Related papers: Three Value Ranges for Symmetric Self-mappings

For $\alpha\ge 0$, let $\mathcal{W}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. This article…

It is shown that smooth maps $f: S^3 \rightarrow S^3$ contain two countable families of harmonic representatives in the homotopy classes of degree zero and one.

Let $w$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $f: \mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give several characterizations…

Matrix-valued covariance functions are crucial to geostatistical modeling of multivariate spatial data. The classical assumption of symmetry of a multivariate covariance function is overlay restrictive and has been considered as unrealistic…

Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article,…

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant…

The joint numerical range $W(F)$ of three hermitian $3$-by-$3$ matrices $F=(F_1,F_2,F_3)$ is a convex and compact subset in $\mathbb{R}^3$. Generically we find that $W(F)$ is a three-dimensional oval. Assuming $\dim(W(F))=3$, every one- or…

Let $F$ be a planar, $3$-connected graph of radius one on $p$ vertices, with $a$ vertices of degree three. We characterise all unigraphic degree sequences for such graphs, when $a\geq 3$ and $p$ is large enough with respect to $a$. This…

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the…

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the…

Let $(M,g)$ be a $n-$dimensional, compact Riemannian manifold. We define the frequency scale $\lambda$ of a function $f \in C^{0}(M)$ as the largest number such that $\left\langle f, \phi_k \right\rangle =0$ for all Laplacian eigenfunctions…

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow…

If $\mathcal{G}$ is the group (under composition) of diffeomorphisms $f : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ of the closed unit disc ${\bar{D}}(0;1)$ which are the identity map $id : {\bar{D}}(0;1) \rightarrow {\bar{D}}(0;1)$ on the…

Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the…

Let $(A, \|\cdot\|)$ be any normed algebra (not necessarily complete nor unital). Let $a \in A$ and let $V_A(a)$ denote the spatial numerical range of $a$ in $(A, \|\cdot\|)$. Let $A_e = A + {\mathbb C} 1$ be the unitization of $A$. If $A$…

We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $\Omega$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $\Omega$. We prove that if $V$ is two analytic disks…

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy \[\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1…

Let $f$ be an entire function and $L(f)$ a linear differential polynomial in $f$ with constant coefficients. Suppose that $f$, $f'$, and $L(f)$ share a meromorphic function $\alpha(z)$ that is a small function with respect to $f$. A…

We propose a generalization of the so-called rational map ansatz on the Euclidean space $\mathbb{R}^3$, for any compact simple Lie group $G$ such that $G/{\widehat K}\otimes U(1)$ is an Hermitian symmetric space, for some subgroup…