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Journal articles on the topic 'Analytical expressions'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 July 2025

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1

Korytov, M. S., V. S. Sherbakov, and I. E. Kashapova. "Analytical analysis of the vibration protection system of a human operator." Nauchno-tekhnicheskiy vestnik Bryanskogo gosudarstvennogo universiteta 9, no. 2 (2023): 145–58. http://dx.doi.org/10.22281/2413-9920-2023-09-02-145-158.

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The task of mathematical description and study of vibration isolation systems for seats of mobile machine operators is relevant, since operators of many construction, road and other mobile machines are exposed to significant vibration effects. An original analytical solution was proposed for the differential equation of forced oscillations of a linear oscillator with kinematic excitation, which describes the vertical oscillations of the seat with the operator, for given sinusoidal oscillations of the seat base. Analytical differentiation in time of the expression for the absolute displacement

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2

Savoskin, Anatoly, and Natalia Lavlinskaya. "Spectral and cross-spectral densities expressions’ refinement for rail vehicle oscillations." E3S Web of Conferences 402 (2023): 06001. http://dx.doi.org/10.1051/e3sconf/202340206001.

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In this paper, we derive analytical expressions describing spectral and mutual spectral densities of random processes by performing analytical Fourier transform of the corresponding correlation functions. On the basis of accepted analytical expressions for correlation functions of differentiable random processes, analytical expressions for spectral density and mutual spectral density components, which is a complex frequency function, are derived. The influence of the accepted analytical expressions on the correlation functions, spectral densities, and mutual spectral density components (real,

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3

Bondarenko, Yakov A., and Galina Ya Menshikova. "EXPLORING ANALYTICAL AND HOLISTIC PROCESSING IN FACIAL EXPRESSION RECOGNITION." Moscow University Psychology Bulletin, no. 2 (2020): 103–40. http://dx.doi.org/10.11621/vsp.2020.02.06.

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Background. The study explores two main processes of perception of facial expression: analytical (perception based on individual facial features) and holistic (holistic and non-additive perception of all features). The relative contribution of each process to facial expression recognition is still an open question. Objective. To identify the role of holistic and analytical mechanisms in the process of facial expression recognition. Methods. A method was developed and tested for studying analytical and holistic processes in the task of evaluating subjective differences of expressions, using com

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4

Rios, Paulo Rangel, Weslley L. S. Assis, Tatiana C. Salazar, and Elena Villa. "Analytical Expressions for Formal Kinetics." Materials Science Forum 715-716 (April 2012): 971–76. http://dx.doi.org/10.4028/www.scientific.net/msf.715-716.971.

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In recent papers Rios and Villa resorted to developments in stochastic geometry to revisit theclassical KJMA theory and generalize it for situations in which nuclei were located in space accordingto both homogeneous and inhomogeneous Poisson point processes as well as according to Materncluster process and surface and bulk nucleation in small specimens. Rigorous mathematical methodswere employed to ensure the reliability of the new expressions. These results are briefly described.Analytical expression for inhomogeneous Poisson point process nucleation gives very good agreementwith Cellular Aut

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5

Zhou, Xinglong, Baichao Chen, Yao Luo, and Runhang Zhu. "Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils." Energies 12, no. 3 (2019): 566. http://dx.doi.org/10.3390/en12030566.

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Mutual inductance between finite-length coaxial helical filaments and tape coils are presented analytically. In this paper, a mathematical model for finite-length coaxial helical filaments is established, and subsequently, the mutual inductance of the filaments is derived in a series form, containing a one-dimensional integral. The mutual inductance expression of the filaments is then generalized for a tape conductor. When the tape conductor of each coil is closely wound, then the inverse Mellin transform is further utilized for transforming the generalized integral in the mutual inductance ex

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6

RASMUSSON, LARS, and MAGNUS BOMAN. "ANALYTICAL EXPRESSIONS FOR PARRONDO GAMES." Fluctuation and Noise Letters 02, no. 04 (2002): L343—L348. http://dx.doi.org/10.1142/s0219477502001019.

