Series Representation Research Articles

A Paradox in the Theory of Prediction

Given the set of past values, [Formula: see text], [Formula: see text], it is known that the conditional mean [Formula: see text] is the best predictor of [Formula: see text], [Formula: see text], where ‘best’ is defined in terms of minimization of mean square error. In this paper, we show that a prediction using the Riemann sum approximation to the spectral (Fourier) representation of a stationary time series produces a smaller mean square error. We attribute the resolution of this apparent paradox to the fact that the Riemann sum approach preserves more information of the spectral (frequency) content of the past time series than does the conditional mean — which effectively represents only the zeroth (constant value) frequency.

Finding on Convergence of the Flint Hills and Cookson Hills Series based on a Summation Formula of Adamchik and Srivastava involving the Riemann Zeta Function

This article showcases significant progress in solving two renowned problems in the calculus of series: the Flint Hills and Cookson Hills series. For almost twenty years, a long-standing question has remained unanswered in regard to their convergence. Mainly, proving the convergence of the Flint Hills series would significantly impact the redefinition of the upper bound for the irrationality measure of the number π. One of the results presented in this article is that the Flint Hills series converges to 30.3144... which leads to a redefinition of the upper bound for the irrationality measure of π, specifically μ(π)≤ 2.5. This work proposes a transformation that solves the mystery of the Flint Hills and Cookson Hills series. It is based on a summation formula developed by mathematicians Adamchik and Srivastava. By leveraging a specialized series supported by the Riemann zeta function, this approach successfully transforms the original Flint Hills and Cookson Hills series into novel convergent versions with unique significance. The resulting sequences linked to these series are positive and bounded and satisfy convergence. Moreover, this article extends the Flint Hills series when the cosecant function has an arbitrary complex argument n+iβ, with i=√(-1), establishing a new series representation based on the polylogarithm 〖Li〗_3 (e^i2k), with k=1,2,3,…, e the Euler’s number, which bears resemblance to the famous integral of the Bose-Einstein distribution as a relevant finding. This is a never-seen-before link between the Flint Hills series and polylogarithms. Furthermore, a relationship between the Apéry constant and the Flint Hills and Cookson Hills series has been established. This article presents a significant breakthrough in the calculus of series by introducing a new method based on the Riemann Zeta function and logarithmical expressions derived from the Adamchik and Srivastava summation formula. The novel approach extends the analysis of convergence criteria for series, addressing ambiguous cases characterized by abrupt jumps. Thus, the Flint Hills series converges to 30.3144... and the Cookson Hills series to 42.9949... as proved in this article.

OM3: An Ordered Multi-level Min-Max Representation for Interactive Progressive Visualization of Time Series

We present a novel multi-level representation of time series called OM3 that facilitates efficient interactive progressive visualization of large data stored in a database and supports various interactions such as resizing, panning, zooming, and visual query. Based on our proposed line-segment aggregation, this representation can produce error-free line visualizations that preserve the shape of a time series in windows of arbitrary sizes. To reduce the interaction latency, we develop an incremental tree-based query strategy to support progressive visualizations, allowing a finer control on the accuracy-time tradeoff. We quantitatively compare OM3 with state-of-the-art methods, including a method implemented on a leading time-series database InfluxDB, in two settings with databases residing either in the local area network or on the cloud. Results show that OM^3 maintains a low latency within 300~ms on the web browser and a high data reduction ratio regardless of the data size (ranging from millions to billions of records), achieving around 1,000 times faster than the state-of-the-art methods on the largest dataset experimented with.

