Symplectic invariants and covariants of matrices

This paper deals with symplectic algebraic invariants and covariants of matrices. We adapt the Aronhold symbolic method to characterize and compute the generators of algebras of symplectic invariants and covariants. New symplectic identities are obtained and applied to construct minimal generator systems of a matrix of order 4 and a Hamiltonian matrix of order 2n.     

A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2-(mn²+1,mn,(n+1)(mn−1)) design starting from a Steiner system S(2,m+1,mn²+1) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.     

On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spaceP ⁿ (n 2). Puttingn = 2, we showed in [3] that the theorem of Desargues inP ⁿ is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inP ⁿ. We only give a few examples and leave it as an open problem which other special cases could be found.     

Orthogonal invariants of skew-symmetric matrices

The algebra of invariants of d-tuples of n?×?n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n?=?3 and d?>?0 a minimal set of generators is established. A homogeneous system of parameters (i.e. an algebraically independent set such that the algebra of invariants is a finitely generated free module over subalgebra generated by this set) is described for n?=?3 and d?>?0, for n?=?4 and d?=?2,?3, for n?=?5 and d?=?2.     

Polynomial modules and a generalized cauchy problem for systems of holomorphic partial differential operators

In this paper, we study holomorphic solutions of overdetermined systems of partial differential equatoins with constant coefficients. We parametrize the solution space, using an ideal of holomorphic functions generated by certain polynomials associated to the principal symbols of the operators, by proving a Cauchy-Kowalevsky type theorem for these systems. In this theorem, we introduce a non-characteristicity condition for systems which is a direct generalization of a condition for single equations introduced by the author and H.S. Shapiro in a previous paper.

As a tool for studying the systems mentioned above, certain estimates, which may be of independent interest, for elements of analytic modules generated by polynomials are obtained.     

Intrinsic normalizations of a nonholonomic hypersurface with <Emphasis Type="Italic">m</Emphasis>-dimensional generators in affine space <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We consider some intrinsic normalizations of a nonholonomic hypersurface with m-dimensional generators in the n-dimensional affine spaced. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 532–562, October–December, 2007.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

Conformally Invariant Operators,Differential Forms,Cohomology and a Generalisation of Q-Curvature

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalize in a natural way Branson's Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nⁿ to N^m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nⁿ to Nⁿ and the implicit function theorem for functions from Nⁿ to N^m with m<n.     

An Analog Characterization of the Grzegorczyk Hierarchy

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n−3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions.     

Differential Calculus on Compact Quantum Group Uθ（2） and its Applications

In this paper, via constructing special matrices, we will show that there exists a differential calculus on Uθ（2）, where θ is an irrational number. Then using the above results, we shall discuss the properties of infinitesimal generators of corepresentations of Uθ（2）. And in the final, we shall discuss its irreducible corepresentations and give the Peter-Weyl theorem explicitly for compact quantum group Uθ（2）.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with n degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each k there exists an explicitly known infinite set ⊂ ℚ such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) calculated at a non-zero d ∈ ℂ ⁿ satisfying V′(d) = d, belong to . The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning V we prove the following fact. For each k and n there exists a finite set such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) belong to . We give an algorithm which allows to find sets . We applied this results for the case n = k = 3 and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.      

Classical invariant theory for finite reflection groups

We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups). As an application of the results we prove a generalization of Chevalley's restriction theorem for the classical Lie algebras. In the interesting case when the group is of Coxeter typeD _n (n4) we use higher polarization operators introduced by Wallach. The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables. We conjecture that this is also true for the exceptional reflection groups and then sketch a proof for the group of typeF ₄.     

An Analogue of the Magnus Problem for Associative Algebras

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n + k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

On webs of maximum rank

This paper contains two results on webs of maximal rank. First, we show that at each point, the web normals for a codimension-r web of maximum r-rank are (r−1)-dimensional generators of a rational normal scroll in the projectivized tangent space to the web domain, an extension of a theorem of Chern and Griffiths in the case r=2 [5]. We use this statement to deduce that webs of maximum rank are almost-Grassmannizable in the sense of Akivis [1]. Second, we show that there are exceptional (that is, maximum rank, but not algebraizable) webs W(2n, n, 2) for all n≥2. The construction relies on the properties of zero-cycles on algebraic K3 surfaces.     

The Bar Construction and Affine Stacks

We discuss the equivalence of two constructions of a unipotent group scheme attached to a differential graded algebra over a ?-algebra. The first construction is using the bar resolution and the second is using Toen's theory of affine stacks. We use this to establish the equivalence of two approaches to the comparison theorem in p-adic Hodge theory for the unipotent fundamental group of varieties defined over p-adic fields.     

Partitioning Sets of Triples into Small Planes

We study partitions of the set of all ₃ ^v triples chosen from a v-set intopairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2,2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions)or copies of some planes of each type (mixed partitions).We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in severalcases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We constructsuch partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, andan affine partition for v = 18. Using these as starter partitions, we prove that Fano partitionsexist for v = 7 ⁿ + 1, 13 ⁿ + 1,27 ⁿ + 1, and affine partitions for v = 8 ⁿ + 1,9 ⁿ + 1, 17 ⁿ + 1. In particular, both Fano and affine partitionsexist for v = 3⁶ⁿ + 1. Using properties of 3-wise balanced designs, weextend these results to show that affine partitions also exist for v = 3²ⁿ .Similarly, mixed partitions are shown to exist for v = 8 ⁿ ,9 ⁿ , 11 ⁿ + 1.     

Affine Isometric Embeddings and Rigidity

The Pick cubic form is a fundamental invariant in the (equi)affine differential geometry of hypersurfaces. We study its role in the affine isometric embedding problem, using exterior differential systems (EDS). We give pointwise conditions on the Pick form under which an isometric embedding of a Riemannian manifold M ³ into is rigid. The role of the Pick form in the characteristic variety of the EDS leads us to write down examples of nonrigid isometric embeddings for a class of warped product M ³'s.