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We present some new analytical expressions for the so-called Parrondo effect, where simple coin-flipping games with negative expected value are combined into a winning game. Parrondo games are state-dependent. By identifying the game state variable, we can compute the stationary game state probabilities. Mixing losing games increases the probability of entering into a game state with positive value, as can be seen clearly from our analytical expressions for the Parrondo game value.

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7

Lang, S. J. "Analytical Expressions for TOSS Sequences." Journal of Magnetic Resonance, Series A 104, no. 3 (1993): 345–46. http://dx.doi.org/10.1006/jmra.1993.1233.

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8

EMAMPOUR, JALAL S., ALI MORSALI, and S. ALI BEYRAMABADI. "THE HEAT CAPACITIES FROM ANALYTICAL DISTRIBUTION FUNCTIONS." Journal of Theoretical and Computational Chemistry 08, supp01 (2009): 943–56. http://dx.doi.org/10.1142/s0219633609005234.

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The heat capacities and Joule–Thomson coefficients have been evaluated using three known analytical expressions for radial distribution functions (RDFs). Using such approach not only tests the power of distribution functions theory in predicting the heat capacities and Joule–Thomson coefficients, but also specifies better expressions in determining these properties. To calculate these quantities, the variation of RDF with density and temperature is required. Therefore, we should have analytical expressions which explicitly present RDF as a function of temperature, density, and interparticle di

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9

Skoblar, Ante, Roberto Žigulić, and Sanjin Braut. "Numerical ill-conditioning in evaluation of the dynamic response of structures with mode superposition method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 1 (2016): 109–19. http://dx.doi.org/10.1177/0954406216653982.

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Dynamic response of structures can be determined by mode superposition method which uses mode shape functions and modal coordinate functions. If the structure can be modelled with a uniform Euler–Bernoulli beam and a combination of clamped, free, pinned or sliding boundary conditions, analytical expressions for mode shape functions and modal coordinate functions are used. If analytical expressions for higher-order mode shapes or analytical expressions for the modal coordinate functions at high value of time are numerically evaluated, errors occur since the programming language cannot recognise

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10

Jung, E. L., M. R. Hwang, D. Park, L. Tamaryan, and S. Tamaryan. "Three-qubit Groverian measure." Quantum Information and Computation 8, no. 10 (2008): 925–42. http://dx.doi.org/10.26421/qic8.10-3.

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The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in t

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11

Auffray, Nicolas. "Analytical expressions for anisotropic tensor dimension." Comptes Rendus Mécanique 338, no. 5 (2010): 260–65. http://dx.doi.org/10.1016/j.crme.2010.04.003.

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12

Amiri, Amir Mohammad, Ali Ghanbari, and Mahdi Derakhshandi. "An Analytical Model for Estimating the Vibration Frequency of Structures Located on the Pile Group in the Case of Floating Piles and End-bearing Pile." Civil Engineering Journal 4, no. 2 (2018): 450. http://dx.doi.org/10.28991/cej-0309105.

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Exact estimation of vibration fundamental period of structures plays a vital role in their designing procedure. The proposition of a relatively exact expression which considers the effects of a pile group on the fundamental period of the structures was of less interest to previous researchers. This study aims to propose an analytical model and expression so as to estimate the free vibration period of the structures located on a pile group. To reach the objectives of this study, several numerical analyses has been carried out using the method of equivalent spring which takes into account the ef

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13

Rajendran, Sathya, Rasi Muthuramalingam, and Rajendran Lakshmanan. "Analytical Expressions for the Concentration of Substrate, Prey and Predator in Chemostat Model." Indian Journal of Applied Research 4, no. 4 (2011): 372–76. http://dx.doi.org/10.15373/2249555x/apr2014/115.

