Volume-3 ~ Issue-2
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Abstract :This paper proposed fast Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L) iterative method for
solving path planning problem for a mobile robot operating in indoor environment model. It is based on the use
of Laplace's Equation to constraint the distribution of potential values in the environment of the robot. Fast
computation with half-sweep iteration is obtained by considering only half of whole points in the configuration
model. The inclusion of SOR and 9-point Laplacian into the formulation further speeds up the computation. The
simulation results show that HSSOR9L performs much faster than the previous iterative methods in computing
the potntial values to be used for generating smooth path from a given initial point to a specified goal position.
Keywords - Robot path planning, Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L), Laplace's Equation, Harmonic functions
Keywords - Robot path planning, Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L), Laplace's Equation, Harmonic functions
[1] Khatib, O. 1985. Real time obstacle avoidance for manipulators and mobile robots. IEEE Transactions on Robotics and Automation
1:500–505.
[2] Koditschek, D.E. 1987. Exact robot navigation by means of potential functions: Some topological considerations. Proceedings of
the IEEE International Conference on Robotics and Automation: 1-6.
[3] Connolly, C. I., Burns, J.B. & Weiss, R. 1990. Path planning using Laplace's equation. Proceedings of the IEEE International
Conference on Robotics and Automation: 2102–2106.
[4] Akishita, S., Kawamura, S. & Hayashi, K. 1990. Laplace potential for moving obstacle avoidance and approach of a mobile robot.
Japan-USA Symposium on flexible automation, A Pacific rim conf.: 139–142.
[5] Connolly, C.I. & Gruppen, R. 1993. On the applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7): 931–
946.
[6] Waydo, S. & Murray, R.M. 2003. Vehicle motion planning using stream functions. In Proc. of the Int. Conf. on Robotics and
Automation (ICRA), 2003, pp.2484-2491.
[7] Szulczyński, P., Pazderski, D. & Kozłowski, K. 2011. Real-Time Obstacle Avoidance Using Harmonic Potential Functions. Journal
of Automation, Mobile Robotics & Intelligent Systems. Volume 5, No 3, 2011.
[8] Sasaki, S. 1998. A Practical Computational Technique for Mobile Robot Navigation. Proceedings of the IEEE International
Conference on Control Applications: 1323-1327.
[9] Daily, R. & Bevly, D.M. 2008. Harmonic Potential Field Path Planning for High Speed Vehicles. In the proceeding of American
Control Conference, Seattle, June 11-13, 4609-4614.
[10] Garrido, S., Moreno, L., Blanco, D. & Monar, F.M. 2010. Robotic Motion Using Harmonic Functions and Finite Elements. Journal
of Intelligent and Robotic Systems archive. Volume 59, Issue 1, July 2010. Pages 57 – 73.
1:500–505.
[2] Koditschek, D.E. 1987. Exact robot navigation by means of potential functions: Some topological considerations. Proceedings of
the IEEE International Conference on Robotics and Automation: 1-6.
[3] Connolly, C. I., Burns, J.B. & Weiss, R. 1990. Path planning using Laplace's equation. Proceedings of the IEEE International
Conference on Robotics and Automation: 2102–2106.
[4] Akishita, S., Kawamura, S. & Hayashi, K. 1990. Laplace potential for moving obstacle avoidance and approach of a mobile robot.
Japan-USA Symposium on flexible automation, A Pacific rim conf.: 139–142.
[5] Connolly, C.I. & Gruppen, R. 1993. On the applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7): 931–
946.
[6] Waydo, S. & Murray, R.M. 2003. Vehicle motion planning using stream functions. In Proc. of the Int. Conf. on Robotics and
Automation (ICRA), 2003, pp.2484-2491.
[7] Szulczyński, P., Pazderski, D. & Kozłowski, K. 2011. Real-Time Obstacle Avoidance Using Harmonic Potential Functions. Journal
of Automation, Mobile Robotics & Intelligent Systems. Volume 5, No 3, 2011.
[8] Sasaki, S. 1998. A Practical Computational Technique for Mobile Robot Navigation. Proceedings of the IEEE International
Conference on Control Applications: 1323-1327.
[9] Daily, R. & Bevly, D.M. 2008. Harmonic Potential Field Path Planning for High Speed Vehicles. In the proceeding of American
Control Conference, Seattle, June 11-13, 4609-4614.
