Geometric Mean Approximation Research Articles

Causal Walk: Debiasing Multi-Hop Fact Verification with Front-Door Adjustment

Multi-hop fact verification aims to detect the veracity of the given claim by integrating and reasoning over multiple pieces of evidence. Conventional multi-hop fact verification models are prone to rely on spurious correlations from the annotation artifacts, leading to an obvious performance decline on unbiased datasets. Among the various debiasing works, the causal inference-based methods become popular by performing theoretically guaranteed debiasing such as casual intervention or counterfactual reasoning. However, existing causal inference-based debiasing methods, which mainly formulate fact verification as a single-hop reasoning task to tackle shallow bias patterns, cannot deal with the complicated bias patterns hidden in multiple hops of evidence. To address the challenge, we propose Causal Walk, a novel method for debiasing multi-hop fact verification from a causal perspective with front-door adjustment. Specifically, in the structural causal model, the reasoning path between the treatment (the input claim-evidence graph) and the outcome (the veracity label) is introduced as the mediator to block the confounder. With the front-door adjustment, the causal effect between the treatment and the outcome is decomposed into the causal effect between the treatment and the mediator, which is estimated by applying the idea of random walk, and the causal effect between the mediator and the outcome, which is estimated with normalized weighted geometric mean approximation. To investigate the effectiveness of the proposed method, an adversarial multi-hop fact verification dataset and a symmetric multi-hop fact verification dataset are proposed with the help of the large language model. Experimental results show that Causal Walk outperforms some previous debiasing methods on both existing datasets and the newly constructed datasets. Code and data will be released at https://github.com/zcccccz/CausalWalk.

A new approximation approach for analytically solving the population balance equation due to thermophoretic coagulation

The analytical solution of the population balance equation (PBE) for coagulation by differential settling velocities remains a challenging issue due to the absolute value in the collision kernel. A geometric mean approximation approach is first proposed in this study to handle the integration of the absolute value and convert the moment equation into an integrable form. Using this approach along with the log-normal method of moments, the PBE is first analytically solved for thermophoretic coagulation in the low Knudsen number limit. The present analytical solution is validated by comparing the size distribution parameters with the sectional method. The maximum relative errors in this study are quite acceptable as compared with previous analytical solutions for Brownian coagulation. Moreover, the self-preserving size distribution predicted by the present analytical solution compares well with the sectional method and previous numerical studies on thermophoretic coagulation. Based on the analytical solution, the characteristics of the size distribution evolution during thermophoretic coagulation are investigated theoretically. The results show that the coagulation process is faster with a larger initial geometric standard deviation and the time required to reach the self-preserving distribution increases with the difference between the initial and asymptotic geometric standard deviations.

Estimation of Surface Free Energy of Poly(lactic acid) During UV‐Grafting with N‐Vinylpyrrolidone

The surface free energy (SFE) of polylactic acid (PLA) during grafting with N‐vinylpyrrolidone (NVP) was determined from contact angle measurements using the vOCG, Fowkes’, and Owens‐Wendt methods (Harmonic and Geometric mean approximation equations). A good correlation between the results of the three methods was found. Polymer film was obtained by melt extrusion and grafted with NVP using UV light and benzophenone (BP) as photoinitiator. Comparison of SFE results with those from the literature for pristine PLA showed that the correlation depends on the method used for contact angle determination and the properties of the liquids used for this measurement. This finding might promote the rational design of surface modification of biocompatible films.

Capacity Enhancement with Joint Subchannel and Power Allocation Scheme for OFDMA Femtocell Networks

Efficient power and subchannel allocation methods are required for orthogonal frequency division multiple access (OFDMA) based femtocell networks to improve the capacity of the system. This paper considers a joint subchannel and power allocation algorithm with capacity maximization for downlink of an OFDMA based femtocell networks. In the proposed algorithm subchannel allocation is first performed based on signal to interference plus noise ratio (SINR) of the channel with equal power distribution. Then for enhancing capacity with optimal power allocation, successive convex approximation (SCA) based power optimization is adopted. The effect of Arithmetic geometric mean (AGM) approximation with SCA on power optimization is also investigated. The optimal power is subsequently distributed by water-filling algorithm.

