By David R. Ferguson, Thomas J. Peters

This number of papers is a unique booklet for SIAM. it truly is, in the beginning, a discussion board for best commercial and govt scientists and engineers to explain their paintings and what they desire to complete over the subsequent decade. The objective is to concentration the mathematical neighborhood on learn that might either enhance arithmetic and supply fast merits to undefined. So this isn't a publication on arithmetic within the strictest experience. it's not geared up round a unmarried or perhaps a number of mathematical disciplines. The papers include few, if any, statements and proofs of theorems. particularly, they concentrate on describing the desires of and govt and on highlighting arithmetic which could play a task in delivering ideas. The papers have been elicited basically from arithmetic for undefined: demanding situations and Frontiers, a convention backed by way of SIAM in October, 2003. This convention was once diverse from different SIAM meetings in that it used to be a discussion board for medical specialists from and executive to fulfill with participants of the mathematical neighborhood to debate the visions they'd for his or her industries and the stumbling blocks they observed to attaining these targets. a particular characteristic of this paintings is its association. the 3 sections: commercial difficulties; Mathematical Responses to commercial difficulties; and the method of Mathematical Modeling, perform, and schooling provide a procedure view, constantly initiated through a realistic challenge, resulting in the necessity for responsive arithmetic either on the examine and academic degrees. All themes are tightly built-in with modern computing expertise. Mathematical issues comprise splines, modeling, computational fluid dynamics, statistical version verification, and arithmetic for layout and simulation. educational mathematicians who desire to increase their view of commercial arithmetic will locate this publication beneficial. commercial mathematicians will locate help for the expanding have to collaborate with educational mathematicians and scientists. scholars who're, or are deliberating, pursuing a profession in business or educational utilized arithmetic will locate this e-book a wealthy resource of fabric at the difficulties dealing with and may strengthen an realizing of the connection among arithmetic and business difficulties. academics of utilized arithmetic can use this e-book to enhance normal educating fabrics.

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Zhu YH, Yang BC. Improvement on Euler’s summation formula and some inequalities on sums of powers. Acta Scientiarum Naturalium Universitis Sunyatseni, 1997, 36(4): 21-26. 15. Xie ZT. A generalization of Stirling formula. Mathematical Practice and Cognition, 2006, 36 (6): 331-333. 16. Yang BC. Some new inequalities on step multiply. Journal of Guangdong Education Institute, 2002, 22(2):1-4. 17. Wei SR, Yang BC. An inequality of Stieltjes coefficients and estimation of their order. Journal of Central Nation University ( Natural Science), 1996,5(2):149~152.

2 n ( ne ) n e 12 n (1 1 ) 30 n2 1 2 n ( ne ) n e 24 . 35), it follows that | n | 4 e1/ 24 n ( en ) n , n 2 . 4. REFERENCES 1. Xu LZ, WANG XH. Methods on mathematical analysis and examples. Beijing: Higher Education Press, 1985. 2. Theory and application of infinite series. Londen: Blackie & Son Limited,1928. 3. Qu WL. Combination mathematics. Beijing: Beijing University Press, 1989. 4. Yang BC. A new formula for evaluating the sum of d-th powers of the first n terms of an arithmetic sequence.

6 If k ( x, y ) m 1 1 m y y 1 1 n ( , then it follows n 1 1 1 1 u u u du sin( / r ) 0 . 4 If k ( x, y ) ( ) dy 1 r , n 1 m 1 1 u r du rs 0 . 2 SOME EXAMPLES 1 r , then it follows n 1 p m 1 0 m y y 1 max{ x , y } amp . 27). By the same way, we can show the best possible property of the constant factor in the case of the reverse equivalent inequalities. 27). k (r ) ) [ p sin( / r ) (r , m) max{1m,n} ( mn ) amp . 28) ( ) r dy 1 ( ) du 0 , ln u 1 r 1 u 1 u ln u 1 1 u 1 u 0 l (r ) 1 ( ) s du ( s, n) ln(mm/nn ) ( mn ) s [ sin( / r ) ]2 .