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1.
We consider an infinite lower triangular matrix L=[?n,k]n,kN0 and a sequence Ω=(ωn)nN0 called the (a,b)-sequence such that every element ?n+1,k+1 except lying in column 0 can be expressed as
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2.
In the space L 2 of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality
$$E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$
, is considered, where E n (f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L 2 of order 1 or 2. The value
$$\mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}}{{\omega (\delta ,f)}}:f \in L^2 } \right\}$$
is found at the points δ = 2π/m (where m ∈ ?) for m ≥ 3n 2 + 2 and ω = ω 1 as well as for m ≥ 11n 4/3 ? 1 and ω = ω 2.
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3.
Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup Hω12 that is an HFD with the following property
(P)
the projection of H onto every partial product I2 for Iω[ω1] is onto.
Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ2, there is an HFD topological group in ω12 which has property (P).  相似文献   

4.
For any real number β > 1, let S n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S n (β) is β ?n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets
$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$
and
$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$
are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) .
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5.
Let Ω⊂{0,1}N be a nonempty closed set with N={0,1,2,…}. For N={N0<N1<N2<?}⊂N and ω∈{0,1}N, define ω[N]∈{0,1}N by and
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6.
Suppose β1 α1 ≥0,β2 α2 ≥ 0 and(k,j) ∈R2. In this paper, we mainly investigate the mapping properties of the operator T_αβf(x,y,z)=∫_Q~2f(x-t,y-s,z-t~ks~j)e~(-2πit-β1_s-β2)t~(-1-α1)s~(-1-α2)dtds on modulation spaces, where Q~2 = [0,1] x [0,1] is the unit square in two dimensions.  相似文献   

7.
Hamiltonian cycles in Dirac graphs   总被引:1,自引:1,他引:0  
We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least
$exp_2 [2h(G) - n\log e - o(n)],$
where h(G)=maxΣ e x e log(1/x e ), the maximum over x: E → ?+ satisfying Σ e?υ x e = 1 for each υV, and log =log2. (A second paper will show that this bound is tight up to the o(n).)
We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) n . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) n , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.  相似文献   

8.
In this paper, we consider the non-autonomous modified Korteweg-de Vries (mKdV) equation
$${u_t} = {u_{xxx}} - 6f\left( {\omega t} \right){u^2}{u_x},x \in \mathbb{R}/2\pi \mathbb{Z}$$
, where f(ωt) is real analytic and quasi-periodic in t with frequency vector ω = (ω1,ω2, · · ·; ω m ). Basing on an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, we obtain the existence of Cantor families of smooth quasi-periodic solutions.
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9.
Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L1-solution of the following random matrix refinement type equation
f(x)=Ω|detL(ω)|C(ω)f(L(ω)x-M(ω))P(dω),  相似文献   

10.
We prove theorems on the exact asymptotic forms as u → ∞ of two functional integrals over the Bogoliubov measure μB of the forms
$$\int_{C[0,\beta ]} {[\int_0^\beta {|x(t){|^p}dt{]^u}d{\mu _B}(x)} } ,\;\int_{C(0,\beta )} {\exp \left\{ {\mu {{(\int_0^\beta {|x(t){|^p}dt} )}^{a/p}}} \right\}d{\mu _B}(x)} $$
for p = 4, 6, 8, 10 with p > p0, where p0 = 2+4π22ω2 is the threshold value, β is the inverse temperature, ω is the eigenfrequency of the harmonic oscillator, and 0 < α < 2. As the method of study, we use the Laplace method in Hilbert functional spaces for distributions of almost surely continuous Gaussian processes.
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11.
We study the operator-valued positive dyadic operator
$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$
where the coefficients {λ Q : CD} QD are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.
In the two-weight case, we prove that the L C p (σ) → L D q (ω) boundedness of the operator T λ( · σ) is characterized by the direct and the dual L testing conditions:
$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$
,
$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$
.
Here L C p (σ) and L D q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < pq < ∞, and locally finite Borel measures σ and ω.
In the unweighted case, we show that the L C p (μ) → L D p (μ) boundedness of the operator T λ( · μ) is equivalent to the end-point direct L testing condition:
$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$
.
This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.  相似文献   

12.
For αβ>−1, stable time periodic solutions A(X,T)=AqeiqX+qT are the locally preferred planform for the complex Ginzburg-Landau equation
  相似文献   

13.
Let −(·,z)D+q be a differential operator in L2(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form
  相似文献   

14.
Let S be the space of functions of regular variation and let ω = (ω1,..., ωn), ωjS. The weighted Besov space of holomorphic functions on polydisks, denoted by B p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U n such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm2n(z) is the 2ndimensional Lebesgue measure on U n and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B p (ω) and L p (ω) (the weighted L p -space).  相似文献   

15.
Given a subset S of Z and a sequence I = (In)n=1 of intervals of increasing length contained in Z, let
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16.
It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed1,…,dn on PN defined as the kernel of a general epimorphism
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17.
Let {Q n (α,β) (x)} n=0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product
$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$
where λ>0 and d μ α,β(x)=(x?a)(1?x)α?1(1+x)β?1 dx, d ν α,β(x)=(1?x) α (1+x) β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q n (α,β) are obtained.
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18.
19.
We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ 2 0 -complete), computable models with ω-categorical theories (Δ ω 0 -complex Π ω+2 0 -set), prime models (Δ ω 0 -complex Π ω+2 0 -set), models with ω 1-categorical theories (Δ ω 0 -complex Σ ω+1 0 -set. We obtain a universal lower bound for the model-theoretic properties preserved by Marker’s extensions (Δ ω 0 .  相似文献   

20.
We consider the random walk on Z+={0,1,…}, with up and down transition probabilities given the chain is in state x∈{1,2,…}:
(1)  相似文献   

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