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

Shifang Yuan  Anping Liao 《Journal of The Franklin Institute》2014

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.  相似文献   

The solution to matrix equation AX+XC=B     

Fengxian Piao  Qingling Zhang 《Journal of The Franklin Institute》2007,344(8):1056-1062

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.  相似文献   

The minimal norm least squares Hermitian solution of the complex matrix equation $AXB+CXD=E$     

Fengxia Zhang  Musheng Wei  Ying Li  Jianli Zhao 《Journal of The Franklin Institute》2018,355(3):1296-1310

In this paper, by applying the real representations of complex matrices, the particular structure of the real representations and the Moore–Penrose generalized inverse, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation $AXB+CXD=E$. And we also derive the minimal norm least squares Hermitian solution of the complex matrix equation $AXB=E$. Our proposed formulas only involve real matrices, and therefore are more effective and portable than those reported in Yuan and Liao (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of complex matrices. Two numerical examples are provided to demonstrate the effectiveness of our algorithms.  相似文献   

An econometric property of the g-index     

L. Egghe 《Information processing & management》2009

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On the geometry of the smallest circle enclosing a finite set of points     

Lance D. Drager  Jeffrey M. Lee 《Journal of The Franklin Institute》2007,344(7):929-940

A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P.  相似文献   

The obstacle problem of integro-partial differential equations with applications to stochastic optimal control/stopping problem     

Lei Zhang  Bin Liu 《Journal of The Franklin Institute》2019,356(3):1396-1423

This paper is devoted to existence and uniqueness of minimal mild super solutions to the obstacle problem governed by integro-partial differential equations. We first study the well-posedness and local Lipschitz regularity of L^p solutions (p?≥?2) to reflected forward-backward stochastic differential equations (FBSDEs) with jump and lower barrier. Then we show that the solutions to reflected FBSDEs provide a probabilistic representation for the mild super solution via a nonlinear Feynman–Kac formula. Finally, we apply the results to study stochastic optimal control/stopping problems.  相似文献   

Existence theorem of the quadruple (P, R, F, M): Precision,recall, fallout and miss     

L. Egghe 《Information processing & management》2007

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Norm estimates for function Lyapunov equations and applications     

Michael Gil’ 《Journal of The Franklin Institute》2018,355(10):4241-4247

We consider the function Lyapunov equation $f^{*}\left(A\right)X+Xf\left(A\right)=C,$ where A and C are given matrices, f(z) is a function holomorphic on a neighborhood of the spectrum σ(A) of A. For a solution X of that equation, norm estimates are established. By these estimates we investigate perturbations of a matrix A whose spectrum satisfies the condition $\inf ?\sigma \left(f\right(A\left)\right)>0$. In the case $f\left(z\right)=z^{\nu }$ with a positive integer ν we obtain conditions that provide localization of the spectrum of a perturbed matrix in a given angle.  相似文献   

Some properties of a class of biased regression estimators     

J.M. Lowerre 《Journal of The Franklin Institute》1977,303(6):517-526

For the linear statistical model y = Xb + e, X of full column rank estimates of b of the form (C + X′X)⁺X′y are studied, where C commutes with X′X and Q⁺ is the Moore-Penrose inverse of Q. Such estimators may have smaller mean square error, component by component than does the least squares estimator. It is shown that this class of estimators is equivalent to two apparently different classes considered by other authors. It is also shown that there is no C such that (C + X′X)⁺X′Y = My, in which My has the smallest mean square error, component by component. Two criteria, other than tmse, are suggested for selecting C. Each leads to an estimator independent of the unknown b and σ². Subsequently, comparisons are made between estimators in which the C matrices are functions of a parameter k. Finally, it is shown for the no intercept model that standardizing, using a biased estimate for the transformed parameter vector, and retransforming to the original units yields an estimator with larger tmse than the least squares estimator.  相似文献   

An overview of the applications and solutions of a fundamental matrix equation pair     

Chia-Chi Tsui 《Journal of The Franklin Institute》2004,341(6):465-475

Equation TA−FT=LC (F is stable) is necessary and sufficient for the output of a feedback compensator (F,L,K_Z,K_y) to converge to a state feedback (SF) signal for a constant K, where (A,B,C,0) is the open loop system and TB is the compensator gain to the open loop system input. Thus, equation TB=0 is (1) the defining condition for this feedback compensator to be an output feedback compensator. Equation TB=0 is also the necessary and sufficient condition to (2) fully realize the critical loop transfer function and robust properties of SF control if K is systematically designed. Furthermore, because B is compatible to the open loop system gain to its unknown inputs and its input failure signals, TB=0 is also necessary for (3) unknown input observers and (4) failure detection and isolation systems. Finally, this equation pair (TA−FT=LC, TB=0) is the key condition of a really systematic and explicit design algorithm for (5) eigenstructure assignment by static output feedback control. This paper reviews the existing solutions of this equation pair, and points out that a general and exact solution is uniquely direct, simple, and decoupled. This paper also points out that these unique features also enable two decisive advantages: (1) the systematic compensator order adjustment and (2) a simple and approximate solution which is general to all systems (A,B,C,0) and which can be simply added to the exact solution whether it exists or not.  相似文献   

