The efficient compiling of arbitrary single qubit gates into a sequence of gates from an inverse-closed finite gate set is of fundamental importance in quantum computation. The exact bounds of this compiling are given by the Solovay-Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse-closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less-efficient. This was recently resolved by a version of the Solovay-Kitaev theorem for inverse-free gate sets, yielding a significant gain.

Considering the recent progress in the direction of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay-Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse-closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work we answer this positively, by obtaining improved Solovay-Kitaev approximations to an arbitrary orthogonal qutrit gate, to an accuracy ε from a sequence of O(log^8.62(1/ε)) orthogonal gates taken from an inverse-free set.

This paper deals with a queuing system with general renewal arrivals, exponential service times, single service channel and infinite number of waiting positions, customers are serviced in the order of their arrival. In the stationary case a new form of the probability generating functions of the number of clients in the system is derived. This new form is written in terms of the probability generating functions of the tail distribution function of the number of customers per group and of the probability generating functions of a embedded discrete time homogeneous Markov chain.

” Quasar” or” Quasi-stellar object” astronomical star-like object having a large ultraviolet flux of radiation accompanied by generally broad emission lines and absorption lines in some cases found at large redshift. Some of the quasars (10 %) are radio loud. The luminosity increases with redshift up to z = 2 after which it slows down and there is a decline toward higher redshifts. The data set we are working on is extracted from Veron Cetti Catalogue of AGN and Quasar (13th Edition). Consisting of Parameters like Color indexes redshift, absolute magnitude, and magnitude. The dimension of this data set is 168940 x 13. The objective of this work is to partition the quasar based on their spectral properties (their absolute Magnitude, redshift, and Color Indexes of luminosity) and classify them with respect to the obtained clusters. To achieve our objectives, we have considered the K-means partitioning method where the optimal number of Clusters is determined by Three methods which are” The Dunn Index”,” Elbow plot”,” Silhouette Method”, Evaluating the Three Mentioned Algorithm we came to the conclusion of considering 2 optimal clusters to Carry forward our analysis. The k-means clustering algorithm, considering two optimal clusters, yields two distinctive clusters of sizes ”44468”, ”124472”. Based on the obtained clusters we applied classification techniques,” Xgboost”,” Linear Discriminant Analysis” for identifying the percentage misclassification. The Xgboost and LDA evaluates a misclassification of around 26% Respectively. So, it is safe to infer that the miss classification rate is around 26 percent for our partitioning.

The Farlie-Gumbel-Morgenstern (FGM) is a flexible family that has been widely used to model weak dependence between the random variables. The family was introduced by Morgenstern (1956), Gumbel (1960) and Farlie (1960). Characteristic properties of FGM family, see Nelsen (1999). In the literature, we can see some studies that investigate the FGM family when marginals have uniform, normal, exponential, logistic or gamma distributions, see Bekci and Bairamov (1999), Scaria and Nair (1999), Smith and Moffatt (1999), Abo-Eleneen and Nagaraja (2002), Ucer and Yildiz (2012), Yildiz and Ucer (2017). Balakrishnan and Lai (2009) provide different bivariate distributions in the FGM family.

Ranked set sampling (RSS) was introduced by McIntyre (1952) as cost effective sampling method. One of its useful properties is that it can be used to collect data when exact measurements of the sampling units are either difficult and/or expensive. In the literature, it can be seen many applications of RSS in areas such as environmental, ecological, agricultural, biological and medical. Using RSS and its modifications, some authors dealt with the estimation of the dependence parameter. Stokes (1980) investigated maximum likelihood (ML) estimator of the association parameter for bivariate normal distribution. Modarres and Zheng (2004) suggested ML estimator of dependence parameter of bivariate normal distribution and bivariate extreme value distribution using RSS. Estimation of the association parameter based on bivariate RSS proposed by Al-Saleh and Samawi (2005) when the ranked set samples are obtained from bivariate normal distribution. Recently, Sevil and Yildiz (2022) considered the estimation problem of the association parameter for Gumbel (Type Ⅱ) bivariate exponential distribution.

In the present work, we consider FGM type bivariate gamma distribution that is developed by D’Este (1981) and Gupta and Wong (1984). We investigate maximum likelihood (ML) estimators based on simple random sampling (SRS) and RSS. We provide algorithms to generate sampling data from FGM type bivariate gamma distribution. Finally, we present biases and efficiency of the ML estimator based on RSS with respect to the ML estimator based on SRS.

Keywords: Ranked set sampling; algorithm for sampling data; order statistics; dependence parameter; FGM family; maximum likelihood estimation

References

Abo-Eleneen Z, Nagaraja H (2002) Fisher information in an order statistic and its concomitant. Annals of the Institute of Statistical Mathematics 54:667-680.

