ARS 2008
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Item Influence of Gender on Academic Performance: A Comparative Study between Management and Commerce Undergraduates in the University of Kelaniya, Sri Lanka(University of Kelaniya, 2008) Weerakkody, W.A.S.; Ediriweera, A.N.Many studies have been carried out on the factors affecting the students' academic performance in university examinations. Some of these factors have been identified as attendance of lectures, knowledge of English, income of the parents, perceptions of learning, attitudes of students and lecturers towards education, teaching aids and method and environmental factors. In countries such as United Kingdom, Australia, New Zealand and USA, it has been recorded that, gender has a significant relationship with the examination performance. Further, these studies have been done for individual subjects. According to this, it can be identified that there are theoretical explanations in respect of influence of gender on university students' academic performance. But there is a lack of comparative studies. Most of them have examined the difference of relationship between gender and academic performance of several categories of university students in Western countries. Therefore, it is important to conduct research in non-Western context. As such, this research tries to indicate the difference of influence of gender on academic performance of various student groups in the Sri Lankan university system by selecting Management & Commerce students who study in the Diversity of Kelaniya. The objective of the research is to investigate whether there is a significant difference of the performance of Bachelor of Business Management (Human Resource, Accountancy, and Marketing) Special Degree Examinations and performance of the Bachelor of Commerce Special Degree Examinations among the university students with respect to their gender. Variables are neither manipulated nor controlled for the study. Hence, the study was conducted in a non contrived setting. The data for this study was collected at a several point in time. Sample for this study was selected from the students of the above mentioned study programmes. The survey was carried out using 5 years data related to the period, from 2002/2003 academic year to 2006/2007 academic year. The sample was consisted of 1200 students ( 600 females and 600 males). Stratified random sampling was used to select the sample. The unit of analysis was at the individual level. For this study, independent sample t-test was used to compare the academic performance of two ��ender categories. Results indicated that there is no significant difference between Commerce and Management students but significant differences could be seen between the academic performance of the male and female students in this study. Exploratory data analysis indicated that in all the course units of Management degrees and Commerce degree, female students tend to perform better at the university examinations than their male counterparts. Hence, the study found that irrespective of the fact that students are Management or Commerce; female students have higher academic performance than the male students.Item An Assessment of the Influence of Fundamental Factors on Share Prices in Sri Lanka(University of Kelaniya, 2008) Fernando, G.W.J.S.This study analyses certain fundamental factors which are determinants of equity share prices in Sri Lanka. These factors are earning, growth, leverage, risk and company size. The literature review of the study have identified various aspects of equity share valuation. The theoretical and empirical studies have been used to formulate the foundation for the study. To assess above mentioned fundamental factors on share prices, multiple regression analyze was