Related papers: The Cantor's First Diagonal Formalized and Extende…

We introduce an analogue of Calder\'on's first commutator along a parabola, and establish its $L^2$ boundedness under essentially sharp hypotheses.

We present a simple a bijection between permutations of $\{1,..., n\}$ with $k$ descents and permutation tableaux of length $n$ with $k$ columns.

We present a method for the explicit diagonalization of some Hankel operators. This method allows us to recover classical results on the diagonalization of Hankel operators with the absolutely continuous spectrum. It leads also to new…

Let $P=\mathbb P^m(e)\times\mathbb P^n(h)$ be a product of weighted projective spaces, and let $\Delta_P$ be the diagonal of $P\times P$. We prove an algebraization result for formal-rational functions on certain closed subvarieties $X$ of…

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of…

In this note, we investigate the regularity of Cantor's one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by…

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying…

This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a…

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can find a unitary transformation such that the transformed quadratic Hamiltonian is diagonal but still written in terms of the original position and…

Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every…

The formula for the dihedral angle of the simplex of n dimensions, arccos(1/n), is derived using classical geometry.

Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…

The famous contradiction of a bijection between a set and its power set is a consequence of the impredicative definition involved. This is shown by the fact that a simple mapping between equivalent sets does also fail to satisfy the…

It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. (1) Cantor's first proof of the uncountability of the set of all real numbers does not apply to the set of irrational numbers alone, and,…

We prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient…

We give a short constructive proof for the existence and uniqueness of the rational normal form of a quadratic matrix.

We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supporting this heuristic.

The Cantor pairing polynomials are extended to larger 2D sub-domains and more complex mapping, of which the most important property is the bijectivity. If corners are involved inside (but not the borders of) domain, more than one connected…

This is the first of three papers dealing with the XX finite quantum chain with arbitrary, not necessarily hermitian, boundary terms. This extends previous work where the periodic or diagonal boundary terms were considered. In order to find…

The form factor of the unitary group U(N) endowed with the Haar measure characterizes the correlations within the spectrum of a typical unitary matrix. It can be decomposed into a sum over pairs of ``periodic orbits'', where by periodic…