The Complexity of Approximating the Matching Polynomial in the Complex Plane

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.

Radially Distributed Values and Normal Families

Let L0 and L1 be two distinct rays emanating from the origin and let ${\mathcal{F}}$ be the family of all functions holomorphic in the unit disk ${\mathbb{D}}$ for which all zeros lie on L0 while all 1-points lie on L1. It is shown that ${\mathcal{F}}$ is normal in ${\mathbb{D}}\backslash \{0\}$. The case where L0 is the positive real axis and L1 is the negative real axis is studied in more detail.

A ZERO-FREE REGION FOR HYPERGEOMETRIC ZETA FUNCTIONS

This paper investigates the location of "trivial" zeros of some hypergeometric zeta functions. Analogous to Riemann's zeta function, we demonstrate that they possess a zero-free region on a left-half complex plane, except for infinitely many zeros regularly spaced on the negative real axis.

General Models for Linear Damping

The standard model for damping is the linear viscous dashpot which produces a force proportional to velocity. Although other sources of linear damping are known to exist, such as that due to viscoelasticity, it is not clear what range of mathematical forms damping models can take. Here it is suggested that there are only three types of damping model. These models are deduced by examining three configurations of mechanical components. These configurations include combinations of springs and dashpots and, most significantly, a semi-infinite beam. It is found that these models are best examined in the Laplace or s-plane so that features of the damping models may be expressed in terms of complex variable theory. The three types of damping model revealed by this analysis correspond to poles lying off the imaginary axis, poles on the negative real axis and pole like forms on the negative real axis that give rise to branch cuts. It is conjectured that these are the complete set of mathematical models that describe damping.

On Transmission Zeros of Mass-Dashpot-Spring Systems

For a noncollocated mass-dashpot-spring system with B=CTΓ, a novel approach is proposed to gain a better insight into the fact that none of its transmission zeros lie in the open right-half of the complex plane. In addition, the transmission zeros have physical meanings and will simply be the natural frequencies of a substructure constrained in the equivalently transformed system. Moreover, it is also shown that transmission zeros interlace with poles along the imaginary axis for a mass-spring system with B=CTΓ. They also interlace with poles along the negative real axis for a mass-dashpot system with B=CTΓ. Finally, two examples are used to illustrate the interlacing property.

Active and passive Damping of Euler-Bernoulli Beams and Their Interactions

Active and Passive damping of Euler-Bernoulli beams and their interactions have been studied using the beam’s exact transfer function model without mode truncation or finite element or finite difference approximation. The combination of viscous and Voigt damping is shown to map the open-loop poles and zeros from the imaginary axis in the undamped case into a circle in the left half plane and into the negative real axis. While active PD collocated control using sky-hooked actuators is known to stabilize the beam, it is shown that the derivative action using proof-mass (reaction-mass) actuators can destabilize the beam.

SUSCEPTIBILITY SINGULARITIES AT FIRST-ORDER PHASE TRANSITIONS

Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<Tc) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.

Infinitely many Stokes smoothings in the gamma function

The Stokes lines for Г( z ) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for In Г( z ) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Г( z ). Corresponding to each small exponential is a separate component asymptotic series in the expansion for In Г( z ). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from In Г( z ), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.

The effective conductivity of strongly heterogeneous 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.