[10] Garrido, S., Moreno, L., Blanco, D. & Monar, F.M. 2010. Robotic Motion Using Harmonic Functions and Finite Elements. Journal
of Intelligent and Robotic Systems archive. Volume 59, Issue 1, July 2010. Pages 57 – 73.
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Paper Type | : | Research Paper |
Title | : | Perimeter of the Elliptical Arc a Geometric Method |
Country | : | India |
Authors | : | Aravind Narayan |
: | 10.9790/5728-0320813 |
Abstract :There are well known formulas approximating the circumference of the Ellipse given in different
periods in history. However there lacks a formula to calculate the Arc length of a given Arc segment of an
Ellipse. The Arc length of the Elliptical Arc is presently given by the Incomplete Elliptical Integral of the Second
Kind, however a closed form solution of the Elliptical Integral is not known. The current solution methods are
numerical approximation methods, based on series expansions of the Elliptical Integral. This paper introduces a
Geometric Method (procedure) to approximate the Arc length of any given Arc segment of the ellipse. An
analytical procedure of the defined geometric method is detailed.
[1] Wikipedia Article: Ellipse
[2] Wikipedia Article: Elliptic integral
[3] Wikipedia Article: Trigonometric functions:
[4] Wikipedia Article: Eccentric Anomaly
[5] Wikipedia Article: Elliptical Integral
[6] Appendices
[7] Appendix A: Determine the side of an Isosceles Triangle given the Angle opposite to the Base and Base length
[8] Appendix B: Approximating the Incomplete Elliptical Integral of Second Kind
[9] Appendix C: Sample Problem & Solution Algorithm
[2] Wikipedia Article: Elliptic integral
[3] Wikipedia Article: Trigonometric functions:
[4] Wikipedia Article: Eccentric Anomaly
[5] Wikipedia Article: Elliptical Integral
[6] Appendices
[7] Appendix A: Determine the side of an Isosceles Triangle given the Angle opposite to the Base and Base length
[8] Appendix B: Approximating the Incomplete Elliptical Integral of Second Kind
[9] Appendix C: Sample Problem & Solution Algorithm
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Paper Type | : | Research Paper |
Title | : | A Mathematical Analysis of Compromising Programming Techniques |
Country | : | India |
Authors | : | Anita, Dr. Sandeep Kumar |
: | 10.9790/5728-0321421 |
Abstract :Agriculture is the back bone of Indian economy and provides livelihood to about 70 percent of the
population and about one third of our national income gets generated in this sector. After independence, in
early years, there was a problem of a food shortage. The position at the food front became a matter of concern
in the early sixties. To meet the situation efforts were made to develop the agriculture sector of the economy.
Toward the mid of sixties the new agriculture technology emphasizing the use of fertilizer and irrigation,
ushered an era popularly could as green revaluation. However it was confined to few states and few crops. In
present state, the nation's objective is not only to increase the food grain production but also to increase the
employment ventures. While individual farmer may be interested in maximizing his cash income risks aversion
etc. The mathematical programming approach to the modeling of agricultural decisions rests on certain basic
assumptions about the situation being modeled and the decision maker himself. One fundamental assumption is
that the decision maker (DM) seeks to optimize a well defined single objective. In reality this is not the case, as
the DM is usually seeking an optimal compromise amongst several objectives, many of which can be in conflict,
or trying to achieve satisfying levels of his goals. For instance, a subsistence farmer may be interested in
securing adequate food supplies for the family, maximizing cash income, increasing leisure, avoiding risk etc.
but not necessarily in that order. Similarly a commercial farmer may wish to maximize gross margin, minimize
his indebtedness, acquire more land, reduce fixed costs etc.
Keywords - Decision Making, Optimization, Mathematical programming, Minimization & maximization
Keywords - Decision Making, Optimization, Mathematical programming, Minimization & maximization
[1]. Bansil P C (2000) Demand for food grains by 2020 AD: Agricultural Incentives and Sustainable development : past trends and
future scenarios. Techno Economics research institute, New Delhi.
[2]. Bansil P C (1990) Agricultural Stastistical Compendium VOL O. Techno Economics Research Institute, New Delhi.
