Takahiro Hasebe |

Faculty of Science Mathematics Mathematics |

Associate Professor |

Last Updated :2019/10/03

- Associate Professor

**Independence and infinite divisibility in non-commutative probability**Takahiro HasebeSugaku 70 (3) 296 - 320 2018 [Refereed][Invited]

Research paper (scientific journal)- Takahiro Hasebe, Yuki UedaALEA Lat. Am. J. Probab. Math. Stat. 15 353 - 374 2018 [Refereed][Not invited]

Research paper (scientific journal)We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are shown for a symmetric stable process with index 1 and a positive stable process with index $1/2$. We also prove that free Brownian motion with initial symmetric unimodal distribution is unimodal, and discuss strong unimodality for free convolution. - Hasebe Takahiro, Sakuma NoriyoshiAnn. Inst. Henri Poincare Probab. Stat. 53 (2) 916 - 936 0246-0203 2017/05 [Refereed][Not invited]
- Takahiro Hasebe, Toshinori Miyatani, Masahiko YoshinagaJournal of Singularities 16 212 - 227 2017 [Refereed][Not invited]

Research paper (scientific journal)The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials. - Takahiro Hasebe, Shuhei TsujieJOURNAL OF ALGEBRAIC COMBINATORICS 46 (3-4) 499 - 515 0925-9899 2017/12 [Refereed][Not invited]

Research paper (scientific journal)Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions. - Benoit Collins, Takahiro Hasebe, Noriyoshi SakumaJ. Math. Soc. Japan 70 (3) 1111 - 1150 2018 [Refereed][Not invited]

Research paper (scientific journal)In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model. - Marek Bożejko, Wiktor Ejsmont, Takahiro HasebeInternat. J. Math. 28 (2017), no. 2, 1750010, 30 pp 0129-167X [Refereed][Not invited]

Research paper (scientific journal)We construct a deformed Fock space and a Brownian motion coming from Coxeter groups of type D. The construction is analogous to that of the $q$-Fock space (of type A) and the $(\alpha,q)$-Fock space (of type B). - Arizmendi Octavio, Hasebe TakahiroTrans. Amer. Math. Soc. 368 (7) 4873 - 4905 0002-9947 2016/07 [Refereed][Not invited]
**Free infinite divisibility for powers of random variables**Hasebe TakahiroALEA Lat. Am. J. Probab. Math. Stat. 13 (1) 309 - 336 1980-0436 2016 [Refereed][Not invited]- Marek Bożejko, Wiktor Ejsmont, Takahiro HasebeJ. Funct. Anal. 269 (6) 1769 - 1795 1096-0783|0022-1236 2015 [Refereed][Not invited]

Research paper (scientific journal)In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(\alpha,q)$-Fock space, which satisfy the commutation relation $$ b_{\alpha,q}(x)b_{\alpha,q}^\ast(y)-qb_{\alpha,q}^\ast(y)b_{\alpha,q}(x)=\langle x, y\rangle I+\alpha\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(\alpha,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{\alpha,q}(x)+b_{\alpha,q}^\ast(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $\alpha=0$ and free Meixner polynomials when $q=0$. - Nobuhiro Asai, Marek Bożejko, Takahiro HasebeJ. Math. Phys. 57 (2) 021702 0022-2488 2016 [Refereed][Not invited]

Research paper (scientific journal)Let $\nu_{\alpha,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(\alpha,q)$-Gaussian process (the Gaussian process of type B) over the $(\alpha,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $\nu_{\alpha,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(\alpha,q)$-operators are presented. - Octavio Arizmendi, Takahiro Hasebe, Franz Lehner, Carlos VargasAdvances in Mathematics 282 56 - 92 0001-8708|1090-2082 2015 [Refereed][Not invited]

Research paper (scientific journal)We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed. - Octavio Arizmendi, Takahiro HasebeComplex Analysis and Operator Theory 10 (3) 581 - 603 1661-8262|1661-8254 2016 [Refereed][Not invited]

Research paper (scientific journal)We realize the Belinschi-Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov-Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated to the Cauchy distribution and show that it is a homomorphism with respect to monotone, Boolean and free additive convolutions. - Arizmendi Octavio, Hasebe TakahiroProc. Amer. Math. Soc. 142 (5) 1621 - 1632 0002-9939 2014/05 [Refereed][Not invited]
- Takahiro Hasebe, Steen ThorbjornsenJournal of Theoretical Probability 29 (3) 922 - 940 1572-9230|0894-9840 2016 [Refereed][Not invited]

Research paper (scientific journal)We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato's result in [Ann. Probab. 6 (1978), 523-531]. - Takahiro Hasebe, Alexey KuznetsovElectronic Communications in Probability 19 1 - 12 1083-589X 2014/04 [Refereed][Not invited]

