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1.
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog ln (4/π), and on the other hand the infinite products of the first author for e, of the second author for π, and of Ser for e
γ
. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization
to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for
the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series.
We also use Ramanujan’s polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations
between certain dilogarithm values.
相似文献
2.
The fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ
d
: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I
γ
which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity,
decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular,
we extend the definition of the classical Riesz potential I
γ
to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I
γ
f is not p-integrable. We then introduce the new unique left-inverse I
γ, p
of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} with the property that I
γ, p
is dilation-invariant (but not translation-invariant) and that I
γ, p
f is p-integrable for any Schwartz function f. We finally apply that linear operator I
γ, p
with p = 1 to solve the stochastic partial differential equation
(-\triangle)g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w. 相似文献
3.
In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley,
C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting
f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional
case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically
convergent to simple roots except Newton’s or a family of Newton’s method.
相似文献
4.
Shiran Rachmilevitch 《International Journal of Game Theory》2011,40(1):63-85
We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an
axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution.
Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class
of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction
of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it
to this axiom. 相似文献
5.
The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values
of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for
γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main
tools are Euler-type integrals and hypergeometric series.
2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15
The work of the second author is supported by an Alexander von Humboldt research fellowship
Dedication: To Leonhard Euler on his 300th birthday. 相似文献
6.
A. N. Tikhomirov 《Siberian Advances in Mathematics》2009,19(4):277-286
The rate of convergence of the expected spectral distribution function of a sample covariancematrix to theMarchenko-Pastur
distribution is studied under the existence of the entries’ moments of order 2 + γ, where 0 < γ ≤ 2. 相似文献
7.
J. H. Wang 《Journal of Optimization Theory and Applications》2011,148(1):125-145
The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points
for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz
condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications
to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the
sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu,
Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005). 相似文献
8.
Yanjun Liu 《Algebras and Representation Theory》2011,14(2):213-215
Our main purpose of this paper is to give π-block forms of Brauer’s k(B) −conjecture and Olsson’s conjecture for finite π −separable groups. 相似文献
9.
Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations.
In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative.
Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s
method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order
2
n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are
competitive with other similar robust methods and very effective in high precision computations. 相似文献
10.
M. H. M. Rashid 《Ukrainian Mathematical Journal》2012,63(8):1256-1267
If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum
of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations
by finite-rank operators. 相似文献
11.
We investigate Newton’s method to find roots of polynomials of fixed degree d, appropriately normalized: we construct a finite set of points such that, for every root of every such polynomial, at least
one of these points will converge to this root under Newton’s map. The cardinality of such a set can be as small as 1.11 d log2
d; if all the roots of the polynomial are real, it can be 1.30 d.
Oblatum 24-II-2000 & 14-II-2001?Published online: 20 July 2001 相似文献
12.
We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer
k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,
this last inequality also holding if z ⩾ 2. We then use these inequalities to obtain probabilistic results and we state a conjecture. Finally, using (*), we show
that the probability that the abc conjecture does not hold is 0. 相似文献
13.
We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including
Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct
explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another
direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement
of Gal’s conjecture, which asserts that such γ-vectors are nonnegative. 相似文献
14.
We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer
k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,
this last inequality also holding if z ⩾ 2. We then use these inequalities to obtain probabilistic results and we state a conjecture. Finally, using (*), we show
that the probability that the abc conjecture does not hold is 0.
Research supported in part by a grant from NSERC.
Re?u le 17 décembre 2001; en forme révisée le 23 mars 2002
Publié en ligne le 11 octobre 2002 相似文献
15.
Nitis Mukhopadhyay 《Methodology and Computing in Applied Probability》2010,12(4):609-622
In this communication, we first compare z
α
and t
ν,α
, the upper 100α% points of a standard normal and a Student’s t
ν
distributions respectively. We begin with a proof of a well-known result, namely, for every fixed
0 < a < \frac120<\alpha <\frac{1}{2} and the degree of freedom ν, one has t
ν,α
> z
α
. Next, Theorem 3.1 provides a new and explicit expression b
ν
( > 1) such that for every fixed
0 < a < \frac120<\alpha < \frac{1}{2} and ν, we can conclude t
ν,α
> b
ν
z
α
. This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere.
A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s
t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the
oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated
well by tn,a/22za/2-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by tn,a/22za/2-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of tn,a/22za/2-2,t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}, namely bn2,b_{\nu }^{2}, or comes incredibly close to bn2b_{\nu }^{2} where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under
Stein’s fixed-width confidence interval method. 相似文献
16.
Let be a union-closed family of subsets of an m-element set A. Let . For b ∈ A let w(b) denote the number of sets in containing b minus the number of sets in not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first
author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2
m
− 2
m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2
m/2 with the upper integer part of 2
m
/3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and then the averaged Frankl’s property holds (i.e., 2
m/2 can be replaced with the lower integer part of 2
m
/3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader
is assumed to be familiar neither with lattices nor with combinatorics.
This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148. 相似文献
17.
We show that a manifold-stratified space X is the interior of a compact manifold-stratified space with boundary if and only if X is tame-ended and a K-theoretic obstruction γ*(X) vanishes. The obstruction γ*(X) is a localization of Quinn's mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7
below. In particular, this explains when a G-manifold is the interior of a compact G-manifold with boundary. One of our methods is a new transversality theorem, Theorem 1.16.
Oblatum 30-VI-1996 & 21-X-1997 / Published online: 14 January 1999 相似文献
18.
Jürgen Pöschel 《Mathematische Annalen》2011,349(2):433-458
We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential q of Hill’s equation, −y′′ + qy = λy, to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths, γ
n
, belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also
recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem. 相似文献
19.
In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of
subsets {A
t
| t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f
t
of G under which #{t∈ℕ | f
t
x∈A
t
} is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical
Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.
Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999 相似文献
20.
Wojciech Połowczuk Piotr Więcek Tadeusz Radzik 《Mathematical Methods of Operations Research》2007,65(1):141-152
This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity
properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria
consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure
of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart
of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous
concave (convex) games with compact convex spaces of players’ pure strategies. 相似文献