On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

On R-robustly entropy-expansive diffeomorphisms

For a homoclinic class H(p _f ) of f ?? Diff¹(M), f?OH(p _f ) is called R-robustly entropy-expansive if for g in a locally residual subset around f, the set ?? _? (x) = {y ?? M: dist(g ⁿ (x), g ⁿ (y)) ?? g3 (?n ?? ?)} has zero topological entropy for each x ?? H(p _g ). We prove that there exists an open and dense set around f such that for every g in it, H(p _g ) admits a dominated splitting of the form E ?? F ₁ ?? ... ?? F _k ?? G where all of F _i are one-dimensional and non-hyperbolic, which extends a result of Pacifico and Vieitez for robustly entropy-expansive diffeomorphisms. Some relevant consequences are also shown.     

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n^1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n^1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n.     

Recurrence relations based on minimization and maximization

Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and g(n) = δ_nI. (2) α + β > 1, min(α, β) > 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and (a) g(n) = δ_n1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases $\text{M(n) = Ω(n}{}^{\mathrm{<mtext>1\; +\; </mtext><mtext>1</mtext><mtext>\gamma </mtext>}}\text{)}$, where γ is defined by α^?γ + β^?γ = 1, and that $\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{M(n)}\text{n}{}^{\mathrm{1\; +\; \gamma }}$ does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even.     

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H²(M)?L^2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.     

Bimodule and twisted representation of vertex operator algebras

In this paper, for a vertex operator algebra V with an automorphism g of order T, an admissible V-module M and a fixed nonnegative rational number n ∈1/T Z_+, we construct an A_(g,n)(V)-bimodule Ag,n(M) and study its properties, discuss the connections between bimodule A_(g,n)(M) and intertwining operators. Especially, bimodule A _(g,n)-1T(M) is a natural quotient of A_(g,n)(M) and there is a linear isomorphism between the space IM~k M Mjof intertwining operators and the space of homomorphisms HomA_(g,n)(V)(A_(g,n)(M)  A_(g,n)(V)M~j(s), M~k(t)) for s, t ≤ n, M~j, M~k are g-twisted V modules, if V is g-rational.     

Recurrence relations based on minimization

This paper investigates solutions of the general recurrence M(0) = g(0), M(n + 1) = g(n + 1) + min_0?k?n(αM(k) + βM(n ? k)) for various choices of α, β, and g(n). In a large number of cases it is possible to prove that M(n) is a convex function whose values can be computed much more efficiently than would be suggested by the defining recurrence. The asymptotic behavior of M(n) can be deduced using combinatorial methods in conjunction with analytic techniques. In some cases there are strong connections between M(n) and the function H(x) defined by $\text{H(x) = 1}\text{for}\text{x < 1, H(x) = H(}\text{(x ? 1)}\text{α}\text{) + H(}\text{(x ? 1)}\text{β}\text{)}\text{for}\text{x ? 1}$. Special cases of these recurrences lead to a surprising number of interesting problems involving both discrete and continuous mathematics.     

The Hamiltonian dynamics over real hypersurfaces in Kaehler manifolds

Let X be a Kaehler manifold with complex dimension n. Let ωX be its Kaehler form. Let M be a strongly pseudo convex real hypersurface in X. For this hypersurface, the deformation theory of CR structures is successfully developed. And we find that H¹(M,T^′) (the T^′-valued Kohn-Rossi cohomology) is the Zariski tangent space of the versal family. In this paper, the geometrical meaning of H¹(M,O) is studied, and we propose to study displacements of the real hypersurface, which preserves the type of the differential form, ωX, over CR structures, on M, infinitesimally.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

A note on scalar curvature type problems in dimension 4

Let (M,g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem Δgu+a(x)u=f(x)u³ always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem.     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

Dirichlet series related to the Riemann zeta function

A study is made of the function H(s, z) defined by analytic continuation of the Dirichlet series H(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z, where s and z are complex variables. For each fixed z it is shown that H(s, z) exists in the entire s-plane as a meromorphic function of s, and its poles and residues are determined. Also, for each fixed s ≠ 1 it is shown that H(s, z) exists in the entire z-plane as a meromorphic function of z, and again its poles and residues are determined. Two different representations of H(s, z) are given from which a reciprocity law, H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), is deduced. For each integer q ≥ 0 the function values H(s, ?q) and H(?q, s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet series T(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z (m + n)^?1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma.     

Collapse of warped foliations

The aim of this note is to generalize the concept of warped product to a foliated manifold (M,F,g) as follows: If is a smooth function constant along the leaves of the foliation F then new metric structure gf on the manifold M is constructed as follows: gf(v,w)=f²g(v,w) if v,w are tangent to F and gf(v,w)=g(v,w) if v or w is perpendicular to F. A foliated manifold (M,F,gf) is called warped foliation while f is called warping function.Next, if is a sequence of warping functions on M, the question of the existence of the limit in Gromov-Hausdorff of a sequence ((M,F,gfn))_n∈N warped foliation is asked. A number of examples is considered such foliations with dense leaf or foliations consisting of finite number of Reeb components. Next, sufficient and necessary condition of converging in Gromov-Hausdorff sense of a Riemannian foliation with all leaves compact to the space of leaves with a metric defined by Hausdorff distance of leaves is developed. Finally some results on Hausdorff foliations with all leaves compact are shown.