On the compression of search trees     

Francisco Claude  Patrick K. Nicholson  Diego Seco 《Information processing & management》2014

Let X=_x1,_x2,…,_xn$X=x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n$ be a sequence of non-decreasing integer values. Storing a compressed representation of X that supports access and search is a problem that occurs in many domains. The most common solution to this problem uses a linear list and encodes the differences between consecutive values with encodings that favor small numbers. This solution includes additional information (i.e. samples) to support efficient searching on the encoded values. We introduce a completely different alternative that achieves compression by encoding the differences in a search tree. Our proposal has many applications, such as the representation of posting lists, geographic data, sparse bitmaps, and compressed suffix arrays, to name just a few. The structure is practical and we provide an experimental evaluation to show that it is competitive with the existing techniques.  相似文献   

Performance analysis for ordered selection combining schemes in Nakagami-m environments     

Joy Iong-Zong Chen 《Journal of The Franklin Institute》2005,342(6):638-656

Ordered statistics is applied in this paper to analyze the performance of ordered selection combining schemes with different modulation receptions operating in Nakagami-m fading environments. The ordered selection combining will be adapted to include the conventional selection combining (SC), the second-order diversity combining (SC-2), and the third-order diversity combining (SC-3). All the results are validated by comparing the special case Rayleigh distribution with the fading figure m=1 in the Nakagami-m distribution. The results show that SC is in performance the worst in both coherent and non-coherent schemes, as expected. The performance differences between SC-2 and MRC, and SC-3 and EGC are not significant when the diversity order L?3, but the differences will increase when L?5.  相似文献   

A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations     

Xingping Sheng 《Journal of The Franklin Institute》2018,355(10):4282-4297

The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations $AX+YB=C,DX+YE=F$. It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X₀ and Y₀. Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form $A_{i1}{X}_1{B}_{i1}+A_{i2}{X}_2{B}_{i2}+?+A_{ip}{X}_p{B}_{ip}=C_i,(i=1,2,\dots ,p)$. Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ₁ and µ₂. Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.  相似文献   

Optimal tensegrity structures in bending: The discrete Michell truss     

Robert E. Skelton  Maurício C. de Oliveira 《Journal of The Franklin Institute》2010,347(1):257-283

This paper provides the closed form analytical solution to the problem of minimizing the material volume required to support a given set of bending loads with a given number of discrete structural members, subject to material yield constraints. The solution is expressed in terms of two variables, the aspect ratio, ρ^-1, and complexity of the structure, q (the total number of members of the structure is equal to q(q+1)). The minimal material volume (normalized) is also given in closed form by a simple function of ρ and q, namely, V=q(ρ^-1/q-ρ^1/q). The forces for this nonlinear problem are shown to satisfy a linear recursive equation, from node-to-node of the structure. All member lengths are specified by a linear recursive equation, dependent only on the initial conditions involving a user specified length of the structure. The final optimal design is a class 2 tensegrity structure. Our results generate the 1904 results of Michell in the special case when the selected complexity q approaches infinity. Providing the optimum in terms of a given complexity has the obvious advantage of relating complexity q to other criteria, such as costs, fabrication issues, and control. If the structure is manufactured with perfect joints (no glue, welding material, etc.), the minimal mass complexity is infinite. But in the presence of any joint mass, the optimal structural complexity is finite, and indeed quite small. Hence, only simple structures (low complexity q) are needed for practical design.  相似文献   

On some n-normed linear space valued difference sequences     

Hemen Dutta 《Journal of The Franklin Institute》2011,348(10):2876-2883

The main aim of this article is to extend the notion of strongly Cesàro summable and strongly lacunary summable real sequences to n-normed linear space valued (n-nls valued) difference sequences. Consequently we introduce the spaces |σ₁|(X,?) and N_θ(X,?), respectively, where X is an n-normed space and ? is a difference operator. We investigate these spaces for completeness as well as for the relationship between these spaces.  相似文献   

Direct projection to latent variable space for fault detection     

Jing Hu  Chenglin Wen  Ping Li  Tianqi Yuan 《Journal of The Franklin Institute》2014

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.  相似文献   

Periodic solutions for third order p-Laplacian equation with a deviating argument     

Shan Jin  Shiping Lu 《Journal of The Franklin Institute》2009,346(2):126-135

By means of Mawhin's continuation theorem, we study a third-order p-Laplacian differential equation