Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds

We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g ₄(x, y),..., g _d (x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g ₄, and d − 4 PDEs of the second order with respect to f and g ₄,..., g _d . For d = 4, this result confirms Blaschke’s conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.     

Conclaves of planes in PG (4,2) and certain planes external to the Grassmannian ${\cal G}_{1,4,2} \subset$ PG(9,2)

We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P} _{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28 intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such conclaves (see [6]) $\mathcal{P}_{8}$ of planes in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit $\operatorname{orb}(2\gamma)$ (see [4]). The fact that, under the action of $\operatorname{GL}(5,2),$ the stabilizer groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a regulus-free partial spread $\mathcal{S}_{8}$ in $\operatorname{PG}(4,2)$ we also give a construction of a conclave of planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$     

Spin invariants of multivectors

LetCl(p, q) be a real universal Clifford algebra which is isomorphic to a full matrix algebra ?(2m). In this paper we show that on the linear subspaceCl ^k(p, q) ofk-vectors the determinant can be written as a product of two polynomialsd _i of degreem and that on the subset ofdecomposable k-vectors we have det=±Q ^m for some quadratic formQ. The polynomialsd _i andQ are examples of a spin invariant, the latter being defined as a functionJ:Cl ^k (p,q) → ? for whichJ(sus^?1)=J(u) for allu ∈Cl ^k(p, q) ands ∈Spin(p, q). In the last section we identify the ‘fundamental’ spin invariants on the bivector spacesCl ²(p, p) forp=2 andp=3.     

The Classification of Flats in {\boldsymbol {PG}}({\bf 9,2}) which are External to the Grassmannian {\cal G}_{\bf 1,4,2}

Constructions are given of different kinds of flats in the projective space $PG(9,2)={\mathbb P}(\wedge^{2}V(5,2))$ which are external to the Grassmannian ${\cal G}_{\bf 1,4,2}$ of lines of PG(4,2). In particular it is shown that there exist precisely two GL(5,2)-orbits of external 4-flats, each with stabilizer group ?31:5. (No 5-flat is external.) For each k=1,2,3, two distinct kinds of external k-flats are simply constructed out of certain partial spreads in PG(4,2) of size k+2. A third kind of external plane, with stabilizer ?2³:(7:3), is also shown to exist. With the aid of a certain ‘key counting lemma’, it is proved that the foregoing amounts to a complete classification of external flats.     

MR Study of Water Distribution in a Beech (Fagus sylvatica) Branch Using Relaxometry Methods

Wood is a widely used material because it is environmentally sustainable, renewable and relatively inexpensive. Due to the hygroscopic nature of wood, its physical and mechanical properties as well as the susceptibility to fungal decay are strongly influenced by its moisture content, constantly changing in the course of everyday use. Therefore, the understanding of the water state (free or bound) and its distribution at different moisture contents is of great importance. In this study, changes of the water state and its distribution in a beech sample while drying from the green (fresh cut) to the absolutely dry state were monitored by 1D and 2D ¹H NMR relaxometry as well as by spatial mapping of the relaxation times T₁ and T₂. The relaxometry results are consistent with the model of homogeneously emptying pores in the bioporous system with connected pores. This was also confirmed by the relaxation time mapping results which revealed the moisture transport in the course of drying from an axially oriented early- and latewood system to radial rays through which it evaporates from the branch. The results of this study confirmed that MRI is an efficient tool to study the pathways of water transport in wood in the course of drying and is capable of determining the state of water and its distribution in wood.     

The Geometry of Lightlike Hypersurfaces of the de Sitter Space

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space Sn+11 is directly connected with the geometry of hypersurfaces of the conformal space Cn. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.     

Eighth International Conference on Geometry

The Conference took place in the Nahsholim Sea Resort, Israel. The work was done in two parallel sections: Research in Geometry and Geometry and School. The Scientific Committee which led the Conference consisted of A. Barlotti (Firenze), W. Benz (Hamburg), A. Bichara (Roma), H. Karzel (München), H.-J. Kroll (München), F. Mazzocca (Napoli), H. Zeitler (Bayreuth), and the local organizers R. Artzy and J. Zaks. The sessions were chaired by M. A. Akivis, G. Becker, J. Böhm, D. Camp, P. V. Ceccherini, V. V. Goldberg, H. Havlicek, W. Heise, P. Herfort, B. Klotzek, H. Martini, R. H. Schulz, H. Siemon, H. Stachel, G. Stanilov, B. Uhrin, H. Walser, H. Wefelscheid, G. Weiss. A total of 71 lectures were delivered. The Conference was supported in part by The European Mathematical Society, The Emmy Noether Center of the Minerva Foundation at Bar Ilan University in Israel, The Israel Mathematical Union, and the University of Haifa. The following are abstracts of talks presented in the Research section of the Conference.     

Linearizability of d-webs, d?≥?4, on two-dimensional manifolds

We find d – 2 relative differential invariants for a d-web, d 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g₄(x, y),..., g_d(x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g₄, and d – 4 PDEs of the second order with respect to f and g₄,..., g_d. For d = 4, this result confirms Blaschkes conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give the Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.