L\"{u}roth展式中数字和的快速增长速度[英文]

我们研究L\"{u}roth展式中数字和的快速增长速度,并证明相关水平集的Hausdorff维数是满维的.     

Lüroth展式相邻字符乘积的部分和的度量性质和相关分形维数

本文研究了Lüroth展式字符乘积的部分和Sn(x)=∑di(x)di+1(x)的度量性质和相关分形集的Hausdorff维数,其中di(x)表示x∈(0,1)的Lüroth展式的第i个字符.利用对部分和序列的修正和适当分形集的构造,获得了Sn(x)/n log2n依勒贝格测度收敛于1/2并且得到了相关例外集的Haus...     

Lüroth误差和函数的若干性质

本文研究了Lüroth展式的误差和函数.利用误差和函数的Perron-Frobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

形式级数域上Engel级数展式中“数字”次数的增长速度

WU在2003年研究了形式级数域上Engel级数展式中"数字"的次数以线性速度增长的例外集,并利用质量分布原理证明了该例外集具有满Hausdorff维数.本文我们主要研究形式级数域上Engel级数展式中"数字"的次数以多项式和指数速度增长的例外集,并给出他们的Hausdorff维数估计.     

Engel连分数中一个例外集的Hausdorff维数

本文研究了Engel连分数展式中部分商以某种速度增长的集合的Hausdorff维数.利用自然覆盖和质量分布原理,得到了集合B(α)={x∈(0,1):lim n→∞ log bn+1(x)/log bn(x)=α}的Hausdorff维数是1/α的结果.     

Lǖroth误差和函数的若干性质

本文研究了L櫣roth展式的误差和函数.利用误差和函数的PerronFrobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

Lüroth误差和函数的若干性质

本文研究了Lüroth展式的误差和函数.利用误差和函数的Perron-Frobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

Engel连分数展式与Huasdorff维数

本文研究了Engel连分数中部分商以某种速度增长的集合,以及Engel连分数展式收敛速度较快的点组成的集合,利用质量分布原理,证明了这些集合的Haus-dorff维数为1.     

由Cantor展式定义的Besicovitch-Eggleston子集

本文研究了由Cantor展式所确定的一类Besicovitch-Eggleston子集.应用Billingsley定理,得到了这类集合的维数.并且表明无穷符号空间和有限符号空间上的Besicovitch-Eggleston子集的性质是有区别的.     

形式级数域中具有某种连分数展式集合的Hausdorff维数

本文研究了形式级数域中若干连分数例外集.利用质量分布原理和构造特殊覆盖,得到了当连分数展式部分商的度分别以多项式速度和指数速度趋向无穷大时,分别对应例外集的Hausdorff维数.     

The Euler-Maclaurin Formula in Presence of a Logarithmic Singularity

In this short note we prove an extension of the Euler-Maclaurin expansion for general rectangular composite quadrature rules in one dimension when the derivative of the integrand has a logarithmic singularity. We show that a correction series has to be added to the formula, but that the asymptotic expansion in powers of the discretization parameter still holds.     

Hausdorff dimension of some kind of alternating Oppenheim series expansion

For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.     

The local structure of characters

Let G be a connected real semisimple Lie group with Lie algebra g. Let g = t? + s be the Cartan decomposition and K the maximal compact subgroup with Lie algebra t?. Let Θ be the character of an irreducible representation. Then Θ has an asymptotic expansion at zero (in the sense of Taylor series). As consequences of this expansion we obtain results about the asymptotic directions in which the K-types occur and about the Gelfand-Kirillov dimension of the representation.     

ON THE ERROR-SUM FUNCTION OF ALTERNATING LüROTH SERIES

The error-sum function of alternating Luroth series is introduced, which, to some extent, discerns the superior or not of an expansion comparing to other expansions. Some elementary properties of this function are studied. Also, the Hausdorff dimension of graph of such function is determined.     

An approach for identifying and predicting economic recessions in real‐time using time–frequency functional models

Identifying periods of recession and expansion is a challenging topic of ongoing interest with important economic and monetary policy implications. Given the current state of the global economy, significant attention has recently been devoted to identifying and forecasting economic recessions. Consequently, we introduce a novel class of Bayesian hierarchical probit models that take advantage of dimension‐reduced time–frequency representations of various market indices. The approach we propose can be viewed as a Bayesian mixed frequency data regression model, as it relates high‐frequency daily data observed over several quarters to a binary quarterly response indicating recession or expansion. More specifically, our model directly incorporates time–frequency representations of the entire high‐dimensional non‐stationary time series of daily log returns, over several quarters, as a regressor in a predictive model, while quantifying various sources of uncertainty. The necessary dimension reduction is achieved by treating the time–frequency representation (spectrogram) as an “image” and finding its empirical orthogonal functions. Subsequently, further dimension reduction is accomplished through the use of stochastic search variable selection. Overall, our dimension reduction approach provides an extremely powerful tool for feature extraction, yielding an interpretable image of features that predict recessions. The effectiveness of our model is demonstrated through out‐of‐sample identification (nowcasting) and multistep‐ahead prediction (forecasting) of economic recessions. In fact, our results provide greater than 85% and 80% out‐of‐sample forecasting accuracy for recessions and expansions respectively, even three quarters ahead. Finally, we illustrate the utility and added value of including time–frequency information from the NASDAQ index when identifying and predicting recessions. Copyright © 2012 John Wiley & Sons, Ltd.     

ON THE ERROR-SUM FUNCTION OF ALTERNATING L(U)ROTH SERIES

The error-sum function of alternating Lüroth series is introduced, which, to some extent, discerns the superior or not of an expansion comparing to other expansions. Some elementary properties of this function are studied. Also, the Hausdorff dimension of graph of such function is determined.     

ON THE ERROR-SUM FUNCTION OF ALTERNATING LǖIROTH SERIES

The error-sum function of alternating Lǖroth series is introduced, which, to some extent, discerns the superior or not of an expansion comparing to other expansions. Some elementary properties of this function are studied. Also, the Hausdorff dimension of graph of such function is determined.     

The Fractional Clifford-Fourier Transform

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.     

An efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.