The problem of a uniformly propagating finite crack in an infinite medium is solved within the linearized couple-stress theory. The self-equilibrated system of pressure is applied to the crack surfaces. The problem is reduced to dual integral equations and solved by a series-expansion method. The dynamic stress-intensity factor is computed numerically.

S.  Itou

The effect of couple-stresses on the stress concentration around a moving crack

Two limitation methods, <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> and <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math>, are said to be consistent for a class <mml:math alttext="$b$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi></mml:math> of sequences, iff, every sequence belonging to <mml:math alttext="$b$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi></mml:math> is limitable both by <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>
 and <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> and that the <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-limit equals the <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math>-limit. Any two regular limitation methods are consistent for the class-<mml:math alttext="$c$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi></mml:math> of convergent sequences. However, this is not true in general and in fact, corresponding to every bounded non-convergent sequence it is possible to determine two <mml:math alttext="$T$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-matrices such that they limit the sequence to two different values. In this paper, we establish the necessary and sufficient conditions for the consistency of two limitation methods, for <mml:math alttext="$\left( {N,p_n } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> summable sequences.

Nand  Kishore

U. K. Misra

On the consistency of limitation methods for <svg style="vertical-align:-4.10045pt" height="17.299999" version="1.1" viewBox="0 0 50.125 17.299999" width="50.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.137)"><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,-.017,5.944,12.137)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.017,-0,0,-.017,20.801,12.137)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,27.498,12.137)"><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><g transform="matrix(.012,-0,0,-.012,37.588,17.1)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,44.188,12.137)"><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> summable sequences

The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let <mml:math alttext="$X$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> be a <mml:math alttext="$p \times n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:math> matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix <mml:math alttext="$\sum $" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>∑</mml:mo></mml:math>. Further, let <mml:math alttext="$S = XX'$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:msup><mml:mi>X</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math> and let <mml:math alttext="$1_1  &gt;  \ldots  &gt; 1_p  &gt; 0$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mo>…</mml:mo><mml:mo>&gt;</mml:mo><mml:msub><mml:mn>1</mml:mn><mml:mi>p</mml:mi></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math> be the characteristic roots of <mml:math alttext="$S$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math>. Thus <mml:math alttext="$S$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> has a noncentral Wishart distribution. In this paper, the exact distribution of <mml:math alttext="$f_p  = 1 - {{1_p } \mathord{\left/  {\vphantom {{1_p } {1_1 }}} \right.  \kern-\nulldelimiterspace} {1_1 }}$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>f</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mn>1</mml:mn><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:math> is derived. The density of <mml:math alttext="$f_p $" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>f</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> is given in terms of zonal polynomials. These results have applications in nuclear physics also.

V. B. Waikar

On the noncentral distribution of the ratio of the extreme roots of wishart matrix

New developments and results in the theory of expectatiors and variances for random variables with range in a topological group are presented in the following order (i) Introduction (2) Basic notions (3) The three series theorem in Banach spaces (4) Moment Conditions (5) Expectations and variances (6) A general three series theorem (7) The special cases of finite groups and Lie groups (8)The strong laws of large numbers on a Lie group (9) Further studies on
moments of probability measures.

Herbert  Heyer

Moments of probability measures on a group

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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