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We give a simple proof of Bourgain's theorem on the singularity of Ornstein's maps.

We provide a proof of the Borwein Conjecture using analytic methods.

In this paper, we introduced the theory of the sieve function transformation. Using the principle of sieve function transformation, we improved sieve method, and obtained the difference range of similar sieve function values. For this, we…

We show that simple modifications to van der Hoeven's forward and inverse truncated Fourier transforms allow the algorithms to be performed in-place, and with only a linear overhead in complexity.

The purpose of this note is to prove the celebrated Discrete Renewal Theorem in a common special case. We use only very elementary methods from real analysis, rather than markov chain theory, complex analysis, or generating functions.…

We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate…

We give a simple proof of Dorronsoro's theorem and use similar ideas to establish an equivalence for embeddings of vector fields.

This article presents simple and easy proofs of the Implicit Function Theorem and the Inverse Function Theorem, in this order, both of them on a finite-dimensional Euclidean space, that employ only the Intermediate Value Theorem and the…

We prove the sufficient conditions for convergence of a certain iterative process of order 2 for solving nonlinear functional equations, which does not require inverting the derivative. We translate and detail our results for a system of…

The better title is "Yet another FALSE proof of the 4-colour theorem." Please consider all versions of this paper as historical material on the way to a non-computer proof of the 4-colour theorem. Interpreted as proofs, all versions are…

We present a new approach to proving non-termination of non-deterministic integer programs. Our technique is rather simple but efficient. It relies on a purely syntactic reversal of the program's transition system followed by a…

Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on…

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…

We present an elementary proof of Fermat's Last Theorem. No ancillary results are used, not even the most basic ones. The proof directly leads to a contradiction of the Fermat equation in the set of integers.

We introduce a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultradistributions we show that this complex plane-generalization overcomes all…

Starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system [cond-mat/0703213] the quantum version of the Crooks fluctuation…

This note presents an elementary and direct proof for the convexity of the Choquet integral when the corresponding set function is submodular.

In this paper we study the inverse of so-called unfair permutations, and explore various properties of them. Our investigation begins with comparing this class of permutations with uniformly random permutations, and showing that they behave…

We present a short new proof of Cobham's theorem without using Kronecker's approximation theorem, making it suitable for generalization beyond automatic sequences.

We establish a simple generalization for the famous theorem of Morley about trisectors in triangle with a purely synthetic proof using only angle chasing and similar triangles. Furthermore, based on the converse construction, another simple…