000092   Soner N D;Chaluvaraju B (Mathematics Dep, University of Mysore, Manasaganogothri, Mysore-750 006) : Connected double domination number of a graph. Acta Cienc Indica-Math 2003, 29(2), 237-42.

Let G = (V, E) be a connected graph. A double dominating set D is said to be a connected double dominating set (cdd-set), if the subgraph of G induced by D is connected. The connected double domination number of G is the size of its smallest cdd-set, and is denoted by Y_dc (G).

1 illus, 11 ref

000090   Singh S P;Sahu D P;Patel R N (Mathematics Dep, Guru Ghasidas University, Bilaspur-495 009) : Extension of Naimpally and Singh's theorem under compatible maps of type (A). Acta Cienc Indica-Math 2003, 29(2), 359-61.

A common fixed point theorem is established for a four self mappings in normal space which extends a result of Naimpally and Singh's with the use of the Mann iterative process.

4 ref

000089   Singh G N (Applied Mathematics Dep, Indian School of Mines, Dhanbad-826 004) : Alternative approach in two phase sampling. Acta Cienc Indica-Math 2003, 29(1), 151-4.

The problem of estimation of the population mean of study variable y using an alternative approach in two-phase sampling has been considered. The conditions for preference of proposed estimators have been derived and compared theoretically with several well-known estimators in two-phase sampling.

5 ref

000088   Singh A P;R M L (Mathematics Dep, Avadh University, Faizabad, U.P.) : k-variable extension of Bernoulli and Euler polynomials of general order. Acta Cienc Indica-Math 2003, 29(2), 353-7.

In the year 1978, Hussain and Singh generalised the Bernoulli, Euler and Eulerian Polynomials due to Karande-Thakare and derived several results. The study of these polynomials has contributed much to the knowledge of theory of numbers. These polynomials have been extended in k-variables and various interesting results such as recurrence relations, complementary arguments theorem, series form and multiplication theorem have been established. The approach also leads to the extension of the generalised Bernuolli and Euler polynomials defined by Milne-Thomson, the Unified Polynomials R_n^(α)(x; a, r, b) studied by Probhaker-Reva and B_n^(α)(x; y) due to crombez.

7 ref

000087   Shyam Lal;Yadav R S (Dep of Math, Faculty of Science, University of Allahabad, Allahabad-211 002) : Almost (N,p,q) summability of conjugte series of a fourier series. Bull Calcutta Mathl Soc 2003, 95(2), 137-46.

Generalizing an earlier result of Singh (1963) a new theorem on almost generalized Norlund summability of conjugate series of Fourier series has been established under a very general condition.

9 ref

000083   Sharma S;Narayan A (Dep of Math, Bhilai Inst of Technology, Durg-491 001) : Some non-linear effects of perturbing forces of general nature and the Solar radiation pressure on the motion and stability of an Inter-connected satellite system in elliptical orbit. Bull Calcutta Mathl Soc 2003, 95(3), 245-58.

The combined effects of the solar radiation pressure and external periodic forces of general nature on the motion and stability of two satellites connected by a light, flexible and inextensible string in the central gravitational field of the earth have been studied. The motion of each of them relative to their center of mass has been studied, while the center of mass of the system orbiting around the earth. The presence of these perturbing forces enable the application of asymptotic methods of the theory of non-linear oscillations due to Bogoliubov and Mitropolsky to the equations of motion of the system in non-resonance case. It is observed that the amplitude of the oscillation varies even in the first approximation.

2 illus, 14 ref

000082   Sharma S;Mishra S (Mathematics Dep, Govt. Kalidas Girls College, Ujjain (M.P.)) : Approximation of a function by its Fourier series. Acta Cienc Indica-Math 2003, 29(1), 1-4.

A theorem on approximation by using generalized harmonic means has been proved.

4 ref

000081   Sharma S;Bagwan A (Mathematics Dep, Central Water and Power Research Station, Khadakwasla, Pune-411 024) : Common fixed point throrems on normed space. Acta Cienc Indica-Math 2003, 29(1), 19-24.

A common fixed point theorem on normed space has been proved.

11 ref

000080   Sharma S (Mathematics Dep, Guru Nanak Dev University, Amritsar, Punjab) : Some remarks on polynomial 2-cocyciles. Acta Cienc Indica-Math 2003, 29(1), 201-4.