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14

Sivakumar, M., M. Mallikarjuna, and R. Senthamarai. "A kinetic non-steady state analysis of immobilized enzyme systems with external mass transfer resistance." AIMS Mathematics 9, no. 7 (2024): 18083–102. http://dx.doi.org/10.3934/math.2024882.

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<abstract><p>The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spheri

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15

Givens, Michael J., George Mylonakis, and Jonathan P. Stewart. "Modular Analytical Solutions for Foundation Damping in Soil-Structure Interaction Applications." Earthquake Spectra 32, no. 3 (2016): 1749–68. http://dx.doi.org/10.1193/071115eqs112m.

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Foundation damping incorporates combined effects of energy loss from waves propagating away from a vibrating foundation (radiation damping) and hysteretic action in supporting soil (material damping). Foundation damping appears in analysis and design guidelines for force- and displacement-based analysis of seismic building response (ASCE-7, ASCE-41), typically in graphical form (without predictive equations). We derive closed-form expressions for foundation damping of a flexible-based single degree-of-freedom oscillator from first principles. The expressions are modular in that structure and f

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16

MAYERGOYZ, I. D., and F. P. EMAD. "Computation of analytical expressions for transfer functions." International Journal of Control 46, no. 6 (1987): 1935–45. http://dx.doi.org/10.1080/00207178708934026.

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17

Baes, M., and G. Gentile. "Analytical expressions for the deprojected Sérsic model." Astronomy & Astrophysics 525 (December 8, 2010): A136. http://dx.doi.org/10.1051/0004-6361/201015716.

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18

Traunmüller, Hartmut. "Analytical expressions for the tonotopic sensory scale." Journal of the Acoustical Society of America 88, no. 1 (1990): 97–100. http://dx.doi.org/10.1121/1.399849.

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19

Baes, M., and E. Van Hese. "Analytical expressions for the deprojected Sérsic model." Astronomy & Astrophysics 534 (October 2011): A69. http://dx.doi.org/10.1051/0004-6361/201117708.

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20

Georgantzinos, S. K., G. I. Giannopoulos, A. Fatsis, and N. V. Vlachakis. "Analytical expressions for electrostatics of graphene structures." Physica E: Low-dimensional Systems and Nanostructures 84 (October 2016): 27–36. http://dx.doi.org/10.1016/j.physe.2016.05.028.

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21

Rao, B. Venkoba. "Analytical expressions for classifier product size distributions." Minerals Engineering 18, no. 5 (2005): 557–60. http://dx.doi.org/10.1016/j.mineng.2004.08.014.

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22

Panda, R. C., and Cheng-Ching Yu. "Analytical expressions for relay feed back responses." Journal of Process Control 13, no. 6 (2003): 489–501. http://dx.doi.org/10.1016/s0959-1524(02)00119-1.

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23

Hunt, F. Y., and R. D. McMichael. "Analytical expressions for Barkhausen jump size distributions." IEEE Transactions on Magnetics 30, no. 6 (1994): 4356–58. http://dx.doi.org/10.1109/20.334086.

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24

Shevchenko, Ivan I., and Nikolay N. Vasiliev. "Algorithms of numeric deduction of analytical expressions." ACM SIGSAM Bulletin 27, no. 1 (1993): 1–3. http://dx.doi.org/10.1145/152379.152380.

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25

Aissing, G., H. J. Monkhorst, and Yu V. Petrov. "Simple analytical expressions for mesomolecular matrix elements." Physical Review A 42, no. 11 (1990): 6894–96. http://dx.doi.org/10.1103/physreva.42.6894.

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26

August, H. J., and J. Wernisch. "Analytical Expressions for the Electron Backscattering Coefficient." Physica Status Solidi (a) 114, no. 2 (1989): 629–33. http://dx.doi.org/10.1002/pssa.2211140225.

