Search Results (23)

In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference $\alpha$, an integer with the Voronoĭ function weight $V_k$. In the case of $V_1\left(x\right)=e^{-x}$, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The $\alpha =0$ case is the divisor function, while the $\alpha =1$ case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan. Full article

Schröder based series for the principal and negative one branches of the Lambert W function are defined; the series are generic and are in terms of an initial, arbitrary, approximating function. Upper and lower bounds for the initial approximating functions, consistent with convergence, are determined. Approximations for both branches of the Lambert W function are proposed which have modest relative error bounds over their domains of definition and which are suitable as initial approximation functions for a convergent Schröder series. For the principal branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹³⁶. For the negative one branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹⁴³. Applications of the approximations for the principal and negative one branches include new approximations for the Lambert W function, analytical approximations for the integral of the Lambert W function, upper and lower bounded functions for the Lambert W function, approximations for the power of the Lambert W function and approximations to the solution of the equations c^c = y and C^C = e^v, respectively, for c and C. Full article

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the Weierstrass elliptic functions and some Lambert series. These identities are used to derive some of the properties of the Lambert series. Full article

“Rejoice, Texandria, for Oda!” Thus begins the series of chants and readings commemorating the virgin St Oda, patron of the village that took her name—Sint-Oedenrode—in the late medieval liturgy of the town of ’s-Hertogenbosch. Overt praise for the surrounding region, Texandria, extending across the northern limits of the duchy of Brabant and diocese of Liège, is a recurring theme in the liturgy inspired by the saint’s legend. Yet how did Oda, of Scottish origin, become so closely associated with this remote region? And what was the significance of her liturgical veneration in ’s-Hertogenbosch, to which Sint-Oedenrode was enfranchised? Exemplifying interactions between central and secondary places within a specific region, this study argues for the relevance of the historical approach to urban–rural dynamics in medieval hagiography and its related liturgy. Recognition that smaller towns and villages played important roles in regional networks prompts more focused attention to regional priorities in the legends and liturgies of local saints. That Oda’s cult is attested by a diversity of extant documentary evidence—historical, hagiographic, and liturgical, including newly discovered liturgical readings—facilitates interpretation of her veneration in ’s-Hertogenbosch and of the intertextual connections between her legend and those of other saints, notably Lambert, associated with the duchy and diocese. As suggested by this example, regionalism merits greater scrutiny as an integral component of civic religion. Full article

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions ${\mathsf{{\rm T}}}_{\mathcal{J}}\left({\alpha }_i\right)$. Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions ${f\in \mathsf{{\rm T}}}_{\mathcal{J}}\left({\alpha }_i\right)$. In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. Full article

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions. Full article

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained. Full article

Dynamic reconfiguration, the monitoring of power production, and the fault diagnosis of photovoltaic arrays, among other applications, require fast and accurate models of photovoltaic arrays. In the literature, some models use the Lambert-W function to represent each module of the array, which increases the calculation time. Other models that use implicit equations to avoid the Lambert-W function do not use the inflection voltages to simplify the system of nonlinear equations that represent the array, increasing the computational burden. Therefore, this paper proposes mathematical models for series-parallel (SP) and total-cross-tied (TCT) photovoltaic arrays based on the implicit equations of the single-diode model and the inflection points of the current–voltage curves. These models decrease the calculation time by reducing the complexity of the nonlinear equation systems that represent each string of SP arrays and the whole TCT. Consequently, the calculation process that solves the model speeds up in comparison with processes that solve traditional explicit models based on the Lambert-W function. The results of several simulation scenarios using the proposed SP model with different array sizes show a reduction in the computation time by $82.97\%$ in contrast with the traditional solution. Additionally, when the proposed TCT model for arrays larger than $2\times 2$ is used, the reduction in the computation time is between $47.71\%$ and $92.28\%$. In dynamic reconfiguration, the results demonstrate that the proposed SP model provides the same optimal configuration but 7 times faster than traditional solutions, and the TCT model is solved at least 4 times faster than classical solutions. Full article

The laser extinction method (LEM) is particularly suitable for measuring particle sizes in flames because this method, which is based on the Beer–Lambert law, is non-intrusive and easy to implement. In the LEM, the interpretation of the extinction data is usually developed under the assumption that light extinction due to scattering is a result of the superposition of single scattering by individual particles; however, this could be violated for flames with dense concentrations of particles in which multiple scattering could occur. Quantifying the effect of multiple scattering under general conditions is still a formidable problem. In this work, we carried out a series of careful measurements of the laser extinction using standard particles of various known sizes, number densities and optical path lengths, all under the condition that the acceptance angle of the detector was limited to nearly zero. Combined with a four-flux model, we quantitatively analyzed the effect of multiple scattering on the size measurement using the LEM. The results show that the effect of multiple scattering could be ignored when the optical thickness is less than two under strict restrictions on the detector acceptance angle. Guided by this, the size distribution of an alumina (${\mathrm{Al}}_2{\mathrm{O}}_3$) particle sample was measured by the LEM with dual wavelengths. Parameterized distributions were solved with the help of graph plotting, and the results compared well with the measurement from the Malvern particle size analyzer. The same method was then used to measure the particle size distribution in the plume of a solid rocket motor (SRM). The use of an off-axis parabolic mirror in the experimental setup could suppress the jitter of light passing through the SRM plume, and the particle size in the plume of the measured SRM was in the order of microns. Full article

The most commonly used model of solar cells is the single-diode model, with five unknown parameters. First, this paper proposes three variants of the single-diode model, which imply the voltage dependence of the series resistance, parallel resistance, and both resistors. Second, analytical relationships between the current and the voltage expressed were derived using the Lambert W function for each proposed model. Third, the paper presents a hybrid algorithm, Chaotic Snake Optimization (Chaotic SO), combining chaotic sequences with the snake optimization algorithm. The application of the proposed models and algorithm was justified on two well-known solar photovoltaic (PV) cells—RTC France solar cell and Photowatt-PWP201 module. The results showed that the root-mean-square-error (RMSE) values calculated by applying the proposed equivalent circuit with voltage dependence of both resistors are reduced by 20% for the RTC France solar cell and 40% for the Photowatt-PWP201 module compared to the standard single-diode equivalent circuit. Finally, an experimental investigation was conducted into the applicability of the proposed models to a solar laboratory module, and the results obtained proved the relevance and effectiveness of the proposed models. Full article

A modification of the two-flux Kubelka-Munk (K-M) model was proposed to describe the energy conservation of scattered light in colored mixed material with a defined scattered photometric, which is applied for the relative quantity distribution of each colored monochrome component in mixed material. A series of systematical experiments demonstrated a higher consistency with the reference quantity distribution than the common Lambert-Beer (L-B) law. Its application in the fibrogram of each component for measuring the cotton fiber’s length was demonstrated to be good, extending its applicability to white and dark colored blended fibers, the length of which is harder to measure using L-B law. Full article

In this paper, we propose a compression-based anomaly detection method for time series and sequence data using a pattern dictionary. The proposed method is capable of learning complex patterns in a training data sequence, using these learned patterns to detect potentially anomalous patterns in a test data sequence. The proposed pattern dictionary method uses a measure of complexity of the test sequence as an anomaly score that can be used to perform stand-alone anomaly detection. We also show that when combined with a universal source coder, the proposed pattern dictionary yields a powerful atypicality detector that is equally applicable to anomaly detection. The pattern dictionary-based atypicality detector uses an anomaly score defined as the difference between the complexity of the test sequence data encoded by the trained pattern dictionary (typical) encoder and the universal (atypical) encoder, respectively. We consider two complexity measures: the number of parsed phrases in the sequence, and the length of the encoded sequence (codelength). Specializing to a particular type of universal encoder, the Tree-Structured Lempel–Ziv (LZ78), we obtain a novel non-asymptotic upper bound, in terms of the Lambert W function, on the number of distinct phrases resulting from the LZ78 parser. This non-asymptotic bound determines the range of anomaly score. As a concrete application, we illustrate the pattern dictionary framework for constructing a baseline of health against which anomalous deviations can be detected. Full article

There are three standard equivalent circuit models of solar cells in the literature—single-diode, double-diode, and triple-diode models. In this paper, first, a modified version of the single diode model, called the Improved Single Diode Model (ISDM), is presented. This modification is realized by adding resistance in series with the diode to enable better power loss dissipation representation. Second, the mathematical expression for the current–voltage relation of this circuit is derived in terms of Lambert’s W function and solved by using the special trans function theory. Third, a novel hybrid algorithm for solar cell parameters estimation is proposed. The proposed algorithm, called SA-MRFO, is used for the parameter estimation of the standard single diode and improved single diode models. The proposed model’s accuracy and the proposed algorithm’s efficiency are tested on a standard RTC France solar cell and SOLAREX module MSX 60. Furthermore, the experimental verification of the proposed circuit and the proposed solar cell parameter estimation algorithm on a solar laboratory module is also realized. Based on all the results obtained, it is shown that the proposed circuit significantly improves current–voltage solar cell representation in comparison with the standard single diode model and many results in the literature on the double diode and triple diode models. Additionally, it is shown that the proposed algorithm is effective and outperforms many literature algorithms in terms of accuracy and convergence speed. Full article

This paper presents new analytical expressions for the maximum power point voltage, current, and power that have an explicit dependence on the series resistance. An explicit expression that relates the series resistance to well-known solar cell parameters was also derived. The range of the validity of the model, as well as the mathematical assumptions taken to derive it are explained and discussed. To test the accuracy of the derived model, a numerical single-diode model with solar cell parameters whose values can be found in the latest installment of the solar cell efficiency tables was used. The accuracy of the derived model was found to increase with increasing bandgap and to decrease with increasing series resistance. An experimental validation of the analytical model is provided and its practical limitations addressed. The new expressions predicted the maximum power obtainable by the studied cells with estimated errors below $0.1\%$ compared to the numerical model, for typical values of the series resistance. Full article