Explicitly Expressed Research Articles

Adaptive filter design for cyclostationary processes associated with sine-wave extraction operation

Cyclostationary signals have been generally used in condition monitoring and fault diagnosis of bearings, for which the adaptive filter is important in obtaining the signal-of-interest cyclostationary process from a disturbed signal. However, due to the nonstationarity of cyclostationary signals, their adaptive filters are usually time-varying and difficult to be implemented. In this study, a new optimum filter is obtained by converting the time-varying optimization function to be time-invariant using a sine-wave extraction (SE) operator, for which the adaptive implementation is designed and analyzed. The time-varying linear minimum mean-square error is converted to be time invariant using the SE operator, and its solution is time invariant but without explicit expression. Subsequently, an adaptive implementation, called the SE least mean square (SE-LMS), is introduced, for which the mean and weighted mean-square-deviation (MSD) are explicitly expressed. The scope of step size is determined to obtain a mean-convergent and mean-square-stable SE-LMS. The steady-state SE-operated excess mean-square-error is analyzed, and its explicit expression is obtained. The SE-LMS algorithm uses only a single cyclic frequency, based on which, the combined SE-LMS algorithm is introduced for a cyclostationary signal with multiple cyclic frequencies. Finally, several simulations are designed to verify the superiority of the proposed estimators over the time-average operator-based estimator.

3D elasticity solutions for stress field around square hole in functionally graded plate

Since the square holes are widely used in many structures and machines, the purpose of this work is to obtain analytical solutions for the three-dimensional (3D) elastic stress field of transversely isotropic functionally graded (FG) infinite plates containing a single square hole with small round corners subjected to far-field loads or on the edges of the holes. Based on the generalized England-Spencer plate theory, material parameters can vary continuously and arbitrarily along the thickness direction of the plate, the solutions are obtained by using the conformal mapping technology combined with the Cauchy integral method. Concentration factors of internal forces Kij around the square hole are explicitly expressed. The obtained explicit expressions of solutions can be validated against the classical two-dimensional (2D) solutions when the material degenerates into an isotropic and homogeneous material. The detailed numerical examples are given to further investigate the influence of several parameters such as the material gradient factor and load conditions on the stress field around the hole. The present 3D analytical solutions can not only provide an in-depth understanding of the 3D stress field of FG plates with holes, but also be utilized as a benchmark to verify the numerical solutions of the same problems.

Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

AbstractBased on polynomial dimensional decomposition (PDD), a novel method for analysing nonstationary random vibration of structures with uncertain parameters is proposed in this paper. The uncertain structural parameters are assumed to be independent random variables, with certain types of probability distributions; thus, the nonstationary random vibration responses of the structures become stochastic functions of these uncertain parameters. A dimensional decomposition and Fourier expansion were applied to calculate the statistical characteristics of the nonstationary random vibration responses of the structures. To calculate the conditional power spectral density (CPSD) and conditional standard deviation (CSD) of the nonstationary random vibration responses, CPSD and CSD were explicitly expressed using expansion coefficients and polynomial basis. Subsequently, the probability distributions of the structural responses can be effectively calculated in terms of these explicit expressions. Dimension‐reduction integration and Gauss numerical integration were applied to calculate expansion coefficients. The validity of the proposed method was verified by numerical examples, including a single‐degree‐of‐freedom system and a multi‐degree‐of‐freedom system (reinforced concrete shear wall). The results indicate that the proposed method is as accurate as Monte Carlo simulation (MCS), whereas the proposed method requires considerably smaller number of nonstationary random vibration analyses than MCS. Moreover, the nonstationary random vibration analysis of a long‐span cable‐stayed bridge demonstrates that the proposed method is feasible for quantifying the uncertainties of the nonstationary random vibration responses of complex engineering structures with uncertain parameters.

CTMC integral equation method for American options under stochastic local volatility models

In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the 4/2 model and the α−hypergeometric models. They illustrate the high accuracy and efficiency of the method.

On Isotypic Decompositions for Non-Semisimple Hopf Algebras

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.

Computation of Kontsevich weights of connection and curvature graphs for symplectic Poisson structures

We give a detailed explicit computation of weights of Kontsevich graphs which arise from connection and curvature terms within the globalization picture for the special case of symplectic manifolds. We will show how the weights for the curvature graphs can be explicitly expressed in terms of the hypergeometric function as well as by a much simpler formula combining it with the explicit expression for the weights of its underlined connection graphs. Moreover, we consider the case of a cotangent bundle, which will simplify the curvature expression significantly.

Nonlinear contact force law for spherical indentation of FGM coated elastic substrate: An extension of Hertz's solution

This paper develops a nonlinear contact force law for the indentation of an elastic substrate with thin functionally graded material (FGM) coating by a rigid smooth sphere. The nonlinear contact force law is explicitly expressed in form of extended Hertz's solution with a correction factor, which is related to the coating thickness, radius of the indenter, contact interference and the modulus ratio. The explicit expression of the correction factor for arbitrary coating modulus gradation can be determined statistically with extensive numerical results. For computational efficiency and accuracy, an analytical contact model based on multilayered half space is developed and the associated mixed boundary value problem is converted to a Fredholm integral equation of the second kind. This model discretizes the thin FGM coating into n dissimilar and fully bonded sub-layers. Each sub-layer is a homogeneous elastic seam of finite thickness and constant elastic modulus and Poisson's ratio. Variation of the coating elastic properties along the thickness direction is accurately approximated by the multilayered system. The non-linear contact force laws in closed-form are specifically given for both linear and exponential modulus gradations. The load-displacement relations predicted by these force laws are shown to be in exact agreement with the numerical results from the Fredholm integral equation. Additionally, this nonlinear contact force law for FGM coating is further incorporated into the Greenwood and Williamson model, which provides a feasible and effective way for modeling the contact of FGM coated rough surfaces.

Accurate modal superposition method for harmonic frequency response sensitivity of non-classically damped systems with lower-higher-modal truncation

Frequency response and their sensitivities analysis are of fundamental importance. Due to the fact that the mode truncation errors of frequency response functions (FRFs) are introduced for two times, the errors of frequency response sensitivities may be larger than other dynamic analysis. Many modal correction approaches (such as modal acceleration methods, dynamic correction methods, force derivation methods and accurate modal superposition methods) have been presented to eliminate the modal-truncation error. However, these approaches are just suitable to the case of un-damped or classically damped systems. The state-space equation based approaches can extend these approaches to non-classically damped systems, but it may be not only computationally expensive, but also lack physical insight provided by the superposition of the complex modes of the equation of motion with original space. This paper is aimed at dealing with the lower-higher-modal truncation problem of harmonic frequency response sensitivity of non-classically damped systems. Based on the Neumann expansion and the frequency shifting technique, the contribution of the truncated lower and higher modes to the harmonic frequency response sensitivity is explicitly expressed only by the available middle modes and system matrices. An extended hybrid expansion method (EHEM) is then proposed by expressing harmonic frequency response sensitivity as the explicit expression of the middle modes and system matrices. The EHEM maintains original-space without having to involve the state-space equation of motion such that it is efficient in computational effort and storage capacity. Finally, a rail specimen is used to illustrate the effectiveness of the proposed method.

Analytical description of ballistic spin currents and torques in magnetic tunnel junctions

In this work we demonstrate explicit analytical expressions for both charge and spin currents which constitute the 2x2 spinor in magnetic tunnel junctions with noncollinear magnetizations under applied voltage. The calculations have been performed within the free electron model in the framework of the Keldysh formalism and WKB approximation. We demonstrate that spin/charge currents and spin transfer torques are all explicitly expressed through only three irreducible quantities, without further approximations. The conditions and mechanisms of deviation from the conventional sine angular dependence of both spin currents and torques are shown and discussed. It is shown in the thick barrier approximation that all tunneling transport quantities can be expressed in an extremely simplified form via Slonczewski spin polarizations and our effective spin averaged interfacial transmission probabilities and effective out-of-plane polarizations at both interfaces. It is proven that the latter plays a key role in the emergence of perpendicular spin torque as well as in the angular dependence character of all spin and charge transport considered. It is demonstrated directly also that for any applied voltage, the parallel component of spin current at the FM/I interface is expressed via collinear longitudinal spin current components. Finally, spin transfer torque behavior is analyzed in a view of transverse characteristic length scales for spin transport.

Moving Dugdale crack along the interface of two dissimilar magnetoelectroelastic materials

In this paper, the concept of Dugdale crack model and Yoffe model is extended to propose a moving Dugdale interfacial crack model, and the interfacial crack between dissimilar magnetoelectroelastic materials under anti-plane shear and in-plane electric and magnetic loadings is investigated considering the magneto-electro-mechanical nonlinearity. It is assumed that the constant moving crack is magneto-electrically permeable and the length of the crack keeps constant. Fourier transform is applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. The explicit expression of the size of the yield zone is derived, and the crack sliding displacement (CSD) is explicitly expressed. The result shows that the stress, electric and magnetic fields in the cracked magnetoelectroelastic material are no longer singular and the CSD is dependent on the loading, material properties and crack moving velocity. The current model can be reduced to the static interfacial crack case when the crack moving velocity is zero.

CERTAIN COMBINATORIAL CONVOLUTION SUMS INVOLVING DIVISOR FUNCTIONS PRODUCT FORMULA

It is known that certain combinatorial convolution sums involving two divisor functions product formulae of arbitrary level can be explicitly expressed as a linear combination of divisor functions. In this article we deal with cases for certain combinatorial convolution sums involving three, four, six and twelve divisor functions product formula and obtain explicit expressions.

Evaluation of a certain combinatorial convolution sum in higher level cases

It is known that certain combinatorial convolution sums involving divisor functions of “levels” 1 and 2 can be explicitly expressed as a linear combination of divisor functions. In this article we deal with a case for arbitrary level and obtain an explicit expression.

Explicit invariant manifolds and specialised trajectories in a class of unsteady flows

A class of unsteady two- and three-dimensional velocity fields for which the associated stable and unstable manifolds of the Lagrangian trajectories are explicitly known is introduced. These invariant manifolds form the important time-varying flow barriers which demarcate coherent fluids structures, and are associated with hyperbolic trajectories. Explicit expressions are provided for time-evolving hyperbolic trajectories (the unsteady analogue of saddle stagnation points), which are proven to be hyperbolic in the sense of exponential dichotomies. Elliptic trajectories (the unsteady analogue of stagnation points around which there is rotation, i.e., the “centre of a vortex”) are similarly explicitly expressed. While this class of models possesses integrable Lagrangian motion since formed by applying time-dependent spatially invertible transformations to steady flows, their hyperbolic/elliptic trajectories can be made to follow any user-specified path. The models are exemplified through two classical flows: the two-dimensional two-gyre Duffing flow and the three-dimensional Hill's spherical vortex. Extensions of the models to finite-time and nonhyperbolic manifolds are also presented. Given the paucity of explicit unsteady examples available, these models are expected to be useful testbeds for researchers developing and improving diagnostic methods for tracking flow structures in genuinely time-dependent flows.

Explicit isospectral flows associated to the AKNS operator on the unit interval. II

Explicit flows associated to any tangent vector fields on any isospectral manifold for the AKNS operator acting in L2 × L2 on the unit interval are written down. The manifolds are of infinite dimension (and infinite codimension). The flows are called isospectral and also are Hamiltonian flows. It is proven that they may be explicitly expressed in terms of regularized determinants of infinite matrix-valued functions with entries depending only on the spectral data at the starting point of the flow. The tangent vector fields are decomposed as ∑ξkTk where ξ ∈ ℓ2 and the Tk ∈ L2 × L2 form a particular basis of the tangent vector spaces of the infinite dimensional manifold. The paper here is a continuation of Amour [“Explicit isospectral flows for the AKNS operator on the unit interval,” Inverse Probl. 25, 095008 (2009)]10.1088/0266-5611/25/9/095008 where, except for a finite number, all the components of the sequence ξ are zero in order to obtain an explicit expression for the isospectral flows. The regularized determinants induce counter-terms allowing for the consideration of finite quantities when the sequences ξ run all over ℓ2.

A Simplified Model and Similarity Solutions for Interfacial Evolution of Two-Phase Flow in Porous Media

For two-phase flows of immiscible displacement processes in porous media, we proposed a simplified model to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface. A new similarity solution for the interfacial evolution in the rectangular coordinate system was derived by postulating a first-order approximation of the velocity distribution in the region that the two-phase fluids co-exist. The interfacial evolution equation can be explicitly expressed as a linear function, where the slope of the interfacial equation is simply related to the mobility ratio of two-phase fluids in porous media. The application of the proposed solutions to predictions of interfacial evolutions in carbon dioxide injected into saline aquifers was illustrated under different mobility ratios and operational parameters. For the purpose of comparison, the numerical solutions obtained by level set method and the similarity solutions based on the Dupuit assumptions were presented. The results show that the proposed solution can give a better approximation of interfacial evolution than the currently available similarity solutions, especially in the situation that the mobility ratio is large. The proposed approximate solutions can provide physical insight into the interfacial phenomenon and be readily used for rapidly screening carbon dioxide storage capacity in subsurface formations and monitoring the migration of carbon dioxide plume.

Mathematical Model for Computing Maximum Power Output of a PV Solar Module and Experimental Validation

The conversion of solar radiation into electricity has been extensively studied by many authors In particular, photovoltaic (PV) cells allow the energy transported by electromagnetic waves (i.e., photons) to be directly converted into electricity. Predicting the performance of PV panels is essential for design engineers. From the characteristic I–V curve of a given PV cell, three key physical quantities are defined: the short-circuit current, the open-circuit voltage, and the values of current and voltage that permit the maximum power to be obtained. These variables correspond to well-defined points in the I–V plane. The determination of these points is essential for the development of appropriate PV cell models. The nonlinear and implicit relationships that exist between them, however, necessitate using iterative numerical calculations. urthermore, most of these parameters depend on both the cell temperature and the solar irradiance; therefore, the knowledge of their behavior is crucial to correctly predict the performance of PV cells and arrays. In terrestrial applications, solar cells are generally exposed to temperatures varying from 10 °C to 50 °C. The performance of a solar cell is influenced by temperature as its performance parameters, namely, open-circuit voltage (Voc), short-circuit current (Isc), curve factor (CF), and efficiency, are temperature dependent. Many mathematical models have been developed to estimate PV efficiency, power, short-circuit current, and open-circuit voltage. An excellent review of different correlations is used to determine these quantities as a function of irradiance and cell temperature [8]. An explicit model was developed in [1] to determine currents and voltages for PV points by using manufacturers’ data, where the electrical current changes linearly with irradiance and temperature, while voltages are expressed by temperature coefficients and a correlation term for the irradiance. An analytical model based on manufacturers’ data was presented in [6]. The authors have explicitly expressed the PV current and power as a function of voltage by using a shading linear factor expressed for open-circuit voltage loses for irradiances from 1,000 W/m2 to 200 W/m2; I–V and P–V curves can thus be produced without calculating explicit expressions of current and voltage at key operational points. It must be pointed out that most of the aforementioned works deal with models that are essentially based on linear temperature and irradiance relationships that require parameters which are not available from manufacturers’ datasheets. In the current study.

Constitutive equations for the Doi–Edwards model without independent alignment

We present two representations of the Doi–Edwards model without Independent Alignment explicitly expressed in terms of the Finger strain tensor, its inverse and its invariants. The two representations provide explicit expressions for the stress prior to and after Rouse relaxation of chain stretch, respectively. The maximum deviations from the exact representations in simple shear, biaxial extension and uniaxial extension are of order 2%. Based on these two representations, we propose a framework for Doi–Edwards models including chain stretch in the memory integral form.

Generalized Ginzburg-Landau functionals for studies of phase transformations between FCC, BCC, and HCP structures in metals and alloys

Generalized Ginzburg-Landau functionals describing transitions between bcc, fcc, and hcp structures are studied using four weakly nonuniform transformation parameters: three tensile strains along the principal crystallographic axes and a “phonon” parameter describing relative slip of close-packed planes. Several versions of transformation paths indicated for each of the three transformations considered, which appear to be the most realistic; explicit expressions for atomic displacements are given for each of these paths. It is shown that the gradient terms in these functionals can be explicitly expressed in terms of the dynamic matrix of the crystal on a transformation path; these dynamic matrices can be determined using interpolations of experimental data on phonon spectra between the initial and final phases. The results are used for estimating the characteristics of interphase boundaries between ferrite and cementite in steels. It is found that the width of such interfaces considerably exceeds the interatomic distance, while the estimated values of the gradient terms are in reasonable agreement with the available experimental data.

Explicit expression on specifications of mask mean to target and mask uniformity

An analytic expression on the relationship of the mask mean to target (MTT) and the mask uniformity specifications is suggested. The MTT and uniformity (M-U) curve defines the boundary of the specification region, and this curve is explicitly expressed as mask error enhancement factors (MEEFs), dose sensitivities, and critical dimension (CD) tolerances of cell and peripheral patterns. This curve shows linear relationships between mask M-U. The decrease of the mask uniformity allows the increase of the mask MTT margin. M-U specifications are suggested for 63, 73, and 90nm design rules. In this experiment, the tolerance of the cell pattern is given as the product of the MEEF and the mask uniformity margin which is specified as 60% of the total CD variation at the given design rule, while that of the peripheral pattern is set to be 10% of the design rule. An isolated pattern is selected as a peripheral pattern because its pitch size is infinite. The mask MTT specification for 63nm is ±2.3nm at the mask uniformity of 2.23nm. For 73nm design rule, specifications of mask MTT are ±1.94 and ±1.48nm at the mask uniformities of 2.32 and 3.31nm at the process constants (k1’s) of 0.295 and 0.322, respectively. For 90nm design rule, the mask MTT specifications are ±3.89 and ±4.13nm at mask uniformities of 3.89 and 4.13nm for k1’s of 0.364 and 0.396, respectively.

Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.