Jungck [1] obtained a fixed-point theorem for a pair of continuous selfmappings on a complete metric space. Recently, Barada K. Ray [2] extended the theorem of Jungck [1] for three self-mappings on a complete metric space. In the present paper we omit the continuity of the mapping used by Ray [2] and replace his four
conditions by a single condition. Our results so obtained generalize and/or unify fixed-point theorems of Jungck [1], Ray [2], Rhoades [3], Ciric [4], Pal and Maiti [5], and Sharma and Yuel [6].

Maibam Ranjit Singh

Remarks on a fixed-point theorem of Gerald Jungck

We establish that, in the sense of the Law of Large Numbers, almost none of the sequences of <mml:math alttext="$0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn></mml:math>&#8217;s and <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>&#8217;s are assigned the same value by every Banach limit.

Jeff  Connor

Almost none of the sequences of <svg style="vertical-align:-0.1638pt" height="11.125" version="1.1" viewBox="0 0 8.2875004 11.125" width="8.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g></svg> &#x27;s and <svg style="vertical-align:-0.0pt" height="10.9125" version="1.1" viewBox="0 0 8.2875004 10.9125" width="8.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g></svg> &#x27;s are almost convergent

In this paper we generalize some well-known commutativity theorems for associative rings as follows: Let <mml:math alttext="$R$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> be a left <mml:math alttext="$s$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi></mml:math>-unital ring. If there exist nonnegative integers <mml:math alttext="$m &#x3E; 1$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>&#x3E;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>, <mml:math alttext="$k \geqslant 0$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, and <mml:math alttext="$n \geqslant 0$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> such that for any <mml:math alttext="$x$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math>, <mml:math alttext="$y$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi></mml:math> in <mml:math alttext="$R$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, <mml:math alttext="$\left[ {x^k y - x^n y^m ,x} \right] = 0$" id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>y</mml:mi><mml:mo>&#x2212;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, then <mml:math alttext="$R$" id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is commutative.

Hamza A. S. Abujabal

A commutativity theorem for left <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 6.5 7.9499998" width="6.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81&#xA;q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" /></g></svg>-unital rings


A simple independent integral equation technique is presented to solve an integral equation with a logarithmic kernel which governs solutions of many two-dimensional Dirichlet boundary value problems involving two coplanar and parallel infinite strips. The technique is further illustrated by an example.

C.  Sampath

D. J. Jain

Solution of an integral equation with a logarithmic kernel

The model for resistive diffusion of a force-free magnetic field in a compressible plasma is analysed. Such a model has been suggested for describing the behavior of the solar outer atmosphere and that of solar flares. Previous analysis of the model involved assumptions of linear plasma velocity and constant magnetic energy. This paper deals with nonconstant magnetic energy in the cases of linear plasma velocity and planar plasma velocity. Some necessary conditions on the plasma direction vectors are derived. Finally, some aspects of the model assumptions are discussed, including nonnormality of initial data vectors to initial surfaces.

David  Eberly

Analysis of a model in solar magnetohydrostatics

Let <mml:math alttext="$F = $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo></mml:mrow></mml:math>{<mml:math alttext="$A\left( i \right):1 \leqslant i \leqslant t$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mn>1</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>i</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$t \geqslant 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>}, be a finite collection of finite, pairwise disjoint subsets of <mml:math alttext="$Z^ +  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>Z</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math>. Let <mml:math alttext="$S \subset R\backslash \left\{ 0 \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>&#8834;</mml:mo><mml:mi>R</mml:mi><mml:mo>\</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math> and <mml:math alttext="$A \subset Z^ +  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mo>&#8834;</mml:mo><mml:msup><mml:mi>Z</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> be finite sets. Denote by <mml:math alttext="$S^A  = $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mi>A</mml:mi></mml:msup><mml:mo>=</mml:mo></mml:mrow></mml:math>{<mml:math alttext="$\sum\limits_{i = 1}^a {s_i } :a \in A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:munderover><mml:mo>&#8721;</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>a</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>:</mml:mo><mml:mi>a</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$S_i  \in S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>&#8712;</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:math>, the <mml:math alttext="$s_i $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math> are not necessarily distinct}. For <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> and <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> as above we say that <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> is <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>-free if for every <mml:math alttext="$A\left( i \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math alttext="$A\left( j \right) \in F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>j</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8712;</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$i \ne j$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi><mml:mo>&#8800;</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$S^{A\left( i \right)}  \bigcap S^{A\left( j \right)}  = \phi $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>&#8898;</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>j</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>&#981;</mml:mi></mml:mrow></mml:math>.We prove that for <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> and <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> as above, <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> contains an <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>-free subset <mml:math alttext="$Q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> such that <mml:math alttext="$\left| Q \right| \geqslant c\left( F \right)\left| S \right|$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>Q</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>&#8805;</mml:mo><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>F</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>S</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math>, when <mml:math alttext="$c\left( F \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>F</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is a positive constant depending only on <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>.This result generalizes earlier results of Erdos [3] and Alon and Kleitman [2], on sum-free subsets. Several possible extensions are also discussed.

Yair  Caro

Generalized sum-free subsets

The structure of a finite group having specified number of second and third maximal subgroups has been investigated in the paper.

N. P. Mukherjee

R. Khazal

A note on finite group structure influenced by second and third maximal subgroups

In this paper we establish a continued fraction represetation for the ratio <mml:math alttext="$qf$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:math> two basic bilateral hypergeometric series <mml:math alttext="$2^\psi  2$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mi>&#x3C8;</mml:mi></mml:msup><mml:mn>2</mml:mn></mml:mrow></mml:math>&#x27;s which generalize Gauss&#x27; continued fraction for the ratio of two <mml:math alttext="$2^F 1$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mi>F</mml:mi></mml:msup><mml:mn>1</mml:mn></mml:mrow></mml:math>&#x27;s.

Remy Y. Denis

On generalization of continued fraction of Gauss

In this note we derive a proof of Pollaczek-Spitzer identity using a generalization of Takacs ballot theorem.

S. Paramasamy

A proof of Pollaczek-Spitzer identity

In this paper we obtain regions containing all the zeros of a class of analytic functions whose coefficients are subject to certain conditions. Our results sharpen some of the results known in this direction. Also we give some examples to show that in some cases the regions obtained by our results are considerably sharper than the regions obtained by known results.

K. K. Dewan

N. K. Govil

On the location of the zeros of analytic functions

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.

Richard A. Mollin

Prime valued polynomials and class numbers of quadratic fields

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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences