Exploring Visualization Methods for Complex Variables

Applications of complex variables and related manifolds appear throughout mathematics and science. Here we review a family of basic methods for applying visualization concepts to the study of complex variables and the properties of specific complex manifolds. We begin with an outline of the methods we can employ to directly visualize poles and branch cuts as complex functions of one complex variable. $CP^2$ polynomial methods and their higher analogs can then be exploited to produce visualizations of Calabi-Yau spaces such as those modeling the hypothesized hidden dimensions of string theory. Finally, we show how the study of N-boson scattering in dual model/string theory leads to novel cross-ratio-space methods for the treatment of analysis in two or more complex variables

Poincaré and the idea of a group

In many different fields of mathematics and physics Poincaré found many uses for the idea of a group, but not for group theory. He used the idea in his work on automorphic functions, in number theory, in his epistemology, Lie theory (on the so-called Campbell–Baker–Hausdorff and Poincaré–Birkhoff–Witt theorems), in physics (where he introduced the Lorentz group), in his study of the domains of complex functions of several variables, and in his pioneering study of 3-manifolds. However, as a general rule, he seldom appealed to deep results in group theory, and developed no more structural analysis of any group than was necessary to solve a problem. It was usually enough for him that there is a group, or that there are different groups

The development and use of variables in mathematics and computer science

There are a wide variety of uses of variables in mathematics which we cope with in practice through conventions and tacit assumptions. Experience with computers has made us articulate, criticise and develop these assumptions much more carefully. Historically the term 'variable quantity' was introduced in the context of describing and calculating changing quantities which corresponded to phenomena in the observable world (e.g. the velocity of fluxion of a body moving under the inverse square law). The evolution of the concept has divorced it from these routes of reference and required us to establish the formal apparatus of interpretation and valuation. While the changes considered are highly structured this may be satisfactory, but computing power invites us to cope with change in vastly more complex, unstructured situations such as in simulation of 'real world' processes. We relate this challenge to the distinctive differences in the use of variables in mathematics and practical computing, and we develop a general framework in which all uses of variables can be described in a unified way

Statistics for traces of cyclic trigonal curves over finite fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.Comment: 30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notice

A Test for the Stability of Networks

A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].Rowinska-Schwarzweller Agnieszka - Chair of Display Technology University of Stuttgart Allmandring 3b, 70596 Stuttgart, GermanySchwarzweller Christoph - Institute of Computer Science University of Gdansk Wita Stwosza 57, 80-952 Gdansk, PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.Christoph Schwarzweller and Agnieszka Rowinska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006. doi:10.2478/v10037-006-0017-9.Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Rolf Unbehauen. Netzwerk- und Filtersynthese: Grundlagen und Anwendungen. Oldenbourg-Verlag, fourth edition, 1993.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.Okazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.Andrzej Nędzusiak. s-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Hiroyuki Okazaki and Yasunari Shidama. Probability on finite set and real-valued random variables. Formalized Mathematics, 17(2):129-136, 2009, doi: 10.2478/v10037-009-0014-x.Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555-561, 1990.Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel's inequality. Formalized Mathematics, 11(2):169-173, 2003.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. The relevance of measure and probability, and definition of completeness of probability. Formalized Mathematics, 14(4):225-229, 2006, doi:10.2478/v10037-006-0026-8