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1.
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
2.
In this paper we introduce an arithmetical function (n), the difference between the number of divisors of n congruent to 1 mod 3 and those congruent to –1 mod 3. This function then is related to the classical function (n) which is the sum of the divisors of n. In particular we prove the identity
相似文献
3.
Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) =
. Ramanujan has given a number of Lambert series expansions such as
A formula is proved which includes this and other expansions as special cases. 相似文献
4.
An examination of the sizable literature on Wythoff pairs, their generalizations, and Beatty sequences shows a considerable emphasis on connections with second order recurrence relations and quadratic irrationalities. However, it does not seem to have been previously noticed that a subsequence of the classical Wythoff pairs satisfies the irreducible fourth order linear recurrence
Thus, while Wythoff pairs are ordinarily associated with the roots of x2 – x – 1 = 0, there is a subset thereof that is associated with the roots of
相似文献
5.
Let K, D be n-dimensional convex bodes. Define the distance between K and D as
where the infimum is taken over all
and all invertible linear operators T. Assume that 0 is an interior point of K and define
where is the uniform measure on the sphere. We use the difference body estimate to prove that K can be embedded into
so that
for some absolute constants C and
. We apply this result to show that the distance between two n-dimensional convex bodies does not exceed
up to a logarithmic factor. 相似文献
6.
M. Skałba 《Archiv der Mathematik》2005,84(6):485-495
Let S(x) denote the number of primes p < x which divide both 2n–3 and 3n–2 for some
We prove that
Received: 8 April 2004 相似文献
7.
Let q 2 be an integer. Then –q gives rise to a number system in
, i.e., each number n
has a unique representation of the form n = c
0 + c
1 (–q) + ... + c
h
(–q)
h
, with c
i
{0,..., q – 1}(0 i h). The aim of this paper is to investigate the sum of digits function –q
(n) of these number systems. In particular, we derive an asymptotic expansion for
and obtain a Gaussian asymptotic distribution result for –q
(n) – –q
(–n). Furthermore, we prove non-differentiability of certain continuous functions occurring in this context. We use automata and analytic methods to derive our results. 相似文献
8.
Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution
of the Hamilton-Jacobi equation
is an action function in the large, i.e.,
for all
Received: 13 June 2003 相似文献
9.
In this paper, we refine work of Beukers, applying results from the theory of Padé approximation to (1 – z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to
(where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have
provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x
2 + D = p
n where D is a nonzero integer and p is an odd rational prime, coprime to D. 相似文献
10.
Using the integral operator that defines a solution of the Cauchy problem for the equation
we find an integral representation for solutions of the equation
in terms of arbitrary functions that are continuously differentiable sufficiently many times. 相似文献
11.
12.
Let n1 and let p be a prime. Expand j[0,p
n
–1]\(p) p-adically as j=
s0
a
s
p
s
with a
s
[0,p–1]. The #([0,j]\(p))th Z
(p)[
p
n
]-linear elementary divisor of the cyclotomic Dedekind embedding
has valuation
at 1–
p
n
. There is a similar result for the related cyclic Wedderburn embedding. 相似文献
13.
Moharram A. Khan 《Czechoslovak Mathematical Journal》2000,50(4):791-801
In this paper we investigate commutativity of rings with unity satisfying any one of the properties:
for some f(X) in
and g(X), h(X) in
where m 0, r 0, s 0, n > 0, t > 0 are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements x and y for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently. 相似文献
14.
Kolitsch and Sellers showed recently that a8(n), the number of 8-core partitions of n, is even when n belongs to certain arithmetic progressions. We prove a similar result for 16-cores. In doing so, we prove the surprising result that the a16(n), given by
satisfy
相似文献
15.
Xia Chen 《Journal of Theoretical Probability》1999,12(2):421-445
Let {X
n
}
n0 be a Harris recurrent Markov chain with state space E, transition probability P(x, A) and invariant measure , and let f be a real measurable function on E. We prove that with probability one,
under some best possible conditions. 相似文献
16.
The value distribution of a normalised sequence of strongly additive arithmetic functions
is approximated by a nearly standard normal law. The remainder is expressed in terms of third and fourth absolute moments and contains the multiplier (1 + |x|)-3. 相似文献
17.
Let u(x) xR
q
be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p
x, (·) be the density of an R
q
valued canonical normal random variable with mean x and variance and let {G
x, ; (x, )R
q
×[0,1 ]} be the mean zero Gaussian process with covariance
A finite positive measure on R
q
is said to be in
with respect to u, if
When
, a multiple Wick product chaos
is defined to be the limit in L
2, as 0, of
where
,
denotes the Wick product of the m
j
normal random variables
.Consider also the associated decoupled chaos processes
,
defined as the limit in L
2, as 0, of
where
are independent copies of G
x,.Define
Note that a neighborhood of the diagonals of
in
is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set
One of the main results of this paper is:
Theorem A. If
is continuous on (R
q
)
r
for all
then
is continuous on
.When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of
on (R
q
)
r
. Also several variants of (i) are considered and related to different types of decoupled processes. These results have applications in the study of intersections of Lévy process and continuous additive functionals of several Lévy processes. 相似文献
18.
Let
We show that for every function
satisfying the conditional equation
either there exists a solution
of the Goab-Schinzel equation
such that
(i.e., f(x) = g(x) for
) or there is x0 > 0 with f(x0) < –1 and f(x) = 0 for x x0 . In particular we determine the solutions
of the conditional equation that are continuous at a point, Lebesgue measurable or Baire measurable (i.e., have the Baire property). In this way we solve some problems raised by the first author.Received: 2 March 2004 相似文献
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19.
Margit Lénárd 《Acta Mathematica Hungarica》1999,84(4):317-327
Let the set of knots
(n 1) be given on the interval [-1, 1]. Find a polynomial Qm(x) of minimal degree satisfying (0, 2)-interpolational conditions at the inner knots and boundary conditions at the endpoints, that is
and
where yi
(s),O
(j), n+1
(j) are arbitrarily given real numbers, and k, l are arbitrary fixed non-negative integers. In this paper the existence and uniqueness of the polynomial Qm(x) is proved if the inner nodal points are the zeros of Jacobi polynomials Pn
2k + 1, 2l – 1 (x) or Pn
2k – 1, 2l + 1 (x). Explicit formulae for the fundamental polynomials of interpolation are also given. 相似文献
20.
Shaun Cooper 《The Ramanujan Journal》2002,6(4):469-490
Let r
k(n) denote the number of representations of an integer n as a sum of k squares. We prove that
where
Here n = 2 p
p
p
is the prime factorisation of n, n is the square-free part of n, the products are taken over the odd primes p, and (
) is the Legendre symbol.Some similar formulas for r
7(n) and r
9(n) are also proved. 相似文献