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

Exploring the Relationship between Geomagnetic Variations and Seismic Energy Release in Proximity to the Vrancea Seismic Zone

Understanding the seismo–ionospheric coupling mechanism requires a quiet geomagnetic condition, as this represents an ideal situation to detect abnormal variations in the geomagnetic field. In reality, continuous interactions between solar wind and Earth’s magnetosphere create many fluctuations in the geomagnetic field that are more related to sun–magnetosphere interactions than to seismotectonic causes. A triaxial magnetometer was installed at the Muntele Rosu Observatory near the Vrancea seismic zone in 1996 to measure the local magnetic field. Since 2002, the data have become more consistent, allowing for the representation of long time series. Since then, variations have been observed on the eastern component (By) of the magnetic field, which sometimes overlaps with significant earthquakes. Previous studies have shown that high decreases in amplitude recorded on the By component of the magnetic field measured at Muntele Rosu have been accompanied by higher seismicity, while small decreases have been accompanied by lower seismic energy release. This research analyzes the geomagnetic data collected between September 2002 and May 2008 from two geomagnetic observatories, one located in the proximity of the Vrancea seismic zone and another one situated 120 km away. For each geomagnetic anomaly identified, the daily seismic energy released was plotted logarithmically, along with seismicity and Kp indices. Additionally, the daily seismic energy released was also plotted logarithmically for all earthquakes with Mw ≥3. To identify variations in the By component, datasets recorded at Muntele Rosu (MLR) were compared with those recorded at Surlari National Geomagnetic Observatory (SUA), to discriminate between global magnetic variations associated with solar activity and possible seismo–electromagnetic variations. The standard deviation (SDBy) was calculated for each anomaly recorded on the By component of the magnetic field and compared with the cumulative seismic energy release. To determine if this type of variation was present in other components of the magnetic field, the following ratios were calculated for all data recorded at Muntele Rosu: Bz/Bx, Bz/By, and Bz/BH. The size of the anomalies resulting from the standard deviation measured on the By component (SDBy) partially validates the relationship between the size of the anomalies and the seismic energy release during the anomaly. The relationship between the released seismic energy and the anomaly magnitude is vaguely respected, but these variations seem to follow two patterns. One pattern is described by smooth decreases, and the other pattern involves decreases where the By component varies significantly over short periods, generating decreases/increases in steps. It was noticed that seismic activity is greater for the second pattern. Additionally, using standard deviation measured on the magnetic field represents a great tool to discriminate external magnetic field variations from local, possibly seismo–magnetic variations.

Evaluation of modified Hantush’s function: A tool for parametric study with the finite line-source model

A solution to the three-dimensional finite line-source (FLS) model for borehole heat exchangers (BHEs) is identical to Hantush’s M function, commonly used in groundwater pumping modeling. A modified Hantush function E appears in the steady-state solution to the moving finite line-source (MFLS) model for borehole heat exchangers (BHEs) with groundwater flow that takes into account heat conduction and advection by Péclet number. An exact series representation is obtained for this two-parametric function E, and from it two-types of approximate expressions for the E are derived. One is suitable for small values of a parameter Pe and the other for its large values. High accuracy of the approximate formulae is verified by comparison with the exact values provided by modified Hantush’s function E in those validity domains.These analytical expressions can be used for designing complex borehole configurations and may be useful in steady-state analysis of partially penetrated wells (PPWs) in a leaky aquifer.

On the bivariate Padovan polynomials matrix

In this paper, we intruduce the bivariate Padovan sequence we examine its various identities. We define the bivariate Padovan polynomials matrix. Then, we find the Binet formula, generating function and exponential generating function of the bivariate Padovan polynomials matrix. Also, we obtain a sum formula and its series representation.

On Petersson norms of generic cusp forms and special values of adjoint L -functions for GSp4

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint $L$-functions and some elementary constants depending only on local representations.

Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

Epsilon Dichotomy for Linear Models: The Archimedean Case

Abstract Let $G=\textrm {GL}_{2n}({\mathbb {R}})$ or $G=\textrm {GL}_n({\mathbb {H}})$ and $H=\textrm {GL}_n({\mathbb {C}})$ regarded as a subgroup of $G$. Here, ${\mathbb {H}}$ is the quaternion division algebra over ${\mathbb {R}}$. For a character $\chi $ on ${\mathbb {C}}^\times $, we say that an irreducible smooth admissible moderate growth representation $\pi $ of $G$ is $\chi _H$-distinguished if $\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$. We compute the root number of a $\chi _H$-distinguished representation $\pi $ twisted by the representation induced from $\chi $. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast (H, \pi \otimes \chi )$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi $ of $H$.

