Let <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> denote the set of all functions <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> which are analytic and univalent in the unit disk <mml:math alttext="$D$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> normalized so that <mml:math alttext="$f\left( z \right) = z + a_2 z^2  +  \ldots $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>&#8230;</mml:mo></mml:mrow></mml:math>. Let <mml:math alttext="$S^ *  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mo>&#8727;</mml:mo></mml:msup></mml:mrow></mml:math> and <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> be those functions <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> in <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> for which <mml:math alttext="$f\left( D \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is starlike and convex, respectively. For <mml:math alttext="$0 \leqslant k \lt 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>k</mml:mi><mml:mo>&#60;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>, let <mml:math alttext="$S_k $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math> denote the subclass of functions in <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> which admit <mml:math alttext="${{\left( {1 + k} \right)} \mathord{\left/  {\vphantom {{\left( {1 + k} \right)} {\left( {1 - k} \right)}}} \right.  \kern-\nulldelimiterspace} {\left( {1 - k} \right)}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#8722;</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math>-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a function <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> belongs to <mml:math alttext="$S_k  \bigcap S^ *  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>&#8898;</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mo>&#8727;</mml:mo></mml:msup></mml:mrow></mml:math> or <mml:math alttext="$S_k  \bigcap C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>&#8898;</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math>. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong to <mml:math alttext="$S_k $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math>. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.

Johnny E. Brown

Quasiconformal extensions for some geometric subclasses of univalent functions

In this paper we study the behavior of dynamical systems in uniform space which are Poisson stable and distal. The main results are that the trajectories of such motions are closed and every almost periodic motion is stable <mml:math alttext="${\mathrm{P}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>P</mml:mtext></mml:math> and distal. In complete uniform space, this motion is periodic, Lagrange stable and recurrent.

S. S. Prasad

Anant  Kumar

Stable <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 9.6999998 11.175" width="9.6999998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x50" d="M46 650h247q112 0 169 -41q64 -49 64 -141q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-271v28q60 5 74 20t14 77v400q0 65 -12.5 79t-69.5 18v28zM212 559v-245q31 -12 73 -12q59 0 102 38t43 124q0 77 -42.5 115.5&#xA;t-108.5 38.5q-44 0 -56 -12q-11 -8 -11 -47z" /></g></svg> and distal dynamical systems

An integral transform involving the associated Legendre function of zero order, <mml:math alttext="$P_{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/  {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}} + i\tau } \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mfrac bevelled="true"><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>&#964;</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math alttext="$x \in \left[ {1,\infty } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>&#8712;</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, as the kernel (considered as a function of <mml:math alttext="$\tau $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#964;</mml:mi></mml:math>), is called Mehler-Fock transform. Some generalizations, involving the function <mml:math alttext="$P_{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/  {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}} + i\tau }^\mu  \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mfrac bevelled="true"><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>&#964;</mml:mi></mml:mrow><mml:mi>&#956;</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, where the order <mml:math alttext="$\mu $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math> is an arbitrary complex number, including the case when <mml:math alttext="$\mu  = 0,1,2, \ldots $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#956;</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo></mml:mrow></mml:math> have been known for some time. In this present note, we define a general Mehler-Fock transform involving, as the kernel, the Legendre function <mml:math alttext="$P_{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/  {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}} + t}^\mu  \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mfrac bevelled="true"><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mi>&#956;</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, of general order <mml:math alttext="$\mu $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math> and an arbitrary index <mml:math alttext="$ - \frac{1} {2} + t$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$t = \sigma  + i\tau $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>&#963;</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>&#964;</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$ - \infty  \lt \tau  \lt \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mi>&#8734;</mml:mi><mml:mo>&#60;</mml:mo><mml:mi>&#964;</mml:mi><mml:mo>&#60;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:math>. Then we develop a symmetric inversion formulae for these transforms. Many well-known results are derived as special cases of this general form. These transforms are widely used for solving many axisymmetric potential problems.

Cyril  Nasim

The Mehler-Fock transform of general order and arbitrary index and its inversion

We use the Lie series averaging method to obtain a complete second order solution for motion of a charged particle in a spatially periodic magnetic field. A comparison is made with the first order solution obtained previously by Coffey.

Sikha  Bhattacharyya

R. K. Roy Choudhury

Lie transforms and motion of a charged particle in a periodic magnetic field

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameter <mml:math alttext="$\varepsilon $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#1013;</mml:mi></mml:math> may be obtained without resorting to secularity conditions if the time scales <mml:math alttext="$t_i  = \varepsilon ^i t\left( {i = 0,1, \ldots } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>&#1013;</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</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:mo>&#8230;</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

L. E. Levine

W. C. Obi

Generalized multitime expansions for equations with slowly varying coefficients

This paper is concerned with sequences that satisfy a class of fourth order linear recurrence equations. Basic properties of such sequences are derived. In addition, we discuss the oscillatory and nonoscillatory behavior of such sequences.

Sui-Sun Cheng

On a class of fourth order linear recurrence equations

We consider the usual wave equation <mml:math alttext="$u_{tt} \left( {x,t} \right) = c^2 u_{xx} \left( {x,t} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> on the real line with some typical initial and boundary conditions. In each case, we establish a natural isometrical identity and inverse formula between the sourse function and the response function.

Saburou  Saitoh

Some isometrical identities in the wave equation

In an open bounded region of <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-space occupied by a homogeneous and isotropic medium, we control the temperature through the boundary. The normal derivative of the temperature (which measures the appropriate heat flux) is restricted to be nonnegative. This gives rise to a free boundary in space-time separating the areas of positive and zero heat flux. Under a natural monotonicity condition, the free boundary is the graph of a function of space. This function is shown to be locally Lipschitz. Moreover for <mml:math alttext="$n = 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> the time derivative of the temperature is proven to be continuous across the free boundary.

Ioannis  Athanasopoulos

A temperature control problem

It is shown that the Poulsen simplex is not a projective tensor product of non-trivial Ghoquet simplexes.

Thomas E. Armstrong

The Poulsen simplex is not a tensor product

Let <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> be a ring with <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>, <mml:math alttext="$\rho $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#961;</mml:mi></mml:math> an automorphism of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> of order <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>. Then a normal extension of the free quadratic extension <mml:math alttext="$R\left[ {x,\rho } \right]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>&#961;</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> with a basis <mml:math alttext="$\left\{ {1,x} \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><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:mrow></mml:math> over <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> with an <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>-automorphism group <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> is characterized in terms of the element <mml:math alttext="$\left( {x - \left( x \right)\alpha } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>&#8722;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>&#945;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for <mml:math alttext="$\alpha $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math> in <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>. It is also shown by a different method from the one given by Nagahara that the order of <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> of a Galois extension <mml:math alttext="$R\left[ {x,\rho } \right]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>&#961;</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> over <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> with Galois group <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> is a unit in <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. When <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> is not a zero divisor, more properties of <mml:math alttext="$R\left[ {x,\rho } \right]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>&#961;</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> are derived.

George  Szeto

On automorphism group of free quadratic extensions over a ring

This study extends the notion of regularity series from convergence spaces to Cauchy spaces, and includes an investigation of related topics such as that <mml:math alttext="$T_2 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math alttext="$T_3 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> modifications of a Cauchy space and their behavior relative to certain types of quotient maps. These concepts are applied to obtain a new characterization of Cauchy spaces which have <mml:math alttext="$T_3 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> completions.

Darrell C. Kent

The regularity series of a Cauchy space

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


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International Journal of Mathematics and Mathematical Sciences | Hindawi

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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