Relative Curvature Research Articles

Bending Curvature of Particulate Composites Undergoing Uniform Linear Heat Flow

For particulate composites undergoing linear heat flow, a general expression for the relative curvature in the inclusions, relative to the radius of curvature in the matrix alone in the absence of inclusions, has been developed in terms of the ratios between the values of thermal and elastic constants of the matrix and the dispersed phase material (i.e., inclusion). Cases where the property ratios may have some specific values or extreme values, giving some particular results, have been identified and discussed. Results presented with the help of plots covering a wide range of ratios of the constituents’ properties are expected to be helpful in assessing the behavior of particulate composites under steady-state heat flow condition.

Influence of Meandering on the Estimation of Velocity and Shear Velocity in Cobble-Bed Channels

Field experiments carried out on five reaches of the meandering cobble-bed Beheshtabad River in central Iran showed that the position of the maximum velocity was independent of the relative submergence ( h/ d50 less or more than 4) and relative curvature ( R/W ) (where h is the flow depth, d50 is the median diameter of sediment, R is the radius of curvature, and W is the river width). A new method, called the boundary-layer characteristic method, was employed for the determination of shear velocity. The shear velocity values estimated with this method were in agreement with those obtained using the parabolic law. The law of the wall was valid for low and high relative curvatures in the inner, outer, and central zones of the meandering reaches. However, this law was not suitable for determining the shear velocity for the relative submergence range of 2.7<h/ d50 <9.6 due to its high sensitivity to near-bed velocities. Interestingly, it was possible to predict the flow velocity in meandering reaches using th...

Two-folds in nonsmooth dynamical systems

In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘two-fold’ singularities are difficult to find amongst physical models. They occur where the relative curvature between the flow field and the switching manifold is nonsymmetric about the discontinuity. Here we motivate their study with a local form model of nonlinear control that exhibits the two-fold singularity, where the flow is constant either side of a curved switching manifold. We describe the local dynamics around general two-fold singularities, then consider their effect on global dynamics via one parameter bifurcations of limit cycles.

Quantitative model for the kinetics of lyotropic phase transitions involving changes in monolayer curvature

We present a kinetic model for transformations between different self-assembled lipid structures. The model shows how data on the rates of phase transitions between mesophases of different geometries can be used to provide information on the mechanisms of the transformations and the transition states involved. This can be used, for example, to gain an insight into intermediate structures in cell membrane fission or fusion. In cases where the monolayer curvature changes on going from the initial to the final mesophase, we consider the phase transition to be driven primarily by the change in the relaxed curvature with pressure or temperature, which alters the relative curvature elastic energies of the two mesophase structures. Using this model, we have analyzed previously published kinetic data on the inter-conversion of inverse bicontinuous cubic phases in the 1-monoolein–30 wt% water system. The data are for a transition between QIIG and QIID phases, and our analysis indicates that the transition state more closely resembles the QIID than the QIIG phase. Using estimated values for the monolayer mean curvatures of the QIIG and QIID phases of −0.123 nm−1 and −0.133 nm−1, respectively, gives values for the monolayer mean curvature of the transition state of between −0.131 nm−1 and −0.132 nm−1. Furthermore, we estimate that several thousand molecules undergo the phase transition cooperatively within one “cooperative unit”, equivalent to 1–2 unit cells of QIIG or 4–10 unit cells of QIID.

Relative Curvature Measure for Heteroscedastic or Non Normal Nonlinear Regression

The Bates–Watts relative curvature measure can assess the validity of the linearized approximation in nonlinear regression models. However, it is developed based on an ordinary nonlinear regression in which the observation is assumed to be homoscedastically and normally distributed. In this article, we extend the original Bates–Watts relative curvature measure to one that can be applicable to nonlinear regression with heteroscedastic or non normal data, based on the transformation-both-sides (TBS) approach. In pharmacokinetic models, a diagnostic use of their measures is illustrated. By means of a simulation experiment, the performance of the relative curvature measure for the TBS approach is evaluated.

Contact of surfaces and contact characteristics of offset surfaces

Based on differential geometry, the contact problems of two surfaces are discussed in this paper. The relationship between the contact status of two surfaces and that of offset surfaces are also analyzed. For a 5-axis NC machining, some research such as optimization of cutter location and calculation of the geometrical cusp height are important. The research results indicate that the relative normal curvature is an important geometrical invariant for describing the contact state of two surfaces. For point contact two surfaces, the calculation equation for the second order remained error is given. For line contact two surfaces, the condition of the second order line contact is that the principal directions and curvatures of the two surfaces are the same along the contact curve. If two surfaces keep the second order line contact, their two offset surfaces will also keep the second order line contact, and their third order remained errors are also uniform with that of the two offset surfaces.

An experimental study on hot round jets impinging a concave surface

An experimental investigation of heat transfer due to a row of air jets impinging on a concave semi-cylindrical surface is presented. Heat transfer characteristics are measured using a heat thin foil technique and infrared thermography. Adiabatic wall temperatures and local heat transfer coefficients are determined by means of a linear regression method. The effect of high relative curvature ( d/ D) is investigated by changing the jet tube diameter (impinging surface diameter remaining constant). Reynolds number, injection temperature, spacing between adjacent jets and jet exit to surface spacing are also made to vary. Curvature has different effects over the adiabatic wall temperature and Nusselt number distributions. First, the curvature increase provokes a small growth of Nusselt number in the impingement region. On the other hand, curvature produces a confinement of the jet’s flow that has two consequences: stagnation of the adiabatic wall temperature and decrease of Nusselt number distribution.

Plasmon-Based Nanolenses Assembled on a Well-Defined DNA Template

The controlled and reproducible synthesis of closely spaced noble metal nanoparticle groupings is an essential step toward the rational design of nanostructures for surface enhanced Raman scattering with single-molecule sensitivity. In this communication, we demonstrate the facile synthesis of 5, 8, and 18 nm gold particle groupings on a well-defined DNA template by hybridizing monoconjugated gold-DNA building blocks. The obtained nanometer interparticle gaps should yield local intensity enhancements up to 4 orders of magnitude as estimated by Generalized Mie Theory. By tuning the particle size and relative curvatures, we calculate that the maximum enhancement can be optimized and localized on a specific part of the nanostructure. These groupings act as plasmon-based nanolenses assembled on a biomolecular backbone that can be functionalized in proximity to the region of highest field.

NUMERICAL APPROACH TO DETERMINING INSTANTANEOUS CONTACT REGION FOR CONJUGATE SURFACES

According to the defect of traditional method of determining instantaneous contact regions for conjugate surfaces, a numerical approach to the determination is proposed. A local coordinate system is built by using the surface unit tangent and unit normal at the contact point. Considering that the gap forming the boundary of instantaneous contact region in the direction of the common normal vectors is given, a system of nonlinear equations is built to fred the instantaneous contact boundary in local coordinate system, a modified Powell's hybrid algorithm of finite-difference approximation to the Jacobian used to solve the system. The new method simplifies the task of determining instantaneous contact regions without calculating curvature and relative curvature. The validity of the proposed approach is verified by an example of hypoid gears.

A Note on Parameterizations in Pharmacokinetic Compartment Models

In pharmacokinetics, compartment models often play an important role in the description of the concentration of the drug in the blood over time after its administration to a subject. In the inference in the compartment models in pharmacokinetics, unlike that of the usual nonlinear regression models, parameterizations with respect to the parameters related to pharmacokinetic indices are frequently applied, for some reasons of the constraint that the parameter takes only the positive value, facilitation of the compartment models, or interpretation of the relationships between the parameters and the demographic or physiological individual attributes. However, in practice no attention is paid to checking if the utilized parameterization is valid. In this article, the relative curvature measure that enables us to assess the intrinsic and parameter-effects nonlinearity in the model is applied for checking it from the latter nonlinearity. From a viewpoint of the relative curvature measure, we reconsider several well-known examples and demonstrate the parameterization checking.

Knuckle walking signal in the manual digits of Pan and Gorilla

This article examines the curvature of the manual proximal and middle phalanges of species belonging to Pan, Gorilla, Ateles, Macaca, Pongo, Hylobates, and Cebus to determine whether middle phalangeal curvature, when considered in conjunction with proximal phalangeal curvature, yields a locomotor signal. Prior studies have demonstrated the discriminatory power of proximal phalanges for separating suspensory species (including knuckle walkers) from pronograde quadrupedal species, but less emphasis has been placed on the distinguishing phalangeal characteristics of taxa within the suspensory category. This study demonstrates, first, that middle phalanges discriminate suspensory from nonsuspensory species, although not as cleanly as proximal phalanges. Finer discrimination of locomotor signals, including subtle differences among animals employing different modes of suspension, is possible through a comparison of the curvatures of the proximal phalanges and corresponding middle phalanges. Their relative curvature differs in quadrupeds, brachiators, and knuckle walkers. Knuckle walkers (Pan and Gorilla) have relatively little curvature of the middle phalanges coupled with marked curvature of the proximal phalanges, whereas brachiators (Ateles and Hylobates) display marked curvature of both proximal and middle phalanges, and pronograde quadrupeds (Cebus and Macaca) have relatively straight proximal and moderately curved middle phalanges. Quadrumanous climbers (Pongo) have a unique combination of traits, whereby curvature is high in both proximal and middle phalanges, but less so in the latter than the former. These differences, predictable on the basis of the biomechanical forces to which digits are subjected, may open a new venue for future research on the locomotor repertoire of prebipedal ancestors of hominins.

