A Method for the Squaring of a Circle

This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm2, it produced a square having an area of 12.7 cm2, using only an unmarked ruler and a compass. This result was a clear demonstration that not only is the construction valid for the squaring of a circle but also for achieving absolute results (independent of the number pi (π) and in a finite number of steps) when carried out with precision.

A Procedure for the Squaring of a Circle (of Any Radius)

This paper presents a graphical procedure for the squaring of a circle of any radius. This procedure, which is based on a novel application of the involute profile, when applied to a circle of arbitrary radius (using only an unmarked ruler and a compass), produced a square equal in area to the given circle, which is 50 cm2. This result was a clear demonstration that not only is the construction valid for the squaring of a circle of any radius, but it is also capable of achieving absolute results (independent of the number pi (π), in a finite number of steps), when carried out with precision.

Squaring the Circle Is Possible When Taking into Consideration the Heisenberg Uncertainty Principle

Squaring the circle is one of the oldest challenges in mathematical geometry. In 1882, it was proven that π was transcendental, and the task of squaring the circle was considered impossible. Demonstrating that squaring the circle was not possible took place before discovering quantum mechanics. The purpose of this paper is to show that it is actually possible to square the circle when taking into account the Heisenberg uncertainty principle. The conclusion is clear: it is possible to square the circle when taking into account the Heisenberg uncertainty principle.

Squaring the Dialectic of Inference and Chance

Instead of merely inferring that our beliefs are in a correlation to the facts, I will contend that what matters is the logic of their inferential structure. This will reveal what is actually true for their relation and not just real for experience. Given one and two-sided inferences, the conditions of their functional relation will be consistent within a square of opposition; for contraries, contradictions and sub-contraries in a way that probability theory is not. Within this model of inferences, the two-sided concept of inferences will also have a modal value for its relation to the one-sided concept of inferences. Proposing that empirical, formal, speculative, and an Intuitionist Speculative understanding of logic will have a modal identity within a dialectical principle.

A Synthesis of Multi-Precision Multiplication and Squaring Techniques for 8-Bit Sensor Nodes＂ State-of-the-Art Research and Future Challenges

Multi-precision multiplication and squaring are the performance-critical operations for the implementation of public-key cryptography, such as exponentiation in RSA, and scalar multiplication in elliptic curve cryptography （ECC）. In this paper, we provide a survey on the multi-precision multiplication and squaring techniques, and make special focus on the comparison of their performance and memory footprint on sensor nodes using 8-bit processors, Different from the previous work, our advantages are in at least three aspects. Firstly, this survey includes the existing techniques for multi- precision multiplication and squaring on sensor nodes over prime fields. Secondly, we analyze and evaluate each method in a systematic and objective way. Thirdly, this survey also provides suggestions for selecting appropriate multiplication and squaring techniques for concrete implementation of public-key cryptography. At the end of this survey, we propose the research challenges on efficient implementation of the multiplication and the squaring operations based on our observation.

On Quadratic Wasserstein Metric with Squaring Scaling for Seismic Velocity Inversion

The quadraticWasserstein metric has shown its power in comparing prob-ability densities.It is successfully applied in waveform inversion by generating ob-jective functions robust to cycle skipping and insensitive to data noise.As an alter-native approach that converts seismic signals to probability densities,the squaring scaling method has good convexity and thus is worth exploring.In this work,we apply the quadratic Wasserstein metric with squaring scaling to regional seismic to-mography.However,there may be interference between different seismic phases in a broad time window.The squaring scaling distorts the signal by magnifying the unbalance of the mass of different seismic phases and also breaks the linear super-position property.As a result,illegal mass transportation between different seismic phases will occur when comparing signals using the quadratic Wasserstein metric.Furthermore,it gives inaccurate Fr´echet derivative,which in turn affects the inver-sion results.By combining the prior seismic knowledge of clear seismic phase sep-aration and carefully designing the normalization method,we overcome the above problems.Therefore,we develop a robust and efficient inversion method based on optimal transport theory to reveal subsurface velocity structures.Several numerical experiments are conducted to verify our method.

融合IMR-WGAN的时序数据修复方法

工业数据由于技术故障和人为因素通常导致数据异常,现有基于约束的方法因约束阈值设置的过于宽松或严格会导致修复错误,基于统计的方法因平滑修复机制导致对时间步长较远的异常值修复准确度较低.针对上述问题,提出了基于奖励机制的最小迭代修复和改进WGAN混合模型的时序数据修复方法.首先,在预处理阶段,保留异常数据,进行信息标注等处理,从而充分挖掘异常值与真实值之间的特征约束.其次,在噪声模块提出了近邻参数裁剪规则,用于修正最小迭代修复公式生成的噪声向量.将其传递至模拟分布模块的生成器中,同时设计了一个动态时间注意力网络层,用于提取时序特征权重并与门控循环单元串联组合捕捉不同步长的特征依赖,并引入递归多步预测原理共同提升模型的表达能力;在判别器中设计了Abnormal and Truth奖励机制和Weighted Mean Square Error损失函数共同反向优化生成器修复数据的细节和质量.最后,在公开数据集和真实数据集上的实验结果表明,该方法的修复准确度与模型稳定性显著优于现有方法.

Simple Formulas of πin Terms of Φ

The paper presents a novel exploration of π through a re-calculation of formulas using Archimedes’ algorithm, resulting in the identification of a general family equation and three new formulas involving the golden ratio Φ, in the form of infinite nested square roots. Some related geometrical properties are shown, enhancing the link between the circle and the golden ratio. Applying the same criteria, a fourth formula is given, that brings to the known Dixon’s squaring the circle approximation, thus an easier approach to this problem is suggested, by a rectangle with both sides proportional to the golden ratio Φ.

Underwater four-quadrant dual-beam circumferential scanning laser fuze using nonlinear adaptive backscatter filter based on pauseable SAF-LMS algorithm

The phenomenon of a target echo peak overlapping with the backscattered echo peak significantly undermines the detection range and precision of underwater laser fuzes.To overcome this issue,we propose a four-quadrant dual-beam circumferential scanning laser fuze to distinguish various interference signals and provide more real-time data for the backscatter filtering algorithm.This enhances the algorithm loading capability of the fuze.In order to address the problem of insufficient filtering capacity in existing linear backscatter filtering algorithms,we develop a nonlinear backscattering adaptive filter based on the spline adaptive filter least mean square(SAF-LMS)algorithm.We also designed an algorithm pause module to retain the original trend of the target echo peak,improving the time discrimination accuracy and anti-interference capability of the fuze.Finally,experiments are conducted with varying signal-to-noise ratios of the original underwater target echo signals.The experimental results show that the average signal-to-noise ratio before and after filtering can be improved by more than31 d B,with an increase of up to 76%in extreme detection distance.

Incorporating empirical knowledge into data-driven variable selection for quantitative analysis of coal ash content by laser-induced breakdown spectroscopy

Laser-induced breakdown spectroscopy(LIBS)has become a widely used atomic spectroscopic technique for rapid coal analysis.However,the vast amount of spectral information in LIBS contains signal uncertainty,which can affect its quantification performance.In this work,we propose a hybrid variable selection method to improve the performance of LIBS quantification.Important variables are first identified using Pearson's correlation coefficient,mutual information,least absolute shrinkage and selection operator(LASSO)and random forest,and then filtered and combined with empirical variables related to fingerprint elements of coal ash content.Subsequently,these variables are fed into a partial least squares regression(PLSR).Additionally,in some models,certain variables unrelated to ash content are removed manually to study the impact of variable deselection on model performance.The proposed hybrid strategy was tested on three LIBS datasets for quantitative analysis of coal ash content and compared with the corresponding data-driven baseline method.It is significantly better than the variable selection only method based on empirical knowledge and in most cases outperforms the baseline method.The results showed that on all three datasets the hybrid strategy for variable selection combining empirical knowledge and data-driven algorithms achieved the lowest root mean square error of prediction(RMSEP)values of 1.605,3.478 and 1.647,respectively,which were significantly lower than those obtained from multiple linear regression using only 12 empirical variables,which are 1.959,3.718 and 2.181,respectively.The LASSO-PLSR model with empirical support and 20 selected variables exhibited a significantly improved performance after variable deselection,with RMSEP values dropping from 1.635,3.962 and 1.647 to 1.483,3.086 and 1.567,respectively.Such results demonstrate that using empirical knowledge as a support for datadriven variable selection can be a viable approach to improve the accuracy and reliability of LIBS quantification.

Natural Convection and Irreversibility of Nanofluid Due to Inclined Magnetohydrodynamics(MHD)Filled in a Cavity with Y-Shape Heated Fin:FEM Computational

This study explains the entropy process of natural convective heating in the nanofluid-saturated cavity in a heated fin andmagnetic field.The temperature is constant on the Y-shaped fin,insulating the topwall while the remaining walls remain cold.All walls are subject to impermeability and non-slip conditions.The mathematical modeling of the problem is demonstrated by the continuity,momentum,and energy equations incorporating the inclined magnetic field.For elucidating the flow characteristics Finite ElementMethod(FEM)is implemented using stable FE pair.A hybrid fine mesh is used for discretizing the domain.Velocity and thermal plots concerning parameters are drawn.In addition,a detailed discussion regarding generation energy by monitoring changes in magnetic,viscous,total,and thermal irreversibility is provided.In addition,line graphs are created for the u and v components of the velocity profile to predict the flow behavior.Current simulations assume the dimensionless representative of magnetic field Hartmann number Ha between 0 and 100 and a magnetic field inclination between 0 and 90 degrees.A constant 4% volume proportion of nanoparticles is employed throughout all scenarios.