Al-Saleh MF, Samawi HM (2005) Estimation of the correlation coefficient using bivariate ranked set sampling with application to the bivariate normal distribution. Communication in Statistics-Theory and Methods 34:875-889.

Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer-Verlag.

Bekci M, Bairamov I (1999) Concomitants of order statistics in FGM type bivariate uniform distributions. ISTATISTIK, Journal of Turkish Statistical Association 2:135-144.

D’Este GA (1981) A Morgenstern-type bivariate gamma distribution. Biometrika 68:339-340.

Farlie DJ (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47:307-323.

Gumbel EJ (1960) Bivariate exponential distributions. Journal of the American Statistical Association 55:698-707.

Gupta AK, Wong C (1984) On a Morgenstern-type bivariate gamma distribution. Metrika 31:327-332.

McIntyre (1952) A method for unbiased selective sampling, using ranked set sampling. Australian Journal of Agricultural Research 3:385-390.

Modarres R, Zheng G (2004) Maximum likelihood estimation of dependence parameter using ranked set sampling. Statistics & Probability Letters 68:315-323.

Morgenstern D (1956) Einfache Beispiele zweidimensionaler Verteilung. Mitteislingsblatt für Mathematische Statistik 8:234-235.

Nelsen RB (2007) An introduction to copulas. Springer Science & Business Media.

Scaria J, Nair U (1999) On concomitants of order statistics from Morgenstern family. Biometrical Journal 41:483-489.

Smith MD, Moffatt PG (1999) Fisher’s information on the correlation coefficient in bivariate logistic models. Australian & New Zealand Journal of Statistics 41:315-330.

Sevil YC, Yildiz TO (2022) Gumbel’s bivariate exponential distribution: estimation of the association parameter using ranked set sampling. Computational Statistics 37:1695-1726.

Stokes SL (1980) Inference on the correlation coefficient in bivariate normal populations from ranked set samples. Journal of the American Statistical Association 75:989-995.

Ucer BH, Yildiz TO (2012) Estimation and goodness-of-fit procedures for Farlie-Gumbel-Morgenstern bivariate copula of order statistics. Journal of Statistical Computation and Simulation 82:137-147.

Yildiz TO, Ucer BH (2017) Fisher information of dependence in progressive type Ⅱ censored order statistics and their concomitants. International Journal of Applied Mathematics & Statistics 56:1-10.

Financial management is important for construction sector as the construction companies contribute to the development of the country. Malaysia encourages the construction sector to develop an advanced infrastructure related to transport and housing. Financial management is a multi-criteria decision making (MCDM) problem since the companies have to consider multiple goals in order to achieve the optimal decision. Therefore, goal programming is proposed in financial management to solve optimization in MCDM problems. According to the past studies, there has been no comprehensive study conducted on the optimization and comparison among the construction companies with goal programming model. Thus, this study aims to propose a goal programming model to optimize and compare the financial management of listed construction companies in Malaysia for benchmarking purpose. Six goals of financial management, namely total assets, total liabilities, equity, profit, earnings and optimum management item of the construction companies are examined in this study. The results of this study show that the goal programming model is able to determine the optimal solution and goal achievement for each construction company. Besides, the target value can be further increased according to the optimal solution of goal programming model. This study provides insights to the listed construction companies in Malaysia to identify the potential improvement based on the benchmarking and optimal solution of goal programming model.

The univariate Toppleone distribution introduced by [12] with close forms of cumulative distribution function i.e [0, 1] is extended to unbounded limit called Log-Toppleone distribution, the shapes of the hazard function can be increase, decrease or constant, therefore, can serve as an alternative distribution to the gamma, Weibull and exponential distributions. Bivariate of this proposed distribution is introduced by joining probability density function using three distinct copulas. The MLE, IFM and Bayesian method of estimation were employed to estimate the parameters, the Plackett copula regarded as the best based on MLE and IFM method of estimation while Clayton copula regarded as the best using Bayesian method.

In this work, we study stochastic problems in acoustics as well as in linear elasticity in two dimensions. We first provide the appropriate variational formulation for the stochastic source Helmholtz equation. By using Wiener-chaos expansion for the stochastic source, we transform the stochastic problem in a hierarchy of deterministic boundary value problems. The latter will be applied in order to establish existence and uniqueness for the finite element approximate solution of the stochastic source Helmholtz equation. Further, following the same concept of decomposing a stochastic problem into an hierarchy of deterministic ones we present an elastic scattering Dirichlet problem where the medium density is a random variable. In particular we consider a modified Navier equation introducing the density with its Wiener-chaos expansion in a combination with the appropriate Wick product. We reconstruct our solution via Wiener-chaos expansion techniques and finally we give some useful results and conclusions.

