Akira Sakai |

Faculty of Science Mathematics Mathematics |

Professor |

Last Updated :2020/08/07

- 50506996

- mathematical physics statistical mechanics probability theory φ^4 model random walk phase transition Ising model percolation lace expansion contact process self-avoiding walk critical behavior

- Natural sciences / Applied mathematics and statistics
- Natural sciences / Basic mathematics
- Natural sciences / Mathematical physics and basic theory
- Natural sciences / Mathematical analysis

- 2011/04 - Today Hokkaido University Department of Mathematics Associate professor
- 2008/03 - 2011/03 Hokkaido University Creative Institution SOUSEI Tenure-track assistant professor
- 2006/04 - 2008/02 University of Bath Department of Mathematical Sciences Lecturer
- 2004/04 - 2006/03 Technische Universiteit Eindhoven Department of Mathematics and Computer Science Postdoctoral fellow
- 2003/01 - 2004/03 Eurandom Postdoctoral fellow
- 2001/01 - 2002/12 University of British Columbia Department of Mathematics Postdoctoral fellow

- 1996/04 - 2000/12 Tokyo Institute of Technology
- 1994/04 - 1996/03 Tokyo Institute of Technology
- 1990/04 - 1994/03 Tokyo Institute of Technology

- Satoshi Handa, Yoshinori Kamijima, Akira SakaiTAIWANESE JOURNAL OF MATHEMATICS 24 (3) 723 - 784 1027-5487 2020/06 [Refereed][Not invited]

The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a d-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent. - Kasho Yamamoto, Kota Ando, Normann Mertig, Takashi Takemoto, Masanao Yamaoka, Hiroshi Teramoto, Akira Sakai, Shinya Takamaeda-Yamazaki, Masato Motomura138 - 140 2020 [Refereed][Not invited]
- Chen, Lung-Chi, Sakai, AkiraCOMMUNICATIONS IN MATHEMATICAL PHYSICS 372 (2) 543 - 572 0010-3616 2019/12 [Refereed][Not invited]

Consider the long-range models on Z(d) of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as |x|(-d-alpha) for some alpha > 0. In the previous work (Chen and Sakai in Ann Probab 43:639-681, 2015), we have shown in a unified fashion for all alpha not equal 2 that, assuming a bound on the "derivative" of the n-step distribution (the compoundzeta distribution satisfies this assumed bound), the critical two-point function G(pc) (x) decays as |x|(alpha boolean AND 2-d) above the upper-critical dimension d(c) = (alpha boolean AND 2)m, where m = 2 for self-avoiding walk and the Ising model and m = 3 for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the n-step distribution, that G(pc) (x) for the marginal case alpha = 2 decays as |x|(2-d)/ log |x| whenever d >= d(c) (with a large spread-out parameter L). This solves the conjecture in Chen and Sakai (2015), extended all the way down to d = d(c), and confirms a part of predictions in physics (Brezin et al. in J Stat Phys 157:855-868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections. - Lung-Chi Chen, Akira SakaiCommunications in Mathematical Physics 372 (2) 543 - 572 1432-0916 2019/12 [Refereed][Not invited]
- Sakai, Akira, Slade, GordonELECTRONIC JOURNAL OF PROBABILITY 24 (65) 1 - 18 1083-6489 2019 [Refereed][Not invited]

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions d > 8. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions d +1 > 4 + 1. We use the lace expansion to prove estimates on all moments for the spread-out critical contact process in dimensions d > 4, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants. - Sakai, AkiraJOURNAL OF STATISTICAL PHYSICS 171 (3) 462 - 469 0022-4715 2018/05 [Refereed][Not invited]

Consider nearest-neighbor oriented percolation in space-time dimensions. Let be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality , which holds for all and is a strict inequality above the upper-critical dimension 4, becomes an equality for , i.e., , provided existence of at least two among . The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6]. - Toshihiro Arae, Shiori Isai, Akira Sakai, Katsuhiko Mineta, Masami Yokota Hirai, Yuya Suzuki, Shigehiko Kanaya, Junji Yamaguchi, Satoshi Naito, Yukako ChibaPLANT AND CELL PHYSIOLOGY 58 (6) 1090 - 1102 0032-0781 2017/06 [Refereed][Not invited]

