Updated on 2024/02/04

写真a

 
SASAKI Toru
 
Organization
Faculty of Environmental, Life, Natural Science and Technology Professor
Position
Professor
External link

Degree

  • 博士 (数理科学) ( 東京大学 )

Research Interests

  • Biomathematics

  • Applied Analysis

  • Functional Equations

Research Areas

  • Natural Science / Applied mathematics and statistics

  • Natural Science / Basic mathematics

  • Natural Science / Mathematical analysis

Education

  • The University of Tokyo   大学院数理科学研究科  

    1992.4 - 1993.3

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    Notes： 博士課程

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  • The University of Tokyo   大学院理学系研究科  

    1986.4 - 1988.3

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    Notes： 修士課程

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  • The University of Tokyo   理学部   数学科

    1981.4 - 1986.3

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Research History

  • Okayama University   Professor

    2023.4

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  • Okayama University   環境生命科学研究科   Professor

    2020.10 - 2023.3

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  • Okayama University   The Graduate School of Environmental and Life Science   Associate Professor

    2012.4 - 2020.9

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  • Okayama University   大学院環境学研究科   Associate Professor

    2007.4 - 2012.3

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  • Okayama University   大学院環境学研究科   Lecturer

    2005.4 - 2007.3

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  • Okayama University   Faculty of Environmental Science and Technology Department of Environmental and Mathematical Sciences   Lecturer

    1994.10 - 2005.3

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  • Okayama University   教養部   Lecturer

    1993.10 - 1994.9

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Professional Memberships

  • THE JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS

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  • THE JAPANESE SOCIETY FOR MATHEMATICAL BIOLOGY

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  • THE ECOLOGICAL SOCIETY OF JAPAN

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  • THE MATHEMATICAL SOCIETY OF JAPAN

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Committee Memberships

  • 日本数理生物学会   会計監事  

    2017.1 - 2018.12   

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    Committee type：Academic society

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  • 日本数理生物学会   運営委員  

    2015.1 - 2016.12   

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    Committee type：Academic society

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  • 日本数理生物学会   幹事長  

    2015.1 - 2016.12   

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    Committee type：Academic society

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  • 日中韓数理生物コロキウム   第5回日中韓数理生物コロキウム実行委員  

    2015.1   

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    Committee type：Other

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  • RIMS 共同研究   第10回生物数学の理論と応用 研究代表者  

    2013.1 - 2013.12   

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    Committee type：Other

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  • 新しい研究の芽を育む会   選考委員  

    2012.4 - 2014.3   

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  • 日本数理生物学会   年会 大会副委員長  

    2012.1 - 2012.12   

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    Committee type：Academic society

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  • 日本数理生物学会   研究奨励賞選考委員  

    2011.10 - 2014.10   

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    Committee type：Academic society

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  • 日本数理生物学会   大久保賞選考委員  

    2011.10 - 2014.9   

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    Committee type：Academic society

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  • 日本数理生物学会   研究奨励賞選考委員  

    2010.1 - 2010.12   

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    Committee type：Academic society

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  • 第2回日中数理生物コロキウム   Local Co-Chair  

    2008.1 - 2008.12   

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    Committee type：Other

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  • 日本数理生物学会   学会サーバー運営委員  

    2007.1 - 2008.12   

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    Committee type：Academic society

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  • 日本数理生物学会   学会サーバー運営委員会 委員長  

    2005.1 - 2006.12   

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    Committee type：Academic society

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  • 日本数理生物学会   幹事  

    2005.1 - 2006.12   

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    Committee type：Academic society

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Papers

  • Global stability of an age-structured infection model in vivo with two compartments and two routes Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki, Yoji Otani

    Mathematical Biosciences and Engineering   19 ( 11 )   11047 - 11070   2022

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    Publishing type：Research paper (scientific journal)   Publisher：American Institute of Mathematical Sciences (AIMS)  

    <p lang="fr">&lt;abstract&gt;&lt;p&gt;In this paper, for an infection age model with two routes, virus-to-cell and cell-to-cell, and with two compartments, we show that the basic reproduction ratio $ R_0 $ gives the threshold of the stability. If $ R_0 &amp;gt; 1 $, the interior equilibrium is unique and globally stable, and if $ R_0 \le 1 $, the disease free equilibrium is globally stable. Some stability results are obtained in previous research, but, for example, a complete proof of the global stability of the disease equilibrium was not shown. We give the proof for all the cases, and show that we can use a type reproduction number for this model.&lt;/p&gt;&lt;/abstract&gt;</p>

    DOI： 10.3934/mbe.2022515

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  • Global stability for an age-structured multistrain virus dynamics model with humoral immunity Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki, Yoji Otani

    Journal of Applied Mathematics and Computing   62 ( 1-2 )   239 - 279   2020.2

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    DOI： 10.1007/s12190-019-01283-w

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    Other Link： http://link.springer.com/article/10.1007/s12190-019-01283-w/fulltext.html

  • Global stability of an age-structured model for pathogen–immune interaction Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki, Yoji Otani

    Journal of Applied Mathematics and Computing   59 ( 1-2 )   1 - 30   2019.2

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Verlag  

    In this paper, we present an age-structured mathematical model for infectious disease in vivo with infection age of cells. The model contains an immune variable and the effect of absorption of pathogens into uninfected cells. We construct Lyapunov functionals for the model and prove that the time derivative of the functionals are nonpositive. Using this, we prove the global stability results for the model. Especially, we present the full mathematical detail of the proof of the global stability.

    DOI： 10.1007/s12190-018-1194-8

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  • Asymptotic behaviour of the solutions to a virus dynamics model with diffusion Reviewed

    Toru Sasaki, Takashi Suzuki

    Discrete and Continuous Dynamical Systems - Series B   23 ( 2 )   525 - 541   2018.3

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    Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：American Institute of Mathematical Sciences  

    Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the L1 boundedness of the orbit. Then the precompactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number R0 is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if R0 &gt
    1, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

    DOI： 10.3934/dcdsb.2017206

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  • LYAPUNOV FUNCTIONALS FOR MULTISTRAIN MODELS WITH INFINITE DELAY Reviewed

    Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   22 ( 2 )   507 - 536   2017.3

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST MATHEMATICAL SCIENCES-AIMS  

    We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

    DOI： 10.3934/dcdsb.2017025

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  • LYAPUNOV FUNCTIONALS FOR VIRUS-IMMUNE MODELS WITH INFINITE DELAY Reviewed

    Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   20 ( 9 )   3093 - 3114   2015.11

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST MATHEMATICAL SCIENCES-AIMS  

    We present a systematic method to construct Lyapunov functionals of delay differential equation models of infectious diseases in vivo. For generality we construct Lyapunov functionals of models with infinitely distributed delay. We begin with simpler models without delay and construct Lyapunov functionals for the complex models progressively. We construct those functionals using our result obtained previously instead of constructing each functional independently. Additionally we discuss some problems that arise from the mathematical requirements caused by the infinitely distributed delay.

