check, the predominate approach taken by these students, is the focus. The goal is to consider the students' use of systematic guess and check reasoning in terms of the broadening perspective of algebra and algebraic thinking by highlighting ways in which this reasoning can provide a basis for developing some of the thinking patterns and discourse of formal algebra. Two perspectives will be highlighted: relationships among quantities and function‐based reasoning.     

An Algebraic Method for Pole Placement in Multivariable Systems

This paper considers the pole placement in multivariable systems involving known delays by using dynamic controllers subject to multirate sampling. The controller parameterizations are calculated from algebraic equations which are solved by using the Kronecker product of matrices. It is pointed out that the sampling periods can be selected in a convenient way for the solvability of such equations under rather weak conditions provided that the continuous plant is spectrally controllable. Some overview about the use of nonuniform sampling is also given in order to improve the system's performance.     

On the Algebraic Properties of Errors

In numerical analysis (absolute) errors can be identified with one‐dimensional intervals symmetric with respect to zero. Addition, multiplication and inclusion of errors are well‐defined (set‐theoretically) in interval analysis. We study axiomatically the algebraic properties of such a system of errors. To this end we introduce and investigate a new abstract algebraic structure called e‐ring.     

Categorical abstract algebraic logic: The Diagram and the Reduction Operator Lemmas

The study of structure systems, an abstraction of the concept of first‐order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first‐order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality‐free first‐order structures are provided in the framework of structure systems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Gauss map of pseudo‐algebraic minimal surfaces in R4

In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces in Euclidean four‐space and give a kind of unicity theorem (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Algebraic Method for Pole Placement in Multivariable Systems

This paper considers the pole placement in multivariable systems involving known delays by using dynamic controllers subject to multirate sampling. The controller parameterizations are calculated from algebraic equations which are solved by using the Kronecker product of matrices. It is pointed out that the sampling periods can be selected in a convenient way for the solvability of such equations under rather weak conditions provided that the continuous plant is spectrally controllable. Some overview about the use of nonuniform sampling is also given in order to improve the system's performance.     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Ion acoustic solitary wave solutions of two‐dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation in quantum plasma

Propagation of two‐dimensional nonlinear ion‐acoustic solitary waves and shocks in a dissipative quantum plasma is analyzed. By applying the reductive perturbation theory, the two‐dimensional ion acoustic solitary waves in a dissipative quantum plasma lead to a nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation. By implementing extended direct algebraic mapping, extended sech‐tanh, and extended direct algebraic sech methods, the ion solitary traveling wave solutions of the two‐dimensional nonlinear KPB equation are investigated. An analytical as well as numerical solution of the two‐dimensional nonlinear KPB equation is obtained and analyzed with the effects of external electric field and ion pressure. Copyright © 2016 John Wiley & Sons, Ltd.     

The tractability index of memristive circuits: branch‐oriented and tree‐based models

The memory‐resistor or memristor is a new electrical element characterized by a nonlinear charge‐flux relation. This device poses many challenging problems, in particular from the circuit modeling point of view. In this paper, we address the index analysis of certain differential‐algebraic models of memristive circuits; specifically, our attention is focused on so‐called branch‐oriented models, which include in particular tree‐based formulations of the circuit equations. Our approach combines results coming from differential‐algebraic equation theory, matrix analysis and theory of digraphs. This framework should be useful in future studies of dynamical aspects of memristive circuits. Copyright © 2012 John Wiley & Sons, Ltd.     

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

Output tracking control of Boolean control networks with impulsive effects

This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state‐space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time‐varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.     

Regulous vector bundles

Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified‐algebraic vector bundles. Furthermore, using new results on curve‐rational functions, we characterize regulous vector bundles among families of vector spaces parametrized by an affine regulous variety. We also study relationships between regulous and topological vector bundles.     

Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, we analyse some algebraic formulations for the approximation of the smallest non‐zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift‐and‐Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings. Copyright © 2002 John Wiley & Sons, Ltd.