Symmetrical Augmented System of Equations for the Parameter Identification of Discrete Fractional Systems by Generalized Total Least Squares

This paper is devoted to the identification of the parameters of discrete fractional systems with errors in variables. Estimates of the parameters of such systems can be obtained using generalized total least squares (GTLS). A GTLS problem can be reduced to a total least squares (TLS) problem. A total least squares problem is often ill-conditioned. To solve a TLS problem, a classical algorithm based on finding the right singular vector or an algorithm based on an augmented system of equations with complex coefficients can be applied. In this paper, a new augmented system of equations with real coefficients is proposed to solve TLS problems. A symmetrical augmented system of equations was applied to the parameter identification of discrete fractional systems. The simulation results showed that the use of the proposed symmetrical augmented system of equations can shorten the time for solving such problems. It was also shown that the proposed system can have a smaller condition number.

A new approach of estimating the galactic thermal dust and synchrotron polarized emission template in the microwave bands

ABSTRACT The Internal Linear Combination (ILC) method has been extensively used to extract the cosmic microwave background (CMB) anisotropy map from foreground contaminated multifrequency maps. However, the performance of simple ILC is limited and can be significantly improved by heavily constraint equations, dubbed constrained ILC (cILC). The standard ILC and cILC work on spin-0 fields. Recently, a generalised version of ILC has been developed, named polarization ILC (PILC), in which Q ± iU at multiple frequencies are combined using complex coefficients to estimate Stokes Q and U maps. A statistical moment expansion method has recently been developed for high-precision modelling of the galactic foregrounds. This paper develops a semiblind component separation method combining the moment approach of foreground modelling with a generalised version of the PILC method for heavily constraint equations. The algorithm is developed in pixel space over a spin-2 field. We demonstrate the performance of the method on three sets of absolutely calibrated simulated maps at WMAP and Planck frequencies with varying foreground models. We apply this component separation technique in simultaneous estimation of Stokes Q and U maps of the thermal dust at 353 GHz and synchrotron at 30 GHz. We also recover both dust and synchrotron maps at 100 and 143 GHz, where separating two components is challenging.

A variety of multivalently analytic functions with complex coefficients and some argument properties of their applications

The principal goal of this scientific note is first to compose various rational types functions directly connecting to a variety of multivalently functions with complex coefficients, which are regular in certain domains in the complex plane, and then to designate numerous argument properties associating with certain applications of those (multivalent) functions

Spectral Transformations and Associated Linear Functionals of the First Kind

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

Real Root Polynomials and Real Root Preserving Transformations

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial p with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to p if we restrict the coefficients to be real. Let n ≥ 1 and P n be the vector space of all polynomials of degree n or less with real coefficients. In this article, we give explicit forms of polynomials in P n such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on P n which preserve real roots of polynomials in a certain subset of P n .