Approximate joint diagonalization and geometric mean of symmetric positive definite matrices.

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.

Distributed resource allocation in cognitive and cooperative ad hoc networks through joint routing, relay selection and spectrum allocation

Cooperative relaying and dynamic-spectrum-access/cognitive techniques are promising solutions to increase the capacity and reliability of wireless links by exploiting the spatial and frequency diversity of the wireless channel. Yet, the combined use of cooperative relaying and dynamic spectrum access in multi-hop networks with decentralized control is far from being well understood.We study the problem of network throughput maximization in cognitive and cooperative ad hoc networks through joint optimization of routing, relay assignment and spectrum allocation. We derive a decentralized algorithm that solves the power and spectrum allocation problem for two common cooperative transmission schemes, decode-and-forward (DF) and amplify-and-forward (AF), based on convex optimization and arithmetic–geometric mean approximation techniques. We then propose and design a practical medium access control protocol in which the probability of accessing the channel for a given node depends on a local utility function determined as the solution of the joint routing, relay selection, and dynamic spectrum allocation problem. Therefore, the algorithm aims at maximizing the network throughput through local control actions and with localized information only.Through discrete-event network simulations, we finally demonstrate that the protocol provides significant throughput gains with respect to baseline solutions.

Biaxiality-induced magnetic field effects in bent-core nematics: Molecular-field and Landau theory

Nematic liquid crystals composed of bent-core molecules exhibit unusual properties, including an enhanced Cotton-Mouton effect and an increasing isotropic (paranematic)-nematic phase transition temperature as a function of magnetic field. These systems are thought to be good candidate biaxial liquid crystals. Prompted by these experiments, we investigate theoretically the effect of molecular biaxiality on magnetic-field-induced phenomena for nematic liquid crystals, using both molecular field and Landau theory. The geometric mean approximation is used in order to specify the degree of molecular biaxiality using a single parameter. We reproduce experimental field-induced phenomena and predict also an experimentally accessible magnetic critical point. The Cotton-Mouton effect and temperature dependence of the paranematic-nematic phase transition are more pronounced with increased molecular biaxiality. We compare our theoretical approaches and make contact with recent relevant experimental results on bent-core molecular systems.

Spectral analysis for wide energy channels

Abstract. For energetic particle measurements whose spectra follow a power law, it is often challenging to define a characteristic ("effective") energy of an energy channel. In order to avoid time-consuming calculations, the geometric mean is often used as an approximation for the effective energy. This approximation is considered to be pretty good. It is, however, potentially inadequate in cases with wide energy channels and soft spectral slopes. In order to determine the limits of the goodness of the approximation, we derive formulas to calculate the deviation of the effective energy, phase space density and energy density based on the geometric-mean approximation from those based on the power law. The results show that the geometric-mean approximation is usually adequate and that corrections are needed only in extraordinary cases.

Effect of composition on thermal conductivity of MgO–Nd2Zr2O7 composites for inert matrix materials

Inert matrix fuels based on magnesium oxide (MgO) as the inert phase have been of historic interest due to its high thermal conductivity. Minor actinide-bearing phases possessing an A2B2O7 pyrochlore structure are also believed to contain a range of favorable attributes, suggesting a possible pairing with MgO to yield a high performance inert matrix fuel. The thermal diffusivity, heat capacity, and thermal expansion of MgO–Nd2Zr2O7 composites were measured from room temperature to 1273K, where the MgO phase content was varied from 40 to 70 volume percent. The thermal conductivity of each composition was calculated using these results and then compared to widely employed methods to approximate the thermal conductivity of composite materials based upon the properties of the constituent phases. Results suggest that use of either a rule of mixtures or geometric mean approximation for the thermal conductivity of composite systems such as this one would be subject to significant uncertainties when the constituent properties widely differ. A sigmoidal average of the upper and lower Hashin–Shtrikman bounds was found to be in good agreement with the thermal conductivity of the composites as determined experimentally.

