Tables of the Inverse Laplace Transform of the Function [... formula ...]

The inverse transform, g(t) = ?^?1(e^?sβ), 0 < β < 1, is a stable law that arises in a number of different applications in chemical physics, polymer physics, solid-state physics, and applied mathematics. Because of its important applications, a number of investigators have suggested approximations to g(t). However, there have so far been no accurately calculated values available for checking or other purposes. We present here tables, accurate to six figures, of g(t) for a number of values of β between 0.25 and 0.999. In addition, since g(t), regarded as a function of β, is uni-modal with a peak occurring at t = t_max we both tabulate and graph t_max and 1/g(t_max) as a function of β, as well as giving polynomial approximations to 1/g(t_max).     

Intrinsic frequency for a systems approach to haemodynamic waveform analysis with clinical applications

The reductionist approach has dominated the fields of biology and medicine for nearly a century. Here, we present a systems science approach to the analysis of physiological waveforms in the context of a specific case, cardiovascular physiology. Our goal in this study is to introduce a methodology that allows for novel insight into cardiovascular physiology and to show proof of concept for a new index for the evaluation of the cardiovascular system through pressure wave analysis. This methodology uses a modified version of sparse time–frequency representation (STFR) to extract two dominant frequencies we refer to as intrinsic frequencies (IFs; ω₁ and ω₂). The IFs are the dominant frequencies of the instantaneous frequency of the coupled heart + aorta system before the closure of the aortic valve and the decoupled aorta after valve closure. In this study, we extract the IFs from a series of aortic pressure waves obtained from both clinical data and a computational model. Our results demonstrate that at the heart rate at which the left ventricular pulsatile workload is minimized the two IFs are equal (ω₁ = ω₂). Extracted IFs from clinical data indicate that at young ages the total frequency variation (Δω = ω₁ − ω₂) is close to zero and that Δω increases with age or disease (e.g. heart failure and hypertension). While the focus of this paper is the cardiovascular system, this approach can easily be extended to other physiological systems or any biological signal.     

Analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol-Duffing oscillators by extended homotopy analysis method

In this study, the extended homotopy analysis method (EHAM) is applied to derive the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled oscillators. The present paper not only introduces the rationale for the EHAM for MDOF oscillators, but also strengthens the availability of the conventional homotopy analysis method (HAM) in solving complex MDOF dynamical systems. Employing the EHAM for the two-degree-of-freedom (TDOF) coupled van der Pol-Duffing oscillator, the explicit analytical solutions of frequency ?? and displacements x ₁(t) and x ₂(t) are formulated for various initial conditions and physical parameters. To verify the accuracy and correctness of this approach, a number of comparisons are conducted between the results of the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the third-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if the time variable t progresses to a comparatively large domain in the time history responses.     

Drag force acting on a neuromast in the fish lateral line trunk canal. II. Analytical modelling of parameter dependencies

In Part I of this two-part study, the coupled flows external and internal to the fish lateral line trunk canal were consecutively calculated by solving the Navier–Stokes (N–S) equations numerically in each domain. With the external flow known, the solution for the internal flow was obtained using a parallelepiped to simulate the neuromast cupula present between a pair of consecutive pores, allowing the calculation of the drag force acting on the neuromast cupula. While physically rigorous and accurate, the numerical approach is tedious and inefficient since it does not readily reveal the parameter dependencies of the drag force. In Part II of this work we present an analytically based physical–mathematical model for rapidly calculating the drag force acting on a neuromast cupula. The cupula is well approximated as an immobile sphere located inside a tube-shaped canal segment of circular cross section containing a constant property fluid in a steady-periodic oscillating state of motion. The analytical expression derived for the dimensionless drag force is of the form |FN|/(|PL−PR|π(D/2)2)=f(d/D,Lt/D,ωD*), where |F_N| is the amplitude of the drag force; |P_L−P_R| is the amplitude of the pressure difference driving the flow in the interpore tube segment; d/D is the ratio of sphere diameter to tube diameter; L_t/D is the ratio of interpore tube segment length to tube diameter; and ωD*=ω(D/2)2/ν is the oscillating flow kinetic Reynolds number (a dimensionless frequency). Present results show that the dimensionless drag force amplitude increases with decreasing L_t/D and maximizes in the range 0.65≤d/D≤0.85, depending on the values of L_t/D and ωD*. It is also found that in the biologically relevant range of dimensionless frequencies 1≤ωD*≤20 and segment lengths 4≤L_t/D≤16, the sphere tube (neuromast–canal) system acts as a low-pass filter for values d/D≤0.75, approximately. For larger values of d/D the system is equally sensitive to all frequencies, but the drag force is significantly decreased. Comparisons with N–S calculations of the drag force show good agreement with the analytical model results. By revealing the parameter dependencies of the drag force, the model serves to guide biological understanding and the optimized design of corresponding bioinspired artificial sensors.     

Standardization of 63Ni by 4πβ Liquid Scintillation Spectrometry With 3H-Standard Efficiency Tracing

The low energy (E_βmax = 66.945 keV ± 0.004 keV) β-emitter ⁶³Ni has become increasingly important in the field of radionuclidic metrology. In addition to having a low β-endpoint energy, the relatively long half-life (101.1 a ± 1.4 a) makes it an appealing standard for such applications. This paper describes the recent preparation and calibration of a new solution Standard Reference Material of ⁶³Ni, SRM 4226C, released by the National Institute of Standards and Technology. The massic activity C_A for these standards was determined using 4πβ liquid scintillation (LS) spectrometry with ³H-standard efficiency tracing using the CIEMAT/NIST method, and is certified as 50.53 kBq ·g⁻¹ ± 0.46 Bq · g⁻¹ at the reference time of 1200 EST August 15, 1995. The uncertainty given is the expanded (coverage factor k = 2 and thus a 2 standard deviation estimate) uncertainty based on the evaluation of 28 different uncertainty components. These components were evaluated on the basis of an exhaustive number (976) of LS counting measurements investigating over 15 variables. Through the study of these variables it was found that LS cocktail water mass fraction and ion concentration play important roles in cocktail stability and consistency of counting results. The results of all of these experiments are discussed.     

An influence function for the intensity factor in tensile fracture

A semi-infinite plane-strain tensile crack occupies the region z = 0, ? ∞ < x < l(t) of an elastic medium. The faces of the crack are loaded symmetrically with an arbitrary time dependent load ($\text{σ}{}_{\mathrm{zz}}\text{= ?(x, t)}$. We derive a closed form expression for a kernel function K(l; t₁; x, t) such that the stress-intensity factor k(L; t₁) at the point (l (t₁), 0, t₁) is given by

$\text{k}\text{(l;t}{}_1\text{)= ∫}\text{K}\text{(l;t}{}_1\text{x, t)?(x, t)}\text{d}\text{x}\text{d}\text{t}$

the domain of integration being a suitable characteristic triangle in the x, t plane. Because of the special form of K we are able to recover Freund's 1973 result that, given f(x, t), k(i;t_l) depends upon/only through l(t_l) and l'(t_l), the latter through a multiplicative factor k(l').     

Theory of Formation of Polymer Crystals with Folded Chains in Dilute Solution

A detailed interpretation of the kinetics of homogeneous nucleation and growth of crystals of a linear homopolymer from dilute solution is given. The probability of forming both nuclei with folded chains, and conventional bundlelike nuclei, from dilute solution is analyzed. It is predicted that at sufficiently high dilution, critical nuclei of length lp* will be formed from single polymer molecules by sharp folding of the chain backbone. The step height of the nucleus is given approximately by lp*=4σe/Δf. Here σ_e is the free energy required to form a unit area of the loop-containing end surfaces, and Δf is the free energy difference per unit volume of crystal between the crystalline and solution states. The quantity Δf is approximately proportional to the degree of supercooling ΔT. The growth of these nuclei is then analyzed. After growth, the resulting crystal is flat and platelike, the loops formed by the chain folds being on the upper and lower surfaces. Kinetic factors determine that the distance between the flat surfaces in the grown crystal will vary over only a narrow range about a value that is in the vicinity of 1*=4σ_e/Δf. (Neglecting effects due to edge free energies, the theoretical upper and lower limits are 1*=4σ_e/Δf and 1*=2σ_e/Δf, respectively.) In some cases the predicted temperature dependence of the step height of the grown crystal, 1* = const./ΔT, may be modified by the existence of a constant term resulting from the presence of an edge free energy ϵ_p. A grown loop-type crystal is predicted to be stable in comparison with a bundlelike crystal of the same shape and volume in a sufficiently dilute solution. The logarithm of the nucleation rate is approximately proportional to 1/(ΔT)² near the melting point. The exponent n in the free growth rate law is predicted under various assumptions. To the extent that comparison is possible, the predictions given agree with the experimental results obtained by Keller and O’Connor and others on single crystals of unbranched polyethylene grown from dilute solution.A survey is given of homogeneous nucleation in bulk polymers, where the conventional bundlelike nucleus containing segments from many different molecules is valid, and the essential results compared with those calculated for the dilute solution case.The theory given for loop nuclei is both general and precise enough at the critical points to suggest that, on crystallization from sufficiently dilute solution, crystals of a definite step height are commonly to be expected for other crystallizable linear polymers than polyethylene, provided loop formation is sterically possible.     

Interpretation of the human skin biotribological behaviour after tape stripping

The present study deals with the modification of the human skin biotribological behaviour after tape stripping. The tape-stripping procedure consists in the sequential application and removal of adhesive tapes on the skin surface in order to remove stratum corneum (SC) layers, which electrically charges the skin surface. The skin electric charges generated by tape stripping highly change the skin friction behaviour by increasing the adhesion component of the skin friction coefficient. It has been proposed to rewrite the friction adhesion component as the sum of two terms: the first classical adhesion term depending on the intrinsic shear strength, τ₀, and the second term depending on the electric shear strength, τ_elec. The experimental results allowed to estimate a numerical value of the electric shear strength τ_elec. Moreover, a plan capacitor model with a dielectric material inside was used to modelize the experimental system. This physical model permitted to evaluate the friction electric force and the electric shear strength values to calculate the skin friction coefficient after the tape stripping. The comparison between the experimental and the theoretical value of the skin friction coefficient after the tape stripping has shown the importance of the electric charges on skin biotribological behaviour. The static electric charges produced by tape stripping on the skin surface are probably able to highly modify the interaction of formulations with the skin surface and their spreading properties. This phenomenon, generally overlooked, should be taken into consideration as it could be involved in alteration of drug absorption.     

Dual reciprocity boundary element analysis of nonlinear diffusion: temporal discretization

This article presents an application of higher order time integration schemes in the dual reciprocity boundary element analysis of nonlinear transient diffusion problems involving nonlinear boundary conditions as well as temperature-dependent material properties. Multistep θ-methods, cubic Hermitian schemes and one-step least squares method have been considered. An error estimate based on the conservation of thermal energy has been presented to assess the accuracy of the numerical solutions. Numerical results are presented for a set of representative test problems to demonstrate the usefulness of the presented time integration algorithms.     

Evaluation of dynamic stress intensity factors and T-stress using the scaled boundary finite-element method

A technique to evaluate the dynamic stress intensity factors and T-stress is developed by extending the scaled boundary finite-element method. Only the boundary of the problem domain is discretized. The inertial effect at high frequencies is modeled by a continued fraction solution of the dynamic stiffness matrix without introducing an internal mesh. Standard time-stepping scheme is applied to perform response history analyses directly in the time domain. The internal displacement field is obtained by numerical integration after removing the stress singularity. The dynamic stress intensity factors and the T-stress are evaluated directly based on their definitions. No asymptotic solution around the singular point is required. Numerical examples of cracks in homogeneous and bi-material plates demonstrate the simplicity and accuracy of this technique.     

High-pressure high-temperature synthesis and structure of α-tetragonal boron

Microcrystals of α-tetragonal (α-t) boron with unit cell parameters a=9.05077(6) and c=5.13409(6) Å and measured density 2.16–2.22 g cm⁻³ were obtained by pyrolysis of decaborane B₁₀H₁₄ at pressures of 8–9 GPa and temperatures of 1100–1600 ^○C. The crystal structure is in good agreement with the model proposed by Hoard et al (1958 J. Am. Chem. Soc. 80 4507). However, compared to the original model, we found small deformations of icosahedra and changes in the interatomic distances within the unit cell of the synthesized α-t boron.     

Compressibility of Natural Rubber at Pressures Below 500 kg/cm2

The specific volumes of unvulcanized natural rubber and of a peroxide-cured vulcanizate of natural rubber were measured at pressures of 1–500 kg/cm² at temperatures from 0 to 25 °C. Observations on mercury-filled dilatometers were made through a window in the pressure system. No time effects or hysteresis phenomena were observed. The specific volume V in cm³/e over the range studied can be represented by V = V_0,25{1 + A(t ? 25)}{1 + [α₂₅ + k₁(t ? 25)]P + [β₂₅ + k₂(t ? 25)]P²}where P is the pressure in kg/cm², and t the temperature in °C. The constants for the unvulcanized and for the peroxide-cured samples are:

• V₀,₂₅= 1.0951 and 1.1032 cm³/g;
• 10⁴A = 6.54 and 6.36 per degree;
• 10⁶α₂₅= −50.5 and −50.4 (kg/cm²)⁻¹;
• 10⁶k₁ = −0.227 and −0.203 per degree;
• 10⁹β₂₅= 10 and 11.5 (kg/cm²)⁻²;
• and 10⁹k₂=0.048 and 0.073 per degree, respectively. The compressibility of unvulcanized natural rubber at 25° and 1 kg/cm² is thus 50.5×10⁻⁶ (kg/cm²)⁻¹ falling to 40.6×10⁻⁶ (kg/cm²) ⁻¹ at a pressure of 500 kg/cm². It is concluded that a low degree of vulcanization produces no significant changes in the constants listed. The values are not far different from those obtained by extrapolating to zero sulfur content the observations of Scott on the rubbersulfur system. Calculations of values of compressibility (and its reciprocal the bulk modulus), “internal pressure”, bulk wave velocity, difference between specific heats, and several other physical properties are in reasonable agreement with those obtained by direct observation by other workers. For the prediction of values at pressures above 500 kg/cm² the use of the Tait equation is recommended.

     

New aspects of π–d interactions in magnetic molecular conductors

The 2 : 1 cation radical salts of bent donor molecules of ethylenedithio-tetrathiafulvalenoquinone-1,3-dithiolemethide (EDT-TTFVO), ethylenedithio-diselenadithiafulvalenoquinone-1,3-dithiolemethide (EDT-DSDTFVO), ethylenedithio-diselenadithiafulvalenothioquinone-1,3-diselenolemethide (EDT-DSDTFVSDS), ethylenedioxy-tetrathiafulvalenoquinone-1,3-dithiolemethide (EDO-TTFVO) and ethylenedioxy-tetrathiafulvalenoquinone-1,3-diselenolemethide (EDO-TTFVODS) with FeX₄⁻ (X = Cl, Br) ions are prepared by electrocrystallization. The crystal structures of these salts are composed of alternately stacked donor molecule and magnetic anion layers. The band structures of the donor molecule layers are calculated using the overlap integrals between neighboring donor molecules and are compared with the observed electronic transport properties. The magnetic ordering of the Fe(III) d spins of FeX₄⁻ ions is determined from magnetization and heat capacity measurements. The magnetic ordering temperatures are estimated by considering a combination of a direct d–d interaction between the d spins and an indirect π–d interaction between the conduction π electron and the d spins, whose magnitudes are separately calculated from the crystal structures with an extended Hückel molecular orbital method. The occurrence of a π–d interaction is proved by the negative magnetoresistance, and the magnitude of magnetoresistance reflects the strength of the π–d interaction. The effect of pressure on the magnetoresistance is studied, and the result indicates that the magnitude of magnetoresistance increases, namely, the π–d interaction is enhanced with increasing pressure. From these experimental results it is shown that (EDT-TTFVO)₂•FeBr₄ is a ferromagnetic semiconductor, (EDT-DSDTFVO)₂•FeX₄ (X = Cl, Br) and (EDT-DSDTFVSDS)₂•FeBr₄ are metals exhibiting antiferromagnetic ordering of the d spins, and (EDO-TTFVO)₂•FeCl₄ and (EDO-TTFVODS)₂•FeBr₄•(DCE)_0.5 (DCE =-dichloroethane) are genuine antiferromagnetic metals. Among them, the (EDT-TTFVO)₂•FeBr₄ salt is the first π–d molecular system where the d spins of FeBr₄⁻ ions are ferromagnetically ordered through antiferromagnetic interaction with the conduction π electrons. Corresponding to this ferromagnetic ordering, an anomalous dielectric slow-down phenomenon toward the ordering temperature is observed. The π–d interaction in (EDT-DSDTFVSDS)₂•FeBr₄ is very large and comparable to that in λ-(BETS)₂•FeCl₄, which has the highest reported value so far, while the d–d interaction is fairly small. Concerning the ratio between the magnitudes of π–d and d–d interactions (J_πd/J_dd), this salt is currently the best π–d molecular system.     

A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption

We give a new and explicit estimate for the asymptotic behavior of the solutions of the problem u_t − u_xx + u₊^P = 0, x > 0, t> 0, with conditions u(0, t) = 1, t > 0 and u(x, 0) = U₀(x) ≥ 0, x > 0, for a class of functions U₀ and parameter 0 < p < 1. We use an approximate solution given by the heat balance integral method with the innovation property which fixes appropriately the asymptotic limit of the corresponding approximate free boundary.     

Estimation of Concentration and Bonding Environment of Water Dissolved in Common Solvents Using Near Infrared Absorptivity

Integrated near infrared (NIR) absorbance has been used to determine the absorptivity of the υ₂ + υ₃ combination band of the asymmetric stretch (υ2) and the bending vibration (υ3) for water in several organic solvents. Absorptivity measured in this way is essentially constant across the absorption envelope and is found to be 336 L mol⁻¹ cm⁻¹ with a standard deviation of 4 L mol⁻¹ cm⁻¹ as estimated from a least squares fit of a straight line to data from water concentrations between 0.01 mol/L and 0.06 mol/L. Absorptivity measured from the peak maximum of the υ2 + υ3 combination band of water varies with the type of hydrogen bonding of the water molecule because the shape of the NIR absorption envelope changes with the hydrogen bonding.Because the integrated NIR absorptivity of the υ₂ + υ₃ combination band of water is essentially constant across the absorption envelope, the NIR absorption envelope reflects the distribution of hydrogen bonding of the water. The shape and location of the absorption envelope appear to be governed mostly by the number of hydrogen bonds from the water molecules to easily polarized atoms. Water that is a donor in hydrogen bonds to atoms which are not easily polarized (such as the oxygen of a typical carbonyl group) absorbs near 5240 cm⁻¹ to 5260 cm⁻¹. Water that donates one hydrogen bond to an easily polarized atom (such as a water molecule oxygen) absorbs near 5130 cm⁻¹ to 5175 cm⁻¹, and water that donates two hydrogen bonds to easily polarized atoms is estimated to absorb near 5000 cm⁻¹ to 5020 cm⁻¹. Water donating two hydrogen bonds to other water molecules may be said to be in a water-like environment. In no case does a small amount of water absorbed in a host material appear to have a water-like environment.     

A Quantum Algorithm Detecting Concentrated Maps

We consider an arbitrary mapping f: {0, …, N − 1} → {0, …, N − 1} for N = 2ⁿ, n some number of quantum bits. Using N calls to a classical oracle evaluating f(x) and an N-bit memory, it is possible to determine whether f(x) is one-to-one. For some radian angle 0 ≤ θ ≤ π/2, we say f(x) is θ − concentrated if and only if e^2πif(x)/N ? e^{i[ψ₀?θ,ψ₀+θ]} for some given ψ₀ and any 0 ≤ x ≤ N − 1. We present a quantum algorithm that distinguishes a θ-concentrated f(x) from a one-to-one f(x) in O(1) calls to a quantum oracle function U_f with high probability. For 0 < θ < 0.3301 rad, the quantum algorithm outperforms random (classical) evaluation of the function testing for dispersed values (on average). Maximal outperformance occurs at θ=12sin−11π≈0.1620 rad.     

Finite element modelling versus classic beam theory: comparing methods for stress estimation in a morphologically diverse sample of vertebrate long bones

Classic beam theory is frequently used in biomechanics to model the stress behaviour of vertebrate long bones, particularly when creating intraspecific scaling models. Although methodologically straightforward, classic beam theory requires complex irregular bones to be approximated as slender beams, and the errors associated with simplifying complex organic structures to such an extent are unknown. Alternative approaches, such as finite element analysis (FEA), while much more time-consuming to perform, require no such assumptions. This study compares the results obtained using classic beam theory with those from FEA to quantify the beam theory errors and to provide recommendations about when a full FEA is essential for reasonable biomechanical predictions. High-resolution computed tomographic scans of eight vertebrate long bones were used to calculate diaphyseal stress owing to various loading regimes. Under compression, FEA values of minimum principal stress (σ_min) were on average 142 per cent (±28% s.e.) larger than those predicted by beam theory, with deviation between the two models correlated to shaft curvature (two-tailed p = 0.03, r² = 0.56). Under bending, FEA values of maximum principal stress (σ_max) and beam theory values differed on average by 12 per cent (±4% s.e.), with deviation between the models significantly correlated to cross-sectional asymmetry at midshaft (two-tailed p = 0.02, r² = 0.62). In torsion, assuming maximum stress values occurred at the location of minimum cortical thickness brought beam theory and FEA values closest in line, and in this case FEA values of τ_torsion were on average 14 per cent (±5% s.e.) higher than beam theory. Therefore, FEA is the preferred modelling solution when estimates of absolute diaphyseal stress are required, although values calculated by beam theory for bending may be acceptable in some situations.     

Solutions for a local equation of anisotropic plant cell growth: an analytical study of expansin activity

This paper presents a generalization of the Lockhart equation for plant cell/organ expansion in the anisotropic case. The intent is to take into account the temporal and spatial variation in the cell wall mechanical properties by considering the wall ‘extensibility’ (Φ), a time- and space-dependent parameter. A dynamic linear differential equation of a second-order tensor is introduced by describing the anisotropic growth process with some key biochemical aspects included. The distortion and expansion of plant cell walls initiated by expansins, a class of proteins known to enhance cell wall ‘extensibility’, is also described. In this approach, expansin proteins are treated as active agents participating in isotropic/anisotropic growth. Two-parameter models and an equation for describing α- and β-expansin proteins are proposed by delineating the extension of isolated wall samples, allowing turgor-driven polymer creep, where expansins weaken the non-covalent binding between wall polysaccharides. We observe that the calculated halftime (t_1/2 = εΦ₀ log 2) of stress relaxation due to expansin action can be described in mechanical terms.