Plants possess a cold acclimation system to acquire freezing tolerance through pre-exposure to non-freezing low temperatures. The transcriptional cascade of C-repeat-binding factors (CBFs)/dehydration response element-binding factors (DREBs) is considered a major transcriptional regulatory pathway during cold acclimation. However, little is known regarding the functional significance of mRNA stability regulation in the response of gene expression to cold stress. The actual level of individual mRNAs is determined by a balance between mRNA synthesis and degradation. Therefore, it is important to assess the regulatory steps to increase our understanding of gene regulation. Here, we analyzed temporal changes in mRNA amounts and half-lives in response to cold stress in Arabidopsis cell cultures based on genome-wide analysis. In this mRNA decay array method, mRNA half-life measurements and microarray analyses were combined. In addition, temporal changes in the integrated value of transcription rates were estimated from the above two parameters using a mathematical approach. Our results showed that several cold-responsive genes, including Cold-regulated 15a, were relatively destabilized, whereas the mRNA amounts were increased during cold treatment by accelerating the transcription rate to overcome the destabilization. Considering the kinetics of mRNA synthesis and degradation, this apparently contradictory result supports that mRNA destabilization is advantageous for the swift increase in CBF-responsive genes in response to cold stress. - Yuki Chino, Akira SakaiJOURNAL OF STATISTICAL PHYSICS 163 (4) 754 - 764 0022-4715 2016/05 [Refereed][Not invited]

Following similar analysis to that in Lacoin (Probab Theory Relat Fields 159: 777-808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on Z(d) is almost surely a constant, which does not depend on the location of the reference point. We provide upper and lower bounds which are valid for all d >= 1. - Akira SakaiCOMMUNICATIONS IN MATHEMATICAL PHYSICS 336 (2) 619 - 648 0010-3616 2015/06 [Refereed][Not invited]

Using the Griffiths-Simon construction of the model and the lace expansion for the Ising model, we prove that, if the strength of nonlinearity is sufficiently small for a large class of short-range models in dimensions d > 4, then the critical two-point function is asymptotically times a model-dependent constant, and the critical point is estimated as , where is the massless point for the Gaussian model. - Lung-Chi Chen, Akira SakaiANNALS OF PROBABILITY 43 (2) 639 - 681 0091-1798 2015/03 [Refereed][Not invited]

We consider long-range self-avoiding walk, percolation and the Ising model on Z(d) that are defined by power-law decaying pair potentials of the form D(x) asymptotic to vertical bar x vertical bar(-d-alpha) with alpha > 0. The upper-critical dimension d(c) is 2(alpha boolean AND 2) for self-avoiding walk and the Ising model, and 3(alpha boolean AND 2) for percolation. Let alpha not equal 2 and assume certain heat-kernel bounds on the n-step distribution of the underlying random walk. We prove that, for d > d(c) (and the spread-out parameter sufficiently large), the critical two-point function G p(c) (X) for each model is asymptotically C vertical bar x vertical bar(alpha boolean AND 2-d), where the constant C is an element of (0, infinity) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between alpha < 2 and alpha > 2. We also provide a class of random walks that satisfy those heat-kernel bounds. - Lung-Chi Chen, Akira SakaiANNALS OF PROBABILITY 39 (2) 507 - 548 0091-1798 2011/03 [Refereed][Not invited]

We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as vertical bar x vertical bar(-d-alpha) with alpha > 0. For random walk in any dimension d and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension d(c) equivalent to 2(alpha boolean AND 2), we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincare Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151-188] and [Probab. Theory Related Fields 145 (2009) 435-458]. - Akira SakaiRIMS Kokyuroku Bessatsu B21 53 - 61 2010/12 [Not refereed][Not invited]
- Remco van der Hofstad, Akira SakaiELECTRONIC JOURNAL OF PROBABILITY 15 801 - 894 1083-6489 2010/06 [Refereed][Not invited]

