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358: １３２人目の素数さん [] 2023/03/13(月) 21:11:52.56 ID:UeELXD7y

>>357
つづき

The claim
The key to the argument is the following

Claim. The set V of all elements a of D such that a2 <= 0 is a vector subspace of D of dimension n - 1. Moreover D = R 〇+ V as R-vector spaces, which implies that V generates D as an algebra.
Proof of Claim: Let m be the dimension of D as an R-vector space, and pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we can write

p(x)=(x-t_{1})\cdots (x-t_{r})(x-z_{1})(x-{\overline {z_{1}}})\cdots (x-z_{s})(x-{\overline {z_{s}}}),\qquad t_{i}\in \mathbf {R} ,\quad z_{j}\in \mathbf {C} \backslash \mathbf {R} .
We can rewrite p(x) in terms of the polynomials Q(z; x):

p(x)=(x-t_{1})\cdots (x-t_{r})Q(z_{1};x)\cdots Q(z_{s};x).
Since zj ∈ C\R, the polynomials Q(zj; x) are all irreducible over R. By the Cayley?Hamilton theorem, p(a) = 0 and because D is a division algebra, it follows that either a ? ti = 0 for some i or that Q(zj; a) = 0 for some j. The first case implies that a is real. In the second case, it follows that Q(zj; x) is the minimal polynomial of a. Because p(x) has the same complex roots as the minimal polynomial and because it is real it follows that

p(x)=Q(z_{j};x)^{k}=\left(x^{2}-2\operatorname {Re} (z_{j})x+|z_{j}|^{2}\right)^{k}

Since p(x) is the characteristic polynomial of a the coefficient of x2k?1 in p(x) is tr(a) up to a sign. Therefore, we read from the above equation we have: tr(a) = 0 if and only if Re(zj) = 0, in other words tr(a) = 0 if and only if a2 = ?|zj|2 < 0.

So V is the subset of all a with tr(a) = 0. In particular, it is a vector subspace. The rank?nullity theorem then implies that V has dimension n - 1 since it is the kernel of

{\displaystyle \operatorname {tr} :D\to \mathbf {R} }. Since R and V are disjoint (i.e. they satisfy

{\displaystyle \mathbf {R} \cap V=\{0\}}), and their dimensions sum to n, we have that D = R 〇+ V.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/358

682: １３２人目の素数さん [] 2023/03/23(木) 13:41:19.56 ID:gtBUMZjM

>>681
つづき

Oka's result has been generalized to domains spread over any Stein manifold: If such a domain D
is a pseudo-convex manifold, then D
is a Stein manifold. The Levi problem has also been affirmatively solved in a number of other cases, for example, for non-compact domains spread over the projective space CPn
or over a Kahler manifold on which there exists a strictly plurisubharmonic function (see ), and for domains in a Kahler manifold with positive holomorphic bisectional curvature [7]. At the same time, examples of pseudo-convex manifolds and domains are known that are not Stein manifolds and not even holomorphically convex. A necessary and sufficient condition for a complex space to be a Stein space is that it is strongly pseudo-convex (see Pseudo-convex and pseudo-concave). Also, a strongly pseudo-convex domain in any complex space is holomorphically convex and is a proper modification of a Stein space (see , [4] and also Modification; Proper morphism).
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1677671318/682

709: １３２人目の素数さん [] 2023/03/23(木) 22:39:15.56 ID:aDRJxbk2

小松玄の弟子↓

Kengo Hirachi (平地 健吾 Hirachi Kengo, born 30 November 1964) is a Japanese mathematician, specializing in CR geometry and mathematical analysis.

Hirachi received from Osaka University his B.S. in 1987, his M.S. in 1989, and his Dr.Sci., advised by Gen Komatsu, in 1994 with dissertation The second variation of the Bergman kernel for ellipsoids.[1] He was a research assistant from 1989 to 1996 and a lecturer from 1996 to 2000 at Osaka University. He was an associate professor from 2000 to 2010 and a full professor from 2010 to the present at the University of Tokyo. He was a visiting professor at the Mathematical Sciences Research Institute from October 1995 to September 1996, at the Erwin Schrödinger Institute for Mathematical Physics from March 2004 to April 2004, at Princeton University from October 2004 to July 2005, and at the Institute for Advanced Study from January 2009 to April 2009.

Awards and honors
Takebe Senior Prize (1999) of the Mathematical Society of Japan
Geometry Prize (2003) of the Mathematical Society of Japan
Stefan Bergman Prize (2006)
Inoue Prize for Science (2012)
Invited lecture at ICM, Seoul 2014

http://rio2016.5ch.net/test/read.cgi/math/1677671318/709

813: １３２人目の素数さん [] 2023/03/28(火) 22:02:48.56 ID:hsF37p1R

Michael Eastwood FAA is a mathematician at the University of Adelaide, known for his work in twistor theory, conformal differential geometry and invariant differential operators. In 1976 he received a PhD at Princeton University in several complex variables under Robert C. Gunning. He was a member of the twistor research group of Roger Penrose at the University of Oxford and he coauthored the monograph The Penrose Transform: Its Interaction with Representation Theory with Robert Baston. After moving to South Australia in 1985 he was the 1992 recipient of the Australian Mathematical Society Medal and made a Fellow of the Australian Academy of Science in 2005. In 2012 he was named to the inaugural (2013) class of fellows of the American Mathematical Society.

Rod Gover
Nationality New Zealander
Known for Invariant theory problems, operator classification problem
Scientific career
Fields Mathematics, differential geometry, theoretical physics
Thesis A Geometrical Construction of Conformally Invariant Differential Operators (1989)
Doctoral advisor Michael Eastwood
Lane P. Hughston

http://rio2016.5ch.net/test/read.cgi/math/1677671318/813

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