    DOI： 10.3934/dcdsb.2015.20.3093

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  • CONSTRUCTION OF LYAPUNOV FUNCTIONS FOR SOME MODELS OF INFECTIOUS DISEASES IN VIVO: FROM SIMPLE MODELS TO COMPLEX MODELS Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi

    MATHEMATICAL BIOSCIENCES AND ENGINEERING   12 ( 1 )   117 - 133   2015.2

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST MATHEMATICAL SCIENCES  

    We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.

    DOI： 10.3934/mbe.2015.12.117

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  • Construction of Lyapunov functionals for delay differential equations in virology and epidemiology Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi

    NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS   13 ( 4 )   1802 - 1826   2012.8

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：PERGAMON-ELSEVIER SCIENCE LTD  

    In the present paper, we present a method for constructing a Lyapunov functional for some delay differential equations in virology and epidemiology. Here some delays are incorporated to the original ordinary differential equations, for which a Lyapunov function is already obtained. We present simple and clear explanation of our method using some models whose Lyapunov functionals are already obtained. Moreover, we present several new results for constructing Lyapunov functionals using our method. (C) 2011 Elsevier Ltd. All rights reserved.

    DOI： 10.1016/j.nonrwa.2011.12.011

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  • Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics Reviewed

    Gang Huang, Hiroki Yokoi, Yasuhiro Takeuchi, Tsuyoshi Kajiwara, Toru Sasaki

    JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS   28 ( 3 )   383 - 411   2011

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

    This paper investigates a class of viral infection models with a nonlinear infection rate and two discrete delays, one of which represents an intracellular latent period for the contacted target cell with virus to begin producing virions, the other of which represents the time needed in cytotoxic T cells (CTLs) response before immune becomes effective after a novel pathogen invades. Since immune system is a complex network of cells and signals that have evolved to respond to the presence of pathogens, we further assume two situations for immune activation delay. When both delays are ignored, the global stability for the ordinary differential equations model are established. While both delays are included, the positivity and boundedness of all solutions of the delay differential equations model are proved. Utilizing Lyapunov functionals and LaSalle invariance principle, the global dynamical properties are also studied. In particular, stability switch is shown to occur as immune delay increasing by bifurcation theory. Our results exhibit that the intracellular delay does not affect the stability of equilibria. However, the immune activation delay is able to destabilize the interior equilibrium and brings periodic solutions. Numerical simulations are performed to verify the theoretical results and display the different impacts of two type delays in two cases. Those analysis give us some useful suggestions on new drugs to fight against viral infection such that it is effective for the drugs to prolong the latent period, and/or to reduce the activation delay of CTLs immune response and/or to inhibit infection.

    DOI： 10.1007/s13160-011-0045-x

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  • Global stability of models of humoral immunity against multiple viral strains Reviewed

    Toru Inoue, Tsuyoshi Kajiwara, Toru Sasaki

    Journal of Biological Dynamics   4 ( 3 )   282 - 295   2010.5

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    Language：English   Publishing type：Research paper (scientific journal)  

    We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state. © 2010 Taylor &amp
    Francis.

    DOI： 10.1080/17513750903180275

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  • Global stability of pathogen-immune dynamics with absorption Reviewed

    Tsuyoshi Kajiwara, Toru Sasaki

    Journal of Biological Dynamics   4 ( 3 )   258 - 269   2010.5

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    Language：English   Publishing type：Research paper (scientific journal)  

    In this paper, we consider the global stability of the models which incorporate humoural immunity or cellmediated immunity.We consider the effect of loss of a pathogen, which is called the absorption effect when it infects an uninfected cells.We construct Lyapunov functions for these models under some conditions of parameters, and prove the global stability of the interior equilibria. It is impossible to remove the condition of parameters for the model incorporating humoural immunity. © 2010 Taylor &amp
    Francis.

    DOI： 10.1080/17513750903051989

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  • On the optimal duration of memory of losing a conflict - a mathematical model approach Reviewed

    Toru Sasaki, Kensuke Okada, Tsuyoshi Kajiwara, Takahisa Miyatake

    Journal of Biological Dynamics   4 ( 3 )   270 - 281   2010.5

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    Language：English   Publishing type：Research paper (scientific journal)  

    In male broad-horned flour beetles, Gnatocerus cornutus, losers of conflicts avoid fighting at subsequent encounters. The loser effect lasts for 4 days. It is considered that the memory of losing remains for 4 days. The duration of the memory is expected to affect the fitness, and the duration, 4 days, is expected to be optimal.We consider the fitness of a mutant in an homogeneous population to obtain the optimal duration. Here we carry out simulations using an individual-based model. The results suggest that the trade-off of getting mating chances and avoiding damage can cause the optimal duration of the memory, and that the decay in time of the female population is an important factor. © 2010 Taylor &amp
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    DOI： 10.1080/17513750903161036

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  • Dynamical properties of autoimmune disease models: Tolerance, flare-up, dormancy Reviewed

    Shingo Iwami, Yasuhiro Takeuchi, Yoshiharu Miura, Toru Sasaki, Tsuyoshi Kajiwara

    JOURNAL OF THEORETICAL BIOLOGY   246 ( 4 )   646 - 659   2007.6

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD  

    The mechanisms of autoimmune disease have remained puzzling for a long time. Here we construct a simple mathematical model for autoimmune disease based on the personal immune response function and the target cell growth function. We show that these two functions are sufficient to capture the essence of autoimmune disease and can explain characteristic symptom phases such as tolerance, repeated flare-ups and dormancy. Our results strongly suggest that a more complete understanding of these two functions will underlie the development of an effective therapy for autoimmune disease. (C) 2007 Elsevier Ltd. All rights reserved.