Relative mean curvature configurations for surfaces in ℝn, n ≥ 5

We define the relative mean curvature directions on surfaces immersed in ℝ n , n ≥ 4, generalizing the concept of mean curvature directions for surfaces in 4-space studied by Mello. We obtain their differential equations and study their corresponding generic configurations.

Geodesic relative curvature of instantaneous conjugation curve on base surface and geometrical differentials of third order

The four independent base geometrical differentials of third order of a surface are introduced. The equations for determining these geometrical differentials for some plain surfaces, which are widely used in gear tooth design and manufacturing, are geometrically deduced. The formula for determining the geodesic relative curvature of the ICC (instantaneous conjugation curve) in terms of parameters of conjugate motion and geometrical differentials up to third order is derived. Also derived are the formulas for determining the geodesic relative curvature of the ICC when the base conjugate surface is one of the plain surfaces. These formulas are instrumental in optimization of the parameters of the gear tooth design, the gear cutting tool design and the machine setting.

Assessing Nonlinearity in Compartment Models via the Relative Curvature Measure

In pharmacokinetics, compartment models often play an important role in the description of the concentration of the drug in the blood over time after its administration to an individual. Statistical inference in these models can be conducted based on a linear approximation with respect to the parameter related to pharmacokinetic indices, in the same way that the usual nonlinear regression models are dealt with. Therefore, it is necessary to assess the degree of nonlinearity in a compartment model and to evaluate its effect on the linear approximation. The relative curvature measure that enables us to assess the intrinsic and parameter-effects (PE) nonlinearity can be used, but in practice it has not been applied to compartment models in pharmacokinetics. One reason may be that the relative curvature measure cannot be directly applied to blood drug concentration data that exhibit heteroscedasticity. Therefore, the relative curvature measure including the heteroscedastic variance function was utilized to assess the nonlinearity in the compartment models, and in particular, the influences of some of the reparameterizations that are empirically used in fitting the compartment models were examined. Several examples showed that the reparameterized compartment model had less PE nonlinearity than the original compartment model, but that several reparameterizations could increase the PE nonlinearity. In addition, by means of a simulation experiment with heteroscedastic blood drug concentration data, the accuracy, and precision of the relative curvature measure with the heteroscedastic variance function were evaluated and compared with those of the original relative curvature measure. The results showed that the relative curvature measure with the variance function was not affected by heteroscedastic blood drug concentration data and could be utilized for the assessment of the nonlinearity in compartment models.

An alternative computational algorithm for calculating the nonlinearity of regression models with two parameters

This paper presents an alternative computational algorithm for finding the root mean square (RMS) curvature as a nonlinearity measure of nonlinear regression model (NLR model) with two parameters. Specially, the suggested algorithm is applied on a NLR model in which there is a conditionally linear parameter. The algorithm is mainly based on the work presented in [ D.M. Bates, D.G. Watts, Relative curvature measures of nonlinearity (with discussion), Journal of the Royal Statistical Society 42 (1) (1980) 1– 25]. The algorithm is suited for implementation using computer algebra systems (CAS) such as MAPLE, MATLAB, MACSYMA and MATHEMATICA. The researchers compare the results with the corresponding output of the S-PLUS Program for the same considering example and the results in some earlier works.

Model based statistical analysis of adsorption equilibrium data

A large group of separation problems can be solved using selective adsorption on suitable solids. A mathematical description of adsorption isotherms, which relate the equilibrium concentrations in the fluid phase to the loadings of the solid, could be used to design, observe and control such processes in an efficient way. However, the determination of the isotherms typically requires the identification of unknown parameters in postulated models from experimental data. While for the estimation of the parameters a number of tools and methods are available, a comprehensive analysis of the quality of the parameters is seldom performed. To estimate and characterize parameters obtained from adsorption measurements in this work a non-linear regression analysis was explored in combination with an extended statistical analysis. Hereby, the non-linearity method (“intrinsic” and “parameter-effect” non-linearity) proposed by Bates and Watts [1980. Relative curvature measures of non-linearity. Journal of the Royal Statistical Society: Series B (Methodological) 42, 1–25] was used to check the quality of parameters and the suitability of model/data combinations. The variances of the parameters are determined with the bootstrap method originally proposed by Efron and Tibshirani [1993. An Introduction to the Bootstrap. Chapman and Hall, CRC Press, London, Boca Raton.]. The later approach clearly overcomes some limitation of classical Fisher-information-matrix (FIM) method. By applying these statistical methods to different adsorption models and data sets, it was found that non-linearity method is a good tool to check the quality of the model/data combination. Furthermore, it was found that the confidence intervals of the parameters determined based on the bootstrap are larger than predicted by traditional methods.

Finite deformation model of simple shear of fault with microrotations: apparent strain localisation and en-echelon fracture pattern

Strain localisation is a widespread phenomenon often observed in shear and compressive loading of geomaterials, for example, the fault gouge. It is believed that the main mechanisms of strain localisation are strain softening and mismatch between dilatancy and pressure sensitivity. Observations show that gouge deformation is accompanied by considerable rotations of grains. In our previous work as a model for gouge material, we proposed a continuum description for an assembly of particles of equal radius in which the particle rotation is treated as an independent degree of freedom. We showed that there exist critical values of the model parameters for which the displacement gradient exhibits a pronounced localisation at the mid-surface layers of the fault, even in the absence of inelasticity. Here, we generalise the model to the case of finite deformations characteristic for the gouge deformation. We derive objective constitutive relationships relating the Jaumann rates of stress and moment stress to the relative strain and curvature rates, respectively. The model suggests that the pattern of localisation remains the same as in the linear case. However, the presence of the Jaumann terms leads to the emergence of non-zero normal stresses acting along and perpendicular to the shear layer (with zero hydrostatic pressure), and localised along the mid-line of the gouge; these stress components are absent in the linear model of simple shear. These additional normal stresses, albeit small, cause a change in the direction in which the maximal normal stresses act and in which en-echelon fracturing is formed.

On Spatial Euler–Savary Equations for Envelopes

Presented are three equations that are believed to be original and new to the kinematics community. These three equations are extensions of the planar Euler–Savary relations (for envelopes) to spatial relations. All three spatial forms parallel the existing well established planar Euler–Savary equations. The genesis of this work is rooted in a system of cylindroidal coordinates specifically developed to parameterize the kinematic geometry of generalized spatial gearing and consequently a brief discussion of such coordinates is provided. Hyperboloids of osculation are introduced by considering an instantaneously equivalent gear pair. These analog equations establish a relation between the kinematic geometry of hyperboloids of osculation in mesh (viz., second-order approximation to the axode motion) to the relative curvature of conjugate surfaces in direct contact (gear teeth). Planar Euler–Savary equations are presented first along with a discussion on the terms in each equation. This presentation provides the basis for the proposed spatial Euler–Savary analog equations. A lot of effort has been directed to establishing generalized spatial Euler–Savary equations resulting in many different expressions depending on the interpretation of the planar Euler–Savary equation. This work deals with the interpretation where contacting surfaces are taken as the spatial analog to the contacting planar curves.

The influence of contact-point position and contact-line trend on entraining velocity and nonzero principal relative normal curvature of cylindrical worm gearing

The object of this paper is to ascertain the influence of contact-point position and contact-line trend on the essential performance indexes (entraining velocity and radius of nonzero principal relative normal curvature) and to find their maximum and minimum values so as to provide criterions for evaluating (or to provide clues for improving) the performances of the existing worm gearings. The formulas for them are derived by the author for the orthogonal worm gearing on condition that the tooth profiles are unknown. The essential findings of this paper are as follows: these performance indexes are functions of both contact-point position and contact-line trend, but their maximum and minimum values are functions of contact-point position only. The maximum entraining velocity vector is equal to a fixed resultant vector in case the contact line is perpendicular to this fixed resultant vector. The minimum entraining velocity vector is equal to zero in case the contact line is tangent to it. The radius of nonzero principal relative normal curvature has its maximum and minimum values of opposite signs in case the contact line is tangent to each of the bisectors of the vertical angles included between the helix and its conjugate curve. It becomes zero in case the contact line is tangent to the helix or its conjugate curve.

Contact characteristics of conical involute gears with crossed axes

Based on the theory of gearing and the gear generating mechanism, this study investigates the contact characteristics of conical involute gear pairs with crossed axes. The meshing model with a reference rack as intermedium is established and tooth contact analysis (TCA) is performed to examine the meshing and bearing contact of the conical involute gear pairs composed of a pinion and a gear. Using the contact line of reference rack tooth surface, the path of contact is confirmed, and the calculation formula of principal direction, relative curvature, path of contact and ellipse of contact are derived. Finally, numerical examples are provided to demonstrate computational results and test gears are made for tooth-bearing tests. The conclusion verifies that the theory is applicable.