Related papers: Half of an inseparable pair

We show that one can force over $L$ that $\Sigma^1_3$-separation holds, while $\Pi^1_3$-reduction fails, thus separating these two principles for the first time. The construction can be lifted to canonical inner models $M_n$ with $n$-many…

Let $\alpha > 0$ be any fixed non-integer, $I$ be any subinterval of $[0; 1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes $p$ satisfying the condition $\{ p^{\alpha} \} \in I$. This strengthens…

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one…

We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are…

We give a classification of semisimple and separable algebras in a multi-fusion category over an arbitrary field in analogy to Wedderben-Artin theorem in classical algebras. It turns out that, if the multi-fusion category admits a…

Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite…

In this work we prove the undecidability (and $\Sigma^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

In these notes we prove two main results: 1) It is well-known that two strongly continuous $E_0$-semigroups on $B(H)$ can be paired if and only if they have anti-isomorphic Arveson systems. For a new notion of pairing (which coincides only…

In this paper, we prove the following: If $n\ge3$, there is a generic extension of $L$ -- the constructible universe -- in which it is true that the Separation principle holds for both effective (lightface) classes $\varSigma^1_n$ and…

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two…

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is…

We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely,…

We generically construct a model in which the $\bf{\Sigma^1_3}$-separation property is true, i.e. every pair of disjoint $\bf{\Sigma^1_3}$-sets can be separated by a $\bf{\Delta^1_3}$-definable set. This answers an old question from the…

We prove congruences for the number of partition pairs $(\pi_1,\pi_2)$ such that $\pi_1$ is non-empty, $s(\pi_1)\le s(\pi_2)$, and $\ell(\pi_2)< 2s(\pi_1)$ where $s(\pi)$ is the smallest part and $\ell(\pi)$ is the largest part of a…

We present a method which forces the failure of $\Pi^1_3$ and $\Sigma^1_3$-separation, while $\mathsf{MA} (\mathcal{I}$) holds, for $\mathcal{I}$ the family of indestructible ccc forcings. This shows that, in contrast to the assumption…

A result of Kaufmann shows that if $L_\alpha$ is countable, admissible and satisfies $\Pi_n\textsf{-Collection}$, then $\langle L_\alpha, \in \rangle$ has a proper $\Sigma_{n+1}$-elementary end extension. This paper investigates to what…

We show that the distance from an element of a C*-algebra to the set of regular elements is the infimum of the $\delta>0$ for which the $\delta$-cut-down of the element admits a polar decomposition within the algebra. This parallels results…

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive…

In this paper, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theorem1.…