Tootfinder

Centific, an AI "data foundry" that works with businesses to develop, train, and deploy their AI models, raised a $60M Series A led by Granite Asia (Nicholas Gordon/Fortune)
https://fortune.com/asia/2025/06/25/centi…

Singapore VC Granite Asia leads $60 million funding round for AI ‘data foundry’ Centific to fund global expansion
“This funding round isn’t about necessity—it’s about ambition,” CEO Venkat Rangapuram said in a statement.

Computability of Separation Axioms in Countable Second Countable Spaces
Andrew DeLapo, David Gonzalez
https://arxiv.org/abs/2507.18564 https://arxiv.org/pd…

Computability of Separation Axioms in Countable Second Countable Spaces
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic completeness result for each of the Tychonoff separation axioms. Beyond this, we prove completeness results for various other topological properties, such as being Polish and having a particular Cantor-Bendixson rank, using tools from computable structure theory. Thi…

Maßnahmen zur #CO2Entfernung aus den Ozeanen könnten das #Klima entlasten, aber zugleich das #Meeresleben gefährden.
Besonders biotische Methoden wie

Unimodular random graphs with Property (T) have cost one
{\L}ukasz Grabowski, H\'ector Jard\'on-S\'anchez, Sam Mellick
https://arxiv.org/abs/2506.19454

Unimodular random graphs with Property (T) have cost one
Hutchcroft and Pete showed that countably infinite groups with Property (T) admit cost one actions, resolving a question of Gaboriau. We give a streamlined proof of their theorem, and extend it both to locally compact second countable groups and unimodular random graphs. We prove unimodular random graph analogues of the Connes--Weiss and Glasner--Weiss theorem characterising Property (T).

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Convergence of the Birkhoff spectrum for nonintegrable observables
Godofredo Iommi, Mike Todd, Boyuan Zhao
https://arxiv.org/abs/2506.21390 https://…

Convergence of the Birkhoff spectrum for nonintegrable observables
We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the polynomial tail exponent. This result extends previous work by Arima on more regular observables and demonstrates how the tail behaviour influences the structure of the Birkhoff spectrum. Our proof relies on techniques from thermodynamic formalism and tail esti…

Self-care has become very commodified https://www.overcomecompulsivehoarding.co.uk/podcast-ep-187-what-if-we-focused-less-on-self-care-and-more-on-self-nurture-to-progress-our-hoarding-reco…

Perturbed infinite-state Markov systems with holes and its application
Haruyoshi Tanaka
https://arxiv.org/abs/2506.20053 https://arxi…

Perturbed infinite-state Markov systems with holes and its application
We consider a perturbed system $(X,φ(ε,\cdot))$, where $X$ is a topological Markov shift with a countably infinite state space, and $φ(ε,\cdot)$ is a real-valued potential on X depending on a small parameter $ε\in (0,1)$. We assume that for each $ε>0$, the system has a unique transitive component and a unique Gibbs measure (or more generally, a Ruelle-Perron-Frobenius (RPF) measure) $μ_ε$, while the unperturbed system possesses multiple transitive components and Gibbs measures on these …

An ergodic Lebesgue differentiation theorem
Aidan Young
https://arxiv.org/abs/2506.21421 https://arxiv.org/pdf/2506.21421

An ergodic Lebesgue differentiation theorem
We show that if $(X, μ, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ is a countable partition of $(X, μ)$, then the limit $$ \lim_{n, k \to \infty} \mathbb{E} \left[ \frac{1}{k} \sum_{j = 0}^{k - 1} f \circ T^j \mid \bigvee_{i = 0}^{n - 1} T^{-i} \mathscr{P} \right] $$ exists almost surely for all $f \in L^p(μ), p > 1$. We prove this as a corollary of a geometric result: that if $(X, μ)$ is a metric measure space on which the Hardy-Littlewood maximal inequalit…

Projective length, phantom extensions, and the structure of flat modules
Matteo Casarosa, Martino Lupini
https://arxiv.org/abs/2506.17982 https://

Projective length, phantom extensions, and the structure of flat modules
We consider the natural generalization of the notion of the order of a phantom map from the topological setting to triangulated categories. When applied to the derived category of the category of countable flat modules over a countable Dedekind domain, this yields a notion of\emph{\ phantom extension} of order $α<ω_{1}$. We provide a complexity-theoretic characterization of the module $\mathrm{Ph}% ^{α}\mathrm{Ext}\left( C,A\right) $ of phantom extensions of order $α$ with respect to the st…