## Recursos de colección

1. #### Motion of Non-Convex Polygons by Crystalline Curvature and Almost Convexity Phenomena

Ishiwata, Tetsuya
The behavior of solution polygons to generalized crystalline curvature flow is discussed. The conditions to guarantee that the solution polygon keeps its admissibility as long as enclosed area of solution polygon is positive are clarified. We also show that the solution polygon becomes almost convex'' before the extinction time.

2. #### Complex-Analytic Approach to the Sinc-Gauss Sampling Formula

Ken'ichiro, Tanaka; Sugihara, Masaaki; Kazuo, Murota
This paper is concerned with theoretical error estimates for a sampling formula with the sinc-Gaussian kernel. Qian et al. have recently given an error estimate for the class of band-limited functions by Fourier-analytic approach. In contrast, we adopt in this paper a complex-analytic approach to derive an error estimate for a wider class of functions including unbounded functions on $\mathbf{R}$. Part of the result of Qian et al. can be derived from ours as an immediate corollary. Computational results show a fairly good agreement with our theoretical analysis.

3. #### Preconditioned GMRES Methods for Least Squares Problems

Ito, Tokushi; Hayami, Ken
For least squares problems of minimizing $\|\boldsymbol{b}-A\boldsymbol{x}\|_2$ where $A$ is a large sparse $m\times n$ ($m\ge n$) matrix, the common method is to apply the conjugate gradient method to the normal equation $A^{\mathrm{T}} A\boldsymbol{x}=A^{\mathrm{T}} \boldsymbol{b}$. However, the condition number of $A^{\mathrm{T}} A$ is square of that of $A$, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning. In this paper, we propose two methods for applying the GMRES method to the least squares problem by using an $n\times m$ matrix $B$. We give the necessary and sufficient condition that $B$ should satisfy in order that the proposed methods give a least squares solution. Then, for implementations for $B$, we propose...

4. #### Numerical Solution of Volterra Integral Equations with Weakly Singular Kernel Based on the DE-Sinc Method

Mori, Masatake; Nurmuhammad, Ahniyaz; Murai, Takefumi
A method for numerical solution of Volterra integral equations of the second kind with a weakly singular kernel based on the double exponential (DE) transformation is proposed. In this method we first express the approximate solution in the form of a Sinc expansion based on the double exponential transformation by Takahasi and Mori in 1974 followed by collocation at the Sinc points. We also apply the DE formula to the kernel integration. In every sample equation a numerical solution with very high accuracy is obtained and a nearly exponential convergence rate $\exp(-cM/{\log M})$, $c>0$ in the error is observed where $M$ is a parameter representing the number of terms in the Sinc...

5. #### Voronoi Diagrams with Respect to Criteria on Vision Information

Asano, Tetsuo; Katoh, Naoki; Tamaki, Hisao; Tokuyama, Takeshi
Voronoi diagrams for a set of geometric objects is a partition of the plane (or space in higher dimensions) into disjoint regions each dominated by some given object under a predetermined criterion. In this paper we are interested in various measures associated with criteria on goodness of an input line segment with respect to each point in the plane as the point of view.'' These measures basically show how well a segment or information displayed on the segment can be seen from the point. Mathematically, the measures are defined in terms of the shapes of the triangle determined by the point and the line segment. We study the combinatorial and...

6. #### Dynamics of Front Solutions in a Specific Reaction-Diffusion System in One Dimension

Ei, Shin-Ichiro; Ikeda, Hideo; Kawana, Takeyuki
In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.

7. #### Asymptotic Behavior of the Solutions for One-Dimensional Equations of a Viscous Reactive Gas

Yanagi, Shigenori
We consider the asymptotic behavior of the complete system of equations governing a heat-conductive, reactive, compressible viscous gas bounded by two infinite parallel plates. The motion is proved to tend towards the corresponding constant state, as time tends to infinity. Moreover, the decay rate is investigated.

8. #### A Polynomial-Time Algorithm for a Stable Matching Problem with Linear Valuations and Bounded Side Payments

Farooq, Rashid
We study an extension of the Gale--Shapley marriage model and the Shapley--Shubik assignment model by considering linear valuations and bounded side payments. Our model includes the Eriksson--Karlander hybrid model as a special case. We propose a polynomial-time algorithm which finds a pairwise-stable outcome.

9. #### Rigorous Proof of Cubic Convergence for the dqds Algorithm for Singular Values

Aishima, Kensuke; Matsuo, Takayasu; Murota, Kazuo
Fernando and Parlett observed that the dqds algorithm for singular values can be made extremely efficient with Rutishauser's choice of shift; in particular it enjoys local'' (or one-step) cubic convergence at the final stage of iteration, where a certain condition is to be satisfied. Their analysis is, however, rather heuristic and what has been shown is not sufficient to ensure asymptotic cubic convergence in the strict sense of the word. The objective of this paper is to specify a concrete procedure for the shift strategy and to prove with mathematical rigor that the algorithm with this shift strategy always reaches the final stage'' and enjoys asymptotic cubic convergence.

10. #### Applications of Operator-Splitting Methods to the Direct Numerical Simulation of Particulate and Free-Surface Flows and to the Numerical Solution of the Two-Dimensional Elliptic Monge--Ampère Equation

Glowinski, R.; Dean, E.J.; Guidoboni, G.; Juárez, L.H.; Pan, T.-W.
The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge--Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.

11. #### An Observation on the Cyclicity of the Group of the $\mathbb{F}_p$-Rational Points of Abelian Surfaces

Yamauchi, Takuya
Let $A$ be a principally polarized Abelian surface defined over $\mathbb{Q}$ with $\End(A)=\mathbb{Z}$ and $\widetilde{A}$ be the reduction at a good prime $p$. In this paper, we study the density of prime numbers $p$ for which $\widetilde{A}(\mathbb{F}_p)$ is a cyclic group and establish a conjecture which relates this density.

12. #### Index Calculus Attack for Jacobian of Hyperelliptic Curves of Small Genus Using Two Large Primes

Nagao, Koh-ichi
This paper introduces a fast algorithm for solving the DLP of Jacobian of hyperelliptic curve of small genus. To solve the DLP, Gaudry first shows that the idea of index calculus is effective, if a subset of the points of the hyperelliptic curve of the base field is taken by the smooth elements of index calculus. In an index calculus theory, a special element (in our case it is the point of hyperelliptic curve), which is not a smooth element, is called a large prime. A divisor, written by the sum of several smooth elements and one large prime, is called an almost smooth divisor. By the use of the...

13. #### All Harmonic Numbers Less than $10^{14}$

Goto, Takeshi; Okeya, Katsuyuki
A positive integer $n$ is said to be \textit{harmonic} if the harmonic mean $H(n)$ of its positive divisors is an integer. Ore proved that every perfect number is harmonic and conjectured that there exist no odd harmonic numbers greater than $1$. In this article, we give the list of all harmonic numbers up to $10^{14}$. In particular, we find that there exist no nontrivial odd harmonic numbers less than $10^{14}$.

14. #### Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five

Harasawa, Ryuichi; Sueyoshi, Yutaka; Kudo, Aichi
In this paper, we consider the Tate and Ate pairings for the genus-$2$ supersingular hyperelliptic curves $y^{2}=x^{5} -\alpha x$ ($\alpha = \pm2$) defined over finite fields of characteristic five. More precisely, we construct a distortion map explicitly, and show that the map indeed gives an input for which the value of the Tate pairing is not trivial. We next describe a computation of the Tate pairing by using the proposed distortion map. We also see that this type of curve is equipped with a simple quintuple operation on the Jacobian group, which leads to an improvement for computing the Tate pairing. We further show the Ate pairing, a variant of the Tate pairing for...

15. #### Computing Certain Fano 3-Folds

Brown, Gavin; Suzuki, Kaori
We use the computer algebra system Magma to study graded rings of Fano 3-folds of index $\ge 3$ in terms of their Hilbert series.

16. #### Efficient Verification of Tunnell's Criterion

Bach, Eric; Ryan, Nathan C.
An integer $n$ is congruent if there is a triangle with rational sides whose area is $n$. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids $2x^2+y^2+8z^2 = n$ and $2x^2+y^2+32z^2=n$. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell's criterion use $O(n)$ operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using time $n^{1/2 + o(1)}$ and space $n^{o(1)}$, and a deterministic algorithm using space and time $n^{1/2 + o(1)}$.

17. #### Preface and Guest Editors' Preface

Sugihara, Kokichi; Nakamula, Ken

18. #### Tighter Bounds of Errors of Numerical Roots

Sasaki, Tateaki
Let $P(z)$ be a monic univariate polynomial over $\mathbf{C}$, of degree $n$ and having roots $\zeta_1,\ldots,\zeta_n$. Given approximate roots $z_1,\ldots,z_n$, with $\zeta_i \simeq z_i$ ($i=1,\ldots,n$), we derive a very tight upper bound of $|\zeta_i - z_i|$, by assuming that $\zeta_i$ has no close root. The bound formula has a similarity with Smale's and Smith's formulas. We also derive a lower bound of $|\zeta_i - z_i|$ and a lower bound of $\min\{|\zeta_j - z_i|\mid j \neq i\}$.

19. #### Asymptotic Analysis of the Model for Distribution of High-Tax Payers

Yamamoto, Hiroshi; Ohtsuki, Toshiya; Fujihara, Akihiro; Tanimoto, Satoshi; Yamamoto, Keizo; Miyazima, Sasuke
The z-transform technique is used to investigate the model for distribution of high-tax payers, which is proposed by two of the authors (K. Y and S. M) and others [12]--[14]. Our analysis shows an asymptotic power-law of this model with the exponent $-5/2$ when a total mass'' has a certain critical value. Below the critical value, the system exhibits an ordinary critical behavior, and scaling relations hold. Above the threshold, numerical simulations show that a power-law distribution coexists with a huge monopolized'' member. It is argued that these behaviors are observed universally in conserved aggregation processes, by analizing an extended model.

20. #### On Testing Hypothesis of Fuzzy Sample Mean

Wu, Berlin; Chang, Shu-Kwang
In many expositions of fuzzy methods, fuzzy techniques are described as an alternative to a more traditional statistical approach. In this paper, we present a class of fuzzy statistical decision process in which testing hypothesis can be naturally reformulated in terms of interval-valued statistics. We provide the definitions of fuzzy mean, fuzzy distance as well as investigation of their related properties. We also give some empirical examples to illustrate the techniques and to analyze fuzzy data. Empirical studies show that fuzzy hypothesis testing with soft computing for interval data are more realistic and reasonable in the social science research. Finally certain comments are suggested for the further studies. We hope...

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