323,718 research outputs found
Grassmann integral representation for spanning hyperforests
No full textGiven a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry
The space of linear maps into a Grassmann manifold
No full textWe show that the space of all holomorphic maps of degree one from the Rie-mann sphere into a Grassmann manifold is a sphere bundle over a flag manifold. Using the notions of "kernel" and "span" of a map, we completely identify the space of unparame-terized maps as well. The illustrative case of maps into the quadric Grassmann manifold is discussed in detail and the homology of the corresponding spaces is computed
Grassmann speciality of Jordan supersystems
No full textAbstractThe Grassmann envelope Γ(S)=S0̄⊗Γ0̄+S1̄⊗Γ1̄ is a useful tool for deciding varietal questions about supersystems S=S0̄+S1̄. It is not useful in deciding simplicity, since the envelope is always fraught with trivial ideals coming from trivial superscalars. At first glance it would not seem useful in deciding superspeciality either, but we will show that it is a more sensitive tool than it seems. We say a Jordan supersystem Js (algebra, triple, or pair) is Grassmann special if its Grassmann envelope is special as an ordinary Jordan system over Γ0̄. Certainly superspeciality Js⊂As+ over Φ implies Grassmann speciality Γ(Js)⊂Γ(As)+ over Γ0̄, but it is not obvious how Grassmann speciality influences superspeciality of Js. Nevertheless, we show how to transform obstacles to speciality of Js over Φ (anti-superspecial elements) into obstacles to speciality of Γ(Js) over Γ0̄ (anti-special elements) using Grassmann boosters, so that non-superspeciality Js over Φ implies non-speciality of Γ(Js) over Γ0̄. Thus Grassmann speciality is the same as superspeciality: a supersystem J is a superspecial Jordan supersystem over Φ iff Γ(Js) is a special Jordan system over Γ0̄. (It is not clear this is the same as speciality of Γ(Js) over Φ, since in general speciality of quadratic Jordan systems depends on the scalars: we give an example of a Jordan Ω-algebra which is not special, but is special over a scalar subring Φ.) As a corollary, any Jordan superalgebra with zero linear extreme radical which is viably evenly 4-interconnected is superspecial; thus exceptional superalgebras must be built over even parts of degree ⩽3
Comments on the CDF 88-89 di-lepton top candidate event
No full textCDF reported a possible top candidate event from their 988/89 run. The event was interpreted and reconstructed as pp→tt→ W + W + jet(b) + + jet(B) + Μ.(B) by some authors. We study the structure of the event to see whether this interpretation is likely. © 1994 Società Italiana di Fisica
Grassmann extensions of Yang-Baxter maps
Get PDFIn this paper we show that there are explicit Yang–Baxter (YB) maps with Darboux–Lax representation between Grassman extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the YB property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional YB maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations
Duals of Affine Grassmann Codes and Their Relatives
Get PDFAffine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes
Non-Parallelizability of Grassmann Manifolds
No full textAbstractThe fact that no real Grassmann manifolds Gk(Rn) are parallelizable (or even stably parallelizable) except for the obvious cases G1R2≅S1, G1(R4)≅G3(R4) ≅ RP3, and G1(R8)≅ G7(R8) ≅ RP7 was first noted by Hiller and Stong. Their work in turn depends on induction and the work of Oproiu, who examined detailed calculations of Stiefel-Whitney classes for k = 2, 3. In this note we give a short proof of this result, using elementary results from K-theory, that also covers the complex and quaternionic Grassmann manifolds.</jats:p
Clustering on Grassmann manifolds via kernel embedding with application to action analysis
Get PDFWith the aim of improving the clustering of data (such as image sequences) lying on Grassmann manifolds, we propose to embed the manifolds into Reproducing Kernel Hilbert Spaces. To this end, we define a measure of cluster istortion and embed the manifolds such that the distortion is minimised. We show that the optimal solution is a generalised eigenvalue problem that can be solved very efficiently. Experiments on several clustering tasks (including human action clustering) show that in comparison to the recent intrinsic Grassmann k-means algorithm, the proposed approach obtains notable improvements in clustering accuracy, while also being several orders of magnitude faster
Dictionary learning and sparse coding on Grassmann manifolds: an extrinsic solution
Get PDFRecent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis.Mehrtash Harandi, Conrad Sanderson, Chunhua Shen, and Brian C. Lovel
Two parameter deformation of Grassmann matrix group and supergroup
No full textThe two parameter quantum deformations of 2X2 Grassmann matrices, Gr(2), and supermatrices, Gr(1\1), are presented. Gr(2) whose matrix elements are all Grassmannian variables is called the superdual of the general linear group GL(2), and Gr(1\1) whose diagonal matrix elements are Grassmannian variables is called the superdual of the supergroup GL(1\1) whose nondiagonal elements are Grassmannian. Noncentral dual superdeterminant for Grassmann supermatrices belonging to Gr(p,q)(1\1) is constructed. As with the 2X2 quantum matrices, the relations satisfied by the matrix elements of the Grassmann matrices and supermatrices are expressed in terms of an (R) over tilde matrix. The properties of the nth power of a Grassmann supermatrix are given as an Appendix. (C) 1996 American Institute of Physics
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