New construction of mutually unbiased bases for odd-dimensional state space

We study the construction of mutually unbiased bases in Hilbert space for composite dimensions d which are not prime powers.We explore the results for composite dimensions which are true for prime power dimensions.We then provide a method for selecting mutually unbiased vectors from the eigenvectors of generalized Pauli matrices to construct mutually unbiased bases.In particular,we present four mutually unbiased bases in C^(15).

Study on the Prediction of Rice Blast Based on the Unbiased GM (1,1) Model

To create a new prediction model, the unbiased GM （1,1） model is optimized by the five-point slide method in this paper. Then, based on the occurrence areas of dce blast in Enshi District during 1995 -2004, the new model and unbiased GM （1, 1 ） model are applied to predict the occurrence areas of rice blast during 2005 -2010. Predicting outcomes show that the prediction accuracy of five-point unbiased sliding optimized GM （1, 1 ） model is higher than the unbiased GM （1,1） model. Finally, combined with the prediction results, the author provides some suggestion for Enshi District in the prevention and control of rice blast in 2010.

Robustness of Minimum Norm Quadratic Unbiased Estimator of Variance Under the General Linear Model

Necessary and sufficient conditions for equalities between a 2 y′(I-P Xx)y and minimum norm quadratic unbiased estimator of variance under the general linear model, where a 2 is a known positive number, are derived. Further, when the Gauss? Markov estimators and the ordinary least squares estimator are identical, a relative simply equivalent condition is obtained. At last, this condition is applied to an interesting example.

Equivalency and unbiasedness of grey prediction models

In order to deeply research the structure discrepancy and modeling mechanism among different grey prediction models, the equivalence and unbiasedness of grey prediction models are analyzed and verified. The results show that all the grey prediction models that are strictly derived from x^（0）（k） ＋az^（1）（k） = b have the identical model structure and simulation precision. Moreover, the unbiased simulation for the homogeneous exponential sequence can be accomplished. However, the models derived from dx^（1）/dt ＋ ax^（1）= b are only close to those derived from x^（0）（k） ＋ az^（1）（k） = b provided that ｜a｜ has to satisfy｜a｜ 0.1; neither could the unbiased simulation for the homogeneous exponential sequence be achieved. The above conclusions are proved and verified through some theorems and examples.

De-correlated unbiased sequential filtering based on best unbiased linear estimation for target tracking in Doppler radar

In target tracking applications,the Doppler measurement contains information of the target range rate,which has the potential capability to improve the tracking performance.However,the nonlinear degree between the measurement and the target state increases with the introduction of the Doppler measurement.Therefore,target tracking in the Doppler radar is a nonlinear filtering problem.In order to handle this problem,the Kalman filter form of best linear unbiased estimation(BLUE)with position measurements is proposed,which is combined with the sequential filtering algorithm to handle the Doppler measurement further,where the statistic characteristic of the converted measurement error is calculated based on the predicted information in the sequential filter.Moreover,the algorithm is extended to the maneuvering target tracking case,where the interacting multiple model(IMM)algorithm is used as the basic framework and the model probabilities are updated according to the BLUE position filter and the sequential filter,and the final estimation is a weighted sum of the outputs from the sequential filters and the model probabilities.Simulation results show that compared with existing approaches,the proposed algorithm can realize target tracking with preferable tracking precision and the extended method can achieve effective maneuvering target tracking.

Unbiased Estimation Formula of Unit Weight Standard Deviation in Regularization Solution

Regularization method is an effective method for solving ill\|posed equation. In this paper the unbiased estimation formula of unit weight standard deviation in the regularization solution is derived and the formula is verified with numerical case of 1 000 sample data by use of the typical ill\|posed equation, i.e. the Fredholm integration equation of the first kind.

Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W,?Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.

