Showing papers in "International Journal of Mathematics and Mathematical Sciences in 2003"

Recent applications of fractional calculus to science and engineering

Abstract: This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.

Two fixed-point theorems for mappings satisfying a general contractive condition of integral type

Abstract: We establish two fixed-point theorems for mappings satisfying a general contractive inequality of integral type. These results substantially extend the theorem of Branciari (2002).

A general scheme for constructing inversion algorithms for cone beam ct

Abstract: Given a rather general weight function n0, we derive a new cone beam transform inversion formula. The derivation is explicitly based on Grangeat’s formula (1990) and the classical 3D Radon transform inversion. The new formula is theoretically exact and is represented by a 2D integral. We show that if the source trajectory C is complete in the sense of Tuy (1983) (and satisfies two other very mild assumptions), then substituting the simplest weight n0 ≡ 1 gives a convolution-based FBP algorithm. However, this easy choice is not always optimal from the point of view of practical applications. The weight n0 ≡ 1 works well for closed trajectories, but the resulting algorithm does not solve the long object problem if C is not closed. In the latter case one has to use the flexibility in choosing n0 and find the weight that gives an inversion formula with the desired properties. We show how this can be done for spiral CT. It turns out that the two inversion algorithms for spiral CT proposed earlier by the author are particular cases of the new formula. For general trajectories the choice of weight should be done on a case-by-case basis.

On conformal dilatation in space

Abstract: We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

On the prime submodules of multiplication modules

Abstract: By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results.

Integral equations of the first kind of sonine type

Abstract: A Volterra integral equation of the first kind Kϕ(x) :≡ � x −∞ k(x −t)ϕ(t)dt = f( x) with a locally integrable kernel k(x) ∈ L loc (R 1) is called Sonine equation if there exists another locally integrable kernel �(x) such thatx 0 k(x − t)�(t)dt ≡ 1( lo- cally integrable divisors of the unit, with respect to the operation of convolu- tion). The formal inversion ϕ(x) = (d/dx) � x 0 �(x − t)f (t)dt is well known, but it does not work, for example, on solutions in the spaces X = Lp(R 1 ) and is not defined on the whole range K(X). We develop many properties of Sonine ker- nels which allow us—in a very general case—to construct the real inverse oper- ator, within the framework of the spaces Lp(R 1 ), in Marchaud form: K −1 f( x)= �( ∞)f (x)+ � ∞ 0 � � (t)(f (x −t)−f (x))dt with the interpretation of the convergence of this "hypersingular" integral in Lp-norm. The description of the range K(X) is given; it already requires the language of Orlicz spaces even in the case when X is

Nonlinear peristaltic transport of mhd flow through a porous medium

Abstract: In order to determine the characteristics of peristaltic transport of magnetohydrodynamic flow through a porous medium, the motion of a hydromagnetic (electrically conducting), viscous, and incompressible fluid in planer channel filled with a homogeneous porous medium and having electrically insulated walls that are transversely displaced by an infinite, harmonic travelling wave of large wavelength was analyzed using a perturbation expansion in terms of a variant wave number. We obtain an explicit form for the velocity field, a relation between the pressure rise and flow rate, in terms of Reynolds number, wave number, Hartmann number, permeability parameter, and the occlusion. The effects of all parameters of the problem are numerically discussed and graphically explained. 2000 Mathematics Subject Classification: 76S05. 1. Introduction. Peristalsis is now well known to the physiologists as one of the major mechanisms for fluid transport in many biological systems. In particular, peristaltic mechanism may be involved in swallowing food through the oesophagus, urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ductus efferents of the male reproductive tracts and in the cervical canal, movement of the ovum in the fallopian tubes, and in the vasomotion of small blood vessels as well as blood flow in arteries. In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems. Also, finger and roller pumps are frequently used for pumps corrosive or very pure materials so as to prevent direct contact of the fluid with the pump’s internal surfaces. A number of analytical [3, 5, 7, 9, 11, 12, 16, 23], numerical, and experimental [2, 10, 19, 20, 21] studies of peristaltic flow of different fluids have been reported. Several review articles have been written [8, 14]. Also a summary of analytical papers up to 1984 has been presented in [18]. Most of the analytical studies use perturbation series in a small parameter such as Reynolds number or a dimensionless wave number, which, unfortunately, limits the range of validity of the results. However, a perturbation method does provide explicit information about the physical effects of that parameter. Also, the analytical results can be used to check the calculations of wider-range numerical methods.