Finitely generated group G are characterized for which P_nH² (G, Z_p) = 0 for all n ≥ 0.

6 ref

000079   Sharma S (Mathematics Dep, Guru Nanak Dev University, Amritsar, Punjab) : Note on circle groups. Acta Cienc Indica-Math 2003, 29(1), 43-4.

Proves that groups of nilpotency class 2 and of exponent pⁿ, p ≠ 2 are circle groups.

10 ref

000078   Sharma A K;Sharma A;Singh S R (Mathematics Dep, Raj Rishi P.G. College, Alwar-301 001) : Order level inventory model for deteriorating items with an exponentially increasing demand with time. Acta Cienc Indica-Math 2003, 29(1), 213-5.

An order level inventory model without backlogging the demand is taken exponentially increasing with time and production rate depends upon demand function. The deterioration is taken constant.

8 ref

000077   Shandil R G;Jagjit Singh (Dep of Math, H.P. Univ, Shimla-171 005) : On Howard's conjecture in heterogeneous shear flow problem. Proc Indian Acad Sci-Mathl Sci 2003, 113(4), 451-6.

Howard's conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy force gβ <60><60> 1, where <60> is the basic heterogeneity distribution function.

5 ref

000076   Sengupta J;Biswas B (Dep of Math, St Xavier's College, 30, Park Street, Kolkata-700 016, West Bengal) : Quarter-symmetric non-metric connection on a Sasakian manifold. Bull Calcutta Mathl Soc 2003, 95(2), 169-76.

A quarter-symmetric non-metric connection on a Sasakian manifold has been defined and studied and also its existence has been proved. The curvature tensor and the Ricci tensor of the quarter-symmetric non-metric connection is also found. It is also proved that if the Ricci tensor of the quarter-symmetric non-metric connection vanishes, then the manifold becomes an η-Einstein manifold.

13 ref

000075   Saxena R K;Jeta Ram;Chauhan A R (Mathematics and Statistics Dep, Faculty of Science, J.N.V. University, Jodhpur-342 005) : Unified form of generalized gamma-type distribution. Acta Cienc Indica-Math 2003, 29(2), 299-304.

A new probability distribution associated with a generalized hypergeometic function _pF_q (.) which provide generalization of the distribution of gamma-type studied by Agarwal and Kalla, Kalla et al., and Saxena et al. has been studied. The r^th order moment at the origin and moment generating functions for this distribution are obtained. The distribution of a linear combination of independent random variables each distributed according to the proposed distribution is also derived. The probability model has been discussed.

22 ref

000074   Sanyal D C;Dasgupta S (Mathematics Dep, University of Kalyani, Kalyani-741 235) : Radiation effect on combined free and forced MHD vertical channel flow. Acta Cienc Indica-Math 2003, 29(2), 363-7.

The radiation effect on the combined free and forced convection flow of an electrically conducting fluid through a non conducting vertical channel having zero heat flux on the wall and permeated by a uniform transverse magnetic field is considered. Assuming the case of optically thin limit, the analytical solutions for temperature, velocity and the induced magnetic field have been obtained and the results are studied numerically in graphical form.

3 illus, 7 ref

000073   Saha P K (Dep of Math, North Bengal University, Dist. Darjeeling, W.B.) : Spare set topology in Romanovski spaces. Bull Calcutta Mathl Soc 2003, 95(3), 229-34.

9 ref

000072   Saha P K (Dep of Math, University of North Bengal, Dist. Darjeeling, West Bengal-734 430) : On some properties of Infinite series. Bull Calcutta Mathl Soc 2003, 95(2), 165-8.

Some more results have been added to the investigation initiated in Lahiri and Ghosh (1995) on a space X in relation to certain infinite series of real terms.

5 ref

000071   Saha A;Das A G (Dep of Math, University of Kalyani, Kalyani Nadia, West Bengal-741 235) : On mixed trilateral generating relation of modified laguerre polynomials. Bull Calcutta Mathl Soc 2003, 95(3), 219-22.

Group theoretic method has been adopted to obtain mixed trilateral generating relations involving modified Laguerre polynomials f^β_n(x) from the existence of a class of bilinear generating relations.