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27

CANET, D., J. BRONDEAU, and C. ROUMESTAND. "Analytical Expressions for the DANTE Pulse Sequence." Journal of Magnetic Resonance, Series A 117, no. 1 (1995): 103–8. http://dx.doi.org/10.1006/jmra.1995.9963.

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28

Rudolph, Michelle, and Alain Destexhe. "On the Use of Analytical Expressions for the Voltage Distribution to Analyze Intracellular Recordings." Neural Computation 18, no. 12 (2006): 2917–22. http://dx.doi.org/10.1162/neco.2006.18.12.2917.

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Different analytical expressions for the membrane potential distribution of membranes subject to synaptic noise have been proposed and can be very helpful in analyzing experimental data. However, all of these expressions are either approximations or limit cases, and it is not clear how they compare and which expression should be used in a given situation. In this note, we provide a comparison of the different approximations available, with an aim of delineating which expression is most suitable for analyzing experimental data.

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29

Wang, Jia Ling, Xin Du, Jia Lv, et al. "Solving Rate Equations Including Background Losses Applicable to Threshold Pumped Fiber Lasers with Three-Energy-Level Dopants." Applied Mechanics and Materials 121-126 (October 2011): 4788–95. http://dx.doi.org/10.4028/www.scientific.net/amm.121-126.4788.

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An analytical expression of the threshold pump power of the fiber laser with three-energy-level dopants has been obtained from rate equations including loss coefficients of the pump and laser radiations. And the rate equations describing the threshold pumped fiber lasers have been analytically solved following the derivation of the implicit expression specifying the pump variation along the fiber and expressions of the position dependent population densities on the three energy levels in terms of the pump variation.

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30

LAPORTA, S. "ANALYTICAL EXPRESSIONS OF THREE- AND FOUR-LOOP SUNRISE FEYNMAN INTEGRALS AND FOUR-DIMENSIONAL LATTICE INTEGRALS." International Journal of Modern Physics A 23, no. 31 (2008): 5007–20. http://dx.doi.org/10.1142/s0217751x08042869.

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In this paper we continue the work began in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the three-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the four-loop on-shell self-mass integral with four equal mas

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31

Granot, Er'el. "Analytical Solutions for the Propagation of UltraShort and UltraSharp Pulses in Dispersive Media." Applied Sciences 9, no. 3 (2019): 527. http://dx.doi.org/10.3390/app9030527.

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Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, Gaussian and Airy pulses, which keep their shape. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the objective of this paper to present a variety of pulse profiles, which have analytical expressions but can simulate real-physical pulses with great accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and

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32

Wang, Jiujiang, Xin Liu, Yuanyu Yu, et al. "A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections." Micromachines 12, no. 6 (2021): 714. http://dx.doi.org/10.3390/mi12060714.

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Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán e

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33

Garza-Rios, Luis O., and Michael M. Bernitsas. "Analytical Expressions for Stability and Bifurcations of Turret Mooring Systems." Journal of Ship Research 42, no. 03 (1998): 216–32. http://dx.doi.org/10.5957/jsr.1998.42.3.216.

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The eight necessary and sufficient conditions for stability of turret mooring systems (TMS) are derived analytically. Analytical expressions for TMS bifurcation boundaries where static and dynamic loss of stability occur are also derived. These analytical expressions provide physics-based means to evaluate the stability properties of TMS, find elementary singularities, and describe the morphogeneses occurring as a parameter (or design variable) or group of parameters are varied. They eliminate the need to compute numerically the TMS eigenvalues. Analytical results are verified by comparison to

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34

Orly, Aviv. "Analytical Approximations as Close as Desired to Special Functions." Axioms 14, no. 8 (2025): 566. https://doi.org/10.3390/axioms14080566.

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We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants—with differential evolution optimization, a modern machine learning technique, we achieve high-accuracy approximations using elegantly simple expressions. This method transforms non-elementary functions, which lack closed-form expressions and are often defined by integrals or infinite series, into simple analytical forms. This transformation enables deeper quali

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35

Hamzehlouia, Sina, and Kamran Behdinan. "Squeeze film dampers supporting high-speed rotors: Fluid inertia effects." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 234, no. 1 (2019): 18–32. http://dx.doi.org/10.1177/1350650119855792.