Research paper (scientific journal)We investigate the analytical properties of free stable distributions and discover many connections with the classical stable distributions. Our main results include an explicit formula for the Mellin transform, series representations for the characteristic function and for the density of the free stable distribution; all of these explicit formulas exhibit close resemblance to the corresponding expressions for classical stable distributions. As applications of these results we obtain a new derivation of the duality law obtained in [1] and a factorization of a classical stable random variable into the independent (in the classical sense) product of a free stable random variable and a power of Gamma(2) random variable. - Takahiro Hasebe, Noriyoshi SakumaDemonstratio Mathematica 48 (3) 424 - 439 0420-1213 2015 [Refereed][Not invited]

Research paper (scientific journal)We give a complete list of the Lebesgue-Jordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal. - Takahiro HasebeElectron. J. Probab. 19 (81) 1 - 33 1083-6489 2014 [Refereed][Not invited]

Research paper (scientific journal)We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1. - Takahiro Hasebe, Hayato SaigoNagoya Math. J. 215 151 - 167 0027-7630 2014 [Refereed][Not invited]

Research paper (scientific journal)We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of Central Limit Theorem for operator-valued case. Moreover, we prove a generalization of Muraki's formula for the sum of independent random variables and a relation between generating functions of moments and cumulants. - Marek Bozejko, Takahiro HasebeProbab. Math. Stat. 33 (2) 363 - 375 0208-4147 2013 [Refereed][Not invited]

Research paper (scientific journal)We prove that symmetric Meixner distributions, whose probability densities are proportional to $|\Gamma(t+ix)|^2$, are freely infinitely divisible for $0 - Octavio Arizmendi, Takahiro Hasebe, Noriyoshi SakumaALEA, Lat. Amer. J. Probab. Math. Stat. 10 (2) 271 - 291 1980-0436 2013 [Refereed][Not invited]

Research paper (scientific journal)We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for $0\leq t\leq 1$, t-th free multiplicative power for $t\geq 1$ and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative convolution of a free Poisson and a free regular measure. This gives two new explicit examples of distributions which are infinitely divisible with respect to both classical and free convolutions: \chi^2(1) and F(1,1). Another consequence is that the free commutator operation preserves free infinite divisibility. **Joint cumulants for natural independence**Hasebe Takahiro, Saigo HayatoElectron. Commun. Probab. 16 491 - 506 1083-589X 2011/09 [Refereed][Not invited]- Takahiro HasebeInternat. J. Math. 23 (3) 1250041 (21 pages) 0129-167X 2012 [Refereed][Not invited]

Research paper (scientific journal)We introduce a class of probability measures whose densities near infinity are mixtures of Pareto distributions. This class can be characterized by the Fourier transform which has a power series expansion including real powers, not only integer powers. This class includes stable distributions in probability and also non-commutative probability theories. We also characterize the class in terms of the Cauchy-Stieltjes transform and the Voiculescu transform. If the stability index is greater than one, stable distributions in probability theory do not belong to that class, while they do in non-commutative probability. - Octavio Arizmendi, Takahiro HasebeBernoulli 19 (5B) 2750 - 2767 1350-7265 2013 [Refereed][Not invited]

Research paper (scientific journal)We consider a class of probability measures $\mu_{s,r}^{\alpha}$ which have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with L\'{e}vy measure a monotone $\alpha$-stable law. This implies the free infinite divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive, $\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free Poisson law and a monotone $\alpha$-stable law. We also investigate the free infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\leq r\leq2$. - Takahiro HasebeJ. Theoret. Probab. 25 (3) 756 - 770 1572-9230|0894-9840 2012 [Refereed][Not invited]

Research paper (scientific journal)We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions. - Octavio Arizmendi, Takahiro HasebeStudia Math. 215 157 - 185 1730-6337|0039-3223 2013 [Refereed][Not invited]

Research paper (scientific journal)Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible or not. In this paper we further investigate this indicator, introduce a multiplicative version of it and are able to show many properties. Specifically, on the first half of the paper, we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bozejko's conjecture which says the Boolean power of a probability measure mu by 0 < t < 1 is freely infinitely divisible if mu is so. In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible concerning the multiplicative free convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bozejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection. - Hasebe TakahiroCOLLOQUIUM MATHEMATICUM 124 (1) 79 - 94 0010-1354 2011 [Refereed][Not invited]
**INDEPENDENCE GENERALIZING MONOTONE AND BOOLEAN INDEPENDENCES**Hasebe TakahiroQUANTUM PROBABILITY AND RELATED TOPICS 27 190 - 201 1793-5121 2011 [Refereed][Not invited]- Hasebe TakahiroINFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS 13 (1) 111 - 131 0219-0257 2010/03 [Refereed][Not invited]
- Hasebe TakahiroSTUDIA MATHEMATICA 200 (2) 175 - 199 0039-3223 2010 [Refereed][Not invited]
- Takahiro HasebeCompl. Anal. Oper. Theory 7 (1) 115 - 134 1661-8262|1661-8254 2013 [Refereed][Not invited]

Research paper (scientific journal)We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed. - Takahiro HasebeInfin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (3) 465 - 516 0219-0257 2011 [Refereed][Not invited]

