Gersten's conjecture for commutative discrete valuation rings

The purpose of this article is to prove that Gersten's conjecture for a commutative discrete valuation ring is true. Combining with the result of \cite{GL87}, we learn that Gersten's conjecture is true if the ring is a commutative regular local, smooth over a commutative discrete valuation ring

Supersymmetric Contributions to the B -> phi K Decays in the PQCD Approach

We study the effects of supersymmetric contribution on both the B_d -> phi K^0 and B^\pm -> phi K^\pm modes using the perturbative QCD approach. We estimate the deviation of mixing-induced and direct CP asymmetries and discuss the strong-phase dependence of them.Comment: 4 pages, 2 figures, to appear in the proceedings of the 12th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY 2004), Tsukuba, Japan, June 17-23, 200

Remarks on modified Ding functional for toric Fano manifolds

We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano manifolds.Comment: 9 page

An introduction to computational algebraic statistics

In this paper, we introduce the fundamental notion of a Markov basis, which is one of the first connections between commutative algebra and statistics. The notion of a Markov basis is first introduced by Diaconis and Sturmfels (1998) for conditional testing problems on contingency tables by Markov chain Monte Carlo methods. In this method, we make use of a connected Markov chain over the given conditional sample space to estimate the P-values numerically for various conditional tests. A Markov basis plays an importance role in this arguments, because it guarantees the connectivity of the chain, which is needed for unbiasedness of the estimate, for arbitrary conditional sample space. As another important point, a Markov basis is characterized as generators of the well-specified toric ideals of polynomial rings. This connection between commutative algebra and statistics is the main result of Diaconis and Sturmfels (1998). After this first paper, a Markov basis is studied intensively by many researchers both in commutative algebra and statistics, which yields an attractive field called computational algebraic statistics. In this paper, we give a review of the Markov chain Monte Carlo methods for contingency tables and Markov bases, with some fundamental examples. We also give some computational examples by algebraic software Macaulay2 and statistical software R. Readers can also find theoretical details of the problems considered in this paper and various results on the structure and examples of Markov bases in Aoki, Hara and Takemura (2012).Comment: Kobe-Lyon Summer School (2015), Lecture Notes, Algorithms and Computation in Mathematics, 28 pages, 5 figure

P-time Completeness of Light Linear Logic and its Nondeterministic Extension

In CSL'99 Roversi pointed out that the Turing machine encoding of Girard's seminal paper "Light Linear Logic" has a flaw. Moreover he presented a working version of the encoding in Light Affine Logic, but not in Light Linear Logic. In this paper we present a working version of the encoding in Light Linear Logic. The idea of the encoding is based on a remark of Girard's tutorial paper on Linear Logic. The encoding is also an example which shows usefulness of additive connectives. Moreover we also consider a nondeterministic extension of Light Linear Logic. We show that the extended system is NP-complete in the same meaning as P-completeness of Light Linear Logic

Distributions of the largest singular values of skew-symmetric random matrices and their applications to paired comparisons

Let $A$ be a real skew-symmetric Gaussian random matrix whose upper triangular elements are independently distributed according to the standard normal distribution. We provide the distribution of the largest singular value $\sigma_1$ of $A$. Moreover, by acknowledging the fact that the largest singular value can be regarded as the maximum of a Gaussian field, we deduce the distribution of the standardized largest singular value $\sigma_1/\sqrt{\mathrm{tr}(A'A)/2}$. These distributional results are utilized in Scheff\'{e}'s paired comparisons model. We propose tests for the hypothesis of subtractivity based on the largest singular value of the skew-symmetric residual matrix. Professional baseball league data are analyzed as an illustrative example

Finiteness theorem on Blow-semialgebraic triviality for a family of 3-dimensional algebraic sets

In this paper we introduce the notion of Blow-semialgebraic triviality consistent with a compatible filtration for an algebraic family of algebraic sets, as an equisingularity for real algebraic singularities. Given an algebraic family of 3-dimensional algebraic sets defined over a nonsingular algebraic variety, we show that there is a finite subdivision of the parameter algebraic set into connected Nash manifolds over which the family admits a Blow-semialgebraic trivialisation consistent with a compatible filtration. We show a similar result on finiteness also for a Nash family of 3-dimensional Nash sets through the Artin-Mazur theorem. As a corollary of the arguments in their proofs, we have a finiteness theorem on semialgebraic types of polynomial mappings from the 2-dimensional Euclidean space to the p-diemnsional Euclidean space.Comment: 38 pages, 1 figur

Logarithmic Chow semistability of polarized toric manifolds

The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability for polarized toric manifolds and its toric divisors. As its application, we show the implication from the asymptotic log Chow semistability to the log K-semistability by combinatorial arguments. Furthermore we give a non-semistable example which has a conical Kahler Einstein metric

Parasupersymmetry in Quantum Graphs

We study hidden parasupersymmetry structures in purely bosonic quantum mechanics on compact equilateral graphs. We consider a single free spinless particle on the graphs and show that the Huang-Su parasupersymmetry algebra is hidden behind degenerate spectra.Comment: 17 pages, 9 eepic figures; typos corrected, a reference added, figures improved, minor changes in terminolog

Bottom-up approach to massive spin-two theory in arbitrary curved spacetime

The linear theory of massive spin-two field in arbitrary curved background is investigated. In flat spacetime, the Fierz-Pauli model is well-known as the unique linear theory describing the massive spin-two field. On the other hand, in order to construct the massive spin-two theory in fixed curved background with arbitrary metric, infinite series of nonminimal coupling terms are necessary. In [Nucl. Phys. B 584 (2000) 615], Buchbinder et al. have derived the condition for the ghost-freeness and they have solved the condition in small curvature approximation. In the leading order approximation, three free parameters of the leading order nonminimal coupling terms are allowed. However, existence of the completion corresponding to all the three parameters is not guaranteed. On the other hand, recently, a class of the completion is obtained by linearizing the dRGT model. However, the leading order nonminimal coupling terms of the linearized dRGT model depend only on one free parameter. Therefore, possibility of a class larger than the linearized dRGT model has not been excluded. In this paper, we develop the method for solving the conditions for ghost freeness in higher order and investigate whether lower order nonminimal coupling terms can be constrained by higher order conditions or not. As a result, we obtain an additional constraint on the leading order nonminimal coupling terms from the fourth order condition. Although the leading order nonminimal coupling terms of the linearized dRGT model is still a subclass of our resulting nonminimal coupling terms in the leading order, a trivial extension of the linearized dRGT model is perfectly equivalent to our resulting action.Comment: 23 pages, 1 figure