An abrasive blasting apparatus comprising: supply means for supplying abrasive particles, and a rotatable impeller for blasting the abrasive particles supplied from this abrasive supply means onto a workpiece, the impeller having a plurality of vanes arranged in mutually spaced relationship around a rotation axis of the impeller, each of the vanes having a surface containing a first axis inclined with an angle of 25 DEG -65 DEG relative to said rotation axis and a second axis inclined with an angle of 40 DEG -80 DEG relative to an axis perpendicular to said rotation axis, each of the vanes having a first end edge and a first side edge which are located close to said rotation axis and also a second end edge and a second side edge which both face said first end edge and said first side edge, respectively, and also having a third edge positioned between said first end edge and said first side edge and crossing these first end and side edges, the crossing point of said first end edge and said second side edge being either in agreement with the cross point of said third edge and said first end edge or located between a crossing point of said first end edge and said third edge and a crossing point of said first side edge and said third edge relative to said rotation axis, said supply means supplying the abrasive particles from said thrid edge onto said surface which is arranged to have either a flat plane, a plurality of planes or a single curved surface.

M. ABRAMOWrrz ANO I. A. STEttJN, EDS., Handbook ofMathematical Functions, Dover, New York, 1965. Also solved by ROBIN CHAPMAN (University of Exeter, Exeter, UK), THOMAS DICKENS (Exxon Production Research, Houston, TX), CARL C. GROSJEAN (University of Ghent, Ghent, Belgium), W. B. JORDAN (Scotia, NY), ALLEN R. MILLER (Washington, DC), NORBERT ORTNER (University of Innsbruck, Austria), and the proposer. Editorial note. MILLER generalizes the integral to

Previous article Next article Integrals of Bessel and Elliptic Functions from Heat Conduction (Ping Hui)Carl C. GrosjeanCarl C. Grosjeanhttps://doi.org/10.1137/1038086PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Integrals of Bessel and Elliptic Functions from Heat Conduction (Ping Hui)." SIAM Review, 38(3), pp. 523–524[1] P. Hui and , H. S. Tan, Temperature distributions in a heat dissipation system using a cylindrical diamond heat spreader on a copper heat sink, J. Appl. Phys., 75 (1994), 748–757 10.1063/1.356480 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 38, Issue 3| 1996SIAM Review367-551 History Published online:12 July 2006 InformationCopyright © 1996 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1038086Article page range:pp. 523-524ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Previous article Next article Expected Type of Triangle (D. J. Newman)Carl C. GrosjeanCarl C. Grosjeanhttps://doi.org/10.1137/1035015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. Hanani, , D. Ornstein and , V. T. Sos, On the lottery problem, Magyar Tud. Akad. Mat. Kutató Int. Közl., 9 (1964), 155–158 29:5749 0139.01005 Google Scholar[2] F. Sterboul, Le problème du Loto, Cahiers Centre Études Rech. Opér., 20 (1978), 443–449 81g:05005 0399.05040 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 35, Issue 1| 1993SIAM Review History Published online:17 February 2012 InformationCopyright © 1993 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1035015Article page range:pp. 139-141ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Previous article Next article Limit of a Definite Integral (C. A. Oster)Peter WagnerPeter Wagnerhttps://doi.org/10.1137/1034013PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Limit of a Definite Integral (C. A. Oster)." SIAM Review, 34(1), pp. 122–123[1] A. W. Warrick and , D. O. Lomen, Time-dependent linearized infiltration: III. Strip and disc sources, Soil Sci. Soc. Amer. J., 40 (1976), 639–643 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 34, Issue 1| 1992SIAM Review1-166 History Published online:18 July 2006 InformationCopyright © 1992 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1034013Article page range:pp. 122-123ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

This paper contains a detailed discussion concerning the validity of ∫+∞0ø(x)⧸xxdx= ∑+∞k=−∞ø(k)⧸kk, which was proposed by S. Ramanujan in his second notebook. The formula, regarded either as a strict equality or as an asymptotic relation, is not valid for every continuous function ø(x) on R for which the integral is convergent. But it is shown that the formula holds asymptotically for ø(x = axf(x) whereby a represents a positive real parameter whose value can become as large as desired and f(x) can be 1, any positive integer power of x, any polynomial x and any nonpolynomial function belonging to a wide class of functions each representable by its Maclaurin series expansion on R. Much attention is paid to the aspect of practical application of the obtained results. The article ends with the derivation of some remarkable identities following as a by-product from the calculations presented.