    DOI： 10.1016/j.jtbi.2007.01.020

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  • Global Dynamics of B Cells and Anti-Idiotipic B Cells and its Application to Autoimmunity Reviewed

    Toru Sasaki, Tsuyoshi Kajiwara

    Japan Journal of Industrial and Applied Mathematics   24 ( 1 )   105 - 118   2007

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    Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：Springer Tokyo  

    Global behavior of B models is discussed. When the source term for new B cells equals zero, the system has a conservation quantity. It implies the structurally unstability. It suggests that lack of the source of new B cells may unstabilize the immune system. When the B model incorporates autoimmunity, it loses symmetry. The asymmetry suggests the transition from a tolerant state to autoimmune state is more likely than the inverse transition. Effect of dose of antigen is also considered.

    DOI： 10.1007/BF03167510

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  • Asymptotic analysis of a chemotactic model of bacteria colonies Reviewed

    S Miyata, T Sasaki

    MATHEMATICAL BIOSCIENCES   201 ( 1-2 )   184 - 194   2006.5

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE INC  

    An estimate of the distance between spots generated by a bacterial colony model is obtained. The model describes the morphogenesis of a spot pattern in colonies of chemotactic strains of Escherichia coli. Asymptotic methods for other cell-chemotaxis models, which have been successfully used by previous researchers, can be applied also to this model. However the calculations and the result is more complicated for this model. The result is verified by comparing it with the results by numerical computations of solutions of the model. (c) 2005 Elsevier Inc. All rights reserved.

    DOI： 10.1016/j.mbs.2005.12.007

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  • Stability analysis of pathogen-immune interaction dynamics Reviewed

    A Murase, T Sasaki, T Kajiwara

    JOURNAL OF MATHEMATICAL BIOLOGY   51 ( 3 )   247 - 267   2005.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

    The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.

    DOI： 10.1007/s00285-005-0321-y

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  • A note on the stability analysis of pathogen-immune interaction dynamics Reviewed

    T Kajiwara, T Sasaki

    DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B   4 ( 3 )   615 - 622   2004.8

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST MATHEMATICAL SCIENCES  

    The stability analysis of the interior equilibria, whose components are all positive, of non linear ordinary differential equation models describing in vivo dynamics of infectious diseases are complicated in general. Liu, "Non-linear oscillation in models of immune responses to persistent viruses, Theor. Popul. Biol. 52(1997), 224-230" and Murase, Sasaki and Kajiwara, "Stability analysis of pathogen-immune interaction dynamics (submitted)" proved the stability of the interior equilibria of such models using symbolic calculation software on computers. In this paper, proofs without using symbolic calculation software of the stability theorems given by Liu and Murase et al. are presented. Simple algebraic manipulations, properties of determinants, and their derivatives are used. The details of the calculation given by symbolic calculation software can be seen clearly.

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  • The effect of local prevention in an SIS model with diffusion Reviewed

    Toru Sasaki

    Discrete and Continuous Dynamical Systems - Series B   4 ( 3 )   739 - 746   2004

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    Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：Southwest Missouri State University  

    The effect of spatially partial prevention of infectious disease is considered as an application of population models in inhomogeneous environments. The area is divided into two ractangles, and the local contact rate between infectives and susceptibles is sufficiently reduced in one rectangle. The dynamics of the infection considered here is that described by an SIS model with diffusion. Then the problem can be reduced to a Fisher type equation, which has been fully studied by many authors, under some conditions. The steady states of the linearized equation are considered, and a Nagylaki type result for predicting whether the infection will become extinct over time or not is obtained. This result leads to some necessary conditions for the extinction of the infection.

    DOI： 10.3934/dcdsb.2004.4.739

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  • Propagation of ultradifferentiability for the solutions of semi-linear hyperbolic equations in one space dimension Reviewed

    Toru Sasaki

    Journal of the Faculty of Science, the Universtity of Tokyo   40 ( 2 )   529 - 547   1993

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  • INTERACTION OF 2 NONLINEAR-WAVES AT THE BOUNDARY Reviewed

    T SASAKI

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES   63 ( 10 )   375 - 378   1987.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：JAPAN ACAD  

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Books

  • 数理モデリング入門 : ファイブ・ステップ法

    Meerschaert, Mark M., 佐藤, 一憲, 梶原, 毅, 佐々木, 徹, 竹内, 康博, 宮崎, 倫子, 守田, 智（ Role： Joint translator）

    共立出版  2015.1  （ ISBN:9784320111004 ）

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    Total pages：xiv, 382p   Language：Japanese

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  • 偏微分方程式

    John, Fritz, 佐々木, 徹, 示野, 信一, 橋本, 義武（ Role： Joint translator）

    丸善出版  2012.3  （ ISBN:9784621065600 ）

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    Total pages：ix, 321p   Language：Japanese

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  • 生物数学入門 : 差分方程式・微分方程式の基礎からのアプローチ

    Allen, Linda J. S., 竹内, 康博, 佐藤, 一憲, 守田, 智, 宮崎, 倫子（ Role： Joint translator）

    共立出版  2011.10  （ ISBN:9784320057159 ）

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    Total pages：xiv, 440p   Language：Japanese

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  • 「数」の数理生物学

    日本数理生物学会, 瀬野, 裕美（ Role： Contributor）

    共立出版  2008.9  （ ISBN:9784320056756 ）

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    Total pages：viii, 224p   Language：Japanese

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  • 感染症の数理モデル

    稲葉, 寿, 西浦, 博, 梶原, 毅, 佐々木, 徹, 竹内, 康博, 細野, 雄三, 増田, 直紀, 今野, 紀雄, 梯, 正之, 加茂, 将史, 佐々木, 顕（ Role： Contributor）

    培風館  2008.7  （ ISBN:9784563011376 ）

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    Total pages：x, 311p   Language：Japanese

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  • 進化のダイナミクス : 生命の謎を解き明かす方程式

    Nowak, M. A. (Martin A.), 竹内, 康博, 佐藤, 一憲, 巌佐, 庸, 中岡, 慎治（ Role： Joint translator）

    共立出版  2008.2  （ ISBN:9784320056657 ）

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    Total pages：xii, 333p   Language：Japanese

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  • 生物集団の数学 : 人口学・生態学・疫学へのアプローチ

    Thieme, Horst R., 齋藤, 保久（ Role： Joint translator）

    日本評論社  2006  （ ISBN:4535784183 ）

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    Total pages：2冊   Language：Japanese

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MISC

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Presentations

  • Stability analysis of a simple cell-pathogen-immune system

    Toru Sasaki

    2023.9.6 

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    Event date： 2023.9.4 - 2023.9.6