Some properties of maximal open sets

Abstract: Some fundamental properties of maximal open sets are obtained, such as decomposition theorem for a maximal open set. Basic properties of intersections of maximal open sets are established, such as the law of radical closure.

On certain means of two arguments and their extensions

Abstract: Inequalities for three means introduced by H.-J. Seiffert are obtained. Generalizations of these means, their basic properties, and inequalities satisfied by the new class of means are also included.

Toeplitz operators with BMO symbols and the Berezin transform

Abstract: We prove that the boundedness and compactness of the Toeplitz operator on the Bergman space with a BMO1 symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends the known results in the cases when the symbol is either a positive L1-function or an L∞ function.

Techniques of the differential subordination for domains bounded by conic sections

Abstract: We solve the problem of finding the largest domain D for which, under given ψ and q, the differential subordination ψ(p(z),zp′(z))∈D⇒p(z)≺q(z), where D and q(𝒰) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q(𝒰). Also, we find the best dominant of the differential subordination p(z)

THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION

Abstract: We derive analytic bounds for the zeros of the third Jackson q -Bessel function J v ( 3 ) ( z ; q ) .

Generalizations of bernoulli numbers and polynomials

Abstract: The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a, b) are generalized to the one Bn(x; a, b, c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a, b) ,a ndBn(x; a, b, c) are established.

Biharmonic curves in Minkowski 3-space

Abstract: We give a differential geometric interpretation for the classification of biharmonic curves in semi-Euclidean 3-space due to Chen and Ishikawa (1991).

Constant scalar curvature metrics on connected sums

Abstract: The Yamabe problem (proved in 1984) guarantees the existence of a metric of constant scalar curvature in each conformal class of Riemannian metrics on a compact manifold of dimension n ≥ 3 , which minimizes the total scalar curvature on this conformal class. Let ( M ′ , g ′ ) and ( M ″ , g ″ ) be compact Riemannian n -manifolds. We form their connected sum M ′ # M ″ by removing small balls of radius ϵ from M ′ , M ″ and gluing together the 𝒮 n − 1 boundaries, and make a metric g on M ′ # M ″ by joining together g ′ , g ″ with a partition of unity. In this paper, we use analysis to study metrics with constant scalar curvature on M ′ # M ″ in the conformal class of g . By the Yamabe problem, we may rescale g ′ and g ″ to have constant scalar curvature 1 , 0 , or − 1 . Thus, there are 9 cases, which we handle separately. We show that the constant scalar curvature metrics either develop small “necks” separating M ′ and M ″ , or one of M ′ , M ″ is crushed small by the conformal factor. When both sides have positive scalar curvature, we find three metrics with scalar curvature 1 in the same conformal class.

Parrondo's paradox

Abstract: Since coming to the attention of the general news media several years ago, the paradoxical combination of two losing games into a winning game by J. M. R. Parrondo has been the subject of numerous numerical investigations and simulations. This note provides a clear statement of the nature of the paradox together with a straightforward analysis and resolution that is accessible to a wide audience.

Matrix measure and application to stability of matrices and interval dynamical systems

Abstract: Using some properties of the matrix measure, we obtain a general condition for the stability of a convex hull of matrices that will be applied to study the stability of interval dynamical systems. Some classical results from stability theory are reproduced and extended. We present a relationship between the matrix measure and the real parts of the eigenvalues that make it possible to obtain stability criteria.

On the equivalence of mann and ishikawa iteration methods

Abstract: We show that certain Mann and Ishikawa iteration schemes are equivalent for various classes of functions.

On some bk spaces

Abstract: We characterize the spaces s α ( Δ ) , s α ∘ ( Δ ) , and s α ( c ) ( Δ ) and we deal with some sets generalizing the well-known sets w 0 ( λ ) , w ∞ ( λ ) , w ( λ ) , c 0 ( λ ) , c ∞ ( λ ) , and c ( λ )

Generalizations of Euler Numbers and Polynomials

Abstract: The concepts of Euler numbers and Euler polynomials are generalized and some basic properties are investigated.

