2018年度秋季総合分科会(於:岡山大学)

SUMMARY: We talk about decidability and undecidability within a general setting. Where decidability or undecidability are studied, there are computable structures (of symbols) and its congruence relations. We call such pairs computable prestructures. Lindenbaum algebras can be viewed as examples of computable prestructures. It is natural to define that a given computable prestructure is decidable if its congruence relation is a computable relation; and undecidable otherwise. In the talk, we give two stronger notions of undecidability: essential undecidability and ultimately undecidability. We characterize ultimately undecidability as essential undecidability if computable prestructures are some kinds of prelattices.

SUMMARY: This paper introduces some Japanese mathematicians’ connections in 17th century who were known as disciples of SEKI Takakazu. One of them, KURU Shigezane, was the first manuscript writer of SEKI’s Kaihohnpen no Ho, 1683. The author points out that KURU was not only SEKI’s disciple, but that of HASHIMOTO Masakazu in Kamigata area (i.e. Kyoto and Osaka). This relation suggests us that KURU should be a candidate who bridged the gap between the both schools of SEKI and HASHIMOTO, giving us the reason of the similarity of their mathematical works. On the other hand, we could identify KURU was a distant relative of TAKEBE Katahiro, referring the description of Kansei Choshu Shokafu, 1812. (TAKEBE’s cousin married with KURU’s older brother.) As a conclusion, these connections could clarify their circulations of mathematical works were relatively narrow than we thought.

SUMMARY: It is known that, under \(\mathsf {MA}(\sigma \)-linked\()\), a separable measure algebra forces \(\mathsf {MA}(\sigma \)-linked\()\). We extend it to MA for Todorcevic orderings \(\mathbb {T}(X)\) for topological spaces \(X\). If \(X\) is a second countable Hausdorff space, \(\mathbb {T}(X)\) is a non-\(\sigma \)-linked ccc forcing. It is proved that, under \(\mathsf {MA}\), a separable measure algebra forces \(\mathsf {MA}\) for all \(\mathbb {T}(X)\) for second countable Hausdorff \(X\).

SUMMARY: Let \(\Bbb N\) denote an infinite countable discrete space. Let \(\Bbb N^\kappa \) denote the products of \(\kappa \) many copies of \(\Bbb N\). We prove that it is undecidable in ZFC that there is an uncountable \(C^*\)-embedded discrete subset in \(\Bbb N^{2^\omega }\). Moreover, we show that the existence of \(C^*\)-embedded discrete subset with cardinality \(\kappa \) in \(\Bbb N^\kappa \) is depends on the choice of \(\kappa \).

SUMMARY: We show that if there is an extendible cardinal, then the mantle, the intersection of all ground models of the universe, must be the minimum ground model.

SUMMARY: J. K. Truss showed that for all automorphism \(g\) on Rado graph, any other automorphisms can be described by product of just 3 conjugates of \(g\) in 2003. We call such number of products a width of automorphism groups. There is several result on width of automorphisms with some conditions obtained by using techniques of descriptive set theory. We talk on the way of the improvements of width of automorphisms of more general structures.

SUMMARY: After introducing the notion of geometric isomorphism of quantum 2-tori and the notion of Morita equivalence of coordinate algebras of quantum 2-tori, we examine the relation between the two notions.

SUMMARY: In this talk, we want to explain some result which says that if Hrushovski’s pseudoplane is model complete then there is a strictly stable generic structure whose theory is not omega-categorical and has finite closures.

SUMMARY: In this presentation, we shall define a BCK-algebra with the condition (I), and prove some properties of this notion, and discuss the relationship between the existence of commutative elements and the existence of infomums in BCK-algebra. Then, we show the two characterizations; a BCK-algebra with Canonical Condition (I) are equivalent to a commutative BCK-algebra; and a BCK-algebra with Condition (I) are a lower semi-lattice with respect to the operation \(\times \).

SUMMARY: The disjunction and existence properties in intermediate predicate logics were revealed to be independent in our previous paper. In that paper, we introduced Z-normality, which is a weak variant of disjunction property in the setting of intermediate predicate logic. In this talk, we introduce another weak variant of disjunction property called sentential disjunction property, and report relations among disjunction property, sentential disjunction property, and Z-normality in intermediate predicate logics.

SUMMARY: We prove that there exists a Rosser provability predicate whose provability logic is the modal logic \(\bf KD\). Also we prove that there exists a Rosser provability predicate whose provability logic includes the logic \({\bf KD} + (\Box p \to \Box \Diamond p)\) which is a proper extension of \(\bf KD\).

SUMMARY: The Frege–Hilbert controversy over the foundations of geometry reveals the difference between Frege’s philosophical standpoints and the basic assumptions of first-order logic, rather than the variance in their views about geometry. I will review the controversy and discuss the relationship between Frege’s philosophical standpoints and some proposals of extensions and modifications of the first-order logic.

SUMMARY: The notion of probability plays a crucial role in quantum mechanics. It appears as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In our former works, based on the toolkit of algorithmic randomness, we presented an alternative rule to the Born rule for specifying the property of results of measurements in an operational way. In this talk, we investigate the consequences of our alternative rule from a physical point of view.

