Research Projects
Research Projects
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Construction of Theory of Digital Analysis
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
2008-2010YAMADA Yoshio, TAKAHASHI Daisuke, MATSUSHIMA Toshiyasu, KASHIWAGI Masahide, NISHIDA Takaaki, OISHI Shin'ichi
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Our research group is composed of scholars working in the areas of discrete mathematics, nonlinear differential equations, information theory and numerical computation. We have organized "Seminar on Digital Analysis" so that members can hold common understanding and insight on the fundamental theories and ideas of digital mathematics. As speakers of this seminar, we have invited 16 researchers who are highly active in the areas of discrete mathematics, mathematical modeling, information theory and numerical computation. We have succeeded in getting common understanding on digital analysis through exciting discussions in each lecture of the seminar.
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Stream Cipher System Based on Chaotic Binary Sequences
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
1997-1999KOHDA Tohru, TSUNEDA Koichi, OOHAMA Yasutada, NISHI Testuo, FUJISAKI Hiroshi
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Stream ciphers provide probably the most important method of modern encipherment. The central problem in stream cipher cryptography is the difficulty of efficiently generating long unpredictable sequencers of binary signals form a short and random key. Although such an unpredictable sequence can be generated in various ways, linear feedback shift register (LFSR) sequences are employed in nearly all the methods. In this study, not using such sequences, we give a simple method to obtain a running key sequence of stream cipher system in which chaotic orbits of nonlinear maps are used. Two type of sequences of balanced I. I. D. (dependent and identically distributed) binary random variables have been defined, referred to as a chaotic binary sequence and a chaotic bit sequence, each of which is obtained from chaotic real-valued orbits generated by nonlinear maps. The stream cipher system proposed in this study has the following characteristics : 1) There are several parameters as short secret key : 2) The linear complexity of the running-key sequence of period N is nearly equal to N/2 : 3) The correlation property of ciphertexts is as good as those of the DES and FEAL
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Research of Analysis Method for Very High Speed Integrated Circuits Composed of Distributed and Lumped Elements
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
1996-1997NISHI Tetsuo, KASHIWAGI Masahide, KOHDA Tohru
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(1) We summarized the state of the art of this research subject and classified systematically various analysis methods so far known.(2) We pointed out that both the characteristics impedance function ZETA_0(s) and a bilinear, functionof the phase characteristics rheta(s) of distributed elements have to bc approximated by rational positive real functions. We proposed the method for approximating these quantities by positive real functions. This resulted in better property of convergence than the conventional Pade approximation method.(3) A new transient analysis method for multi-conductor transmission line terminated with lumped-constant elements was proposed and the validity by numerical simulation was shown. The method utilizes multi-dimensional WDF (Wave Digital Filter), which Fettweis proposed for partial differential equation. Our method solved the difficulty arising from the treatment of mixed lumped- and distributed systems.(4) fi new equivalent circuit for a dielectric Alter was proposed, where the equivalent circuit consists.i multiconductor transmission line and lumped constant elements and its element values are determined from experimental results. The total characteristics of the circuit are much coincident with physical results over wide frequency than usual model.(5) The properties of lossless bandpass ladder network composed of only two kinds of resonant sections consisting of lumped- and distributed-elements were investigated and the necessary and sufficient conditions for realization were given.(6) A theoretical lower bound for the ratio of the maximum error of the minimal approximation to that of the the least pth approximation by a rational function is given. Numerical examples on various kind of functions verified that the above lower bound is a good estimation for the actual values
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Research of Computer aided Nonlinear Analysis with Flexibility
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
1995-1997HORIUCHI Kazuo, KASHIWAGI Masahide, YAMAMURA Kiyotaka, OISHI Shin'ichi, MATSUMOTO Takashi, KAWASE Takehiko
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We had the purpose to establish fundamental theories and to constitute the elements of computer aided mathematical analyzing software. We developed our research as we planned as the following :
(1) We proposed the fixed point theorem for fuzzy map which is obtained by modeling the system with uncertain property.
