Polynomial Distribution Research Articles

Multivariate Network Meta-Analysis of Survival Function Parameters

Recently, network meta-analysis of survival data with a multi-dimensional treatment effect was introduced using fractional polynomial (FP) distribution. With these models, the hazard ratio is not assumed to be constant over time, thereby reducing the possibility of violating consistency in indirect comparisons. However, beyond the FP models it is challenging to assess parametric distributions often used for cost-effectiveness models in health technology assessments (HTA). We aim to develop a two-step network meta-analysis (NMA) for time-to-event data. First, for each arm of every randomized controlled trial (RCT) connected in the network of evidence simulated patient data were fit to alternative parametric distributions, including exponential, Weibull, Gompertz, log-normal, log logistic, gamma, and generalized gamma. For each distribution, the resulting scale and shape parameters per arm were then included in a multivariate NMA, which preserved randomization and accounted for the correlation between the parameters. Results were compared to FP models to validate results and evaluate any differences. An illustrative analysis is presented for a network of RCTs evaluating interventions for advanced melanoma. The NMA was assessed using alternative distributions, which were compared using Akaike information criterion, which would facilitate model averaging to propagate structural uncertainty in a cost-effectiveness analysis. Based on the generalized gamma distribution, the difference in mu, Q, and sigma parameters for each treatment versus dacarbazine (DTIC) were: Non-DTIC: 0.41 (-0.19,1.01), 0.12 (-0.08,0.32), 0.39 (-0.7,1.49); DTIC+ Interferon (IFN): 0.24 (-0.15,0.69), -0.07 (-0.24,0.11), 0.45 (-0.87,1.66); DTIC+non-IFN: 0.22 (-0.13,0.65), -0.05 (-0.21,0.11), -0.17 (-1.13,1.1). A two-step NMA of survival data allows for a straightforward comparison of alternative parametric distributions in terms of goodness of fit by avoiding the need to specify each distribution in WinBUGS. This approach provides a more generalizable evidence synthesis framework for HTA.

Theory and numerical simulation for a design of broadband constant beam pattern transducer

The theory and numerical simulations for the constant beam pattern (CBP) broadband transducer design are introduced. For a large spherical transducer, if the normal direction radial velocity distribution on the surface of the sphere transducer is a single Legendre polynomial, the far field angular beam pattern shows the same Legendre polynomial distribution due to a spherical Hankel function asymptotic approximation. Arbitrary radial velocity shading functions, including the equal sidelobe classic Dolph-Chebyshev shading, can be expanded by the Legendre series per Sturm-Liouville theory, such that the converging acoustic beam pattern displays the same angular shape as the original shading function itself for the broadband CBP transducer.

Elastic field of a spherical inclusion with non-uniform eigenfields in second strain gradient elasticity

In this paper, within the complete form of Mindlin’s second strain gradient theory, the elastic field of an isolated spherical inclusion embedded in an infinitely extended homogeneous isotropic medium due to a non-uniform distribution of eigenfields is determined. These eigenfields, in addition to eigenstrain, comprise eigen double and eigen triple strains. After the derivation of a closed-form expression for Green’s function associated with the problem, two different cases of non-uniform distribution of the eigenfields are considered as follows: (i) radial distribution, i.e. the distributions of the eigenfields are functions of only the radial distance of points from the centre of inclusion, and (ii) polynomial distribution, i.e. the distributions of the eigenfields are polynomial functions in the Cartesian coordinates of points. While the obtained solution for the elastic field of the latter case takes the form of an infinite series, the solution to the former case is represented in a closed form. Moreover, Eshelby’s tensors associated with the two mentioned cases are obtained.

High order sub-cell finite volume schemes for solving hyperbolic conservation laws I: basic formulation and one-dimensional analysis

In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with $\mathrm{P_NP_M}$ technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.

PROBABILISTIC SPACE FOR CALCULATION OF PROBABILISTIC CHARACTERISTICS OF A THREE-PARAMETER QUEUEING SYSTEM MODEL

A model of queueing theory is proposed that describes a queueing system with three parameters, which has important practical applications. The model is based on the continuous time Markov process with a discrete number of states. The model is formalized by a probabilistic space in which the space of elementary events is a set of inconsistent states of the queueing system; and the probabilistic measure is a probability distribution corresponding to a set of elementary events, that is, each elementary event is associated with the probability of the system staying in this state, for each fixed time moment. The model is represented by a system of ordinary differential equations, compiled by methods of queueing theory (Kolmogorov equations). To find the solution of the system of equations, the method of generating functions is used. For the generating function, a partial differential equation is obtained. Finding the generating function completes the construction of a probability space. The latter means that for any random variables and functions defined on the resulting probability space, one can find their probabilistic characteristics. In particular, analytical expressions of the moments (mathematical expectations and variances) of random functions that depend on time are obtained. The peculiarity of finding a solution is that it is obtained not from the probability distribution, but directly from the partial differential equation, which represents a system of ordinary differential equations. For the probability distribution, the solution was found by a combinatorial method, which made it possible to significantly reduce the computations. To apply the formulas in engineering calculations, we consider the stationary case, to which a considerable simplification of the calculations corresponds. A relationship between a system of differential equations and a polynomial distribution known in probability theory is shown. The results are used in the analysis of the reliability of the operation of scalable computing systems; graphical implementation is shown

Propagation of input uncertainties in particle transport and the distribution of the sum of n independent stochastic variables a generalization of the Irwin–Hall distribution

We show, under wide hypothesis, that the problem of forward propagation of uncertainties in a physical model reduces to a well known problem in probability theory: the determination of the weighted sum of N independent stochastic variables obeying arbitrary distributions supported on bounded intervals. We determine various exact expressions for the probability distribution of the sum, possibly weighted, of N independent stochastic variables obeying arbitrary polynomial distributions on different intervals. These exact results provide the mathematical basis for the treatment of the given problem. We discuss the relevance of these findings for parameter uncertainty quantification in the context of particle transport computations. As a byproduct an explicit formula for the convolution of an arbitrary number of polynomials supported on bounded intervals is obtained.

Polynomial probability distribution estimation using the method of moments.

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation.

Solving the Quantity Element Using New Numerical Techniques on the Discontinues Boundary Element Method

This paper deals with solving the quantity element using new numerical techniques on discontinues boundary element method (DBEM). The common practice in getting solution with BEM is using constant element and for that, in a Sub-parametric element, quantity has a constant value along the element and geometry discretization is supposed to have a linear variation. But using higher order (polynomial) distribution of quantity over elements could have a better description of physical process. For this, the corresponding discretized expressions based on new techniques are derived and used for solution of Laplace equation. Many results for the quantity elements are presented and discussed for the ellipse at various diameters and mesh numbers.

Polynomial logistic distribution associated with a cubic polynomial

ABSTRACTLet P(x) be a polynomial monotone increasing on ( − ∞, +∞). The probability distribution possessing the distribution function is called the polynomial logistic distribution with associated polynomial P. This has recently been introduced by Koutras et al., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided.

High order sub-cell finite volume schemes for solving hyperbolic conservation laws II: Extension to two-dimensional systems on unstructured grids

In this paper, the second in a series, the high order sub-cell finite volume method is extended to two-dimensional hyperbolic systems on unstructured quadrilateral grids. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume method is applied to each of the sub-cells. The average values on the sub-cells belonging to current and face neighboring main cells are used to reconstruct a common polynomial distribution of a dependent variable on current main cell. This method can achieve high order accuracy using a compact stencil. The focus of this paper is to study the performance of the sub-cell finite volume method on two-dimensional unstructured quadrilateral grids and to verify that high order accuracy can be achieved using face neighboring sub-cells only. Fourier analysis is performed to analyze the dispersion and dissipation properties of the two-dimensional sub-cell finite volume schemes. To capture the discontinuities, the paper proposes a cell-based multi-dimensional limiting procedure using only face-neighboring main cells. Several benchmark test cases are simulated to validate the proposed sub-cell finite volume schemes and the multi-dimensional limiting procedure.

