Stability criteria for certain classes of nonlinear difference equations

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.     

Event-triggered H∞ state estimation for state-saturated complex networks subject to quantization effects and distributed delays

This paper is concerned with the event-triggered H_∞ state estimation problem for a class of discrete-time complex networks subject to state saturations, quantization effects as well as randomly occurring distributed delays. A series of Bernoulli distributed random variables is utilized to model the random occurrence of distributed delays. For the energy-saving purpose, an event-triggered mechanism is proposed to decide whether the current quantized measurement should be transmitted to the estimator or not. For the state-saturated complex networks, our aim is to design event-triggered state estimators that guarantee both the exponential mean-square stability of and the H_∞ performance constraint on the error dynamics of the state estimation. Stochastic analysis is conducted, in combination with the Lyapunov functional approach, to derive sufficient conditions for the existence of the desired estimators whose gain matrices are obtained by solving a set of matrix inequalities. An illustrative example is exploited to show the usefulness of the estimator design algorithm proposed.     

On Systematic Procedures for Constructing Magic Squares

In the first part of the paper, a systematic procedure for constructing high-order magic squares as an extension of the lower-order basic magic squares is developed and demonstrated. For a 2N x 2N magic square, one can start with a basic N x N magic square, say, N = 3,4,5 or 7,11,13. Using small 2 x 2 squares, like (1,2,3,4) or (5,6,7,8), to fill the elements of the basic N x N magic square according to the ordering of the magic square, one develops the 2N x 2N magic square automatically. For a 3N x 3N magic square, one can choose another basic N x N magic square, say, the 4 x 4 magic square. Then, inserting 3 x 3 small squares in the 16 empty spaces according to the ordering of the 4 x 4 magic square, one develops the 12 x 12 magic square. There may be small adjustments in the numbering of (1,2,3,4) in the 2 x 2 small square, or of (1,2,3,4,5,6,7,8,9) in the 3 x 3 small square to form the proper 2N x 2N or 3N x 3N magic squares. In most cases, the correct results are obtained without adjustments. For N = 4, 4N x 4N produces the 16 x 16 magic square. For N = 5 or 7, using 2 x 2 small squares produces 10 x 10 or 14 x 14 magic squares. Part II of the paper gives a review of other known procedures for constructing odd and even order magic squares with illustrative examples for 4 x 4, 5 x 5, 7 x 7, 10 x 10, 11 x 11, 13 x 13, 15 x 15, 17 x 17, 19 x 19 and 20 x 20 magic squares. One 18 x 18 magic square is given in Part I. In the Appendix, the G and H squares, given by Moschopulus of Constantinople nearly five centuries ago, are discussed concerning symmetrical and pandiagonal squares. It is shown that H can be modified to form H₁ which is both symmetrical and pandiagonal.     

Processing truncated terms in document retrieval systems

In a typical inverted-file full-text document retrieval system, the user submits queries consisting of strings of characters combined by various operators. The strings are looked up in a text-dictionary which lists, for each string, all the places in the database at which it occurs. It is desirable to allow the user to include in his query truncated terms such as X¹, ¹X, ¹X¹, or X¹Y, where X and X are specified strings and ¹ is a variable-length-don't-care character, that is, ¹ represents an arbitrary, possibly empty, string. Processing these terms involves finding the set of all words in the dictionary that match these patterns. How to do this efficiently is a long-standing open problem in this domain.In this paper we present a uniform and efficient approach for processing all such query terms. The approach, based on a “permuted dictionary” and a corresponding set of access routines, requires essentially one disk access to obtain from the dictionary all the strings represented by a truncated term, with negligible computing time. It is thus well suited for on-line applications. Implementation is simple, and storage overhead is low: it can be made almost negligible by using some specially adapted compression techniques described in the paper.The basic approach is easily adaptable for slight variants, such as fixed (or bounded) length don't-care characters, or more complex pattern matching templates.     

