We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ${ℝ}^4$ around a $𝕐$-point and the existence of a point of particular type of a Mumford-Shah minimal set in ${ℝ}^4$, which is very close to a $𝕋$. This will give a local description of minimal sets of dimension 3 in ${ℝ}^4$ around a singular point and a property of Mumford-Shah minimal sets in ${ℝ}^4$.

The aim of this paper is to show that for every Banach space $(X,\parallel ·\parallel )$ containing asymptotically isometric copy of the space $c_0$ there is a bounded, closed and convex set $C\subset X$ with the Chebyshev radius $r\left(C\right)=1$ such that for every $k\ge 1$ there exists a $k$-contractive mapping $T:C\to C$ with $\parallel x-Tx\parallel >1-1/k$ for any $x\in C$.

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

We show that if n is a positive integer and $2^{\aleph ₀}\le \aleph ₙ$, then for every positive integer m and for every real constant c > 0 there are functions $f₁,...,f_{n+m}:ℝⁿ\to ℝ$ such that $(f₁,...,f_{n+m})\left(ℝⁿ\right)={ℝ}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i₁},...,f_{iₙ})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {ℝ}^{n-1}$.

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f{\in }_m$ and f̂ be its Taylor series at $0\in {ℝ}^m$. Split the set ${\mathbb{N}}^m$ of exponents into two disjoint subsets A and B, ${\mathbb{N}}^m=A\cup B$, and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g,h{\in }_m$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_1,P_2,P_3)$ with the property ${\overline{\varphi }}_3=1$. In fact, our method generalizes to provide $n$-systems with ${\overline{\varphi }}_n=1$, for arbitrary $n\ge 3$. We visualize our constructions with graphics. We further present explicit examples of numbers ${\xi }_1,...,{\xi }_{n-1}$ that induce the $n$-systems in question.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $F$ be a closed subset of ${ℝ}^n$ and let $P\left(x\right)$ denote the metric projection (closest point mapping) of $x\in {ℝ}^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in ${ℝ}^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty$ and prove that the $i$th component $P_i\left(x\right)$ of $P\left(x\right)$ is differentiable a.e. if $P_i\left(x\right)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i\left(x\right)=x_i$.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.