    Language：Japanese   Presentation type：Oral presentation (general)  

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  • ウイルスダイナミクス基本モデルの安定性解析

    佐々木徹, 梶原毅, 應谷洋二, 石丸優希

    第16回生物数学の理論とその応用  2020.1.30 

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    Event date： 2020.1.27 - 2020.1.31

    Language：Japanese   Presentation type：Oral presentation (general)  

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  • クマの食べ残しの競争系への効果について

    佐々木徹, 坪田一輝

    数理生物学会年会  2021.9.13 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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Research Projects

  • Analysis of functional equations describing dynamics of infectious disease

    Grant number：17K05365  2017.04 - 2023.03

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    佐々木 徹

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    Grant amount：\3510000 （ Direct expense: \2700000 、 Indirect expense：\810000 ）

    ウイルスダイナミクス・数理モデルにおいて安定な平衡点を不安定化させうる要素として，(1) 時間遅れ (感染からウイルス放出までのタイムラグ), (2) 吸収効果 (感染時にウイルスが細胞内に吸収され、その後新しい細胞に感染しなくなる効果), (3)免疫変数の採用 (免疫の強さを未知変数として取り入れる), の三つに焦点をあてる研究を, 前年度に引き続き行なった. 上記 3 つの効果の他に, 未知変数として感染細胞密度を取り入れる事も平衡点を不安定化させうる (Murase, Kajiwara, Sasaki 2005). これは, 細胞が感染した後感染細胞に変化し, その感染細胞がウイルスを放出するという意味では時間遅れと考えられる. しかし, 今まで検討したのは, いわゆる離散時間遅れと呼ばれる場合で, 感染細胞経由による時間遅れとは意味が異なる. これに関して, 時間遅れがガンマ分布に従う場合の考察を行なった.
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    ウイルスダイナミクス・数理モデルにおいて, 2 つの感染経路と 2 つのコンパーメント, 齢構造を考慮した微分方程式系の解析を進めた. 前年度に得た平衡点の安定性に関する結果を進めるとともに, 時間大域解の存在と一意性について詳細に検討したり，解の正値性に関しては先行研究を調査し研究を進めた. また, 先行研究の基礎再生産数に関する議論に対して, タイプ別再生産数の概念を利用してその意味に関する検討を進めた. 更にパーシステンスなどについて詳細に解析を行ない, 研究を進めた. これらの結果を詳細に検討し, まとめる作業を行ない, 論文の作成に着手した.

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  • Qualitative theory of differential equations describing dynamics of infectious disease

    Grant number：18540122  2006 - 2009

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    SASAKI Toru, KAJIWARA Tsuyoshi

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    Grant amount：\2750000 （ Direct expense: \2300000 、 Indirect expense：\450000 ）

    We obtained some results on qualitative properties of differential equation models describing the dynamics of infectious agents in a host. Here we dealt with the interaction among viruses, target cells, infected cells, and immunity. We showed for some models that the interior equilibrium, which corresponds to the infected state, is, under some conditions, globally asymptotically stable. In this connection, we considered some mathematical models of immunity, and established some properties of systems of differential equations describing autoimmunity.

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  • Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models

    Grant number：16540154  2004 - 2006

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    HORA Akihito, YAMADA Hiro-Fumi, MURAI Joshin, SASAKI Toru

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    Grant amount：\3600000 （ Direct expense: \3600000 ）

    The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.
    1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.
    2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.
    3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.

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  • Research for Hilbert C*-bimodules and its application to analysis of discrete dynamical systems

    Grant number：15540207  2003 - 2006

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    KAJIWARA Tsuyoshi, WATATANI Yasuo, SASAKI Toru

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    Grant amount：\2500000 （ Direct expense: \2500000 ）

    1. We proved that the fact a countably generated Hilbert C^*-bimodule if of finite index and the fact that it has a conjugation is equivalent. Moreover, we constructed many examples of countably generated Hilbert C^*-bimodules of finite index. We clarified some properties of bases of Hilbert C^*-modules. These results has been published in "Jones index theory for Hilbert C^*-bimodules and its equivalence with conjugation theory".
    2. We constructed Hilbert C^*-bimodules from the dynamical systems on the Riemannian sphere given by rational functions, and constructed C^*-algebras using Pimsner construction. We proved simplicity and pure infiniteness of these C^*-algebras. We calculated K-groups for some examples. These results has been published in "C^*-algebras associated with complex dynamical systems".
    3. We constructed Hilbert C^*-bimodules from self-similar sets given by families of proper contractions, and constructed C^*-algebras using Pimsner construction. Under appropriate condition, we proved simplicity and pure infiniteness of theseC^*-algebras. We showed that two C^*-algebras differently constructed for SierPinski Gasket, which is a typical fractal, are different using K-group. We also calculated K-group of the C^*-algebra constructed from Koch curve, which is also a typical example. These results has been published in "C^*-algebras associated with self-similar sets".
    4. We constructed countable basis explicitly for the Hilbert C^*-modules constructed from complex dynamical system and self-similar sets. This construction is a generalization of that given for the Hilbert C^*-module constructed from tent map using the idea of wavelet basis. This seems the first explicit example of countable bases. Although this construction is not contained in the papers which is already published, it gives some help for research of KMS states of C^*-algebras constructed from rational functions and self-similar sets. This research continues in the next period.
    5. We constructed C^*-algebras for transcendental functions and studied them. We showed simplicity for exponential map case. But there exist a difficulty arising from the existence of pure singularity, and this research also continues.

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  • Study of equivariant homology surgery theory and its applications

    Grant number：15540076  2003 - 2005

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    MORIMOTO Masaharu, SHIMAKAWA Kazuhisa, NAKAJIMA Atsushi, IKEHATA Shuichi, SASAKI Toru

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    Grant amount：\3500000 （ Direct expense: \3500000 ）

    (1)Let G be a finite group, Y a homology disk with a smooth G-action, f : X → Y a G-framed map, H a subgroup of G. Suppose that Y has a G-fixed point and a point of isotropy subgroup H. Then we constructed G-connected sum f#_G G x_H D(f) : X #_G G x_H D(X) → Y.
    (2)We generalized the notion of quadratic forms of Cappell-Shaneson and their group. We proved the generalized group is isomorphic to the original one. Using this new notion, we proved a sum formula of surgery obstructions for the G-connected sum above.
    (3)For Z_{(p)} where p is a prime such that the order of fundamental group of Y, we proved that the equivariant homology obstruction group is isomorphic to the Wall group. So, we could develop induction-restriction theory of equivariant homology surgery obstruction groups.
    (4)Using above results, we proved a deleting-inserting theorem of G-fixed point sets for gap Oliver group G.
    (5)We decided manifolds appearing as the G-fixed point sets of smooth G-actions on spheres where G was a nilpotent Oliver group or a nontrivial perfect group.

