Tootfinder

Aleph Filter: To Infinity in Constant Time
Niv Dayan, Ioana Bercea, Rasmus Pagh
https://arxiv.org/abs/2404.04703 https://arxiv.org/pd…

Aleph Filter: To Infinity in Constant Time
Filter data structures are widely used in various areas of computer science to answer approximate set-membership queries. In many applications, the data grows dynamically, requiring their filters to expand along with the data that they represent. However, existing methods for expanding filters cannot maintain stable performance, memory footprint, and false positive rate at the same time. We address this problem with Aleph Filter, which makes the following contributions. (1) It supports all oper…

Aleph Filter: To Infinity in Constant Time
Niv Dayan, Ioana Bercea, Rasmus Pagh
https://arxiv.org/abs/2404.04703 https://arxiv.org/pd…

Aleph Filter: To Infinity in Constant Time
Filter data structures are widely used in various areas of computer science to answer approximate set-membership queries. In many applications, the data grows dynamically, requiring their filters to expand along with the data that they represent. However, existing methods for expanding filters cannot maintain stable performance, memory footprint, and false positive rate at the same time. We address this problem with Aleph Filter, which makes the following contributions. (1) It supports all oper…

Weak diamond and pcf theory
Shimon Garti
https://arxiv.org/abs/2405.03142 https://arxiv.org/pdf/2405.03142

Weak diamond and pcf theory
We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from $\aleph_δ$ for which $2^{\aleph_δ}<\aleph_{(|δ|^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.

This https://arxiv.org/abs/2207.03659 has been replaced.
link: https://scholar.google.com/scholar?q=a

$Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups
Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic, $Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the other hand, there are non-empty closed subsets of $\mathcal G_n$ that admit no $Out(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the group $Aut(F_n)$ plays a crucial role in the proof of both results. We also discuss model theoretic implications of the existe…

Weak diamond and pcf theory
Shimon Garti
https://arxiv.org/abs/2405.03142 https://arxiv.org/pdf/2405.03142

Weak diamond and pcf theory
We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from $\aleph_δ$ for which $2^{\aleph_δ}<\aleph_{(|δ|^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.

On the singular abelian rank of ultraproduct II$_1$ factors
Patrick Hiatt, Sorin Popa
https://arxiv.org/abs/2402.18559 https://arxiv.…

On the singular abelian rank of ultraproduct II$_1$ factors
We prove that, under the continuum hypothesis $\frak c=\aleph_1$, any ultraproduct II$_1$ factor $M= \prod_ω M_n$ of separable finite factors $M_n$ contains more than $\frak c$ many mutually disjoint singular MASAs, in other words the {\it singular abelian rank of} $M$, $\text{\rm r}(M)$, is larger than $ \frak c$. Moreover, if the strong continuum hypothesis $2^{\frak c}=\aleph_2$ is assumed, then ${\text{\rm r}}(M) = 2^{\frak c}$. More generally, these results hold true for any II$_1$ factor…

Sources: Canada and the UAE have lobbied leading European and UK AI startups, including Aleph Alpha, Synthesia, and Stability AI, to move their headquarters (Financial Times)
https://t.co/qRAFJut3Cp

The Aleph & Other Metaphors for Image Generation
Gonzalo Ramos, Rick Barraza, Victor Dibia, Sharon Lo
https://arxiv.org/abs/2402.07104 https://<…

The Aleph & Other Metaphors for Image Generation
In this position paper, we reflect on fictional stories dealing with the infinite and how they connect with the current, fast-evolving field of image generation models. We draw attention to how some of these literary constructs can serve as powerful metaphors for guiding human-centered design and technical thinking in the space of these emerging technologies and the experiences we build around them. We hope our provocations seed conversations about current and yet-to-be developed interactions w…

Detecting and Deterring Manipulation in a Cognitive Hierarchy
Nitay Alon, Lion Schulz, Joseph M. Barnby, Jeffrey S. Rosenschein, Peter Dayan
https://arxiv.org/abs/2405.01870

Detecting and Deterring Manipulation in a Cognitive Hierarchy
Social agents with finitely nested opponent models are vulnerable to manipulation by agents with deeper reasoning and more sophisticated opponent modelling. This imbalance, rooted in logic and the theory of recursive modelling frameworks, cannot be solved directly. We propose a computational framework, $\aleph$-IPOMDP, augmenting model-based RL agents' Bayesian inference with an anomaly detection algorithm and an out-of-belief policy. Our mechanism allows agents to realize they are being deceiv…

This https://arxiv.org/abs/2110.11614 has been replaced.
link: https://scholar.google.com/scholar?q=a

Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals
Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $ω^ω$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from arXiv:1805.11005.

This https://arxiv.org/abs/2110.11614 has been replaced.
link: https://scholar.google.com/scholar?q=a

Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals
Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $ω^ω$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from arXiv:1805.11005.

This https://arxiv.org/abs/2308.00299 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Full Souslin trees at small cardinals
A $κ$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $κ$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be $\aleph_3$ many full $\aleph_2$-trees such that the product of any countably many of them is an $\aleph_2$-Souslin tree.