We consider the critical spread-out contact process in Z(d) with d >= 1, whose infection range is denoted by L >= 1. In this paper, we investigate the higher-point functions tau((r))((t) over right arrow)((x) over right arrow) for r >= 3, where tau((r))((t) over right arrow)((x) over right arrow) is the probability that, for all i = 1,...,r-1, the individual located at x(i) is an element of Z(d) is infected at time t(i) by the individual at the origin o is an element of Z(d) at time 0. Together with the results of the 2-point function in [16], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper critical dimension 4. We also prove partial results for d <= 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation in Z(d) x epsilon Z(+), where epsilon is an element of (0,1] is the time unit. For ordinary oriented percolation (i.e., epsilon = 1), we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in epsilon is an element of (0,1], which allows us to establish the scaling results also for the contact process (i.e., epsilon down arrow 0). We also show that the main term of the vertex factor V, which is one of the non-universal constants in the scaling limit, is 2 - epsilon (= 1 for oriented percolation, = 2 for the contact process), while the main terms of the other non-universal constants are independent of epsilon. The lace expansion we develop in this paper is adapted to both the r-point function and the survival probability. This unified approach makes it easier to relate the expansion coefficients derived in this paper and the expansion coefficients for the survival probability, which will be investigated in [18]. - Lung-Chi Chen, Akira SakaiPROBABILITY THEORY AND RELATED FIELDS 145 (3-4) 435 - 458 0178-8051 2009/11 [Refereed][Not invited]

We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha > 0 converges to e(-C)vertical bar k vertical bar(alpha boolean AND 2) for some C is an element of (0, infinity) above the upper- critical dimension d(c) equivalent to 2(alpha boolean AND 2). This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151-188, 2008). Moreover, we show that the constant C exhibits crossover at alpha = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients. - Markus Heydenreich, Remco van der Hofstad, Akira SakaiJOURNAL OF STATISTICAL PHYSICS 132 (6) 1001 - 1049 0022-4715 2008/09 [Refereed][Not invited]

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d > 2(alpha boolean AND 2) for self-avoiding walk and the Ising model, and d > 3(alpha boolean AND 2) for percolation, where d denotes the dimension and alpha the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886-1944, 2005). - Lung-Chi Chen, Akira SakaiPROBABILITY THEORY AND RELATED FIELDS 142 (1-2) 151 - 188 0178-8051 2008/09 [Refereed][Not invited]

We consider oriented percolation on Z(d) x Z(+) whose bond-occupation probability is pD(center dot), where p is the percolation parameter and D is a probability distribution on Z(d). Suppose that D(x) decays as vertical bar x vertical bar(-d-alpha) for some alpha > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension d(c) = 2(alpha boolean AND 2). We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to e-c vertical bar k vertical bar(alpha boolean AND 2) for some c > 0. - M. Holmes, A. SakaiSTOCHASTIC PROCESSES AND THEIR APPLICATIONS 117 (10) 1519 - 1539 0304-4149 2007/10 [Refereed][Not invited]

We consider random walks with transition probabilities depending on the number of consecutive traversals n of the edge most recently traversed. Such walks may get stuck on a single edge, or have every vertex recurrent or every vertex transient, depending on the reinforcement function f (n) that characterizes the model. We prove recurrence/transience results when the walk does not get stuck on a single edge. We also show that the diffusion constant need not be monotone in the reinforcement. (C) 2007 Elsevier B.V. All rights reserved. - Akira Sakai2007/08 [Not refereed][Not invited]
- Akira SakaiCOMMUNICATIONS IN MATHEMATICAL PHYSICS 272 (2) 283 - 344 0010-3616 2007/06 [Refereed][Not invited]

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with d >> 4 and for the spread-out model with d > 4 and L >> 1, without assuming reflection positivity. - R van der Hofstad, A SakaiPROBABILITY THEORY AND RELATED FIELDS 132 (3) 438 - 470 0178-8051 2005/07 [Refereed][Not invited]

We consider self-avoiding walk and percolation in Z(d), oriented percolation in X-d x Z(+), and the contact process in Z(d), with pD(center dot) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y-x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p(c). We investigate the value of p(c) when d > 6 for percolation and d > 4 for the other models, and L >> 1. We prove in a unified way that p(c)=1+C(D)+O(L-2), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L-d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p(c)=1+cL(-d) +O(L-d-1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models. - A SakaiJOURNAL OF STATISTICAL PHYSICS 117 (1-2) 111 - 130 0022-4715 2004/10 [Refereed][Not invited]

