## A Fourier series representation of the probability density function (PDF) of the subarray phase estimate is used to simplify evaluation of the PDF of the UCA estimator and its circular variance. 子阵列相位估计的概率密度函数 (PDF) 的傅里叶级数表示用于简化对 UCA 估计器的 PDF 及其循环方差的评估。

Signal-Phase Estimation for Product Array Processors

## Some important properties are demonstrated, such as the series representation of the probability density function, quantile function, moments, stress-strength reliability, and Rényi entropy. 展示了一些重要的性质，例如概率密度函数、分位数函数、矩、应力强度可靠性和 Rényi 熵的级数表示。

A New Generator of Probability Models: The Exponentiated Sine-G Family for Lifetime Studies

## Further time complexity reduction is achieved using time series representation with low computational needs. 使用具有低计算需求的时间序列表示可以进一步降低时间复杂度。

Smart Meters Time Series Clustering for Demand Response Applications in the Context of High Penetration of Renewable Energy Resources

## In this paper, we propose two new data structures named interval table (ITable) and extend interval table (EITable) for time series representation to capture more original information. 在本文中，我们提出了两个新的数据结构，称为区间表（ITable）和扩展区间表（EITable），用于时间序列表示，以捕获更多的原始信息。

ADET: anomaly detection in time series with linear time

## In addition, we demonstrate that Fourier series representations can also be expressed as a special class of SPINN and propose generalized neural network analogues of Fourier representations. 此外，我们证明傅里叶级数表示也可以表示为特殊类别的 SPINN，并提出了傅里叶表示的广义神经网络类似物。

SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs

## The Fourier series representation is commonly used to compute the dispersion diagrams via the plane wave expansion (PWE) method. 傅里叶级数表示通常用于通过平面波展开 (PWE) 方法计算色散图。

Widening wave band gaps of periodic plates via shape optimization using spatial Fourier coefficients

## The R-groups R χ ˜ and R χ of the corresponding principal series representations fit in the exact sequence 0 → R χ ˜ → R χ → R χ / R χ ˜ → 0. 对应的主级数表示的 R 组 R χ ~ 和 R χ 符合精确序列 0 → R χ ~ → R χ → R χ / R χ ~ → 0。

R-groups for unitary principal series of Spin groups

## In the present work we perform such a mode expansion for massive scalar fields using the unitary principal series representations of SO(1, 3) with a view to developing a holographic approach to gravity in asymptotically flat spacetime. 在目前的工作中，我们使用 SO(1, 3) 的单一主级数表示对大规模标量场进行这种模式扩展，以期在渐近平坦的时空中开发一种全息方法来处理重力。

Conformal wave expansions for flat space amplitudes

## Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. 此外，我们建立了各种梅林变换公式，并获得了包含拉盖尔多项式的某些无限级数表示。

Extended elliptic-type integrals with associated properties and Turán-type inequalities

## By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved “log-summed” approximation to the principal eigenvalue. 通过使用 Ewald 求和技术首先获得该规则部分的快速收敛无限级数表示，实现了超越方程的简单牛顿迭代方案，以数值评估改进的“对数求和”近似主特征值。

Asymptotics of the principal eigenvalue of the Laplacian in 2D periodic domains with small traps

## Additionally, we derive an iterative formula for the coefficients of this power series representation. 此外，我们推导出这个幂级数表示的系数的迭代公式。

Persistence exponents via perturbation theory : autoregressive and moving average processes

## Our goal has been fulfilled by extending the residual power series method into the two-dimensional time and space, with time and space endowed with fractional derivative orders $$\alpha$$ and $$\gamma$$ , respectively (simply denoted by fractional $$(\alpha ,\gamma )$$ -space), by virtue of a new $$(\alpha ,\gamma )$$ -fractional power series representation ( $$(\alpha ,\gamma )$$ -FPS). 我们的目标是通过将剩余幂级数方法扩展到二维时间和空间来实现的，时间和空间具有分数导数阶 $$\alpha$$ 和 $$\gamma$$ ，分别（简单地表示为分数 $$(\alpha ,\gamma )$$ -space)，凭借新的 $$(\alpha ,\gamma )$$ -分数幂级数表示（ $$(\alpha ,\gamma )$$ -FPS）。

