Open Access Publications

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Open Access
On -transform
(1981-01-01) Nasim, C.
Using a combination of infinite order linear differential operators and integral operators, the inversion of K-transform is established. Inversion procedures for Laplace transform and Potential transform are derived as special cases.
Open Access
Inversion of the Poisson-Hankel transform
(1982-01-01) Nasim, C.
The Poisson-Hankel transform is defined as an integral transform of the initial temperature function, with the kernel as the source solution of the generalized heat equation. In this paper a technique involving integral and differential operators has been used to effect the inversion of the Poisson-Hankel transform.
Open Access
Identities involving iterated integral transforms
(1983-01-01) Nasim, Cyril
A number of identities involving iterated integral transforms are established,making use of the fact that a function which is a linear combination of the Macdonald's function Kν(z), where z is a complex variable, is a Fourier kernel.
Open Access
Admissible groups, symmetric factor sets, and simple algebras
(1984-01-01) Mollin, R. A.
Let K be a field of characteristic zero and suppose that D is a K-division algebra; i.e. a finite dimensional division algebra over K with center K. In Mollin [1] we proved that if K contains no non-trivial odd order roots of unity, then every finite odd order subgroup of D* the multiplicative group of D, is cyclic. The first main result of this paper is to generalize (and simplify the proof of) the above. Next we generalize and investigate the concept of admissible groups. Finally we provide necessary and sufficient conditions for a simple algebra, with an abelian maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets.
Open Access
More on the Schur group of a commutative ring
(1985-01-01) Mollin, R. A.
The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphic image of a group ring RG for some finite group G. It is the purpose of this article to continue an investigation of this group which was introduced in earler work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.
Open Access
More on the Schur group of a commutative ring
(1985-01-01) Mollin, R. A.
The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphic image of a group ring RG for some finite group G. It is the purpose of this article to continue an investigation of this group which was introduced in earlier work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.
Open Access
Uniform distribution of Hasse invariants
(1985-01-01) Mollin, R. A.
I. Schur's study of simple algebras around the turn of the century, andsubsequent investigations by R. Brauer, E. Witt and others, were later reformulated in terms of what is now called the Schur subgroup of the Brauer group. During the last twenty years this group has generated substantial interest and numerous palatable results have ensued. Among these is the discovery that elements of the Schur group satisfy uniform distribution of Hasse invariants. It is the purpose of this paper to continue an investigation of the latter concept and to highlight certain applications of these results, not only to the Schur group, but also to embeddings of simplealgebras and extensions of automorphisms, among others.
Open Access
The general chain transform and self-reciprocal functions
(1985-01-01) Nasim, Cyril
A theory of a generalized form of the chain transforms of order n is developed, and various properties of these are established including the Parseval relation. Most known cases of the standard theory are derived as special cases. Also a theory of self-reciprocal functions is given, based on these general chain transforms; and relations among various classes of self-reciprocal functions are established.
Open Access
On powerful numbers
(1986-01-01) Mollin, R. A.; Walsh, P. G.
A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2)=1. Golomb (op.cit.) conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.
Open Access
On dual integral equations with Hankel kernel and an arbitrary weight function
(1986-01-01) Nasim, C.
In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.
Open Access
The rational canonical form of a matrix
(1986-01-01) Devitt, J. S.; Mollin, R. A.
The purpose of this paper is to provide an efficient algorithmic means of determining the rational canonical form of a matrix using computational symbolic algebraic manipulation packages, and is in fact the practical implementation of a classical mathematical method.
Open Access
The power of powerful numbers
(1987-01-01) Mollin, R. A.
In this note we discuss recent progress concerning powerful numbers, raise new questions and show that solutions to existing open questions concerning powerful numbers would yield advancement of solutions to deep, long-standing problems such as Fermat's Last Theorem. This is primarily a survey article containing no new, unpublished results.
Open Access
On the solution of reaction-diffusion equations with double diffusivity
(1987-01-01) Aggarwala, B. D.; Nasim, C.
In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.
