Related papers: Borel Conjecture and Dual Borel Conjecture

In this short paper we will show, via elementary arguments, the equivalence of the Twin Prime Conjecture to a problem which might be simpler to prove. Some conclusions are drawn, and it is shown that proving the Twin Prime Conjecture is…

We present a conjecture about partitions, with a very elementary formulation.

In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.

We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups.

We present the basic theory of calculus on dual real numbers, and prove the counterpart of the ordinary fundamental theorem of calculus in the context of dual real numbers.

We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…

A vector variational principle is proved.

In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [arXiv:1508.01064], in all dimensions, for principal solenoidal tori.

The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.

By introducing a new point of view in Algebraic Topology relating elliptic curves in $\mathbb{R}^2$ and suitable bordism groups, the congruent number problem is solved showing that the Tunnell's theorem is also sufficient. This could be…

After reviewing the entropy power, the McKean, and the Gaussian completely monotone conjectures, we prove that the first implies the second, for each order of the time-derivative. The proof is elementary and is based on manipulating the…

The main aim of this paper is to make a remark about the relation between (i) dualities between theories, as `duality' is understood in physics and (ii) equivalence of theories, as `equivalence' is understood in logic and philosophy. The…

An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.

We formulate the Hauptvermutung of Causal Set Theory in two mathematically well-defined but different ways one of which turns out to be wrong and the other one turns out to be true. A further result is that the Hauptvermutung is true if we…

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

Generalized uncertainty relations may depend not only on the commutator relation of two observables considered, but also on mutual correlations, in particular, on entanglement. The equivalence between the uncertainty relation and Bohr's…

According to some discussions based on syllogism, we present results on the binary Goldbach conjecture in three categories: results that are weaker than the Goldbach conjecture, sufficient conditions for the Goldbach conjecture, and results…

We prove the equality of three conjectural formulas for the Brumer--Stark units. The first formula has essentially been proven, so the present paper also verifies the validity of the other two formulas.