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1.
Let n,p and k be three non negative integers. We prove that the apparently rational fractions of q:
are actually polynomials of q with non negative integer coefficients. This generalizes a recent result of Lassalle (Ann. Comb. 6(3–4), 399–405, 2002),
in the same way as the classical q-binomial coefficients refine the ordinary binomial coefficients.
相似文献
2.
In this paper we establish results on the existence of nontangential limits for weighted
-harmonic functions in the weighted Sobolev space
, for some q>1 and w in the Muckenhoupt A
q
class, where
is the unit ball in
. These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted
-harmonic functions are weak solutions of the partial differential equation
where
for some fixed q∈(1,∞), where 0<α≤β<∞, and w(x) is a q-admissible weight as in Chap. 1 of Heinonen et al., Nonlinear Potential Theory, 2006.
Later, we apply these results to improve on results of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996 and Martio and Srebro, Math. Scand. 85, 49–70, 1999 on the existence of radial limits for bounded quasiregular mappings in the unit ball of
with some growth restriction on their multiplicity function.
相似文献
3.
4.
We prove the following statement.
Let , and let . Suppose that, for all and , the sequence satisfies the relation
where e(u) : = e2πiu
.
Then
where
q is the set of q-multiplicative functions g such that . 相似文献
5.
Glyn Harman 《Archiv der Mathematik》2008,90(3):239-245
Let P
+(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that
for . Here improves on the previous exponent .
Received: 20 April 2007 相似文献
6.
Irrationality measures are given for the values of the series
, where
and Wn is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality
of all the numbers
where fn and ln are the Fibonacci and Lucas numbers, respectively.
2000 Mathematics Subject Classification Primary—11J82, 11B39 相似文献
7.
J.-P. Allouche 《The Ramanujan Journal》2007,14(1):39-42
We answer a question of Berndt and Bowman, asking whether it is possible to deduce the value of the Ramanujan integral I from the value of the Ramanujan integral J, where
and
We also show that the second integral can be deduced from a classical expression of the ψ function due to Dirichlet and from
the classical equality
which is a simple consequence of Frullani-Cauchy’s theorem.
2000 Mathematics Subject ClassificationPrimary—33B15
Partially supported by MENESR, ACI NIM 154 Numération. 相似文献
8.
Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups
Wen-Bin Zhang 《The Ramanujan Journal》2008,15(1):47-75
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic
semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let
be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in
with “degree” ∂(a)=m satisfies
with constants q>1, ρ
1<ρ
2<⋅⋅⋅<ρ
r
=ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ
ρ≥1, and τ
α
ρ≥1. Then for a multiplicative function f(a) on
the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ
ρ mean-value
exists with m
f
≠0 and the limit
exists with M
v
(α)>0.
相似文献
9.
Choonkil Park 《Acta Appl Math》2008,102(1):71-85
This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C
*-algebras:
The concept of Hyers–Ulam–Rassias stability originated from the Th.M. Rassias’ stability theorem (Rassias in Proc. Am. Math.
Soc. 72:297–300, [1978]).
This work was supported by the research fund of Hanyang University (HY-2007-S). 相似文献
10.
In this work the Dirichlet series
associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case
i=0
q–1
f(i) = 0. It is known that in this case the function
f
(s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function
f
(t) satisfying the integral equation with delayed argument
for some nonzero real
f
naturally appears in the representation of the function
f
(s). In this article we find some asymptotic properties of the function
f
(s), prove that
f
(s) is an entire function of order 2, and also prove that in the region
the function
f
(s) has only trivial zeros which are simple. 相似文献
11.
László Losonczi 《Aequationes Mathematicae》2007,74(3):262-281
Summary. Let be given continuous functions on the interval I such that g ≠ 0, and is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant function μ on [0, 1]
is a mean value of . Here we solve the homogeneity equation
for two important special cases of symmetric means of this type: for the quasi-arithmetic means weighted by a weight function
and for the Cauchy means. We assume that is open, , and f, g satisfy strong differentiability conditions.
Research supported by OTKA grants T 043080, T 047373. 相似文献
12.
Mohammad Sal Moslehian 《Bulletin of the Brazilian Mathematical Society》2007,38(4):611-622
In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation
相似文献
13.
We consider autonomous integrals
in the multidimensional calculus of variations, where the integrand f is a strictly W
1,p
-quasiconvex C
2-function satisfying the (p,q)-growth conditions
with exponents 1 < p ≤ q < ∞. Under these assumptions we establish an existence result for minimizers of F in provided . We prove a corresponding partial C
1,α
-regularity theorem for . This is the first regularity result for autonomous quasiconvex integrals with (p,q)-growth. 相似文献
14.
Wang Lei Pan Ting Dept. of Math. Zhejiang Univ. Hangzhou China. Univ. of International Relation Hangzhou China. 《高校应用数学学报(英文版)》2004,19(2):212-222
Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(… 相似文献
15.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈R+ /Q, h(t) ∈L∞ [0, 2π ] is 2π-periodic, x±=max {±x, 0 }.
Received: 23 September 2004 相似文献
16.
Using measure-capacity inequalities we study new functional inequalities, namely L
q
-Poincaré inequalities
and L
q
-logarithmic Sobolev inequalities
for any q ∈ (0, 1]. As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media
equation
for m ≥ 1, in terms of L
2-norms and entropies.
相似文献
17.
Walter Allegretto Yanping Lin Zhiyong Zhang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,22(5):848-868
We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations,
. for initial data with different end states,
which displays the complexity in between ellipticity and dissipation. Due to smoothing effect of the parabolic operator,
we detail the regularity property and estimates when t > 0 for the higher order spatial derivatives despite its relatively
lower regularity of the initial data. Also we discuss the decay estimates without the restriction of L
1 bound as in Tang and Zhao [17], Wang [20]. Related to recent work by [15], our derivation may also establish the same estimates
directly if under the same condition.
Work supported by NSERC (Canada). 相似文献
| (1) |
| (2) |
18.
Cristinel Mortici 《Archiv der Mathematik》2009,93(1):37-45
We prove in this paper that for every x ≥ 0,
where and α = 1.072042464..., then
where and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger
than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.
相似文献
19.
Iosif Pinelis 《Probability Theory and Related Fields》2007,139(3-4):605-635
Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter
. Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in .
A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales
with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications
to generalized self-normalized sums and t-statistics are given.
相似文献
20.
I. Kátai 《Lithuanian Mathematical Journal》2007,47(1):24-31
We characterize the q-multiplicative functions which belong to
or
= the set of uniformly summable functions on the set of primes.
Dedicated to Professor Jonas Kubilius on his 85th anniversary
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 29–38, January–March, 2007. 相似文献