(2) We make the theory to prove numerically the existence of solution for nonlinear operator equations.
(3) Based on C++ and an object oriented language in which rational number arithmetic is implemented, we constituted 3 prototypes of object oriented software which can deal with various objects corresponding to interval analysis, automatic differentiation, function representation and so on.
(4) By extending the methods to prove numerically the existence of bifurcation point, we developed the theory to cancel singular points. We also applied our theory to various types of bifurcation phenomena and indicated that we can prove the existence of actual bifurcation phenomena.
(5) We proposed the theory to prove the existence of homoclinic orbits or heteroclinic orbits and prove their existence for actual examples.
(6) We proposed an algorithm to prove the existence of all solutions in a bounded region for finite dimensional nonlinear equations and proved that this algorithm stops within finite steps under the certain conditions.
(7) We proposed a method to prove the existence of all solutions with high speed in a bounded region for finite dimensional nonlinear equations with separability, whose example is VLSI circuit. (8) We could change the speed of calculation by the accuracy. Concretely, We proposed the method in which we can obtain the calculated results with super high speed when we demand its low accuracy and in which we can obtain the results with high speed even when we demand its high accuracy.
(9) We realized the obtained techniques of numerical method with guaranteed accuracy on our prototypes of the software. We applied our software to various nonlinear functional equations and indicated its usefulness.
(10) We combined the numerical method in the case that we demand its low accuracy and the numerical one in the case that we demand its high accuracy, by which we proposed the numerical method with high speed at our request of accuracy. We realized this method on our software and indicated its usefulness by the example of nonlinear circuit systems.
(11) We integrated the above organized investigations and remade a prototype of the software for computer aided nonlinear analysis which can correspond to the changes of the problem or the accuracy. We also indicated its usefulness by applying it to nonlinear circuit problems. -
STUDY ON MODELLING OF NONLINEAR SYSTEM AND SELF-VALIDATING MUMERICAL METHOD
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research
Project Year :
1992-1994HORIUCHI Kazuo, KASHIWAGI Masahide, YAMAMURA Kiyotaka, OISHI Shin'ichi, MATSUMOTO Takashi, KAWASE Takehiko
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Recently, the study of nonlinear systems and nonlinear technologes has made a great advance. For example, the studies of optical fiber soliton communications, neural networks, fuzzy systems, analogue VLSI and so on have achieved much interests. Since such systems essentially use nonlinearity , new modelling techniques and reliable simulation techniques are required. Moreover, in the field of computer assisted design of nonlinear systems, it is very important to guarantee the accuracy of the result of calculation. For example, in the VLSI design, if we can validate the accuracy of modeling and numerical simulation, it enables to reduce the cost and period of design.
In this study, taking the accuracy of modeling into consideration, and guarantteng the accuracy of numerical simulation of the modeling, we have develpoed a numerical validation method through the total simulation process of nonlinear system.
In this year, we have improved the theory, algorithm and system developed until last year, and have made them more practical.
1.Effectiveness of our fuzzy modeling theory is confirmed by numerical simulation.
2.Automatic numerical validation method for general ordinary differential equations is developed. It provides a rigorous method for transient analysis.