A Novel Fractional Dengue Transmission Model in the Presence of Wolbachia Using Stochastic Based Artificial Neural Network

The purpose of this research work is to investigate the numerical solutions of the fractional dengue transmission model(FDTM)in the presence of Wolbachia using the stochastic-based Levenberg-Marquardt neural network(LM-NN)technique.The fractional dengue transmission model(FDTM)consists of 12 compartments.The human population is divided into four compartments;susceptible humans(S_(h)),exposed humans(E_(h)),infectious humans(I_(h)),and recovered humans(R_(h)).Wolbachia-infected and Wolbachia-uninfected mosquito population is also divided into four compartments:aquatic(eggs,larvae,pupae),susceptible,exposed,and infectious.We investigated three different cases of vertical transmission probability(η),namely when Wolbachia-free mosquitoes persist only(η=0.6),when both types of mosquitoes persist(η=0.8),and when Wolbachia-carrying mosquitoes persist only(η=1).The objective of this study is to investigate the effectiveness of Wolbachia in reducing dengue and presenting the numerical results by using the stochastic structure LM-NN approach with 10 hidden layers of neurons for three different cases of the fractional order derivatives(α=0.4,0.6,0.8).LM-NN approach includes a training,validation,and testing procedure to minimize the mean square error(MSE)values using the reference dataset(obtained by solving the model using the Adams-Bashforth-Moulton method(ABM).The distribution of data is 80% data for training,10% for validation,and,10% for testing purpose)results.A comprehensive investigation is accessible to observe the competence,precision,capacity,and efficiency of the suggested LM-NN approach by executing the MSE,state transitions findings,and regression analysis.The effectiveness of the LM-NN approach for solving the FDTM is demonstrated by the overlap of the findings with trustworthy measures,which achieves a precision of up to 10^(-4).

Algorithms and statistical analysis for linear structured weighted total least squares problem

Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.

Consistent and Accurate TDOA Localization in Dynamic NLOS Environments via Prior Information Independent Estimation

In the time-difference-of-arrival(TDOA)localization,robust least squares(LS)problems solved by mathematical programming were proven to be superior in mitigating the effects of non-line-of-sight(NLOS)propagation.However,the existing algorithms still suffer from two disadvantages:1)The algorithms strongly depend on prior information;2)The approaches do not satisfy the mean square error(MSE)optimal criterion of the measurement noise.To tackle the troubles,we first formulate an MSE minimization model for measurement noise by taking the source and the NLOS biases as variables.To obtain stable solutions,we introduce a penalty function to avoid abnormal estimates.We further tackle the nonconvex locating problem with semidefinite relaxation techniques.Finally,we incorporate mixed constraints and variable information to improve the estimation accuracy.Simulations and experiments show that the proposed method achieves consistent performance and good accuracy in dynamic NLOS environments.

Numerical simulation of inlet placement on sewage characteristics in the rounded square aquaculture tank with single inlet

To improve the self-cleaning ability of aquaculture tank and the efficiency of circulating water,physical and numerical experiments were conducted on the influence of inlet structure on sewage discharge in a rounded square aquaculture tank with a single inlet.Based on the physical model of the tank,analysis of how inlet structure adjustment affects sewage discharge efficiency and flow field characteristics was conducted to provide suitable flow field conditions for sinkable solid particle discharge.In addition,an internal flow field simulation was conducted using the RNG k-εturbulence model in hydraulic drive mode.Then a solid-fluid multiphase model was created to investigate how the inlet structure affects sewage collection in the rounded square aquaculture tank with single inlet and outlet.The finding revealed that the impact of inlet structure is considerably affecting sewage collection.The conditions of C/B=0.07-0.11(the ratio of horizontal distance between the center of the inlet pipe and the tank wall(C)to length of the tank(B))andα=25°(αis the angle between the direction of the jet and the tangential direction of the arc angle)resulted in optimal sewage collection,which is similar to the flow field experiment in the rounded square aquaculture tank with single inlet and outlet.An excellent correlation was revealed between sewage collection and fluid circulation stability in the aquaculture tank.The present study provided a reference for design and optimization of circulating aquaculture tanks in aquaculture industry.

