‎A universal schema for diagonalization was popularized by N.S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fi...

Restriction on the scope of diagonal argument will be set using two absolutely different proof techniques. One of the proof techniques will analyze contradictory equivalence (R ∈ R ↔ R ∉ R) in a rather unconventional way.

We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means a complification argument, we obtain an asymptotic formula for the number integral solutions this system bounded height that resolves associated paucity problem.

Of fundamental significance in many problems of additive number theory are estimates for mean values of exponential sums over polynomial functions. In this paper we shall show that the exponential sums of greatest interest in additive number theory demonstrate quasi-diagonal behaviour, which is to say that by taking the degree of the polynomial argument sufficiently large, we can obtain upper b...

An algebraic argument based on a series of generalized Dirac equations, truncated by an ”intrinsic Pauli principle”, shows that there should exist two sterile neutrinos as well as three families of leptons and quarks. Then, the influence of these additional neutrinos on neutrino oscillations is studied. As an example, a specific model of effective five–neutrino texture is proposed, where only t...

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument αx + βx. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess nontrivial integral solutions whenever the number of variables exceeds 10.

We study an integro-differential parabolic problem modelling a process with jumps and stochastic volatility in financial mathematics. Under suitable conditions, we prove the existence of solutions in a general domain using the method of upper and lower solutions and a diagonal argument.

For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner’s argument showing the convergence of the density of states to the sem...

We analyze the Brandenburger-Keisler paradox in epistemic game theory, which is a ‘two-person version of Russell’s paradox’. Our aim is to understand how it relates to standard one-person arguments, and why the ‘believes-assumes’ modality used in the argument arises. We recast it as a fixpoint result, which can be carried out in any regular category, and show how it can be reduced to a relation...