Coefficients of multivalent symmetric functions of bounded boundary rotation

In this paper the sharp coefficient estimate problem for the classesC _p(β, m) andV _p(k,m) of multivalent close-to-convex functions of order β and multivalent functions of bounded boundary rotation of at mostkπ, whose functions are given bym-fold symmetric gap series, have been discussed respectively for β≥1?p/m>0 andk≥2(m/p). Moreover, it is shown that every function inV _p(k,m) arep-valent close-to-convex; hencep-valent; ifk<2 (1+m/p).     

On Linearly Isometric Extensions for 1-Lipschitz Mappings Between Unit Spheres of ALP-spaces （p 〉 2）

In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.     

Asymptotics of orthogonal L-polynomials for log-normal distributions

Asymptotic properties of orthogonal Laurent (L-) polynomialsV _n (z), associated with log-normal distributions, are derived by constructive methods. It is shown that the sequences {V _2n (z)} and {V _2n+1 (z)} converge separately (asn→∞) to functionsV ⁽⁰⁾ (z) andV ⁽¹⁾ (z), respectively, both holomorphic in 0<|z|<∞. Explicit Laurent series expansions are obtained, from which it follows that each limit function has essential, isolated singularities atz=0 andz=∞.     

Commutators of Riesz Transforms Related to?Schr?dinger Operators

In this work we obtain boundedness on L ^p , for 1<p<??, of commutators T _b f=bTf?T(bf) where T is any of the Riesz transforms or their conjugates associated to the Schr?dinger operator ???+V with V satisfying an appropriate reverse H?lder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=??, under a slightly stronger condition on?b.     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

Spectral analysis of a Dirac operator with a meromorphic potential

We consider an operator Q(V) of Dirac type with a meromorphic potential given in terms of a function V of the form V(z)=λV₁(z)+μV₂(z), z∈C?{0}, where V₁ is a complex polynomial of 1/z, V₂ is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q(V) acts in the Hilbert space L²(R²;C⁴)=^⊕4L²(R²). The main results we prove include: (i) the (essential) self-adjointness of Q(V); (ii) the pure discreteness of the spectrum of Q(V); (iii) if V₁(z)=z−p and 4?degV₂?p+2, then kerQ(V)≠{0} and dimkerQ(V) is independent of (λ,μ) and lower order terms of ∂V₂/∂z; (iv) a trace formula for dimkerQ(V).     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.     

Phase transitions with semi-diffuse interfaces

In this paper, we examine new “phase-field” models with semi-diffuse interfaces. These models have the property that the −1/+1 planar phase transitions take place over a finite interval. The models also support multiple interface solutions with interfaces centered at arbitrary points L₁<L₂<?<L_N. These solutions correspond to local minima of an entropy functional (see (3.3) and (3.7)) rather than saddle points and are dynamically stable. The classical models have no such exact solutions but they do support solutions with N equally spaced transition points where the order parameter transitions between values p_min(N) and p_max(N) satisfying −1<p_min(N)<0<p_max(N)<1. These solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable.     

On the structural properties of the weight space L p(x),ω for 0 < p(x) < 1

The main purpose of this paper is to study the weight space L _p(x),ω for 0 < p(x) < 1 as well as the topology of this space. Embeddings between different Lebesgue spaces with variable exponent of summability are established. In particular, it is proved that the set of all linear continuous functionals over L _p(x),ω for 0 < p(x) < 1 consists only of the zero functional.     

On density of interpolation points,a Kadec-type theorem,and Saff's principle of contamination inL p-approximation

Letf ∈L _p (I) and denote byB _n,p (f) the polynomial of bestL _p-approximation tof of degreen (1<p<∞,I=[?1,1], the norm is weightedL _p-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈L _p (I) (not a polynomial) points of sign change of the error functionf-B _n,p (f) are dense inI asn→∞.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μ_π, on the unit hypersurface, V₁(π), for the Teichmüller metric. Μ_π is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be L^p(Μ_π), 0 <p < 1, over the moduli space V₁(π)/γ (g, n). In particular, Μ_π(V₁(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur. Research supported by NSF-MCS-8219148 and NSF-DMS-8521620.     

Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

Multilinear Calderón-Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H¹(μ)×H¹(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp₁(μ)×Lp₂(μ) into Lp(μ) with 1<p₁,p₂<∞ and p satisfying 1/p=1/p₁+1/p₂.     

Convergent sieve sequences in arithmetic progressions

Let V be a set of pairwise coprime integers not containing 1 and suppose, there is a 0?δ<1, such that _∑v∈Vv−1+δ<∞ holds. Let χV(n)=1 if v?n for all v∈V and χV(n)=0 elsewhere. We study the behavior of χV in arithmetic progressions uniformly in the modulus, both individually and in the quadratic mean over the residue classes. As an application, new bounds for the mean square error of squarefree numbers in arithmetic progressions are obtained.     

Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) fQ_| has 2g+2 critical points p₀,p₁,…,p2g+1 with f(p₀)<f(p₁)<?<f(p2g+1) where p₀ is a minimum and p2g+1 is a maximum, and p₁,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri<f(pi), then f−1(ri)∩Q consists of a circle if i is odd, and f−1(ri)∩Q consists of two circles if i is even.     

Factors and Connected Induced Subgraphs

 Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg _G (x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑ _{x ∈ V ( G )} f(x) is odd. Received: June 29, 1998?Final version received: July 30, 1999     

A note on balanced bipartitions

A balanced bipartition of a graph G is a bipartition V₁ and V₂ of V(G) such that −1≤|V₁|−|V₂|≤1. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V₁,V₂ such that max{e(V₁),e(V₂)}≤m/3, where e(V_i) denotes the number of edges of G with both ends in V_i. In this note, we prove this conjecture for graphs with average degree at least 6 or with minimum degree at least 5. Moreover, we show that if G is a graph with m edges and n vertices, and if the maximum degree Δ(G)=o(n) or the minimum degree δ(G)→∞, then G admits a balanced bipartition V₁,V₂ such that max{e(V₁),e(V₂)}≤(1+o(1))m/4, answering a question of Bollobás and Scott in the affirmative. We also provide a sharp lower bound on max{e(V₁,V₂):V₁,V₂ is a balanced bipartition of G}, in terms of size of a maximum matching, where e(V₁,V₂) denotes the number of edges between V₁ and V₂.