GEO-MIX-SELF calculations of the elastic properties of a textured graphite sample at different hydrostatic pressures

The recently developed GEO-MIX-SELF approximation (GMS) is applied to interpret the pressure dependence of the longitudinal ultrasonic wave velocities in a polycrystalline graphite sample that has already been investigated in a wide range of experimental contexts. Graphite single crystals have extremely anisotropic elastic properties, making this sample a challenging test to demonstrate the potential of the GMS method. GMS combines elements of well known self-consistent algorithms and of the geometric mean approximation. It is able to consider mixtures of different polycrystalline phases, each with its own nonspherical grain shape and preferred orientation (texture). Pores and `cracks', typical for bulk graphite, are modeled as phases with `empty' grains. The pressure dependence (up to 150 MPa) of the experimental wave velocities can be well explained using the known texture of the sample by fitting the shape parameters and volume fractions of the graphite grains, cracks and spherical pores. The pressure dependence of these parameters describes a reasonable scenario for the closing of the cracks and pores with increasing pressure.

The Influence of a Potential Diffusion Control on the Outcome of Modular Polymer–Polymer Click Conjugations

AbstractThe only reliable factors for assessing if a polymer–polymer click conjugation is successful are an assessment of the number average $\overline {M} _{{\rm n}} $ of the resulting molecular weight distribution (MWD) and the shape of the number, respectively, concentration distribution as was demonstrated earlier. The present contribution explores if a potential chain length dependence (CLD) of the conjugation reaction—caused by a potential diffusion control—can affect the outcome of the polymer–polymer click ligation process. The CLD of the click rate coefficient, $k_{{\rm click}}^{{\rm i},{\rm j}} $, was implemented via a geometric mean approximation for chain length dependent termination, employing α‐values from 1.0 to 0.2. The number averages $\overline {M} _{{\rm n}} $ and weight averages $\overline {M} _{{\rm w}} $ (and thus the PDIs) of the initial distributions were varied widely to explore under which conditions CLD effects have the largest impact. The present study clearly demonstrates that—if diffusion effects and thus a CLD of $k_{{\rm click}}^{{\rm i},{\rm j}} $—play a role in polymer–polymer ligation, the MWDs generated through conjugation display a marked conversion dependence, i.e., $M_{{\rm n}}^{{\rm click}} (x)$, with the maximum $M_{{\rm n}}^{{\rm click}} $ being reached at quantitative conversions. While the $\overline {M} _{{\rm n}} $ reached at quantitative conversions coincides with the $\overline {M} _{{\rm n}} $ obtained under CL‐independent conditions, the final $\overline {M} _{{\rm w}} $ of both the CLD and non‐CLD differs significantly. While the present theoretical study cannot replace the actual experiment, it unambiguously demonstrates that polymer–polymer click conjugations can be influenced by CLD effects. Given the larger α‐values for smaller chain length, broad distributions in the range up to approximately 3 000 Da are especially affected.magnified image

On the use of the geometric mean approximation in estimating the effective hygro-elastic behavior of fiber-reinforced composites

A geometric averaging is employed in a scale transition model dedicated to determining the effective hygro-elastic macroscopic properties of a fiber-reinforced composites from the properties of its constituents (i.e., the fiber and the matrix), for the first time. Calculations of macroscopic and local stresses due to hygroscopic loads are also carried out. Numerical computations performed according to the geometric mean are compared to the corresponding predictions given by the arithmetic average, classically used in the field of micro-mechanical modeling.

Dependence of termination kinetics in methyl acrylate-dodecyl acrylate free-radical copolymerization on comonomer composition, pressure, temperature and conversion

The termination kinetics of the free-radical bulk copolymerisation of dodecyl acrylate (DA) and methyl acrylate (MA) has been investigated at various monomer mole fractions between 15 and 50°C and up to 2000 bar. The ratio of termination to propagation rate coefficients, (k t /k p ) copo , is measured via the single pulse-pulsed laser polymerisation (SP-PLP) technique. Chain-length averaged k t,copo are deduced from (k t /k p ) copo in conjunction with k t,copo data that are estimated via a simplified version of the implicit penultimate unit effect (IPUE) model. At low and moderate degrees of monomer conversion extended ranges of almost constant k t are observed where termination is controlled by segmental diffusion with important contributions of steric effects. These plateau k t values are significantly - by almost two orders of magnitude - different for MA and DA. The increase with MA content of k t,copo is adequately described by a penultimate unit model which uses the geometric mean approximation for estimating rate coefficients of cross-termination between radicals of different free-radical terminus. The model applies within the entire pressure and temperature range of the present study. At high degrees of monomer conversion, at and above 50%, homo-k t and k t,copo are almost insentive toward the composition of the monomer mixture. The termination rate under these conditions is essentially controlled by reaction diffusion.

