共查询到20条相似文献,搜索用时 15 毫秒
1.
R. Nair 《Israel Journal of Mathematics》2009,171(1):197-219
We consider a system of “generalised linear forms” defined at a point x = (x
(i, j)) in a subset of R
d
by
for k ≥ 1. Here d = d
1 + ⋯ + d
l
and for each pair of integers (i, j) ∈ D, where D = {(i, j): 1 ≤ i ≤ l, 1 ≤ j ≤ d
i
} the sequence of functions (g
(i, j), k
(x))
k=1∞ are differentiable on an interval X
ij
contained in R. We study the distribution of the sequence on the l-torus defined by the fractional parts X
k
(x) = ({ L
1(x)(k)}, ..., {L
l
(x)(k)}) ∈ T
l
, for typical x in the Cartesian product . More precisely, let R = I
1 × ⋯ × I
l
be a rectangle in T
l
and for each N ≥ 1 define a pair correlation function
and a discrepancy , where the supremum is over all rectangles in T
l
and χ
R
is the characteristic function of the set R. We give conditions on (g
(i, j), k
(x))
k=1∞ to ensure that given ε > 0, for almost every x ∈ T
l
we have Δ
N
(x) = o(N(log N)
l+∈). Under related conditions on(g
(i, j), k
(x))
k =1∞ we calculate for appropriate β ∈ (0, 1) the Hausdorff dimension of the set {x : lim sup
N→∞
N
β Δ
N
(x > 0)}. Our results complement those of Rudnick and Sarnak and Berkes, Philipp, and Tichy in one dimension and M. Pollicott
and the author in higher dimensions. 相似文献
2.
ForX a set the expression Prt(X) denotes the composition monoid of all functionsf X ×X. Fork a positive integer the letterk denotes also the set of all nonnegative integers less thank. Whenk > 1 the expression rk denotes the connected injective element {<i, i + 1>i k – 1} in Prt (k). We show for every word w=w(x,y) in a two-letter alphabet that if the equation w(x, y)=rk has a solution =y) 2Prt(k) then ¯w(x,y)=rk also has a solution in2Prt(k), where ¯w is the word obtained by spelling the wordw backwards. It is a consequence of this theorem that if for every finite setX and for everyf Prt(X) the equation w(x,y)=f has a solution in2Prt(X) then for every suchX andf the equation ¯w(x, y)=f has a solution in2Prt(X).Presented by J. Mycielski. 相似文献
3.
U. Luther 《Integral Equations and Operator Theory》2006,54(4):541-554
We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N1, t ∈N2, or x=t, where N1,N2 are sets of measure zero. It is shown that such operators map weighted L∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel
k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors
of best weighted uniform approximation by algebraic polynomials. 相似文献
4.
Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL
λ(x)
x∈R, satisfying the interpolatory conditionsL
k = δ0k,k∈Z
. One objective of this paper is to derive several additional properties ofL
λ. For example, it is shown thatL
λ possesses the signregularity property sgn[L
λ(x)]=sgn[sin(πx)/(πx)],x∈R, and that |L
λ (x)|≤2e
8 min {(⌊|x|⌋+1)-1,exp(-λ⌊|x|⌋)},x∈R. The analysis is based on a simple representation formula forL
λ and employs some methods from classical function theory. A second consideration in the paper is the Gaussian cardinal-interpolation
operatorL
λ, defined by the equation (L
λy)(x):=
,x∈R, y=(yk)k∈Z. On account of the exponential decay of the cardinal functionL
λ,L
λ is a well-defined linear map froml
∞ (Z) intoL
∞ (R). Its associated operatornorm ‖L
λ‖ is called the Lebesgue constant ofL
λ. The latter half of the paper establishes the following estimates for the Lebesgue constant: ‖L
λ‖≍1, λ→∞, and ║Lλ║≍log(1/λ), λ→0+. Suitable multidimensional analogues of these results are also given.
For Carl de Boor, on the occasion of his sixtieth birthday 相似文献
5.
Bounds for the 3G-expression∫
Ω
G(x,z)G(z,y)d,z/G(x,y) play a fundamental role in potential theory. Here,G(x,y) is the Green function for the Laplace problem with zero dirichlet boundary conditions on Ω. The 3G-formula equals
, the expected lifetime for a Brownian motion starting in
that is killed on exiting ω and conditioned to converge to and to be stopped at
. Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx ε δΩ, then the supremum ofy \at E
x
y
is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed E
x
y
(\gt\om) is maximized on
by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the
positivity-preserving property of some elliptic systems. In particular, it confirms a, relationE
x
y
(\gt\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers. 相似文献
6.
Abstract
we prove that the operator
maps
into itself for
where
and k(x,y)=ϕ(x,y) eig(x,y), ϕ(x,y) satisfies (5), (e.g. ϕ(x,y)=|x–y|iτ,τ real) and the phase g(x,y)=xa⋅ yb +Φ**(xa,yb). We obtain Lp estimates for operators with more general phases than in [5] and for these operators we require that b1 b2>1, and
and al≥ bl≥ 1, which remained open from [4].
Keywords Oscillatory integrals, Lp mappings
Mathematics Subject Classification (2000) Primary 42B20, Secondary 46B70, 47G10 相似文献
7.
Yusup Kh. Eshkabilov 《Central European Journal of Mathematics》2008,6(1):149-157
Let Ω= [a, b] × [c, d] and T
1, T
2 be partial integral operators in (Ω): (T
1
f)(x, y) =
k
1(x, s, y)f(s, y)ds, (T
2
f)(x, y) =
k
2(x, ts, y)f(t, y)dt where k
1 and k
2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT
1, τ ∈ ℂ and E−τT
2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T
1, T
2, and T = T
1 + T
2 are proved.
相似文献
8.
