Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL

The argument principle is one of the basic theorems from the classical complex analysis and it's crucial role in complex analysis is very nicely underlined in the book of A. F. Beardon Complex Analysis – the Argument Principle in Analysis and Topology, John Wiley and Sons, 1979. We shall generalize a result of T. Radó and P.V. Reichelderfer from Continuous Transformations in Analysis, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955, p. 399, and a recent paper of P. Duren, w. Hengartner and R. S. LaugesenAmer. Math. Monthly 103(5) (1996), 411-415, showing that the argument principle is an entirely topological theorem (because it holds for continuous, open, discrete mappings in the plane) and hence this theorem belongs more to topology than to complex analysis. We also present as an application a generalization of Rouché's theorem.

Equivalence transformations are widely used in practical designs of VLSI circuits using Ruby. This paper demonstrates how proofs of these equivalences easily may be performed within a formal framework by a theorem prover. The proof tool used is called RubyZF and contains a semantical embedding of Ruby within Zermelo-Fraenkel set theory using the Isabelle theorem prover. The use of the system is exemplified by a concrete example taken from the T-Ruby design system.

Manual placement of components is often used in FPGA circuit design in order to achieve better results than would be generated by automatic place and route algorithms. However, explicit placement of basic elements in parametrized hardware descriptions is tedious and error-prone. We describe a framework for the description and verification of parametrized hardware libraries with layout information, supporting both placing components with explicit symbolic coordinates and `neighboring' placement directives such as A beside B. The correctness of generated layouts is established by proof in higher-order logic, automated by using the Isabelle theorem prover. We have developed an extensive library of theorems describing properties of layouts that are combined by our compiler and the theorem prover to achieve a high level of automation in the verification of complete circuit layouts, making formal verification of circuit layouts practical with minimal user effort. Our system has been used to verify layout descriptions for a range of circuits that have been mapped to Xilinx FPGAs.

A novel systematic approach to the design of directivity patterns of higher order differential microphones is proposed. The directivity patterns are obtained by optimizing a cost function which is a convex combination of a front-back energy ratio and uniformity within a frontal sector of interest. Most of the standard directivity patterns—omnidirectional, cardioid, subcardioid, hypercardioid, supercardioid—are particular solutions of this optimization problem with specific values of two free parameters: the angular width of the frontal sector and the convex combination factor. More general solutions of practical use are obtained by varying these two parameters. Many of these optimal directivity patterns are trigonometric polynomials with complex roots. A new differential array structure that enables the implementation of general higher order directivity patterns, with complex or real roots, is then proposed. The effectiveness of the proposed design framework and the implementation structure are illustrated by design examples, simulations, and measurements.

From classical complex analysis it is well-known that the index of the Riemann-Hilbert problem is connected with the winding number. How to generalize the winding number to a rotational number or rotation is explained in [nf] and also in [krza]. How to get a connection between the index and the rotation can be found in [dei]. For matrices of singular integral operators a rotation can be defined which is connected with the index of this system of singular integral operators. Because Clifford analysis is a higher dimensional analog to complex analysis there should be some analogous behavior of singular integrals. In fact, we will show that the index of singular integral operators with paravector-valued symbols in Clifford algebras can be connected with the so-called wrapping number which is the higher dimensional analog to the winding number. The proof is rather elementary.

We review the different techniques known for doing exact computations on polynomial systems. Some are based on the use of Grobner bases and linear algebra, others on the more classical resultants and its modern counterparts. Many theoretical examples of the use of these techniques are given. Furthermore, a full set of examples of applications in the domain of artificial vision, where many constraints boil down to polynomial systems, are presented. Emphasis is also put on very recent methods for determining the number of (isolated) real and complex roots of such systems.

A new method for mode determination of multilayer planar waveguides with complex-valued dielectric permeability is proposed. The method is based on the combination of an argument principle in complex analysis and a quasi-Newton method. The argument principle allows reliable determination of the number of modes within any finite region of the complex plane representing the propagation constants. This makes it possible to locate the propagation constants to the precision sufficient for the usage of the quasi-Newton method. The method involves 'tracking of the phase' of a complex function instead of conventional integration along a contour.

On using high-order polynomial curve fits in the quasi-steady theory for square-cylinder galloping

The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal capstone course in mathematics.

