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We give a proof of the Marker-Steinhorn Theorem which fills a gap in previous proofs of the result.
A vector variational principle is proved.
We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.
With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of…
In this paper we prove the WALA conjecture.
We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional…
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
We prove a new version of Hall's Harem Theorem, where the final matching is realized by a unary function with additional conditions on behavior of cycles. The present paper can be considered as a helpful companion of the paper of the…
This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.
We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an…
Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why…
We consider several kinds of quintessence models in the framework of scale factor duality. We show that this symmetry exists only for a very small number of quintessence potentials. We then apply the duality transformations found to several…
The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.
We give a new proof of Lucas' Theorem in elementary number theory.
We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations…
We present another proof for the well-known {\em small model property} of two-variable logic. As far as we know, existing proofs of this property rely heavily on model theoretic concepts. In contrast, ours is purely combinatorial and uses…
We give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger's theorem.
We consider the positions of occurrences of a factor $x$ and its binary complement $\overline{x}$ in the Thue-Morse word ${\bf t} = {\tt 01101001} \cdots$, and show that these occurrences are "intertwined" in essentially two different ways.…
This article will prove a theorem for the existence of k-factor for k>1 ,and present an efficient algorithm for computing k-factor for all values of k based on this theorem.