The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>d over <mml:math alttext="$GF\left( q \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, where <mml:math alttext="$q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> and <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> are odd. If the stabilizer of the zero vector is non-solvable, let <mml:math alttext="$G_0 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> be a minimal normal non-solvable subgroup. We suspect that <mml:math alttext="$G_0 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> must be isomorphic to some <mml:math alttext="$SL\left( {2,u} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> or homomorphic to <mml:math alttext="$A_6 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>A</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math> or <mml:math alttext="$A_7 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>A</mml:mi><mml:mn>7</mml:mn></mml:msub></mml:math>. Our main result is that this is the case when <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> is the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-groups when <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> and <mml:math alttext="$q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> are both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order <mml:math alttext="$ 27 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>27</mml:mn></mml:math> (i.e., <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> and <mml:math alttext="$q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> are both equal to <mml:math alttext="$ 3 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>) which admits <mml:math alttext="$SL\left( {2,13} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>13</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.

T. G. Ostrom

Translation planes of odd order and odd dimension

We begin with the basic definition and some very simple examples from the theory of univalent functions. After a brief look at the literature, we survey the progress that has been made on certain problems in this field. The article ends with a few open questions.

A. W. Goodman

An invitation on the study of univalent and multivalent functions

This is a summary of the lectures delivered on Special Functions and Linear Representation of Lie Groups at the NSF-CBMS Research Conference at East Carolina University in March 5-9, 1979. The entire lectures will be published by the American Mathematical Society as a conference monograph in Mathematics.

Jean Dieudonn&#xE9;

Gelfand pairs and spherical functions

A Bloch function is exhibited which has radial limits of modulus one almost everywhere but fails to belong to <mml:math alttext="$H^p $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math> for each <mml:math alttext="$0 &#60; p \leqslant \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#60;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math>.

Richard M. Timoney

An example of a Bloch function

For an ordinary differential equation invariant under a one-parameter group of scale transformations <mml:math alttext="$x \to \lambda x$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>→</mml:mo><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:math>, <mml:math alttext="$y \to \lambda ^\alpha  y$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>λ</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mi>y</mml:mi></mml:math>, <mml:math alttext="$y' \to \lambda ^{\alpha  - 1} y'$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>λ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>, <mml:math alttext="$y'' \to \lambda ^{\alpha  - 2} y''$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>y</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>λ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mo>″</mml:mo></mml:msup></mml:math>, etc., it is shown by example that an explicit analytical general solution may be obtainable in parametric form in terms of the scale-invariant variable <mml:math alttext="$\xi  = \int_{}^x {y^{ - 1/\alpha } dx} $" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ξ</mml:mi><mml:mo>=</mml:mo><mml:mo>∫</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math>. This alternative integration may go through, as it does for the example equation <mml:math alttext="$y'' = kxy^{ - 2} y'$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>y</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mi>x</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math>, in cases for which the customary dependent and independent variables <mml:math alttext="$\left( {x^{ - \alpha } y} \right)$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math alttext="$\left( {\ell nx} \right)$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>n</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> do not yield an analytically integrable transformed equation.

Gerald Rosen

Alternative integration procedure for scale-invariant ordinary differential equations

In literature, it is generally asserted that the &#147;principle of objectivity&#148; is valid for constitutive equations but not valid for general field equations. In this paper, the principle of objectivity is formulated clearly to emphasize the correct significance, and necessity of the principle and to discuss its validity in physical theories. A discussion is made to show how successfully the principle of objectivity is also applicable to non-relativistic classical physics.

M.  Dutta

U.  Basu

Principle of objectivity and classical physics

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="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> be a nonnegative integer. The ring <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is called a <mml:math alttext="$\theta _n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>-ring if the number of elements in <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> which are not in <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> is at most <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>. The following theorem is proved: If <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is a <mml:math alttext="$\theta _n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>-ring, then <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is nil or <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is finite. Conversely, if <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is a nil ring or a finite ring, then <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is a <mml:math alttext="$\theta _n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>-ring for some <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