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This work represents closed-form analytical expressions for the operating parameters for short-length open-ended squeeze film dampers, including the lubricant velocity profiles, hydrodynamic pressure distribution, and lubricant reaction forces. The proposed closed-form expressions provide an accelerated calculation of the squeeze film damper parameters, specifically for rotordynamics applications. In order to determine the analytical solutions for the squeeze film damper parameters, the thin film equations for lubricant are introduced in the presence of the influence of lubricant inertia. Subs

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36

Garza-Rios, Luis O., and Michael M. Bernitsas. "Analytical Expressions of the Stability and Bifurcation Boundaries for General Spread Mooring Systems." Journal of Ship Research 40, no. 04 (1996): 337–50. http://dx.doi.org/10.5957/jsr.1996.40.4.337.

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Spread mooring systems (SMS) are labeled as general when they are not restricted by conditions of symmetry. The six necessary and sufficient conditions for stability of general SMS are derived analytically. The boundaries where static and dynamic loss of stability occur also are derived in terms of the system eigenvalues, thus providing analytical means for defining the morphogenesis that occurs when a bifurcation boundary is crossed. The equations derived in this paper provide analytical expressions of elementary singularities and routes to chaos for general mooring system configurations. Cat

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37

NICULESCU, Valentin I. R., Dumitru POPESCU, and Alin Gabriel POPESCU. "A versatile analytical representation of the different aircraft wings." INCAS BULLETIN 16, no. 3 (2024): 61–65. http://dx.doi.org/10.13111/2066-8201.2024.16.3.6.

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This paper presents an analytical method to describe the shape of an airplane wing. The analytical shape of the wing section implies analytical expressions for the wing parameters: maximum thickness, maximum camber, maximum camber position, minimum thickness position, and so on. The analytical expressions of the wings in a first representation allow simple relationships with the geometric parameters of the wing. The defined expressions for the wings parameters relevant for aircraft design are versatile.

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38

Agrawal, Om P. "Analytical Solution for Stochastic Response of a Fractionally Damped Beam." Journal of Vibration and Acoustics 126, no. 4 (2004): 561–66. http://dx.doi.org/10.1115/1.1805003.

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This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations, each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green’s function and a Duhamel integral-type expression for the system’s response. The response expression c

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39

Alvi, Sajid Hussain, Bakhtiar Ali, Jawad Mirza та ін. "Performance Analysis of IRS-Assisted THz Communication Systems over α-μ Fading Channels with Pointing Errors". Sensors 23, № 16 (2023): 7028. http://dx.doi.org/10.3390/s23167028.

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In this paper, we analyze the performance of an intelligent reflecting surface (IRS)-aided terahertz (THz) wireless communication system with pointing errors. Specifically, we derive closed-form analytical expressions for the upper bounded ergodic capacity and approximate expression of the outage probability. We adopt an α-μ fading channel model for our analysis that is experimentally demonstrated to be a good fit for THz small-scale fading statistics, especially in indoor communication scenarios. In the proposed analysis, the statistical distribution of the α-μ fading channel is used to deriv

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40

Wei, Qun. "Analytical Investigations of Spin-Spin, Spin-Other-Orbit and Combined Mechanisms of Zero-Field Splitting for 3d3 Ions at Trigonal Symmetry." Zeitschrift für Naturforschung A 64, no. 9-10 (2009): 646–52. http://dx.doi.org/10.1515/zna-2009-9-1017.