Subspace design of low-rank estimators for higher-order statistics

Higher-order statistics (HOS) are well known for their robustness to additive Gaussian noise and ability to preserve phase. HOS estimates, on the other hand, have been criticized for high complexity and the need for long data in order to maintain small variance. Since rank reduction offers a general principle for reduction of estimator variance and complexity, we consider the problem of designing low-rank estimators for HOS. We propose three methods for choosing the transformation matrix that reduces the mean-square error (MSE) associated with the low-rank HOS estimates. We also demonstrate the advantages of using low-rank third-order moment estimates for blind system estimation. Results indicate that the full rank MSE corresponding to some data length N can be attained by a low-rank estimator corresponding to a length significantly smaller than N.     

The solution to matrix equation AX+XC=B

In this note, the problem of solution to the matrix equation AX+X^TC=B is considered by the Moore-Penrose generalized inverse matrix. A general solution to this equation is obtained. At the same time, some useful conclusions are made, which play important roles in the linear system theories and applications.     

A simplified stability test for 1-D discrete systems

This paper shows that the stability tests for 1-D discrete systems using the transformation p=(z+z⁻¹) and properties of Chebyshev polynomials developed previously can be directly obtained from the z-domain continued fraction expansion based on the functions (z+1) and (z⁻¹+1) on an alternate basis. Furthermore, it is shown that the root distribution of a polynomial with real coefficient can be determined by the same algorithm.     

The inverse lure problem of optimal control

Given the linear system $\text{x}\text{}$ = Ax - bu, y = c^Tx, it is shown that, for a certain non-quadratic cost functional, the optimal control is given by u_opt(x) = h(c^Tx), where the function h(y) must satisfy the conditions ky²?h(y)y>0 for y≠0, h(0) = 0 and existence of h^-1 everywhere. The linear system considered must satisfy the Popov condition 1/k + (1 +?ωβ) G(?ω)>0 for all ω, G(s) being the y(s)/u(s) transfer function.     

On the linear combination, product and ratio of normal and Laplace random variables

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this note, the exact distributions of αX+βY, |XY| and |X/Y| are derived when X and Y are independent normal and Laplace random variables.     

Least squares Hermitian solution of the complex matrix equation AXB+CXD=E with the least norm

In this paper, a new method is presented to solve the least squares Hermitian problem of the complex matrix equation AXB+CXD=E. The explicit expression of least squares Hermitian solution with the least norm is given. The least squares Hermitian solution with the least norm of complex matrix equation AXB=E is also derived. Numerical algorithms and numerical examples show the feasibility of our method.     

Light-inadequacy of the ultraviolet theory of ionization in the E-layer

From the reports of the CRPL, average yearly values of h′_E, h′_F1,f °_E and N_F1, for the months of January, March, June, July, September, and December of the years 1944 to 1949 were computed and plotted against the magnetic latitude. Altogether some 25,000 individual records were averaged. Seven curves were plotted. They showed that: (1) all curves are symmetrical with respect to the magnetic equator; (2) the h′_E, h′_F1, f⁰_E and the N_F, curves all swing up and down as the magnetic latitude, north and south, increases; (3) the h′_E is definitely periodic with constant period and constant amplitude; (4) the h′_E curve rises only slightly (on the average) with latitude. These results appear to be inconsistent with the ultraviolet light theory of ionization.     

A state estimation H∞ issue for discrete-time stochastic impulsive genetic regulatory networks in the presence of leakage,multiple delays and Markovian jumping parameters

In this work, we probes the stability results of H_∞ state estimation for discrete-time stochastic genetic regulatory networks with leakage, distributed delays, Markovian jumping parameters and impulsive effects. Here, we focus to evaluate the true absorption of mRNAs and proteins by calculating the H_∞ estimator in such a way that the estimation error dynamics is stochastically stable during the completion of the prescribed H_∞ disturbance attenuation level. In favor of decreasing the data communion in trouble, the H_∞ system accept and evaluate the outputs that are only transferred to the estimator when a certain case is acroses. Further, few sufficient conditions are formulated, by utilizing the Lyapunov–Krasovskii functional under which the estimation error system is stochastically stable and also satisfied the H_∞ attainment constraint. The estimator is obtained in terms of linear matrix inequalities (LMIs) and these LMIs are attainable, only if the estimator gains can be absolutely given. In addition to that, two numerical examples are exposed to establish the efficiency of our obtained results.     