[3]. Bhalla G S and Hazell P (1997) Food Grains demand in India to 2020 - A preliminary exercise Economics and Political weekly
32:A150- A154.
[4] Damodaran H (1998) Coarse Cereals to outstrip staples in demand. Business Line (daily) December 1st pp4.
[5]. Dantawala M L (1987) Growth and equity agriculture. Indian journal of Agricultural Economics .42(1): 149-59.
[6]. Das and Purnendo Shekhar (1978) Growth and instability in Crop output in Eastern India. Economics and Political weekly 13)41):1741-48.
[7]. Das P (2000) 50 yeasr of frontline Agricultural extension programmes. ICAR New Delhi.
[8]. Desai D K and Patel N T (1983) Improving Growth of food grains production in the western region in India. Indian Journal of
Agricultural Economics 38(4):539-56.
[9]. Easter K W (1972) Regions of Indian Agricultural Planning and management. The Ford Foundation, New Delhi.
[10]. Gadgil Sulochana, Abrol Y P and Rao P R Seshagiri (1999) On Growth and fluctuation of Indian foodgrain production. Current
Science. 76. (4): 151 – 59.
future scenarios. Techno Economics research institute, New Delhi.
[2]. Bansil P C (1990) Agricultural Stastistical Compendium VOL O. Techno Economics Research Institute, New Delhi.
[3]. Bhalla G S and Hazell P (1997) Food Grains demand in India to 2020 - A preliminary exercise Economics and Political weekly
32:A150- A154.
[4] Damodaran H (1998) Coarse Cereals to outstrip staples in demand. Business Line (daily) December 1st pp4.
[5]. Dantawala M L (1987) Growth and equity agriculture. Indian journal of Agricultural Economics .42(1): 149-59.
[6]. Das and Purnendo Shekhar (1978) Growth and instability in Crop output in Eastern India. Economics and Political weekly 13)41):1741-48.
[7]. Das P (2000) 50 yeasr of frontline Agricultural extension programmes. ICAR New Delhi.
[8]. Desai D K and Patel N T (1983) Improving Growth of food grains production in the western region in India. Indian Journal of
Agricultural Economics 38(4):539-56.
[9]. Easter K W (1972) Regions of Indian Agricultural Planning and management. The Ford Foundation, New Delhi.
[10]. Gadgil Sulochana, Abrol Y P and Rao P R Seshagiri (1999) On Growth and fluctuation of Indian foodgrain production. Current
Science. 76. (4): 151 – 59.
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- Abstract
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Paper Type | : | Research Paper |
Title | : | On qpI-Irresolute Mappings |
Country | : | India |
Authors | : | Mandira Kar, S. S. Thakur |
: | 10.9790/5728-0322224 | |
Abstract :TIn the present paper the concept of qpI-Irresolute mappings have been introduced and studied.
Keywords - Ideal bitopological spaces, qpI- closed sets, qpI- open sets and qpI- Irresolute mappings AMS Subject classification 54A05, 54C08
Keywords - Ideal bitopological spaces, qpI- closed sets, qpI- open sets and qpI- Irresolute mappings AMS Subject classification 54A05, 54C08
[1] M.C. Datta, Contributions to the theory of bitopological spaces, Ph.D. Thesis, B.I.T.S. Pilani, India., (1971)
[2] S. Jafari. and N. Rajesh, On qI open sets in ideal bitopological spaces, University of Bacau, Faculty of Sciences, Scientific
Studies and Research, Series Mathematics and Informatics., Vol. 20, No.2 (2010), 29-38
[3] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc.,13(1963), 71-89
[4] K. Kuratowski, Topology, Vol. I, Academic press, New York., (1966)
[5] S. N. Maheshwari, G. I. Chae and P. C. Jain On quasi open sets, U. I. T. Report., 11 (1980), 291-292.
[6] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb On precontinuous and weak precontinuous mappings, Proc. Math. Phys.