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  • Harmonic analysis on graphs and discrete groups, and scaling limit for probability models

    Grant number：13640175  2001 - 2003

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    HORA Akihito, MURAI Joshin, SASAKI Toru

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    Grant amount：\3500000 （ Direct expense: \3500000 ）

    The aim of this project is to read out statistical properties of huge systems characterized by a certain symmetry from the viewpoint of asymptotic spectral analysis and scaling limits by using the methods of harmonic analysis and representation theory. We obtained concrete results as follows.
    1. We computed scaling limits for the spectral distributions of adjacency operators on graphs in the framework of quantum central limit theorem. We introduced Gibbs states as well as vacuum states on distance-regular graphs and investigated the limit picture in low temperature and high degrees especially for Johnson graphs. The result is described in terms of the interacting Fock space associated with Meixner polynomials. Interesting distributions are derived in the limit by using combinatorial structure of creators and annihilators.
    2. We established a general theory for spectral analysis of graphs by the method of quantum decomposition. We revealed a connection of asymptotic characteristic values of regular graphs with the parameters of interacting Fock spaces. The limit distributions are systematically described by using methods of orthogonal polynomials and Green functions beyond computation of individual spectral limits. The item here is closely related to a joint work with Nobuaki Obata at Tohoku University.
    3. We obtained an extension (a quantization) of Kerov's central limit theorem for irreducible characters and the Plancherel measure as an asymptotic aspect of representations of the symmetric groups. Since the result goes out of the framework of interacting Fock spaces, we introduced a modification of the usual Young graph as well as creators and annihilators on it.

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  • Studies on the transmission model for vivax malaria and its simulations

    Grant number：13640116  2001 - 2003

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    ISHIKAWA Hirofumi, SASAKI Toru

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    Grant amount：\2600000 （ Direct expense: \2600000 ）

    We have proposed a mathematical model describing the transmission of Plasmodium vivax malaria quantitatively, which is adjusted to the infected region, Guadalcanal, in the Solomon Islands. The simulation of a transmission model will be instrumental in planning the malaria control strategy. A characteristic of the life cycle of P. vivax parasites is that a sporozoite injected into the blood stream by a mosquito bite may sometimes stay in a hepatocyte as a hypnozoite. Therefore we have incorporated a phenomenon of renewed infections caused by a relapse into the transmission model. Also through the simulations we have attempted to evaluate the decline in prevalence caused by the programs of selective mass drug administration (MDA) and vector control such as the distribution of permethrin-treated bednets. The simulations have indicated that the concentrated repetition of MDA at one week intervals would reduce the prevalence of vivax malaria swiftly in the beginning and would keep the parasite rate below 1% for a few years but the prevalence would increase thereafter. In contrast the parasite rate would remain below 1% for a long time if a trial of one or two times MDA is accompanied with some reduction of the vectorial capacity by the enforcement of vector control. In any case, it is important to beware of relapse cases because even after the execution of MDA it takes a long time to decrease the proportion of hypnozoite carriers.

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  • Harmonic analysis on discrete structures and its applications to classical and quantum probability models

    Grant number：11640168  1999 - 2000

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    HORA Akihito, MURAI Joshin, SASAKI Toru, HIROKAWA Masao

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    Grant amount：\2900000 （ Direct expense: \2900000 ）

    We studied asymptotic behavior of probability models by using the methods of harmonic analysis. Main results are included in 1. the cut-off phenomenon in random walks and 2. central limit theorems in algebraic probability.
    1. The cut-off phenomenon is a sort of critical phenomenon widely observed in the process of convergence to the equilibrium for Markov chains. It is known that the multiplicities of eigenvalues of a transition matrix, which are caused by symmetry of the system, play an important role. In this project, we seeked a rigorous and practical criterion for the cut-off phenomenon beyond verification in individual models and intuitive understanding based on the degeneration of the second eigenvalue. Focusing on distance-regular graphs, we obtained a criterion described in terms of spectral data of the graph. This enables us to find models of the cut-off phenomenon systematically.
    2. Quantum central limit theorems form a main stream in algebraic probability. In this project, we studied important relations between independence of noncommutative random variables and central limit theorems, making much of their algebraic and combinatorial aspects. As a concrete result, we mention the asymptotic spectral distribution of the Laplacian operator on a Johnson graph with respect to the Gibbs state under a low temperature and infinite volume limit. This result leads us to the consideration of creators and annihilators on a nontrivial interacting Fock space and hence gives a good working example in this direction.

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  • Study of control theory of the fixed point sets on spheres

    Grant number：09640110  1997 - 1999

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    MORIMOTO Masaharu, NAKAJIMA Atsushi, NODA Ryuzaburo, SHIMOKAWA Kazuhisa, TANAKA Katsumi, IKEHATA Shuichi

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    Grant amount：\3500000 （ Direct expense: \3500000 ）

    The purpose of this research was to study the following three : (1) (P(G), L(G))-controlled equivariant surgery, cobordism and representation theories and a theory to control isotropy subgroups appearing on manifolds ; (2) Dress'induction of the equivariant cobordism theory of equivariant framed normal maps ; (3) the injection maps IndィイD3G(/)HィエD3 among various finite groups H ⊂ G ; and determine the G-fixed point manifolds of smooth G-actions on spheres for Oliver groups G. We obtained the following results in the research. (1) We proved a deleting-inserting theorem of fixed point components on disks and spheres for Oliver Groups. In a joint work K. Pawalowski, we proved an extension theory of (P(G), L(G))-vector bundles on finite G-CW complexes. Using the equivariant thickening theory with this extension theory, we developed a theory to control isotropy subgroups on disks. (2) We proved that Bak-Morimoto's surgery obstruction group is a Mackey functor on which a Green functor acts, and algebraic Dress'induction works for the obstruction group. In addition, we proved that the cobordism invariance of the surgery obstruction and show that geometric Dress'induction works. (3) In joint works with T. Sumi and M. Yanagihara, we studied the induction maps IndィイD3G(/)HィエD3 for various finite groups H ⊂ G, we constructed (P(G), L(G))-matched pairs and (P(G), L(G))-gap modules for many G. Putting all this together, we determined the G-fixed point manifolds of smooth G-actions on spheres for various Oliver groups G.