used with a log linear model as a cross section analyses related to the period 1993-2001. The sample for the study was selected from companies registered and listed in the Colombo Stock Exchange. The sample consisted of 40 companies from a group of239 companies in all industries. Result show that dividends appear to be a powerful influence in determining share prices than growth and retained earnings. Business risk and financial risk cannot be assessed because they are redundant variables. Company size is having a considerable influence on share prices and, accordingly large companies enj oy higher values for shares. On the basis of findings the following observation was made about the stock market in Sri Lanka. "The determinants of valuation of the share prices are not very clear. It goes mainly by considering dividend and company size factors which are readily measurable. However, factors like growth and risk cannot be measured with more certainty"Item Structure of primitive Pythagorean triples and the proof of a Fermat's theorem(University of Kelaniya, 2008) Jayasinghe, W.J.M.L.P.; Piyadasa, R.A.D.In a short survey of survey of primitive Pythagorean triples (x,y,z) 0 < x < y < z , we have found that one of x, y, z is divisible by 5 and z is not divisible by 3, there are Pythagorean triples whose corresponding element are equal , but there cannot be two Pythagorean triples such that (x10 y��" z1 ), (x1, zP z 2 ) , where z1 and z 2 hypotenuses of the corresponding Pythagorean triples. This is due to a Fermat's theorem [1] that the area of a Pythagorean triangle cannot be a perfect square of an integer, which can directly be used to prove Fermat's last theorem for n = 4. Therefore the preceding theorem is proved using elementary mathematics, which is the one of the main objectives of this contribution. All results in this contribution are summarized as a theorem. Theorem If (x, y, z) is a primitive Pythagorean triangle, where z is the hypotenuse, then z is never divisible by 3, andJ? = O(mod3) , xyz = O(mod5) ,and there are Pythagorean triangles whose corresponding one side is the same. But there are no two Pythagorean triangles such that (x1,y"z1) , (x"z"z 2 ) ), where z"z 2 are hypotenuses. Proof of the theorem Pythagoras' equation can be written as z2 =y2+x2 ,(x,y)= 1 (1) and if z = O(mod 3) ,then since J? is not divisible by 3, z2= y2 -1+ x2 -1+ 2 .Now, it follows at once from Fermat's little theorem that z cannot be divisible by 3. If xyz is not divisible by 5, squaring (1), one obtains z4 = y4 + x4 + 2x2 y2 and hence z4 -1 = y4 -1+ x 4 -1+ 2(x2 y2 ± 1)+ t, where t =- 1 or 3. Therefore xyz = O(mod5). It is easy to obtain two Pythagorean triples whose corresponding two elements are equal, from the pair-wise disjoint sets which have recently been obtained in Ref.2. For example 3652 =3642 +2 2 3652 =3642 +2 2 3652 =3572 +�� 2 .Now, assume that there exists ' ' two primitive Pythagorean triples of the form a z = bz + cz dz =az +cz (1) (2) It is clear that a is odd and c = O(mod 3) .F rom these two equations, one obtains immediately d2 - b2 = 2c2, d2 + b2 = 2a2, and therefore d4 -b4 = 4c2a2 = w�� (3) It has been proved by Fermat, after obtaining the representation of the primitive Pythagorean triples as x = 2rs,y = r2 -s2 ,z = r2 + s2, where 0 < s < r and r,s are of opposite parity, that (3) has no non trivial integral solution for d,b, w.To prove the same in an easy manner consider the equation d2 + b2 = 2a2 in the form d2 -a2 = a2 -b2 and writing it as ( d -a)( d + a) = (a -b)( a +b) use the technique used in Ref.3 to obtain the parametric solution for d andb If d -a= a -b., then d + b = 2a , from we deduce db= a2 .This never holds since (d,a)= 1 = ( b,a) by (1) and (2).1f (d -a)!!