Discussion on Paper entitled ``Synthesis of Transfer Functions with Poles Restricted to the Negative Real Axis''

1953 ◽  
Vol 24 (12) ◽  
pp. 1525-1526
Author(s):  
A. D. Fialkow ◽  
Irving Gerst
Keyword(s):  
Real Axis ◽  
Transfer Functions ◽  
Negative Real Axis

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 Remarks on Stability Conditions for the Differential Equation x″ + a(t)ƒ(x) = 0

1969 ◽  
Vol 9 (3-4) ◽  
pp. 496-502 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Continuous Function ◽  
Nonlinear Differential Equation ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Second Order ◽  
Stability Conditions ◽  
Negative Real Axis ◽  
All Solutions ◽  
Image Position

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).

On the invariance of the essential spectrum of an arbitrary operator. III

1968 ◽  
Vol 64 (4) ◽  
pp. 975-984 ◽  
Author(s):  
Martin Schechter
Keyword(s):  
Essential Spectrum ◽  
Real Axis ◽  
Limit Point ◽  
Sufficient Conditions ◽  
Energy Operator ◽  
Hydrogen Energy ◽  
Negative Real Axis ◽  
Negative Eigenvalues ◽  
Arbitrary Operator ◽  
Image Position

The spectrum of the hydrogen energy operator(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operatorin En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of(a) the non-negative real axis,(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.

The effective conductivity of strongly heterogeneous composites

1991 ◽  
Vol 433 (1888) ◽  
pp. 353-381 ◽  
Keyword(s):  
Real Axis ◽  
Complex Variable ◽  
Effective Conductivity ◽  
Numerical Calculations ◽  
Variable Method ◽  
High Contrast ◽  
Negative Real Axis ◽  
Two Phase ◽  
Periodic Arrays ◽  
Random Composites

We deal with the effective conductivity m = m(z) of two phase, ordered or disordered mixtures consisting of particles of material of conductivity z inserted in a matrix of conductivity 1. We focus on finding bounds on the set of values of z for which the function m is singular or vanishes, and we apply our results to the estimation of the effective conductivity of high contrast mixtures ( z = 0 or z = ∞). We find that the zeroes and singularities of the function m lie on an interval of the negative real axis, which depends on the shape of the particles and the interparticle distances. Our results agree with previous numerical calculations for periodic arrays of spheres. In some cases we show that our estimates are optimal. We apply our results about the zeroes and singularities together with the complex variable method, and find bounds on the effective conductivity of matrix-particle random composites. These bounds give good estimations even in cases of high contrast, and, in many cases, they improve substantially over the bounds obtained by other methods, for the same types of high contrast mixtures.

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