Simulating the joint impact of temporal and spatial memory indices via a novel analytical scheme

## This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. 本文考虑离散序列表示的晶格轨道的跨越性质与相关晶格共体积之间的关系。

The density theorem for discrete series representations restricted to lattices

## In this thesis, we construct a residue map and a Poisson kernel between holomorphic discrete series representations on the Drinfeld period domain and harmonic cocycles with certain non-trivial coefficients on the Bruhat-Tits building for GL3 over a local field of any characteristic. 在本论文中，我们在 Drinfeld 周期域上的全纯离散序列表示和具有某些非平凡系数的调和 cocycle 之间构建了残差图和泊松核，用于 GL3 在任何特征的局部场上构建的 Bruhat-Tits。

Boundary Distributions for GL3 over a Local Field andSymmetric Power Coefficients

## As examples, \begin{aligned} \zeta (3)=\frac{1}{7}\sum _{k=1}^{\infty }\frac{H_{k-1}4^k}{k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \quad \text{ and }\quad \zeta (3)=\frac{8}{7}+\frac{1}{7}\sum _{k=1}^{\infty } \frac{H_{k-1}4^k}{k^2(2k+1)\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \end{aligned} ζ ( 3 ) = 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 2 k k , and ζ ( 3 ) = 8 7 + 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 ( 2 k + 1 ) 2 k k , which are new series representations for the Apéry constant $$\zeta (3)$$ ζ ( 3 ). 例如，\begin{aligned} \zeta (3)=\frac{1}{7}\sum _{k=1}^{\infty }\frac{H_{k-1}4^k} {k^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) }, \quad \text{ 和 }\quad \zeta (3)=\frac{8 }{7}+\frac{1}{7}\sum _{k=1}^{\infty } \frac{H_{k-1}4^k}{k^2(2k+1)\left ( {\begin{array}{c}2k\\ k\end{array}}\right) }, \end{aligned} ζ ( 3 ) = 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 2 k k 和 ζ ( 3 ) = 8 7 + 1 7 ∑ k = 1 ∞ H k - 1 4 k k 2 ( 2 k + 1 ) 2 k k ，它们是 Apéry 常数 $$\zeta ( 3)$$ ζ ( 3 )。

Parametric binomial sums involving harmonic numbers

## A new series representation of the modified Bessel function of the second kind K0(x) in terms of simple elementary functions (Kummer’s function) is obtained. 得到了用简单初等函数（Kummer 函数）表示的第二类 K0(x) 修正贝塞尔函数的新级数表示。

On new approximations for the modified Bessel function of the second kind $$K_0(x)$$

## The numerical results show that the n-term approximation is a rapidly convergent series representation and they present also the high accuracy and effectiveness of this method. 数值结果表明，n项近似是一种快速收敛的级数表示，它们也展示了该方法的高精度和有效性。

Solving Duffing-Van der Pol Oscillator Equations of Fractional Order by an Accurate Technique

## As a corollary, we give explicit evaluation formulas or rapidly convergent series representations for T(s, t, u) in terms of series of the gamma function and the Riemann zeta function. 作为推论，我们根据伽马函数和黎曼zeta函数的级数给出了T(s, t, u)的明确评估公式或快速收敛级数表示。

Symmetric Tornheim double zeta functions

## Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. 设计/方法/方法基于局部泰勒级数表示的无方向数值方法是为 ADR 方程设计的。

An implicit-explicit local method for parabolic partial differential equations

## We then show that Taylor series representations of smooth functionals on S ≈ Φ , g $\mathcal {S}^{\Phi ,g}_{\approx }$ provide a new analytic description for solutions of combinatorial Dyson–Schwinger equations. 然后我们展示了 S ≈ Φ , g $\mathcal {S}^{\Phi ,g}_{\approx }$ 上的泰勒级数表示为组合 Dyson-Schwinger 方程组的解提供了新的解析描述。

The Analytic Evolution of Dyson–Schwinger Equations via Homomorphism Densities

## Of particular importance will be the contractive nature of this map, which is needed to show that the closed-loop system has a Chen-Fliess series representation. 特别重要的是这张地图的收缩性质，需要证明闭环系统具有 Chen-Fliess 级数表示。