Polynomial and multivariate mapping-based triple-key approach for secure key distribution in wireless sensor networks

Distribution of cryptographic key plays a vital role to improve the security and integrity in Wireless Sensor Networks (WSN). Due to high resilience to the attacks, the pairwise key distribution schemes are highly preferred than other schemes. However, due to the resource constraints on the nodes and the threat of node capture, pairwise key establishment in the sensor networks is a challenging task. The security performance of these schemes decreases with the increase in the number of compromised nodes. To overcome these issues, a Polynomial and Multivariate Mapping-Based Triple-Key (PMMTK) distribution approach is proposed in this paper. It involves calculation of individual key of the nodes and common triple-key. Our proposed method achieves higher data freshness, throughput, average number of clusters, overall network lifetime, network connectivity, better packet delivery ratio and lower end-to-end delay, average number of key establishment and total authentication overhead.

A design of broadband constant beam pattern transducers

A constant beam pattern (CBP) transducer is an acoustic transducer, where its beam patterns are independent of frequency. The theory and numerical simulations for constant beam pattern transducer design are introduced. For a hemispherical design, if the radial velocity distribution on the surface of a conventional hemisphere transducer is a single complete Legendre polynomial, the far field angular beam pattern shows the same Legendre polynomial distribution for all frequencies above its cut-off frequency, based on spherical Hankel function asymptotic approximation to the solutions from Helmholtz wave equation. Because of orthogonality, Legendre polynomials form a complete set, per Sturm-Liouville theory that an arbitrary velocity shading function can be expanded by Legendre series. Each polynomial within this Legendre series contributes its share to the far field, such that the converging acoustic beam pattern displays the same shape as the original shading function itself. Numerical simulations with various samples include (cos(theta))^3 and Gaussian, a well as equal sidelobe suppression classic Dolph-Chebyshev functions used as shading and achievable as beam patterns for a broadband frequency range.

An approach for bending and transient dynamic analysis of integrated thermal protection system with temperature-dependent material properties

A C0 higher-order layerwise finite element model combined with homogenization techniques for bending and transient dynamic analysis of integrated thermal protection system (ITPS) panel is presented. The ITPS panel is homogenized as an equivalent sandwich plate, and a reduced temperature field for the entire panel is derived. An advanced analytical equivalent model based on energy approach and Castigliano’s theorem for the mechanical properties of the trapezoidal corrugated core is presented, which considers the effects of temperature-dependent material properties. The proposed layerwise theory which is developed for a sandwich plate, assumes linear, quadratic and cubic polynomial distributions of displacements for middle layer and linear polynomial distributions of displacements for top and bottom layers. And the effect of transverse normal strains is taken into account in the layerwise theory. A four-noded isoparametric element with 16 degrees of freedom per node is employed to model the equivalent sandwich plate. The accuracy of the proposed method is verified by comparison with the results by three-dimensional (3D) finite element analysis. It has been shown that the proposed method requires significantly less computational effort and agrees well with the finite element results.

A 3rd-order polynomial temperature profile model for the heating and evaporation of moving droplets

Droplet heating and evaporation is of fundamental importance in various engineering applications. Based on the assumption that droplets’ internal temperature follows a third-order polynomial distribution, a new liquid phase model is proposed, taking into account both temperature gradient and internal circulation inside the droplet in terms of Hill’s spherical vortex. The new liquid model and other commonly used liquid models are compared with the experimental data outlined in the literature. The internal temperature predicted by the new model matches the experimental data of an n-Decane droplet heating best, whereby the enhancement of heat transport by internal circulation is correctly reflected. In comparison, other models always predict the monotonic variation of internal temperature during the initial heating period, which deviates widely from the experimental data. The new model displays the best performance with regard to predicting droplet evaporation in the later vaporizing period, whereas the parabolic temperature profile model and conduction limit model significantly underestimate and overestimate the droplet evaporation rate, respectively. The CPU requirement for the new model is far less than that for the conduction limit model and the effective thermal conductivity model with an order of magnitude, which shows its potential for implementation into CFD codes.

Fluctuation and arching formation of very dense and slow pebble flow in a silo bed

ABSTRACTImproved particle tracking velocimetry (improved PTV) is proposed for experimental study of very slow dense pebble flow in a silo bed by combining the relaxation method and Voronoï diagram. The improved PTV method is validated by comparison with DEM (discrete element method) simulation and experimental result. Velocity and fluctuation characteristics on different streamlines are investigated to show the macroscopic kinematics of pebble flow. Bulk arches on different locations of the pebble bed are identified. Two local arching characteristics including size distribution and horizontal span are analyzed, and their effects on whole flow regime are discussed. It is found that the arch size distribution follows second-order polynomial distribution in semi-logarithmic scale. The horizontal span of arches provides evidence for transition between the ordered and disordered regimes. The probability distribution of successive angles between the neighbors of arching particles is a good indicator of arch stability. The relation between arch breakdown and contact network change is demonstrated by correlation analysis. High correlation between arching structure and mean velocity in targeted regions is indicated. The characteristic lifetime of arches and autocorrelation time of fluctuation velocity present positive correlation. The bulk arching dynamics is the main reason for fluctuations of particle velocities.

Development of a quasi-dimensional vaporization model for multi-component fuels focusing on forced convection and high temperature conditions

A new quasi-dimensional multi-component vaporization model considering the finite thermal conductivity and mass diffusivity within the droplet was constructed. First, the heat flux of conduction, enthalpy diffusion, and radiation absorption in the gas phase were calculated based on the Fourier’s law, a multi-diffusion sub-model, and a simplified analytical solution, respectively. The phase equilibrium at the gas–liquid interface was calculated by the ideal and real gas approaches according to the ambient pressure. For the liquid phase, the assumption of the quadratic polynomial distributions of the temperature and component concentration within the droplet was proposed in the quasi-dimensional model. Then, the proposed vaporization model was extensively validated by the experimental measurements, and good agreements were observed. Based on the computational results, the vaporization and movement behaviors of fuel droplets under forced convection conditions were further understood. Finally, by comparing with the zero-dimensional vaporization model with uniform temperature and component concentration distributions within the droplet and the one-dimensional vaporization model with finite thermal conductivity and mass diffusivity in the radical direction of the droplet, it is found that the quasi-dimensional model agrees better with the one-dimensional model than the zero-dimensional model, especially for the conditions with high ambient temperature and velocity. Sensitivity analysis indicates that the temperature gradient within the droplet plays a significantly important role in the droplet vaporization process.

Differential Evolutionary Selection and Natural Evolvability Observed in ALT Proteins of Human Filarial Parasites.

The abundant larval transcript (ALT-2) protein is present in all members of the Filarioidea, and has been reported as a potential candidate antigen for a subunit vaccine against lymphatic filariasis. To assess the potential for vaccine escape or heterologous protection, we examined the evolutionary selection acting on ALT-2. The ratios of nonsynonymous (K(a)) to synonymous (K(s)) mutation frequencies (ω) were calculated for the alt-2 genes of the lymphatic filariasis agents Brugia malayi and Wuchereria bancrofti and the agents of river blindness and African eyeworm disease Onchocerca volvulus and Loa loa. Two distinct Bayesian models of sequence evolution showed that ALT-2 of W. bancrofti and L. loa were under significant (P<0.05; P < 0.001) diversifying selection, while ALT-2 of B. malayi and O. volvulus were under neutral to stabilizing selection. Diversifying selection as measured by ω values was notably strongest on the region of ALT-2 encoding the signal peptide of L. loa and was elevated in the variable acidic domain of L. loa and W. bancrofti. Phylogenetic analysis indicated that the ALT-2 consensus sequences formed three clades: the first consisting of B. malayi, the second consisting of W. bancrofti, and the third containing both O. volvulus and L. loa. ALT-2 selection was therefore not predictable by phylogeny or pathology, as the two species parasitizing the eye were selected differently, as were the two species parasitizing the lymphatic system. The most immunogenic regions of L. loa and W. bancrofti ALT-2 sequence as modeled by antigenicity prediction analysis did not correspond with elevated levels of diversifying selection, and were not selected differently than predicted antigenic epitopes in B. malayi and O. volvulus. Measurements of ALT-2 evolvability made by χ2 analysis between alleles that were stable (O. volvulus and B. malayi) and those that were under diversifying selection (W. bancrofti and L. loa) indicated significant (P<0.01) deviations from a normal distribution for both W. bancrofti and L. loa. The relationship between evolvability and selection in L. loa followed a second order polynomial distribution (R2 = 0.89), indicating that the two factors relate to one another in accordance with an additional unknown factor. Taken together, these findings indicate discrete evolutionary drivers acting on ALT-2 of the four organisms examined, and the described variation has implications for design of novel vaccines and diagnostic reagents. Additionally, this represents the first mathematical description of evolvability in a naturally occurring setting.

