arXiv subjects
Publications and source records attributed to Eberhard H. -A. Gerbracht.
In this paper, by way of three examples - a fourth order low pass active RC filter, a rudimentary BJT amplifier, and an LC ladder - we show, how the algebraic capabilities of modern computer algebra systems can, or in the last example, might be brought to use in the task of designing analog circuits.
In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the point-line incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the vertices. These together with the defining algebraic equations suffice to calculate arbitrarily exact approximations for the exact embeddings, and so to show that the Heawood graph indeed is a unit-distance graph. Thus we refute a "suspicion" of V. Chvatal from 1972.
In this paper we show how to extend the known algorithm of nodal analysis in such a way that, in the case of circuits without nullors and controlled sources (but allowing for both, independent current and voltage sources), the system of nodal equations describing the circuit is partitioned into one part, where the nodal variables are explicitly given as linear combinations of the voltage sources and the voltages of certain reference nodes, and another, which contains the node variables of these reference nodes only and which moreover can be read off directly from the given circuit. Neither do we need preparational graph transformations, nor do we need to introduce additional current variables (as in MNA). Thus this algorithm is more accessible to students, and consequently more suitable for classroom presentations.
In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit. In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on differential algebra.
Two letters from William Burnside have recently been found in the Nachlass of Robert Fricke that contain instances of Burnside's Problem prior to its first publication. We present these letters as a whole to the public for the first time. We draw a picture of these two mathematicians and describe their activities leading to their correspondence. We thus gain an insight into their respective motivations, reactions, and attitudes, which may sharpen the current understanding of professional and social interactions of the mathematical community at the turn of the 20th century.
Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.
The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furthermore some algebraic properties of these polynomials, and consequences to the structure of the Harborth graph are determined.