Formal Power Series Approach to Nonlinear Systems with Static Output Feedback

## Given a single-input, single-output (SISO) system with a Chen-Fliess series representation whose generating series has a well defined relative degree, it is shown that there is a notion of universal zero dynamics that describes a set of dynamics evolving on a locally convex (infinite dimensional) Lie group so as to render the system's output exactly zero. 给定一个具有 Chen-Fliess 级数表示的单输入单输出 (SISO) 系统，其生成级数具有明确定义的相对度数，它表明存在一个通用零动力学概念，它描述了一组在一个局部凸的（无限维）李群，以使系统的输出正好为零。

Universal Zero Dynamics: The SISO Case

## First, a model for data loss is proposed using a finite series representation of a set of periodic continuous pulses. 首先，使用一组周期性连续脉冲的有限序列表示提出了数据丢失模型。

Stability Analysis and Performance Evaluation of Delayed Bilateral Telerobotic Systems over a Lossy Communication Channel

## Here, we assume a finite series representation of the DF, constructed from the spectrum of the system’s Koopman operator. 在这里，我们假设 DF 的有限级数表示，由系统的 Koopman 算子的谱构成。

Inferring time-dependent distribution functions from kinematic snapshots

## The thermo-mechanical properties of the adhesive were then characterised using time–temperature superposition parameters and a Prony series representation for generalised Maxwell creep. 然后使用时间-温度叠加参数和广义麦克斯韦蠕变的 Prony 级数表示来表征粘合剂的热机械性能。

Linear creep of bonded FRP-strengthened metallic structures at warm service temperatures

## Storage and loss modulus were obtained from white and grey matter, of bovine brains, using dynamic mechanical analysis and time domain material functions were derived based on a Prony series representation. 使用动态力学分析从牛脑的白质和灰质获得储能模量和损耗模量，并基于 Prony 级数表示推导出时域材料函数。

Investigation of the Compressive Viscoelastic Properties of Brain Tissue Under Time and Frequency Dependent Loading Conditions.

## In general, the frequency analysis of nonlinear systems is conducted based on the Volterra series representation of nonlinear systems, and as basic issues, the evaluation of the Volterra series representation and its convergence are discussed in Chapters 3 and 4, respectively. 一般来说，非线性系统的频率分析是基于非线性系统的沃尔泰拉级数表示进行的，作为基本问题，沃尔泰拉级数表示的评估及其收敛性分别在第3章和第4章讨论。

Analysis and Design of Nonlinear Systems in the Frequency Domain

## We obtain a Volterra series representation for these solutions which is norm convergent and uniformly convergent on compact subsets of the time scale. 我们获得了这些解决方案的 Volterra 级数表示，它在时间尺度的紧凑子集上是范数收敛和一致收敛的。

Bilinear state systems on an unbounded time scale

## In order to capture the shapelet dynamics and evolutions, in this paper, we propose a novel framework of bridging time series representation learning and graph modeling, with two different implementations. 为了捕捉 shapelet 的动态和演变，在本文中，我们提出了一种新的框架，用于连接时间序列表示学习和图建模，具有两种不同的实现。

Time2Graph+: Bridging Time Series and Graph Representation Learning via Multiple Attentions

## This study proposes a unified, flexible time series representation learning framework for both univariate and multivariate time series called Rand-TS. 本研究为单变量和多变量时间序列提出了一个统一、灵活的时间序列表示学习框架，称为 Rand-TS。

Randomized trees for time series representation and similarity

## The generating functions, series representations, and certain other significant relations and identities of this class are established. 建立了这个类的生成函数、序列表示和某些其他重要的关系和恒等式。

A Note on q -Fubini-Appell Polynomials and Related Properties

## The DCRL learns the price time-series representation by adaptively selecting different price features depending on the current state. DCRL 通过根据当前状态自适应地选择不同的价格特征来学习价格时间序列表示。

DCRL: Approach for Pattern Recognition in Price Time Series using Directional Change and Reinforcement Learning

## Moreover, we propose a simple method that can be used to generate other continued fractions using their series representations. 此外，我们提出了一种简单的方法，可以使用它们的序列表示来生成其他连分数。

