The Dieudonn&#233;-Schwartz Theorem on bounded sets in a strict inductive limit is investigated for non-strict inductive limits. Its validity is shown to be closely connected with the problem of whether the projective limit of the strong duals is a strong dual itself. A counter-example is given to show that the Dieudonn&#233;-Schwartz Theorem is not in general valid for an inductive limit of a sequence of reflexive, Fr&#233;chet spaces.

C.  Bosch

J.  Kucera

K.  McKennon

Dual characterization of the Dieudonne-Schwartz theorem on bounded sets

The authors introduce the notions of Ritt order and lower order to
functions defined by the series <mml:math alttext="$\sum\limits_1^\infty {f_n \left( s \right)\exp \left( { - \lambda _n s} \right)} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munderover><mml:mo>&#8721;</mml:mo><mml:mn>1</mml:mn><mml:mi>&#8734;</mml:mi></mml:munderover><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>exp</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#955;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> where <mml:math alttext="$\left( {\lambda _n } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>&#955;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> is a <mml:math alttext="$D$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>-sequence and <mml:math alttext="$f_n \left( s \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> are entire functions of bounded index.

R.  Parvatham

J.  Thangamani

On the Ritt order of a certain class of functions

The purpose of this paper is to study a class of polynomials <mml:math alttext="$F_n^m \left( {x,\lambda ,\nu } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>F</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>&#955;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#957;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> associated with a class of <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> differential equations, where <mml:math alttext="$\lambda $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#955;</mml:mi></mml:math> is fixed, <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> is a variable parameter, <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> is fixed positive integer and <mml:math alttext="$\nu $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#957;</mml:mi></mml:math> is a non-negative integer <mml:math alttext="$ &#60; m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#60;</mml:mo><mml:mi>m</mml:mi></mml:math>. We also obtain all the important properties of this class of polynomials including Rodrigues' formulas, generating function, and recurrence relations. Several special cases of
interest are obtained from our analysis.

M.  Dutta

Kanchan Prabha Manocha

A class of orthogonal polynomials of a new type

Given that <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> and <mml:math alttext="$P$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math> as nonlinear onto and into self-mappings of a complete metric space <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, we offer here a constructive proof of the existence of the unique solution of the operator equation <mml:math alttext="$Au = Pu$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mi>u</mml:mi></mml:math>, where <mml:math alttext="$u \in R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>R</mml:mi></mml:math>, by considering the iterative sequence <mml:math alttext="$Au_{n + 1} = Pu_n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:msub><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> (<mml:math alttext="$u_0 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> prechosen, <mml:math alttext="$n = 0,1,2, \ldots $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo></mml:math>). We use Kannan's criterion [1] for the existence of a unique fixed point of an operator instead of the contraction mapping principle as employed in [2]. Operator equations of the form <mml:math alttext="$A^n u = P^m u$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>P</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mi>u</mml:mi></mml:math>, where <mml:math alttext="$u \in R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>R</mml:mi></mml:math>, <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> and <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> positive integers, are also treated.

Rabindranath  Sen

Sulekha  Mukherjee

On iterative solution of nonlinear functional equations in a metric space

The purpose of this paper is to define the continuous Jacobi transform as an extension of the discrete Jacobi transform. The basic properties including the inversion theorem for the continuous Jacobi transform are studied. We also derive an inversion formula for the transform which maps <mml:math alttext="$L^1 \left( {R^ + } \right)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> into <mml:math alttext="$L_w^2 \left( { - 1,1} \right)$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>L</mml:mi><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> where <mml:math alttext="$w\left( x \right) = \left( {1 - x} \right)^\alpha \left( {1 + x} \right)^\beta $" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>w</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>β</mml:mi></mml:msup></mml:math>.

E. Y. Deeba

E. L. Koh

The continuous Jacobi transform

An inverse sampling procedure is proposed for the problem of selecting the better of two treatments when the responses are dichotomous. This procedure is particularly useful when it is desired to limit the number of failures during the decision making stage. The regret function of the procedure is derived and it is shown that this procedure has a minimax regret property when compared to a fixed sample procedure studied by Pradhan and Sathe [2]. Numerical evidence indicates that this procedure dominates the fixed sample procedure of Pradhan and Sathe over the entire parameter space.

S. S. Chitgopekar

A Note on Selecting the Better Binominal Population

The tree polynomials [1] of the basic graphs with cyclomatic number 2 are
derived. From these polynomials, results about forest decompositions are deduced. Explicit formulae are given for the number of decompositions of the basic graphs into forest with specified finite cardinalities.

E. J. Farrell

Forest decompositions of graphs with cyclomatic number 2

Let <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> be a ring and let <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> denote the set of nilpotent elements of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. Let <mml:math alttext="$Z$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> denote the center of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. Suppose that (i) <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> is commutative, (ii) for every <mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math> in <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> there exists <mml:math alttext="$x'\varepsilon &#60; x &#62; $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mi>&#1013;</mml:mi><mml:mo>&#60;</mml:mo><mml:mi>x</mml:mi><mml:mo>&#62;</mml:mo></mml:math> such that <mml:math alttext="$x - x^2 x'\varepsilon N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>&#8722;</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mi>&#1013;</mml:mi><mml:mi>N</mml:mi></mml:math>, where <mml:math alttext="$ &#60; x &#62; $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#60;</mml:mo><mml:mi>x</mml:mi><mml:mo>&#62;</mml:mo></mml:math> denotes the subring generated by <mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math>, (iii) for every <mml:math alttext="$x,y$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:math> in <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, there exists an integer <mml:math alttext="$n = n\left( {x,y} \right) \geqslant 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8805;</mml:mo><mml:mn>1</mml:mn></mml:math> such that both <mml:math alttext="$\left( {xy} \right)^n - \left( {yx} \right)^n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>&#8722;</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>y</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> and <mml:math alttext="$\left( {xy} \right)^{n + 1} - \left( {yx} \right)^{n + 1} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>&#8722;</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>y</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> belong to <mml:math alttext="$Z$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math>. Then <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is commutative and, in fact, <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is isomorphic to a subdirect sum of nil commutative rings and local commutative rings. It is further shown that both conditions in hypothesis (iii) are essential. The proof uses the structure theory of rings along with some earlier results of the authors.

Hazar  Abu-Khuzam

Adil  Yaqub

Commutativity and structure of rings with commuting nilpotents

The fundamental results of <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. Connes which determine a complete set of isomorphism classes for most injective factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the investigation in the classification of factors is presented, culminating in the Connes-Takesaki structure theory of type <mml:math alttext="$III$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:math> factors. After a discussion of injectivity for finite factors, the main result of the paper, the uniqueness of the injective <mml:math alttext="$II_1 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> factor, is deduced, and the structure of <mml:math alttext="$II_\infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mi>&#8734;</mml:mi></mml:msub></mml:math> and type <mml:math alttext="$III$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:math> injective factors is then obtained as corollaries of the main result.

Steve  Wright

Classification of injective factors: The wok of Alain Connes

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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