Soc. Egypt., 53 (1982), 47-53
[7] U. D. Tapi, S. S. Thakur and Alok Sonwalkar On quasi precontinuous and quasi preirresolute mappings, Acta Ciencia Indica.,
21(14) (2)(1995), 235-237
[8] U. D. Tapi, S. S. Thakur and Alok Sonwalkar, Quasi preopen sets, Indian Acad. Math., Vol. 17 No.1, (1995), 8-12
[9] M. Kar, and S.S. Thakur, Quasi Pre Local Functions in Ideal Bitopological Spaces Book: International Conference on
Mathematical Modelling and Soft Computing 2012, Vol. 02 (2012),143-150
[10] R. Vaidyanathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20 (1945), 51-61
[2] S. Jafari. and N. Rajesh, On qI open sets in ideal bitopological spaces, University of Bacau, Faculty of Sciences, Scientific
Studies and Research, Series Mathematics and Informatics., Vol. 20, No.2 (2010), 29-38
[3] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc.,13(1963), 71-89
[4] K. Kuratowski, Topology, Vol. I, Academic press, New York., (1966)
[5] S. N. Maheshwari, G. I. Chae and P. C. Jain On quasi open sets, U. I. T. Report., 11 (1980), 291-292.
[6] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb On precontinuous and weak precontinuous mappings, Proc. Math. Phys.
Soc. Egypt., 53 (1982), 47-53
[7] U. D. Tapi, S. S. Thakur and Alok Sonwalkar On quasi precontinuous and quasi preirresolute mappings, Acta Ciencia Indica.,
21(14) (2)(1995), 235-237
[8] U. D. Tapi, S. S. Thakur and Alok Sonwalkar, Quasi preopen sets, Indian Acad. Math., Vol. 17 No.1, (1995), 8-12
[9] M. Kar, and S.S. Thakur, Quasi Pre Local Functions in Ideal Bitopological Spaces Book: International Conference on
Mathematical Modelling and Soft Computing 2012, Vol. 02 (2012),143-150
[10] R. Vaidyanathaswamy, The localization theory in set topology, Proc. Indian Acad. Sci., 20 (1945), 51-61
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Abstract :Various kinds of deterministic models for the spread of infectious disease have been analyzed
mathematically and applied to control the epidemic. A vaccine is a biological preparation that improves
immunity to a particular disease. In this paper, a deterministic model for the dynamics of an infectious disease
in the presence of a preventive vaccine and natural death rate is formulated. The model has various kinds of
parameter. In this paper, we try to present a model for the transmission dynamics of an infectious disease In
order to control the epidemic by changing the value of the parameters.
AMS: 92C60, 92D30, 92B05, 92D25.
Keywords - Basic reproduction number, diseases free equilibrium, Infectious diseases, Stability analysi
Keywords - Basic reproduction number, diseases free equilibrium, Infectious diseases, Stability analysi
[1] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of
London. Series A, Vol. 115 (1927), pp. 700–721.
[2] P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental
models of disease transmission, Mathematical. Bioscience 180 (2002) pp. 29-48.
[3] Herbert W. Hethcote, The Mathematics of Infectious Diseases, Society for Industrial and Applied Mathematics. SIAM Rev. Vol. 42, No. 4 (2000), pp. 599-653.
[4] J. Arino, C.C. McCluskey, P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward
bifurcation, SIAM J. Appl.Math. vol. 64, No. 1 (2003), pp. 260–276.
[5] B. Buonomo, A. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination,
Mathematical Biosciences. Vol. 216 (2008), pp. 9–16.
[6] O. Diekmann, J. A. P. Heesterbeek, and J.A.J. Metz, On the definition and the Computation of the basic reproduction number ratio
0 R in models for infectious diseases in heterogeneous population, journal of Mathematical Biology, Vol. 28 (1990), pp. 365- 382.
[7] Nakul Chitnis, J. M. Cushing And J. M. Hyman, Bifurcation Analysis of a Mathematical Model for Malaria Transmission, Society
for Industrial and Applied Mathematics, Vol. 67, No. 1(2006), pp. 24–45.
London. Series A, Vol. 115 (1927), pp. 700–721.
[2] P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental
models of disease transmission, Mathematical. Bioscience 180 (2002) pp. 29-48.
[3] Herbert W. Hethcote, The Mathematics of Infectious Diseases, Society for Industrial and Applied Mathematics. SIAM Rev. Vol. 42, No. 4 (2000), pp. 599-653.
[4] J. Arino, C.C. McCluskey, P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward
bifurcation, SIAM J. Appl.Math. vol. 64, No. 1 (2003), pp. 260–276.