Penultimate unit effects in free radical copolymerization

The penultimate unit effects (PUEs) on the propagation, termination, and reversible addition-fragmentation chain transfer (RAFT) processes in free radical copolymerization are discussed on the basis of recent publications. The propriety of the implicit and explicit PUE models in propagation and chain transfer processes is commented. The penultimate termination model with the geometric-mean approximation and the related rate equation are highlighted.

Termination Kinetics of Free-Radical Methyl Methacrylate - Dodecyl Acrylate and Dodecyl Methacrylate - Methyl Acrylate Copolymerizations

Copolymerization termination rate coefficients (kt) of the methyl methacrylate–dodecyl acrylate (MMA–DA) and dodecyl methacrylate–methyl acrylate (DMA–MA) systems at 40°C and 1000 bar have been measured using the single pulse (SP)–pulsed laser polymerization (PLP) technique. Plateau regions of kt are observed in the initial polymerization period. The region of constant kt increases with the size of the alkyl ester group, e.g. it extends up to at least 60% monomer conversion in DA and DMA homopolymerizations. The plateau kt values of MA and MMA are significantly above the corresponding DA and DMA values. The kt penultimate unit effect model, which uses the so-called geometric mean approximation, is well suited for representation of the dependence on monomer composition of the plateau kt values. The dependence of kt on monomer composition is quite different at low and at high degrees of monomer conversion. The reason behind this is seen in different types of diffusion control being operative at low and at high degrees of monomer conversion. The plateau-type behaviour is assigned to segmental diffusion control, whereas high-conversion kt is controlled by reaction diffusion.

Modeling the Crystal Shape of Polar Organic Materials: Prediction of Urea Crystals Grown from Polar and Nonpolar Solvents

The shape of a crystal is an important characteristic which can influence fundamental properties of the material, including the rate of dissolution, solubility, stability in storage, and compressibility. Understanding the mechanism by which a material with distinct internal structure crystallizes and forms different shapes will enable the engineer to manipulate the crystallization process, so that the desired shape is obtained. There are numerous models for predicting the shape of vapor-grown compounds. Three of the most prominent models: the Bravais−Friedel−Donnay−Harker (BFDH) model, the equilibrium model, and the attachment energy model, have been explored in this article. In addition, we have developed a model based on a detailed analysis of the BCF growth mechanism supplemented by additional terms to account for the adhesion surface energy at the solid−liquid interface. Polar solute−nonpolar solvent interactions were incorporated into the model by a geometric mean approximation for the adhesion surf...

Free-Radical Copolymerization of Styrene and Diethyl Fumarate. Penultimate-Unit Effects on Both Propagation and Termination Processes

The propagation and termination processes of the bulk copolymerization of styrene (1) and diethyl fumarate (2) at 40 °C were examined on the basis of a complete set of experimental data obtained by rotating sector and other experiments. The composition curve conformed to the terminal model with a standard deviation of about 1.5%. However, an exceptionally strong penultimate-unit effect (PUE) for the propagation of the terminal radical 1 (s1 = k211/k111 = 0.055) was observed. The measured termination rate constants were describable by the penultimate termination model with a geometric mean approximation, showing that a strong PUE exists also on the termination of the terminal radical 1 (δ1 = (kt21·21/kt11·11)1/2 = 0.067). The geometric mean approximation was justified by the shielding effect arguments. These analyses suggest that the frequency factors of the propagation and termination rate constants are somehow correlated with each other. On the basis of these results, a general, yet simple two-parameter ...

