Related papers: Non-vanishing implies numerical dimension one abun…

Quantum physics, despite its observables being intrinsically of a probabilistic nature, does not have a quantum entropy assigned to them. We propose a quantum entropy that quantify the randomness of a pure quantum state via a conjugate pair…

We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $\text{Refl}_n(\kappa)$ if and only if $\kappa$ is a $\Pi^1_n$-indescribable cardinal and every $\Pi^1_n$-indescribable subset of $\kappa$ has a…

Sets with many additive quadruples are guaranteed to have many additive octuples, by H\"{o}lder's inequality. Sets with not many more than this are said to be additively nonsmoothing. We give a new proof of a structural theorem for…

Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the set of real numbers $x$ such that $|nx -a| < \psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\psi)$ is of…

We present a counter-example to the recent claim that supermultiplets of N-extended supersymmetry with no central charge and in 1-dimension are specified unambiguously by providing the numbers of component fields in all available…

We prove the analogue of Schanuel's conjecture for raising to the power of an exponentially transcendental real number. All but countably many real numbers are exponentially transcendental. We also give a more general result for several…

Let $(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent and identically distributed random variables satisfying $P(\epsilon_i=1)=P(\epsilon_i=-1)=1/2$. Set $S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$ for…

For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of the number of Dedekind cuts in linear orders of size $\kappa$. It is known that $\kappa<ded\kappa\leq 2^{\kappa}$ for all $\kappa$ and that $ded\kappa<2^{\kappa}$ is…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a…

The conjecture of Kosniowski asserts that if the circle acts on a compact unitary manifold $M$ with a non-empty fixed point set and $M$ does not bound a unitary manifold equivariantly, then the dimension of the manifold is bounded above by…

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Renyi entropies of channels are not generally additive for p>1, we demonstrate here by a careful random selection…

We give a short proof that there exists a countable state Bernoulli measure maximizing the dimension of their images under the continued fraction expansion.

For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…

In the simplest cosmological models consistent with General Relativity, the total volume of the Universe is either finite or infinite, depending on whether or not the spatial curvature is positive. Current data suggest that the curvature is…

Measurement incompatibility is the most basic resource that distinguishes quantum from classical physics. Contextuality is the critical resource behind the power of some models of quantum computation and is also a necessary ingredient for…

If S is a smooth compact surface in $\mathbb{R}^{3}$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3$, $\left\|E_S f\right\|_{L^p\left(\mathbb{R}^3\right)}…