The general complex bounded case of the strict weighted approximation problem

1971 ◽  
Vol 192 (2) ◽  
pp. 90-98 ◽  
Author(s):  
W. H. Summers
Keyword(s):  
Weighted Approximation ◽  
Approximation Problem ◽  
General Complex

scholarly journals The general complex case of the Bernstein-Nachbin approximation problem

1978 ◽  
Vol 28 (1) ◽  
pp. 193-206 ◽  
Author(s):  
S. Machado ◽  
Joao Bosco Prolla
Keyword(s):  
Approximation Problem ◽  
Complex Case ◽  
General Complex

scholarly journals Optimising the Complex Refractive Index Model for Estimating the Permittivity of Heterogeneous Concrete Models

2021 ◽  
Vol 13 (4) ◽  
pp. 723
Author(s):  
Hossain Zadhoush ◽  
Antonios Giannopoulos ◽  
Iraklis Giannakis
Keyword(s):  
Refractive Index ◽  
Shape Factor ◽  
Complex Refractive Index ◽  
Fdtd Method ◽  
Air Voids ◽  
Water Fraction ◽  
Index Model ◽  
General Complex ◽  
Ground Penetrating ◽  
Fine Tune

Estimating the permittivity of heterogeneous mixtures based on the permittivity of their components is of high importance with many applications in ground penetrating radar (GPR) and in electrodynamics-based sensing in general. Complex Refractive Index Model (CRIM) is the most mainstream approach for estimating the bulk permittivity of heterogeneous materials and has been widely applied for GPR applications. The popularity of CRIM is primarily based on its simplicity while its accuracy has never been rigorously tested. In the current study, an optimised shape factor is derived that is fine-tuned for modelling the dielectric properties of concrete. The bulk permittivity of concrete is expressed with respect to its components i.e., aggregate particles, cement particles, air-voids and volumetric water fraction. Different combinations of the above materials are accurately modelled using the Finite-Difference Time-Domain (FDTD) method. The numerically estimated bulk permittivity is then used to fine-tune the shape factor of the CRIM model. Then, using laboratory measurements it is shown that the revised CRIM model over-performs the default shape factor and provides with more accurate estimations of the bulk permittivity of concrete.

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals The core inverse and constrained matrix approximation problem

2020 ◽  
Vol 18 (1) ◽  
pp. 653-661 ◽  
Author(s):  
Hongxing Wang ◽  
Xiaoyan Zhang
Keyword(s):  
Unique Solution ◽  
Approximation Problem ◽  
Frobenius Norm ◽  
Matrix Approximation ◽  
The Core ◽  
Core Inverse

Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

STABILITY OF TWO TYPICAL COMPLEX DYNAMICAL NETWORKS

2008 ◽  
Vol 22 (05) ◽  
pp. 553-560 ◽  
Author(s):  
WU-JIE YUAN ◽  
XIAO-SHU LUO ◽  
PIN-QUN JIANG ◽  
BING-HONG WANG ◽  
JIN-QING FANG
Keyword(s):  
Numerical Simulation ◽  
Dissipative System ◽  
Small World ◽  
Complex Dynamical Networks ◽  
Dynamical Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
General Complex ◽  
The Stability ◽  
Theoretical Results

When being constructed, complex dynamical networks can lose stability in the sense of Lyapunov (i. s. L.) due to positive feedback. Thus, there is much important worthiness in the theory and applications of complex dynamical networks to study the stability. In this paper, according to dissipative system criteria, we give the stability condition in general complex dynamical networks, especially, in NW small-world and BA scale-free networks. The results of theoretical analysis and numerical simulation show that the stability i. s. L. depends on the maximal connectivity of the network. Finally, we show a numerical example to verify our theoretical results.

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