Polynomial-time computation of the degree of a dominant morphism in zero characteristic. III

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in (dd′)ⁿ and the size of the input. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 203–239.     

Polynomial-Time Computation of the Degree of a Dominant Morphism in Characteristic Zero. I

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables over the field of characteristic zero. We show how to compute the degree of a dominant rational morphism from W to W′. The morphism is given by homogeneous polynomials of degree d′.This algorithms is deterministic and polynomial in (dd′)ⁿ and the size of the input. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2003, pp. 189–235.     

Polynomial-time computation of the degree of a dominant morphism in zero characteristic. IV

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in (dd′) ⁿ and the size of the input. This work concludes a series of four papers. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 260–294.     

Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

Perturbations of regularizing maximal monotone operators

We consideru′(t)+Au(t)∋f(t), whereA is maximal monotone in a Hilbert spaceH. AssumeA is continuous or A=ϱφ or intD(A)≠∅ or dimH<∞. For suitably boundedf′s, it is shown that the solution mapf→u is continuous, even if thef′s are topologized much more weakly than usual. As a corollary we obtain the existence of solutions ofu′(t)+Au(t)∋B(u(t)), whereB is a compact mapping inH. An erratum to this article is available at .     

Searches for <Emphasis Type="Italic">W′</Emphasis> and <Emphasis Type="Italic">Z′</Emphasis> in models with large extra dimensions

We discuss the characteristic features of processes mediated by intermediate gauge bosons in the framework of theories with large extra dimensions and show that if gauge bosons propagate in the entire multidimensional space, then a destructive interference arises not only between W and W′ (or Z and Z′) but also between W′ and Z′ and the respective Kaluza-Klein towers of higher excitations of W and Z bosons. We perform and graphically present calculations for the LHC with the center-of-mass energy 14 TeV.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.     

To what extent does the dual Banach spaceE′ determine the polynomials overE?

We show that under conditions of regularity, ifE′ is isomorphic toF′, then the spaces of homogeneous polynomials overE andF are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.     

Compact complex homogeneous manifolds with large automorphism groups

Let X be a compact complex homogeneous manifold and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. It is well-known that the dimension of Aut(X) is bounded by an integer that depends only on n=dim X. Moreover, if X is K?hler then dimAut (X)≤n(n+2) with equality only when X is complex projective space. In this article examples of non-K?hler compact complex homogeneous manifolds X are given that demonstrate dimAut(X) can depend exponentially on n. Let X be a connected compact complex manifold of dimension n. The group of holomorphic automorphisms of X, Aut(X), is a complex Lie group [3]. For a fixed n>1, the dimension of Aut(X) can be arbitrarily large compared to n. Simple examples are provided by the Hirzebruch surfaces F _m , m∈N, for which dimAut(F _m )=m+5, see, e.g. [2, Example 2.4.2]. If X is homogeneous, that is, any point of X can be mapped to any other point of X under a holomorphic automorphism, then the dimension of the automorphism group of X is bounded by an integer that depends only on n, see [1, 2, 6]. The estimate given in [2, Theorem 3.8.2] is roughly dimAut(X)≤(n+2) ⁿ . For many classes of manifolds, however, the dimension of the automorphism group never exceeds n(n+2). For example, it follows directly from the classification given by Borel and Remmert [4], that if X is a compact homogeneous K?hler manifold, then dimAut(X)≤n(n+2) with equality only when X is complex projective space P ⁿ . It is an old question raised by Remmert, see [2, p. 99], [6], whether this same bound applies to all compact complex homogeneous manifolds. In this note we show that this is not the case by constructing non-K?hler compact complex homogeneous manifolds whose automorphism group has a dimension that depends exponentially on n. The simplest case among these examples has n=3m+1 and dimAut(X)=3m+3 ^m , so the above conjectured bound is exceeded when n≥19. These manifolds have the structure of non-trivial fiber bundles over products of flag manifolds with parallelizable fibers given as the quotient of a solvable group by a discrete subgroup. They are constructed using the original ideas of Otte [6, 7] and are surprisingly similar to examples found there. Generally, a product of manifolds does not result in an automorphism group with a large dimension relative to n. Nevertheless, products are used in an essential way in the construction given here, and it is perhaps this feature that caused such examples to be previously overlooked. Oblatum 13-X-97 & 24-X-1997     

Normal Hopf subalgebras in cocycle deformations of finite groups

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = H ^σ of the Hopf algebra H = k ^G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra corresponds to a Hopf subalgebra . Our main result is an explicit necessary and sufficient condition for the normality of L′ in L. This work was partially supported by CONICET, Fundación Antorchas, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC).     

