A necessary and sufficient condition that a vector 
<mml:math alttext="\[ f \]" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> is an antieigenvector of a
strictly accretive operator <mml:math alttext="\[ A \]" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is obtained. The structure of antieigenvectors of selfadjoint and certain
class of normal operators is also found in terms of eigenvectors. The Kantorovich inequality for
selfadjoint operators and the Davis's inequality for normal operators are then easily deduced. A
sort of uniqueness is also established for the values of 
<mml:math alttext="\[ \operatorname{Re} \left( {Af,f} \right) \]" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>Re</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math alttext="\[ {\left\| {Af} \right\|} \] " id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>&#x2016;</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>f</mml:mi></mml:mrow><mml:mo>&#x2016;</mml:mo></mml:mrow></mml:mrow></mml:math> if the first antieigenvalue, which is equal to min <mml:math alttext="\[ \operatorname{Re} {{\left( {Af,f} \right)} \mathord{\left/  {\vphantom {{\left( {Af,f} \right)} {\left( {\left\| {Af} \right\|\left\| f \right\|} \right)}}} \right.  \kern-\nulldelimiterspace} {\left( {\left\| {Af} \right\|\left\| f \right\|} \right)}} \] " id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>Re</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mi>f</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:mrow><mml:mo>&#x2016;</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>f</mml:mi></mml:mrow><mml:mo>&#x2016;</mml:mo></mml:mrow><mml:mrow><mml:mo>&#x2016;</mml:mo><mml:mi>f</mml:mi><mml:mo>&#x2016;</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math> is attained at the unit vector <mml:math alttext="\[ f \]" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>.

K. C. Das

M. Das Gupta

K. Paul

Structure of the antieigenvectors of a strictly accretive operator

In this paper a collection of efficient algorithms are described for solving an algebraic system
with a symmetric Toeplitz coecient matrix. Systems of this form arise when approximating the solution of
boundary value Volterra integro-differential equations with finite difference methods. In the nonlinear case,
an iterative procedure is required and is incorporated into the algorithms presented. Numerical examples
illustrate the results.

L. E. Garey

R. E. Shaw

Efficient algorithms for solving nonlinear Volterra
			integro-differential equations with two point boundary conditions

Under weaker conditions of probability,we discuss in this paper the complete convergence
for the partial sums and the randomly indexed partial sums of <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math>-valued <mml:math alttext="$\mathcal{L}^p$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math>-mixingale
sequences.

Liang  Hanying

Ren  Yaofeng

Complete convergence for <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125" 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="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g></svg>-valued <svg style="vertical-align:-0.0pt" height="14.45" version="1.1" viewBox="0 0 17.512501 14.45" width="17.512501" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,14.387)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g><g transform="matrix(.012,-0,0,-.012,9.763,6.225)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-mixingale sequences

New identities are given, based on ideas related  to visible (from the origin) point vectors. Each result was found from summing on vpv lattices dividing space 
into radial regions from the origin. This is 
related to recent work by the author in which new partition type infinite products were derived. Also
recently, Baake et al [3] and Mosseri [14] considered the 2-<mml:math alttext="$D$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>
 visible lattice points as part of an optical
experiment in which the so-called Optical Fourier Transform was applied. Many of  the techniques
espoused in Glasser and Zucker [11], and in Ninham et al [15] involving Mellin and M&#246;bius inversions
are applicable also to the current paper.

Geoffrey B. Campbell

Visible point vector summations from hypercube and hyperpyramid lattices

The aim of this paper is to study the class of <mml:math alttext="$N$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-scattered spaces, i.e. the spaces
whose nowhere dense subsets are scattered. The concept was recently used in a decomposition of
scatteredness &#x97; a topological space 
<mml:math alttext="\[ \left( {X,\tau } \right) \]" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x3C4;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>

 is scattered if and only if <mml:math alttext="$X$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> is <mml:math alttext="$\alpha$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B1;</mml:mi></mml:math>-scattered (= its <mml:math alttext="$\alpha$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B1;</mml:mi></mml:math>-topology
is scattered) and <mml:math alttext="$N$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-scattered.

Julian Dontchev

David Rose

On spaces whose nowhere dense subsets are scattered

A mapping <mml:math alttext="\[ f:X \to Y \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>Y</mml:mi></mml:math>
 between continua <mml:math alttext="\[ X \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>
 and <mml:math alttext="\[ Y \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math>
 is said to be atomic at a
<em>subcontinuum</em><mml:math alttext="\[ K \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math>
 of the domain <mml:math alttext="\[ X \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>
 provided that <mml:math alttext="\[ {f\left( K \right)} \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>K</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is nondegenerate and <mml:math alttext="\[ K = f^{ - 1} \left( {f\left( K \right)} \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>K</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. The set
of subcontinua at which a given mapping is atomic, considered as a subspace of the hyperspace of all
subcontinua of <mml:math alttext="\[X\]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>, is studied. The introduced concept is applied to get new characterizations of atomic
and monotone mappings. Some related questions are asked.

Janusz J. Charatonik

W&#322;odzimierz J. Charatonik

Atomoicity of mappings

This paper examines smoothness attributes of probability measures on lattices which
indicate regularity, and then discusses weaker forms of regularity; specifically, weakly regular and
vaguely regular. They are obtained from commonly used outer measures, and we study them mainly for
the case of 
<mml:math alttext="\[ M\left(\mathcal{L} \right) \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> or for those components of <mml:math alttext="\[ M\left(\mathcal{L} \right) \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> with added smoothness prerequisites. This is a
generalization of many concepts presented in my earlier paper (see [1]).

Dale  Siegel

Weak regularity of probability measures

Let <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> be a <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-torsion free semiprime ring with derivation <mml:math alttext="$d$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>. Supposed <mml:math alttext="$d^{2n}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>

is a derivation of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, where <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> is a positive integer. It is shown that if <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is <mml:math alttext="$(4n- 2)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>4</mml:mn><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-torsion free
or if <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is an inner derivation of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, then <mml:math alttext="$d^{2-1}=0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>.

Jiang  Luh

Youpei  Ye

Derivations of higher order in semiprime rings

In order to have a topological category that contains both pMET and Unif in a
nice way (and therefore combines quantitative and uniform concepts), approach uniformities are
introduced. Approach uniformities uniformize the so-called approach spaces, introduced in Lowen
[8]. Different characterizations of approach uniformities are formulated. Some natural examples
are presented, such as function spaces, hyperspaces, spaces of measures, and an example concerning
theoretical computer science.

R. Lowen

B. Windels

A Unif: A common supercategory of pMET and Unif

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International Journal of Mathematics and Mathematical Sciences - Table of contents 1998

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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