Generalized continued fraction expansions for π and e

## We provide a series representation for the inverse of this censored fractional derivative. 我们为这种删失分数导数的逆提供了一系列表示。

Censored stable subordinators and fractional derivatives

## In this work, we investigate whether DNN-based models can be used to forecast these TS conjointly by learning a joint representation of the series instead of computing the forecast from the raw time-series representations. 在这项工作中，我们研究了基于 DNN 的模型是否可以通过学习序列的联合表示而不是从原始时间序列表示计算预测来联合预测这些 TS。

Spatiotemporal adaptive neural network for long-term forecasting of financial time series

## In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. 为了检验分数阶 Fisher 类型方程的近似结果，IETM 解以序列表示形式获得。

Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform

## The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. 傅里叶级数表示在许多情况下具有精确的优点，但会随着先验的每次更新而增加复杂性。

Efficient Bayesian phase estimation using mixed priors

## A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. 开发了用于正非整数幂的离散分数拉普拉斯算子的级数表示。

A series representation of the discrete fractional Laplace operator of arbitrary order

## Since only feed-forward structure is embodied in the series representation, the coefficients of the proposed compensator are easy to access with adaptive algorithms of low complexity. 由于序列表示中仅包含前馈结构，因此所提出的补偿器的系数易于使用低复杂度的自适应算法访问。

LED Nonlinearity Post-compensator with Legendre polynomials in Visible Light Communications

## This series representation extends the one of bivariate Gaussian. 该级数表示扩展了二元高斯的一种。

Series Representation of Jointly S $$\alpha$$ α S Distribution via Symmetric Covariations

## augmenting the aggregated time-series representation). 增加聚合的时间序列表示）。

Augmenting Transferred Representations for Stock Classification

## The computational technique of N -fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. N 倍梅林-巴恩斯 (MB) 积分的计算技术，由同一作者在一篇配套论文中提出，用于推导各种配置的大规模单环保形 3 点费曼积分的系列表示集。

Massive one-loop conformal Feynman integrals and quadratic transformations of multiple hypergeometric series

## We analyze the characteristics of adversarial time-series examples and normal time-series samples and propose a time-series representation method. 我们分析了对抗性时间序列样本和正常时间序列样本的特征，并提出了一种时间序列表示方法。

An Adversarial Examples Identification Method for Time Series in Internet-of-Things System

## The series representation of this approximate expression is also developed, which enables us to avoid dealing with the integration. 这个近似表达式的级数表示也得到了发展，这使我们能够避免处理积分。

Novel approximate distribution of the sum of Gamma–Gamma variates with pointing errors and applications in MIMO FSO links

## Applying the clustering to various time-series representation allows us to gain confidence about the redundant spatial patterns. 将聚类应用于各种时间序列表示使我们能够获得对冗余空间模式的信心。

Time-series clustering approaches for subsurface zonation and hydrofacies detection using a real time-lapse electrical resistivity dataset

## In an alternative approach, we show that the α-sine transform of a function f admits a series representation for all α > −1, which involves the Fourier transform of f and coefficients which can all be explicitly computed with the Gauss hypergeometric theorem. 在另一种方法中，我们表明函数 f 的 α-正弦变换允许所有 α > -1 的级数表示，这涉及 f 的傅里叶变换和系数，这些都可以用高斯超几何定理显式计算。

Inversion of α-sine and α-cosine transforms on R

## Furthermore, we provide some novel results about the underlying vector of subordinators such as a series representation and relevant moments. 此外，我们提供了一些关于从属子的基础向量的新结果，例如序列表示和相关矩。

Compound vectors of subordinators and their associated positive Lévy copulas

## Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q-derivative and q-integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. 接下来，我们将介绍这些多项式的许多性质和公式，例如显式表达式、级数表示、求和公式、加法公式、q-导数和 q-积分公式，以及新多项式的许多特殊情况及其相关公式两张桌子。

q-Generalized Tangent Based Hybrid Polynomials

## Time-series representations of tweet storms are constructed using post metadata, and the DWFP converts these into binary images using a wavelet transform. 推文风暴的时间序列表示是使用发布元数据构建的，DWFP 使用小波变换将这些转换为二进制图像。