A transfer morphism for Witt cogroups

Let A be a ring with involution in which 2 is invertible, ε be 1 or −1, and s(∈ A) be a central regular element such that s* = s. A transfer homomorphism _εW′₀(A/s) → _ε W′₁(A) for Witt cogroups is constructed and a projection formula is proved. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 149–158.     

Efficient construction of local parameters for irreducible components of an algebraic variety in nonzero characteristic

Consider an (n-s)-dimensional algebraic variety W defined over an infinite field k of nonzero characteristic p and irreducible over this field. Let W be a subvariety of the projective space of dimension n. We prove that the local ring of W has a sequence of local parameters represented by s nonhomogeneous polynomials with the product of degrees less than the degree of the variety multiplied by a constant depending on n. This allows us to prove the existence of an effective smooth cover and a smooth stratification of an algebraic variety in the case of the ground field of nonzero characteristic, extending the analogous results of the author obtained earlier for the ground field of zero characteristic. Bibliography: 6 titles. To A. M. Vershik with sincere gratitude and respect __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 248–278.     

Monodromy and irreducibility criteria with algorithmic applications in characteristic zero

Consider a projective algebraic variety V, which is the set of all common zeros of homogeneous polynomials of degrees less than d in n + 1 variables over a field of characteristic zero. We suggest an algorithm that decides whether two (or more) given points of V belong to the same irreducible component of V. We also show how to construct, for each s < n, an (s + 1)-dimensional plane in the projective space such that the intersection of every irreducible component of dimension n — s of V with the constructed plane is transversal and is an irreducible curve. These algorithms are deterministic and polynomial in dⁿ and the input size. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 130–152.This revised version was published online in April 2005 with a corrected cover date and article title.     

On nulls of perturbed Fredholm operators and degenerate homoclinic bifurcations

It is known that small perturbations of a Fredholm operator L have nulls of dimension not larger than dim N (L). In this paper for any given positive integer k≤dim N(L) we prove that there is a perturbation of L which has an exactly κ-dimensional null. Actually, our proof gives a construction of the perturbation. We further apply our result to concrete examples of differential equations with degenerate homoclinic orbits, showing how many independent homoclinic orbits can be bifurcated from a perturbation.     

Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

 Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂ ⁿ , n > 1, and a real analytic subset M′ ⊂ ℂ ^n′ . We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M ^′ . Received: 7 August 2001 Mathematics Subject Classification (2000): 32V25, 32V40, 32H99     

Trajectories of the zeros of hypergeometric polynomials F(−n, b; 2b; z) for b < − 1/2

In a previous paper [2] we studied the zeros of hypergeometric polynomials F(−n, b; 2b; z), where b is a real parameter. Making connections with ultraspherical polynomials, we showed that for b > − 1/2 all zeros of F(−n, b; 2b; z) lie on the circle |z − 1| = 1, while for b < 1 − n all zeros are real and greater than 1. Our purpose now is to describe the trajectories of the zeros as b descends below the critical value − 1/2 to 1 − n. The results have counterparts for ultraspherical polynomials and may be said to “explain” the classical formulas of Hilbert and Klein for the number of zeros of Jacobi polynomials in various intervals of the real axis. These applications and others are discussed in a further paper [3].     

Taylor formula for homogenous groups and applications

In this paper, we provide a Taylor formula with integral remainder in the setting of homogeneous groups in the sense of Folland and Stein (Hardy spaces on homogeneous groups. Mathematical notes, vol 28. Princeton University Press, Princeton, 1982). This formula allows us to give a simplified proof of the so-called ‘Taylor inequality’. As a by-product, we furnish an explicit expression for the relevant Taylor polynomials. Applications are provided. Among others, it is given a sufficient condition for the real-analiticity of a function whose higher order derivatives (in the sense of the Lie algebra) satisfy a suitable growth condition. Moreover, we prove the ‘L-harmonicity’ of the Taylor polynomials related to a ‘L-harmonic’ function, when L is a general homogenous left-invariant differential operator on a homogeneous group. (This result is one of the ingredients for obtaining Schauder estimates related to L).     

Linear groups with minimality condition for some infinite-dimensional subgroups

Let F be a field, let A be a vector space over F, and let GL(F, A) be the group of all automorphisms of the space A. If H is a subgroup of GL(F, A), then we set aug dim_F (H) = dim_F (A(ωFH)), where ωFH is the augmentation ideal of the group ring FH. The number aug dim_F (H) is called the augmentation dimension of the subgroup H. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1476–1489, November, 2005.