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We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3\le n\le 5$ by Chodosh-Li,…
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in…
Motivated by the Forelli--Rudin projection theorem we give in this paper a criterion for boundedness of an integral operator on weighted Lebesgue spaces in the interval $(0,1)$. We also calculate the precise norm of this integral operator.…
Our purpose in this article is first, following [14], to find the topological upper limits of projections of secant planes to $C^{1}$ surfaces and the topological upper limits of projections of secant hyperplanes to $C^{1}$ hypersurfaces…
Extending results of Hershberger and Suri for the Euclidean plane, we show that ball hulls and ball intersections of sets of $n$ points in strictly convex normed planes can be constructed in $O(n \log n)$ time. In addition, we confirm that,…
In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hyperplanes with truncated counting functions. Our results are improvements of the previous…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…
The point-to-set principle \cite{LutLut17} characterizes the Hausdorff dimension of a subset $E\subseteq\R^n$ by the \textit{effective} (or algorithmic) dimension of its individual points. This characterization has been used to prove…
We use the measurable Hall's theorem due to Cie\'sla and Sabok to prove that (i) if two measurable sets $A,B \subset \mathbb{R}^d$ of the same measure are bounded remainder sets with respect to a given irrational $d$-dimensional vector…
We study the generalized analogues of conics for normed planes by using the following natural approach: It is well known that there are different metrical definitions of conics in the Euclidean plane. We investigate how these definitions…
The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $ I^d. $ This assertion provides an upper estimate for the…
In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically…
According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…
A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…
Let $(X,\left\Vert \cdot \right\Vert )$ be a real normed space of dimension $N\in \mathbb{N}$ with a basis $(e_{i})_{1}^{N}$ such that the norm is invariant under coordinate permutations. Assume for simplicity that the basis constant is at…
In this article we prove martingale type pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space ($d$ is a positive integer), where conditional expectations are replaced by their…
We established a fixed-point theorem for mapping satisfying a general contractive inequality of integral type depended an another function. This theorem substantially extend the theorem due to Branciari (2003) and Rhoades (2003)
We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace…
Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be given functions. We provide necessary and sufficient conditions for the existence of a convex function $F\in…
The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the…