=( a -b) , where (u, v) =1 , V then V ( d + a)-= (a +b) . From these two relations, one derives the simultaneous u equations vd -ub = a( u -v) ud+vb=a(u+v) (4a) (4b) From ( 4a),( 4b ),it is easy to deduce the relations that we need to prove the theorem as (v2 + u2 )d = [2uv + u2 -v2 ]a, (v2 + u2 )b = [2uv -(u2 -v2 )]a, (v2 +u2)(d+b))=4uva, v2 +u2 =2a, assuming that uand v are odd. Hence d -b=(u2 -v2),(d+b)=2uv. Therefore d2 -b2 =2(u2 -v2)uv=2c2 and hence u, v are perfect squares and we can find two integers g,h such that. g4 -h4 = w21 < w�� .Now, proof of the last part of the theorem follows from the method of infinite descends of Fermat. Even if u and v are of opposite parity proof of the theorem can be done in the same way. To complete the proof of a Fermat's theorem that g4 -h4 = w�� is not satisfied by any non-trivial integers, we write (g2 + h2 )(g2 -h2) = w�� , where g,h are of opposite parity, to obtain g2 + h2 = x2, g2 -h2 = y2 and x4 -y4 = 4g2 h2 = z�� , where x and y are odd and eo-prime. But, in the case of the main theorem, we have shown that this is not satisfied by any non-trivial odd x, y and even z0 numbers . This completes the proof of the Fermat's theorem we mentioned above.Item Exact formula for the sum of the squares of spherical Bessel and Neumann function of the same order(University of Kelaniya, 2008) Jayasinghe, W.J.M.L.P.; Piyadasa, R.A.D.The sum of the squares of the spherical Bessel and Neumann function of the same order (SSSBN)is the square of the modulus of the Hankel function when the argument of all function are real, and is very important in theoretical physics. However, there is no exact formula for SSSBN.Corresponding formula, which has been derived by G.N.Watson[l] is an approximate formula[!], [2] valid for Re(z) > 0 ,and it can be eo (2 k -l)!! r(v + k + !) written as J ,; (z) + N; (z) �� 2 L ( 2 ) and the error term RP satisfies JrZ k=O 2k z 2k k! f V- k + !_ 2 IR I cosvJr p! I (R(v) ,p) I 2 2P • h ?p < sm - t P cosR(vJr) (2p)! p-1 where cosh2vt = �� m! (v,m) 2m sinh2m t+ R cosht �� (2m)! P m=O Upper bound of RP in the important case when v = n + !_ , is undefined since 2 r(n+l+ m cos R(vJr) =cos VJZ' = 0 ,where R stands for the real part and m!(v, m)= ) r (n +l-m ) The same formula has been derived [l]by the method called Barne's method but the error tern is very difficult to calculate. In this contribution, we will show that an exact formula exists for SSSBN when the order of the Bessel and the Neumann function is 1 . . 2 () 2 () 2 L n (2k-l) !r (n+ k+ l) n + - ,and It can be wntten as J 1 z + N 1 z = - k 2k ( ) • 2 n+-- n+- JrZ 2 z k' r n - k + 1 Proof of the formula 2 2 k=O ' In order to show that the above formula is exact, one has to establish the identity, cosh(2n+ l)t = I r (n+l+m) 2 2111sinh2111t (1) . cosht m��of(n+ 1-m) 2m! It is an easy task to show that the equation (1) holds for n= 0 and n= 1. Now, assume that the equation ( 1) is true for n �� p. It can be easily shown that cosh(2 p + 3)t = 4 cosh(2 p + 1)t. sinh 2 t + 2 cosh(2 p + 1)- cosh(2 p -I)t (2) and hence the following formula holds. cosh(2 p + 3)t " r(p +I+ m) 2 2111+2 sinh 2111+2 t " r(p +I +m) 2 2111 sinh 2111 t p-I r(p +m) 2 2111 sinh 2111 t _ ::_:_ = :L + 2 :L - :L ----7- ---7 - - - cosh t lll=o r(p + I -m) 2m! lll=o r(p + I -m) 2m! lll=o r(p -m) 2m! =l+L...J +LJ +LJ m=l r(p-m+2) (2m-2). m=l r(p+l-m) 2m! m=l r(p-m+l) (2m-1) +P 2r(2p + 1) )22P sinh2P t r(2p) 22P sinh2p t 22(p+l) sinh2(p+l) t where p = + (p v +r(2p+1) 2p! r(2) 2 -1} 2p! It can be shown that Q = I22m sinh2m t (p +m+ 1). and P = (2p + 1)22P sinh 2P t + 22(p+I) sinh 2(p+t) t m=l (p-m+ 1) !