The Picard group of a coarse moduli space of vector bundles in positive characteristic

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M _r,L ^ss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M _r,L ^ss) = ℤ, identify the ample generator, and deduce that M _r,L ^ss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.     

On the integral of mean curvature of a flattened convex body in space forms

Under the assumptions that E _λ ⁿ is an n-dimensional, simply connected Riemannian manifold of constant sectional curvature λ and L _λ ^r is an r-dimensional, totally geodesic submanifold of E _λ ⁿ , the paper investigates the q-th integral of the mean curvature M _q ⁿ of a convex body K ^r in E _λ ⁿ and gives the expression of M _q ⁿ in the terms of M _p ^r , where M _p ^r is the p-th integral of the mean curvature of K ^r > in L _λ ^r . A result of L. A. Santaló [2] holds in particular.     

On relative widths of classes of differentiable functions. II

We obtain an upper bound for the least value of the factor M for which the Kolmogorov widths d _n (W _C ^r , C) are equal to the relative widths K _n (W _C ^r , MW _C ^j , C) of the class of functions W _C ^r with respect to the class MW _C ^j , provided that j > r. This estimate is also true in the case where the space L is considered instead of C.     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Minors of a random binary matroid

Let A be an n × m matrix over GF ₂ where each column consists of k ones, and let M be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant C_M such that m ≥ C_Mn² implies that the binary matroid induced by A contains M as a minor. We prove that if the columns of A = A _n,m,k are chosen randomly, then there are constants k_M,L_M such that k ≥ k_M and m ≥ L_Mn implies that A contains M as a minor with high probability .     

The Atiyah bundle and connections on a principal bundle

Let M be a C ^∞ manifold and G a Lie a group. Let E _G be a C ^∞ principal G-bundle over M. There is a fiber bundle C(E _G ) over M whose smooth sections correspond to the connections on E _G . The pull back of E _G to C(E _G ) has a tautological connection. We investigate the curvature of this tautological connection.     

Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary

We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somer ^th mean curvature functionH _r a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0H _r1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H ₁ is the mean curvature) we obtain results whenC is convex.     

QUASI-FROBENIUS BIMODULES OF FUNCTIONS ON A SEMIGROUP

Let _AM_B be a QF-bimodule, A a left Artinian ring, B a right Artinian ring, G a semigroup with a unit element (a monoid). Let M^G be the set of all functions on G with values in M. Consider M^G as an (AG, BG)-bimodule over the semigroup rings AG and BG. It is proved that the annihilator maps I → r_M^G (I) and R → l_AG (R) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated left ideals I ? AG and the set of right BG-submodules R ? M^G finitely generated over B. The maps J → l_M^G (J) and L → r_AG (L) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated right ideals J ? AG and the set of left AG-submodules L ? M^G finitely generated over A. This result also makes it possible, starting from a given QF-bimodule _A M_B , to construct new QF-bimodules _AG/IS_BG/J as bimodules of functions on a semigroup with values in M.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

The Category of Directed Systems in a Category

We present two related categorical constructions. Given a category C, we construct a category C^[d], the category of directed systems in C. C embeds into C^[d], and if C has enough colimits, then C is monadic over C^[d]. Also, if E,M is a factorization structure for C, then C^[d] has a related factorization structure E_d M_d such that if E consists entirely of monic arrows, then so does E_d and the E_d-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does M_d and the M_d-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C^[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C^[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

On the L2n 1– Extension Properties

We have proved that for all compact linear operator u from R into an L_p ([0,1], ν) (0 < p < 1) extends to L ₁ ([0,1], ν), where R denotes the closed linear subspace in L ₁ ([0,1], ν) of the Rademacher functions {r_n }_{n ? N}. In this paper, we study this type of extension for E_n ? L_2n ¹ where E_n is the n–dimensional subspace which appears in Kasin's theorem such that L_2n ¹ = E_n ⊕ E ^⊥ _n and the L_2n ¹ , L_2n ² norms are universally equivalent on both E_n , E ^⊥ _n. We show that, the precedent extension fails for the pair (E_n , L_2n ¹ ) and we generalize this to any E in an L ₁(Ω, A, P) by giving some conditions on E.     

On approximation of functions from below by splines of the best approximation with free nodes

Let M be the set of functions integrable to the power β=(r+1+1/p)^-1. We obtain asymptotically exact lower bounds for the approximation of individual functions from the set M by splines of the best approximation of degree rand defect k in the metric of L _p.     

Existence of contractive projections on preduals of <Emphasis Type="Italic">JBW</Emphasis>*-triples

In 1965, Ron Douglas proved that if X is a closed subspace of an L ¹-space and X is isometric to another L ¹-space, then X is the range of a contractive projection on the containing L ¹-space. In 1977 Arazy-Friedman showed that if a subspace X of C ₁ is isometric to another C ₁-space (possibly finite dimensional), then there is a contractive projection of C ₁ onto X. In 1993 Kirchberg proved that if a subspace X of the predual of a von Neumann algebra M is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of M onto X.     

Sharpening of the estimates for relative widths of classes of differentiable functions

We improve the earlier obtained upper estimates for the least value of the coefficient M for which the Kolmogorov widths d _n (W _C ^r , C) of the function class W _C ^r are equal to the relative widths K _n (W _C ^r , MW _C ^j , C) of the class W _C ^r with respect to the class MW _C ^j , j < r.     

Gauge-Invariant Characterization of Yang–Mills–Higgs Equations

Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n⁺, n⁻) and let E → M be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n⁺, n⁻) × G-invariant quadratic Lagrangians on J¹(C × _M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n⁺, n⁻) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory. Submitted: May 19, 2005; Accepted: April 25, 2006     

A remark on a maximal function over a Cantor set of directions

LetM _e ⁰ be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL ². We consider a sequence of operators over finite sets of directions converging toM _e ⁰ . We improve the previous estimate for the (L ²,L ²)-norm of these particular operators. We also prove thatM _e ⁰ is bounded from some subsets ofL ² toL ². These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip. Partially supported by Spanish DGICYT grant no. PB90-0187.     

On the theta completions for maximal subalgebras

Let L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and M_L?C and C∕M_L contains no non-zero ideal of L∕M_L, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.     

Inequalities for scalar-valued linear operators that extend to their vector-valued analogues

A theorem of Marcinkiewicz and Zygmund asserts that a linear operator satisfying a strong type (L^r, L^q) inequality with norm M automatically extends to a vector-valued operator satisfying a strong type (L^r(l^v), L^q(l^v)) inequality with norm not exceeding C_{r, q}^(γ)M. In this paper, this theorem is proved in a more general context by replacing the L^q metric with a more general class of metrics. In doing so, the theorem of Marcinkiewicz and Zygmund is not only extended to more general contexts, but improvements of that theorem are also realized. In particular, our results show that operators satisfying weak type inequalities automatically extend to their vector-valued analogues; also the constant C_{r, q}^(γ) may be taken as one in the theorem of Marcinkiewicz and Zygmund whenever q ? r, and this includes most cases of interest.