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  • Research for Hilbert C^*-bimodules and associated C^*-algebras

    Grant number：09640189  1997 - 1998

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    KAJIWARA Tsuyoshi, HIRAI Yasuhisa, SASAKI Toru, IKEHATA Shuichi, ISHIKAWA Hirofumi, NAKAJIMA Atushi

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    Grant amount：\2500000 （ Direct expense: \2500000 ）

    1. We define crossed product C*-bimodule by finite group, and obtain some fundamental results. More-over, we study A-D duality in operator algebra theory by this method. These results are published in "Crossed products of Hilbert *C-bimodules by countable discrete groups".
    2. We define crossed product C*-bimodule by bundles over finite groups, and prove the associativity law for multiple crossed products. These results are published in "Crossed products of Hilbert C*-bimodule by bundles.
    3. We investigate ideal structures of C*-algebras associated with finitely generated Hilbert C*-bimodules over unital C*-algebras A.We define the condition (I) and show the simplicity under this condition. Moreover we define the condition (11) and under this condition we present the correspondence the ideals in the bimodules algebras and the ideals of A.We present some examples satisfying these conditions. These results are published in "Ideal structure and simplicity of the C*-algebra generated b9Hilbert bimodules".
    4. We define coaction on finitely generated Hilbert C*-bimodule by finite groups. We show that resulting crossed product is made into a finitely generated Hilbert C*-bimodule. This results is published in "Coaction crossed products of Hubert C*-bimodule by finite groups".
    5. We define Hilbert C*-bimodules for countable continuous graphs whose components are 1-dimensional torus, and study the structure of associated C*-algebras. We obtain simplicity and ideal structure of these algebras. These results are published in "Hilbert C*-bimodules and countably infinite continuous graphs"
    6. We are studying singular dynamical system using Hilbert C*-bimodule method. We construct natural basis for the bimodule associated with the tent map. We are plan to. define conjugacy invariant for the above dynamical systems using our method.

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  • Models for Rhythmic Phenomena for Living Bodies and Their Applications

    Grant number：08640285  1996 - 1998

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)

    WATANABE Masaji, SASAKI Tooru, KAJIWARA Tsuyoshi, ISHIKAWA Hirofumi

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    Grant amount：\2200000 （ Direct expense: \2200000 ）

    Sleep-wake rhythm and periodic body temperature change persist with period close to 24-hour even in isolation from 24-hour environmental change, which shows that these periodic changes are generated by some endogenous mechanism. These periodic changes observed in living things are called circadian rhythms. In order to help finding the mechanism of circadian rhythms, we set up mathematical models, analyze them, and verify analytical results with numerical analysis. Besides circadian rhythms, oscillations of some glycolytic intermediates are other periodic phenomena observed in living things. The period of biochemical oscillations ranges a few minutes to 10 minutes, and the period of circadian rhythms are more than 100 times as long. In spite of the significant difference, one can not deny some sort of relation between circadian rhythms and biochemical reactions, because circadian rhythms are observed in variety of living things, and because biochemical reactions are essential for living things to obtain energy. In this study, we suppose that a part of a living thing is surrounded by another part. In case there is no transport of substances between them, the change of concentrations of substances in the inner part is governed by a nonlinear oscillator. Then we assume that there is transport of the substances between the parts. The assumption leads to a system in which a system governing the concentrations in the inner part and a system governing the concentrations in the outer part are coupled. Assuming that a non oscillator has some characteristics, we show that periodic solutions of the coupled system can exist. Moreover, we analyze the system to find conditions under which oscillations of long period can be generated from oscillations of relatively short period. We analyze a model for a electrical circuit anda model for a biochemical reactions, numerically, and verify the results of analysis.

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  • ヒルベルト双加群によるC^*環の指数理論の研究

    Grant number：08640205  1996

    日本学術振興会  科学研究費助成事業  基盤研究(C)

    梶原 毅, 曽布川 拓也, 佐々木 徹, 平井 安久, 洞 彰人, 実方 宣洋

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    Grant amount：\1500000 （ Direct expense: \1500000 ）

    ヒルベルト双加群の離散群による接合積の研究を進めた。これはCombたちによるimprimitivity双加群の接合積の概念の拡張にあたり、ヒルベルト双加群の新たな例を構成する有力な手段の一つである。合わせて、換離散群接合積に関するTakesaki双対性、テンソル積などのカテゴリー的な性質を証明した。この結果は、Crossed Products of Hilbert C^*-Bimodules by Countable Discrete Groupsにおいて、刊行される。
    有限群上のバンドルによるヒルベルトC^*双加群の接合積を定義し、基本的な性質に引き続き、複数のバンドル、双加群の接合積における結合法則を示し、それによって興味ある例を構成した。この結果は、Crossed Products of Hilbert C^*-bimodule by bundlesにまとめ、フーリエ変換によるKac環の積法則の計算など、さらに研究中である。
    ヒルベルト双加群から作られるC^*環は、共変表現環の拡張にあたると考えられるが、これについて、Cuntzが与えた条件(I)と類似の条件のもとで、単純性を証明した。この結果は、Ideal Structure and Simplicity of the C^*-Algebras Generated by Hilbert Bimodulesにまとめている。
    連続群に接合積双加群を定義し、加算生成ヒルベルト双加群の公理を満たすことを示し、単位元がない場合のノルムの同値性を導いた。これにより、無限コンパクト群の表現環からなるDoplicher-Roberts環のヒルベルトC^*双加群による実現が可能となり、K-理論など双加群C^*環の理論の適用が可能となる。
    一般化されたクンツ環において、双加群による構成とgropoidによる構成の関係を研究した。特にトーラス上の関数環加群について詳細に計算している。トーラスのゲージ作用とともに、自由群の余作用のスペクトル部分空間を考えることが、この研究において重要であることがわかった。

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  • 心臓の拍動に関する数学的理論における興奮系を通しての複数振動子の同調

    Grant number：07640308  1995

    日本学術振興会  科学研究費助成事業  一般研究(C)

