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 matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

This paper presents the design and analysis of fractional slot multilayer interior permanent magnet synchronous machines (PMSMs) with concentrated winding for electric vehicles applications. The major advantages of concentrated winding will be analyzed first. The significant design consideration of multilayer interior permanent magnet machines will be illustrated, as well. Finally, the advanced...

A new integrated magnetic full wave dc/dc converter1 that provides flexible transformer design by incorporating an independent output inductor winding is introduced. The transformer is implemented on a traditional three-leg magnetic core. The inductor winding can be separately designed to control the output current ripple. The cross-sectional area of the inductor core leg can be reduced dramati...

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure mo...

We show that, in the context of dilaton gravity, a recently proposed democratic principle for intersection possibilities of branes winding around extra dimensions yield stabilization, even with the inclusion of momentum modes of the wrapped branes on top of the winding modes. The constraints for stabilization massaged by string theory inputs forces the number of observed dimensions to be three....

Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings ofM with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in M . In particular, we prove a Gaussian type central limit theorem showing t...

Contents Preface v Chapter 1. From i to z: the basics of complex analysis 1 1. The field of complex numbers 1 2. Differentiability and conformality 3 3. Möbius transforms 7 4. Integration 12 5. Harmonic functions 19 6. The winding number 21 7. Problems 24 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis 27 1. The winding number version of Cauchy's th...

Smart grid technology is a new approach for promoting the energy efficiency and for accomplishing the environment objectives. In this context, the lifetime of transformers have a great impact on the quality of the delivered energy. The paper deals with a mathematical statistic method for estimation of winding insulation lifetime of the power transformers, based on Rayleigh distribution. The con...

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...