002084   Amardeep Singh;Rajput A;Arya S (NO, Govt Motilal Vigyan Mahavidyalaya, Bhopal-462 003) : Coincidentally commuting maps in D-metric space for fixed point. Vikram mathl J 1999, 19, 40-7.

11 ref

001173   Vasantha W B;Neelakantan N R;Ramathilgam S (Department of Mathematics and Chemical Engineering, Indian Institute of Technology, Madras) : Maximize the production of cement industries by the maximum satisfaction of employees using fuzzy matrix. Ultra Scient phys Sci 2003, 15(1M), 45-56.

Maximization of the production of cement industries with maximum satisfaction to employees using fuzzy matrix has been attempted. Using experts opinion this problem into a relational map. This relational map relates both ways the attributes of the employee and the employer, which is described by the directed graph. This directed graph has been converted given a relational matrix representation. As the next stage of the problem an average matrix for this relational matrix has been defined. Average matrix is converted into a fuzzy matrix by the method of membership functions. Finally using various α - cut values. The graphical representation has been given to the fuzzy matrices. Combined effects of the α - cuts and have been obtained.

1 illus, 7 ref

001172   Vasantha W B;Mary John (Mathematics Dep, Indian Institute of Technology, Madras, Chennai-600 036) : Fuzzy analysis to study the pollution and the disease caused by hazardous waste from textile industries. Ultra Scient phys Sci 2002, 14(2), 248-51.

Diseases caused by hazardous waste from the textile industries have been studied. A person staying in the neighbourhood of a textile industry may suffer from one or more diseases in different degrees. So Fuzzy Analysis has been used in general and Fuzzy Associative Memories (FAM) in particular to study the problem. A brief account about the hazardous waste of a textile industry has been given. The diseases caused by the hazardous waste which pollute the atmosphere using FAM have been analysed. The problem of the hazardous waste which pollute the soil and waste using FAM has been studied.

2 tables, 3 ref

001171   Tunc E (Gaziosmanpasa University, Faculty of Arts and Science, Department of Mathematics, 60100, Tokat., Turkey) : Nonoscillation of solutions of a class of third order functional differential equations. Pure appl Math Sci 2003, 58(1-2), 1-6.

Sufficient conditions have been obtained for nonoscillation of all solutions of third order functional differential equations

14 ref

001170   Tijjani A A (Mathematics Dep, Ahmadu Bello University, Zaria Federal Republic of Nigeria, ) : Averaging operators on non commutative L<. Ultra Scient phys Sci 2003, 15(1M), 91-6.

Theory of averaging operators on Haagerup L^p-spaces is considered and relations between averaging operators and conditional expectations are studied.

24 ref

001169   Tijjani A A (Mathematics Dep, University of Maiduguri, Maiduguri Federal Republic of Nigeria) : Convergence of supermartingales on Haagerup's L<. Ultra Scient phys Sci 2002, 14(2), 266-75.

Norm convergence of supermartingale on Haagerup's L^p-spaces has been studied. The results extend the case^3,4 where A is a σ-finite von Neumann algebra with an expected increasing full right continuous net (B_α)_α∈ R⁺ of von Neumann sub-algebras and the associated τ-measurable processes.

36 ref

001168   Thengane K D;Zade V T;Karade T M (R S BIdkar Arts, Commerce & Science College, , Hinganghat-442 301) : Plane gravitational waves in four dimensional space-time (I). Math Educ 2002, 36(3), 159-61.

The plane 1/√3 (x+y+z)-t] type wave solutions of the field equations Rij=o are obtained in the form L^dash₂/γ^dash₄+L2Y4+L1/4=0 from QYij+P<100>ij=0.

3 ref

001167   Tak S S;Nain S (Mathematics and Statistics Dep, Jai Narain Vyas University, Jodhpur) : MHD free convection mass transfer flow through a porous medium of fluctuating permeability bounded by vertical plate in slip flow regime. Vikram mathl J 2000, 20, 42-54.

A theoretical analysis of unsteady two dimensional free convection and mass transfer flow in slip flow regime of a porous medium of variable permeability bounded by vertical infinite plate with constant suction velocity and constant heat and mass flux in the presence of uniform magnetic field and heat sources/sink is presented. The permeability of porous medium fluctuates with time about a constant mean, which causes unsteadiness in the flow. Using perturbation method, the expressions for velocity, temperature, concentration and skin-friction are obtained. The effect of permeability parameter, Grashof number for heat transfer, Grashof number for mass transfer, Schmidt number and heat source/sink parameter on velocity, temperature and concentration profiles are discussed numerically and shown graphically. The effects of the mentioned parameters on the skin friction coefficient are also discussed. The effects of the skin friction coefficient are also presented.

7 illus, 18 ref

001166   Tailor M R;Mehta M N (Applied Math Dep, S V R College of ENgg and Technology, Surat-395 007) : Solution of a unidimensional vertical groundwater recharge through porous media. Math Educ 2002, 36(3), 137-40.

Analytical solution for a one dimensional vertical groundwater recharge has been obtained by using separable variable technique. An alternative closed form of solution in terms of exponential function is derived. The average diffusivity co-efficient over the whole range of moisture content is regarded as constant a parabolic variation of permeability with moisture content of assumed.

8 ref

001165   Syam Babu M;Jayaraj B (Dep of Mathematics, S V U College of Arts & Science, Tirupati) : Axisymmetric free convection flow of viscous incompressible fluid on a horizontal infinite plate. Ultra Scient phys Sci 2003, 15(1M), 117-22.

Axisymmetric free convection flow of viscous incompressible fluid on a horizontal infinite plate subjected to a mixed thermal boundary conditions under the influence of a uniform transverse magnetic field is considered. Similarity analysis is used and the non linear ordinary differential equations are solved by numerical integration using Range-Kutta Gill method with a systematic guessing of conditions of the problem. The effect of magnetic field on velocity, tempearture and pressure for different flow parameters is studied.

3 illus, 3 tables, 16 ref

001164   Sunder Lal;Goyal A K (Math Dep, Institute of Basic Science, Khandari, Agra-282 002) : Some common fixed point theorems in 2-uniform spaces. Math Educ 2002, 36(2), 63-70.

Some results on common fixed points for a pair of operators defined on a sequentially complete Hausdroff 2-uniform space have been obtained.

6 ref

001163   Somasundaram S;Ponraj R (Department of Mathematics, Manonmaniam Sundaranar University, Tirunelvii-627 012) : Pure and applied mathematika sciences. Pure appl Math Sci 2003, 58(1-2), 29-35.

It has been proved that P_m X P_n, P_m XC_n are mean graphs and that the friendship graph C^t₃ is not a mean graph for t<62> 2. Also the total number of possible mean labelings for a given graph have been found.

4 illus, 5 ref

001162   Sinha S K (P G Dep Math, P G Centre, Chaibasa-833 202) : Orthogonality of polynomials like Gegenbauer polynomials. Math Educ 2002, 36(3), 132-6.

Orthogonality of seven polynomials life Gegnbauer polynomials are discussed by the help of differential equations.

2 ref

001161   Sinha A K (Chemistry Dep, L S College, Muzaffarpur-842 001) : An example of cantor's set. Math Educ 2002, 36(3), 162-3.

4 ref

001160   Singh M;Sharma R K;Jain A (Mathematics and Computer Science Dep, Bundelkhand University, Jhansi) : Compatible maps of type (A) and common fixed points in Menger spaces. Vikram mathl J 2000, 20, 68-78.

Some common fixed point theorems for compatible maps of type (A) in Menger spaces have been proved.

20 ref

001159   Singh J R;Shahnaj S;Mishra S (School of Studies in Statistics, Vikram University, Ujjain-456 010) : Procedure for detecting level shifts of first order autocorrelated processes with known coefficient of variation. Vikram mathl J 2000, 20, 85-90.

Recent advances in manufacturing automation make the collection of data for gauging process condition fast and economical. Statistical process control procedures cannot be directly implemented due to the inherent autocorrelation. The effect of autocorrelation with known coefficient of variation on the operating characteristics (OC) function and type-1 error of the control chart for mean has been studied.

3 tables, 2 ref

001158   Singh J R;Sayyed M;Soni D (School of Studies in Statistics, Vikarm University, Ujjain) : Commulative-sum control chart for proportion under inspection error. Ultra Scient phys Sci 2002, 14(2), 252-61.

Comulative-Sum Control Chart (CSCC) under inspection error has been considered. CSCC with special emphasis on the practical problems of inspection error has been described. The angles of the mask and lead distance of CSCC for proportion have been tabulated under inspection error.