Mohan S. Putcha

Adil  Yaqub

Rings with a finite set of nonnilpotents

Let <mml:math alttext="$\pi _{b,c} \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#960;</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> denote the number of primes <mml:math alttext="$ \leqslant x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#8804;</mml:mo><mml:mi>x</mml:mi></mml:math> and <mml:math alttext="$ \equiv c\left( {\bmod b} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#8801;</mml:mo><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>mod</mml:mo><mml:mi>b</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, and for positive integers <mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math> let <mml:math alttext="$\Delta _b \left( {x,c,l} \right) = \pi _{b,c} \left( x \right) - \pi _{b,l} \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>&#960;</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#960;</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. Negative values of <mml:math alttext="$\Delta _4 \left( {x,3,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> less than <mml:math alttext="$10^{12} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math> occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thousand negative values of <mml:math alttext="$\Delta _4 \left( {x,3,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. However, the authors [1] have recently discovered <mml:math alttext="$3$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math> new regions, the sixth occurring before <mml:math alttext="$20$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>20</mml:mn></mml:math> billion and containing more than half a billion negative values of <mml:math alttext="$\Delta _4 \left( {x,3,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus <mml:math alttext="$8$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>8</mml:mn></mml:math> are presented. Previously, no negative values have been found for <mml:math alttext="$\Delta _8 \left( {x,c,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$c = 3,5$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn></mml:math>, or <mml:math alttext="$7$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>7</mml:mn></mml:math> and our search to <mml:math alttext="$10^{12} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math> reveals no such values for <mml:math alttext="$\Delta _8 \left( {x,3,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> or <mml:math alttext="$\Delta _8 \left( {x,7,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. For <mml:math alttext="$\Delta _8 \left( {x,5,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> we have discovered the first two regions of negative values. The first of these regions, beginning at <mml:math alttext="$x = 588067889$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>588067889</mml:mn></mml:math>, contains <mml:math alttext="$422,500$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>422</mml:mn><mml:mo>,</mml:mo><mml:mn>500</mml:mn></mml:math> negative values of <mml:math alttext="$\Delta _8 \left( {x,5,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>; the second occurs in the vicinity of <mml:math alttext="$35$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>35</mml:mn></mml:math> billion and contains more than a billion negative values of <mml:math alttext="$\Delta _8 \left( {x,5,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#916;</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.

Carter  Bays

Richard H. Hudson

Numerical and graphical description of all axis crossing regions for moduli <svg style="vertical-align:-0.0pt" height="10.8625" version="1.1" viewBox="0 0 8.2875004 10.8625" 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,.062,10.8)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g></svg> and <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="x38" d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5&#xA;t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" /></g></svg> which occur befor <svg style="vertical-align:-0.1638pt" height="16.049999" version="1.1" viewBox="0 0 28.487499 16.049999" width="28.487499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,15.775)"><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><g transform="matrix(.017,-0,0,-.017,8.222,15.775)"><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><g transform="matrix(.012,-0,0,-.012,16.375,7.613)"><use xlink:href="#x31"/></g><g transform="matrix(.012,-0,0,-.012,22.086,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg> 

Restricted isomorphism between two queueing systems implies that they have equivalent distribution function for at least one (but not all) output elements (e.g. waiting time, queue size, idle time, etc.). Quasi-isomorphism implies an approximate equivalence. Most of the single-server queueing systems can be approximated by a quasi-isomorphic system which has a gamma inter-arrival and gamma service distributions <mml:math alttext="$\left( {{{{{E_p } \mathord{\left/  {\vphantom {{E_p } {E_q }}} \right.  \kern-\nulldelimiterspace} {E_q }}} \mathord{\left/  {\vphantom {{{{E_p } \mathord{\left/  {\vphantom {{E_p } {E_q }}} \right.  \kern-\nulldelimiterspace} {E_q }}} 1}} \right.  \kern-\nulldelimiterspace} 1}} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.This paper deals with the derivation of simpler isomorphs (in quasi-restricted sense) for such gamma-gamma <mml:math alttext="$\left( {{{{{E_p } \mathord{\left/  {\vphantom {{E_p } {E_q }}} \right.  \kern-\nulldelimiterspace} {E_q }}} \mathord{\left/  {\vphantom {{{{E_p } \mathord{\left/  {\vphantom {{E_p } {E_q }}} \right.  \kern-\nulldelimiterspace} {E_q }}} 1}} \right.  \kern-\nulldelimiterspace} 1}} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> systems.

A.  Ghosal

Sudhir  Madan

Isomorphs of a class of Queueing systems

There are three kinds of results. First we extend and sharpen a convexity inequality of Agmon and Nirenberg for certain differential inequalities in Hilbert space. Next we characterize the bounded solutions of a differential equation in Hilbert space involving and arbitrary unbounded normal operator. Finally, we give a general sufficient condition for a bounded solution of a differential equation in Hilbert space to be almost periodic.

Jerome A. Goldstein

Convexity, boundedness, and almost periodicity for differential equations in Hillbert space

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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