Quartic Variation Research Articles

Blue-shift in the optical band gap of sol-gel derived Zn(1-x)SrxO nanoparticles

Zn(1-x)SrxO nanoparticles (x = 0.00, 00.01, 0.02, 0.03, 0.05) are prepared by sol-gel co-precipitation. Strontium (Sr) doping causes small but highly significant band gap enhancement that amounts 0.58%. This band gap engineering, following a quartic variation with respect to dopant concentration (x), may find potential applications in the fabrication of various devices based on heterojunctions. The structural analysis of the nanoparticles is done using XRD. The randomly oriented polycrystalline nanoparticles (NPs) notice a decrease in lattice parameters for initial two doping levels, thereafter an increment. The crystallite sizes range between 10.54 and 27.76 nm, along various crystallographic planes. Due to Sr doping, dislocations are also caused whose density vary between 1.2 × 1015 and 9.0 × 1015 lines/m2. FTIR analysis confirms the presence of Zn–O bond prominently in tetrahedral mode. UV–Vis spectra reveal a dip at the wavelength corresponding to the band gap of NPs with different Sr concentration. Tauc's plot yields the band gap of undoped ZnO as 3.281 eV which gets augmented up to 3.30 eV by 5 at.% Sr doping. This band gap enhancement is an outcome of infiltration of Sr into the ZnO matrix, rather than substitution. PL emission spectra contain the peaks in UV and visible (violet-blue) range. The UV peak, arising from band-to-band exciton recombination, gets blue-shifted with Sr incorporation into ZnO. Violet-blue emissions, mainly originating from defect levels, see the changes only in their intensities. The surface morphology, through FESEM, presents elongated spherical, hexagonal and platelet kind of structures with interlinking granular chains. Interlinks densify with initial doping, whose role has been comprehended in governing the violet-blue emission. Sr doping of 3 at.% is found to have larger voids in the granular distribution. For highest doping interlinkages appear to resume. For compositional analysis EDX is performed, which confirms the presence of Sr in all doped ZnO NPs.

Stress in a polymer brush

We study the stress distribution in a polymer brush material over a range of graft densities, using theory and molecular dynamics (MD) simulations. MD simulations confirmed the quartic variation of the normal stress within the bulk of the brush, as predicted in our previous work. However, in the high graft density regime, further improvements to the theory are needed, as it is known that finite extensibility effects (force-extension divergence) invalidate Gaussian elasticity of chains, and the restriction to binary interaction among monomers is insufficient. This motivated us to extend a semi-analytical strong stretching theory (SST) for polymer brushes that use Langevin force-extension relation for chains and a modified Carnahan–Starling (CS) equation of state to model monomer interactions. Our extended theory elucidates the stress profiles obtained from MD simulations. It reproduces Gaussian chain results for small graft densities, and shows a good qualitative agreement with MD results for monomer density profile, end density profile and stress profile for high graft densities without the need for fitting parameters (virial coefficients). Quantitative comparisons of MD results with various available theories suggest that excluded volume correlations may be important.

Correlated quartic variation of band gap and NBE energy in sol-gel derived Zn1−Co O nanoparticles

Abstract Zn1−xCoxO (x = 0.0, 0.01, 0.03, 0.05, 0.07) nanoparticles were synthesized via sol-gel method. For structural parameters like lattice constants, bond length and particle size, XRD patterns have been analyzed. The particle sizes lie between 15.75 and 35.31 nm. The Williamson-Hall analysis has been performed for strain calculation. The UV–Visible spectra show high transparency in visible range with a dip in near UV region. The band gap calculation has been done using Tauc's plot method. The band gap values show a quartic variation with x. PL emission spectra exhibit NBE emission that too undergoes quartic variation with x. The correlation between both the quartic variations has been analyzed and it has been established that the B-M effect and sp-d exchange interaction both play their role alternately. M-H curves of the Co doped nanoparticles show room temperature ferromagnetism whereas the undoped one have the diamagnetic character. The electrical resistivity has been found to be high; in the range of 4.81 × 104 to 7.26 × 104 Ω-m. A increasing trend in resistivity is noticed with increasing the Co concentration in ZnO matrix.

Mass Sensitivity of Nonuniform Microcantilever Beams

Microelectromechanical systems (MEMS) based cantilever beams have been widely used in various sensing applications. Previous studies have aimed at increasing the sensitivity of biosensors by reducing the size of cantilever beams to nanoscale. However, the influence of nonuniform cantilever beams on mass sensitivity has rarely been investigated. In this paper, we discuss the mass sensitivity with respect to linear and nonlinear response of nonuniform cantilever beam with linear and quartic variation in width. To do the analysis, we use the nonlinear Euler–Bernoulli beam equation with harmonic forcing. Subsequently, we derive the mode shape corresponding to linear, undamped, free vibration case for different types of beams with a tip mass at the end. After applying the boundary conditions, we obtain the resonance frequencies corresponding to various magnitudes of tip mass for different kinds of beams. To do the nonlinear analysis, we use the Galerkin approximation and the method of multiple scales (MMS). Analysis of linear response indicates that the nondimensional mass sensitivity increases considerably by changing the planar geometry of the beam as compared to uniform beam. At the same time, sensitivity further increases when the nonuniform beam is actuated in higher modes. Similarly, the frequency shift of peak amplitude of nonlinear response for a given nondimensional tip mass increases exponentially and decreases quadratically with tapering parameter, α, for diverging and converging nonuniform beam with quartic variation in width, respectively. For the converging beam, we also found an interesting monotonically decreasing and increasing behavior of mass sensitivity with tapering parameter α giving an extremum point at α=0.5. Overall analysis indicates a potential application of the nonuniform beams with quartic converging width for biomass sensor.

Trends in Atomic Parameters for Crystals and Free Ions across the Lanthanide Series: The Case of LaCl3:Ln(3+).

Analyses of the crystal field energy levels of the series LaCl3:Ln(3+) using a semiempirical Hamiltonian shows that only five ions (Pr, Nd, Pm, Dy, Ho) meet the criteria to avoid overfitting of the atomic part. A new parameter (SNES) has been introduced to represent the strength of the normalized electrostatic repulsion for these ions. This parameter varies linearly (R(2)adj = 0.9994, N = 5) with the reciprocal of the radius of the tripositive lanthanide ion, as expected from the form of repulsive Coulomb interaction. The Slater parameters from the crystal field analyses, F(k)(corr) (i.e., corrected for the effects of the two-particle component of the three-body operator associated with the T(2) parameter), exhibit an exponential variation with the number of electrons, n, in 4f(n). This is explained by reference to the radial part of a hydrogen-like wave function. The ratio of F(k)(corr) with the ab initio free ion Slater parameter F(k)(ab initio) varies linearly with n. Fitted parameters F(k)(corr: free ion) from the free ion data for Pr(3+) and Nd(3+) show that the corresponding ab initio values are between 14 and 27% too high. The spin-orbit coupling constant from crystal field analyses (ζ4f) exhibits a quartic variation with atomic number, and the ratio ζ4f/ζ4f(ab initio) follows an exponential growth model with n. The results serve to confirm the hypothesis that smooth trends can be observed across the Ln(3+) series for the fitted parameters despite the fact that the majority of experimental data is lacking.

Multidimensional Riemann problem with self-similar internal structure. Part I – Application to hyperbolic conservation laws on structured meshes

Multidimensional Riemann solvers have been formulated recently by the author (Balsara, 2010, 2012) [6,7]. They operate at the vertices of a two-dimensional mesh, taking input from all the neighboring states and yielding the resolved state and fluxes as output. The multidimensional Riemann problem produces a self-similar strongly interacting state which is the result of several one-dimensional Riemann problems interacting with each other. The prior work was restricted to the use of one-dimensional HLLC Riemann solvers as building blocks. In this paper, we formulate the problem in similarity variables. As a result, any self-similar one-dimensional Riemann solver can be used as a building block for the multidimensional Riemann solver. This paper focuses on the structure of the strongly-interacting state. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)In this work the strongly-interacting state is expanded in a set of basis functions that depend on the similarity variables. Consequently, the resolved state and the fluxes can be endowed with considerably richer sub-structure compared to prior work. Unlike the multidimensional HLLC Riemann solver, the need to independently specify a direction for the evolution of the contact discontinuity is eliminated. The richer sub-structure in the strongly-interacting state naturally accommodates waves that may be moving in any direction relative to the mesh, thereby minimizing mesh-imprinting. Two formulations are presented. The first formulation does not linearize the problem around a favorable state. Its derivation takes a few cues from the derivation of the multidimensional HLL Riemann solver. The second formulation identifies such a state and carries out a linearization of the fluxes about that state.This paper is the very first time that a series solution of the multidimensional Riemann problem has been presented. Explicit formulae are presented for up to quartic variation in the self-similar variables. While linear variations are sufficient for numerical work, the higher order terms in the series solution could prove useful for analytical studies of the multidimensional Riemann problem.The formulation presented here is general enough to accommodate any hyperbolic conservation law. It can also accommodate any one-dimensional Riemann solver and yields a multidimensional version of the same. It has been incorporated in the author's RIEMANN code. As examples of the different types of hyperbolic conservation laws, we use Euler flow, Magnetohydrodynamics (MHD) and relativistic MHD. As examples of different types of Riemann solvers, we show multidimensional formulations of HLL, HLLC and HLLD Riemann solvers for MHD all working fluently within this formulation. Several stringent test problems are presented.

Traces of the discrete Hilbert transform with quadratic phase

For the function \(H:\mathbb{R}^2 \mapsto \mathbb{C}\), \(H: = (p.v.)\sum\nolimits_{n \in \mathbb{Z}\backslash \{ 0\} } {\tfrac{{\exp \left\{ {\pi i\left( {tn^2 + 2xn} \right)} \right\}}} {{2\pi in}}}\) of two real variables (t, x) ∈ ℝ2, we study the uniform moduli of continuity and the variations of the restrictions H|t and H|x onto the lines parallel to the coordinate axes x = 0 and t = 0. Smoothness of such restrictions is primarily determined by the Diophantine approximation of the fixed parameter. Generalized (weak) variations are also studied, and it is shown in particular that supx w4[H|x] < ∞ where w4 denotes the weak quartic variation. Previously it was known that uniformly in the parameter t ∈ ℝ, the restriction H|t is a function of bounded weak quadratic variation in the variable x, i.e., supt w2[H|t] < ∞. The function H has multiple applications: in the study of the spectra of uniform convergence (P.L. Ul’yanov’s problem), in the incomplete Gaussian sums (where it plays the role of the generating function), in the partial differential equations of mathematical physics (in the Cauchy problem for the Schrodinger equation), and in quantum optics (Talbot’s phenomenon).

The Roughness and Smoothness of Numerical Solutions to the Stochastic Heat Equation

The stochastic heat equation is the heat equation driven by white noise. We consider its numerical solutions using the finite difference method. Its true solutions are Holder continuous with parameter $(\frac{1}{2}-\epsilon)$ in the space variable, and $(\frac{1}{4}-\epsilon)$ in the time variable. We show that the numerical solutions share this property in the sense that they have non-trivial limiting quadratic variation in x and quartic variation in t. These variations are discontinuous functionals on the space of continuous functions, so it is not automatic that the limiting values exist, and not surprising that they depend on the exact numerical schemes that are used; it requires a very careful choice of scheme to get the correct limiting values. In particular, part of the folklore of the subject says that a numerical scheme with excessively long time-steps makes the solution much smoother. We make this precise by showing exactly how the length of the time-steps affects the quadratic and quartic variations.

A change of variable formula with Itô correction term

We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical It\^{o} sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\int g(F(t),t)\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary It\^{o} integral with respect to a Brownian motion that is independent of $F$.

Variations of the solution to a stochastic heat equation

We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical It\^{o} calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of $t$, converge weakly to Brownian motion.

Dynamic analysis of laminated composite beams using higher order theories and finite elements

Free vibration analysis of laminated composite beams is carried out using two higher order displacement based shear deformation theories and finite elements based on the theories. Both theories assume a quintic and quartic variation of in-plane and transverse displacements in the thickness coordinates of the beams respectively and satisfy the zero transverse shear strain/stress conditions at the top and bottom surfaces of the beams. The difference between the two theories is that the first theory assumes a non-parabolic variation of transverse shear stress across the thickness of the beams whereas the second theory assumes a parabolic variation. The equations of motion are derived using Hamilton’s principle. Further two-node C 1 finite elements of eight degrees of freedom per node, based on the theories, are presented for the free vibration analysis of the beams in this paper. Numerical results have been computed for various length to thickness ratios and for various boundary conditions of the beams and compared with the results of other theories and finite elements available in literature. The comparison study shows that the theories and the finite elements predict the natural frequencies of the laminated composite beams better than the other theories and the finite elements considered.

Parameter Estimates and Exact Variations for Stochastic Heat Equations Driven by Space-Time White Noise

In this article we calculate the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise. We use the knowledge of exact variations to estimate the drift parameter appearing in the equation.

Comparisons of displacement-based theories for waves and vibrations in laminated and sandwich composite plates

A semi-analytical method incorporating various displacement-based formulations has been developed to investigate propagation of time harmonic waves and vibrations in fiber reinforced polymer composite laminated and sandwich plates. Various displacement-based models starting from the first order shear deformation theory to the fourth order theory have been developed using combinations of linear, quadratic, cubic and quartic variation of axial and transverse displacements through the thickness of a lamina or a mathematical sub-layer. These displacement-based formulations have been validated by comparing their results with the analytical solutions available in the literature. Results of all the displacement models have been compared with those obtained by displacement model using quartic variation of in-plane and transverse displacements for vibration problem. Higher order displacement-based theory using cubic variation of in-plane and transverse displacements through the thickness of sub-layer has been found to yield converging results for wave propagation in laminated composite plates as well as for vibration problems. All the investigations performed indicate the importance of higher order theories for analysis of wave propagation and vibrations in composite laminated and sandwich plates.

Finite element analysis of thick homogeneous plates

Two alternative four noded, C 1 continuous, 52 degrees of freedom rectangular plate elements based on two higher order shear deformation theories are presented. These elements are developed for the flexural analysis of thick plates for which the through-thickness variation of displacement and stress may be of high order and in which a single element is to be used through the thickness of the plate. A non-parabolic variation of the transverse shear strain/stress across the thickness of the plate is assumed for the first element whereas it is assumed to be parabolic for the second element. In both elements, quintic and quartic variation in the thickness co-ordinate of the plate for inplane and out-of-plane displacements are, respectively, assumed. Numerical results are obtained using these elements for a simply supported plate under sinusoidal load for various ratios of width to length of the plate. The exact elasticity solution (Pagano) and other higher order theory solutions are used to evaluate the performance of these two elements. The interesting features of these two elements are: (1) the transverse normal stress across the thickness of the plate is estimated accurately using constitutive law unlike other displacement based two dimensional finite elements in which equilibrium equations are used to compute the transverse normal stress; and (2) a single element is used across the thickness of the plate to accurately predict all stresses.

Shear Lag Effect in Single Plane Cable-Stayed Bridge

The shear lag effect in thin-walled trapezoidal box sections with inclined stiffeners within the cell is analyzed using the principle of minimum potential energy. Due to the combined action of axial force and moment in a cable-stayed bridge, the normal stress in the pylon is given by [Formula: see text]. The coefficient of shear lag can be written as [Formula: see text], where σ is the actual stress taking shear deformation of slabs into account and [Formula: see text] is the stress evaluated by elementary beam theory. In this paper, the longitudinal displacement under axial force is assumed to vary parabolically transversely across the section. Due to moment, a quartic variation is adopted. Two sets of differential equations with boundary conditions are theoretically derived. An example is illustrated by theoretical analysis and test results.

A stochastic model of neural response

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.

A stochastic model of neural response

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.