(2m) ! Hence , cosh(2p + 3)t = f r(p +m+ 2)2m sinh2m t COSht v=O r(p + 2-m) 2(m)! Since (1) is now true for n �� p + 1 , by the mathematical induction , the equation (1) is true for all n .By Nicholson's formula[1], Jv2(z)+Nv2(z)= :2 J K0(2zsinht)cosh 2wd t (3) 0 where K0 (z) = �� Je-zcosht dt is the modified Bessel function of the second kind of the -00 zero order. Substituting for cosh(2v.t) from (1) and using we obtain n (2k-1J!r(v+k+J-) J 12(z)+ N /(z)= 2 �� 2 n+-2 n+-2 71Z L..J k 2k ( 1 ) k=O 2 Z k!r V -k + 2 from which the square of the modulus of the Hankel function follows immediately.Item Singularities of the elastic S-matrix element(University of Kelaniya, 2008) Jayasinghe, W.J.M.L.P.; Piyadasa, R.A.D.It is well known that the standard conventional method of integral equations is not able to explain the analyticity of the elastic S-matrix element for the nuclear optical potential including the Coulomb potential. It has been shown[1],[2] that the cutting down of the potential at a large distance is essential to get rid of the redundant poles of the S-matrix element in case of an attractive exponentially decaying potential. This method has been found [3] to be quite general and it does not change the physics of the problem. Using this method , analiticity and the singularities of the S-matrix element is discussed. Singularities of the elastic S-matrix element Partial wave radial wave equation of angular momentum l corresponding to elastic scattering is given by, [ d2 - 2 /(/+ 1)] 2p [ . ] 2 + k - 2 u1(k,r)=-2 V(r)+ Vc(r)+ zW(r) u1(k,r) M r n (1) where V (r) is the real part of nuclear potential, W (r) is the imaginary part of the optical potential ��· (r) is the Coulomb potential, and k is the incident wave number. Energy dependence of the optical potential is usually through laboratory energy E1ah and hence it depend on k2 and therefore k2- 2 �� [V(r) + Vc(r) + i W(r)] is depending on k n through e . It is Well known that ��. (r) is independent Of k Jn Order tO make U 1 ( k, r) an entire function of k , we impose k independent boundary condition at the origin . Now, we can make use of a well known theorem of Poincare to deduce that the wave function is an entire function of k2 and hence it is an entire function of k as well. We cut off the exponential tails of the optical potential at sufficiently large Rm and use the relation -1 --d u1 =.:u ;<--l -(k,-r)--s--'-r (--k,R--m----') u-'-;(+l--- ( k,-r) u, dr u/-l(k ,r)-sr(k,Rm) uj+l(k,r) (2) to define St(k,Rm),where u,<-l(k,r) and u,<+l(k,r) stand for incoming and outgoing Coulomb wave functions respectively which are given by I (±l(k )-+· [r(/+ l+i 17) ]2 [J[2"+iU+n��J w (-2 'u1 ,r - _l . e 1 1k r ) r(Z+I-z17) +i,,/+2 (3) where Ware the Whittaker functions. In the limit Rm �� oo St (k,Rm) ,the nuclear part of the S-matrix element , becomes St { k) and the redundant poles removed[1 ],[2].Now, the nuclear S-matrix element , in terms of the Whittaker functions is given by where w' 1 (2ikr)-��(k,r) W I (2ikr) IIJ, 1+-2 in'" 1+-2 , ,r 2 Rm W. 1 (-2ikr)-��(-k,r)W 1 (-2ikr) -IIJ, I+-2 -ill ' 1+-2 P1(k,r) = u;(k,r) ,and St (k) has an essential singularity at k = 0, which u1(k,r) , (4) is apparent from the Wister's definition of the Gamma function l(z) smce z= l+ 1 ±i lJ .However, this singularity has no any physical meaning and is an outcome of treating 21Jk as well defined quantity for all k including k = 0 in the corresponding r Schrodinger equation .The infinite number of zeros and poles of S- matrix element due to the Gamma functions associated with S - matrix element have to be interpreted 1 carefully. S;'(k)=O at the zeros of ---- f(l+1+i1J ) and then the total wave function reduces to [ . ]�� I J( '7 +i(/+I)ZZ"j uj-l(k,r)=-i f(l+1-��'7) e l 2 2 W (2ikr) f(l+1+zlJ) i11,1+l2 which is also zero. Even though the corresponding energies of these states are negative since the corresponding wave number is given by ? k= z · z,z2 e- 2 n= 0' 1' 2 , ... 