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Tatsuya Goto

Publications and source records attributed to Tatsuya Goto.

12 recordsLinked to original sources
arXiv2026

Two variants of minimality in forcing extensions

We introduce d-minimal and b-minimal extensions, two weakenings of minimality based respectively on eventual domination and infinitely often domination. We prove that the generic extensions obtained by the full Mathias forcing and by Mathias forcing relative to Ramsey ultrafilters are d-minimal, whereas Hechler extensions are b-minimal. We also construct a d-minimal extension that is weakly $ω^ω$-bounding but neither minimal nor $ω^ω$-bounding, and realize every constellation of these properties compatible with their natural implications.

math.LO↗
arXiv2026

Forcing and Cardinal Invariants of Universally Null Sets

We study universally null sets in forcing extensions and the cardinal invariants of their ideal $\mathcal{UN}$. We prove that the Miller model contains a nonmeager universally null set of cardinality $\aleph_2=\mathfrak{c}$, answering a question of Brendle and Larson. The proof uses preservation of universal nullity by countable support iterations of Miller forcing and shows that the set of Miller reals added during the iteration is universally null. We also prove that every universally null set in the $\mathbb{PT}_{f,g}$ model has cardinality at most $\aleph_1$, and give a consistent example of a universally null set that acquires a perfect subset after an $ω_1$-preserving forcing. For the cardinal invariants, we show that $|X|<\operatorname{cof}(\mathcal{UN})$ for every $X\in\mathcal{UN}$, and that $\operatorname{add}(\mathcal{N})=\mathfrak{c}$ implies $\operatorname{cof}(\mathcal{UN})=\mathfrak{d}_{\mathfrak{c}}$. Finally, we prove the consistency of $\operatorname{add}(\mathcal{UN})<\operatorname{cov}(\mathcal{UN}) <\operatorname{non}(\mathcal{UN})<\operatorname{cof}(\mathcal{UN})$.

math.LO↗
arXiv2026

Goldstern's Principle with respect to Hausdorff Measures

This paper is a continuation of the paper [Got25] and studies Goldstern's principle, a principle about unions of continuum many null sets, further. The main result is that the Hausdorff measure version of Goldstern's principle for $\boldsymbolΠ^1_1$ sets fails in $L$, despite the fact that the Lebesgue measure version is true. Moreover, we show that this version holds provided that the measurable cardinal exists. Other various results regarding Goldstern's principle are established.

math.LO↗
arXiv2025

Extended Factorization Machine Annealing for Rapid Discovery of Transparent Conducting Materials

The development of novel transparent conducting materials (TCMs) is essential for enhancing the performance and reducing the cost of next-generation devices such as solar cells and displays. In this research, we focus on the (Al$_x$Ga$_y$In$_z$)$_2$O$_3$ system and extend the FMA framework, which combines a Factorization Machine (FM) and annealing, to search for optimal compositions and crystal structures with high accuracy and low cost. The proposed method introduces (i) the binarization of continuous variables, (ii) the utilization of good solutions using a Hopfield network, (iii) the activation of global search through adaptive random flips, and (iv) fine-tuning via a bit-string local search. Validation using the (Al$_x$Ga$_y$In$_z$)$_2$O$_3$ data from the Kaggle "Nomad2018 Predicting Transparent Conductors" competition demonstrated that our method achieves faster and more accurate searches than Bayesian optimization and genetic algorithms. Furthermore, its application to multi-objective optimization showed its capability in designing materials by simultaneously considering both the band gap and formation energy. These results suggest that applying our method to larger, more complex search problems and diverse material designs that reflect realistic experimental conditions is expected to contribute to the further advancement of materials informatics.

cond-mat.mtrl-sci↗
arXiv2025

Goldstern's principle about unions of null sets

Goldstern showed in his 1993 paper that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbolΣ^1_1$. Our aim is to study to what extent we can drop the $\boldsymbolΣ^1_1$ assumption. We show Goldstern's principle for the pointclass $\boldsymbolΠ^1_1$ holds. We show that Goldstern's principle for the pointclass of all subsets is consistent with $\mathsf{ZFC}$ and show its negation follows from $\mathsf{CH}$. Also we prove that Goldstern's principle for the pointclass of all subsets holds both under $\mathsf{ZF} + \mathsf{AD}$ and in Solovay models.

math.LO↗
arXiv2024

Game-theoretic variants of splitting number

We consider combining the definition of a cardinal invariant and the notion of an infinite game. We focus on the splitting number $\mathfrak{s}$ since the corresponding cardinal invariants behave in an interesting way. We introduce three kinds of games as reasonable realizations of the combination of the notions of splitting and infinite games. Then, we consider two cardinal invariants for each game, so we define six numbers. We prove that three of them are equal to the size of the continuum $\mathfrak{c}$ and one of them is equal to the $σ$-splitting number $\mathfrak{s}_σ$, which is defined as the minimum size of a $σ$-splitting family. On the other hand, we show that the remaining two numbers are consistently different from $\mathfrak{c}$, $\mathfrak{s}$ and $\mathfrak{s}_σ$. Moreover, though the two numbers share almost the same rule of the game, we prove that they can take distinct values from each other, and hence the slight difference of the rule is actually crucial in this sense.

math.LO↗
arXiv2024

Game-theoretic variants of cardinal invariants

We investigate game-theoretic variants of cardinal invariants of the continuum. The invariants we treat are the reaping number $\mathfrak{r}$, the bounding number $\mathfrak{b}$, the dominating number $\mathfrak{d}$, and the additivity number of the null ideal $\operatorname{add}(\mathsf{null})$. We also consider games, called tallness games, defined according to ideals on $ω$ and characterize that each of Player I and Player II has a winning strategy.

math.LO↗
arXiv2023

The comparability numbers and the incomparability numbers

We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as $ω^ω$, $\mathcal{P}(ω)/\mathrm{fin}$, the Turing degrees $\mathcal{D}$, the quotient algebra $\mathsf{Borel}(2^ω)/\mathsf{null}$, the ideals $\mathsf{meager}$ and $\mathsf{null}$. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on $ω$. We also consider these invariants for ideals on $ω$ and on $ω_1$.

math.LO↗
arXiv2022

Keisler's Theorem and Cardinal Invariants

We consider several variants of Keisler's isomorphism theorem. We separate these variants by showing implications between them and cardinal invariants hypotheses. We characterize saturation hypotheses that are stronger than Keisler's theorem with respect to models of size $\aleph_1$ and $\aleph_0$ by $\mathrm{CH}$ and $\operatorname{cov}(\mathsf{meager}) = \mathfrak{c} \land 2^{<\mathfrak{c}} = \mathfrak{c}$ respectively. We prove that Keisler's theorem for models of size $\aleph_1$ and $\aleph_0$ implies $\mathfrak{b} = \aleph_1$ and $\operatorname{cov}(\mathsf{null}) \le \mathfrak{d}$ respectively. As a consequence, Keisler's theorem for models of size $\aleph_0$ fails in the random model. We also show that for Keisler's theorem for models of size $\aleph_1$ to hold it is not necessary that $\operatorname{cov}(\mathsf{meager})$ equals $\mathfrak{c}$.

math.LO↗
arXiv2021

Cardinal invariants associated with Hausdorff measures

We consider cardinal invariants determined from Hausdorff measures. We separate many cardinal invariants of Hausdorff measure $0$ ideals using two models that separate many cardinal invariants of Yorioka ideals at once from earlier work. Also we show the uniformity numbers of $s$-dimensional Hausdorff measure $0$ ideals for $0 < s < 1$ and that of Lebesgue null ideal can be separated using the Mathias forcing.

math.LO↗