Related papers: Nonlinear mappings preserving at least one eigenva…

We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\mathbb B^n\to \mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\mathcal S\subset \partial \mathbb B^n$ be…

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables…

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some…

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally…

Let $F$ be a field, and $\mathcal{M}$ be a linear subspace of $n$-by-$n$ matrices with entries in $F$ that have at most two eigenvalues in $F$ (respectively, at most one non-zero eigenvalue in $F$). In a previous article, we have determined…

A matrix is homogeneous if all of its entries are equal. Let $P$ be a $2\times 2$ zero-one matrix that is not homogeneous. We prove that if an $n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$ has an $cn\times…

Let $\gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $\gamma$, within a suitable distance $\delta$ of $\gamma$.…

Let $A$ be the algebra of all $n \times n$ matrices with entries from $\RR[x_1,\ldots,x_d]$ and let $G_1,\ldots,G_m,F \in A$. We will show that $F(a)v=0$ for every $a \in \RR^d$ and $v \in \RR^n$ such that $G_i(a)v=0$ for all $i$ if and…

Let $\Gamma$ be a closed subset of a complete Riemannian manifold $M$ of dimension $\geq 2$, let $f: M \to N$ be a Lipschitz map to a complete Riemannian manifold $N$, and let $\psi$ be a continuous function which dominates the local…

We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that…

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

This paper mainly concerns the KAM persistence of the mapping $\mathscr{F}:\mathbb{T}^{n}\times E\rightarrow \mathbb{T}^{n}\times \mathbb{R}^{n}$ with intersection property, where $E\subset \mathbb{R}^{n}$ is a connected closed bounded…

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map…

Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all…

Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of…

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings.…

For a complex $n$-by-$n$ matrix $A$, the numerical range $F(A)$ is the range of the map $f_A(x) = x^*A x$ acting on the unit sphere in $\C^n$. We ask whether the multivalued inverse numerical range map $f_A^{-1}$ has a continuous…

It is shown that the ultimate version of the nonlinear Pietsch Domination Theorem remains true, even in a stronger presentation, if one of its hypotheses is removed. More precisely, we show that no trace of (sub)-homogeneity assumption is…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…

Let $k$ be a field of characteristic zero. Let $F = X + H$ be a polynomial mapping from $k^n \to k^n$, where $X$ is the identity mapping and $H$ has only degree two terms and higher. We say that the Jacobian matrix $JH$ of $H$ is strongly…