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1.
With the notation
, we prove the following result. Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies||p||L2(K)An1/2 and ||p′||L2(K)Bn3/2. ThenwithWe also prove that and for every
, where
denotes the collection of all trigonometric polynomials of the form 相似文献
2.
Let F be any field. Let p( F) be the characteristic of F if F is not of characteristic zero, and let p( F)=+∞ otherwise. Let A1,…, An be finite nonempty subsets of F, and let with k{1,2,3,…}, a1,…, anF{0} and deg g< k. We show that When kn and | Ai| i for i=1,…, n, we also have consequently, if nk then for any finite subset A of F we have In the case n> k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction. 相似文献
3.
Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation where B, C and α, β, γ are positive, k {1, 2, 3, … }, and the initial conditions x−2k+1, … , x−1, x0 are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples. 相似文献
4.
In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian (φp(u′))′+f(t,u)=0, t(0,1), 相似文献
5.
Let
be the radial subspace of the Besov space
. We prove the independence of the asymptotic behavior of the entropy numbers from the difference s0− s1 as long as the embedding itself
is compact. In fact, we shall show that This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces Bp,qs(Ω) on bounded domains Ω. 相似文献
6.
We consider a system of heat equations ut= Δu and vt= Δv in Ω×(0, T) completely coupled by nonlinear boundary conditions We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with for p, q>0, 0≤ α<1 and 0≤ β< p. 相似文献
7.
Let B denote the unit ball of . For 0< p<∞, the holomorphic function spaces Qp and Qp,0 on the unit ball of are defined as and In this paper, we give some derivative-free, mixture and oscillation characterizations for Qp and Qp,0 spaces in the unit ball of . 相似文献
8.
Let
be a probability space and let Pn be the empirical measure based on i.i.d. sample ( X1,…, Xn) from P. Let
be a class of measurable real valued functions on
For
define Ff( t):= P{ ft} and Fn,f( t):= Pn{ ft}. Given γ(0,1], define n,γ(δ):=1/( n1−γ/2δ γ). We show that if the L2( Pn)-entropy of the class
grows as −α for some α(0,2), then, for all
and all δ(0,Δ n), Δ n=O( n1/2), and where
and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define Then for all
uniformly in
and with probability 1 (for
the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory. 相似文献
9.
Let, for example, where α>0, k1, and exp k=exp(exp(…exp())) denotes the kth iterated exponential. Let { An} denote the recurrence coefficients in the recurrence relation xpn(x)=Anpn+1(x)+An-1pn-1(x) | for the orthonormal polynomials {pn} associated with W2. We prove that as n→∞,where logk=log(log(…log())) denotes the kth iterated logarithm. This illustrates the relationship between the rate of convergence to of the recurrence coefficients, and the rate of decay of the exponential weight at ±1. More general non-even exponential weights on a non-symmetric interval (a,b) are also considered. 相似文献
10.
Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.
相似文献
11.
We prove that every positive solution to the difference equation
where
,
i=1,…,
k, converges to the following quantity
, confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all
αi,
i=1,…,
k belong to the interval (0, 1).
相似文献
12.
Let dλ(
t) be a given nonnegative measure on the real line
, with compact or infinite support, for which all moments
exist and are finite, and μ
0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes
where σ=σ
n=(
s1,
s2,…,
sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness
dmax=2∑
ν=1nsν+2
n−1 if and only if
The proof of the uniqueness of the extremal nodes τ
1,τ
2,…,τ
n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term
R(
f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ
ν, ν=1,2,…,
n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.
相似文献
13.
For any
, let
Pk denote the natural projections on
ℓ1. Let |||||| be an equivalent norm of
ℓ1 that satisfies all of the following four conditions:
- (1) There are α>4 and a positive (decreasing) sequence (αn) in (0,1) such that for any normalized block basis {fn} of (ℓ1,||||||) and xℓ1 with Pk−1(x)=x and |||x|||<αk,
- (2) There are two strictly decreasing sequences {βk} and {γk} with such that for any normalized block basis {fn} of (ℓ1,||||||) and x with (I−Pk)(x)=x,
- (3) For any , I−Pk=1.
- (4) The unit ball of (ℓ1,||||||) is σ(ℓ1,c0)-closed.
In this article, we prove that the space (
ℓ1,||||||) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author.
Keywords: Renorming; Fixed point property
相似文献
14.
Let
TR be a time-scale, with
a=inf
T,
b=sup
T. We consider the nonlinear boundary value problem
where
λR+:=[0,∞), and satisfies the conditions
We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (
λ,
u) of (1)–(2),
u is positive on
T a,
b . In addition, we show that there exists
λmax>0 (possibly
λmax=∞), such that, if 0
λ<
λmax then (1)–(2) has a unique solution
u(
λ), while if
λλmax then (1)–(2) has no solution. The value of
λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).
相似文献
15.
Let
be the complex polynomial ring in
d variables. A contractive
-module is Hilbert space
equipped with an
action such that for any
,
||z1ξ1+z2ξ++zdξd||2||ξ1||2+||ξ2||2++||ξd||2.
Such objects have been shown to be useful for modeling
d-tuples of mutually commuting operators acting on a Hilbert space. There is a subclass of the category of contractive
modules whose members play the role of free objects. Given a contractive
-module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form:
where
is a free module for each
i0. The notion of a localization of a free resolution will be defined, in which for each λ
Bd there is a vector space complex of linear maps derived from (*):
We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the
d-tuple (
1,
2,…,
d), of where
i is the
ith coordinate function of a Möbius transform on
Bd such that (λ)=0.
相似文献
16.
For all integers
m3 and all natural numbers
a1,
a2,…,
am−1, let
n=
R(
a1,
a2,…,
am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,
n} there exists a monochromatic solution to
Let
t=min{
a1,
a2,…,
am−1} and
b=
a1+
a2++
am−1−
t. In this paper it is shown that whenever
t=2,
R(a1,a2,…,am−1)=2b2+9b+8.
It is also shown that for all values of
t,
R(a1,a2,…,am−1)tb2+(2t2+1)b+t3.
相似文献
17.
We present a new approach to the variational relaxation of functionals
of the type:
where
is a continuous function with growth conditions of order
p≥1 but not necessarily convex. We essentially study the case when μ is the
k-dimensional Hausdorff measure restricted to a suitable piece of a
k-dimensional smooth submanifold of
.
相似文献
18.
In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞< n < ∞}is a sequence of independent, identically distributed random variables with zero mean, μ>0 is a constant and the coefficients {ψi;-∞< i <∞} satisfy 0 <∞∑j=-∞|jψj| <∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) > x}is discussed. Then the result is applied to ultimate ruin probability.
相似文献
19.
By means of the abstract continuation theory for
k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation:
相似文献
20.
We prove the Marcinkiewicz–Zygmund Strong Law of Large Numbers for
U-statistics of strictly stationary, absolutely regular observations (
ξi)
i≥1. Under suitable moment conditions and conditions on the mixing rate, we show that
for some
γ≥0, in the non-degenerate case, and
in the degenerate case.
相似文献