Elliptical Assumption Research Articles

Reconsidering assumptions in the analysis of muscle fibre cross-sectional area.

Cross-sectional area (CSA) is a fundamental variable in characterizing muscle mechanical properties. Typically, the CSA of a single muscle fibre is assessed by measuring either one or two diameters, and assuming the cross-section is either circular or elliptical in shape. However, fibre cross-sections have irregular shapes. The accuracy and precision of CSAs determined using circular and elliptical shape assumptions are unclear for mammalian skinned muscle fibres. Second harmonic generation imaging of skinned rabbit soleus fibres revealed that the circular assumption overstated real CSA by 5.3±25.9% whereas the elliptical assumption overstated real CSA by 2.8±6.9%. A preferred rotational alignment can bias the circular assumption, as real CSA was overstated by 22.1±24.8% when using the larger fibre diameter and understated by 11.4±13% when using the smaller fibre diameter. With 73% lower variable error and reduced bias, the elliptical assumption is superior to the circular assumption when assessing the CSA of skinned mammalian fibres.

Improved Gravity Wave Drag to Enhance Precipitation Simulation: A Case Study of Typhoon In-Fa

Traditional gravity wave drag parameterizations produce wind stresses that are insensitive to changing horizontal resolution in numerical weather prediction (NWP), partly due to the idealized elliptical assumption. This study employs the modified subgrid-scale orography scheme based on the Fourier transform into gravity wave drag scheme of the China Meteorological Administration Global Forecast System (CMA-GFS) to assess its impacts on simulating precipitation during the slow-moving period of Typhoon In-Fa after its landfall in Zhejiang Province, China. The simulation with the updated scheme can effectively reduce the accumulated precipitation bias of the control one and improve the simulation of precipitation distribution and intensity, especially in the hourly precipitation simulation. The improved scheme primarily influences the wind field of the low-level troposphere and also changes the convergence of the integrated water vapor transport and ascending motions related to the reduced precipitation biases. The modified scheme enhances the tendencies of the horizontal winds caused by the varying horizontal resolutions in the model, strengthening the sensitivity of the gravity wave drag across the horizontal scales. Results from medium-range forecasts indicate the modified scheme benefits the statistics scores of precipitation over China and also reduces root-mean-square errors of 2 m temperature and 10 m winds.

The PVD-Accelerated Soil Deposit Consolidation Based on Elliptic Cylindrical Model

One method to deal with the problem of soft soil is to accelerate consolidation by preloading and prefabricating a vertical drain (PVD). Consolidation analysis was based on a one-dimensional theory that required PVD as an equivalent circular well. Further studies on a simple approximate for consolidated soil were represented by equivalent permeability coefficients, kve. The equivalent conductivity coefficient is influenced by the soil and PVD permeability coefficients. The formulation of kve based on the influence area in cylindrical has been applied to a lot of construction projects. According to the comparative analysis of the classical consolidation theory, it is considered that the diameter of the circle is less representative. This study proposed a simple formulation of kve based on the elliptical assumption of influence area. The kvewas derived based on an equal average degree of consolidation in one dimension, which applied the elliptical coordinate for degree of consolidation in the radial direction. The formulation is based on an elliptical cross-section and a cylindrical coordinate formulation. The validation of this formula is conducted with numerical calculations using 2D FEM. The results show that the consolidation time in the elliptical discharge area is shorter than that in the circular discharge area. Doi: 10.28991/CEJ-2023-09-07-08 Full Text: PDF

Reduced order modeling for elliptic problems with high contrast diffusion coefficients

We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is com- monly made on parametric elliptic PDEs. We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar es- timates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.

Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

In this paper, we focus on nonasymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (nonconstant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called unadjusted Langevin algorithm in the Monte Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for total variation and L1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using stochastic analysis adapted to decreasing step setting.

Beyond Simpson's Rule: Accounting for Orientation and Ellipticity Assumptions.

Simpson's biplane rule (SBR) is considered the gold standard method for left ventricle (LV) volume quantification from echocardiography but relies on a summation-of-disks approach that makes assumptions about LV orientation and cross-sectional shape. We aim to identify key limiting factors in SBR and to develop a new robust standard for volume quantification. Three methods for computing LV volume were studied: (i) SBR, (ii) addition of a truncated basal cone (TBC) to SBR and (iii) a novel method of basal-oriented disks (BODs). Three retrospective cohorts representative of the young, adult healthy and heart failure populations were used to study the impact of anatomical variations in volume computations. Results reveal how basal slanting can cause over- and underestimation of volume, with errors by SBR and TBC >10 mL for slanting angles >6°. Only the BOD method correctly accounted for basal slanting, reducing relative volume errors by SBR from -2.23 ± 2.21% to -0.70 ± 1.91% in the adult population and similar qualitative performance in the other two cohorts. In conclusion, the summation of basal oriented disks, a novel interpretation of SBR, is a more accurate and precise method for estimating LV volume.

Projected flame length and flame extension length of horizontal buoyant jet flames

Abstract Jet fires are a type of dangerous fire caused by the sudden release of flammable materials. In this work, propane was used as fuel and released through a 7.15mm and 9.8 mm circular nozzle with the exit velocities were set ranging between 27 and 65 m/s. The data of this study are compared against theoretical measurement and prediction using previously suggested correlation for projected flame length and flame extension length. It was determined, the projected flame length gives a good prediction by using theoretical measurement while for the prediction of flame extension length, an elliptical shape assumption was suitable to be used after impinging point for a horizontal buoyant jet fire.

SLLN and annealed CLT for random walks in I.I.D. random environment on Cayley trees

We consider the random walk in an independent and identically distributed (i.i.d.) random environment on a Cayley graph of a finite free product of copies of Z and Z2. Such a Cayley graph is readily seen to be a regular tree. Under a uniform ellipticity assumption on the i.i.d. environment we show that the walk has positive speed and establish the annealed central limit theorem for the graph distance of the walker from the starting point.

Resolutivity and invariance for the Perron method for degenerate equations of divergence type

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation divA(x,∇u)=0 in a bounded open set Ω⊂Rn. The vector-valued function A satisfies the standard ellipticity assumptions with 1<p<∞ and a p-admissible weight w. We show that arbitrary perturbations on sets of (p,w)-capacity zero of continuous (and certain quasicontinuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p,w)-capacity zero.

On a class of variational problems with linear growth and radial symmetry

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.

On minimizers of an anisotropic liquid drop model

We consider a variant of Gamow’s liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape inC1-norm and quantify the rate of convergence. We also obtain a quantitative expansion of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.

Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

This paper considers the inverse boundary value problem for the equation ∇ ⋅ (σ∇u + a|∇u| p−2∇u) = 0. We give a procedure for the recovery of the coefficients σ and a from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case.

In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Holder continuity of the flow from $G^{\sigma}$ to $L^2$, where $\sigma\in(0,1)$ depends on the initial spectrum. Building on the analysis carried out by G. Metivier [\textit{Remarks on the well-posedness of the nonlinear Cauchy problem}, Contemp. Math. 2005], we show that ill-posedness follows from a long-time Cauchy-Kovalevskaya construction of a family of exact, highly oscillating, analytical solutions which are initially close to the null solution, and which grow exponentially fast in time. A specific difficulty resides in the observation time of instability. While in Sobolev spaces, this time is logarithmic in the frequency, in Gevrey spaces it is a power of the frequency. In particular, in Gevrey spaces the instability is recorded much later than in Sobolev spaces.

Anisotropic liquid drop models

Abstract We introduce and study certain variants of Gamow’s liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.

Functional central limit theorem for random walks in random environment defined on regular trees

We study Random Walks in an i.i.d. Random Environment (RWRE) defined on b-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on b-regular trees, with b≥4, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).

A unified complex noncentral Wishart type distribution inspired by massive MIMO systems

The eigenvalue distributions from a complex noncentral Wishart matrix S=XHX has been the subject of interest in various real world applications, where X is assumed to be complex matrix variate normally distributed with nonzero mean M and covariance Σ. This paper focuses on a weighted analytical representation of S to alleviate the restriction of normality; thereby allowing the choice of X to be complex matrix variate elliptically distributed for the practitioner. New results for eigenvalue distributions of more generalised forms are derived under this elliptical assumption, and investigated for certain members of the complex elliptical class. The distribution of the minimum eigenvalue enjoys particular attention. This theoretical investigation has proposed impact in communications systems (where massive datasets can be conveniently formulated in matrix terms), in particular the case where the noncentral matrix has rank one which is useful in practice.

Linear collective collocation approximation for parametric and stochastic elliptic PDEs

Consider the parametric elliptic problem where is a bounded Lipschitz domain, , , and the diffusion coefficients satisfy the uniform ellipticity assumption and are affinely dependent on . The parameter can be interpreted as either a deterministic or a random variable. A central question to be studied is as follows. Assume that there is a sequence of approximations with a certain error convergence rate in the energy norm of the space for the nonparametric problem at every point . Then under what assumptions does this sequence induce a sequence of approximations with the same error convergence rate for the parametric elliptic problem in the norm of the Bochner spaces ? We have solved this question using linear collective collocation methods, based on Lagrange polynomial interpolation on the parametric domain . Under very mild conditions, we show that these approximation methods give the same error convergence rate as for the nonparametric elliptic problem. In this sense the curse of dimensionality is broken by linear methods. Bibliography: 22 titles.

Modeling and simulation for cross-sectional ovalization of thin-walled tubes in continuous rotary straightening process

Cross-sectional ovalization of thin-walled circular steel tube caused by large plastic bending usually occurs at initial large bending stage in tube's continuous rotary straightening process. When the bending loads have been removed, there must be residual ovalization kept on the tube's cross-section during the whole straightening process, to induce the poor roundness and the reduction of the tube flexural stiffness. So the maximal residual cross-sectional ovalization should be predict accurately in order to be controlled under the allowable limit during the straightening process. In this paper, this issue is studied by the approaches of simulations and theoretical modeling. Firstly the finite element (FE) simulations for thin-walled tube straightening process are carried out on stainless steel tubes with different geometric parameters under different working rolls' bending radii, to measure the residual cross-sectional ovalization along the circumferential direction. The results imply that the profile of deformed cross-section is not standard ellipse and the appearance of the maximum residual ovalization is usually found in the direction perpendicular to the tube's centroid axis. Secondly the theoretical modeling procedure is carried out as follows. First of all, the normal strain and stress components are derived using the thin-shell large deformation kinematics. Then the cross-sectional ovalization in loading process is analyzed by the principle of virtual work. And then using the general geometric description without the elliptical assumption, a rational model is derived by the classic unloading rule to predict the maximal residual cross-sectional ovalization of thin-walled tube in straightening process. It is solved iteratively using Matlab ode23s function. The theoretical results are compared with the simulation ones. The relative errors are lower than 10%, which validates the effectiveness of the new model. Meanwhile the influences of tube's material, thickness and bending radius on maximum residual ovalization are found.

On multivariate separating Hill estimator under estimated location and scatter

ABSTRACTWe consider estimating the tail-index of a distribution under the assumption of multivariate ellipticity. Recently, a separating Hill estimator for multivariate elliptical distributions was proposed. This estimator is an affine invariant alternative to using the marginal observations in tail-index estimation and is hence unaffected by, e.g. change of units of measurement. However, the separating Hill estimator depends on the location and scatter of the elliptical distribution, which, in practice, have to be estimated. The effect of replacing the true location and scatter of the distribution by estimates has previously been only examined through simulations. In this article we show that the error caused by replacing the location and scatter of the distribution by estimates indeed is asymptotically negligible. This fact is essential for the practicality of the separating Hill estimator. In addition to providing the theoretical results, we present simulation results on the asymptotic behaviour of the estimators.

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006).The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.