The answer depends on your lecturer

This article is concerned with the approaches to the root concept that lecturers in calculus, linear algebra and complex analysis employ in their instruction. Three highly experienced university lecturers participated in the study. In the individual interviews the participants referred to roots of real numbers, roots of complex numbers, roots as real functions and roots as complex functions. However, participants’ approaches to the root concept were mathematically not equivalent and the lecturers resisted to the approaches of their peers. The choices of approach were shaped by different types of pedagogical considerations, such as the mathematical ideas that appear in the courses that lecturers teach, textbook approaches and lecturers’ interpretations of students’ further academic needs. Several cases were indicated where the participants consciously compromised mathematical rigour and limited the scope of their instruction when teaching general mathematics courses.     

How real is your matrix?

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have applications, for instance, in diffractive optics, preconditioning and fast Fourier expansions.     

High-order splitting methods for separable non-autonomous parabolic equations

We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time-dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.     

Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge-Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can‘t reach order more than s 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s 1 when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.     

基于价值工程的房地产项目管理应用研究

房地产项目投资巨大,过程繁多,而房地产项目管理的方法和水平如何直接影响到房地产企业效益的高低.因此,通过对房地产项目全过程的分析,提出了房地产项目全过程功能指标体系,并结合价值工程理论提出了房地产项目全过程运用价值工程的思想,通过运用三家房地产公司房地产项目的成本数据进行实证分析,验证了指标体系和房地产全过程价值工程理论的合理性,为房地产企业全过程运用价值工程理论提供了指导.     

Khovanskii–Rolle Continuation for Real Solutions

We present a new continuation algorithm to find all real solutions to a nondegenerate system of polynomial equations. Unlike homotopy methods, the algorithm is not based on a deformation of the system; instead, it traces real curves connecting the solutions to one system of equations to those of another, eventually leading to the desired real solutions. It also differs from homotopy methods in that it follows only real paths and computes no complex solutions to the original equations. The number of curves traced is essentially bounded above by the fewnomial bound for real solutions, and the method takes advantage of any slack in that bound.     

Splitting methods with complex times for parabolic equations

Using composition procedures, we build up high order splitting methods to solve evolution equations posed in finite or infinite dimensional spaces. Since high-order splitting methods with real time are known to involve large and/or negative time steps, which destabilizes the overall procedure, the key point of our analysis is, we develop splitting methods that use complex time steps having positive real part: going to the complex plane allows to considerably increase the accuracy, while keeping small time steps; on the other hand, restricting our attention to time steps with positive real part makes our methods more stable, and in particular well adapted in the case when the considered evolution equation involves unbounded operators in infinite dimensional spaces, like parabolic (diffusion) equations. We provide a thorough analysis in the case of linear equations posed in general Banach spaces. We also numerically investigate the nonlinear situation. We illustrate our results in the case of (linear and nonlinear) parabolic equations.     

Invariant generalized functions supported on an orbit

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of GL₃(C). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16].     

Deformations of Real Rational Dynamics in Tropical Geometry

In this paper we study analytic properties of orbits given by real rational functions. We introduce some comparison methods which allow us to compare the real rational dynamics with automata given by (max, +) functions, passing through a kind of scale transform in tropical geometry. Such a scale transform gives a one-to-one correspondence of presentations between automata and real rational functions. We study invariant properties of the real rational dynamics under change of presentations of automata.     

Efficient fourth orderP-stable formulae

In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.     

Reelle algebraische Funktionen mit vorgegebenen Null- und Polstellen

This note continues the investigations of Knebusch on algebraic curves over real closed fields and was initiated by reading [3]. Especially we ask for the existence of real algebraic functions with given zeroes and poles, a question going back to Witt [4]. We study the real nature of coverings of real algebraic curves, and if the covering has degree two, we get algebraic proofs for results, which in the classical case have been obtained by topological methods in [2].     

Comparison of various statistical methods for detecting disease outbreaks

In this article, we compared seven statistical methods for detecting outbreaks of infectious disease; Historical limits, English model, SPOTv2, CuSums, Bayesian predictive model, RKI method and Serfling model. We used simulated data and real data to compare those seven methods. Simulated data have parameters such as trend, seasonality, mean and standard deviation. Among these methods, SPOTv2 shows the best performance with a balance between sensitivity and positive predictive value and short time lag. But in datasets having strong trends, Bayesian predictive model, English model and Serfling model perform better than SPOTv2. These methods are also compared through real numerical example.     

Detecting exponential dichotomy on the real line: SVD and QR algorithms

In this paper we propose and implement numerical methods to detect exponential dichotomy on the real line. Our algorithms are based on the singular value decomposition and the QR factorization of a fundamental matrix solution. The theoretical justification for our methods was laid down in the companion paper: “Exponential Dichotomy on the real line: SVD and QR methods.”     

基于Hartley变换的三维真振幅偏移算子研究

在实际应用中，以快速Fourier变换为基础的偏移方法，将本来是实数的地震道转化为复数参加运算，导致了计算机内存的增加。本文把只有纯实数运算的Hartley变换引入到基于Fourier变换的偏移算法，再利用三维真振幅偏移单程波方程，结合Fourier变换与Hartley变换的内在关系，经过数学推理，具体导出了裂步Hartley变换真振幅偏移算子。与一般裂步Fourier法相比，裂步Hartley变换真振幅偏移算法既提高了计算效率又对球面扩散问题进行了振幅补偿。     

Comparing multicriteria methods in the context of environmental problems

The present paper gives an analysis of the use of ELECTRE III, PROMETHEE I, II, and SMART decision-aids in the context of four different real applications to environmental problems in Finland. These methods are widely used decision-aids in the real planning processes. The purpose of this study is to define the differences of these methods and the results obtained with them, and in this way also to consider their applicability in aiding environmental decision-making. Furthermore, a comparison of the methods based on a set of randomly generated problems is carried out. The choice in practice will not be easy; PROMETHEE does not differ much from SMART with linear value functions, and in many cases this holds for ELECTRE III, too. However, in a particular problem the `best alternatives' obtained with these methods may differ greatly. Therefore, the conclusion is that it is better to use several methods for the same problem when possible; when not, we recommend the use of ELECTRE III.     

Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.     

Bounds on a polynomial

Methods are given for isolating and approximating the maxima, minima, and real roots of a polynomial with real coefficients. The methods are based on a variation diminishing property of the Bernstein coefficients of the polynomial and use of a recursive bisection technique.     

Nonstandard decision methods for the solvability of real polynomial equations

For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binaly polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Unifying fourth-order family of iterative methods

In this work, we develop a new two-parameter family of iterative methods for solving nonlinear scalar equations. One of the parameters is defined through an infinite power series consisting of real coefficients while the other parameter is a real number. The methods of the family are fourth-order convergent and require only three evaluations during each iteration. It is shown that various fourth-order iterative methods in the published literature are special cases of the developed family. Convergence analysis shows that the methods of the family are fourth-order convergent which is also verified through the numerical work. Computations are performed to explore the efficiency of various methods of the family.     

On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.