Related papers: One cannot hear the winding number

Let $K$ be the graph on vertices $\{1, 2, 3, 4, 5\}$, and having all edges except $(4, 5)$. A continuous map $f:K\to \R^2$ is called an \emph{almost embedding} if $f$-images of non-adjacent edges are disjoint. Take the winding numbers of…

Let $K_4$ be the complete graph on four vertices. Let $f$ be a continuous map of $K_4$ to the plane such that $f$-images of non-adjacent edges are disjoint. For any vertex $v \in K_4$ take the winding number of the $f$-image of the cycle…

This is an expository talk on a topic of classical analysis, arising from the VMO theory of the topological degree due to Br\'ezis and Nirenberg (1995). We sketch the history of the subject and some of its recent developments. The paper is…

We show that the spectrum of the Schrodinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel…

We introduce Feynman-like rules to compute quivers for two loops and higher for the coloured planar $\phi^3$ theory for winding number zero. We demonstrate this for a few cases.

We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm…

Injectivity of the continuous wavelet transform acting on a square integrable signal is proved under weak conditions on the Fourier transform of the wavelet, namely that it is nonzero somewhere in almost every direction. For a bounded…

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces $B^s_p(\mathbb{R}^n) = B^s_{p,p}(\mathbb{R}^n)$, $1\le p \le \infty$, and between Sobolev spaces…

The aim of this paper is to show that, in various situations, the only continuous linear map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups $\Z/nZ$, the integers $\Z$,…

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more…

The main result of this paper is that one cannot hear orientability of a surface with boundary. More precisely, we construct two isospectral flat surfaces with boundary with the same Neumann spectrum, one orientable, the other…

For every $g\ge 2$ and $n\ge4$, we provide an $n-$manifold $M$ and a continuous $2-$sided map $f\colon S\longrightarrow M$, where $S$ is a closed genus $g$ surface, such that no simple loop is contained in $\text{ker}(\,f_*\,)$. This…

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for…

The discrete Fourier transform and the FFT algorithm are extended from the circle to continuous graphs with equal edge lengths.

Gravitational-wave memory is characterized by a signal component that persists after a transient signal has decayed. Treating such signals in the frequency domain is non-trivial, since discrete Fourier transforms assume periodic signals on…

Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD…

This paper describes a method of calculating the transforms, currently obtained via Fourier and reverse Fourier transforms. The method allows calculating efficiently the transforms of a signal having an arbitrary dimension of the digital…

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…

A Brownian loop is a random walk circuit of infinitely many, suitably infinitesimal, steps. In a plane such a loop may or may not enclose a marked point, the origin, say. If it does so it may wind arbitrarily many times, positive or…

The Fourier transform of the indicator function of arbitrary polygons and polyhedra is computed for complex wavevectors. Using the divergence theorem and Stokes' theorem, closed expressions are obtained. Apparent singularities, all…