In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝ
n+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation
is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of the new geometrical
characteristics of the domain, the upper estimate of L
2-norm ∥u(t)∥ of the solution to the problem is established. In particular, in domains {(x, y) ∈ ℝ
n+1 | x > 0, |y
1| < x
a
}, 0 < a < q/l, under the assumption that the upper and lower symbols of the operator L are separated from zero, this estimate takes the form
. This estimate is determined by minor terms of the equation. The sharpness of the estimate for a wide class of unbounded
domains is proved in the case k = l = m = 1.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 113–132, 2006. 相似文献
9.
V. V. Karachik 《Siberian Advances in Mathematics》2008,18(2):103-117
Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ
n
. We study the representation of this function in the form of a series u(x) = u
0(x) + |x|2
u
1(x) + |x|4
u
2(x) + …, where u
k
(x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula.
Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162. 相似文献
10.
Reinhard Wolf 《Arkiv f?r Matematik》1997,35(2):387-400
One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR
n
andr(X, d
2) the rendezvous number ofX, whered
2 denotes the Euclidean distance inR
n
. (The rendezvous numberr(X, d
2) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x
1,x
2,...,x
n
inX, there exists somex inX such that
.) Then there exists some regular Borel probability measure μ0 onX such that the value of ∫
X
d
2(x, y)dμ0 (y) is independent of the choicex inX, if and only ifr(X, d
2) = supμ ∫
X
∫
X
d
2(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX. 相似文献
11.
Entropy bounds for perfect matchings and Hamiltonian cycles 总被引:1,自引:1,他引:0
For a graph G = (V,E) and x: E → ℜ+ satisfying Σ
e∋υ
x
e
= 1 for each υ ∈ V, set h(x) = Σ
e
x
e
log(1/x
e
) (with log = log2). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and x
e
=Pr(e∈f),
(where H is binary entropy). This implies a similar bound for random Hamiltonian cycles.
Specializing these bounds completes a proof, begun in [6], of a quite precise determination of the numbers of perfect matchings
and Hamiltonian cycles in Dirac graphs (graphs with minimum degree at least n/2) in terms of h(G):=maxΣ
e
x
e
log(1/x
e
) (the maximum over x as above). For instance, for the number, Ψ(G), of Hamiltonian cycles in such a G, we have
.
Supported by NSF grant DMS0200856. 相似文献
12.
An algorithm of searching a zero of an unknown function ϕ: ℝ → ℝ is considered: x
t
= x
t−1 − γ
t−1
y
t
, t = 1, 2, ..., where y
t
= ϕ(x
t−1) + ξ
t
is the value of ϕ measured at x
t−1 and ξ
t
is the measurement error. The step sizes γ
t
> 0 are modified in the course of the algorithm according to the rule: γ
t
= min{uγ
t−1, } if y
t−1
y
t
> 0, and γ
t
= dγ
t−1, otherwise, where 0 < d < 1 < u, > 0. That is, at each iteration γ
t
is multiplied either by u or by d, provided that the resulting value does not exceed the predetermined value . The function ϕ may have one or several zeros; the random values ξ
t
are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on ϕ, ξ
t
, and , the conditions on u and d guaranteeing a.s. convergence of the sequence {x
t
}, as well as a.s. divergence, are determined. In particular, if P(ξ
1 > 0) = P (ξ
1 < 0) = 1/2 and P(ξ
1 = x) = 0 for any x ∈ ℝ, one has convergence for ud < 1 and divergence for ud > 1. Due to the multiplicative updating rule for γ
t
, the sequence {x
t
} converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with,
but instead, approximate one of the zeros of ϕ. By adjusting the parameters u and d, one can reach arbitrarily high precision of the approximation; higher accuracy is obtained at the expense of lower convergence
rate.
相似文献
13.
Kyung Joong Kim Ronald Cools L. Gr. Ixaru 《Journal of Computational and Applied Mathematics》2002,140(1-2)
We consider the integral of a function
and its approximation by a quadrature rule of the formi.e., by a rule which uses the values of both y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form y(x)=f1(x) sin(ωx)+f2(x) cos(ωx) with smoothly varying f1 and f2. In the latter case, the weights wk and αk are ω dependent. We establish some general properties of the weights and present some numerical illustrations. 相似文献
14.
We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.
相似文献
15.
It is possible to formulate the polynomial Szemerédi theorem as follows: Let q
i
(x) ∈ Q[x] with q
i
(Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density, then there are a, n ∈ N such that
(n) - q_1 (0),...,a + q_k (n) - q_k (0) E. |
#xA;
\{ a,a + q_1 (n) - q_1 (0),...,a + q_k (n) - q_k (0)\} \subset E.
相似文献
16.
P. C. Allaart 《Acta Mathematica Hungarica》2008,121(3):243-275
This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable
function. Let F(x):=∑
n=1∞
ε
n
ϕ(2
n−1
x), x ∈ R, where ɛ
1, ɛ
2, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ
1 = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ
n
}, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa
n − 1
ɛ
n
ϕ(2
n − 1
x), where 0 < a < 1.
相似文献
17.
Wenguang Zhai 《中国科学A辑(英文版)》1999,42(11):1173-1183
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). 相似文献
18.
Z. Ditzian 《Israel Journal of Mathematics》1985,52(4):341-354
Equivalences between the condition |P
n
(k)
(x)|≦K(n
−1√1−x
2+1/n
2)
k
n
-a, whereP
n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x
2)
k
f
(2k)(x)∈C[−1,1] is established. A similar result is shown forE
n(f)=
‖f−P
n‖
C[−1,1]. Rates other thann
-a are also discussed.
Supported by NSERC grant A4816 of Canada. 相似文献
19.
We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic
0. We prove that a variety V has polynomial growth if and only if the condition
20.
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