Comparing the expression of human DNA topoisomerase I in KM71H and X33 strains of Pichia pastoris

BackgroundHuman is an essential cellular enzyme that is found in all human cells. As this enzyme is upregulated in cancer cells exceedingly, it is used as a target for cancer chemotherapeutic drug development. As such, producing the in-house enzyme for the purpose to speed up the search for more cost-effective and target specific hTopoI inhibitors is warranted. This study aims to compare the optimised conditions for the expression of hTopoI in KM71H (Mut^S) and X33 (Mut⁺) strains of Pichia pastoris. P. pastoris transfected with an hTopoI recombinant vector was used for the optimization of a higher level of hTopoI expression.ResultsIn the process, fed-batch cultivation parameters that influence the expression of hTopoI, such as culture temperature, methanol induction and feeding strategy, were optimised in the transfected KM71H and X33 P. pastoris strains in a shake flask system. The cell density and total protein concentration (protein level) of transfected P. pastoris were compared to determine the optimum culture conditions for each transfected P. pastoris strain. A higher hTopoI level was observed in the transfected KM71H culture supernatant (2.26 ng/mL) when the culture was incubated in the optimum conditions.ConclusionsThis study demonstrated that Mut^S strain (KM71H) expressed and secreted a higher level of hTopoI heterologous protein in the presence of methanol compared to the Mut⁺ strain; X33 (0.75 ng/mL). However, other aspects of optimization, such as pH, should also be considered in the future, to obtain the optimum expression level of hTopoI in P. pastoris.     

Role of prostaglandin in the regulation of gastric H+—Transporting system

Prostaglandins and (PG) have been reported to be an important gastric acid suppressive factor. However, the mechanism underlying is yet to be clearly established. In vitro study with gastric microsomes in presence of both PGE₂ and PGI₂ shows a stimulation of gastric H⁺ K⁺-ATPase activity below 1X10⁻⁶M and 2.5X10⁻⁷M concentrations respectively. However, with further increase in concentrations of both PGE₂ and PGI₂, H⁺, K⁺-ATPase activity shows an inhibition but PGI₂ completely obliterates the K⁺ stimulated part of H⁺, K⁺-ATPase activity at higher concentration. The H⁺-ion transport study using chambered frog gastric mucosa shows that both PGE₂ and PGI₂ inhibit H⁺-ion transport at 5X10⁻⁶ M and 10X10⁻⁶M concentrations respectively but the effect of PGI₂ is reversible. These differential effects of PGE₂ and PGI₂ on microsomal H⁺, K⁺-ATPase and on H⁺ transport my be caused by the differential effects of these phospholipid mediators with the gastric mucosal cell membrane. This in vitro investigation shows the role of prostaglandin (s) as a physiological switch/regulator of gastric H⁺ ion transport leading to the cessation of gastric acid secretion.     

Finite-horizon state estimation for time-varying complex networks with random coupling strengths under Round-Robin protocol

In this paper, the problem of finite-horizon H_∞ state estimation is investigated for a class of discrete time-varying complex networks with multiplicative noises and random coupling strengths. The network nodes and estimators are connected via a constrained communication network which allows only one node to send measurement data at each transmission instant. The Round-Robin protocol is introduced to determine which node obtains the access to the network at certain transmission instant. The aim of the addressed problem is to design a set of time-varying estimator parameters such that the prescribed H_∞ performance is guaranteed over a finite horizon. By using the stochastic analysis approach and completing-the-square method, sufficient conditions are derived for the existence of the desired estimators in terms of the solution to backward recursive Riccati difference equations. Finally, a numerical example is provided to validate the feasibility and effectiveness of the proposed results.     

Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

平衡损失函数下广义Pareto分布参数的Bayes估计的可容许性

对于一个参数进行估计来讲,不仅要考虑估计的精度,还要考虑所得到的估计对于模型拟合的优良程度。通常的损失函数无能为力,而平衡损失函数可以考虑到这两点,本文将在平衡误差损失函数下研究广义Pareto分布参数的Bayes估计问题。在平衡损失函数下导出了参数的Bayes估计并讨论了一类线性形式估计的可容许性和不可容许性。     

Identification and characterization of a novel 2R,3R-Butanediol dehydrogenase from Bacillus sp. DL01

Background2R,3R-butanediol dehydrogenase (R-BDH) and other BDHs contribute to metabolism of 3R/3S-Acetoin (3R/3S-AC) and 2,3-butanediol (2,3-BD), which are important bulk chemicals used in different industries. R-BDH is responsible for oxidizing the hydroxyl group at their (R) configuration. Bacillus species is a promising producer of 3R/3S-AC and 2,3-BD. In this study, R-bdh gene encoding R-BDH from Bacillus sp. DL01 was isolated, expressed and identified.ResultsR-BDH exerted reducing activities towards Diacetyl (DA) and 3R/3S-AC using NADH, and oxidizing activities towards 2R,3R-BD and Meso-BD using NAD⁺, while no activity was detected with 2S,3S-BD. The R-BDH showed its activity at a wide range of temperature (25°C to 65°C) and pH (5.0–8.0). The R-BDH activity was increased significantly by Cd²⁺ when DA, 3R/3S-AC, and Meso-BD were used as substrates, while Fe²⁺ enhanced the activity remarkably at 2R,3R-BD oxidation. Kinetic parameters of the R-BDH from Bacillus sp. DL01 showed the lowest K_m, the highest V_max, and the highest K_cat towards the racemic 3R/3S-AC substrate, also displayed low K_m towards 2R,3R-BD and Meso-BD when compared with other reported R-BDHs.ConclusionsThe R-BDH from Bacillus sp. DL01 was characterized as a novel R-BDH with high enantioselectivity for R-configuration. It considered NAD⁺ and Zn²⁺ dependant enzyme, with a significant affinity towards 3R/3S-AC, 2R,3R-BD, and Meso-BD substrates. Thus, R-BDH is providing an approach to regulate the production of 3R/3S-AC or 2,3-BD from Bacillus sp. DL01.How to cite: Elmahmoudy M, Elfeky N, Zhongji P, et al. Identification and characterization of a novel 2R,3R-Butanediol Dehydrogenase from Bacillus sp. DL01. Electron J Biotechnol 2021;49. https://doi.org/10.1016/j.ejbt.2020.11.002     

Direct projection to latent variable space for fault detection

Partial least squares (PLSs) often require many latent variables (LVs) T to describe the variations in process variables X correlated with quality variables Y, which are obtained via the traditional nonlinear iterative PLS (NIPALS) optimal solution based on (X, Y). Total projection to latent structures (T-PLSs) performs further decomposition to extract LVs T_y directly related to Y from T, which are obtained by the PCA optimal solution based on the predicted value of Y. Inspired by T-PLS, combined with practical process characteristics, two fault detection approaches are proposed in this paper to solve problems encountered by T-PLS. Without the NIPALS, (X, Y) are projected into the latent variable space determined by main variations of Y directly. Furthermore, the structure and characteristics of several modified methods in statistical analysis are studied based on calculation procedures of solving PCA, PLS and T-PLS optimization problems, and the geometric significance of the T-PLS model is demonstrated in detail. Simulation analysis and case studies both indicate the effectiveness of the proposed approaches.