4 tables, 18 ref

001157   Singh J P;Roy S R (Mathematical Sciences Dep, A.P.S. University, Rewa-486 003) : Bianchi type - VI_h cosmological models with viscous fluid. Vikram mathl J 2000, 20, 14-21.

The possibility of Bianchi type VI_h cosmological models representing viscous fluid distribution have been discussed. The solutions obtained represent expansion scalar θ bearing a constant ratio to the anisotropy in the direction of a space-like unit vector.

3 ref

001156   Singh H P;Tailor R (School of Studies in Statistics, Vikram University, Ujjain-456 010) : Predictive estimation of finite population mean in two phase sampling with known coefficient of variation of second auxiliary variable. Vikram mathl J 2000, 20, 5-13.

Using predictive approach some ratio-type estimators of population mean of the study variable y have been formulated in the presence of two auxiliary variables under a two-phase sampling procedure when the coefficient of variation C₂ of the second auxiliary variable z is known. Conditions are obtained in which newly constructed estimators are better than the usual two-phase sampling ratio-estimator and others.

8 ref

001155   Singh A K;Thaplital D R (Department of Mathematics, H N B Garhwal University, Campus Badshahi Thaul-249199) : Some theorems on special, union and hyper-asymptotic curves of a tachibana hypersurface. Pure appl Math Sci 2003, 58(1-2), 19-27.

The union curve of a Riemannian hypersurface has been defined and studied by Springer; Mishra has investigated the properties of these curves in a subspace of a Riemannian space. Union, special and hyper-asymptotic curves of a Tachibana space have been studied and several theorems have been investigated.

4 ref

001154   Shyam Lal;Nigam H K (Mathematics Dep, Harcourt Butler Technological Institute, Nawabganj, Kanpur) : On degree of approximation of a lipschitz function by (N,p_n) (E, 1) means of its Fourier series. Ultra Scient phys Sci 2002, 14(1), 53-8.

A new theorem on the degree of approximation of a function belonging to the Lipschitz class by (N,p_n) (E, 1) product means of its Fourier series has been established.

9 ref

001153   Sharma P R;Mishra U (Mathematics Dep, University of Rajasthan, Jaipur-302 004) : Steady heat transfer through laminar flow between parallel plates with different permeability. Ultra Scient phys Sci 2002, 14(1), 79-82.

Steady heat transfer through laminar flow of an incompressible viscous fluid between parallel plates of different permeability is investigated. The temperature distribution is derived by using the expressions of velocity derived by Verma and Bhatt, discussed numerically and shown through graph. The expressions of rate of heat transfer at the plates are derived, discussed numerically and their values are presented through table.

1 illus, 1 table, 6 ref

001152   Shah D V (P T S College of Science, , Athwalines, Surat-395 001) : Distribution of pellian triplets. Math Educ 2002, 36(2), 71-82.

Triplet (m, D, V) is defined to be a `Pellian Triplet' if m+DV² is a square. The problem is related with that of Pell's Equation. First the sequence of Pellian Triplets for fixed integera m and square-free D is considered. It is proved that the terms of this sequence increases by a fixed ratio (depending on m and D). Also the total number of members of the sequence of V's not exceeding any given real number Z. Next, in the above sequence of Pellian Triplets, D is allowed to be any positive integer. The total number of Pellian Triplets in the given range of values for m, D and V under certain necessary conditions has been found.

2 ref

001151   Selvaraj C (Math Dep, Loganatha Narayanasamy Govt College, Ponneri-601 204) : On generalised alpha-quasi-uniform convexity-I. Math Educ 2002, 36(3), 149-53.

The normalised, analytic and univalent function f (z) is said to belong to the class QUCV*(α) if Re[{f¹(z)/g1(z)}^{(1-d)(D>{(zf'(z)}α]>0,] [z belong to the unit disc E, α≥0 and g(z) is uniformly convex and powers appearing are meant as principal values. This class extends the class of quasi-uniformly convex functions.}

7 ref

001150   Saka B (Osmangazi University, Mathematics Department, 26480, Eskisehir, Turkiye) : Finite difference methods for numerical solutions of the RLW equation. Pure appl Math Sci 2003, 58(1-2), 43-52.