Numerical simulation of time fractional Benjamin-Bona-Mahony-Burger equation describing propagation of long waves on the water surface

The unidirectional propagation of long waves in a certain nonlinear dispersive system is explained in the Benjamin-Bona-Mahony-Burger equation. The motivation of this manuscript is to examine the Benjamin-Bona-Mahony-Burger equation numerically and analytically by employing reproducing kernel functions. The implementation of reproducing kernel Hilbert space method produces an analytical solution in the form of a series representation, and after truncating the infinite series, an approximate solution is attained. With the support of numerical examples, the accuracy of this method has been shown through graphs and error tables. The convergence analysis of the introduced scheme has been explained by some useful theorems and lemmas. Finally, it has been observed that the approximate solution consistently converges to the exact solution with less computational overhead. The proposed method would be very effective in solving various fractional partial differential equation models.

Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R3/Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UGΛ is revised, reorganized, improved and extended. In particular we construct the so far missing UGΛHex for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b-manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b-deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that the most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions Wr, each associated with a spherical layer of quantized radius r in the momentum lattice. Each Wr is the superposition of a Beltrami field Wr+ plus an anti-Beltrami field Wr−. These latter have a priori exactly the same decomposition into irreps of the group UGΛ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories.

An adaptive spatio-temporal neural network for PM2.5 concentration forecasting

Accurate PM2.5 concentration prediction is essential for environmental control management, therefore numerous air quality monitoring stations have been established, which generate multiple time series with spatio-temporal correlation. However, the statistical distribution of data from different monitoring stations varies widely, which needs to provide higher flexibility in the feature extraction stage. Moreover, the spatio-temporal correlation and mutation characteristics of the time series are difficult to capture. To this end, an adaptive spatio-temporal prediction network (ASTP-NET) is proposed, in which the encoder adaptively extracts the input data features, then captures the spatio-temporal dependencies and dynamic changes of the time series, the decoder part maps the encoded features into a predicted future time series representation, while an objective function is proposed to measure the overall fluctuations of the model’s multi-step prediction. In this paper, ASTP-NET is evaluated based on the Xi'an air quality dataset, and the results show that the model outperforms other baseline methods.

The shadow formalism of Galilean CFT2

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT2). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the GCFT2 can be regarded as null defect in higher dimensional CFTs.

A new deep learning architecture with inductive bias balance for transformer oil temperature forecasting

Ensuring the optimal performance of power transformers is a laborious task in which the insulation system plays a vital role in decreasing their deterioration. The insulation system uses insulating oil to control temperature, as high temperatures can reduce the lifetime of the transformers and lead to expensive maintenance. Deep learning architectures have been demonstrated remarkable results in various fields. However, this improvement often comes at the cost of increased computing resources, which, in turn, increases the carbon footprint and hinders the optimization of architectures. In this study, we introduce a novel deep learning architecture that achieves a comparable efficacy to the best existing architectures in transformer oil temperature forecasting while improving efficiency. Effective forecasting can help prevent high temperatures and monitor the future condition of power transformers, thereby reducing unnecessary waste. To balance the inductive bias in our architecture, we propose the Smooth Residual Block, which divides the original problem into multiple subproblems to obtain different representations of the time series, collaboratively achieving the final forecasting. We applied our architecture to the Electricity Transformer datasets, which obtain transformer insulating oil temperature measures from two transformers in China. The results showed a 13% improvement in MSE and a 57% improvement in performance compared to the best current architectures, to the best of our knowledge. Moreover, we analyzed the architecture behavior to gain an intuitive understanding of the achieved solution.