Ecoregional height-diameter models for Scots pine in Turkiye

Ecoregion-based height-diameter models were developed in the present study for Scots pine(Pinus sylves-tris L.)stands in Turkiye and included several ecological factors derived from a pre-existing ecoregional classification system.The data were obtained from 2831 sample trees in 292 sample plots.Ten generalized height–diameter models were developed,and the best model(HD10)was selected according to statistical criteria.Then,nonlinear mixed-effects modeling was applied to the best model.The R2 for the generalized height‒diameter model(Richards function)modified by Sharma and Parton is 0.951,and the final model included number of trees,dominant height,and diameter at breast height,with a random parameter associated with each ecoregion attached to the inverse of the mean basal area.The full model predictions using the nonlinear mixed-effects model and the reduced model(HD10)predictions were compared using the nonlinear sum of extra squares test,which revealed significant differences between ecore-gions;ecoregion-based height–diameter models were thus found to be suitable to use.In addition,using these models in appropriate ecoregions was very important for achieving reliable predictions with low prediction errors.

滑坡涌浪产生过程中动力学效应对涌浪规模的影响规律

为厘清滑坡涌浪产生过程中的力学效应对涌浪规模的影响,基于体积抬升(LU)、冲击挤压(PA)和界面摩擦(DA)耦合作用下的滑坡涌浪产生过程动力学模型,补充阐述体积抬升效应修正模型及其应用范围,以三维散粒体滑坡涌浪物理模型试验中的若干工况为研究对象,结合Tsunami Squares数值模拟方法,分析LU、PA和DA 3种力学效应分别对涌浪波幅和波能的贡献,揭示滑坡涌浪产生过程中各力学效应的特征和作用机制。结果表明:当水深与块体边长的比值小于1/5时,可以忽略BT修正;PA较LU对水体加速效果更显著,PA和DA对首浪波高的贡献率分别为达到29%和9%,且均随着传播距离的增大而降低;PA对涌浪势能、动能和总能量的贡献分别为23%~30%、28%~29%和25%~30%,DA对3种能量的贡献分别为17%~25%、14%~16%和16%~23%,剩余能量由LU&BT贡献。研究结果揭示了滑坡涌浪产生的动力学机理,为滑坡涌浪灾害预测评价提供动力学模型和参数选取依据。

A Deep Residual PLS for Data-Driven Quality Prediction Modeling in Industrial Process

Partial least squares(PLS)model is the most typical data-driven method for quality-related industrial tasks like soft sensor.However,only linear relations are captured between the input and output data in the PLS.It is difficult to obtain the remaining nonlinear information in the residual subspaces,which may deteriorate the prediction performance in complex industrial processes.To fully utilize data information in PLS residual subspaces,a deep residual PLS(DRPLS)framework is proposed for quality prediction in this paper.Inspired by deep learning,DRPLS is designed by stacking a number of PLSs successively,in which the input residuals of the previous PLS are used as the layer connection.To enhance representation,nonlinear function is applied to the input residuals before using them for stacking highlevel PLS.For each PLS,the output parts are just the output residuals from its previous PLS.Finally,the output prediction is obtained by adding the results of each PLS.The effectiveness of the proposed DRPLS is validated on an industrial hydrocracking process.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Characteristics of laser-induced breakdown spectroscopy of liquid slag

Rapid online analysis of liquid slag is essential for optimizing the quality and energy efficiency of steel production. To investigate the key factors that affect the online measurement of refined slag using laser-induced breakdown spectroscopy(LIBS), this study examined the effects of slag composition and temperature on the intensity and stability of the LIBS spectra. The experimental temperature was controlled at three levels: 1350℃, 1400℃, and 1450℃. The results showed that slag composition and temperature significantly affected the intensity and stability of the LIBS spectra. Increasing the Fe content and temperature in the slag reduces its viscosity, resulting in an enhanced intensity and stability of the LIBS spectra. Additionally, 42 refined slag samples were quantitatively analyzed for Fe, Si, Ca, Mg, Al, and Mn at 1350℃, 1400℃, and 1450℃.The normalized full spectrum combined with partial least squares(PLS) quantification modeling was used, using the Ca Ⅱ 317.91 nm spectral line as an internal standard. The results show that using the internal standard normalization method can significantly reduce the influence of spectral fluctuations. Meanwhile, a temperature of 1450℃ has been found to yield superior results compared to both 1350℃ and 1400℃, and it is advantageous to conduct a quantitative analysis of the slag when it is in a “water-like” state with low viscosity.