It is shown that (i) a diffeomorphism of manifolds with indefinite metrics preserving degenerate <mml:math alttext="$r$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>-plane sections is conformal, (ii) a sectional curvature-preservlng diffeomorphism of manifolds with indefinite metrics of dimension <mml:math alttext="$ \geqslant 4$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> is generically an isometry.

R. S. Kulkarni

Maps of manifolds with indefinite metrics preserving
certain geometrical entities

Arthur Lubin

On von Neumann's inequality

The standard definition of a close-to-convex function involves a complex numerical factor <mml:math alttext="$e^{i\beta } $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>&#946;</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> which is on occasion erroneously replaced by <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>. While it is known to experts in the field that this replacement cannot be made without essentially changing the class, explicit reasons for this fact seem to be lacking in the literature. Our purpose is to fill this gap, and in so doing we are lead to a new coefficient problem which is solved 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>, but is open for <mml:math alttext="$n &#62; 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#62;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>.

A. W. Goodman

E. B. Saff

On the definition of a close-to-convex function

Bradd Evans Clark

Crosscaps and knotes

W. C. Sisarcick

The space of entire functions of two variables as a  metric
space

It is known that every Riesz operator <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> on a Hilbert space can be written <mml:math alttext="$R = Q + C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mi>Q</mml:mi><mml:mo>+</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math>, where <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> is compact and both <mml:math alttext="$Q$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> and <mml:math alttext="$CQ$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:math>-<mml:math alttext="$QC$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Q</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:math> are quasinilpotent. This result is extended to a general <mml:math alttext="$C^* $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math>-algebra setting.

David  Legg

A note on Riesz elements in <svg style="vertical-align:-0.23206pt" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.688)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g></svg>-algebras

For a topological space <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>, and a topological ring <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>, let <mml:math alttext="$C\left( {X,A} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be the ring of all continuous functions from <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> into <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> under the pointwise multiplication. We show that the theorem &#147;there is a completely regular space <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> associated with a given topological space <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> such that <mml:math alttext="$C\left( {Y,R} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is isomorphic to <mml:math alttext="$C\left( {X,R} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>&#148; may be extended to a fairly large class of topologlcal
rings, and that, in the study of algebraic structure of the ring <mml:math alttext="$C\left( {X,A} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, it is sufficient to study <mml:math alttext="$C\left( {X,R} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> if <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is path connected.

J. S. Yang

A note on rings of continous functions

Several representations for a complete residue system in the Euclidean domain <mml:math alttext="$Z\left( \omega  \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#969;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> are presented in this paper.

Gerald E. Bergum

Complete residue systems in the quadratic domain <svg style="vertical-align:-2.3205pt" height="19.5375" version="1.1" viewBox="0 0 58.412498 19.5375" width="58.412498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,16.588)"><path id="x1D44D" d="M698 636l-541 -596q60 -5 176 -5q91 0 138.5 5t69.5 24q44 36 85 124l29 -15q-38 -125 -64 -173h-559l-9 16l545 598h-182q-81 0 -109 -8t-48 -31q-26 -29 -55 -103l-29 3q23 86 42 200h22q11 -16 21 -20.5t34 -4.5h428z" /></g><g transform="matrix(.017,-0,0,-.017,12.319,16.588)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,18.2,16.588)"><path id="x1D452" d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350&#xA;q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" /></g><g transform="matrix(.012,-0,0,-.012,25.237,8.425)"><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><g transform="matrix(.012,-0,0,-.012,30.949,8.425)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" /></g><g transform="matrix(.012,-0,0,-.012,37.981,8.425)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30&#xA;q16 0 76 48z" /></g><g transform="matrix(.012,-0,0,-.012,41.229,8.425)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.012,-0,0,-.012,46.132,8.425)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g><g transform="matrix(.017,-0,0,-.017,52.462,16.588)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> 

Qualitative behavior of second order nonlinear differential equations with variable coefficients is studied. It includes properties such as positivity, number of zeroes, oscillatory behavior, boundedness and monotonicity of the solutions.

Rina  Ling

Behavior of second order nonlinear differantial  equations

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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