Open Access
On permutation polynomials over finite fields
(1987-01-01) Mollin, R. A.; Small, C.
A polynomial f over a finite field F is called a permutation polynomial if the mapping F→F defined by f is one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integer m, the cardinality of finite fields admitting permutation polynomials of degree m is bounded.
Open Access
An operational procedure for Hankel type integrals
(1987-01-01) Nasim, C.
In this paper, an operational procedure is established to evaluate Hankel type integrals. First, an operator L(θ), θ≡−xddx is constructed, which defines the integral. Then making use of some basic properties of this operator, an elementary procedure is developed for evaluating integrals for a special class of analytic functions. A few example are given to illustrate the technique.
Open Access
On generalized heat polynomials
(1988-01-01) Nasim, C.
We consider the generalized heat equation of nth order ∂2u∂r2+n−1r∂u∂r−α2r2u=∂u∂t. If the initial temperature is an even power function, then the heat transform with the source solution as the kernel gives the heat polynomial. We discuss various properties of the heat polynomial and its Appell transform. Also, we give series representation of the heat transform when the initial temperature is a power function.
Open Access
Modified boundary integral method for pressure driven MHD duct flow
(1989-01-01) Aggarwala, B. D.; Ariel, P. D.
In this paper, we investigate the flow of a viscous, Incompressible,electrically conducting fluid through a rectangular duct in the presence of a magneticfield, when one of the boundaries perpedicular to the magnetic field is partlyconducting and partly Insulating, by a modified Boundary Integral Method.Three problems are considered (i) flow through an infinite channel, (ii) flowthrough a rectangular duct when the conducting part is symmetrically situated, and(iii) flow through a rectangular duct when the conducting part is arbltrarilypositioned.Such problems have been studied before by asymptotic means for large values of M,the Hartmann number. Hoverer, the present modification of the Boundary IntegralMethod renders the problem computationally efficient and provides a reliable numericalsolution for all values of M. For large M, our coputation time decreasessignificantly.
Open Access
Prime valued polynomials and class numbers of quadratic fields
(1990-01-01) Mollin, Richard A.
It is the purpose of this paper to give a survey of the relationship between the class number one problem for real quadratic fields and prime-producing quadratic polynomials; culminating in an overview of the recent solution to the class number one problem for real quadratic fields of Richaud-Degert type. We conclude with new conjectures, questions and directions.
Open Access
Meningitis Due to Ampicillin-and Chloramphenicol-Resistant Haemophilus influenzae Type B in Canada. Case Report and Review
(1990-01-01) Kabani, Amin; Joffe, Ari; Cadrain, Gisele; Jadavji, Taj
The first report of a case of ampicillin- and chloramphenicol-resistant Haemophilus influenzae type b invasive infection in Canada is described in a four-month-old male with meningitis. He was treated with cefotaxime 200 mg/kg/day divided every 6 h and dexamethasone 0.6 mg/kg/day divided every 6 h, eventually recovering after a complicated course. Follow-up at 21 months showed mild to moderate global developmental delay. While chloramphenicol resistance is rare in North America, a case of meningitis initially unresponsive to ampicillin and chloramphenicol must be considered suspect for resistance. Third generation cephalosporins should be used for resistant cases.
Open Access
Accumulation of Trospectomycin by Strains of Salmonella typhimurium, escherichia coli and Haemophilus influenzae
(1990-01-01) Wong, S; Bryan, LE
Trospectomycin, unlike aminoglycosidic aminocyclitols, is accumulated by a nonsaturable, energy-independent, diffusional process in Salmonella typhimurium, Escherichia coli and Haemophilus influenzae. A deep rough mutant of S typhimurium was more susceptible and accumulated the drug faster, and F porin deficient mutants of E coli were more resistant than parental strains. Trospectomycin likely uses both porin and nonporin pathways to cross the outer membrane. An E coli strain effectively accumulated the drug anaerobically, explaining its anaerobic activity. An H influenzae strain accumulated trospectomycin at concentrations below those for which detectable uptake could be observed with E coli or S typhimurium strains, consistent with greater activity in Haemophilus species.