The numerical solution of the splitted RLW equation is obtained by usign the classical finite difference method. Results of solitary wave solution of the RLW equation are compared with that of the direct application of the finite difference method to the RLW equation

1 illus, 5 tables, 8 ref

001149   Sah K M;Mandal B L (NO, , At.+P.O.-Dhamoun-844 506) : Stability of a gyroscope on gimbal in the newtonian force field. Math Educ 2002, 36(3), 154-8.

The motion and stability of a Gyroscope on gimbals in the central Newtonian force field has been discussed. It has been assumed that the direction of the axis of the outer gimbal ring coincides with the direction of the line from the alteraction centre to the point of intersection of the exes of the gimbal. The equation of motion is obtained on the line of Lagrange's equations of second kind. The condition of stability of the stationary solutions of the motion on the line of cheataev has been investigated by using second method of Lyapunos V-function.

5 ref

001148   Sah C P;Ramawater Prasad (Math Dep, L N M U, Darbhanga-846 004) : Implications of gaussian measures on the L_p-spaces. Math Educ 2002, 36(2), 112-5.

Establishes the equivence and singularity of Gaussian measures on a real separable Banach space. L. Gross and I.E. Segal studied the characteristic features considering the minimal sigma algebra containing Borels sets. The equivalence of two Gaussian measures on a Banach space has been established This has been applied in particular to the space l_p (1≤p<60>∞)spaces.

4 ref

001147   Ray A K (Math Dep, Jadavpur University, Calcutta-700 032) : On what may be called the dwarf curves in a E_n. Math Educ 2002, 36(2), 87-91.

4 ref

001146   Rathore M S;Deepak Singh;Naval Singh (Mathematics Dep, Govt. P.G. College, Sehore, M.P.) : Common fixed point theorems for compatible maps in D-metric space. Vikram mathl J 2000, 20, 31-41.

Some common fixed point theorems for compatible maps in D-metric space have been proved.

3 ref

001145   Rathore G P S (Soil & Water Engineering Dep, N.M. College of Agriculture, G.A.U., Navsari-396 450) : Almost precontinuous mappings. Vikram mathl J 2000, 20, 1-4.

A new class of mappings which contains the class of precontinuous mappings has been introduced and some of its properties have been obtained.

5 ref

001144   Ram Naresh;Omar S (Mathematics Dep, Harcour Butler Technological Institute, Kanpur-208 002) : Analysis of a nonlinear AIDS epidemic model with standard incidence. Ultra Scient phys Sci 2003, 15(1M), 63-70.

A simple nonlinear mathematical model is proposed and analyzed to study the spread of HIV/AIDS in a population with varying size. The population, unrde consideration, is divided into four subclasses. The dynamics of these classes is assumed to be governed by ordinary differential equations with immigration, nonlinear interaction and natural death terms. The interaction between susceptibles and infectives is taken to be of standard mass action type. The model has been studied qualitatively using stability theory of differential equations and some inference have been drawn regarding the spread of the disease by establishing local and global stability results. It is shown that the positive non-trivial equilibrium is always locally stable and it many becomes endemic due to constant migration of the population into the community.

20 ref

001143   Rajput A;Saluja A S;Sharma G P (Mathematics Dep, Sadhu Vaswani College, Sant Hirdaram Nagar, Bhopal-462 030) : Fixed point theorem for expansion mappings in Quasi - Gauge function space. Ultra Scient phys Sci 2002, 14(1), 107-10.

A fixed point theorem for Housdorff sequentially complete Quasi-Gauge function space generated by family of pseudo metric has been proved.

4 ref

001142   Raj Reddy P (Chaitanya Bharathi Institute of Technology, , Gandipet, Hyderabad-500 075) : Kinetic and kinematic properties of plane magneto fluid flows. Math Educ 2002, 36(2), 100-7.

Defining an orthogonal net in a plane of flow, related to the Magnetic lines and their orthogonal trajectories `C' in Riemanninan Space, various Kinetic and Kinematic properties of Magneto gas Dynamic Studies.

14 ref

001141   Raghuraj Singh;Yogesh Singh (Computer Science & Engineering Dep, Harcourt Butler Technological Institute, Kanpur-208 002) : On the degree of approximation of a function belonging to lip α class by K<. Ultra Scient phys Sci 2002, 14(1), 83-8.