A Sublaminate Generalized Unified Formulation for the analysis of composite structures

This paper presents a very flexible variable kinematics modeling technique for composite structures. The key point is the subdivision of the multilayered cross-section into numerical sublaminates, each consisting of one or more physical plies, and the formulation of model assumptions independently in each sublaminate according to the compact index notation known as Generalized Unified Formulation. Reference is made to the classical displacement-based and the mixed RMVT variational frameworks. For each sublaminate, the user may thus freely choose the order of the through-thickness polynomial distribution of each unknown as well as whether the unknown is described in an Equivalent Single Layer (ESL) or Layer-Wise (LW) approach. Sublaminates are always assembled in a Layer-Wise manner. This Sublaminate-GUF (S-GUF) is generally applicable to multilayered structures, but is particularly useful for sandwich panels, in which different models may be used for the thick, soft core and the thin, stiff skins. A Navier-type solution is used for demonstrating the validity of this approach by means of benchmark problems concerning the static response of sandwich plates.

Beyond the excised ensemble: modelling elliptic curve L-functions with random matrices

The ‘excised ensemble’, a random matrix model for the zeros of quadratic twist families of elliptic curve L-functions, was introduced by Dueñez et al (2012 J. Phys. A: Math. Theor. 45 115207) The excised model is motivated by a formula for central values of these L-functions in a paper by Kohnen and Zagier (1981 Invent. Math. 64 175–98). This formula indicates that for a finite set of L-functions from a family of quadratic twists, the central values are all either zero or are greater than some positive cutoff. The excised model imposes this same condition on the central values of characteristic polynomials of matrices from . Strangely, the cutoff on the characteristic polynomials that results in a convincing model for the L-function zeros is significantly smaller than that which we would obtain by naively transferring Kohnen and Zagier’s cutoff to the ensemble. In this current paper we investigate a modification to the excised model. It lacks the simplicity of the original excised ensemble, but it serves to explain the reason for the unexpectedly low cutoff in the original excised model. Additionally, the distribution of central L-values is ‘choppier’ than the distribution of characteristic polynomials, in the sense that it is a superposition of a series of peaks: the characteristic polynomial distribution is a smooth approximation to this. The excised model did not attempt to incorporate these successive peaks, only the initial cutoff. Here we experiment with including some of the structure of the L-value distribution. The conclusion is that a critical feature of a good model is to associate the correct mass to the first peak of the L-value distribution.

Effect of Filling Interval Time on Long-Term Mechanical Strength of Layered Cemented Tailing Backfill

To explore the influence that filling interval has on the mechanical strength of layered cemented tailing backfill (LCTB), uniaxial compression test is conducted on LCTB samples with a concentration of 70%, 72%, and 75% and a filling interval of 12 h, 24 h, 36 h, and 48 h. From the tests above, mechanical properties and failure modes of LCTB are acquired. The results are as follows: (1) The peak compressive strength of LCTB decreases as the filling interval increases, and it increases when the concentration grows with a certain length of filling interval. At the same time, the peak compressive strength and filling interval show polynomial distribution. (2) There are four stages during the loading process of cemented backfill specimen, that is, compression stage, linear elastic stage, crack extension stage, and undermines development stage. As the filling interval increases, CTB shows a failure mode of tensile failure-tensile shear failure transition-tensile and shear mixing destruction, which provides a theoretical basis for strength design and stability control of backfill.