R2-planes with 3-dimensional automorphism group fixing precisely a line

It is shown that the class of these R²-planes consists of those listed by STRAMBACH 1970, those added by OSTMANN 1975, and those added by GROH 1981. This last class consists of sums of two hyperbola planes. Furthermore, the isomorphism types are determined. All investigations are carried out for the much more general situation of an R²-plane with Δ-invariant screen, Δ ? R².     

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

The Shuffle Pasting

The graded set of n-dimensional (p, q)-shuffles is endowed with the structure of well-formed loop-free pasting scheme. In the process, well-formed subpasting schemes and their sources and targets are characterized, using a higher Bruhat type order.     

Pasting and semimodular lattices

Presented by George Grätzer.     

Cutting and Pasting of Elliptic Operators

If A : C∞E → C∞F is an elliptic operator between sections of vector bundles E, F over a closed smooth n-manifold X, Y a smooth (n – 1)-submanifold of X with trivial normal bundle, and Ψ = (ΨE, ΨF) a pair of automorphisms of E | Y and F | Y inducing a diffeomorphism f of Y and commuting with the principal symbol σ A of A over Y, then an elliptic operator A^Ψ is (uniquely up to operators of lower order) defined between sections of vector bundles E^Ψ, F^Ψ over a closed manifold X^f, all obtained by cutting and pasting the respective objects along Y. The difference index A^Ψ – index A is investigated and the relations with additivity properties of topological invariants and with classical transmission problems are explained.     

Pasting of two processes with independent increments

A terminating stochastically continuous strictly Markov process is obtained as a result of pasting two nonterminating homogeneous stochastically continuous Markov processes with independent increments, one of which is semicontinuous. It is shown that this process can be extended to a complete homogeneous stochastically continuous strictly Markov Feller process. Previously, this problem has been solved by the author under stronger restrictions-both pasting processes were semicontinuous.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 487–491, April, 1993.     

The Almost Pasting Property of Digital Continuity

The pasting property plays a significant role in studying a continuous map in general topology. Unlike the pasting property, digital continuity has some intrinsic features. In this paper we prove that digital continuity has the almost pasting property instead of the pasting one. Furthermore, as a digital version of the straightness of a metric space, we introduce the notion of k-straightness of a digital space (X,k) and study its properties which can be used in image synthesis and digital image weaving.     

Pasting topological spaces at one point

Let {X _α } _α∈Λ be a family of topological spaces and x _α ∈ X _α , for every α ∈ Λ. Suppose X is the quotient space of the disjoint union of X _α ’s by identifying x _α ’s as one point σ. We try to characterize ideals of C(X) according to the same ideals of C(X _α )’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2) Is there any set of prime ideals of cardinality m in a ring R such that the intersection of these prime ideals can not be obtained as an intersection of fewer than m prime ideals in R? Finally, we answer an open question in [11].     

Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

We classify the cohomology classes of Lagrangian 4-planes ?⁴ in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = ?2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = ?γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ? of a line in a smooth Lagrangian n-plane ? ⁿ must satisfy (?,?) = ?(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.     

Observations about the Lie algebra g2⊂so(7), associative 3-planes,and so(4) subalgebras

We make several observations relating the Lie algebra ${\mathfrak{g}}_2?\mathfrak{so}\left(7\right)$, associative 3-planes, and $\mathfrak{so}\left(4\right)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in {\mathfrak{g}}_2$ cannot have rank 2, and if it has rank 4 then its kernel is an associative subspace. We prove a canonical form theorem for elements of ${\mathfrak{g}}_2$. Given an associative 3-plane $P$ in ${\mathbb{R}}^7$, we construct a Lie subalgebra $\Theta \left(P\right)$ of $\mathfrak{so}\left(7\right)={\Lambda }^2\left({\mathbb{R}}^7\right)$ that is isomorphic to $\mathfrak{so}\left(4\right)$. This $\mathfrak{so}\left(4\right)$ subalgebra differs from other known constructions of $\mathfrak{so}\left(4\right)$ subalgebras of $\mathfrak{so}\left(7\right)$ determined by an associative 3-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.     

关于2R（w2R）范数的一个注记

证明了自反Banach空间X上的等价w2R范数全体构成一个剩余集;同时证明了X闭子空间上等价的w2R范数均可延拓为X上等价的w2R范数.特别地,当X是可分时,上述w2R范数可替换为2R范数.     

Zur Differentialgeometrie im totalisotropen R2 und R3 eines komplexen euklidischen R4 und R6

Ohne ZusammenfassungGuido Hoheisel zum 65. Geburtstag     

Huygenssche Differentialgleichungen im R2

Ohne Zusammenfassung