1,721,003 research outputs found
Two problems in the theory of differential equations
Get PDF1) The differential equation considered in terms of exterior differential forms, as É.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet?
2) Why criteria of formal integrability of differential equations are never used in practice
Inverses of Cartan matrices of Lie algebras and Lie superalgebras
Get PDFThe inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras, except for the 5 Lie superalgebras whose Cartan matrices have 0 on the main diagonal: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic).
We interpret most of the results of Wei Yangjiang and Zou Yi Ming, Inverses of Cartan matrices of Lie algebras and Lie superalgebras, Linear Alg. Appl., 521 (2017) 283--298 as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin\u27s paper in 1952, see also Table 2 in Springer\u27s book by Onishchik and Vinberg (1990).We reproduce definition of root spaces from arXiv:0710.5149 and arXiv:0906.1860. Version 2 contains inadvertently forgotten subsection 8.3 and accordingly edited Introduction and Abstrac
On odd parameters in geometry
Get PDF1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic.
2) Any supermanifold which is a ringed space of the form (a manifold , the sheaf of sections of the exterior algebra of a vector bundle over ) is called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension is split. I\u27ll show that there are non-split supermanifolds of superdimension ; for example, certain -dimensional superstrings, the obstructions to their splitness correspond to odd parameters.44 pages; the strange words in Theorem 5.1 are striken out, a references updated; otherwise coincides with the published versio
On Integrations and Cross Ratios on Supermanifolds
No full text(A) The conventional integration theory on supermanifolds had been constructed in order to have (an analog of) the Stokes formula in which a sub-supermanifold is of codimension 1 = (1|0). I review other integrations and formulate related open problems:(1) On the 1|1-dimensional superstring associated with the trivial bundle, in presence of a contact structure there is a special integration useful in describing super versions of elliptic functions. It is needed to construct a~particular spinor representation of the Neveu-Schwarz superalgebra.(2) Versions of the Stokes formula with "over-supermanifold" of codimension (0|-1) due to Shander and Palamodov should be developed further.(3) Apply Shander's integration with odd parameters over chains to inverse problems.(4) Establish existence of conjectural integrations (apparently, not leading to any analog of the Stokes formula) related to various (super)traces on various Lie superalgebras and the corresponding (super)determinants.(B) I offer analogs of the cross ratio for "classical superspaces'', including infinite-dimensional versions. Open problem: apply these invariants to the matrix-valued Riccati equations.</p
New simple Lie superalgebras as queerified associative algebras
Get PDFOver , Montgomery superized Herstein\u27s construction of simple Lie algebras from finite-dimensional associative algebras, found obstructions to the procedure and applied it to -graded associative algebra of differential operators with polynomial coefficients.
Since the 1990s, Vasiliev and Konstein with their co-authors constructed (via the Herstein--Montgomery method, having rediscovered it) simple Lie (super)algebras from the associative (super)algebra such as Vasiliev\u27s higher spin algebras (a.k.a. algebras of observables of the rational Calogero model) and algebras of symplectic reflections.
The queerification is another method for cooking a~simple Lie superalgebra from the simple associative (super)algebra. The above examples of associative (super)algebras, and Lie (super)algebras of matrices of complex size can be queerified by adding new elements resembling Faddeev--Popov ghosts.
Conjectures: 1) a queerified Hamiltonian describes a version of the Calogero model with -dimensional time; 2) metabelean algebras and inhomogeneous subalgebras of Lie superalgebras naturally widen supersymmetries in future theories; 3) only graded-commutative algebras can imitate algebras of functions in a reasonably rich non-commutative Geometry.11 pages. The title shortened, text edited and corrected, conjectures expounde
On Odd Parameters in Geometry
No full text1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic.2) Any supermanifold which is a ringed space of the form (a manifold M, the sheaf of sections of the exterior algebra of a vector bundle over M) is called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension m|1 is split. I'll show that there are non-split supermanifolds of superdimension m|1; for example, certain 1|1-dimensional superstrings, the obstructions to their splitness correspond to odd parameters.</p
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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