Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.

In this paper, we consider a particular class of Kazhdan–Lusztig cells in the symmetric group $$S_n$$ , the cells containing involutions associated with compositions $$\lambda $$ of n. For certain families of compositions we are able to give an explicit description of the corresponding cells by obtaining reduced forms for all their elements. This is achieved by first finding a particular class of diagrams $${\mathcal {E}}^{(\lambda )}$$ which lead to a subset of the cell from which the remaining elements of the cell are easily obtained. Moreover, we show that for certain cases of related compositions $$\lambda $$ and $$\hat{\lambda }$$ of n and $$n+1$$ respectively, the members of $${\mathcal {E}}^{(\lambda )}$$ and $${\mathcal {E}}^{(\hat{\lambda })}$$ are also related in an analogous way. This allows us to associate certain cells in $$S_n$$ with cells in $$S_{n+1}$$ in a well-defined way, which is connected to the induction and restriction of cells.

We give a new proof for the following result of Petty: Every 3-dimensional normed space contains a regular 3-simplex.

In this paper, we present the notion of perfect ideal of a seminearring S and prove that the kernel of a seminearring homomorphism is a perfect ideal. We show that the quotient structure S/I is isomorphic to the structure $$S_{T(I)}.$$ Finally, we prove isomorphism theorems in seminearrings by using tame condition.

In this paper, the families of orthocentric, circumscriptible, isodynamic, and isogonic (or rather, tetra-isogonic) $$d$$ -simplices, $$d \ge 3$$ , are considered, and it is proved that the intersection of any two of them is precisely the family of $$d$$ -kites. Here, a $$d$$ -simplex is called a $$d$$ -kite if $$d$$ of its vertices form a regular $$(d-1)$$ -simplex whose vertices are equidistant from the remaining vertex.

After a few claims about lunes and convex sets on the d-dimensional sphere $$S^d$$ we present some relationships between the diameter, width and thickness of reduced convex bodies and bodies of constant diameter on $$S^d$$. These relationships are applied for proving the final theorem, which permits to recognize if a Wulff shape in the Euclidean space is self-dual.

Trong bài báo này, chúng tôi sẽ nghiên cứu các z-đại lý bậc cao của một số vành đặc biệt. Chúng tôi sẽ mô tả các z-đại lý của vành $$\mathcal {A}[[X]]$$. Hơn nữa, chúng tôi sẽ đưa ra một kết quả tương tự cho vành lý tưởng hóa của một mô-đun R, $$R(+)M$$.

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by F. Santos and D. Bochiş, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 “full” cubic groups cannot have more than 25 facets. Here we study the remaining “quarter” cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.