Winding numbers are fundamental objects arising in algebraic topology, with many applications in non-constructive complex analysis. We present a formalization in Coq of the winding numbers and their main properties. As an application of this development, we also give non-constructive proofs of the following theorems: the Fundamental Theorem of Algebra, the 2-dimensional Brouwer Fixed-Point theo...

For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrixvalued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made. AMS Sub...

This paper presents an improved model of uniform transformer winding for frequency response analysis (FRA), based on traveling wave theory. The accurate representation of the losses and detailed consideration of measurement chains are the main features of the model. The transfer function expression of the transformer winding and equivalent distributed-parameter circuit for FRA have been derived...

The topological constraints on Brownian walks are a long standing subject of interest in mathematics and polymer physics. A particularly fruitful example is the study of the winding angle θ, the continuous angle swept by the movement around a set of points (curves) in two (three) dimensions . In a plane, the probability distribution at large time or length l for the winding angle of a Brownian ...

In the last paper, we proposed a winding and reaching control technique using a physical index of horizontal constraint force for a 3D snake-like robot. Using a snake like robot called SMA, the validity of the methods was shown experimentally. In this paper control methods in the last paper are summarized and a control method of an active impedance control of vertical joints is proposed for win...

In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are inte...

We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the ...

Winding numbers are fundamental objects arising in algebraic topology, with many applications in non-constructive complex analysis. We present a formalization in Coq of the winding numbers and their main properties. As an application of this development, we also give non-constructive proofs of the following theorems: the Fundamental Theorem of Algebra, the 2-dimensional Brouwer Fixed-Point theo...