We consider the critical survival probability (up to time t) for oriented percolation and the contact process, and the point-to-surface (of the ball of radius t) connectivity for critical percolation. Let theta(t) denote both quantities. We prove in a unified fashion that, if theta(t) exhibits a power law and both the two-point function and its certain restricted version exhibit the same mean-field behavior, then theta(t) asymptotic to t(-1) for the time-oriented models with d > 4 and theta(t) asymptotic to t(-2) for percolation with d > 7. - R van der Hofstad, A SakaiELECTRONIC JOURNAL OF PROBABILITY 9 710 - 769 1083-6489 2004/10 [Refereed][Not invited]

We consider the critical spread-out contact process in Z(d) with d greater than or equal to 1, whose infection range is denoted by L greater than or equal to 1. The two-point function tau(t)(x) is the probability that x is an element of Z(d) is infected at time t by the infected individual located at the origin o is an element of Z(d) at time 0. We prove Gaussian behaviour for the two-point function with L greater than or equal to L(o) for some finite L(o) = L(o)(d) for d > 4. When d less than or equal to 4, we also perform a local mean-field limit to obtain Gaussian behaviour for tau(tT) (x) with t > 0 fixed and T --> infinity when the infection range depends on T in such a way that L(T) = LT(b) for any b > (4 - d)/2d. The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on their respective mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper. The results in this paper also apply to oriented percolation, for which we reprove some of the results in [20] and extend the results to the local mean-field setting described above when d less than or equal to 4. - M Holmes, AA Jarai, A Sakai, G SladeCANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 56 (1) 77 - 114 0008-414X 2004/02 [Refereed][Not invited]

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on Z(d) in dimensions d > 4. - A SakaiJOURNAL OF STATISTICAL PHYSICS 106 (1-2) 201 - 211 0022-4715 2002/01 [Refereed][Not invited]

The contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension. Above their upper critical dimension d(c) they exhibit the same critical behavior as the branching process. Assuming existence of the critical exponents, we prove a pair of hyperscaling inequalities which, together with the results of refs. 16 and 18, implies d(c) = 4. - A SakaiJOURNAL OF STATISTICAL PHYSICS 104 (1-2) 111 - 143 0022-4715 2001/07 [Refereed][Not invited]

The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d > 4: theta(lambda) approximate to lambda - lambda (c) as lambda down arrow lambda (c) and chi(lambda) approximate to (lambda (c) - lambda)(-1) as lambda up arrow lambda where theta(lambda) and chi(lambda) are the spread probability and the susceptibility of the infection respectively, and lambda (c) is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.

- Satoshi Handa, Markus Heydenreich, Akira Sakai (ContributorMean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions)
**Sojourns in Probability Theory and Statistical Physics - I**

Springer Singapore 2019 (ISBN: 9789811502941) 183-198 - Akira Sakai (ContributorApplications of the lace expansion to statistical-mechanical models)
**Analysis and Stochastics of Growth Processes and Interface Models**