When one formally solves the initial-value problem y′ = {x − (αx + 1) y}/x2, y(0) = 0, α∈] − 1, + ∞[,by inserting into the differential equation an infinite series of positive integer powers of x, one obtains an asymptotic expansion (around x = 0) of the solution y(x). The corresponding Stieltjes continued fraction can be transformed into a Jacobi continued fraction whose convergents have as denominators the generalized Laguerre polynomials and as numerators the corresponding (first) associated polynomials of the second kind, both proportionally reduced to monic polynomials. In the above-mentioned infinite Stieltjes continued fraction development of y(x), all the elements can be shifted upward over one position and in this manner one obtains the development of y1(x) ≔ (x/y(x))− 1 which is the solution of a nonlinear initial-value problem. To y1(x), there also belongs a sequence of monic orthogonal polynomials {p1,n(x)/n ∈ N}. This process may be repeated an unlimited number of times. In this way one finds infinitely many related sequences of monic orthogonal polynomials {pk,n(x) / n ∈ N}, ∀ k ∈ N0, all satisfying Favard's conditions as a consequence of the elements in the Stieltjes continued fraction expansion of y(x) being positive. These sets can be expressed in terms of the generalized Laguerre polynomials and their associated polynomials of the second kind of all orders. For all the sets involved, the recurrence formula, the weight function and its moments, as well as the orthogonality relation are obtained. Each sequence {pk,n(x) / n ∈ N} can be regarded as stemming from a first-order initial-value problem y′k = fk(x, yk), yk(0) = 0, with known fk-function, and via the associated fu second kind with subscript zero qk,0(x), a Stieltjes-type integral representation of yk(x) is found.

An (alternative) elementary proof of a certain hypergeometric identity, which was posed recently as a problem, leads naturally first to a multiple-series extension of the problem and then also to its related multivariable hypergeometric transformation and summation formulas. A systematic analysis of several general multiple-series identities analogous to these hypergeometric transformations is presented. Many of the general results obtained are shown to unify and extend numerous transformation and reduction formulas for various classes of double and multiple hypergeometric series.

Previous article Next article A Complement of Weber’s Integral (J. W. Miles)Norbert Ortner and M. L. GlasserNorbert Ortner and M. L. Glasserhttps://doi.org/10.1137/1032137PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"A Complement of Weber’s Integral (J. W. Miles)." SIAM Review, 32(4), pp. 683–684[1] John Miles, Parametrically excited standing edge waves, J. Fluid Mech., 214 (1990), 43–57 91d:76007 0699.76018 CrossrefISIGoogle Scholar[2] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1945 Google Scholar[3] A. Erdélyi, , W. Magnus, , F. Oberhettinger and , F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw–Hill, New York, 0058.34103 Google Scholar[4] Milton Abramowitz and , Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964xiv+1046 29:4914 0171.38503 Google Scholar[5] L. Badii and , F. Oberhettinger, Tables of Laplace transforms, Springer-Verlag, New York, 1973, 133–, Berlin 50:5375 0285.65079 Google Scholar[6] Fritz Oberhettinger, Tabellen zur Fourier Transformation, Springer-Verlag, Berlin, 19573, 112, 212, New York 18,481a 0077.12002 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 32, Issue 4| 1990SIAM Review History Published online:18 July 2006 InformationCopyright © 1990 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1032137Article page range:pp. 683-684ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics