Various Forms Of Equations Research Articles

Inherent Optical Properties‐Reflectance Relationships Revisited

AbstractUnderstanding the relationship between remote sensing reflectance, Rrs(λ) and the inherent optical properties (IOPs) of natural waters is potentially a key to improving our ability to determine biogeochemical constituents from radiometric measurements. These relationships are usually described as a function of absorption, a(λ), and backscattering, bb(λ), coefficients, with the literature providing various forms of equation operating on either bb(λ)/a(λ) or bb(λ)/[a(λ)+bb(λ)] to represent the impact of variations in light field geometries and changes in sea‐water composition. The performance of several IOP‐reflectance relationships is assessed using HydroLight radiative transfer simulations covering a broad range of Case 1 and Case 2 water conditions. While early versions of IOP‐reflectance relationships assigned variability to associated proportionality factors (e.g., f/Q) or low‐order polynomial functions, recent studies have demonstrated relationships between Rrs(λ) and bb(λ)/[a(λ)+bb(λ)] are well‐characterized by nonlinear (high‐order polynomial), monotonic functions. This study demonstrates that this approach is also valid for relationships operating on bb(λ)/a(λ) and that there is no intrinsic benefit to functions operating on bb(λ)/[a(λ)+bb(λ)] compared to bb(λ)/a(λ) for Case 2 waters, contrary to recent suggestions in the literature. In all cases it is necessary to carefully consider the performance of best fit relationships across the full range of variability of IOPs and Rrs(λ), with higher order polynomials required to enable equivalent performance across the range of natural variability. The analysis further demonstrates insignificant wavelength sensitivity across the visible region, limited sensitivity to changes in solar zenith angle and extends to relationships for below surface remote sensing reflectance, rrs(λ).

Связь коэффициентов шестиполюсника с двумя входными и четырьмя выходными выводами

Problem Statement (Relevance): This paper describes one of the possible variants of the quantitative relationship between the coefficients of the B-form equations and the equations of other forms that evaluate the state of a passive six-terminal network with two input and four output terminals. Such six-terminal network can replace devices, elements or parts of electrical circuits or electric power systems. The coefficients of the B-form equations, as well as the coefficients of the A-form equations, can be determined experimentally. In principle, the coefficients of equations of other forms can also be determined experimentally. However, such experiments are usually difficult to set up and conduct. Thus, it seems to be more reasonable to determine these coefficients from the established quantitative relationship with the previously determined coefficients of the B-form. Objectives: To establish a quantitative relationship between the coefficients of the B-form equations describing the state of the six-terminal network with two input and four output terminals and the coefficients of the G-, H-, Y- and Z-form equations describing the state of the same six-terminal network. Methods Applied: Mathematical modelling and some elements of the theory of multi-terminal networks. Originality: The originality of this research lies in the proposed method of establishing a quantitative relationship between the coefficients of the A-form equations and the equations of other forms describing the state of the passive six-terminal network with two input and four output terminals. Findings: This paper examines one of the possible variants of the quantitative relationship between the coefficients of the B-form equations and the coefficients of the G-, H-, Y- and Z-form equations describing the state of the passive six-terminal network with two input and four output terminals. Some mathematical statements are presented which can help establish such relationship. Practical Relevance: If one knows the values of the B-form equation coefficients, the proposed quantitative relationship between the coefficients of various forms of equations will help build equations of other forms and establish various types of dependencies between the input and output characteristics of an electric power unit, which can be replaced with a six-terminal network with two input and four output terminals. This technique can be used to establish a quantitative relationship between the coefficients of the G-, H-, Y- or Z-form equations and the coefficients of equations of other forms describing the state of different modifications of passive six-terminal networks.

Discovery of the data-driven differential equation-based models of continuous metocean process

Data-driven models are widely used in cases when the model structure is not known a priori. However, most contemporary data-driven modeling methods like neural networks and most types of simple regression cannot be interpreted. Decision trees are on contrary, interpretable, however, the model structure remains simple and thus interpretable complex models are barely obtained with decision trees. The possible tradeoff between the model complexity and interpretability is the data-driven differential equations discovery methods. At the current time, most of the methods are not able to handle the data with a significant noise level. In the paper, a new approach to the problem is proposed. The approach involves evolutionary algorithm and sparse regression and allows one to obtain various forms of equations, defined only by the number of meta-parameters instead of the pre-defined library of terms. The application of PDE discovery tool to obtain continuous metocean process equation is described.

Some issues concerning a version of the theory of thin solids based on expansions in a system of Chebyshev polynomials of the second kind

We consider various forms of equations of motion and heat influx for deformable solids as well as various forms of Hooke’s law and Fourier’s heat conduction law under the nonclassical parametrization [1–5] of the domain occupied by a thin solid, where the transverse coordinate ranges in the interval [0, 1]. We write out several characteristics inherent in this parametrization. We use the above-mentioned equations and laws to derive the corresponding equations and laws, as well as statements of problems, for thin bodies in moments with respect to Chebyshev polynomials of the second kind. Here the interval [0, 1] is used as the orthogonality interval for the systems of Chebyshev polynomials. For this interval, we write out the basic recursion relations and, in turn, use them to obtain several additional recursion relations, which play an important role in constructing other versions of the theory of thin solids. In particular, we use the recursion relations to obtain the moments of the first and second derivatives of a scalar function, of rank one and two tensors and their components, and of some differential operators of these variables. Moreover, we give the statements of coupled and uncoupled dynamic problems in moments of the (r, N)th approximation in moment thermomechanics of thin deformable solids. We also state the nonstable temperature problem in moments of the (r, N)th approximation.

Modeling synthetic unit hydrograph parameters with geomorphologic parameters of watersheds

This study was undertaken in two medium sized agricultural watersheds and their sub watersheds located in Midnapore and Bankura districts of West Bengal state in India. Runoff hydrographs were generated from flow data and unit hydrographs (UH) were obtained for 1 and 2 h duration. UH parameters such as time to peak ( t p), time base ( t b), and peak discharge were modeled with geomorphologic parameters of the watershed such as channel parameters as well as basin parameters. Basin parameters were obtained from the Survey of India maps using geographical information system (GIS) techniques and drainage network were updated from Indian remote sensing satellite digital images. A correlation matrix between UH parameters and geomorphologic parameters was generated and geomorphologic parameters having higher degree of correlation with UH parameters were selected. Regression equations were developed between individual UH parameters and one of the geomorphologic parameters obtained in the previous step. Out of various forms of equations, power law equations gave the highest values of the coefficient of determination ( R 2). Based on higher values of R 2 geomorphologic parameters were selected. Then non-linear regression models were developed between UH parameters and the above-selected geomorphologic parameters based on Aikaike's information criterion (AIC) and higher R 2 value. All developed models for Tarafeni watershed and its sub watersheds were tested using different statistical tests for different rainfall events. Also these models were validated for nearby Bhairabbanki watershed and its sub watersheds. These models can be suitable for small and medium agricultural sub tropical sub humid basins having similar geohydrological conditions.

Estimation of the particle–wall interaction energy in sedimentation field flow fractionation

A new sedimentation field flow fractionation (SdFFF) method is presented for the estimation of the total potential energy of interaction between colloidal particles and the channel wall. The method is based on the variation of the mean cloud thickness in SdFFF due to the variation of the suspension's ionic strength. It requires only two SdFFF experiments at two different ionic strengths and at a constant acceleration field. The found values are compatible with those calculated from the various forms of equations of the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory.

DEVELOPMENT OF FRESHNESS INDEX OF EGGPLANT

Surface gloss, stiffness, and density were considered prime contributors to the freshness of eggplant fruit (Solanummelongena L.). An index to measure freshness was defined as the ratio of product of relative spectral reflectance (surfacegloss) and stiffness ratios to the density ratio. Surface gloss was measured in terms of relative spectral reflectance using acomputerized spectral radiometer system. Stiffness was determined as an initial slope of force deformation curve obtainedby uniaxial compression tests. Weight and volume for computing eggplant density were determined by standard methods.These parameters were changed to consumers acceptable limit by storing the eggplants for different periods at 25.2.Ctemperature and 905 % relative humidity. Experiments were replicated five to eight times, and the average values were usedto compute dimensionless ratios (values at any time divided by initial value) that were later used to determine a freshnessindex. Various forms of equations were tested for best fitting the data to establish the best relationship with price variation in local market.<br><br>The freshness index of eggplant fruit decreased exponentially with storage period while it varied in form of power equationof weight and density ratios. It attained the lowest value (5.6%) at acceptable limit of weight ratio of about 0.87. Freshnessindex correlated linearly with stiffness ratio and quadratically with surface gloss (correlation coefficient 0.99). Price ofeggplant varied linearly with the freshness index with good correlation coefficient (0.98), which indicated that the developedfreshness index might be useful in estimating the price of eggplants.

Vestibular Toxicity of Gentamicin in the Presence of Renal Disease

A minoglycoside antibiotics are widely used in treatment of serious Gram-negative infections. In addition, they are used in combination with penicillins to treat enterococcal infections and with penicillinase-resistant penicillins or cephalosporins for staphylococcal infections. Currently available aminoglycoside antibiotics include streptomycin, neomycin, paromomycin, gentamicin, tobramycin, and amikacin. Several newer compounds are still under investigation. Unlike other antibiotics, such as penicillin, therapeutic levels of the aminoglycosides are perilously close to toxic levels—the therapeutic index is small. Further, selection of the optimal dosage is difficult because of considerable variation from subject to subject, both in peak blood levels produced by a given dose and also in serum half-life of the antibiotic. 1-3 This problem is magnified for patients with renal insufficiency and even further exaggerated for those receiving long-term hemodialysis. A number of approaches have been proposed for dosage determination in the presence of reduced renal function. Most utilize various forms of equations

Synchronizing out of Phase

Synchronous-machine torques due to out-of-phase synchronizing may be quite large; larger in fact than those encountered during a 3-phase short circuit. Previous works have presented various forms of equations to calculate these torques.1-3,6 With the advent of the widespread use of systems analyzers and computers, it becomes convenient to use more exact equations to calculate synchronous machine transient performance. The objectives of this paper are to present the derivation of the equations for the currents and electrical torques which occur when turbine generators are synchronized out of phase; to show how these electrical torques arise; to illustrate the application of these equations to the problem of calculating the transient mechanical torques in a large, multiunit turbine-generator set under these conditions.

A Least-Square Application to Relaxation Methods

A least-square application to relaxation techniques is presented wherein the mean-square error is minimized in considering any corrections to a given set of values for the unknown function. A systematic procedure is, therefore, devised for weighting errors at various points of the lattice. The process has general applicability to various forms of equations and definitions of corresponding difference equations. As illustrations, Laplace's and the biharmonic equations are considered.

Negative Liquid Pressure at High Temperatures

IT must have been remarked in the discussions of the various forms of equation of state for vapour-liquid (cf. K. Onnes and Keesom, “Ency. der Math., or in Leyden Communications, xi., 1912, p. 727) that this equation should determine the range of possible negative pressures in liquid. If we could assume the van der Waals form of equation to hold over the wide range that is concerned, it would readily follow that negative pressure could subsist only at absolute temperatures below 27 3/2 of the critical point of the substance. For water the latter is 365° C.; thus in that substance internal tension could (theoretically) persist up to 538° absolute, which is 265° C. Such an order of magnitude appears at first sight surprisingly high, though really there is nothing to compare it with. By an oversight I have recently (Proc. Lond. Math. Soc., 1916, p. 191) quoted the critical point of water as 365° absolute, and so obtained the much lower limit 35° C.; and it was a reference to experiments by Prof. H. H. Dixon (Proc. R. Dublin Soc., 1914, p. 233), realising, for vegetable sap, tensions of the order of a hundred atmospheresat temperatures around 80° C., that has given rise to this correction.