Gas sorption, diffusion, and permeation in poly(dimethylsiloxane)

The permeability of poly(dimethylsiloxane) [PDMS] to H2, O2, N2, CO2, CH4, C2H6, C3H8, CF4, C2F6, and C3F8, and solubility of these penetrants were determined as a function of pressure at 35 °C. Permeability coefficients of perfluorinated penetrants (CF4, C2F6, and C3F8) are approximately an order of magnitude lower than those of their hydrocarbon analogs (CH4, C2H6, and C3H8), and the perfluorocarbon permeabilities are significantly lower than even permanent gas permeability coefficients. This result is ascribed to very low perfluorocarbon solubilities in hydrocarbon-based PDMS coupled with low diffusion coefficients relative to those of their hydrocarbon analogs. The perfluorocarbons are sparingly soluble in PDMS and exhibit linear sorption isotherms. The Flory–Huggins interaction parameters for perfluorocarbon penetrants are substantially greater than those of their hydrocarbon analogs, indicating less favorable energetics of mixing perfluorocarbons with PDMS. Based on the sorption results and conventional lattice solution theory with a coordination number of 10, the formation of a single C3F8/PDMS segment pair requires 460 J/mol more energy than the formation of a C3H8/PDMS pair. A breakdown in the geometric mean approximation of the interaction energy between fluorocarbons and hydrocarbons was observed. These results are consistent with the solubility behavior of hydrocarbon–fluorocarbon liquid mixtures and hydrocarbon and fluorocarbon gas solubility in hydrocarbon liquids. From the permeability and sorption data, diffusion coefficients were determined as a function of penetrant concentration. Perfluorocarbon diffusion coefficients are lower than those of their hydrocarbon analogs, consistent with the larger size of the fluorocarbons. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 415–434, 2000

Termination Kinetics in Free-Radical Bulk Copolymerization: The Systems Dodecyl Acrylate−Dodecyl Methacrylate and Dodecyl Acrylate−Methyl Acrylate

The termination kinetics in free-radical bulk copolymerization of the dodecyl acrylate (DA)−dodecyl methacrylate (DMA) and DA−methyl acrylate (MA) systems has been studied by exclusively applying laser-assisted techniques. The ratio of termination to propagation rate coefficients, kt/kp, is measured by the SP−PLP technique, and kp is deduced from PLP−SEC. kt is determined up to fairly large degrees of overall monomer conversion, at least up to 40%. The discussion of the present paper focuses on the initial plateau region of constant kt, which extends up to about 15% in MA homopolymerizations and up to 70% in DA homopolymerizations. Among the existing models for copolymerization kt, the treatment that considers penultimate units and uses the geometric mean approximation to estimate cross-termination turns out to be excellently applicable. It is particularly noteworthy that plotting log kt vs monomer mole fraction (fi) yields a linear correlation for each of the two copolymerization systems. The excellent fit of kt,copo by the penultimate unit model provides some evidence that steric effects significantly contribute to termination rate control in (meth)acrylate free-radical polymerization at low and moderate degrees of monomer conversion. The entire set of experiments has been carried out at high pressure (1000 bar, 40 °C) where the signal-to-noise ratio of the SP−PLP experiments is very satisfactory. No reason is seen why the conclusions derived from these measurements should not be valid at other reaction conditions including ambient pressure.

Thermodynamics of n-Alkane Mixtures and Polyethylene−n-Alkane Solutions: Comparison of Theory and Experiment

We have applied the lattice Born-Green-Yvon theory to the study of hydrocarbon mixtures. In doing so we have fit pure component data in order to determine characteristic parameters for the pure fluids and have used the geometric mean approximation for the mixed-segment interaction energy. As a result, we have been able to make predictions for a wide variety of solution properties without fitting any data on the mixtures and have compared our predictions with experimental results. For binary alkane solutions made from combinations of n-hexane, n-octane, n-decane, and n-hexadecane we have found excellent agreement between theory and experiment for pressure-volume isotherms, volume changes on mixing and its pressure dependence, and equilibrium solution vapor pressures as a function of composition and temperature. For polyethylene-n-alkane solutions we have been able to predict the lower critical solution temperature and its dependence on the chain length of the alkane solvent, as well as the full coexistence curve and its variation with the polymer molecular weight.