Research Progress of the Sampling Theorem Associated with the Fractional Fourier Transform

Sampling is a bridge between continuous-time and discrete-time signals,which is import-ant to digital signal processing.The fractional Fourier transform(FrFT)that serves as a generaliz-ation of the FT can characterize signals in multiple fractional Fourier domains,and therefore can provide new perspectives for signal sampling and reconstruction.In this paper,we review recent de-velopments of the sampling theorem associated with the FrFT,including signal reconstruction and fractional spectral analysis of uniform sampling,nonuniform samplings due to various factors,and sub-Nyquist sampling,where bandlimited signals in the fractional Fourier domain are mainly taken into consideration.Moreover,we provide several future research topics of the sampling theorem as-sociated with the FrFT.

Optimization of Well Position and Sampling Frequency for Groundwater Monitoring and Inverse Identification of Contamination Source Conditions Using Bayes’Theorem

Coupling Bayes’Theorem with a two-dimensional(2D)groundwater solute advection-diffusion transport equation allows an inverse model to be established to identify a set of contamination source parameters including source intensity(M),release location(0 X,0 Y)and release time(0 T),based on monitoring well data.To address the issues of insufficient monitoring wells or weak correlation between monitoring data and model parameters,a monitoring well design optimization approach was developed based on the Bayesian formula and information entropy.To demonstrate how the model works,an exemplar problem with an instantaneous release of a contaminant in a confined groundwater aquifer was employed.The information entropy of the model parameters posterior distribution was used as a criterion to evaluate the monitoring data quantity index.The optimal monitoring well position and monitoring frequency were solved by the two-step Monte Carlo method and differential evolution algorithm given a known well monitoring locations and monitoring events.Based on the optimized monitoring well position and sampling frequency,the contamination source was identified by an improved Metropolis algorithm using the Latin hypercube sampling approach.The case study results show that the following parameters were obtained:1)the optimal monitoring well position(D)is at(445,200);and 2)the optimal monitoring frequency(Δt)is 7,providing that the monitoring events is set as 5 times.Employing the optimized monitoring well position and frequency,the mean errors of inverse modeling results in source parameters(M,X0,Y0,T0)were 9.20%,0.25%,0.0061%,and 0.33%,respectively.The optimized monitoring well position and sampling frequency canIt was also learnt that the improved Metropolis-Hastings algorithm(a Markov chain Monte Carlo method)can make the inverse modeling result independent of the initial sampling points and achieves an overall optimization,which significantly improved the accuracy and numerical stability of the inverse modeling results.

APPROXIMATE SAMPLING THEOREM FOR BIVARIATE CONTINUOUS FUNCTION

An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .

Sampling theorem for multiwavelet subspaces

Unlike scalar wavelets, multiscaling functions can be orthogonal, regular and symmetrical, and have compact support and high order of approximation simultaneously. For this reason, even if multiscaling functions are not cardinal, they still hold for perfect A/D and D/A. We generalize the Walter＇s sampling theorem to multiwavelet subspaces based on reproducing kernel Hilbert space. The reconstruction function can be expressed by multiwavelet function using the Zak transform. The general case of irregular sampling is also discussed and the irregular sampling theorem for multiwavelet subspaces established. Examples are presented.

EXACT EVALUATIONS OF FINITE TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli＇s numbers play a role similar to that of Euler＇s in the old ones. Our technique differs from that of Byrne-Smith （1997） and Berndt-Yeap （2002）.

Vector sampling theorem for wavelet subspaces

The vector sampling theorem has been investigated and widely used by multi-channel deconvolution, multi-source separation and multi-input multi-output （MIh40） systems. Commonly, for most of the results on MIMO systems, the input signals are supposed to be band-limited. In this paper, we study the vector sampling theorem for the wavelet subspaces with reproducing kernel. The case of uniform sampling is discussed, and the necessary and sufficient conditions for reconstruction are given. Examples axe also presented.

How do the landslide and non-landslide sampling strategies impact landslide susceptibility assessment? d A catchment-scale case study from China

The aim of this study is to investigate the impacts of the sampling strategy of landslide and non-landslide on the performance of landslide susceptibility assessment(LSA).The study area is the Feiyun catchment in Wenzhou City,Southeast China.Two types of landslides samples,combined with seven non-landslide sampling strategies,resulted in a total of 14 scenarios.The corresponding landslide susceptibility map(LSM)for each scenario was generated using the random forest model.The receiver operating characteristic(ROC)curve and statistical indicators were calculated and used to assess the impact of the dataset sampling strategy.The results showed that higher accuracies were achieved when using the landslide core as positive samples,combined with non-landslide sampling from the very low zone or buffer zone.The results reveal the influence of landslide and non-landslide sampling strategies on the accuracy of LSA,which provides a reference for subsequent researchers aiming to obtain a more reasonable LSM.

