Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows.     

On spectral characterizations of amenability

We show that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satisfied: the associated unitary representationπ _X ofG intoL ²(X, μ) is weakly contained into the regular representationλ _G and there exists aG-equivariant norm one projection fromL∞(X×X) ontoL∞(X). We give examples of ergodic discrete group actions which are not amenable, althoughπ _X is weakly contained intoλ _G.     

Some results on distance two labelling of outerplanar graphs

Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.     

Complemented copies of ℓ<Subscript>p</Subscript> spaces in tensor products

We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗_π Y, their injective tensor product X ⊗_ɛ Y, or the dual (X ⊗_π Y)^* contain complemented copies of ℓ_p.     

Multilinear Operators on C(K, X) Spaces

Given Banach spaces X, Yand a compact Hausdorff space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K, X) into Yto be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for Xto have the Schur property (resp. to contain no copy of c ₀), and for Kto be scattered. This extends results concerning linear operators.     

Separable functors for the category of Doi-Hopf <Emphasis Type="Italic">π</Emphasis>-modules

Let π be a discrete group. Given a Doi-Hopf π-datum (H,A,C) and α∈π, we give necessary and sufficient conditions for the functor F ^(α) from the category of Doi-Hopf π-modules to the category of right A _α -modules to be separable. This leads to a generalized notion of integrals. As an application, we prove a Maschke type theorem for Doi-Hopf π-modules and relative Hopf π-modules.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

A(α)-acceptability of pade approximations to function exp (Q)

In this paper, a necessary and sufficient condition of A(a)-acceptability for rational approximations to the functionexp(q) and some sufficient conditions to guarantee A(α)-acceptability of Padé approximations[^(R)]_mⁿ (q)\hat R_m^n (q) to the functionexp(q) are given, where a ∈(0,π/2), n≤m,m≥2. Furthermore, it is proved that the condition of A(α)-acceptability of rational approximations toexp(q) is equivalent to the nonnegatively of a real polynomial in interval (−∞,0). Finally, we prove that[^(R)]₄¹ (q)\hat R_4^1 (q) is A(π/3)-acceptable. Based on this conclusion, two A(π/3)-stable multiderivative (hybrid) one-step methods are constructed.     

Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.     

The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ⁻⁴ dx²+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ²C^∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productX _b ² (the blowup ofX ² about the corner, (ϖX)²). The structure of the resolvent is only slightly more complicated. As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t)=Au(t)+f(t)(0 ≤t≤2π),where A is a closed operator in a Banach space X,0α1 and Dα is the fractional derivative in the sense of Weyl.Although this problem is not always well-posed in Lp(0,2π;X) or periodic continuous function spaces Cper([0,2π];X),we show by using the method of sum that it is well-posed in some subspaces of L p(0,2π;X) or C per([0,2π];X).     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Characterizing hyperbolic spaces and real trees

Let X be a geodesic metric space. Gromov proved that there exists ε ₀ > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε ₀ · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε ₀. We also show that if all the triangles D í X{\Delta \subseteq X} satisfy the Rips condition with constant ε ₀ · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.     

A remark on the range of elementary operators

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ _A : L(H) → L(H) by Δ _A (X) = AXA − X. In this paper we study the class of operators A ∈ L(H) which have the following property: ATA = T implies AT*A = T* for all trace class operators T ∈ C ₁(H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ _A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.