A theorem on the degree of approximation of function belonging to Lipschitz class by K^λ - summability of it's Fourier series has been proved.

15 ref

001140   Pramood Kumar;Choudhary K K;Prasad R (Math Dep, R N College, Pandaul, Madhubani) : On non-Archimedean semi-normed linear spaces. Math Educ 2002, 36(3), 178-81.

Examines the properties of semi-normed linear spaces over fields with valuation in related to given semi-norm whose valuations are non-Archimedean. A.F. Monna introduced stronger type of triangle inequality in which each triangle is isosceies. It was known as non-Archimedean structure. The basis of characterisation of non-Archimedean valueation of a field or non-Archimedean linear space which extends the domain of its applications to semi-norms in the context of linear space has been introduced.

5 ref

001139   Pathak G;Tak S S (Mathematics and Statistics Dep, JNV University, Jodhpur-342 001) : Heat and momentum transfer in free convection flow due to hot vertical accelerated plate in presence of transverse magnetic field. Ultra Scient phys Sci 2002, 14(2), 240-7.

Heat and momentum transfer in unsteady free convection flow along an infinite hot vertical accelerated plate, moving with velocity proportional to some power to time, in infinite mass of fluid at rest in presence of uniform transverse magnetic field fixed to the fluid, has been investigated, taking different cases of uniform and non-uniform acceleration of the plate. Similar solution of energy boundary layer equation, for a particular form of suction velocity v_w(t), is obtained. For solution of momentum boundary layer equation, a series expansion of velocity function in powers of product of magnetic field parameter and time (mt) is assumed. The analytical solutions of resulting ordinary differential equations have been obtained in terms of complementary error functions and repeated integrals of complementary error function.

4 illus, 1 table, 18 ref

001138   Pant R P;Jha K (Mathematics Dep, Kumaon University, D S B Campus, Nainital-263 002) : A generalization of meir-Keeler type fixed point theorem for four mappings. Ultra Scient phys Sci 2003, 15(1M), 97-102.

Introduces a common fixed point principle which includes all the known contrative definition as particular cases and employs a Lipschitz type analogue of known contractive definitions.

19 ref

001137   Nayal H S (Department of Mathematics, Govt P G College, Ranikhet-263 645) : On a distributional complex inversion theorem for Laplace Hardy transform. Pure appl Math Sci 2003, 58(1-2), 57-62.

6 ref

001136   Namdeo R K;Jain S (NO, Dr H.S. Gour University, Sagar) : Common fixed point theorem in Banach space. Ultra Scient phys Sci 2002, 14(1), 139-41.

A common fixed point theorem for three selfmaps in a Banach space which extends the results of Abboui and Rhoades have been obtained.

^ssc3 ref

001135   Namdeo R K;Chourasia V K (Mathematics Dep, Dr. H.S. Gour Vishwavidyalaya, Sagar-470 003) : Related fixed point theorem for set valued mappings on three metric spaces. Ultra Scient phys Sci 2002, 14(2), 186-91.

3 ref

001134   Namdeo R K;Chourasia V K (Math & Statistics Dep, Dr Hari Singh Gour Vishwavidyalaya, Sagar-470 003) : Some fixed point theorems for involutory mappings in complete metrizable linear topological spaces. Math Educ 2002, 36(3), 123-7.

1 ref

001133   Muddebihal M H;Usha P;Milind S C (Math Dep, Central College Campus, Bangalore University, Bangalore-560 001) : Image neighbourhood graph of a graph. Math Educ 2002, 36(2), 92-9.

Characterization of graphs whose image neighbourhood graphs are planar, outerplanar, minimally nonouterplanar and crossing numbers have been characterized. In addition the graphs whose image neighbourhood graphs are planar, outerplanar outer planar graph interms of forbidden subgraphs.

6 illus, 2 ref

001132   Mandal R (M L Academy, , Laheriasarai-846 001, Darbhanga) : On algebraic structures of projective space. Math Educ 2002, 36(2), 108-11.

Exanubes the nature of vector bundles on algebraic surface. A.V. Gerameta derived, some results on projective modules and showed that vector bundles over sphere are algebraic. The comparative structures of algebraic and topological behaviour of groups over projective space have been discussed.

