Computing transitive closure of bipolar weighted digraphs

We define a bipolar weighted digraph as a weighted digraph together with the sign function on the arcs such that the weight of each arc lies between 0 and 1, and no two parallel arcs have the same sign. Bipolar weighted digraphs are utilized to model so-called fuzzy cognitive maps, which are used in science, engineering, and the social sciences to represent the causal structure of a body of knowledge. It has been noted in the literature that a transitive closure of a bipolar weighted digraph contains useful new information for the fuzzy cognitive map it models.In this paper we ask two questions: what is a sensible and useful definition of transitive closure of a bipolar weighted digraph, and how do we compute it? We give two answers to each of these questions, that is, we present two distinct models. First, we give a review of the fuzzy digraph model, which has been, in a different form and less rigorously, studied previously in the fuzzy systems literature. Second, we carefully develop a probabilistic model, which is related to the notion of network reliability.This paper is intended for a mathematical audience.     

Definition and properties of a particular notion of convexity

In an attempt to develop a general theory of nonlinear equations with exactly three solutions in general Hilbert spaces, we have noticed that a particular notion of convexity plays a key role. This paper is devoted to the study of some of its basic properties and we show how it generalizes homogeneous convex functionals on the one hand and some special convex functions of one real variable on the other hand. The important case of functionals associated with Nemytskii's operators is treated as an example.     

Markov interest rate models

A general procedure for creating Markovian interest rate models is presented. The models created by this procedure automatically fit within the HJM framework and fit the initial term structure exactly. Therefore they are arbitrage free. Because the models created by this procedure have only one state variable per factor, twoand even three-factor models can be computed efficiently, without resorting to Monte Carlo techniques. This computational efficiency makes calibration of the new models to market prices straightforward. Extended Hull- White, extended CIR, Black-Karasinski, Jamshidian's Brownian path independent models, and Flesaker and Hughston's rational log normal models are one-state variable models which fit naturally within this theoretical framework. The ‘separable’ n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian - which require n(n + 3)/2 state variables - are degenerate members of the new class of models with n(n + 3)/2 factors. The procedure is used to create a new class of one-factor models, the ‘β-η models.’ These models can match the implied volatility smiles of swaptions and caplets, and thus enable one to eliminate smile error. The β-η models are also exactly solvable in that their transition densities can be written explicitly. For these models accurate - but not exact - formulas are presented for caplet and swaption prices, and it is indicated how these closed form expressions can be used to efficiently calibrate the models to market prices.     

Numerical analysis of wave propagation in fluid-filled deformable tubes

The theory of Biot describing wave propagation in fluid saturated porous media is a good effective approximation of a wave induced in a fluid-filled deformable tube. Nonetheless, it has been found that Biot's theory has shortcomings in predicting the fast P-wave velocities and the amount of intrinsic attenuation. These problems arises when complex mechanical interactions of the solid phase and the fluid phase in the micro-scale are not taken into account. In contrast, the approach proposed by Bernabe does take into account micro-scopic interaction between phases and therefore poses an interesting alternative to Biot's theory. A Wave propagating in a deformable tube saturated with a viscous fluid is a simplified model of a porous material, and therefore the study of this geometry is of great interest. By using this geometry, the results of analytical and numerical results have an easier interpretation and therefore can be compared straightforward. Using a Finite Difference viscoelastic wave propagation code, the transient response was simulated. The wave source was modified with different characteristic frequencies in order to gain information of the dispersion relation. It was found that the P-wave velocities of the simulations at sub-critical frequencies closely match those of Bernabe's solution, but at over-critical frequencies they come closer to Biot's solution. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A principal ideal theorem analogue for modules over commutative rings

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.     

Free zeropotent left distributive systems

We prove a result conjectured by J.Je[zcirc]ek in [9],namely that a zeropotent left self-distributive system need not be 3-trivial, i.e.,the identity x(yz)=0 does not follow from the identities x(yz)=(xy)(xz),xx=0 and x0=0x=0.The argument is a general scheme possibly working for various identities in the context of self-distributivity.     

Thermal noise as a spectroscopic tool to determine transport properties

The utilization of thermal fluctuations or Johnson/Nyquist noise as a spectroscopic method to determine transport properties in conductors or semiconductors is developed. The autocorrelation function is obtained from power spectral density measurements thus enabling electronic transport property calculation through the Green–Kubo formalism. This experimental approach is distinct from traditional numerical methods such as molecular dynamics simulations, which have been used to extract the autocorrelation function and directly related physics only. This work reports multi-transport property measurements consisting of the electronic relaxation time, resistivity, mobility, diffusion coefficient, electronic contribution to thermal conductivity and Lorenz number from experimental data. Double validation of the experiment was accomplished through the use of a standard reference material and a standard measurement method, i.e. four-probe collinear resistivity technique. Thermal noise measurements resulted in a 1.1 and 5% agreement with the reference material and four-probe values, respectively. Additional measurements were also taken on nanoscale Au and Cu thin films. Specifically, the validated spectroscopic methodology was applied to 30 nm Au and Cu thin films to obtain transport property data that was again compared to four-probe resistivity measurements. Comparative analysis of the resistivity data showed agreement within 13.6 and 4.8% for the Au and Cu samples, respectively, thus lending further credibility to the experimental method and theory.     

Dissociation of dimethyl ether at high temperatures

The decomposition of dimethyl ether (CH₃OCH₃) has been investigated behind incident shock waves in a diaphragmless shock tube using laser schlieren densitometry, LS (T = 1500–2450 K, P = 57 ± 4, 125 ± 5 and 253 ± 12 Torr). The LS density gradient profiles were simulated and excellent agreement was found between the simulations and experimental profiles. Rate coefficients for CH₃OCH₃ → CH₃O + CH₃ were obtained. They showed strong fall-off, and at the lower end of the experimental temperature range are close to the low pressure limit. First order rate coefficient expressions were determined over 1500 < T < 2450 K. k_57Torr = (3.10 ± 1.0) × 10⁷⁹T^?19.03 exp(?54417/T) s^?1, k_125Torr = (1.12 ± 0.3) × 10⁸³T^?19.94 exp(?55554/T) s^?1and k_253Torr = (1.02 ± 0.3) × 10⁷³T^?17.09 exp(?51500/T) s^?1. The effect of a roaming channel for decomposition of dimethyl ether was assessed and the best agreement was obtained with 1% dissociation of DME via the roaming path.