We offer a new asymptotic expansion with explicit remainder estimate in the central limit theorem. The results obtained are essentially based on the ideas of the paper. We also present a more accurate estimation of the CLT-expansions remainder which is rigorously proved and backed up numerically. It is shown that our approach can be used for further refinement of allied asymptotic expansions

COVID-19 is one of the worst pandemics ever, it is spreading rapidly creating a health crisis around the world. This disease, which continues to seriously threaten human life, has caused more than 664 million confirmed cases and 6.7 million deaths worldwide.
In this work, we propose a discrete mathematical model to predict the evolution of the dynamics of Covid-19, calculating the base reproduction number R₀ and the equilibrium points, then we perform the stability analysis and the sensitivity analysis. Finally, we end with a numerical analysis.

As is known, the theory of quantum transitions in quantum mechanics is based on the convergence of a series of time-dependent perturbation theory. This series converges in atomic and nuclear physics. In molecular physics, the series of time-dependent perturbation theory converges only if the Born-Oppenheimer adiabatic approximation and the Franck-Condon principle are strictly observed. Obviously, in real molecular systems there are always at least small deviations from the adiabatic approximation. Within the framework of quantum mechanics, these deviations lead to singular dynamics of molecular quantum transitions. The only way to eliminate this singularity is to introduce chaos into the electron-nuclear dynamics of the transient state. As a result of the introduction of chaos, we no longer have quantum mechanics, but quantum-classical mechanics, in which the initial and final states are quantum in the adiabatic approximation, and the transient chaotic electron-nuclear(-vibrational) state is classical due to chaos, and the transitions themselves are no longer quantum, but quantum-classical [1,2]. This procedure for introducing chaos into the transient state was done in the simplest case of quantum-classical mechanics, namely, in the case of quantum-classical mechanics of elementary electron transfers in condensed media. Chaos is introduced by replacing the infinitely small imaginary additive in the energy denominator of the total Green's function of the "electron + nuclear environment" system with a finite value [1,2]. This chaos is called dozy chaos, and quantum-classical mechanics is also called dozy-chaos mechanics. The analytical results obtained in this new fundamental physical theory make it possible to explain a large number of experimental data, for example, on the shape of the optical bands of polymethine dyes and their aggregates [2]. The simplicity of the case of quantum-classical mechanics of elementary electron transfers in condensed media and the possibility of obtaining the corresponding analytical result are connected, in particular, with the possibility of neglecting local oscillations of nuclei and taking into account only non-local oscillations in the theory. There is another "simple" problem in the quantum-classical mechanics of complex physical systems, where a similar success in the application of analytical methods can be achieved. This problem is the problem of molecular collisions in gases, which has applications to monomolecular reactions at low pressures. If in the problem of elementary electron transfers in condensed media the electronic state changes significantly, then during such molecular collisions in gases, the electronic states of the molecules do not change, and it is only necessary to take into account the redistribution of vibrational energy between local vibrations in colliding polyatomic molecules. In this case, the transient chaotic state of the motion of nuclei that occurs during molecular collisions can be described by statistical methods based on the use of the microcanonical distribution for molecular collisions [3]. Whereas in the problem of elementary electron transfer in condensed media the singular dynamics of the transient state is damped by dozy chaos, in this statistical approach to molecular collisions in gases, the dynamics of energy redistribution between local vibrations in colliding polyatomic molecules is taken into account by separating all modes into active and passive modes. Active modes include low-frequency vibrational modes and rotational modes that rapidly exchange energy at the moment of collision. Passive modes include high-frequency vibrational modes, which are effectively included in the process of energy redistribution after the elementary act of molecular collision has already been completed. Analytical results are obtained for the distribution function of the probability of energy transfer in collisions of molecules (canonical distribution for molecular collisions), as well as for all moments of the N-th order of the distribution function, which have the form of certain polynomials of the N-th order.

[1] Egorov, V.V. Quantum–classical mechanics as an alternative to quantum mechanics in molecular and chemical physics. Heliyon Phys. 2019, 5, e02579-1–e02579-27.

[2] Egorov, V.V. Quantum–classical mechanics: Nano-resonance in polymethine dyes. Mathematics 2022, 10(9), 1443-1–1443-25.

[3] Lifshitz, E.M.; Pitaevskii, L.P. Physical Kinetics (Course of Theoretical Physics, Volume 10); Butterworth-Heinemann: Oxford, 2012.