Research paper (scientific journal)We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions. - Takahiro Hasebe, Izumi Ojima, Hayato SaigoInfin. Dimens. Anal. Quantum. Probab. Relat. Top. 11 (2) 307 - 311 0219-0257 2008 [Refereed][Not invited]

Research paper (scientific journal)In White Noise Analysis (WNA), various random quantities are analyzed as elements of $(S)^{\ast}$, the space of Hida distributions ([1]). Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On $(S)^{\ast}$, the Wick product is defined in terms of the $\mathcal{S}$-transform. We have found such a remarkable property that the Wick product has no zero devisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the \textquotedblleft operational calculus\textquotedblright in WNA along the line of Mikusi\'{n}ski's version for solving differential equations. - Takahiro Hasebe, Hayato SaigoAnn. Inst. Henri Poincare Probab. Stat. 47 (4) 1160 - 1170 0246-0203 2011 [Refereed][Not invited]

Research paper (scientific journal)In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define ``monotone cumulants'' in the sense of generalized cumulants and we obtain quite simple proofs of central limit theorem and Poisson's law of small numbers in monotone probability theory. Moreover, we clarify a combinatorial structure of moment-cumulant formula with the use of ``monotone partitions''. - Takahiro HasebeInfinite Dimensional Analysis, Quantum Probability and Related Topics 13 (4) 619 - 627 0219-0257 2010 [Refereed][Not invited]

Research paper (scientific journal)The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility for compact Riemannian manifold, using white noise analysis. In this paper we extend its technique to noncompact Riemannian manifolds which have differential operators satisfying some conditions.

- Takahiro Hasebe, Thomas Simon, Min Wang 2018/05 [Not refereed] [Not invited] Research paperWe investigate certain analytical properties of the free $\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha\le 1$, and that they belong to the extended Thorin class when $\alpha \leq 3/4.$ The L\'evy measure is explicitly computed for $\alpha =1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\alpha > 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.
- Octavio Arizmendi, Takahiro Hasebe 2017/11 [Not refereed] [Not invited] Research paperWe consider different limit theorems for additive and multiplicative free L\'evy processes. The main results are concerned with positive and unitary multiplicative free L\'evy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean L\'evy processes at small time, obtaining log Boolean stable laws in the limit.
- Takahiro Hasebe, Franz Lehner 2017/11 [Not refereed] [Not invited] Institution technical report and pre-print, etc.We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and M\"obius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faa di Bruno's theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell-Baker-Hausdorff series in this context. In a final section we exhibit an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.
- Takahiro Hasebe, Noriyoshi Sakuma, Steen Thorbjørnsen 2017/01 [Not refereed] [Not invited] Institution technical report and pre-print, etc.The class of selfdecomposable distributions in free probability theory was introduced by Barndorff-Nielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution $\mu_c$ is freely selfdecomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.
- Takahiro Hasebe, Hao-Wei Huang, Jiun-Chau Wang 2017/05 [Not refereed] [Not invited] Institution technical report and pre-print, etc.In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery.
- Yinzheng Gu, Takahiro Hasebe, Paul Skoufranis 2017/08 [Not refereed] [Not invited] Institution technical report and pre-print, etc.In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined and a moment-cumulant formula is derived in the bi-monotonic setting. In general the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. We include errata to previous papers of the first- and third-named authors, pointing out that bi-Boolean convolution and thus conditionally bi-free convolution in general do not preserve probability measures either.
- Takahiro Hasebe, Kamil Szpojankowski 2017/10 [Not refereed] [Not invited] Institution technical report and pre-print, etc.We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables $X,Y$ where $Y$ has a free Poisson distribution one has $X\stackrel{d}{=}\frac{1}{X+Y}$ if and only if $X$ has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.
- Takahiro Hasebe 2010/02 [Not refereed] [Not invited] Institution technical report and pre-print, etc.This article focuses on properties of monotone convolutions. A criterion for infinite divisibility and time evolution of convolution semigroups are mainly studied. In particular, we clarify that many analogues of the classical results of L\'{e}vy processes hold such as characterizations of subordinators and strictly stable distributions.
- Takahiro Hasebe 2010/09 [Not refereed] [Not invited] Institution technical report and pre-print, etc.We define a new independence in three states called indented independence which unifies many independences: free, monotone, anti-monotone, Boolean, conditionally free, conditionally monotone and conditionally anti-monotone independences. This unification preserves the associative laws. Therefore, the central limit theorem, cumulants and moment-cumulant formulae for indented independence also unify those for the above seven independences.

- Analytic Studies開講年度 : 2017課程区分 : 修士課程開講学部 : 理学院キーワード : グラフ、半順序集合、不変量、同型
- Advanced Mathematical Analysis開講年度 : 2017課程区分 : 学士課程開講学部 : 理学部キーワード : グラフ、半順序集合、不変量、同型
- Calculus I開講年度 : 2017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
- Calculus II開講年度 : 2017課程区分 : 学士課程開講学部 : 全学教育キーワード : 原始関数, 積分, 重積分, リ－マン和, 変数変換