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A new method of obtaining approximate analytical expressions of zero-field splitting (ZFS) has been proposed. By analyzing the magnetic interaction mechanisms which affect the properties of ZFS of ground 4A2 state for 3d3 ions at trigonal symmetry, and comparing with the calculated results of complete diagonalization method, the approximate analytical expressions of the contribution to zero-field splitting of 4A2 state from spin-spin (SS), spin-other-orbit (SOO), and combined mechanisms have been obtained. Also the applicability of these approximate analytical expressions has been analyzed. It

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41

Ananthaswamy, V., and M. Subha. "ANALYTICAL EXPRESSIONS FOR EXOTHERMIC EXPLOSIONS IN A SLAB." International Journal of Research -GRANTHAALAYAH 1, no. 2 (2014): 22–37. http://dx.doi.org/10.29121/granthaalayah.v1.i2.2014.3076.

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Analytical solutions for the strongly exothermic decomposition of combustible material are discussed in this paper. Combustible material uniformly distributed between symmetrically heated parallel plates under Sensitized, Arrhenius and Bimolecular reaction rates are also discussed neglecting consumption of material. Approximate analytical expressions of steady state temperature fields are derived by using Homotopy analysis method (HPM) for various values of relevant dimensionless parameters. Analytical results are compounded with perturbation technique and numerical simulation. Analytical resu

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42

Sarabia, Jose Maria, and Enrique Calderin-Ojeda. "Analytical expressions of risk quantities for composite models." Journal of Risk Model Validation 12, no. 3 (2018): 75–101. http://dx.doi.org/10.21314/jrmv.2018.194.

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43

Sharaf, M., and A. Sendy. "New Analytical Expressions for the Intrinsic Color Index." Journal of King Abdulaziz University-Science 22, no. 1 (2010): 267–74. http://dx.doi.org/10.4197/sci.22-1.18.

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44

Uzunidis, Dimitris, Chris Matrakidis, and Alexandros Stavdas. "Analytical FWM Expressions for Coherent Optical Transmission Systems." Journal of Lightwave Technology 35, no. 13 (2017): 2734–40. http://dx.doi.org/10.1109/jlt.2017.2667722.

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45

Cheng, Li, Jiaming Shi, and Bo Xu. "Analytical Expressions of Dual-Frequency Plasma Diagnostic Theory." Plasma Science and Technology 14, no. 1 (2012): 37–39. http://dx.doi.org/10.1088/1009-0630/14/1/09.

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46

Ou, Ziqiang, and Karl-Rudolf Koch. "Analytical expressions for Bayes estimates of variance components." manuscripta geodaetica 19, no. 5 (1994): 284–93. http://dx.doi.org/10.1007/bf03655458.

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47

Pécseli, H. L., and J. Trulsen. "Analytical expressions for conditional averages: a numerical test." Physica Scripta 43, no. 5 (1991): 503–7. http://dx.doi.org/10.1088/0031-8949/43/5/012.

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48

Fernández, F. M., and R. H. Tipping. "Analytical expressions for the energies of anharmonic oscillators." Canadian Journal of Physics 78, no. 9 (2000): 845–50. http://dx.doi.org/10.1139/p00-066.

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We propose a systematic construction of algebraic approximants for the bound-state energies of anharmonic oscillators. The approximants are based on the Rayleigh-Schrödinger perturbation series and take into account the analytical behavior of the energies at large values of the perturbation parameter. A simple expression obtained from a low-order perturbation series compares favorably with alternative approximants. Present approximants converge in the large-coupling limit and are suitable for the calculation of the energy of highly excited states. Moreover, we obtain some branch points of the

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49

Lam, S., F. Lesage, and X. Intes. "Time Domain Fluorescent Diffuse Optical Tomography: analytical expressions." Optics Express 13, no. 7 (2005): 2263. http://dx.doi.org/10.1364/opex.13.002263.

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Srivastava, Sangeeta, and Enakshi Khular Sharma. "Analytical expressions for power exchange in multiwaveguide systems." Journal of the Optical Society of America A 13, no. 8 (1996): 1683. http://dx.doi.org/10.1364/josaa.13.001683.

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