3 ref

000069   Rekha A;Chenchu Raju V C (Mathematics Dep, University of Bostwana, Gaborone, Botswana) : Reliability of an n-cascade system under random stress attenuation. Acta Cienc Indica-Math 2003, 29(1), 13-8.

The reliability of a system under random stresses is considered where the stress of a component is attenuated and the attenuation factors are random variables. Expressions for Exponentially distributed stress and strength variables and Beta distributed attenuation factors have been obtained and numerical results are evaluated. The results show that a high degree of reliability could be achieved even when the attenuation factors are random variables.

8 tables, 9 ref

000068   Rawat K S (Mathematics Dep, H.N.B. Garhwal University, S.R.T. Campus, Badsahi Thaul, Tehri Garhwal-249 199) : Kaehlerian bi-recurrent and bi-symmetric manifold. Acta Cienc Indica-Math 2003, 29(2), 233-6.

7 ref

000067   Rawat K S (Mathematics Dep, H.N.B. Garhwal University, S.R.T. Campus, Badshahi Thaul, Tehri Garhwal-249 199) : Study of affine motion in the Hermite space. Acta Cienc Indica-Math 2003, 29(2), 221-4.

The existence of affine motion in Hermite space has been studied by taking an infinitesimal transformation and some important theorems have been investigated on it.

10 ref

000065   Rashmi Kumari (Mathematics Dep, S.V. College, Aligarh) : MHD rotational motion of a dusty viscous fluid and porous medium contained in the semi-infinite circular cylinder due to an infinitely applied impulse on the surface. Acta Cienc Indica-Math 2003, 29(2), 243-8.

The effect of Magnetic field and permeability of the medium have been studied in the presence of dust particles, when circular cylinder is rotated by impulsive force. Two cases have been discussed. First, when applied impulsive force distributed within a circular area (i.e., 0≤r≤b). Second, applied impulsive force distributed over the circumference of circle r=b; b<60>a. It is observed that velocity of fluid is affected by permeability of medium and mass concentration of dust particles. Velocity of the fluid decreases as the permeability of medium and magnetic field increase. This system is very important, where the pipes of wood, cork rubber, clay etc. are used. System is also used in a rocket motor exhaust.

8 ref

000064   Rao K P R;Hari Prasad B V S N;Srinivasa Rao N (Applied Mathematics Dep, N.U.P.G. Centre, Nuzvid-521 201) : Generalisations of some fixed point theorems in complete metric spaces. Acta Cienc Indica-Math 2003, 29(1), 31-4.

1 ref

000061   Rajput U S;Varshney N K;Rajput D (Mathematics Dep, S.V. College, Aligarh, U.P.) : Thermo-diffusion effect on MHD free convection flow of a viscous fluid with dissipation and mass transfer. Acta Cienc Indica-Math 2003, 29(2), 251-60.

The effects of thermal diffusion and transpiration on MHD free convection flow of an electrically conducting fluid through porous medium past an infinite porous plate in a rotating frame of reference is investigated. The heat, due to viscous dissipation and mass transfer has also been considered. Analytical solutions for the velocity, temperature and concentration fields have been obtained. The velocity and temperature profiles with cooling and heating effects (transpiration effects) of the plate for different values of M, S₀, Sc and K have been drawn and discussed. The numerical values of rate of heat transfer (Nu) and skin friction at the plate in x- and y-directions have been tabulated.

12 illus, 5 tables, 12 ref

000060   Rajput A;Arya S;Saluja A S (Dep of Math, Sadhu Vaswani College, Bairagarh, Bhopal) : Some results for fixed point theorems of expansion type maps in D-metric space. Bull Calcutta Mathl Soc 2003, 95(3), 193-8.

7 ref

000059   Rajput A;Arya S (Mathematics Dep, Sadhu Vaswani P.G. College, Bairagarh, Bhopal, M.P.) : Fixed point theorems in Dhage metric spaces. Acta Cienc Indica-Math 2003, 29(1), 187-94.

Obtains some result on fixed points for Contraction mappings in D-metric space with rational inequality.

4 ref

000058   Rajput A;Arya S (Mathematics Dep, Sadhu Vaswani College, Bairagarh, Bhopal, M.P.) : Some results for fixed point theorems of coincidentally commuting maps in d-metric space. Acta Cienc Indica-Math 2003, 29(1), 97-104.