11 (n+l+1) they are not physically meaningful bound states as found in[1],[2] long ago. These states are unphysical since poles are redundant poles. This fact is clearly understood by the fact that all these poles are absent in the physically meaningful total S - matrix element. For large 1k1, Sin' (k) �� (- ) { e-21k r S(k), where S(k) = [-ik+��(k)] • +2k [ik-��(-k)]. smce W = e- " for large k. Therefore the S-matrix element has an essential singularity at infinity, which is on the imaginary axis. It is clear that there are no redundant poles in the total S-matrix element is free from redundant poles sinceSJ (k) =SeS t , where Se = f(l + 1 + i7J) f(l+1-i1]) .Item On the Schwarzschild singularity(University of Kelaniya, 2008) Rajapaksha, R.L.R.A.S.; de Silva, N.The Schwarzschild ABSTRACT metric d<2�� (1- 2; }'dt' (I-�� t'-r' (d02+sin2 Od��' ) appears to behave badly near r = 2m, where gtt becomes zero, and grr tends to infinity1• There is a pathology in the line element that is due to a pathology in the space-time geometry itself. The womsome region of Schwarzschild metric, r = 2m , IS called the "event horizon". It is also called the "Schwarzschild singularity"1• There are many coordinate systems that have been found to overcome the Schwarzschild singularity1,2. By using the Schwarzschild metric in Schwarzschild coordinates, m Eddington-Filkelstein coordinates and m Kruslal-Szekeres coordinates, we have obtained some expressions for geodesics to check the behavior of a test particle at r = 2m , and in the two regions, the region outside r = 2m and the region inside r = 2m . W h h h · 11 h d . · · · ak dr e ave s own t at m. a t e coor mate systems It IS consistent to t e - < 0 ds when r > 2m and dr > 0 when r < 2m. The coefficient of dr2 becomes negative ds when r < 2m , making r a time like coordinate in that region. Thus r has to increase in this region. Further I: I becomes greater than c , the speed of light when r = 2mk , where k is a constant that depends on the initial condition, in the case of Schwarzschild coordinates and Eddington-Finkelstein coordinates, and there is jump at 2 . dr r = m, In - ds from -cl to cl , where l is a constant. These results suggest that once the particle crosses the event horizon at r = 2m it tends to remain there as dr > 0 , when r < 2m in all the three coordinate systems. ds A transformation of coordinate does not change this fact and we may suggest that the particle does not cross the event horizon, making it more than a mere coordinate singularity. The fact that I: I becomes greater than c m the neighborhood of r = 2m at least m two coordinate systems also suggest that the particle 1s changed physically around r = 2m . Hence we may say that the singularity at r = 2m is a physical singularity and not merely a coordinate singularity.Item Structure of Fermat triples(University of Kelaniya, 2008) Piyadasa, R.A.D.The structure of Fermat's triples can be immensely useful in finding a simple proof of Fermata's Last Theorem. In this contribution, the structure of Fermat's triples 1s determined using Fermat's little theorem producing a new lower bound for the triples. Fermat's last theorem can be stated as the equation zn =yn +xn (x,y)=1 (1) has no non-trivial integral solutions for x,y,z any prime n > 2 .Due to the famous work of Germain Sophie , if we assume the existence of non trivial integral triples (x,y, z) for any prime n > 2 satisfying (1),there may be two kinds of solutions , namely, one of (x,y,z) is divisible by n and none of (x,y,z) is divisible by n ,and the well known lower bound for positive x, y, z is n , that is, if x is the least, then x > n [ 1]. Let us first consider the triples satisfying xyz :;t: O(modn). Then (z - x) = y��, z -y = x��, x + y = z�� , where xa, Ya, za are the factors of x, y, z respectively. z�� -y�� - x�� = x + y - (z - x) - (z -y) = 2(x + y - z) (2) x+y - z=z�� - za��=za(z��-1 - ��)=za(z��-1 -1+1 - ��) (3) ,where z=za��·Since x+y - z=O(modn) ,which follows from (1) and Fermat's little theorem, and also z��-l -1 = O(modn) .Hence 1 - �� = O(modn)and �� = (nk + 1) , where k is an integer which is non negative since �� :;t: 0 .Therefore , we conclude in a similar manner that z = za(nk + 1) y = Ya(nl + 1) x = xa(nm+ 1) where k,l, m are positive integers and xa :?: 1 , m particular. Also, x + y- z = x - x�� = O(modn) , from which it follows at once that x.:?: n + x�� > n , which first obtained in a different manner by Grunert in 1891[1].In this contribution , it is shown that x very well greater than n2. Proof. 2( X +y - z)= z�� -y; -X��= ( z - za n k) 2 - (y -ya n tr - (X - X a nm) 2 = n 2 L ( 4) due to zn = yn + xn and hence 2(x + y - z) = n2 L. where L is an integer, Hence, x + y - z = O(modn2) . It is easy to check that (5) But. x+ y-z = x-x�� and all numbers za,Ya,xa,n eo-prime to one another, and hence (6) and note also that z a , y a > 1 which guarantees that x is very well greater than n 2 • We deduce that zn-z = O(modn2) yn-y = O(modn2) xn -X = O(modn2) (a) (b) (c) since x + y- z = (z-nkza)-z = zn-z + n2 H, where H is an integer, from which (a) follows, and (b) and (c) follows in a similar manner. Now it is easy to deduce that (x + y)n-zn = O(modn3) (7) In case of (7), one has to use the simple result that if ab *- O(mod n) and a -b = O(mod n11) ,then an -bn = O(mod n11+1) , where n 2:: 3 is a prime. Now assume that xyz = O(modn) , and suppose that y = O(modn) , for example. Then, since y is of the nnflanyn, it follows from the above result thatz-x = nnfl-Ian, where a may takes positive values including a= 1. Now the equation (4)takes the form z�� -nnfl-Ian -x�� = 2(x+ y-z) (8) Now, since x + y- z = O(mod n2) , it follows that z�� -x�� = O(mod n2) , and it is easy to deduce (9) Hence x-5n = O(mod za.nnfla. xa) and from which we deduce that x > za .nnfla. xa , where fJ;;::: 2. The equations (a), (b),(c) can be obtained exactly in the same manner as before. It is easy to understand that above equations hold even if one assumes z = O(modn) .Item Coupling Shallow Water Equation with Navier-Stokes Equations: A viscous shallow water model(University of Kelaniya, 2008) Karunathilake, N.G.A.The general characteristic of shallow water flows is that the vertical characteristic scale D is essentially smaller than and typical horizontal scale L .i.e. £ := D << 1 . L In many classical derivations, in order to obtain the shallow water approximation of the Navier-Stokes's Equations, the molecular viscosity effect is neglected and a posteriori is added into the shallow water model to represent the efficient-viscosity ( a friction term through the Chezzy formula which involes empirical constants) at the bottom topography. However, the validity of this approach has been questioned in some applications as the models lead to different Rankine-Hugoniot curves (see e.g. [1]). Therefore, it can be useful to consider the molecular viscosity effect directly in the derivation of the shallow water model. On the other hand the classical shallow water models are derived under the assumption of slowly varying bottom topographies. Hence, for the description of incompressible shallow water laminar flow in a domain with a free boundary and highly varying bottom topography, the classical Shallow Water Equations are not applicable. The remedy consist of dividing the flow domain into two sub-domains namely, near field (sub domain with the bottom boundary) and far field (sub domain with the free boundary) with a slowly varying artificial interface and employ the Navier-Stokes Equations and Viscous Shallow Water Equations in the near field and far field, respectively. In this work, we derive a two-dimensional Viscous Shallow Water model for incompressible