3.Extending the technique used in all solution method for nonlinear equations, we have developed a inclusion method for solution sets of set-valued functions. It gives a more rigorous system analysing method together with modeling method with guaranteed accuracy . -
非線形常微分方程式の精度保証付き数値計算
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本研究は、計算機自身に数値計算を行うと同時に結果の精度の保証を行わせるという発想に基づく精度保証付き数値計算の分野において、特に非線形常微分方程式の精度保証付き計算の新しい方法を開発することを目的として行なわれたものである。Lohnerが提案した区間解析とTaylor展開を用いる方法について、ベキ級数展開に対する演算を定義することにより新しく捉えなおし,またそれを大幅に改良した方法を構築し,詳細な理論的解析を行うことによりこの方法の有効性を明らかにした。また,本研究の方法は,自動微分法と呼ばれる新しい数値計算法と深い関連があるが,そこで得られた知見を導入することにより更に改良できることを示した。また、この方法を実際に計算機上に実現し、それが用意に実装可能であることを示した。このシステムは高い汎用性を持ち、問題の記述は極めて用意である。更に、特に初期値問題における長時間の積分について、平均値の定理を用いた誤差補償法の導入によって大幅な改良に成功し、実用的問題に対しても適用可能であることを示した。また、本研究で得られたベキ級数に対する演算法は、常微分方程式に限らずより多くの問題に適応可能であることが分かり、それについても検討した。これにより、高次導関数の計算、関数の値域の評価、精度保証付き数値積分が効率的に行なえることが分かった
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中級数演算による常微分方程式の精度保証付き数値計算
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本研究は、計算機自身に数値計算を行うと同時に結果の精度の保証を行わせるという発想に基づく精度保証付き数値計算の分野において、特に非線形常微分方程式の精度保証付き計算の新しい方法を開発することを目的として行なわれたものである。Lohnerが提案した区間解析とTaylor展開を用いる方法について、ベキ級数展開に対する演算を定義することにより新しく捉えなおし,またそれを大幅に改良した方法を構築し、詳細な理論的解析を行うことによりこの方法の有効性を明らかにした。また、この方法を実際に計算機上に実現し、それが用意に実装可能であることを示した。このシステムは高い汎用性を持ち、問題の記述は極めて用意である。更に、特に初期値問題における長時間の積分について、平均値の定理を用いた誤差補償法の導入によって大幅な改良に成功し、実用的問題に対しても適用可能であることを示した。更に、Shooting MethodとKrawczykの方法を組み合わせて2点境界値問題に対しても適用可能であることを示した。また、本研究で得られたベキ級数に対する演算法は、常微分方程式に限らずより多くの問題に適応可能であることが分かり、それについても検討した。これにより、高次導関数の計算、関数の値域の評価、精度保証付き数値積分が効率的に行なえることが分かった
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線形計画法を援用した新しい区間演算方式に関する研究
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計算機自身に数値計算を行うと同時に結果の精度の保証を行わせるという発想とその研究は古くから行われてきた。精度保証付き数値計算と呼ばれるようになったこの分野は、ドイツを中心として近年急速に発展し、今後の数値計算法のあるべき姿として世界的に注目を集めている。区間演算とは、一つの実数値を[下限,上限]のように2つの計算機で表現可能な数で挟んで演算を行う方式であり、計算誤差の把握のみならず一種の集合値演算を行うことが出来、精度保証付き数値計算を支える最も重要な技法の一つである。ところが、一方で評価が悲観的に過ぎ、確かに真の値を含む区間を得られるもののその幅はしばしば非現実的に大きくなってしまうという問題点を抱えている。本研究では、入力変数の線形和をデータメンバとする特殊な演算を定義し、線形計画法を用いることにより区間演算の過大評価の問題を解決し、一般的かつ超高性能な区間演算法を確立した。また、本手法で用いる線形計画法として一般的な単体法を用いた場合、本手法の中で解かれる複数の線形計画問題は共通の制約を数多く持つことから、計算方法を工夫することにより大幅に計算量を削減できることを示した。これにより、線形計画問題の許容解の一頂点を求めるいわゆるPhase-1を完全に省略できる。また、本手法を多くのプログラムに対して素直に適用可能な様に計算機上に実装し、その有効性を確認した。上記のライブラリを用いて、区間演算による値域の評価を基礎とした数多くのアルゴリズムに適用し、その効果を確認した。これにより、理論的には問題なくとも区間演算の過大評価が大き過ぎて実用には難しいと思われていた多くのアルゴリズムが救済されたと考えられる。また、本手法内で使用される線形計画法のアルゴリズム自身の精度保証の問題についても、新たな精度保証付き線形計画法のの開発によって解決することが出来た
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