Oxford University Press 2008/07 (ISBN: 9780199239252) 352 123-147

- Japan Science and Technology Agency：Strategic Basic Research Programs CRESTDate (from‐to) : 2018/10 -2024/03Author : Akira Sakai
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grant-in-Aid for Scientific Research(基盤研究(C))Date (from‐to) : 2018/04 -2022/03Author : Akira Sakai
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grant-in-Aid for Challenging Exploratory ResearchDate (from‐to) : 2015/04 -2018/03Author : Akira Sakai半径ｒのｄ次元球をプラス境界条件下におき，その中心のスピンの期待値を考える．この１スピン期待値は相転移を示し，臨界温度直上ではｒ→∞でゼロに収束することが知られていた．とくに上部臨界次元である４次元より上では，ｒの（ｄ－２）／２乗の逆数より遅く減衰することが，ハイパースケーリング不等式から厳密に知られていた． 半田氏とHeydenreich氏との共同研究において，確率幾何的表現である「ランダムカレント表示」を用いて２次モーメント法を精密に評価し，臨界１スピン期待値が４次元より上では１／ｒよりもゆっくり減衰することを厳密に証明した．Newman氏のfestschriftに掲載されることが決定．
- Ministry of Education, Culture, Sports, Science and Technology - Japan：Grants-in-Aid for Scientific Research(基盤研究(C))Date (from‐to) : 2012/04 -2015/03Author : Akira Sakaiスピン系数理モデルの臨界現象を「確率幾何学的な解析」によって厳密に理解するべく，以下の3つの課題に取り組んでいる．（１）冪的に減衰する長距離２体相互作用で定義されたイジング模型を考え，その冪指数（＝ｄ＋α）に依存して，高次元における臨界２点関数が漸近的にリース核になったり（α＜２），ニュートン核になったり（α＞２），それに対数補正がつく（α＝２）という予想を解決．（２）φ４乗モデルに対するレース展開の手法を確立し，「繰り込み群」による従来の結果を凌駕する精密な結果を導出．（３）「＋境界条件」が課されたｄ次元球の中心に位置するイジングスピンの期待値が，球の半径の或る冪（この冪指数を１-arm指数と呼び，ｄ＞４の最近接格子モデルでは１）で減衰するという予想の解決．平成24年度のそれぞれの課題に対する実績は次の通り．（１）8月に共同研究者のChen教授が訪日，11月と3月に坂井が訪台し，α≠２の結果をまとめて論文を執筆，確率論トップジャーナル“Annals of Probability”に投稿，受理された．α＝２については現在進行中．（２）現在までに得られている解析結果を基に，Universita di Modena e Reggio Emilia（イタリア，6月），Mathematisches Forschungsinstitut Oberwolfach（ドイツ，9月），National Center for Theoretical Sciences（台湾，3月）で開催された国際研究集会で招待講演を行ない，確率論シンポジウム（京都，12月）で発表．論文は現在執筆中．（３）共同研究者のHeydenreich教授とMathematisches Forschungsinstitut Oberwolfach（ドイツ，9月）で研究打ち合わせを行ない，課題が山積していることを再確認した．
- Ministry of Education, Culture, Sports, Science and Technology：Grants-in-Aid for Scientific Research(若手研究(B))Date (from‐to) : 2009/04 -2012/03Author : Akira SAKAIThe lace expansion has been one of the few mathematically rigorous approaches to investigate critical behavior in high dimensions. We have extended this methodology to obtain a universal sharp asymptotic expression of the 2-point functions for long-range self-avoiding walk and long-range oriented percolation which are defined by power-law decaying pair potentials. We have also investigated the finite-range(but sufficiently spread-out) critical contact process and proved that the n-point function under the Brownian scaling converges to the(n-1)-point function for the canonical measure of super-Brownian motion.

- Inter-Graduate School Classes(General Subject):Natural and Applied Sciences開講年度 : 2018課程区分 : 修士課程開講学部 : 大学院共通科目キーワード : ＜正宗＞ 均質化法，材料工学，偏微分方程式 ＜坂井＞ 有向パーコレーション，相転移，臨界現象
- Overview of Mathematical Sciences開講年度 : 2018課程区分 : 修士課程開講学部 : 理学院キーワード : ＜正宗＞ 均質化法，材料工学，偏微分方程式 ＜坂井＞ 有向パーコレーション，相転移，臨界現象
- Linear Algebra I開講年度 : 2018課程区分 : 学士課程開講学部 : 全学教育キーワード : 行列, 連立１次方程式, 基本変形, 階数, 行列式, 逆行列
- Calculus I開講年度 : 2018課程区分 : 学士課程開講学部 : 全学教育キーワード : 数列，収束，関数，極限，微分，偏微分，テイラ－の定理
- Basic Mathematical Science開講年度 : 2018課程区分 : 学士課程開講学部 : 理学部キーワード : ＜栄＞ 微分方程式，散逸構造，形態形成，進行波． ＜行木＞ 離散力学系，カオス，時系列データ解析． ＜坂井＞ グラフ上の調和関数，ランダムウォーク表示，脱出確率，再帰性．