laminar flows with free moving boundaries and slowly varying bottom topographies to employ in the far field. In this approach, the effect of the molecular viscosity is retained and thereby corrections to the velocities and the hydrostatic pressure approximations are established. Coupling modified shallow water model with NSE has been carried out in a separate work. In order to derive the viscous shallow water model the two-dimensional Incompressible Navier-Stokes equations in usual notations au + au2 + auw + ap = �� ( 2v au) + �� (v au +V 8 w) , at ax az ax ax ax az az az aw + auw + 8w2 + ap =-g+�� (vau +j.law) +�� ( 2vau), ---------------------------(1) at ax az az . ax az ax az az a-w+ a-w= 0. ax az are employed in the far field with the suitable boundary conditions. On the free surface, we assume that the fluid particle does not leave the free surface and we neglect the wind effect and the shear stress. On the artificial boundary we set the conditions according with the Navier-Stokes solution at the interface. On the lateral boundaries inflow and outflow conditions are employed. Rescaling the variables with the typical characteristic scales L and D, the dimensionless form of the Navier-Stokes's equations for shallow water flows are obtained. Similarly, assuming that the bottom boundary is regular and the gradient of the free surface remains bounded we obtain the dimensionless boundary conditions. The second order terms with respect to & in the system are neglected and asymptotic analysis is carried out under the assumptions, the flow quantities admit linear asymptotic expansion to the second order with respect to & and the molecular viscosity of the water is very small. Then, rescaling the depth averaged first momentum equation of the resulting system and substituting the zeroth order solution for the velocity and the pressure in it the zeroth order first momentum equation which include the interface conditions is obtained. Again integrating the continuity equation of the dimensionless system from z1 to H(t, x ), a more detailed view of the vertical velocity component is established. Similarly, integrating the vertical momentum equation the dimensionless system from z1to H(t, x ) and replacing boundary conditions, the second order correction to the hydrostatic pressure distribution is derived. Then, dropping o(s2) in the system and switching to the variables with dimensions, the following results are established. Proposition: The formal second order asymp t ot ic ex p ansion of t he Navier-St okes Equat ions for t he shallow wat er laminarfl ow is given by ( z -z I ) ou . I ou 2 u(t, x, z) = u(t,x,z1) + I--- -(x, z1 ,t)(z-z1)---(x, z1 ,t)(z-z1) 2h oz 2h oz h(t,x)+z1 OU w(t,x,z)=w(t,x,z1)- f -d1] OX Z=Zt - ou ou p(t,x,z) = g(h+ z1 -z)-v-(t,x,z)-v-(x,t) ox ox wit h t he viscous shallow wat er equations ah + £(��h)= ( w _ u az 1 ) , at ax ax z=Zt �� (�� h) +£ {��z h) + £( gh2 J = £( 4v h a��J -rl ' at ax �� 8x 2 ax 8x where r, �� [ p : +v: +v: -2v : : +u(u: -w) L,, and z �� z,(x, t ) is t he interface. Concluding remarks In the zeroth order expansion as well as in many classical shallow water models, the horizontal velocity does not change along with the vertical direction. In contrast, our first order correction gives a quadratic expansion to the vertical velocity components retaining more details of the flow. As many classical models we do not neglect the viscosity effect but just assume that it is very small. Also, the zeroth order hydrostatic pressure approximation has been upgraded to the first order giving a parabolic correction to the pressure distribution.Item A metric which represents a sphere of constant uniform density comprising electrically counterpoised dust(University of Kelaniya, 2008) Wimaladharma, N.A.S.N.; de Silva, N.Following the