2017年度秋季総合分科会(於:山形大学)

SUMMARY: The \(n\)-dependent property is one of model theoretic dividing lines between first order theories. Recent studies of the property add evidence that model theory has a deep connection with finite combinatorics, such as hyper graph, structural Ramsey theory and Vapnik–Chervonenkis theory. In this talk it will be explained how they interact with each other.

SUMMARY: We discuss the solubilities in the Study of Mechanics of Poisson 1833, which Poisson issued again in about 20 years after the first publications in 1811, in which he discusses statics, dynamics, the hydrostatics and the hydrodynamics, relating topics, which compose of mechanics. Poisson introduces the methods of solving mathematically the problems in mechanics, in which we have a strong interest. We discuss, in introducing these methods, how he did handle his big scope, since the year 1811, standing on the basis composed of capillary action, mechanics and heat theory, and so on, including his last conclusions of problems reserving before.

SUMMARY: We discuss two points on the Study of Mechanics of Poisson 1833, which Poisson issued in his last period of his life of learing, in which he discusses the hydrostatics and the hydrodynamics. Previously, he discuss in the precedings (1829). We are considering this as the origin of the equations of the Navier–Stokes owing to Stokes’ referring in 1859. The other point is his conjecture on the defect of the preceding proofs of exact differential. We aim to discuss his process of theoretical convergence in this arena.

SUMMARY: A technical term “Suan” is used as the amount of calculations in the annotation for the problem 18 of the chapter “Fangcheng” of the “Nine Chapters on the mathematical art.” However, the way of counting it has been misunderstood, so we will correct it in this talk.

SUMMARY: In recent years, construction of digital image databases from historical documents of Japanese mathematics (wasan) is progressing. Wasan researchers widely utilize these databases, for example, they search images from titles, authors and years. On the other hand, existing databases don’t have enough functions such as searching from shapes or geometric elements. In this study, the authors propose a system that helps us to tag on wasan images automatically. This system can recognize geometric elements such as triangles, squares and circles from images of geometric problems in wasan documents and it can analyze relationship such as tangency of circles and number of elements from these images. Finally, it can tag each wasan image with these information automatically.

SUMMARY: The Bologna manuscript (Archiginnasio Library) of Euclid’s Elements (Heiberg’s codex b) represents a tradition of Greek text totally different from other manuscripts. Though this tradition has turned out to be closer to the original thanks to W. R. Knorr’s study in 1996, very little research has been done since then. There are quite a few errors of copyist both in text and diagram. Tentative edition of diagrams (with labels added or corrected, lines suppressed or supplied) will be shown.

SUMMARY: To approach a longstanding open problem on Todorcevic’s fragments of Martin’s Axiom, we present the consistent assertions with the existence of a non-special Aronszajn tree. In this talk, we explain the assertions from not only Martin’s Axiom but also Proper Forcing Axiom.

SUMMARY: Galvin proved that, under the Continuum Hypothesis, there are two ccc posets \(P_0\) and \(P_1\) such that \(P_0\times P_1\) is not ccc. Roitman proved that Cohen forcing and random forcing adds such Galvin’s example respectively. It is proved that it is consistent that (coherent) Suslin tree adds Galvin’s example. In this talk, I will explain the motivation of this research and idea of the proof.

SUMMARY: We discuss relationships between \(\Pi ^1_n\)-indescribable cardinals and the reflection of \(\Pi ^1_n\)-indescribable sets. Among other things, we generalize the classical result of Jensen to show that in the constructible universe \(L\), a \(\Pi ^1_n\)-indescribable cardinal \(\kappa \) is \(\Pi ^1_{n+1}\)-indescribable if and only if every \(\Pi ^1_n\)-indescribable subset of \(\kappa \) reflects.

SUMMARY: We show that the inclusion reduct of any countable model of ZFC is countable saturated. It follows that the structures arising as the inclusion relation of a countable model of ZFC are all isomorphic, and that they are exactly the countable saturated models of the theory of set-theoretic mereology: an unbounded atomic relatively complemented distributive lattice.

SUMMARY: Let \(A\) be a graph. Put \(\delta (A) = |A|-\alpha e(A)\) where \(|A|\) is the number of vertices in \(A\) and \(e(A)\) the number of edges in \(A\). Let \(f\) be an unbounded concave function on the set of all non-negative real numbers. \(\mathbf {K}_f\) is the class of all graphs \(A\) such that \(B \subseteq A\) implies \(\delta (B) \geq f(|B|)\). There are some conditions on \(f\) that imply the free amalgamation property of \(\mathbf {K}_f\). We discuss relations between those conditions, and see how some constructions work out in \(\mathbf {K}_f\) under those conditions.

SUMMARY: K. Eda raised a question concerning a graph structure and its automorphism groups. We don’t know the exact answer to this question. But we give an affirmative answer to it under an additional model theoretic assumption.

SUMMARY: A type \(p\in S(T)\) is said to be special, if there are \(a,b\models p\) such that \({\rm tp}(b/a)\) is isolated and non-algebraic, and \({\rm tp}(a/b)\) is non-isolated. I proved that a stable Ehrenfeucht theory had a special type. In this talk, I will introduce a result, which says that every Ehrenfeucht theory has a special type.

SUMMARY: In this talk, we would like to present about the simple history, basic logical background, main results and impacts to mathematics and human beings of the division by zero.

SUMMARY: In this talk, as the representation of the point at infinity on the Riemann sphere by the zero z = 0, we will show some delicate geometric relations between 0 and infinity which show a strong discontinuity at the point of infinity on the Riemann sphere:

SUMMARY: Semilattice semantics has been considered in relevant logics. In this semantics, completeness for implicational fragment of familiar relevant logics R and E can be proved but cannot for full R and E. In this talk, we consider semantics in which completeness of weaker relevant logics (and their neighbours) can be proved.

SUMMARY: It is known that there are two hypersequent calculi, \(\mathbf {GLCW}\) and \(\mathbf {GLC}\), characterized by the class of all totally ordered Kripke frames (equivalent to Dummett’s \(\mathbf {LC}\)). However, it is also known that \(\mathbf {GLCW}\) is strictly weaker than \(\mathbf {GLC}\) if part of the propositional logical symbols, \(\to ,~\land ,~\lor \) and \(\lnot \), are restricted. We correct a mistake of the proof of cut-elimination theorem of the hypersequent calculi obtained by restricting the logical symbols of \(\mathbf {GLCW}\) claimed by Avron. Also, we generalize \(\mathbf {GLCW}\) for all \(n \geq 1\) by Jankov’s characteristic formula.

SUMMARY: We introduce a formal system of reduction paths, based on which paths can be generated from a quiver by means of three operators. Next, we define reduction rules on paths and then show that the rules on paths are terminating and confluent, so that we can obtain normal paths. Following this, linearly ordered quivers \(Q'\) and \(Q''\) can be generated by the path operators from a quiver \(Q\) called the Dynkin diagram of type An, such that the constructed \(Q'\) and \(Q''\) provide witness on behalf of the Church–Rosser property for \(Q\).

SUMMARY: We introduce second order function symbols into first order predicate logic. Syntax and semantics are naturally defined. The equality axioms are extended, and the completeness theorem is proved.

SUMMARY: We prove that for each \(n \geq 0\), if the set of all theorems \({\rm Th}(T)\) of a consistent theory \(T\) is \(\Pi _{n+1}\)-definable, then there exists a true \(\Pi _n\) sentence which is not provable in \(T\). This improves Jeroslow’s extension and Hájek’s generalization of the first incompleteness theorem.

SUMMARY: We prove that for every recursively axiomatized consistent extension \(T\) of Peano Arithmetic and \(n \geq 1\), there exists a \(\Sigma _2\) numeration \(\tau (v)\) of \(T\) such that whose provability logic is exactly Sacchetti’s logic \(\mathsf {K} + \Box (\Box ^n p \to p) \to \Box p\). This settles Sacchetti’s problem.

SUMMARY: We will reformulate the forcing construction by Takeuti and Yasumoto by two-sort bounded arithmetic. As a result, we can construct models for various complexity classes below PTIME. We will also give an alternative proofs of some theorems in Takeuti–Yasumoto’s paper. Finally, we will discuss some open problems.

SUMMARY: This research is part of a project to answer “how hard is it to show that an infinite game is determined?” In terms of the foundational program “Reverse Mathematics”, the strength of determinacy is measured by the complexity of a winning strategy required by the determinacy of a given game. In this talk, we will discuss infinite games whose winning sets are defined by deterministic 2-stack visibly pushdown automata (2DVPA), nondeterministic pushdown automata (NPDA) and some others with various acceptance.

SUMMARY: We review the recent progress on the definition of random set with respect to conditional probabilities and a generalization of van Lambalgen theorem (Takahashi 2006, 2008, 2009, 2011). In addition we generalize Kjos Hanssen theorem (2010) when the consistency of the posterior distributions holds. We propose a definition of random sequences with respect to conditional probabilities as the section of the Martin–Löf random set at the random parameters and argue the validity of the definition from the Bayesian statistical point of view.

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 make an application of our framework to the BB84 quantum key distribution protocol in order to demonstrate how properly our framework works in practical problems in quantum mechanics.