• Volume 103, Issue 2

August 1993,   pages  103-207

• Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, pv) and the projective special linear groups PSL(2, pv) are realized for allp, and the Suzuki groups Sz(22v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.

• A footnote to mean square value of DirichletL-series

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for$$s(\sigma + it) = \sum\limits_{x^{(modq)} } {|L(\sigma + it,x} )|^2$$} Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t5/8 q−1/8, which is an improvement in the range q ¦t¦ &lt; q11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for ¦t¦q9/5.

• Matrix summability of a factored Fourier series

This paper reports a result for proving a triangular matrix summability of a factored Fourier series by extending the theorem on Nörlund summability of a factored Fourier series att =x when ϕ(t)∈B.V in (0, π) due to Singh [4] (Indian J. Math.9 227–236). The result generalizes the theorem of Varshney [5] (Proc. Am. Math. Soc.10, 784–789) and that of Singh.

• Absolute matrix summability of a factored Fourier series

A theorem for a factored Fourier series is proved.

• Multipliers for ¦N,pn;δ¦k summability of infinite series

In this paper a theorem on ¦N,pn;δ¦ksummability factors, which generalizes a theorem of Bor [3] on ¦N,pn¦ksummability factors, has been proved.

• Bishop decompositions for vector function spaces

Various Bishop type decompositions for vector function spaces are introduced and discussed. Conditions are given under which some of them coincide. Several examples and counter examples are also given.

In this paper, we introduce a notion calledM2-graded hypergroup, which extends the notion of hypergroup and is motivated by the example of a ‘paragroup’ in the context of the inclusion of a pair ofII1factors. After discussing the example of the ‘paragroup’ we derive certain consequences of the definition and then prove that every finite irreducibleM2-graded hypergroup possesses a unique dimension function, in analogy with a result for hypergroups.

• On Fatou’s lemma and parametric integrals for set-valued functions

In this paper we present new versions of the set-valued Fatou’s lemma for sequences of measurable multifunctions and their conditional expectations. Then we use them to study the continuity and measurability properties of parametrized set-valued integrals.

• Banerjeeet al’s characterization theorem in thermohaline convection and its magnetorotatory extensions

The paper mathematically establishes that magnetorotatory thermohaline convection of the Veronis [7] type cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermohaline Rayleigh numberRs, the Lewis number τ, the thermal Prandtl number σ, the magnetic Prandtl number σ1, the Chandrasekhar numberQ and the Taylor numberT satisfy the inequality$$R_s 4\pi ^4 \left[ {1 + \frac{\tau }{{\sigma \pi ^2 }}\left\{ {\pi ^2 - \left( {O\sigma _1 + \frac{T}{{\pi ^2 }}} \right)} \right\}} \right]$$ when both the boundary surfaces are rigid thus achieving magnetorotatory extension of an important characterization theorem of Banerjeeet al (1992) on the corresponding hydrodynamic problem. A similar characterization theorem is mathematically established in the context of the magnetorotatory thermohaline convection of the Stern (1960) type.

• # Proceedings – Mathematical Sciences

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