authors who have worked on this problem such Bonnor et.al 1•2 , Wickramasuriya3 and we write the metric which represents a sphere of constant density p = -1-, with suitable units, as ds2 = 47Z" (e(: ))2 c2 dt2 - ( e(r )Y ( dr2 + r2 dQ 2) ds2 = ( 1 B)' c'dT' - ( D + !)' (dR' + R2dQ') D+-R O��r��a A .!!! = e(a) dT (1+ ��) (i) -2 ( ) -2 ( B (e(a) )3 B' a cdt = ) ( B )3 -7 cdT 1+-A => _dt = _-_B--'(,e(-'-a��-- )Y---=----�(ii) dT A'B'(a{l + ��)' (1+ B => dr A ) -=-dR --B(a) _____ (iii) From (i) and (ii), we have e(a1 = - B (B(a )Y 3 (t+ A) A'll'(a{l+ ��) (1+ B ) From (iv), ( ) = !!_ (vi) Ba A __ (v) Using equation (vi) in equation (v), we have B �� -A'(:: } '(a)�� -a2ll'(a ) . Substituting the value of B in equation (iv), B(a )a = ( 1 + ��) A = A+ B =A- a2B'(a) =>A= aB(a)+ a2B'(a) . Then the metric becomes ds2 = 1 c2 dt2 - (e( )Y (dr2 + r2 dQ2) (e(r )Y r dsz = 1 cz dT z - (1- (a2B '(a))J 2 ( dRz + R z dQ z ) (t _(a'��(a))J' R where A=(ae(a )+ a2B'(a ))Item A general relativistic solution for the space time generated by a spherical shell with constant uniform density(University of Kelaniya, 2008) Wimaladharma, N.A.S.N.; de Silva, N.In this paper we present a general relativistic solution for the space time generated by a spherical shell of uniform density. The Einstein's field equations are solved for a distribution of matter in the form of a spherical shell with inner radius a and outer radius b and with uniform constant density p . We first consider the region which contains matter (a < r < b ). As the metric has to be spherically symmetric we take the metric in the form ds2 =ev c2dt2 -eA.dr2 -r2d0.2, where d0.2 = (dB2 +sin2B drjJ2 ), A and v are functions of r as in Adler, Bazin and Schiffer1 where the space time metric for a spherically symmetric distribution of matter in the form of sphere of uniform density has been worked out. Solving the field equations, we o btain eA. = 1 ( r2 1 EJ --+ R2 -r and 2 Here R 2 = �� , where c and K are the velocity of the light and the gravitational 87rKp constant respectively and A , B and E are constants to be determined. Let the metric for the matter free regions be ds2 =ev c2dt2 -eA.dr2 -r2d0.2, where as before from spherical symmetry A and v are functions of r . Solving the field equations, we o btain, e" and e'' in the form e' �� (I : 7) and e" �� n(l + ��), for the regions 0 < r b. where D and G are constants. For the region 0 < r b, the metric should be Lorentzian at in finity. So D = 1. Hence the metric for the exterior matter free region is ds' = ( 1 + ��}'dt'- (I +l��r 2 -r2dn2 • Then we can write the metric for the space-time as ds2 = D c2 dt2 - dr2 -r2dQ2 , whenO < r b. - !! - - (b3 - a3) E- 2 G R - R 2 ' (i ) __ (ii) 157 where r 2(a3 - r3 + rR2 Y ( - 9a 6 J;- 3a3rYz R2 + 2rh R 4 ) f Yz dr = --------,-- Yz.,--------'------------'----- (1 - C + _a_3 -) 2 r% ( a3 - r3 + rR2 ) 2 (-27a9 + 27a6r3 - 2 7a6rR2 + 4a3 R6 - 4r3 R6 + 4rR6) R2 R2 r rR2 tP -J; a3 + r3 (-I + :: ) ] F(f/Jim)= fV - msin2 e) dB tP ( )Yz ff ff and E(f/J I m)= f 1 -m sin 2 e dB , -- < fjJ <- 0 2 2 0 are the Elliptic integrals of first kind and second kind respectively, where fjJ =Arcsin (- ;+r3) (r3 - r2) and Here r1 =The first root of ( - 1 + R2 r2 + a3r3 )r2 =The second root of ( - 1+R2 r2 +a3r3. ). r3 =The third root of ( - 1+R2 r2 +a3r3). Furthermore we know that the potential fjJ of a shell of inner radius a1 and outer radius b1 and constant uniform density in Newtonian gravitation is given by fjJ = 2ffKp(a12 -b12) ,;. 2ffKp 2 4ffKp 3 2 b 2 'f'=-3-r +�� a1 - ffKP 1 fjJ = _ 4ffKp (a13 - b13) Using the fact that g00 = ( 1 + ����). (for example in Adler, Bazin and Schiffer1 )we find that the constants a,, b1 in Newtonian gravitation and D can be written in the form __ (iii) __ (iv) D =(I+ 3(a�;,b/ l} Hence the final form of the metric is O