Some liftings of poisson structures to Weil bundles

We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.     

A New Approach to the Representation Theory of the Symmetric Groups. II

The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous modifications to the text were made by the first author for this publication. Bibliography: 35 titles. To the memory of D. Coxeter __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 57–98.     

The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1-\mathbbE)(1-\mathbb{E})-transform, where \mathbbE\mathbb{E} is the “exponential alphabet,” whose elementary symmetric functions are e_n=\frac1n!e_{n}=\frac{1}{n!}. In the case of noncommutative symmetric functions, we recover Schocker’s idempotents for derangement numbers (Schocker, Discrete Math. 269:239–248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon–Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.     

Various products for Lebesgue densities

In Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) various products for primitive liftings in the factors of a product of probability spaces have been considered. In this paper we settle for the d-dimensional Lebesgue densities open problems from Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) by applying results relying on the metrical group structure of \mathbb R^d{{\mathbb R}^d}, if d ? \mathbb N{d\in{\mathbb N}}. In particular, a lifting problem from Musial et al. (Arch Math 83:467–480, 2004), Question 3.3, is decided to the negative for the Lebesgue densities. The relation of the Lebesgue density in the product space and the results of the products taken for the Lebesgue densities in the factors under order is discussed. The results can be carried over to densities and liftings dominating Lebesgue densities and to multiplicative and positive linear liftings on function spaces.     

Combinatorial Groupoids,Cubical Complexes,and the Lovász Conjecture

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature”, etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial “Theorema Egregium” for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend Babson–Kozlov–Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs. The author was supported by grants 144014 and 144026 of the Serbian Ministry of Science and Technology.     

Ploucquet’s “Refutation” of the Traditional Square of Opposition

In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”. My thanks are due to Hanno von Wulfen for helpful discussions and for transforming the word-document into a Latex-file.     

Realizability of greedy algorithms

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.     

Complete Upper Angle Around a Point on the Minkowski Plane

The dependence of the complete upper angle in the sense of A. D. Aleksandrov about a point on the Minkowski plane on the form of the “unit circle” (the centrally symmetric convex curve Φ determining the Minkowski metric ρ_Φ) is studied.The complete upper angle is computed in three cases: if Φ is a square, a “cut circle,” or a “rounded rhombus.” Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 42–53.     

Natural transformations of symmetric affine connections on manifolds to metrics on linear frame bundles: a classification

We find all second order natural transformations of symmetric affine connections on manifolds into metrics on linear frame bundles. The complete family of naturally resulting metrics (to a fixed symmetric affine connection) is a real vector space generated by some generalization of two classical liftings.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

Linear Liftings of Skew-Symmetric Tensor Fields to Weil Bundles

We define equivariant tensors for every non-negative integer p and every Weil algebra A and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type (p, 0) on an n-dimensional manifold M to tensor fields of type (p, 0) on T ^A M if 1 ≤ p ≤ n. Moreover, we determine explicitly the equivariant tensors for the Weil algebras , where k and r are non-negative integers.     

The divergence of Banach space valued random variables on Wiener space

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows. Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05 An erratum to this article is available at .     

On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 99–115, January, 2009.     

Rigidity problems in toric topology: A survey

Several rigidity problems in toric topology are addressed in the survey paper by the second and third authors, “Classification Problems of Toric Manifolds via Topology” (in Toric Topology, Am. Math. Soc., Providence, RI, 2008, pp. 273–286). In the present paper, we survey the results on those problems including recent developments.     

“Naturadeus”, a metaphor of the Perfect Diamond

In this essay, the author outlines his re-construction of Spinoza’s ontological monism by re-presenting the system of Ethica, ordine geometrico demonstrata, in an “intuitive” model of the Perfect Diamond, called NATURADEUS. So, for example, ordo et connexio idearum et rerum, is presented to the inner eye in the forms of two parallel structures, of rays and of facets within the NATURADEUS, respectively. The conceptual background of the proposed model is mostly analytic, the author essays to develop some ideas of Jonathan Bennett’s Spinoza’s metaphysics (especially “transattributive mode identity”), with strong emphasis on ethical issues of ontological monism or pantheism. This essay is written as a dialogue between master Bruno and his student John (physically absent at the moment). More philosophical dialogues of this kind can be found on the author’s web page and in his book Four Seasons (in Slovenian, 2002).     

Graph theoretic method for determining hurwitz equivalence in the symmetric group

Motivated by the problem of Hurwitz equivalence of Δ² factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of 1s _n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the braid group. This paper is part of the author’s PhD thesis. This work was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and by EAGER (EU network, HPRN-CT-2009-00099). Received December 31, 2001 and in revised form August 6, 2002     

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s _λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 14–46, April, 2006.     

Polyhedral surfaces in wedge products

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated “subdirect product” as introduced by McMullen (Discrete Math 14:347–358, 1976); it is dual to the “wreath product” construction of Joswig and Lutz (J Combinatorial Theor A 110:193–216, 2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces “of unusually large genus” that first appeared in works by Coxeter (Proc London Math Soc 43:33–62, 1937), Ringel (Abh Math Seminar Univ Hamburg 20:10–19, 1956), and McMullen et al. (Israel J Math 46:127–144, 1983). Via “projections of deformed wedge products” we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in \mathbb R³{{\mathbb R}^3} . As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (“moduli”) for the surfaces in \mathbb R³{{\mathbb R}^3} . In order to prove that there are many moduli, we introduce the concept of “affine support sets” in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in \mathbb R³{{\mathbb R}^3} via dual 4-polytopes.     

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.