Related papers: An Elementary Proof of a Minimax Theorem
The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the…
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
A short and elementary proof is given of a celebrated eigenvalue-perturbation result due to Alfred Brauer.
We prove a stronger version of a termination theorem appeared in the paper "On existence of log minimal models II". We essentially just get rid of the redundant assumptions so the proof is almost the same as in there. However, we give a…
The main result of this paper is to prove some type of Real Jacobian Conjecture. It is proved by the Minimax Principle and asserts if the eigenvalues of $F'(x)$ are bounded from zero and all the eigenvalues of $F'(x)+F'(x)^T$ are strictly…
The small object argument is a method for transfinitely constructing weak factorization systems originally motivated by homotopy theory. We establish a variant of the small object argument that is enriched over a cofibrantly generated weak…
The symplectic analogues of Schur's theorem and Ky Fan's minimum principle are shown to be equivalent. Moreover, the symplectic Schur's and Horn's theorems are extended to generalized means.
Many models of physics beyond the Standard Model predict minicharged particles to which current and near future low-energy experiments are highly sensitive. Such minicharges arise generically from kinetic-mixing in theories containing at…
Lower bounds involving $f$-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Our proofs just use simple convexity facts. Special cases and straightforward corollaries of our…
Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case…
The aim of this paper is to clarify and generalize techniques of works alg-geom/9711024 (see also math.AG/9810097 and math.AG/9901004). Roughly speaking, we prove that for local Fano contractions the existence of complements can be reduced…
Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…
We give a short and self-contained proof of Levi's Extension Lemma for pseudoline arrangements.
The purpose of this paper is to provide a proof of James' weak compactness theorem that is able to be taught in a first year graduate class in functional analysis.
Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued…
We show that the well-known Konig's Min-Max Theorem (KMM), a fundamental result in combinatorial matrix theory, can be proven in the first order theory $\LA$ with induction restricted to $\Sigma_1^B$ formulas. This is an improvement over…
We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.
We give a simple proof of the increasing strengthening of Arhangel'skii's Theorem. Our proof naturally leads to a refinement of this result of Juh\'asz.
In this note, we combine ideas of several previous proofs in order to obtain a quite short proof of Gr\"otzsch theorem.
The main result of this paper is to prove the strong real Jacobian conjecture under the symmetric assumption and reveals the link between it and the Jacobian conjecture. Precisely, we assume that $F: \mathbb{R}^n \to \mathbb{R}^n$ is of…