This https://arxiv.org/abs/2404.04703 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csDB_…

Aleph Filter: To Infinity in Constant Time
Filter data structures are widely used in various areas of computer science to answer approximate set-membership queries. In many applications, the data grows dynamically, requiring their filters to expand along with the data that they represent. However, existing methods for expanding filters cannot maintain stable performance, memory footprint, and false positive rate at the same time. We address this problem with Aleph Filter, which makes the following contributions. (1) It supports all oper…

Interpolation in H\'ajek's Basic Logic
Wesley Fussner, Simon Santschi
https://arxiv.org/abs/2403.13617 https://arxiv.org/pdf/…

Interpolation in Hájek's Basic Logic
We completely classify the varieties of BL-algebras with the amalgamation property, showing that there are only $\aleph_0$ of them and solving an open problem of Montagna. As a consequence of this classification, we obtain a complete description of which axiomatic extensions of Hájek's basic fuzzy logic BL have the deductive interpolation property. Along the way, we provide similar classifications of varieties of basic hoops with the amalgamation property and axiomatic extensions of the negati…

Distinguishing Internally Club and Approachable on an Infinite Interval
Hannes Jakob, Maxwell Levine
https://arxiv.org/abs/2404.15230 https://

Distinguishing Internally Club and Approachable on an Infinite Interval
Krueger showed that PFA implies that for all regular $Θ\ge \aleph_2$, there are stationarily many $[H(Θ)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n<ω$ and $Θ\ge \aleph_{n+1}$, there is a stationary subset of $[H(Θ)]^{\aleph_n}$ consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers…

On a theorem due to Murray
A. Barbosa, A. Raposo Jr., G. Ribeiro
https://arxiv.org/abs/2403.05806 https://arxiv.org/pdf/2403.05806

On a theorem due to Murray
In this paper, we introduce the notions of $α$-quasicomplemented and totally $α$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }α$-quasicomplemented if, and only if, $α<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right) $. We also show that …

This https://arxiv.org/abs/2307.10692 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The construction principle and non homogeneity of uncountable relatively free groups
In [11] Sklinos proved that any uncountable free group is not $\aleph_1$-homogenenous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek's question in the negative. Our methods are general and they also applications in varieties with tor…

This https://arxiv.org/abs/2208.09380 has been replaced.
link: https://scholar.google.com/scholar?q=a

On compactness of weak square at singulars of uncountable cofinality
Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at $\aleph_ω$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<ω$ while $\square_{\aleph_ω}$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle $\square_κ^*$ is in fact compact at singulars of uncountable cofinality, and that an even …

On a theorem due to Murray
A. Barbosa, A. Raposo Jr., G. Ribeiro
https://arxiv.org/abs/2403.05806 https://arxiv.org/pdf/2403.05806

On a theorem due to Murray
In this paper, we introduce the notions of $α$-quasicomplemented and totally $α$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }α$-quasicomplemented if, and only if, $α<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right) $. We also show that …

Owings-like results for infinitely many colours or finite monochromatic sets
David J. Fern\'andez-Bret\'on, Eliseo Sarmiento Rosales, Germ\'an Vera
https://arxiv.org/abs/2402.13124

Owings-like results for infinitely many colours or finite monochromatic sets
Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $κ,θ$, every colouring $c:G\longrightarrowθ$ yields a subset $X\subseteq G$ with $|X|=κ$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=\mathbb Z$, $θ=2$ and $κ=\aleph_0$; while Hindman proved it false for the same $G$ and $κ$ but $θ=3$.) We completely settle the question (in the affirmative whenever $G$ is infinite) for finite $κ$ and $θ$, and al…

On conjugacy and perturbation of subalgebras
David Gao, Srivatsav Kunnawalkam Elayavalli, Gregory Patchell, Hui Tan
https://arxiv.org/abs/2403.08072 https:…

On conjugacy and perturbation of subalgebras
We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II$_1$ factor $N_0$ we construct a highly indecomposable non Gamma II$_1$ factor $N$ such that $N_0 \subset N$ and moreover every von Neumann subalgebra of $N$ with Haagerup's property admits a unique embedding up to unitary conjugation. Such a factor necessarily has to be non separable, but we show that it can be taken of density character $2^{\aleph_0}$. On the other hand we are able t…

This https://arxiv.org/abs/2110.11614 has been replaced.
link: https://scholar.google.com/scholar?q=a

Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals
Combining creature forcing approaches from arXiv:1003.3425 and arXiv:1402.0367, we show that, under CH, there is a proper $ω^ω$-bounding poset with $\aleph_2$-cc that forces continuum many pairwise different cardinal characteristics, parametrised by reals, for each one of the following six types: uniformity and covering numbers of Yorioka ideals as well as both kinds of localisation and anti-localisation cardinals, respectively. This answers several open questions from arXiv:1805.11005.

This https://arxiv.org/abs/2310.14474 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Building models in small cardinals in local abstract elementary classes
There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that stability is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{Theorem.}$ Suppose $λ<2^{\aleph_0}$. Let $\mathbf{K}$ be an abstract elementary class with $λ\geq \operatorname{LS}(\mathbf{K})$. Assume $\mathbf{K}$ has amal…