[5] B. Buonomo, A. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination,
Mathematical Biosciences. Vol. 216 (2008), pp. 9–16.
[6] O. Diekmann, J. A. P. Heesterbeek, and J.A.J. Metz, On the definition and the Computation of the basic reproduction number ratio
0 R in models for infectious diseases in heterogeneous population, journal of Mathematical Biology, Vol. 28 (1990), pp. 365- 382.
[7] Nakul Chitnis, J. M. Cushing And J. M. Hyman, Bifurcation Analysis of a Mathematical Model for Malaria Transmission, Society
for Industrial and Applied Mathematics, Vol. 67, No. 1(2006), pp. 24–45.
- Citation
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Abstract :The aim of this work is to find the Ehrhart polynomial for a certain type of a convex polytope and the
Ehrhart polynomial for the dual of these polytopes together with a comparison between them; two theorems that
related with the number of lattice points and the volume are also given. Different examples are presented in
order to demonstrate our results.
Keywords - Ehrhart polynomial, dual,polytope.
Keywords - Ehrhart polynomial, dual,polytope.
[1] A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, new perspectives in geometric
combinatorics, MSRI publications, 38, (1999), 91-147.
[2] M. Beck, The Reciprocity law for Dedekind sums via the constant Ehrhart coefficient, AMM, 106, (5), (1999), 459-462.
[3] B. Braun, An Ehrhart series formula for reflexive polytope, the electronic journal of combinatorial, Vol.13, No.15, 2006.
[4] K.Fukuda,Frequently asked questions in polyhedral computation, www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html(Access
date: 30 October 2008.
[5] P. Gailiunasy and J. Sharpz, Duality of polyhedral ,International Journal of Mathematical Education in Science and Technology,
Vol. 36, No. 6, 2005, 617–642.
[6] G.Hegedus,The surface of a lattice polytope, arXiv:1002.1908v4 [math.CO] 26 Feb, 2010.
[7] B.Poonen and F.Rodriguez, Lattice polygons and the number 12, American Mathematical Monthly, Vol. 107, No.3, Mar 2000.
[8] A. S. Shatha, On the volume and integral points of a polyhedron in Rn, Ph.D thesis in Mathematics, University of Al-Nahrain, 2005.
[9] R. P. Stanley, Enumerative combinatorics , Wadsworth & Brooks / Cole Advanced Books & software, California, 1986.
[10] Dual polyhedron - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Dual_polytope , 2011.
combinatorics, MSRI publications, 38, (1999), 91-147.
[2] M. Beck, The Reciprocity law for Dedekind sums via the constant Ehrhart coefficient, AMM, 106, (5), (1999), 459-462.
[3] B. Braun, An Ehrhart series formula for reflexive polytope, the electronic journal of combinatorial, Vol.13, No.15, 2006.
[4] K.Fukuda,Frequently asked questions in polyhedral computation, www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html(Access
date: 30 October 2008.
[5] P. Gailiunasy and J. Sharpz, Duality of polyhedral ,International Journal of Mathematical Education in Science and Technology,
Vol. 36, No. 6, 2005, 617–642.
[6] G.Hegedus,The surface of a lattice polytope, arXiv:1002.1908v4 [math.CO] 26 Feb, 2010.
[7] B.Poonen and F.Rodriguez, Lattice polygons and the number 12, American Mathematical Monthly, Vol. 107, No.3, Mar 2000.
[8] A. S. Shatha, On the volume and integral points of a polyhedron in Rn, Ph.D thesis in Mathematics, University of Al-Nahrain, 2005.
[9] R. P. Stanley, Enumerative combinatorics , Wadsworth & Brooks / Cole Advanced Books & software, California, 1986.
[10] Dual polyhedron - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Dual_polytope , 2011.