The aim of the study is to determine the optimal geometry and magnitude of the applied load to ensure safety and prevent delamination in a three-layered nanocomposite structure under axial mechanical loading. The structure consists of a layer of the nanomaterial tungsten disulfide (WS2) and a substrate layer of poly(methyl methacrylate) (PMMA), which are adhesively bonded using SU-8 glue. To achieve this aim, a multi-parameter optimization problem (MOP) is formulated. It includes a two-dimensional stress function model that describes stress transfer in the considered three-layer structure. Two types of analytical solutions for the interface shear stress (ISS) are derived, featuring real and complex roots. The decision variables in MOP include the external load, layer thicknesses, and structure length. The optimization criterion is defined as the minimization of the difference between the model ISS and ultimate shear stress (USS) in the adhesive layer to assure no delamination occurs in the nanocomposite structure. A genetic algorithm and alternative optimization approach developed within the framework of “Mathematica” are implemented for the optimization of both model solutions. As a result, optimal values for the given external load, layer thicknesses, and structure length are obtained for considered nanostructure. For the case of an ISS model solution with real roots, the optimization procedures ensure optimal geometries that physically correspond to thinner structural layers, but they are limited at lower possible loads. In contrast, for the case of a model solution for the ISS with complex roots, solutions for the optimal geometries of the nanostructures were obtained with thicker layers requiring higher loads, than thinner ones, but delamination does not occur.

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

A winding number algorithm for closed polygonal paths (not necessarily simple) is derived using classical complex analysis results and techniques.The algorithm is designed specifically to handle large cases efficiently.The performance of a computer program based on the algorithm is discussed and compared with the performance of a computer program which obtains the winding number directly by antidifferentiation. Introduction.The algorithm does not involve the division operation, inverse trigonometric functions, or integral approximation techniques, making it quite suitable for computer programs which must process any combination of many polygonal paths, polygonal paths with many sides, and/or compute many winding numbers.In addition, if all the complex numbers in a given application are Gaussian integers, then a computer program based on the algorithm can be written completely in fixed point mode.The algorithm has proven to be computationally efficient.Results of efficiency tests of a FORTRAN program based on the algorithm are given in the final section.To avoid ambiguity, we define briefly the mathematical terminology which is used.By a curve, we mean a continuous function C from a closed real interval [a, b] (called the parameter interval) into the complex plane.C(a) is called the initial point, C() the terminal point of C. The inverse C of C is given by C(/) = C(a + b -t), a ^ t ^ b.C is closed provided C(a) = C().C* denotes the range of C; i.e., C* = {C(/) | a ; t ^ b}.A path is a piecewise continuously differentiable curve.If C is a path and o ( C*, then the winding number of z0, Wciz0), with respect to C is given by 2xiWciz0) = c(z -Zp)~l dz.Given complex numbers z and w, the directed line from z to w is defined by [z, w] = {(1 -t)z + tw | 0 ^ ^ 1}.The distinction between a closed real interval and a directed line in the complex plane is always clear from context.A path P with parameter interval [a, b] is called a polygonal path provided there exists a subdivision a = U < < tN = b of [a, b] such that P([t"-U tn]) = [Pitn-i), P(Q], for each n = 2, 3, , N. The turn-points of P are Pitn), n = 1, 2, , N. For convenience of notation, we identify a polygonal path by its turn-points; P : pn = xn + iyn, n = 1,2, , N, where pn = P{tn), for each n.

This paper concerns the location of roots of polynomials with respect to the principal coordinate axes (referred to generally as the Routh–Hurwitz criteria and the Sturm criteria). It provides a new proof of the results established by Wall and Frank which relate the number of positive and negative coefficients in a continued fraction to the number of roots of a polynomial in the left half plane and right half plane. It extends these results to locate the number of real roots of real polynomials on the positive or negative real axis. The paper also provides some miscellaneous criteria so that selected parts of a polynomial may be analyzed in order to determine the number of roots in a half plane.

In study of the structure of shock waves in bubbly liquids we need to find integral curve connecting two singular points of the system of differential equations. We linearized the system of basic equations around equilibrium state. The condition of existence of nontrivial solution of the linear equations leads to the characteristic transcendental equation with respect to exponential index. The structure of roots of such equation is studied by using argument principle [1]. It is shown that for compression waves such equation has one real root.

Program index to volume 29