SUMMARY: Modal \(\mu \)-calculus, introduced by Kozen, is an extension of propositional modal logic by adding the least and greatest fixpoint operators. In this talk, we review the historical and latest studies on the (semantical) strictness of several alternation hierarchies with respect to various transition systems, and then investigate the variable hierarchies. Among others, we show that the simple alternation hierarchy of the one-variable fragments of modal \(\mu \)-calculus is strict over finitely branching transition systems.

SUMMARY: We present Takeuti–Yasumoto style generic models of theories for classes ALOGTIME, LOGSPACE and NLOGSPACE.

SUMMARY: We investigate an AND-OR tree \(T\) and a probability distribution \(d\) on the truth assignments to the leaves. Tarsi (1983) showed that if \(d\) is an independent and identical distribution (IID) such that probability of a leaf having value 0 is neither 0 nor 1 then there exists an optimal algorithm that is depth-first. We investigate the case where \(d\) is an independent distribution (ID) and probability depends on each leaf. It is known that in this general case, if height is greater than or equal to 3, Tarsi-type result does not hold. It is also known that for a complete binary tree of height 2, Tarsi-type result certainly holds. We ask whether Tarsi-type result holds for an AND-OR tree of height 2.

SUMMARY: We discuss the “sloughs” in A Study of Mathematical Physics by Poisson. In his books, he explodes the established theories in rivalry to Lagrange, Laplace, Fourier, Gauss, etc. who are the trailblazers of each arena and the leading scholars of mathematical physics. We think, these trials are “sloughs”, and look for Poisson’s thinking for newness of learning.

SUMMARY: We discuss the verification of scientific newness on A Study of Mathematical Physics by Poisson. He belongs to the Laboratory of Longitude, and discusses the universal mechanics of about two hundred years ago, in which he emphasizes his standing arena not on the astronomy but on the mathematical physics. We introduce the then scientific-common-sense on some topics comparing with today’s data.

SUMMARY: We discuss Poisson’s integral of ellipsoid using the Legendre’s formulae in his works. He tries his method of integral beginning with the paper 1802, and until at the last work, he appreciates fully Legendre’s method. Without his method, Poisson says, he couldn’t have succeeded in these sort of integrals, although he tries his own of ellipsoid. He adopts Lagrange’s formulae only after five years since Legendre’s table 1826.

SUMMARY: The fraction \(1/(z^2+1)\) has two singularities \(z=i, -i\). Expand in \(z=2i\). (1) Devide the domain into A: \(0 \le |z-2i| < 1\), B: \(1 < |z-2i| < 3\), C: \(3 < |z-2i|\). (2) Decomposite it into partials: \((i/2) \{1/(z+i)-1/(z-i)\}\). Ex. \(1/(z+i)=1/\{(z-2i)+3i\}\). Wada’s formula: \(1/(\square - \triangle )^{p+1}=\Sigma (k=1 \rightarrow \infty ) d_p(k) \square ^{-p-k}\triangle ^{k-1}(0<|\triangle /\square |<1)\), where integers \(p \ge 0\); \(k \ge 1\); \(d_p(k) \equiv (k+p-1)!/p!(k-1)!\), named Suida expansion. If \(1/\{(z-2i)+3i\}=1/\{(z-2i)-(-3i)\}^{0+1}=\Sigma (k=1 \rightarrow \infty ) d_0(k)(z-2i)^{-0-k} (-3i)^{k-1} (0<|-3i/(z-2i)|<1)\), then we have \(\Sigma (k=1\rightarrow \infty )\) \((-3i)^{k-1}(z-2i)^{-k} (3<|z-2i|)\). If \(1/\{3i-(-(z-2i))\}^{0+1}=\Sigma (k=1 \rightarrow \infty )d_0(k)(3i)^{-0-k}(-(z-2i))^{k-1} (0<|-(z-2i)/3i|<1)\), then we have \(\Sigma (k=1\rightarrow \infty )d_0(k)(-1)^{k-1}(1/3i)^k(z-2i)^{k-1}\) \((0<|z-2i|<3)\).

SUMMARY: This is a progress report of the research presented in September 2016. Though the text of the Elements is extremely simple and monotonous Greek prose, it is not easy to provide all the necessary grammatical rules so that the syntactic tree of all the sentences can be constructed. We will show the results hitherto obtained.

SUMMARY: The Taisei Sankei (1683–1711) is a monograph of 20 volumes written by Seki Takakazu, Takebe Kata’akira and Takebe Katahiro. Volume 1, called Five Techniques, treats five fundamental operations on numbers, that is, addition, subtraction, multiplication, division and extraction of root. Both Volumes 2 called Miscellaneous Techniques and 3 called Various Techniques treat numbers and their calculations. Besides, Preparatory Volume which is placed in front of Volume 1, considers, among others, numbers. Because of elementary character, these volumes have been overlooked by many scholars. We shall discuss the inter-dependence among these volumes.

SUMMARY: In this talk, we will give some equations in “The Sea Island Mathematical Manual” by the calculation of rectangles instead of the similarity of triangles.