    渡辺 雅二, 佐々木 徹, 森本 雅治, 長畑 秀和, 野田 隆三郎, 石川 洋文

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    Grant amount：\1600000 （ Direct expense: \1600000 ）

    本研究は、恒温動物の心臓の活動に観察される振動、同調等の現象を解析するための常微分方程式モデルを対象とし、特に、興奮系を通しての複数振動子の同調に関する解析を行なうことを目標とした。先ず、基礎的問題として、外部からの周期的な影響に対する振動系の反応を解析するためのモデルとして考えられる、周期的外力の加えられた振動子を取り扱った。そこでは、周期的外力に対する振動子の同調の特質を数値を用いて表すために次の方法を考案した。周期的外力の加えられた振動子を、振動子の持つ周期解の軌道の接線方向と法線方向の成分の時間的変化が従う常微分方程式系に変換する。この常微分方程式系において、外力の周期の回数に対して、接線方向の成分が振動子の周期を何回増加させるかという、接線方向の成分の増加回数の割合の極限として得られる実数を“rotation number"と呼ぶ。この実数は、1サイクルの外力に対する振動子の平均反応回数と考えられる。このrotation numberにより同調を特徴づける方法の実用化に向け、階段関数による周期的外力の加えられたBonhoeffer-van der Pol方程式に対するフォートランプログラムを作成し、外力の周期に対するratation numberの変化等の解析を行なった。次に、振動子と興奮系の結合における同調に関する問題を取り扱った。このような状態の同調は複数の同一振動子と興奮系の結合における同調とも考えられる。ここでは、二つのBonhoeffer-van der Pol方程式を結合させることによって得られる常微分方程式系に対するフォートランプログラムを作成し、数値解析を行った。二つのBonhoeffer-van der Pol方程式の一つは振動子、もう一つは興奮系となるようにパラメータの値を設定した。数値実験の結果、興奮系の変数が閾値を超えるように振動し、興奮系の変数と振動子の変数が1対1に同調するような周期解が存在することが分かった。

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  • 非線型双曲型偏微分方程式の解の特異性の伝播の研究

    Grant number：06740120  1994

    日本学術振興会  科学研究費助成事業  奨励研究(A)

    佐々木 徹

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    Grant amount：\900000 （ Direct expense: \900000 ）

    1.空間次元1の場合の一般の非線型2階双曲型方程式の解の超可微分性が,その線型化方程式の特性曲線に沿って伝播するかどうかを考察した.これは,半線型方程式に帰着させるとよい.この半線型方程式については,既に超可微分性の伝播について研究してあるので,いかにこの方程式に帰着させるかが問題である.H.LewyやK.Friedrichsの方法を研究した.
    2.空間1次元の半線型双曲型方程式系および3階以上の方程式の超可微分関数のカテゴリーにおける研究を、J.RauchやM.Reedが無限回微分可能関数のカテゴリーにおいて行なった超局所解析を参考に行なった.
    3.変則的な特異性が生ずる条件を,空間次元が1の場合を中心に行なっている.空間次元が1の時には2階の単独方程式では変則的な特異性が現われないので,ここでは方程式系を考察している.まず,J.RauchとM.Reedによって得られた例を中心に検討した.J.RauchとM.Reedの例は,線型の非斉次方程式に帰着して考察しているものである.さらに非線型項のフーリェ変換を精密に考察している.
    4.J.F.Colombeauの一般関数を用いて解の特異性の伝播を考察している.まず,一般関数のクラスや同値関係の定義を整理し,理論を展開し易くするよう試みている.ここでは,パラディストリビューションの理論や超準解析を参考にしつつ,超局所解析の理論などにのりやすいクラスや同値関係を考察している.

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  • Internship （2023academic year） Summer concentration  - その他

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  • Fourier Analysis and Partial Differential Equations 1 （2023academic year） Second semester  - 月5,木1

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  • Partial Differential Equations and Their Applications 1 （2023academic year） Third semester  - 火7,木7

  • Partial Differential Equations and Their Applications 2 （2023academic year） Third semester  - 火8,木8

  • Dynamical Systems and Modeling 1 （2023academic year） 1st semester  - 月3,木5

  • Dynamical Systems and Modeling 2 （2023academic year） 1st semester  - 月4,木6

  • Mathematical Analysis of Dynamics （2023academic year） Prophase  - その他

  • Mathematical Analysis of Dynamics （2023academic year） Prophase  - その他

  • Technical English （2023academic year） 3rd and 4th semester  - [第３学期]水3～4, [第４学期]水1～2

  • Ordinary Differential Equations and Mathematical Models （2023academic year） 1st semester  - 月3～4,木5～6

  • Calculus （2023academic year） 1st and 2nd semester  - 火1～2

  • Calculus II-2 （2023academic year） Second semester  - 木7～8

  • Calculus II-2 Exercise （2023academic year） Second semester  - 月7～8

  • Advanced Calculus and Exercises 1 （2023academic year） 1st semester  - 月7～8,木7～8

  • Advanced Calculus and Exercises 2 （2023academic year） Second semester  - 月7～8,木7～8

  • Calculus I-1 （2023academic year） 1st semester  - 火1～2

  • Calculus I-2 （2023academic year） Second semester  - 火1～2

  • Advanced Mathematical Science in Liberal Arts （2023academic year） Third semester  - 水3～4

  • Topics in Mathematical Modeling A （2023academic year） Summer concentration  - その他

  • Seminar on Mathematical Analysis of Models （2023academic year） Prophase  - その他

  • Seminar in Mathematical Analysis of Model （2023academic year） Late  - その他

  • Seminar on Mathematical Analysis of Models （2023academic year） Late  - その他

  • Seminar in Mathematical Analysis of Model （2023academic year） Prophase  - その他

  • Seminar in Mathematical Modeling and Analysis A （2023academic year） Year-round  - その他

  • Seminar in Mathematical Modeling and Analysis A （2023academic year） Year-round  - その他

  • Seminar in Mathematical Modeling and Analysis B （2023academic year） Year-round  - その他

  • Seminar in Mathematical Modeling and Analysis B （2023academic year） Year-round  - その他

  • Advanced Seminar in Mathematical Modeling and Analysis （2023academic year） Year-round  - その他

  • Advanced Study （2023academic year） Other  - その他

  • Special Research （2023academic year） Year-round  - その他

  • Mathematical Analysis of Phenomena （2023academic year） Late  - 月7～8

  • Mathematical Analysis of Phenomena （2023academic year） Late  - 月7～8

  • Topics in Mathematical Modeling for Environmental Science A （2023academic year） Summer concentration  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2023academic year） Prophase  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2023academic year） Late  - その他