Polynomial NARX-based nonlinear model predictive control of modular chemical systems

The design of control systems for modular chemical systems typically requires the identification of nonlinear dynamic models. Mechanistic models for modular chemical systems are typically of high order, which results in high online computational cost when the models are incorporated into the nonlinear model predictive control (NMPC) formulations developed for explicitly taking constraints into account. This article proposes the use of a particular class of nonlinear input–output models, polynomial nonlinear-autoregressive-with-exogenous-inputs (NARX) models, in the NMPC formulations. A machine learning algorithm, elastic net, is used to select which terms to include within the NARX polynomial series representation. The approach for constructing sparse predictive models and their use in real-time implementable NMPC are demonstrated in a two-input two-output chemical reactor case study. The Julia programming language is used to solve the NMPC optimization problem, resulting in low online computational cost.

Time2State: An Unsupervised Framework for Inferring the Latent States in Time Series Data

Time series data from monitoring applications reflect the physical or logical states of the objects, which may produce time series of distinguishable characteristics in different states. Thus, time series data can usually be split into different segments, each reflecting a state of the objects. These states carry rich high-level semantic information, e.g., run, walk, or jump, which helps people better understand the behaviour of the monitored objects. Nevertheless, these states are latent and hard to discover, because the characteristic of time series is complicated and the computational cost is high. This paper develops an efficient and effective unsupervised approach for inferring the latent states of massive multivariate time data. To reduce the computational cost, we present Time2State, a scalable framework that utilizes a sliding window and an encoder to greatly reduce the length of raw time series. To train the encoder, we propose a novel unsupervised loss function, LSE-Loss. Extensive experiments show that compared to the state-of-the-art time series representation learning methods of the same kind, LSE-Loss brings a performance improvement of up to 15% in accuracy.

Boundary-to-solution mapping for groundwater flows in a Toth basin

In this paper, the authors propose a new approach to solving the groundwater flow equation in the Toth basin of arbitrary top and bottom topographies using deep learning. Instead of using traditional numerical solvers, they use a DeepONet to produce the boundary-to-solution mapping. This mapping takes the geometry of the physical domain along with the boundary conditions as inputs to output the steady state solution of the groundwater flow equation. To implement the DeepONet, the authors approximate the top and bottom boundaries using truncated Fourier series or piecewise linear representations. They present two different implementations of the DeepONet: one where the Toth basin is embedded in a rectangular computational domain, and another where the Toth basin with arbitrary top and bottom boundaries is mapped into a rectangular computational domain via a nonlinear transformation. They implement the DeepONet with respect to the Dirichlet and Robin boundary condition at the top and the Neumann boundary condition at the impervious bottom boundary, respectively. Using this deep-learning enabled tool, the authors investigate the impact of surface topography on the flow pattern by both the top surface and the bottom impervious boundary with arbitrary geometries. They discover that the average slope of the top surface promotes long-distance transport, while the local curvature controls localized circulations. Additionally, they find that the slope of the bottom impervious boundary can seriously impact the long-distance transport of groundwater flows. Overall, this paper presents a new and innovative approach to solving the groundwater flow equation using deep learning, which allows for the investigation of the impact of surface topography on groundwater flow patterns.

On the density arising from the domain of attraction of an operator interpolating between sum and supremum: The α-Sun operator

We explore the analytic properties of the density function h(x;γ,α), x∈(0,∞), γ>0, 0<α<1 which arises as a normed limit from the domain of attraction problem for an operator interpolating between the supremum and sum as applied to a sequence of i.i.d. non-negative random variables. The parameter α controls the interpolation between these two cases, while γ further parametrises the type of distribution from which the underlying random variables are drawn. It is known that in the normed limit, for α=0 the Fréchet density arises, whereas for α=1 the limit is a stable random variable which has the well-known identification with a particular Fox H-function. It is known [21] that for intermediate α an entirely new distribution function appears, which is not one of the extensions to the hypergeometric function considered to date. Here we derive series, integral and continued fraction representations of this latter function.

A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.