- Citation
- Abstract
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Abstract :The governing equations for the description of turbulence with reacting and mixing n-chemical
elements described by R. S. Brodkey 1 and E. E. O'Brien 2 are considered. For convenience, we consider a
turbulent reaction mixture with two reacting and mixing chemical elements A and B. The reactions are
irreversible, isothermic, second order and the type of reactions is A+B→ Product. A brief account of derivation
of the equations describing the dynamics of the energy spectrum density functions for the velocity field and the
concentration fields A and B derived on using Lewis-Kraichnan 3 space-time version of the Hopf 4 functional
formalism and multiple-scale-cumulant-expansion method is given. The equations for spectrum functions of the
velocity field and the concentration fields derived on employing multiple-scale-cumulant-expansion method by
Joshi N. E. and Meshram M. C. 5 are considered. The equations for spectrum functions are rewritten by using a
modified-zero-fourth-order-cumulant-expansion (MZFOCE) approximation method which are first written in
dimensionless form and then integrated numerically. The numerical values of the energy of velocity field as well
as that of concentration fields are obtained. These values are used to evaluate the statistical quantities
describing the turbulence with reacting and mixing chemical elements for large Reynolds numbers up to R=106
.The quantities include energy transfer functions of concentration fields , enstropy of the concentration fields,
skewness of concentration fields, dissipation energy of concentration fields and Taylor's micro-scales for
concentration fields. We present these in the form of graphs for the representative values of Reynolds number
R=104 and R=106.The analysis of the numerical values of statistical parameters thus obtained is carried out and
the laws governing these quantities are investigated. Also, we discuss the merits and scope of the present
closure scheme for studying similar types of turbulent flows.
Keywords - Hopf functional formalism, Cumulants, Multiple scales, Turbulence with reacting and mixing chemical elements
Keywords - Hopf functional formalism, Cumulants, Multiple scales, Turbulence with reacting and mixing chemical elements
[1] Brodkey R . S. (1975) Turbulence in Mixing Operations: Theory and Applications to Mixing and Reaction, New York: Academic
Press.
[2] O'Brien E. E. (1971) Turbulent Mixing of Two Rapidly Chemical Species, Phys. Fluids 14, pp. 1326-1330.
[3] Lewis R. M. and Kraichnan R. H. (1962) A Space-Time Functional Formalism for Turbulence, Comm. Pure Appl. Math.
15,pp.397-411
[4] Hopf E. (1952) Statistical Hydrodynamics and Functional Calculus, J. Rat.Mech. Anal., 1, pp. 87-123.
[5] Joshi N. E. and Meshram M. C. (1988) The space-time functional formalism for turbulent multicomponent mixture with second
order reactions, J.Math. Phy.Sci. 22(6) , pp.711-737.
[6] Rosner D. (2000) Transport Processes in Chemically Reacting Flow, Dover, New York.
[7] Peters H. and Bokhorst R. (2000) Theoretical and Numerical Combustion, (R.T. Adverts Inc.).
[8] Poinsot and Veynante (2001) Theoretical and Numerical Combustion, (R.T. Adverts Inc.).
[9] Veynante and Vervisch (2002) Turbulent Combustion Modelling, Progressing Engg. Sci., 28, pp. 193-266.
[10] Meshram M.C. and Junghare J.R., 2004, Investigation of atmospheric turbulence with mixing and reacting elements of the type
Press.
[2] O'Brien E. E. (1971) Turbulent Mixing of Two Rapidly Chemical Species, Phys. Fluids 14, pp. 1326-1330.
[3] Lewis R. M. and Kraichnan R. H. (1962) A Space-Time Functional Formalism for Turbulence, Comm. Pure Appl. Math.
15,pp.397-411
[4] Hopf E. (1952) Statistical Hydrodynamics and Functional Calculus, J. Rat.Mech. Anal., 1, pp. 87-123.
[5] Joshi N. E. and Meshram M. C. (1988) The space-time functional formalism for turbulent multicomponent mixture with second
order reactions, J.Math. Phy.Sci. 22(6) , pp.711-737.
[6] Rosner D. (2000) Transport Processes in Chemically Reacting Flow, Dover, New York.
[7] Peters H. and Bokhorst R. (2000) Theoretical and Numerical Combustion, (R.T. Adverts Inc.).
[8] Poinsot and Veynante (2001) Theoretical and Numerical Combustion, (R.T. Adverts Inc.).
[9] Veynante and Vervisch (2002) Turbulent Combustion Modelling, Progressing Engg. Sci., 28, pp. 193-266.
[10] Meshram M.C. and Junghare J.R., 2004, Investigation of atmospheric turbulence with mixing and reacting elements of the type