Unbiased parameter estimation of continuous-time system based on modulating functions with input and output white noises

An efficient unbiased estimation method is proposed for the direct identification of linear continuous-time system with noisy input and output measurements.Using the Gaussian modulating filters,by numerical integration,an equivalent discrete identification model which is parameterized with continuous-time model parameters is developed,and the parameters can be estimated by the least-squares (LS) algorithm.Even with white noises in input and output measurement data,the LS estimate is biased,and the bias is determined by the variances of noises.According to the asymptotic analysis,the relationship between bias and noise variances is derived.One equation relating to the measurement noise variances is derived through the analysis of the LS errors.Increasing the degree of denominator of the system transfer function by one,an extended model is constructed.By comparing the true value and LS estimates of the parameters between original and extended model,another equation with input and output noise variances is formulated.So,the noise variances are resolved by the set of equations,the LS bias is eliminated and the unbiased estimates of system parameters are obtained.A simulation example by comparing the standard LS with bias eliminating LS algorithm indicates that the proposed algorithm is an efficient method with noisy input and output measurements.

Unbiased Fuzzy Global Optimization of Morse Clusters with Short-Range Potential for N≤400

Global optimization of Morse clusters with shortrange potential is a great challenge.Here,we apply our recently developed unbiased fuzzy global optimization method to systematically study Morse clusters with the potential rangeρ=14 and the number of atoms N up to 400.All the putative global minima reported in the literature have been successfully reproduced with relatively high success ratios.Compared to the available results for N≤240 and several larger Morse clusters,new global minima(and local minima)with lower energies have been found out for N=164,175,188,193,194,197,239,246,260,318,and 389.Clusters with magic numbers are figured out through fitting the size-dependent global minimum energies.The cluster structures tend to be close-packed for short-range potential with large N.

Autonomous Unbiased Study Group Formation Algorithm for Rapid Knowledge Propagation

Knowledge propagation is a necessity,both in academics and in the industry.The focus of this work is on how to achieve rapid knowledge propaga-tion using collaborative study groups.The practice of knowledge sharing in study groupsfinds relevance in conferences,workshops,and class rooms.Unfortu-nately,there appears to be only few researches on empirical best practices and techniques on study groups formation,especially for achieving rapid knowledge propagation.This work bridges this gap by presenting a workflow driven compu-tational algorithm for autonomous and unbiased formation of study groups.The system workflow consists of a chronology of stages,each made of distinct steps.Two of the most important steps,subsumed within the algorithmic stage,are the algorithms that resolve the decisional problem of number of study groups to be formed,as well as the most effective permutation of the study group participants to form collaborative pairs.This work contributes a number of new algorithmic concepts,such as autonomous and unbiased matching,exhaustive multiplication technique,twisted round-robin transversal,equilibrium summation,among others.The concept of autonomous and unbiased matching is centered on the constitution of study groups and pairs purely based on the participants’performances in an examination,rather than through any external process.As part of practical demon-stration of this work,study group formation as well as unbiased pairing were fully demonstrated for a collaborative learning size of forty(40)participants,and partially for study groups of 50,60 and 80 participants.The quantitative proof of this work was done through the technique called equilibrium summation,as well as the calculation of inter-study group Pearson Correlation Coefficients,which resulted in values higher than 0.9 in all cases.Real life experimentation was carried out while teaching Object-Oriented Programming to forty(40)under-graduates between February and May 2021.Empirical result showed that the per-formance of the learners was improved appreciably.This work will therefore be of immense benefit to the industry,academics and research community involved in collaborative learning.

ASP-Based Programs of Best Linear Unbiased Prediction-Estimated Breeding Values in Breeding Stock

In order to improve the breeding effect of livestock, the data were read from an Excel file with Active Server Page （ASP） programs, and the breeding values of breeding stock were calculated by best linear unbiased prediction （BLUP） method.

Asymptotic Expansion for the Distribution of a Median Unbiased Estimator of ARCH（0,1） Coefficient

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH（0,1） coefficient. The exact distribution of the estimator can be easily derived, however its practical calculations are too heavy to implement, even though the middle range of sample sizes. Since the estimator is shown to have asymptotic normality, asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements. Accuracies of expansion formulas are evaluated numerically, and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations. Derived expansion are applied to testing hypothesis of stationarity, and an implementation for a real data set is illustrated.

A Geometric Approach to Conditioning and the Search for Minimum Variance Unbiased Estimators

Our purpose is twofold: to present a prototypical example of the conditioning technique to obtain the best estimator of a parameter and to show that this technique resides in the structure of an inner product space. The technique uses conditioning of an unbiased estimator on a sufficient statistic. This procedure is founded upon the conditional variance formula, which leads to an inner product space and a geometric interpretation. The example clearly illustrates the dependence on the sampling methodology. These advantages show the power and centrality of this process.

