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Shuzo Izumi

Publications and source records attributed to Shuzo Izumi.

6 recordsLinked to original sources
arXiv2026

Rotation of a simplex within another one

We are interested in the naive problem whether we can move a solid object in a solid box or not. We may restrict the movement to rotation. In the case we can rotate, the centre and the ``direction'' of rotation may be restricted. Simplifying, we consider possibility of rotation of a polytope within another one of the same dimension and give a criterion for the possibility. Consider the particular case of simplices of the same dimension assuming that the vertices of the inner simplex are contained in different facets of the outer one. Premising further that simplices are even dimensional, rotation is possible in a very general situation. However, in dimension 3, the possible case is not not general. Even in these elementary phenomena, the parity of the dimension seems to yield difference.

math.MG↗
arXiv2015

Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Let $\mathcal{O}(U)$ denote the algebra of holomorphic functions on an open subset $U\subset\mathbb{C}^n$ and $Z\subset\mathcal{O}(U)$ its finite-dimensional vector subspace. By the theory of least space of de Boor and Ron, there exists a projection $T_b$ from the local ring $\mathcal{O}_{n,b}$ onto the space $Z_b$ of germs of elements of $Z$ at $b$. At general $b\in U$, its kernel is an ideal and induces a structure of an Artinian algebra on $Z_b$. In particular, it holds at points where $k$-th jets of elements of $Z$ form a vector bundle for each $k\le\dim_{\mathbb{C}}Z_b-1$. Using $T_b$ we define the Taylor projector of order $d$ on an embedded curve $X\subset\mathbb{C}^m$ at a general point $\boldsymbol{a}\in X$, generalising results of Bos and Calvi. It is a retraction of $\mathcal{O}_{X,a}$ onto the set of the polynomial functions on $X_a$ of degree up to $d$. For an embedded manifold $X\subset\mathbb{C}^m$, we introduce a set of higher order tangents following Bos and Calvi and show a zero-estimate for a system of generators of the maximal ideal of $\mathbb{C}\{t-b\}$ at general $b\in X$. It means that $X$ is embedded in $\mathbb{C}^n$ in not very highly transcendental manner at a general point.

math.CV↗
arXiv2015

Sufficiency of simplex inequalities

Let z_0,...,z_n be the (n-1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: z_p < z_0+...+\check{z}_p+...+z_n (0 =< p =< n), generalizations of triangle inequalities. Conversely, suppose that numbers z_0,...,z_n > 0 satisfy these inequalities. Does there exist an n-simplex the volumes of whose facets are them? Kakeya solved this problem affirmatively in the case n = 3 and conjectured that the assertion is affirmative also for all n >= 4. We prove that his conjecture is affirmative. To do this, we define three kinds of spaces of loops associated to n-simplices and study relations among them systematically. In particular, we show that the space of edge loops corresponds to the space of facet loops bijectively under a certain condition of positivity.

math.MG↗
arXiv2006

A formula for ideal lattices of general commutative rings

Let S be a set of n ideals of a commutative ring A. Let G_{even} (respectively G_{odd}) denote the product of all the sums of even (respectively odd) number of ideals of S. If n<7 the product of G_{even} and the intersection of all ideals of S is included in G_{odd}. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.

math.AC↗
arXiv2004

Restrictions of smooth functions to a closed subset

We first provide an approach to the recent conjecture of Bierstone-Milman-Pawlucki on Whitney's old problem on smooth extendability of functions defined on a closed subset of a Euclidean space, using higher order paratangent bundle they introduced. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek's theorem on controllability of flatness by the values on a closed set. The multi-dimensional Vandermonde matrix plays an important role in both cases.

math.CA↗