12 ref

000057   Rajeswara Rao U;Sreenivasa Reddy B (Mathematics Dep, S.K. University, Anantapur-515 003) : Transient convective heat transfer through a porous medium in a horizontal wavy channel. Acta Cienc Indica-Math 2003, 29(1), 117-34.

18 illus, 12 tables, 15 ref

000056   Rajagopal R;Selvaraj C (Dep of Math, Loyola College, Chennai-600 034) : On a class of uniformly quasi-convex functions. Bull Calcutta Mathl Soc 2003, 95(3), 199-206.

A new class U Q C V of uniformly quasi-convex functions f(z> analytic in the open disc E = {z : <157>z<157> <60> 1} has been introduced and a necessary and sufficient condition for a function of the form f(z) = z + Σ^∞_n=2 a_nzⁿ to be in the class UQCV has been obtained. Also the class of quasi-uniformly convex functions QUCV has been deduced and certain properties of functions belonging to the two classes have been studied.

6 ref

000055   Ponnusamy N (Statistics Dep, Periyar E.V.R. College (Autonomous), Tiruchirapalli-620 023) : Study on infinite source preemptive priority queueing model. Acta Cienc Indica-Math 2003, 29(1), 205-12.

A single source queueing model with infinite source has been considered and preemptive priority service discipline is assumed. In this model two types of consumers arrive at the system according to the Poisson process and the services times are assumed to be negative exponential with different parameters depending on the priority. Using generating function technique, various statistical quantitites like mean, variance, covariance are presented. Some special cases are obtained and compared with the earlier models.

1 illus, 13 ref

000054   Pescar V (Mathematics Dep, Faculty of Science, Transilvania University of Brasov, 2200 Brasov, Romania) : Univalence criteria of certain integral operators. Acta Cienc Indica-Math 2003, 29(1), 135-8.

Certain integral operators are studied and conditions for the univalence of these integral operators have been determined.

8 ref

000053   Panigrahi U K;Sahu R C (Dep of Math and Science, UCP Engineering School, Berhampur, Orissa-760 010) : Plane symmetric cosmological macro models in modified theory of general relativity. Bull Calcutta Mathl Soc 2003, 95(3), 183-92.

Homogeneous plane symmetric consmological models are constructed in Barber's second self creation theory when the source of the gravitational field is a macro matter field represented by perfect fluid. Some interesting properties of the macro models are discussed.

18 ref

000052   Pandey D;Kalra B S (Mathematics Dep, Ch. Charan Singh University, Meerut, U.P.) : Warm standby system with imperfect switching under mechanical and critical human errors. Acta Cienc Indica-Math 2003, 29(1), 71-80.

Studies the reliability characteristics of a two unit warm standby system under the effect of mechanical failure and critical human error with the assumption of imperfect switching device. Supplementary variable technique has been used to obtain the various reliability characteristics.

5 illus, 3 tables, 3 ref

000051   Panda T C;Mishra S K;Misra A (Mathematics Dep, Berhampur University, Berhampur-760 007) : Modelling electrification of suspended particulate matter (SPM) on induced flow due to horizontally oscillating plane wall. Acta Cienc Indica-Math 2003, 29(1), 173-86.

The problem under study envisages modelling of electrification of SPM on induced flow due to horizontally oscillating plane wall. The Crank-Nicholson implicit finite difference technique has been employed for solution of the governing equations. The study exhibits the relaxation from a frozen diffusion process to equilibrium-diffusion process. For different values of non-dimensional parameters viz, finite volume fraction, diffusion parameter, concentration parameter, Reynolds number, magnetic parameter, the interdependence of fluid flow characteristics has been established through figures and tables. It has been observed that electrification of particles reduces the magnitude of the fluid velocity as well as particle velocity but increases the temperature of both the fluid and particles.

15 illus, 5 tables, 16 ref

000050   Nigam S K;Bawa N P S;Shrivastava P K (NO, , Prabhat Vihar Colony, Satna-485 001) : Fixed point theorems in Hilbert space. Acta Cienc Indica-Math 2003, 29(2), 261-6.