On the Restricted Almost Unbiased Ridge Estimator in Logistic Regression

In this article, the restricted almost unbiased ridge logistic estimator (RAURLE) is proposed to estimate the parameter in a logistic regression model with exact linear re-strictions when there exists multicollinearity among explanatory variables. The performance of the proposed estimator over the maximum likelihood estimator (MLE), ridge logistic estimator (RLE), almost unbiased ridge logistic estimator (AURLE), and restricted maximum likelihood estimator (RMLE) with respect to different ridge parameters is investigated through a simulation study in terms of scalar mean square error.

Theorem of Logarithm Expectation and Its Application to Prove Sample Correlation Coefficient as Unbiased Estimate

In statistical theory, a statistic that is function of sample observations is used to estimate distribution parameter. This statistic is called unbiased estimate if its expectation is equal to theoretical parameter. Proving whether or not a statistic is unbiased estimate is very important but this proof may require a lot of efforts when statistic is complicated function. Therefore, this research facilitates this proof by proposing a theorem which states that the expectation of variable x 〉 0 is u if and only if the limit of logarithm expectation of x approaches logarithm of u. In order to make clear of this theorem, the research gives an example of proving correlation coefficient as unbiased estimate by taking advantages of this theorem.

Unbiased Reporting Required

Global media has recently reported on several negative incidents emerging from Chinese enterprises investing in Africa. Li Anshan, Director of the Center for African Studies at peking University, noted in a commentary written for ChinAfrica that the incidents, on the

Security of quantum key distribution with virtual mutually unbiased bases

In a perfect quantum key distribution(QKD)protocol,quantum states should be prepared and measured with mutually unbiased bases(MUBs).However,in a practical QKD system,quantum states are generally prepared and measured with imperfect MUBs using imperfect devices,possibly reducing the secret key rate and transmission distance.To analyze the security of a QKD system with imperfect MUBs,we propose virtual MUBs to characterize the quantum channel against collective attack,and analyze the corresponding secret key rate under imperfect state preparation and measurement conditions.More generally,we apply the advantage distillation method for analyzing the security of QKD with imperfect MUBs,where the error tolerance and transmission distance can be sharply improved.Our analysis method can be applied to benchmark and standardize a practical QKD system,elucidating the security analysis of different QKD protocols with imperfect devices.

Best Linear Unbiased Estimators of Location and Scale Ranked Set Parameters under Moving Extremes Sampling Design

In the current paper,the best linear unbiased estimators(BLUEs)of location and scale parameters from location-scale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling(MERSS).Explicit mathematical expressions of these estimators and their variances are derived.Their relative efficiencies with respect to the minimum variance unbiased estimators(MVUEs)under simple random sampling(SRS)are compared for the cases of some usual distributions.The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.

Experimental investigation of measurement incompatibility of mutually unbiased bases

Incompatible measurements are of fundamental importance to re-vealing the peculiar features of quantum theory,and are also use-ful resources in various quantum information tasks.In this work,we investigate the quantum incompatibility of mutually unbiased bases(MUBs)within the operational framework of quantum resource the-ory,and report an experimental validation via the task of state dis-crimination.In particular,we construct an experimentally friendly witness to detect incompatible MUBs,based on the probability of cor-rectly discriminating quantum states.Furthermore,we prove that the noise robustness of MUBs can be retrieved from violating the above witness.Finally,we experimentally test the incompatibility of MUBs of dimensionality ranging from 2 to 4,and demonstrate that it is more robust to noise,as either the dimensionality of measurements or the number of MUBs increases.Our results may aid the exploration of the essential roles of incompatible measurements in both theoretical and practical applications in quantum information.

Robustness of Minimum Norm Quadratic Unbiased Estimator of Variance for the General Linear Model

In this paper, necessary and sufficient conditions for equalities betweenα~2y^1(I-P_X)y and under the general linear model, whereand α~2 is a known positive number, are derived. Furthermore, when the Gauss-Markovestimators and the ordinary least squares estimators are identical, we obtain a simpleequivalent condition.