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The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
A simple solution of Witten's monopole equations is given.
We prove some injectivity, torsion-free, and vanishing theorems for simple normal crossing pairs. Our results heavily depend on the theory of mixed Hodge structures on compact support cohomology groups. We also treat several basic…
This article proves hypersurfaces of degree d in projective n-space are "rationally simply-connected" if $d^2 \leq n$. In a forthcoming paper, de Jong and I prove a slightly weaker result when $d^2 \leq n+1$.
This note is devotes to some remarks regarding the use of variational methods, of minimax type, to establish continuity type results
We give a simple proof of the Fourier Inversion Theorem, using the methods of nonstandard analysis.
We associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them.
Using geometric homology and cohomology we give a simple and conceptual proof of the Thom isomorphism theorem.
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
We give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students…
We prove a connectedness result for products of weighted projective spaces.
This supplement to the article "Approximating multi-dimensional Hamiltonian flows by billiards" contains the proofs of C$^{0}$ and C$^{r}$ - closeness theorems stated in the article.
We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.
We sketch several proofs of F\'ary--Milnor theorem.
We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture,…
A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented.
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
The paper presents a new short proof of one of Adams's theorems and a $t$-trace-class theorem for parabolic Morrey spaces.
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…
Probably we have observed a new simple phenomena dealing with approximations to two real numbers.