Using more general contractive condition, some fixed point theorems in Hilbert space have been proved. The results generalize some important results of Browder and Petryshyn, Koparde and Waghmode and others.

11 ref

000049   Murty M S N;Apparao B V (Applied Mathematics Dep, Nagarjuna University, Post Graduate Centre, Nuzvid-521 201) : Two point boundary value problems existence and uniqueness. Acta Cienc Indica-Math 2003, 29(2), 229-32.

6 ref

000048   Murthy N V E S (Mathematics Dep, , 5-4-6/1, Punyapu Veedhi SBI-6, Srikakulam-532 001) : Correspondence and isomorphism theorems for fuzzy rings. Acta Cienc Indica-Math 2003, 29(1), 67-70.

Proves correspondence and isomorphism theorems for fuzzy rings.

27 ref

000047   Murthy N V E S (NO, , 5-4-6/1, Punyapu Veedhi SBI-6, Srikakulam-532 001) : Fuzzy homomorphisms between fuzzy rings with truth values in different complete lattices. Acta Cienc Indica-Math 2003, 29(1), 45-66.

The notions of fuzzy subring and crisp homomorphism between crisp rings with fuzzy direct and inverse images are generalised to, respectively, fuzzy ring and fuzzy homomorphism between fuzzy rings with truth values in different complete lattices and with fuzzy direct and inverse images, and their important properties are shown to have extended.

9 ref

000046   Mukherjee M C (Mathematics Dep, Netajinagar Vidyamandir, Kolkata-700 092) : On partial semi bilinear generating function involving hypergeometric polynomial-II. Acta Cienc Indica-Math 2003, 29(2), 325-7.

A new theorem on partial semi bilinear generating function involving Hypergeometric polynomial has been obtained by means of group-theoretic approach. Some special cases of interest are also pointed out.

2 ref

000045   Mukherjee M C (Mathematics Dep, , 72, Parnasree Pally, Calcutta-700 060) : On the extension of mixed trilateral generating relation of extended Jacobi polynomials. Acta Cienc Indica-Math 2003, 29(2), 283-6.

An extension of the trilateral generating relation involving extended Jacobi polynomial derived by the author is well presented from group theoretic view point. Several particular cases of interest are also discussed.

2 ref

000043   Meshram S A;Deshmukh K C (Dep of Math, P G T D University Campus, Nagpur University, Nagpur-440 010) : Quasi static thermal deflection of a Thin circular plate. Bull Calcutta Mathl Soc 2003, 95(2), 101-6.

Deals with the determination of the transient quasi-static thermal deflection on the upper surface of a thin circular plate defined as : 0 ≤ r ≤ a, -h/2 ≤ z ≤ h/2. The expressions of temperature distribution and thermal deflection are obtained in terms of Bessel's Functions and the results are illustrated numerically.

1 illus, 5 ref

000042   Majedur Rahman M;Zulfikar Ali M (Mathematics Dep, BIT, Rajshahi, Bangladesh) : On convergence of Newton's method with generalized bound. Acta Cienc Indica-Math 2003, 29(2), 349-51.

A new proof of L.B. Rall's theorem with generalized bound for the open ball W* = {x: <218><218> x - x* <218><218> <60>(x-^r√n)/(nβ*<108>)}, (where n is a positive real number and r is the positive integer 2) is given. The radius is generalized by (n - ^r<182n)/(nβ*λ) for the best possible bound for h, n = 2.

5 ref

000041   Lanzhe L (Dep of Appl Math, Hunan Univ, Changsha-410 082, People's Republic of China) : Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc Indian Acad Sci-Mathl Sci 2003, 113(4), 379-93.

The continuity for some multilinear operators related to certain non-convolution operators on the triebel-Lizorkin space has been obtained. The operators include Littlewood-Paley operator and Marcinkiewicz operator.

14 ref

000040   Lal M;Gautam I K (Mathematics and Computer Science Dep, R.D. University, Jabalpur-482 001) : Annular disc under torsion in an elastic layer bounded to an elastic layer. Acta Cienc Indica-Math 2003, 29(2), 317-23.