On some properties of ⊕-supplemented modules

Abstract: Am oduleM is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.

Two-machine flowshop scheduling problem to minimize makespan or total completion time with random and bounded setup times

Abstract: This paper addresses the two-machine flowshop scheduling problem with separate setup times to minimize makespan or total completion time (TCT). Setup times are relaxed to be random variables rather than deterministic as commonly used in the OR literature. Moreover, distribution-free setup times are used where only the lower and upper bounds are given. Global and local dominance relations are developed for the considered flowshops and an illustrative numerical example is given.

A survey of limit laws for bootstrapped sums

Abstract: Concentrating mainly on independent and identically distributed (i.i.d.) real-valued parent sequences, we give an overview of first-order limit theorems available for bootstrapped sample sums for Efron’s bootstrap. As a light unifying theme, we expose by elementary means the relationship between corresponding conditional and unconditional bootstrap limit laws. Some open problems are also posed.

B.-Y. Chen inequalities for semislant submanifolds in Sasakian space forms.

Abstract: Chen (1993) established a sharp inequality for the sectional curvature of a submanifold in Riemannian space forms in terms of the scalar curvature and squared mean curvature. The notion of a semislant submanifold of a Sasakian manifold was introduced by J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and M. Fernandez (1999). In the present paper, we establish Chen inequalities for semislant submanifolds in Sasakian space forms by using subspaces orthogonal to the Reeb vector field ξ.

Cohomologically Kähler manifolds with no Kähler metrics.

Abstract: We show some examples of compact symplectic solvmanifolds, of dimension greater than four, which are cohomologically Kahler and do not admit Kahler metric since their fundamental groups cannot be the fundamental group of any compact Kahler manifold. Some of the examples that we study were considered by Benson and Gordon (1990). However, whether such manifolds have Kahler metrics was an open question. The formality and the hard Lefschetz property are studied for the symplectic submanifolds constructed by Auroux (1997) and some consequences are discussed.

Critical global asymptotics in higher-order semilinear parabolicequations

Abstract: We consider a higher-order semilinear parabolic equation u t = − ( − Δ ) m u − g ( x , u ) in ℝ N × ℝ + , \! 1$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"> m > 1 . The nonlinear term is homogeneous: g ( x , s u ) ≡ | s | p − 1 s g ( x , u ) and g ( s x , u ) ≡ | s | Q g ( x , u ) for any s ∈ ℝ , with exponents 1$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"> P > 1 , and -2m$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"> Q > − 2 m . We also assume that g satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponent P = 1 + ( 2 m + Q ) / N such that the asymptotic behavior as t → ∞ of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution b ( x , t ) = t − N / 2 m f ( x t − 1 / 2 m ) of the parabolic operator ∂ / ∂ t + ( − Δ ) m , so that for t ≫ 1 , u ( x , t ) = C 0 ( ln t ) − N / ( 2 m + Q ) [ b ( x , t ) + o ( 1 ) ] , where C 0 is a constant depending on m , N , and Q only.

Pseudoinversion of degenerate metrics

Abstract: Let ( M , g ) be a smooth manifold M endowed with a metric g . A large class of differential operators in differential geometry is intrinsically defined by means of the dual metric g ∗ on the dual bundle T M ∗ of 1-forms on M . If the metric g is (semi)-Riemannian, the metric g ∗ is just the inverse of g . This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for which g ∗ is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian space ℝ 1 n + 2 .

Isomorphism of generalized triangular matrix-rings and recovery of tiles

Abstract: We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents 0and 1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

Asymptotic expansions and positivity of coefficients for large powers of analytic functions

Abstract: We derive an asymptotic expansion as n →∞ for a large range of coefficients of (f (z)) n ,w here f( z) is a power series satisfying | f( z)|

The hull number of an oriented graph

Abstract: We present characterizations of connected graphs G of order n ≥ 2 for which h + ( G ) = n . It is shown that for every two integers n and m with 1 ≤ n − 1 ≤ m ≤ ( n 2 ) , there exists a connected graph G of order n and size m such that for each integer k with 2 ≤ k ≤ n , there exists an orientation of G with hull number G .