Aliasing Error of Sampling Theorem

Let f∈Lpr（R）,1【p【∞,r∈N),the bound for aliasing of function fin Lp—metricesis given in[1].From embedding Theorem and Stein inequality,it follows,that if f∈Lpr（R）,1【p【∞,then f∈Lqr（R）,1【p【q≤∞ and f（k）∈Lp（R）∩Lq（R）,k=0,1,…,r-1.In this paper,we con-tinued Fang’s work in[1],and obtain the bound for aliasing error of function f∈Lpr（R）,1【p【∞,inLq-metrices（p≤q≤∞）.And the order is exact when 1【p≤q≤2.

Research on a Monte Carlo global variance reduction method based on an automatic importance sampling method

Global variance reduction is a bottleneck in Monte Carlo shielding calculations.The global variance reduction problem requires that the statistical error of the entire space is uniform.This study proposed a grid-AIS method for the global variance reduction problem based on the AIS method,which was implemented in the Monte Carlo program MCShield.The proposed method was validated using the VENUS-Ⅲ international benchmark problem and a self-shielding calculation example.The results from the VENUS-Ⅲ benchmark problem showed that the grid-AIS method achieved a significant reduction in the variance of the statistical errors of the MESH grids,decreasing from 1.08×10^(-2) to 3.84×10^(-3),representing a 64.00% reduction.This demonstrates that the grid-AIS method is effective in addressing global issues.The results of the selfshielding calculation demonstrate that the grid-AIS method produced accurate computational results.Moreover,the grid-AIS method exhibited a computational efficiency approximately one order of magnitude higher than that of the AIS method and approximately two orders of magnitude higher than that of the conventional Monte Carlo method.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

Multivariate form of Hermite sampling series

In this paper,we establish a new multivariate Hermite sampling series involving samples from the function itself and its mixed and non-mixed partial derivatives of arbitrary order.This multivariate form of Hermite sampling will be valid for some classes of multivariate entire functions,satisfying certain growth conditions.We will show that many known results included in Commun Korean Math Soc,2002,17:731-740,Turk J Math,2017,41:387-403 and Filomat,2020,34:3339-3347 are special cases of our results.Moreover,we estimate the truncation error of this sampling based on localized sampling without decay assumption.Illustrative examples are also presented.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Modified DS np Chart Using Generalized Multiple Dependent State Sampling under Time Truncated Life Test

This study presents the design of a modified attributed control chart based on a double sampling(DS)np chart applied in combination with generalized multiple dependent state(GMDS)sampling to monitor the mean life of the product based on the time truncated life test employing theWeibull distribution.The control chart developed supports the examination of the mean lifespan variation for a particular product in the process of manufacturing.Three control limit levels are used:the warning control limit,inner control limit,and outer control limit.Together,they enhance the capability for variation detection.A genetic algorithm can be used for optimization during the in-control process,whereby the optimal parameters can be established for the proposed control chart.The control chart performance is assessed using the average run length,while the influence of the model parameters upon the control chart solution is assessed via sensitivity analysis based on an orthogonal experimental design withmultiple linear regression.A comparative study was conducted based on the out-of-control average run length,in which the developed control chart offered greater sensitivity in the detection of process shifts while making use of smaller samples on average than is the case for existing control charts.Finally,to exhibit the utility of the developed control chart,this paper presents its application using simulated data with parameters drawn from the real set of data.

Enhancing Deep Learning Semantics:The Diffusion Sampling and Label-Driven Co-Attention Approach

The advent of self-attention mechanisms within Transformer models has significantly propelled the advancement of deep learning algorithms,yielding outstanding achievements across diverse domains.Nonetheless,self-attention mechanisms falter when applied to datasets with intricate semantic content and extensive dependency structures.In response,this paper introduces a Diffusion Sampling and Label-Driven Co-attention Neural Network(DSLD),which adopts a diffusion sampling method to capture more comprehensive semantic information of the data.Additionally,themodel leverages the joint correlation information of labels and data to introduce the computation of text representation,correcting semantic representationbiases in thedata,andincreasing the accuracyof semantic representation.Ultimately,the model computes the corresponding classification results by synthesizing these rich data semantic representations.Experiments on seven benchmark datasets show that our proposed model achieves competitive results compared to state-of-the-art methods.

Recent advances in protein conformation sampling by combining machine learning with molecular simulation

The rapid advancement and broad application of machine learning(ML)have driven a groundbreaking revolution in computational biology.One of the most cutting-edge and important applications of ML is its integration with molecular simulations to improve the sampling efficiency of the vast conformational space of large biomolecules.This review focuses on recent studies that utilize ML-based techniques in the exploration of protein conformational landscape.We first highlight the recent development of ML-aided enhanced sampling methods,including heuristic algorithms and neural networks that are designed to refine the selection of reaction coordinates for the construction of bias potential,or facilitate the exploration of the unsampled region of the energy landscape.Further,we review the development of autoencoder based methods that combine molecular simulations and deep learning to expand the search for protein conformations.Lastly,we discuss the cutting-edge methodologies for the one-shot generation of protein conformations with precise Boltzmann weights.Collectively,this review demonstrates the promising potential of machine learning in revolutionizing our insight into the complex conformational ensembles of proteins.

FPSblo:A Blockchain Network Transmission Model Utilizing Farthest Point Sampling

Peer-to-peer(P2P)overlay networks provide message transmission capabilities for blockchain systems.Improving data transmission efficiency in P2P networks can greatly enhance the performance of blockchain systems.However,traditional blockchain P2P networks face a common challenge where there is often a mismatch between the upper-layer traffic requirements and the underlying physical network topology.This mismatch results in redundant data transmission and inefficient routing,severely constraining the scalability of blockchain systems.To address these pressing issues,we propose FPSblo,an efficient transmission method for blockchain networks.Our inspiration for FPSblo stems from the Farthest Point Sampling(FPS)algorithm,a well-established technique widely utilized in point cloud image processing.In this work,we analogize blockchain nodes to points in a point cloud image and select a representative set of nodes to prioritize message forwarding so that messages reach the network edge quickly and are evenly distributed.Moreover,we compare our model with the Kadcast transmission model,which is a classic improvement model for blockchain P2P transmission networks,the experimental findings show that the FPSblo model reduces 34.8%of transmission redundancy and reduces the overload rate by 37.6%.By conducting experimental analysis,the FPS-BT model enhances the transmission capabilities of the P2P network in blockchain.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

A Path Planning Algorithm Based on Improved RRT Sampling Region

For the problem of slow search and tortuous paths in the Rapidly Exploring Random Tree(RRT)algorithm,a feedback-biased sampling RRT,called FS-RRT,is proposedbasedon RRT.Firstly,toimprove the samplingefficiency of RRT to shorten the search time,the search area of the randomtree is restricted to improve the sampling efficiency.Secondly,to obtain better information about obstacles to shorten the path length,a feedback-biased sampling strategy is used instead of the traditional random sampling,the collision of the expanding node with an obstacle generates feedback information so that the next expanding node avoids expanding within a specific angle range.Thirdly,this paper proposes using the inverse optimization strategy to remove redundancy points from the initial path,making the path shorter and more accurate.Finally,to satisfy the smooth operation of the robot in practice,auxiliary points are used to optimize the cubic Bezier curve to avoid path-crossing obstacles when using the Bezier curve optimization.The experimental results demonstrate that,compared to the traditional RRT algorithm,the proposed FS-RRT algorithm performs favorably against mainstream algorithms regarding running time,number of search iterations,and path length.Moreover,the improved algorithm also performs well in a narrow obstacle environment,and its effectiveness is further confirmed by experimental verification.

Influence of sampling on three-dimensional surface shape measurement

In order to accurately measure an object’s three-dimensional surface shape,the influence of sampling on it was studied.First,on the basis of deriving spectra expressions through the Fourier transform,the generation of CCD pixels was analyzed,and its expression was given.Then,based on the discrete expression of deformation fringes obtained after sampling,its Fourier spectrum expression was derived,resulting in an infinitely repeated"spectra island"in the frequency domain.Finally,on the basis of using a low-pass filter to remove high-order harmonic components and retaining only one fundamental frequency component,the inverse Fourier transform was used to reconstruct the signal strength.A method of reducing the sampling interval,i.e.,reducing the number of sampling points per fringe,was proposed to increase the ratio between the sampling frequency and the fundamental frequency of the grating.This was done to reconstruct the object’s surface shape more accurately under the condition of m>4.The basic principle was verified through simulation and experiment.In the simulation,the sampling intervals were 8 pixels,4 pixels,2 pixels,and 1 pixel,the maximum absolute error values obtained in the last three situations were 88.80%,38.38%,and 31.50%in the first situation,respectively,and the corresponding average absolute error values are 71.84%,43.27%,and 32.26%.It is demonstrated that the smaller the sampling interval,the better the recovery effect.Taking the same four sampling intervals in the experiment as in the simulation can also lead to the same conclusions.The simulated and experimental results show that reducing the sampling interval can improve the accuracy of object surface shape measurement and achieve better reconstruction results.