Cramer's Rule Research Articles

Determinantal representation of the generalized inverses over the quaternion skew field with applications

In this paper, we establish some new determinantal representations of the generalized inverses ArT1,S1(2),AlT2,S2(2) and A(T1,T2),(S1,S2)(2) over the quaternion skew field by the theory of the column and row determinants. In addition, we derive a condensed Cramer rule for the unique solution of some restricted quaternion matrix equations. The findings of this paper extend some known results in the literature.

Wearable sensors in intelligent clothing for measuring human body temperature based on optical fiber Bragg grating

Measuring body temperature is considerably important to physiological studies as well as clinical investigations. In recent years, numerous observations have been reported and various methods of measurement have been employed. The present paper introduces a novel wearable sensor in intelligent clothing for human body temperature measurement. The objective is the integration of optical fiber Bragg grating (FBG)-based sensors into functional textiles to extend the capabilities of wearable solutions for body temperature monitoring. In addition, the temperature sensitivity is 150 pm/°C, which is almost 15 times higher than that of a bare FBG. This study combines large and small pipes during fabrication to implant FBG sensors into the fabric. The law of energy conservation of the human body is considered in determining heat transfer between the body and its clothing. The mathematical model of heat transmission between the body and clothed FBG sensors is studied, and the steady-state thermal analysis is presented. The simulation results show the capability of the material to correct the actual body temperature. Based on the skin temperature obtained by the weighted average method, this paper presents the five points weighted coefficients model using both sides of the chest, armpits, and the upper back for the intelligent clothing. The weighted coefficients of 0.0826 for the left chest, 0.3706 for the left armpit, 0.3706 for the right armpit, 0.0936 for the upper back, and 0.0826 for the right chest were obtained using Cramer's Rule. Using the weighting coefficient, the deviation of the experimental result was ± 0.18 °C, which favors the use for clinical armpit temperature monitoring. Moreover, in special cases when several FBG sensors are broken, the weighted coefficients of the other sensors could be changed to obtain accurate body temperature.

Partially decoupled approach of extended non-minimal state space predictive functional control for MIMO processes

This paper presents a partially decoupled design of the state space predictive functional control for MIMO processes. The multivariable process is first treated into MISO process by a simple Cramer's rule solution to linear equations which provides a balance between model complexity and control system design, and then the derived MISO process based extended state space predictive functional control is presented. The overall design of the controller enables the controller to consider both the process state dynamics and the output dynamics, thus improved control performance for tracking set-points and disturbance rejection is resulted. The proposed controller is tested on both model match and model mismatch cases to demonstrate its superiority. In addition, a closed-form of transfer function representation that facilitates frequency analysis of the control system is provided to give further insight into the proposed method.

영뉵(盈朒)과 영부족술(盈不足術)에 관한 최근 동서양의 연구 분석

영뉵 또는 영부족(盈不足)은 그 글자의 의미에서 보듯이 넘치는 것과 부족한 것에 관계된 '과부족'문제를 나타낼 때 사용되는 유용한 개념으로, 영부족술(盈不足術)은 과부족문제를 푸는 일종의 산법이다. 본 논문은 먼저 지금까지 영부족에 관하여 소개된 최근의 모든 논문을 분석하고, <구장산술 九章算術> 권칠(卷七) "영부족" 장의 대표문제를 통하여 영부족술의 내용과 산법으로써의 의미를 쉽게 이해할 수 있도록 새롭게 서술하였다. 그리고 서양에서 이중가정법(rule of double false position)으로 알려진 영부족에 관한 최근의 동서양 연구결과를 분석하여, 영부족술과 Cramer's Rule과의 관계 및 <산학보감 算學寶鑑>에 소개된 진화된 영부족술의 특징에 대하여 논하였다. 더 나아가 영부족술의 기원과 중국의 영부족술이 아랍을 거쳐 유럽으로 전파된 배경을 구체적으로 밝혔다. In this paper, we deal with recent researches on Ying N<TEX>$\ddot{u}$</TEX> and Ying Buzu(盈不足) which were addressed in the book Jiu Zhang Suan Shu(九章算術, The Nine Chapters on the Mathematical Art). Ying N<TEX>$\ddot{u}$</TEX>(Ying Buzu) is a concept on profit and loss problems. Ying Buzu Shu(盈不足術, the method of excess and deficit) represents an algorithm which has been used for solving many mathematical problems. It is known as a rule of double false position in the West. We show the importance of Ying Buzu Shu via an analysis of some problems in 'Ying Buzu' chapter. In 1202, Fibonacci(c.1170-c.1250) used Ying Buzu Shu in his book. This shows some of Asian mathematics were introduced to the West even before the year 1200. We present the origin of Ying Buzu Shu, and its relationship with Cramer's Rule. We have discovered how Asia's Ying Buzu Shu spread to Europe via Arab countries. In addition, we analyze some characters of Ying N<TEX>$\ddot{u}$</TEX>(Ying Buzu) in the book Suan Xue Bao Jian(算學寶鑑).