  • Linear Algebra III-2 （2023academic year） special  - その他

  • Fourier Analysis and Partial Differential Equations 1 （2022academic year） 1st semester  - 金3～4

  • Fourier Analysis and Partial Differential Equations 2 （2022academic year） Second semester  - 金3～4

  • Partial Differential Equations （2022academic year） Prophase  - 木5～6

  • Partial Differential Equations and Their Applications 1 （2022academic year） Third semester  - 木5～6

  • Partial Differential Equations and Their Applications 2 （2022academic year） Fourth semester  - 木5～6

  • Mathematical Analysis of Dynamics （2022academic year） Prophase  - その他

  • Foundation of Geometry A （2022academic year） 1st semester  - 木3～4

  • Foundation of Geometry B （2022academic year） Second semester  - 木3～4

  • Keys in Geometry A （2022academic year） 1st semester  - 金1～2

  • Keys in Geometry B （2022academic year） Second semester  - 金1～2

  • Calculus II-2 （2022academic year） Second semester  - 木7～8

  • Calculus II-2 Exercise （2022academic year） Second semester  - 月7～8

  • Advanced Calculus and Exercises 1 （2022academic year） 1st semester  - 月7～8,木7～8

  • Advanced Calculus and Exercises 2 （2022academic year） Second semester  - 月7～8,木7～8

  • Calculus I-1 （2022academic year） 1st semester  - 火1～2

  • Calculus I-2 （2022academic year） Second semester  - 火1～2

  • Seminar on Mathematical Analysis of Models （2022academic year） Prophase  - その他

  • Seminar in Mathematical Analysis of Model （2022academic year） Late  - その他

  • Seminar on Mathematical Analysis of Models （2022academic year） Late  - その他

  • Seminar in Mathematical Analysis of Model （2022academic year） Prophase  - その他

  • Special Research （2022academic year） Year-round  - その他

  • Mathematical Biology for Environmental Problems 1 （2022academic year） Third semester  - 木3～4

  • Mathematical Biology for Environmental Problems 2 （2022academic year） Fourth semester  - 木3～4

  • Seminar in Mathematical Analysis for Environmental Studies （2022academic year） Prophase  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2022academic year） Late  - その他

  • Probability Theory 1 （2022academic year） 1st semester  - 木5～6

  • Probability Theory 2 （2022academic year） Second semester  - 木5～6

  • Linear Algebra III-1 （2022academic year） special  - その他

  • Linear Algebra III-2 （2022academic year） special  - その他

  • Fourier Analysis and Partial Differential Equations （2021academic year） 1st and 2nd semester  - 金3～4

  • Fourier Analysis and Partial Differential Equations 1 （2021academic year） 1st semester  - 金3～4

  • Fourier Analysis and Partial Differential Equations 2 （2021academic year） Second semester  - 金3～4

  • Partial Differential Equations and Their Applications （2021academic year） 3rd and 4th semester  - 金3～4

  • Partial Differential Equations and Their Applications 1 （2021academic year） Third semester  - 金3～4

  • Partial Differential Equations and Their Applications 2 （2021academic year） Fourth semester  - 金3～4

  • Mathematical Analysis of Dynamics （2021academic year） Prophase  - その他

  • Advanced Mathematical Science in Liberal Arts （2021academic year） Third semester  - 月5～6

  • Seminar on Mathematical Analysis of Models （2021academic year） Prophase  - その他

  • Seminar in Mathematical Analysis of Model （2021academic year） Late  - その他

  • Seminar on Mathematical Analysis of Models （2021academic year） Late  - その他

  • Seminar in Mathematical Analysis of Model （2021academic year） Prophase  - その他

  • Special Research （2021academic year） Year-round  - その他

  • Mathematical Analysis of Phenomena （2021academic year） Prophase  - 木5～6

  • Seminar on Mathematical Analysis for Environmental Studies （2021academic year） Prophase  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2021academic year） Late  - その他

  • Seminar on Mathematical Analysis for Environmental Studies （2021academic year） Late  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2021academic year） Prophase  - その他

  • 線形代数II-2 （2021academic year） 第４学期

  • 線形代数II-2演習 （2021academic year） 第４学期

  • Linear Algebra III-1 （2021academic year） 1st semester  - 木7～8

  • Linear Algebra III-2 （2021academic year） Second semester  - 木7～8

  • Linear Algebra III （2021academic year） 1st and 2nd semester  - 木7～8

  • Seminar on Applied Mathematical Science （2020academic year） Fourth semester  - 火3,火4

  • Fourier Analysis and Partial Differential Equations （2020academic year） 1st and 2nd semester  - 金2,金3

  • Fourier Analysis and Partial Differential Equations 1 （2020academic year） 1st semester  - 金2,金3

  • Fourier Analysis and Partial Differential Equations 2 （2020academic year） Second semester  - 金2,金3

  • Partial Differential Equations （2020academic year） Prophase  - 木4,木5

  • Partial Differential Equations and Their Applications （2020academic year） 3rd and 4th semester  - 金2,金3

  • Partial Differential Equations and Their Applications 1 （2020academic year） Third semester  - 金2,金3

  • Partial Differential Equations and Their Applications 2 （2020academic year） Fourth semester  - 金2,金3

  • Mathematical Analysis of Dynamics （2020academic year） Prophase  - その他

  • Calculus I （2020academic year） 1st and 2nd semester  - 火4,火5

  • Calculus I-1 （2020academic year） 1st semester  - 火4,火5

  • Calculus I-1 Exercise （2020academic year） 1st semester  - 金6,金7

  • Calculus I-2 （2020academic year） Second semester  - 火4,火5

  • Calculus I-2 Exercise （2020academic year） Second semester  - 金6,金7

  • Calculus I Exercise （2020academic year） 1st and 2nd semester  - 金6,金7

  • Seminar on Applied Mathematical Science （2020academic year） Fourth semester  - 火3,火4

  • Special Research （2020academic year） Year-round  - その他

  • Seminar on Mathematical Analysis for Environmental Studies （2020academic year） Prophase  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2020academic year） Late  - その他

  • Seminar on Mathematical Analysis for Environmental Studies （2020academic year） Late  - その他

  • Seminar in Mathematical Analysis for Environmental Studies （2020academic year） Prophase  - その他

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