Bayesian identification of bots using temporal analysis of tweet storms

## In an alternative approach, we show that the $\alpha$-sine transform of a function $f$ admits a series representation for all $\alpha>-1$, which involves the Fourier transform of $f$ and coefficients which can all be explicitly computed with the Gauss hypergeometric theorem. 在另一种方法中，我们证明了函数 $f$ 的 $\alpha$-sine 变换允许所有 $\alpha>-1$ 的级数表示，这涉及到 $f$ 的傅里叶变换和所有系数用高斯超几何定理显式计算。

Inversion of $\alpha$-sine and $\alpha$-cosine transforms on $\mathbb{R}$

## In the paper, the authors establish nice Maclaurin series expansions and series identities for powers of the inverse sine function, for powers of the inverse hyperbolic sine function, for composites of incomplete gamma functions with the inverse hyperbolic sine function, for powers of the inverse tangent function, and for powers of the inverse hyperbolic tangent function, in terms of the first kind Stirling numbers, binomial coefficients, and multiple sums, apply the nice Maclaurin series expansion for powers of the inverse sine function to derive an explicit formula for special values of the second kind Bell polynomials and to derive a series representation of the generalized logsine function, and deduce several combinatorial identities involving the first kind Stirling numbers. 在本文中，作者为反正弦函数的幂、反双曲正弦函数的幂、不完全伽马函数与反双曲正弦函数的复合、反正切的幂建立了很好的 Maclaurin 级数展开和级数恒等式函数，并且对于反双曲正切函数的幂，根据第一类斯特林数、二项式系数和多重和，对反正弦函数的幂应用漂亮的 Maclaurin 级数展开，以推导出特殊值的显式公式第二类贝尔多项式并导出广义对数函数的级数表示，并推导出涉及第一类斯特林数的几个组合恒等式。

Maclaurin series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions and for incomplete gamma functions with applications

## We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. 我们简要介绍了无限可分定律和相关过程的散粒噪声表示，并讨论了将此类序列表示截断以模拟无限可分随机向量、Lévy 过程、无限可分过程和场以及 Lévy 驱动的随机微分方程。

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

## The modes are given in terms of a family of complex “Hankel-Airy” functions, along with their asymptotic expansions, orthogonality relations and series representations. 这些模式是根据一系列复杂的“Hankel-Airy”函数以及它们的渐近展开、正交关系和级数表示来给出的。

Dynamics of heavy beams: Closed-form vibrations of gravity-loaded Rayleigh–Timoshenko columns

## A bijective transformation of space was carried out to formulate a series representation of the potential in the interior region. 进行了空间的双射变换，以制定内部区域潜力的一系列表示。

Theory of perturbation of electrostatic field by an anisotropic dielectric sphere

## In contrast to time-series representation of the RF signals, with spectrograms, it is possible to focus only on the desired frequency interval, i. 与 RF 信号的时间序列表示相比，使用频谱图，可以只关注所需的频率间隔，即。

RF-based low-SNR classification of UAVs using convolutional neural networks

## Interestingly, new closed-form and series representations for the PDF and CDF of the composite distributions are derived. 有趣的是，导出了复合分布的 PDF 和 CDF 的新封闭形式和系列表示。

Revisiting the Lognormal Modelling of Shadowing Effects during Wireless Communications by Means of the α-μ/α-μ Composite Distribution

## In this paper, we propose an unsupervised Time-Series representation learning framework via Temporal and Contextual Contrasting (TS-TCC), to learn time-series representation from unlabeled data. 在本文中，我们提出了一种通过时间和上下文对比 (TS-TCC) 的无监督时间序列表示学习框架，以从未标记的数据中学习时间序列表示。

Time-Series Representation Learning via Temporal and Contextual Contrasting

## We start with the concept of evolution function, whose subzero-level set can be used to describe reachable set, and find a series representation of the evolution function with its Lie derivatives. 我们从进化函数的概念入手，它的次零水平集可以用来描述可达集，并找到进化函数及其李导数的级数表示。

Over- and Under-Approximations of Reachable Sets With Series Representations of Evolution Functions