Cramer's rule applied to flexible systems of linear equations

Systems of linear equations, called flexible systems, with coefficients having uncer- tainties of type o(.) or O (.) are studied. In some cases an exact solution may not exist but a general theorem that guarantees the existence of an admissible solution, in terms of inclusion, is presented. This admissible solution is produced by Cramer's Rule; depending on the size of the uncertainties appearing in the matrix of coefficients and in the constant term vector some adaptations may be needed.

Analogs of Cramer’s rule for the minimum norm least squares solutions of some matrix equations

The least squares solutions with the minimum norm of the matrix equations AX = B , XA = B and AXB = D are considered in this paper. We use the determinantal representations of the Moore–Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the minimum norm least squares solutions of these matrix equations.

On the Analysis of Localization Accuracy of Wireless Indoor Positioning Systems using Cramer's Rule

On the Analysis of Localization Accuracy of Wireless Indoor Positioning Systems using Cramer's Rule

Condensed Cramer rule for some restricted quaternion linear equations

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.

Cramer rule for the unique solution of restricted matrix equations over the quaternion skew field

In this paper, we establish the determinantal representations of the generalized inverses A r T 1 , S 1 ( 2 ) , A l T 2 , S 2 ( 2 ) and A ( T 1 , T 2 ) , ( S 1 , S 2 ) ( 2 ) over the quaternion skew field by the theory of the column and row determinants. In addition, we derive some generalized Cramer rules for the unique solution of some restricted quaternion matrix equations. The findings of this paper extend some known results in the literature.

Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

Linearized multiparameter inversion is a model-driven variant of amplitude-versus-offset analysis, which seeks to separately account for the influences of several model parameters on the seismic response. Previous approaches to this class of problems have included geometric optics-based (Kirchhoff, GRT) inversion and iterative methods suitable for large linear systems. In this paper, we suggest an approach based on the mathematical nature of the normal operator of linearized inversion—it is a scaling operator in phase space—and on a very old idea from linear algebra, namely, Cramer's rule for computing the inverse of a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

Teaching Tip: Correcting Cramer's Rule

Teaching Tip: Correcting Cramer's Rule

Proof of the Combinatorial Nullstellensatz over Integral Domains, in the Spirit of Kouba

It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J. Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michałek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.

Essential math skills for engineers

Preface. 1 What Do Engineers Do? 2 Miscellaneous Math Skills. 2.1 Equations of Lines, Planes, and Circles. 2.2 Areas and Volumes of Common Shapes. 2.3 Roots of a Quadratic Equation. 2.4 Logarithms. 2.5 Reduction of Fractions and Lowest Common Denominators. 2.6 Long Division. 2.7 Trigonometry. 2.7.1 The Common Trigonometric Functions: Sine, Cosine, and Tangent. 2.7.2 Areas of Triangles. 2.7.3 The Hyperbolic Trigonometric Functions: Sinh, Cosh, and Tanh. 2.8 Complex Numbers and Algebra, and Euler's Identity. 2.8.1 Solution of Differential Equations Having Sinusoidal Forcing Functions. 2.9 Common Derivatives and Their Interpretation. 2.10 Common Integrals and Their Interpretation. 2.11 Numerical Integration. 3 Solution of Simultaneous, Linear, Algebraic Equations. 3.1 How to Identify Simultaneous, Linear, Algebraic Equations. 3.2 The Meaning of a Solution. 3.3 Cramer's Rule and Symbolic Equations. 3.4 Gauss Elimination. 3.5 Matrix Algebra. 4 Solution of Linear, Constant-Coeffi cient, Ordinary Differential Equations. 4.1 How to Identify Linear, Constant-Coeffi cient, Ordinary Differential Equations. 4.2 Where They Arise: The Meaning of a Solution. 4.3 Solution of First-Order Equations. 4.3.1 The Homogeneous Solution. 4.3.2 The Forced Solution for Nice f ( t ). 4.3.3 The Total Solution. 4.3.4 A Special Case. 4.4 Solution of Second-Order Equations. 4.4.1 The Homogeneous Solution. 4.4.2 The Forced Solution for Nice f ( t ). 4.4.3 The Total Solution. 4.4.4 A Special Case. 4.5 Stability of the Solution. 4.6 Solution of Simultaneous Sets of Ordinary Differential Equations with the Differential Operator. 4.6.1 Using the Differential Operator to Verify Solutions. 4.7 Numerical (Computer) Solutions. 5 Solution of Linear, Constant-Coeffi cient, Difference Equations. 5.1 Where Difference Equations Arise. 5.2 How to Identify Linear, Constant-Coeffi cient Difference Equations. 5.3 Solution of First-Order Equations. 5.3.1 The Homogeneous Solution. 5.3.2 The Forced Solution for Nice f ( n ). 5.3.3 The Total Solution. 5.3.4 A Special Case. 5.4 Solution of Second-Order Equations. 5.4.1 The Homogeneous Solution. 5.4.2 The Forced Solution for Nice f ( n ). 5.4.3 The Total Solution. 5.4.4 A Special Case. 5.5 Stability of the Solution. 5.6 Solution of Simultaneous Sets of Difference Equations with the Difference Operator. 5.6.1 Using the Difference Operator to Verify Solutions. 6 Solution of Linear, Constant-Coeffi cient, Partial Differential Equations. 6.1 Common Engineering Partial Differential Equations. 6.2 The Linear, Constant-Coeffi cient, Partial Differential Equation. 6.3 The Method of Separation of Variables. 6.4 Boundary Conditions and Initial Conditions. 6.5 Numerical (Computer) Solutions via Finite Differences: Conversion to Difference Equations. 7 The Fourier Series and Fourier Transform. 7.1 Periodic Functions. 7.2 The Fourier Series. 7.3 The Fourier Transform. 8 The Laplace Transform. 8.1 Transforms of Important Functions. 8.2 Useful Transform Properties. 8.3 Transforming Differential Equations. 8.4 Obtaining the Inverse Transform Using Partial Fraction Expansions. 9 Mathematics of Vectors. 9.1 Vectors and Coordinate Systems. 9.2 The Line Integral. 9.3 The Surface Integral. 9.4 Divergence. 9.4.1 The Divergence Theorem. 9.5 Curl. 9.5.1 Stokes' Theorem. 9.6 The Gradient of a Scalar Field. Index.

False Position, Double False Position and Cramer's Rule

False Position, Double False Position and Cramer's Rule

False Position, Double False Position and Cramer's Rule

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsEugene C. BomanEugene Boman (ecb5@psu.edu) received his Ph.D. from the University of Connecticut in 1993. While in graduate school he began collecting antique slide rules, a hobby which he continued to pursue after he was hired at Penn State's DuBois campus in 1996 and continues today at Penn State's Harrisburg campus which he joined in 2006. He hopes someday to own enough slide rules to require adding a room to his home in which to store his collection. His wife and two daughters are very patient.

Cramer's rule for quaternionic systems of linear equations

New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer's rules for right and left quaternionic systems of linear equations have been obtained.

Incremental analysis of the nonlinear behavior of thin shells

The paper proposes a method of incremental loading for solving nonlinear problems of shell theory. A feature of this method is that the same algorithm is used before buckling, at the limit point, and after buckling. The method is based on the assumption that all the unknowns, including the load parameter, are on an equal footing. It is shown that the method can be used to solve algebraic equations with Cramer's rule involved to avoid numerical instability in the neighborhood of the limit point. Test problems confirm the validity of the approach

Several representations of generalized inverse and their application

In this paper we first introduce a full-rank decomposition representation of the generalized inverse of a given complex matrix A, which is based on an arbitrary full-rank decomposition of G, where G is a matrix such that R(G)=T and N(G)=S. Using this representation, we obtain two maximum rank minor representations of the generalized inverse . As an application we give the Cramer's rule of the general restricted linear system. Finally, a numerical example shows that these representations are correct.

A Unified Approach to Dual Gabor Windows

In this paper, we describe a new method for studying the invertibility of Gabor frame operators. Our approach can be applied to both the continuous (on R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> ) and the finite discrete setting. In the latter case, we obtain algorithms for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature. The framework we propose can also be used to derive other (known) results in Gabor theory in a unified way such as the Zibulski-Zeevi representation. The approach we suggest is based on an adequate splitting of the twisted convolution, which, in turn, provides another twisted convolution on a finite cyclic group. By analogy with the twisted convolution of finite discrete signals, we derive a mapping between the sequence space and a matrix algebra which preserves the algebraic structure. In this way, the invertibility problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. Using Cramer's rule and proving Wiener's lemma for this special class of matrices, we obtain an invertibility criterion that can be applied to a variety of different settings. This alternative approach provides further insight into Gabor frames, as well as a unified framework for Gabor analysis

Analytical solutions for a vertically loaded pile in multilayered soil

Explicit elastic solutions for a vertically loaded single pile embedded in multilayered soil are presented. Solutions are also provided for the case in which the pile base rests on a rigid material. Energy principles are used in the derivation of the governing differential equations. The solutions, which satisfy compatibility of displacement in the vertical and radial directions, were obtained by determining the unknown integration constants using the boundary conditions, Cramer's rule, and a recurrence formula. The solutions provide the pile vertical displacement as a function of depth, the load-transfer curves, and the vertical soil displacement as a function of the radial distance from the pile axis at any depth. The use of the analysis is illustrated for a pile that was load-tested under well-documented conditions by obtaining load–transfer and load–settlement curves that are then compared with those obtained from the load test.