5 ref

001131   Mamat M B;Lola M S B;Ahmad S B (Mathematics Dep, Faculty of Science and Technology University College of Science and Te, 20310 Kuala Terengganu, Terengganu, Malaysia) : A correlation coefficient based on M-estimator. Ultra Scient phys Sci 2003, 15(1M), 57-62.

2 tables, 6 ref

001130   Kar B;Basu D K (Patipukur Pallisree Vidyalaya, , 3 No, Colony, S K Deb Road, Calcutta-700 048) : An extension of a bilateral generating functions for the generalised legendre polynomials by lie group theoretic method. Math Educ 2002, 36(3), 128-31.

A new general class of generating function involving a generalised Legandre and Laguerre Polynomials are obtained. Known generating functions are obtained as a particulare case.

3 ref

001129   Kannan S R (Mathematics Dep, Indian Institute of Technology, Madras) : Fuzzy theory method in the problem of unknown flow-rates in chemical plant. Ultra Scient phys Sci 2003, 15(1M), 79-90.

Experimental study of chemical plants is time consuming expensive and need intensive labour, researchers and engineers prefer only theoretical approach, which is inexpensive and effective. Estimation of unknown flow-rates in splitter mixer unit of a chemical plant plays a vital role in describing the state of the plant. Research approached this problem theoretically using energy and material balance method and has given solution for some unknown flow rates of splitter-mixer unit. But did not give solution to all unknown flow-rates and in their method, error occurred between the measured and the predicted value. Fuzzy relational neural network method has been used in the estimtion of unknown flow-rates of chemical plant. The difference between the measured and predicted values is made very close to zero. Estimation of unknown flow-rates is carried out in two stages. In the first stage, a fuzzy relational equation representation to the data given. If solutions cannot be obtained by this method then in the second stage a fuzzy neural network method is used to estimate the unknown flow-rates and this method guarantees a solution to all unknown flow-rates. It exits between measured and predicted value then by varying the membership that is weightages the process repeated a finite number of times until the difference between the measured and predicted value is made very close to zero. A generalized result for a chemical plant with n-flow-rates has been described.

3 illus, 4 ref

001128   Kandasamy W B V (NO, Indian Institute of Technology, Madras-600 036) : On semi-idempotent matrices. Ultra Scient phys Sci 2002, 14(2), 294-6.

A new notion in matrices called semi-idempotents has been studied. These matrices have been defined and some results about them has been obtained.

^ssc1 ref

001127   Kandasamy W B V (Mathematics Dep, Indian Institute of Technology, Madras, Chennai-600 036) : BI - loops. Ultra Scient phys Sci 2002, 14(1), 127-30.

A new algebraic structure called bi-loops. The bi-loop which are introduced here are non-associative structures which may find itself well placed in industries. Algebraic aspect of bi-loops has been studied.

5 ref

001126   Jha J;Verma P P;Mishra B K (Math Dep, Markham College of Commerce, Hazaribag) : Temperature distribution of pseudoplastic fluid past in infinite porous flat plate. Math Educ 2002, 36(3), 170-2.

Magnetohydrodynamic flow of paeudoplastic fluid (n<60>1) past an infinite porous flat plate whose temperature and transpiration velocity at the plate fluctuates harmonically with time about a constant mean has been investigated. Expression for mean temperature is obtained and numerically computed for various values of magnetic parameter and flow-index number. It is observed that mean temperature decreases as strength of magnetic field increases.

4 tables, 7 ref

001125   Jagdish Singh;Deshwal M (Mathematics Dep, Maharshi Dayanand University, Rohtak-124 001) : Scattering of Rayleigh wave by a rigid plane strip at the bottom of a shallow ocean. Ultra Scient phys Sci 2003, 15(1M), 1-14.

The problem of Rayleigh wave scattering due to a rigid plane strip at the bottom of a shallow ocean has been discussed. A thin rigid plane strip is fixed at the bottom (z=0, x≤0) of an oceanic layer (-H≤z≤O) overlying a solid half-space (z≥O). Fourier transformation and Wiener-Hopf technique are used to solve the problem. The results are derived in terms of integrals whose evaluation along suitable contours in the complex plane gives the reflected, transmitted and the scattered waves. The scattered wave behaves as a decaying cylindrical wave at distant points. The numerical calculations for the scattered waves show that their amplitude decreases rapidly for a small increase in the value of the wave number.

2 illus, 11 ref