On thek dimensional Piatetski-Shapiro prime number theorem

Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x ₁ c ₁,⋯,c _k ) denote the number of primesp withp⩽x, , where 1<c ₁<⋯<c _k are fixed constants. It is proved that π(x;c ₁,⋯,c _k ) has an asymptotic formula ifc ₁ ⁻¹ +⋯+c _k ⁻¹ >k−k/(4k ²+2). Project supported by the National Natural Science Foundation of China (Grant No. 19801021) and the Natural Science Foundation of Shandong Province (Grant No.Q98A02110).     

Cycles with a given number of vertices from each partite set in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d ⁺(x) and d ⁻(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p ∈ ℕ such that d ⁺(x) = d ⁻(x) = p for all vertices x of D. A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether c-partite tournaments with r vertices in each partite set contain a cycle with exactly r − 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c ⩾ 3, then we will show that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a cycle with exactly r − 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2.     

Nouvelle demonstration d’un theoreme de J. L. Krivine sur la finie representation del p dans un espace de Banach

We prove that if (x _n) is a sequence in a Banach space with infinite dimensional span, thenc _o orl _p for al<=p<∞ is block finitely represented in (x _n).     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality |p₁^c+p₂^c+p₃^c+p₄^c+p₅^c-N| < log^-1 N \vert p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N\vert < \log^{-1} N is solvable, where p ₁,···,p ₅ are primes.     

Weighted Weak Behaviour of Fourier-Jacobi Series

Let w(x) = (1 - x)^α (1 + x)^β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators S_n are uniformly bounded from L^p,1 to L^p,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uS_n(u^?1f), where u is also Jacobi weight.     

The limiting angle of certain riemannian brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ? M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let Sect_M(x) denote the sectional curvature of any plane section in M_x. We prove that for each c > 2, there is a 0 < β < 1 such that if - L²r(x)^2β ≦ Sect_M(x) ≦ -cr(x)^?2 for all x in the complement of a compact set, then lim_{t → ∞} θ(X_t) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

On Hilbert''s Integral Inequality

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L²[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π²is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫^∞₀(c(t²x)/(1 + t²)) dt − c(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Small solutions of polynomial congruences

Let p be prime and q|p − 1. Suppose x ^q ≡ a(mod p) has a solution. We estimate the size of the smallest solution x ₀ with 0 < x ₀ < p. We prove that |x ₀| ≪ p ³/² q ⁻¹ log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and Konyagin, we derive refinements of our result.     

Low degree rational spline interpolation

For a strictly monotone functionf on [a,b] we describe the possibility of finding an interpolating rational splineS of the formS(x)=c ₀ +c ₁ x/(1+d ₁ x) on each subinterval of the grida=x ₀ <x ₁ <...<x _n =b. This leads to a nonlinear system for which we get the local existence and uniqueness of a solution. We prove that ‖S−f‖_∞=O(h ³). Numerical test shows good approximation properties of these splines.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

On numbers badly approximable by dyadic rationals

We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle we give an elementary proof that the set {x ∈ : 2 ⁿ x ≥ c (mod 1) n ≥ 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c ↦ dim _H {x ∈ [0; 1]: x − m/2 ⁿ < c/2 ⁿ (mod 1) finitely often}.     

Cp

The spacec _p is the class of operators on a Hilbert space for which thec _p norm |T| _p =[trace(T*T) ^p/2]^1/p is finite. We prove many of the known results concerningc _p in an elementary fashion, together with the result (new for 1<p<2) thatc _p is as uniformly convex a Banach space asl _p. In spite of the remarkable parallel of norm inequalities in the spacesc _p andl _p, we show thatp ≠ 2, noc _p built on an infinite dimensional Hilbert space is equivalent to any subspace of anyl _p orL _p space. The author was supported by National Science Foundation Grant GP-5707.     

Schur multiplier characterization of a class of infinite matrices

Let B _w (ℓ ^p ) denote the space of infinite matrices A for which A(x) ∈ ℓ ^p for all x = {x _k } _k=1^∞ ∈ ℓ ^p with |x _k | ↘ 0. We characterize the upper triangular positive matrices from B _w (ℓ ^p ), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.