Third-order Function Research Articles

Free Vibration Analysis of Composite Structures Using Semi-Analytical Finite Strip Method

In this paper, the linear eigen buckling and vibration analysis of various composite laminated structures such as flat plates, curved panels and cylindrical shell are investigated individually. The method of analysis is the semi-analytical finite strip approach which is based on the full-energy methods. The first-order Lagrange and the third-order Hermitian shape functions are utilized for the transverse and longitudinal directions, respectively, to estimate in-plane and out-of-plane displacements. By using layer-wise theory, thick structures can be modeled with good agreement with finite element methods as presented in this work. In addition, multiple nodes are considered along the thickness of structure, so displacement can be estimated more precisely in comparison with classic and first-order models as compared with other references.

Several new product identities in relation to Rogers–Ramanujan type sums and mock theta functions

Product identities in two variables x, q expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi’s triple product identity, Watson’s quintuple identity, and Hirschhorn’s septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers–Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers–Ramanujan identities; this amounts to an unexpected proportionality of canonical basis vectors, two of which can be viewed as two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers–Ramanujan identities, related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized eta functions and mock theta functions.

Dielectric effect on the optical properties of exciton quantum dot with Gaussian plus generalized Hulthen potential as Pseudo-Harmonic potential

The dielectric effect on the intersubband optical properties for a core/shell quantum dots system is investigated with the Gaussian plus generalized Hulthen potential as a Pseudo-Harmonic potential model. The analytical solutions of the bound states energy and eigenfunctions were calculated using the Nikiforov–Uvarov (N–U). A detailed investigation of the linear dielectric function, third-order nonlinear dielectric function, total absorptions coefficient, and the transmission coefficient has been examined using the density matrix formalisms. Our result reveals that the dielectric constant affects the core materials and enhances the spectra shift, while the shell materials for both materials examined remains constant at zero photon-energy levels.

Stability analysis of concrete block retaining wall based on a scaled laboratory

A concrete block retaining wall is supported by its weight. This block retaining wall structure has some advantages, including lower construction costs, a water-permeable construction that produces less water pressure behind the wall, and a more flexible construction because it can follow the ground's contours. Rankine's theory is usually used to design this block retaining wall. According to this theory, the failure pattern behind the wall forms an angle of 450 + • /2 with the horizontal plane. The laboratory tests indicate that the pattern of failure or the pattern of sand movement behind the wall is similar to the letter S. From the sandy soil of failure pattern curve, look for a functional equation approximating the pattern. The equation of the function obtained is an equation of the third-order function. An analysis of the sliding, overturning, and overall stability block retaining wall is based on this equation of the cube function. Analysis for overall stability using the method of slices, dividing the failure area by several slices. These function equation order three is needed to get the area and length failure.

Combinatorics of two second-order mock theta functions

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

Variations in patterns of internode and branch lengths for several bamboo species

Bamboos are an important plant resource both economically and ecologically and it is, therefore, important to fully understand their biosystem characteristics. In this study, the variations in patterns of internode and branch lengths for twelve bamboo species were examined. The results indicated that node distribution along the culm is not always dense near the ends of the culm and sparse in the middle part of the culm. There were variations in the position of maximum internode length among the 12 bamboo species. For most bamboo species, maximum internode lengths correlated with total culm heights. Similar relationships existed between log (node number) and log (accumulated internode length) at the individual and species levels, and these relationships can be fitted by a linear or sigmoid function of the third-order function. Scaling exponents changed gradually during internode growth for individuals. The relationships between log (average of internode length) and log (variance of internode length) were complicated between individuals and species. The frequency distribution of bamboo branches followed power laws, but the exponents varied with species and individuals. The results may provide a new understanding of the optimal design in the unique biosystem of bamboos.

On features of starting earthquakes at rigid contacts of lithospheric plates with the base

The paper reports the results of a study of the strength of fragments of a lithospheric plate with a fault under the assumption that a granite lithospheric plate is rigidly connected to a basalt base along the Conrad border. In this case, one vertical and two horizontal contact stress components arise in the contact zone. Two types of faults are considered. In the first case, the fault banks are at a certain distance, whereas in the second case they are in contact. When solving the boundary value problem by the block element method, the problem of differential factorization of a third-order matrix function is solved exactly. In the process of solving the posed boundary problem, it is possible to accurately implement all the steps of the block element method, which allows us to obtain an exact analytical solution of the boundary problem. As a result, the problem of concentration of contact stresses for two types of faults is accurately solved. In the first case of nonzero distance between the plates, the concentration of contact stresses results in a finite energy of the system. In the second case of contact of the plates, the stress concentrations in each component of the vector of contact stresses leads to infinite energy in the system that is treated as an earthquake. Calculation of the Earth surface displacement in the epicenter zone in the case of a starting earthquake shows that surface motion occurs simultaneously in three directions. This explains the fact that such earthquakes cause greatest seismic damage on the Earth surface.

Four identities related to third-order mock theta functions

Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function $$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^{2}}}{(x;q)_{n+1}(qx^{-1};q)_{n}}\right) , \end{aligned}$$we provide different proofs of these four identities.

Microwave Power Sensors With Integrated Filtering Function for Transfer Power Standards

This letter presents a bolometric power sensor for transfer power standards in a waveguide. This is a novel design of the microwave power sensor with an integrated filter function. An analytical power sensor synthesis technique using the coupling matrix is presented here for the first time. An ${X}$ -band power sensor with the integrated third-order Chebyshev filter function was designed and manufactured. The filtering sensor uses a small bead thermistor as a sensing element. Measurements are in good agreement with simulation and theoretical expectations.

Predict the thermal conductivity of SiO2/water–ethylene glycol (50:50) hybrid nanofluid using artificial neural network

In the current work, after generating experimental data points for different volume fraction of nanoparticles ( $$\phi$$ ) and different temperatures, an algorithm to find the best neuron number in the hidden layer of artificial neural network (ANN) method is proposed to find the best architecture and then to predict the thermal conductivity ( $$k_{\text{nf}}$$ ) of SiO2/water–ethylene glycol (50:50) nanofluid. This ANN is a feed-forward network with Levenberg–Marquardt for the learning algorithm. Regarding the experimental data points, a third-order function is obtained. In the fitting method, the mean square error is 2.7547e−05, and the maximum value of error is 0.0125. The correlation coefficient of the fitting method is 0.9919. This surface also shows the behavior of nanofluid based on the $$\phi$$ and temperatures, and finally, the results of these methods have been compared. It can be seen that for 8 neuron numbers, the correlation coefficient for all outputs of ANN is 0.993861.

Decomposition of mixed pixels in MODIS data using Bernstein basis functions

The decomposition of mixed pixels in Moderate Resolution Imaging Spectroradiometer (MODIS) images is essential for the application of MODIS data in many fields. Many existing methods for unmixing mixed pixels use principal component analysis to reduce the dimensionality of the image data and require the extraction of endmember spectra. We propose the pixel spectral unmixing index (PSUI) method for unmixing mixed pixels in MODIS images. In this method, a set of third-order Bernstein basis functions is applied to reduce the dimensionality of the image data and characterize the spectral curves of the mixed pixels in a MODIS image, and then the derived PSUIs (i.e., the coefficients of the basis functions) are calibrated by means of the abundance values of the ground features from the Landsat Enhanced Thematic Mapper Plus (ETM+)/Operational Land Imager (OLI) classification images corresponding to the date and region of the MODIS image. The proposed method was tested on MODIS and ETM+/OLI images, and it obtained satisfying unmixing results. We compared the PSUI method with conventional methods, including the pixel purity index, the N-finder algorithm, the sequential maximum angle convex cone, and vertex component analysis and found that the PSUI method outperformed the other four methods.

Third-order mock theta functions

In [G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009), Entry 3.4.7, p. 67; Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24(3) (2011) 345–386; B. Chen, Mock theta functions and Appell–Lerch sums, J. Inequal Appl. 2018(1) (2018) 156; E. Mortenson, Ramanujan’s radial limits and mixed mock modular bilateral [Formula: see text]-hypergeometric series, Proc. Edinb. Math. Soc. 59(3) (2016) 1–13; W. Zudilin, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc. 143(4) (2015) 1471–1476], the authors found the bilateral series for the universal mock theta function [Formula: see text]. In [Choi, 2011], the author presented the bilateral series connected with the odd-order mock theta functions in terms of Appell–Lerch sums. However, the author only derived the associated bilateral series for the fifth-order mock theta functions. The purpose of this paper is to further derive different types of bilateral series for the third-order mock theta functions. As applications, the identities between the two-group bilateral series are obtained and the bilateral series associated to the third-order mock theta functions are in fact modular forms. Then, we consider duals of the second type in terms of Appell–Lerch sums and duals in terms of partial theta functions defined by Hickerson and Mortenson of duals of the second type in terms of Appell–Lerch sums of such bilateral series associated to some third-order mock theta functions that Chen did not discuss in [On the dual nature theory of bilateral series associated to mock theta functions, Int. J. Number Theory 14 (2018) 63–94].

On generalizations of theorems involving the third-order mock theta functions

In this paper, a transformation of bilateral series is again studied. Using this transformation, the generalizations of theorems involving some third-order mock theta functions are derived.

Partitions with Parts Separated by Parity

There have been a number of papers on partitions in which the parity of parts plays a central role. In this paper, the parts of partitions are separated by parity, either all odd parts are smaller than all even parts or vice versa. This concept first arose in a study related to the third-order mock theta function $$\nu (q)$$ . The current study also leads back to one of the Ramanujan’s more mysterious functions.

Bilateral series and Ramanujan radial limits of mock (false) theta functions

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

On Third-Order Linear Recurrent Functions

A function ψ:R→R is said to be a Tribonacci function with period p if ψ(x+3p)=ψ(x+2p)+ψ(x+p)+ψ(x), for all x∈R. In this paper, we present some properties on the Tribonacci functions with period p. We show that if ψ is a Tribonacci function with period p, then limx→∞ψ(x+p)/ψ(x)=β, where β is the root of the equation x3-x2-x-1=0 such that 1<β<2.

Higgs boson production at the LHC using the qT subtraction formalism at N3LO QCD

We consider higher-order QCD corrections to Higgs boson production through gluon-gluon fusion in the large top quark mass limit in hadron collisions. We extend the transverse-momentum (qT ) subtraction method to next-to-next-to-next-to-leading order (N3LO) and combine it with the NNLO Higgs-plus-jet calculation to numerically compute differential infrared-safe observables at N3LO for Higgs boson production in gluon fusion. To cancel the infrared divergences, we exploit the universal behaviour of the associated qT distributions in the small-qT region. We document all the necessary ingredients of the transverse-momentum subtraction method up to N3LO. The missing third-order collinear functions, which contribute only at qT = 0, are approximated using a prescription which uses the known result for the total Higgs boson cross section at this order. As a first application of the third-order qT subtraction method, we present the N3LO rapidity distribution of the Higgs boson at the LHC.

Numerical analysis of density flows in adiabatic two-phase fluids through characteristic finite element method

ABSTRACTThe purpose of this study is to analyze the density flow in adiabatic two-phase fluids through the characteristic finite element method. The fluids are assumed to be liquids. The equations of conservations of mass and momentum for the adiabatic flows and the Birch–Murnaghan equation of state are employed as the governing equations. The employed finite element method is a combination of the characteristic method and the implicit method. The governing equations are divided into two parts: the advection part and the non-advection part. The characteristic method is applied to the advection part. The Hermite interpolation function, which is based on the complete third-order polynomial interpolation using triangular finite element is employed for the interpolation of both velocity and density. Using the discontinuity conditions, an interface translocation method can be derived. The interface of the two flow densities are interpolated through the third-order spline function, using which the curvature of the interface can be directly computed. For the numerical study, the development of density flow over the Tokyo bay is presented. It is detected out that high density area is abruptly diffused over the whole area. According to the differences in the two densities, various flow patterns are computed.

Robust control of regenerative chatter in uncertain milling process with weak nonlinear cutting forces: A comparison with linear model

For various types of materials, milling process is extensively used to generate complex shapes with high quality. During the process and to achieve high removal rate, precision and better surface finish, chatter suppression is of great importance. An extended model of the milling process is presented in which the cutting forces are described as a third-order nonlinear function of chip thickness. Uncertainties associated with the process and tool parameters are also included to achieve a more realistic model. To suppress regenerative chatter, an H∞ robust control is designed based on µ-synthesis with DK-iteration algorithm. The controller guarantees the robust stability and performance of both linear and nonlinear models in the presence of uncertainties. In both models, as the amount of uncertainty increases, more actuator efforts are required to suppress the chatter. The linear model needs less actuation and time to suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed.

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

In this paper, relying on the Volterra series nonlinear system model and the high-order kernel Hilbert’s reconstructed kernel fast solved algorithm, a fault feature frequency domain identification method based on Volterra high-order kernel generalized frequency response graph analysis is proposed. Firstly, the method uses the system input and output vibration signals to determine the Volterra model. Then, the Volterra high-order kernel function is solved quickly by reproducing kernel Hilbert space method, and the generalized frequency response function is used to identify the model. Finally, multidimensional high-order spectral pattern analysis is used to separate and extract the fault and degree characteristic information implied by frequency and phase coupling in the third-order kernel function. Following the theoretical approach, in the experimental part, this paper uses the planetary gearbox fault loading test rig to complete the data collection and establishes the Volterra experimental model through the measured data. The generalized frequency responses of each order kernel function are compared and analyzed and the capability of distinguishing and the adaptability of different order kernel functions for the degree of crack failure are discussed. The effects of changing the memory length of the Volterra model and the order of the kernel function on the recognition result are verified. The final experimental results show that the use of reproducing kernel Hilbert space can effectively avoid the dimension disaster problem that occurs in the high-order kernel solution process. Moreover, the third-order kernel can describe more intuitively the nonlinear system model under multifactor coupling than the second-order kernel. Finally, Volterra series model the third-order kernel’s generalized frequency response can effectively distinguish between nondefective and faulty gears, and its resolution is enough to distinguish the degree of failure of gear cracks.