Deals with the torsion by an annular disc for an elastic layer bounded to an elastic layer. The problem is reduced into triple integral equations, which are further reduced to the solution of a Fredholm integral equation of the first kind. By using iterative method, the expression for a simple rotation of the disc through a small angle have been derived. Also, the expressions for the required torque to rotate the disc through an angle φ have been derived.

10 ref

000039   Kulwinder Kaur;Bhatia S S;Babu Ram (Sch of Math and Computer Applications, Thapar Ins of Eng and Techno, , Post Box No. 32, Patiala-147 004) : L<. Proc Indian Acad Sci-Mathl Sci 2003, 113(4), 355-63.

It is proved that the complex double Fourier series of an integrable function f(x,y) with coefficients {c_jk} satisfying certain conditions, will converge in L¹-norm. The conditions used here are the combinations of Tauberian condition of Hardy-Karamata kind and its limiting case. The result of Bray has been extended to complex double Fourier series.

10 ref

000038   Kamna Rani;Pal A K (Mathematics Dep, G.B.P.U.A. and T, Pant Nagar, Uttaranchal) : Single-server bulk service queue with accessible and non-accessible batches in steady state. Acta Cienc Indica-Math 2003, 29(2), 267-70.

Deals with a single-server poisson queue with a new bulk service rule. The rule admits each-batch served to have not less than a and not more than b units such that the arriving in its can enter service, without affecting the service time, if the size of the batch served is less than some fixed integer d(a≤d≤b). The steady state probabilities are obtained for three cases (1) a - 1 ≥ n ≥0, (2) d - 1 ≥ n ≥ a and, (3) n≥ d, where a is minimum numbers of units when server starts service and b is maximum capacity of units.

10 ref

000037   Kamana Rani;Srivastava P N;Shukla A K (Mathematics, Statistics & Computer Science Dep, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttar Pradesh) : Single-server bulk-service queue with varying capacity and Erlang input using Laplace transform. Acta Cienc Indica-Math 2003, 29(2), 217-20.

Deals with a bulk service queueing model denoted by E_k/M^b-x/1. The random variable X denotes the number of units already present with the server at a service epoch, and if X = m(≤b), the server takes (b-m) more units or queue length. Assume that the interarrival times are independent identically distributed random variable (I, i, d, r, v's) each having the k-phase Erlangian distribution the service is in bulk and there is a single exponential server of capacity b. The random-point probability and busy period distribution are obtained.

6 ref

000036   Kalita M;Choudhury R;Sarma G C (Mathematics Dep, Abhayapuri College, Abhayapuri-783 384) : Population models of single microbial species: algal dynamics in Tunia river, Bongaigaon, Assam, India. Acta Cienc Indica-Math 2003, 29(1), 105-14.

A mathematical model to study the population dynamics of single species is proposed and analysed. In modeling the system, it is assumed that the relative growth rate (RGR) of the population under consideration decreases hyperbolically over distance according to the increasing load of pollution. The model developed have been applied to the algal population of Tunia river, Bongaigaon, Assam, India to obtain the projected population of the river.

7 tables, 21 ref

000035   Joshi N K;Dube K K (Mathematics, Statistics and Computer Science Dep, G.B. Pant University of Agriculture & Technology, Pantnagar-263 145) : Semi-invariant submanifold of an ε-framed metric manifold. Acta Cienc Indica-Math 2003, 29(1), 139-44.

The study of semi-invariant submanifolds of an almost contact metric manifold was initiated by A. Bejancu and N. Papaghiuc and later developed by several authors. The semi invariant submanifold of an ε-framed metric manifold and the integrability of the distributions of such submanifolds have been studied.

8 ref

000034   Joshi D C (Dep of Math, South Gujrat University, Surat-395 007, Gujrat) : Hypergeometric functions ₀F₁ solution of a one dimensional groundwater recharge through porous media. Bull Calcutta Mathl Soc 2003, 95(2), 177-82.

Discusses a general solution of a one-dimensional groundwater recharge through porous media by using a theory of generalized hypergeometric functions. The mathematical formulation gives a nonlinear partial differential equation, which is transformed into hypergeometric differential equation by using similarity variables and transformations. An alternative closed form solution is obtained in terms of the hypergeometric functions. ₀F₁. The average diffusivity coefficient over the whole range of moisture content is regarded as constant and a prescribed variation of permeability with moisture content is assumed.

9 ref