Related papers: A streamlined proof of the Kakeya set conjecture i…
A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this…
A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case
We prove that all bounded subsets of $\mathbb{Q}_p^n$ containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension $n$. This is the analogue of the classical Kakeya conjecture with $\mathbb{R}$ replaced…
This note presents an elementary and direct proof for the convexity of the Choquet integral when the corresponding set function is submodular.
We provide a reduction in the classification problem for non-compact, homogeneous, Einstein manifolds. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.
In this paper, we provide the necessary and sufficient conditions for the connected sum of knots in $S^3$ to be Legendrian simple.
We provide an alternative and self contained proof of the main result of Bennett, Carbery, Tao regarding the multilinear restriction estimate. The approach is inspired by the recent result of Guth about the Kakeya version of multilinear…
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
We compare the smooth Alpert testing condition for the paraboloid Fourier extension conjecture in <cite>RiSa3</cite> to the modulated testing condition for the Kakeya conjecture in <cite>RiSa2</cite>. To this end, the modulated testing…
This is a summary of the proof of BAB conjecture. All material are taken from the two BAB paper in the reference. The aim of this summary is to help reader to understand the more technical side of the proof of BAB.
In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.
The purpose of this article is to present my new proof of the the construction and the convergence theorem of spectral sequences of filtered complexes, which is much shorter and cleaner than the "standard" proof.
We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.
We prove that if a circular arc has angle short enough, then it can be continuously moved to any prescribed position within a set of arbitrarily small area.
We expose here a short proof of Cramer's theorem in R based on convex duality.
A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives…
We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.
We propose a new refinement of the McKay conjecture and we prove it for symmetric groups.
We provide a simple and short proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints. The proof relies on an elementary linear algebra lemma and the local inverse theorem.
In this work we resolve several conjectures stated in the On-Line Encyclopedia of Integer sequences.