The American Mathematical Monthly; The Official Journal of the Mathematical Association of America Volume 5

The American Mathematical Monthly; The Official Journal of the Mathematical Association of America Volume 5

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 edition. Excerpt: ...of t's. It follows: We may write the determinant for 4n-i with u's instead of Its, provided that only v goes over into - r-r and vyr into uv, and obtain exactly the same expansion as before. Writing n7, r-ri, "ni-JM -nr-r--i foi' the rth line instead of -n, i'r2 -m-ii al"J wi7i-u2n "i-i- for the rthcolumn instead of vir, v-ir "n-ir, we have for--/' r-i, after taking out the factor unr, "n-11 "n-12 "n-ln Wn-lJi-l In this determinant pass the rth column over the remaining n--r--1 col umns. The entire sign factor becomes (--1)" r-1+i--(--l)"-r combine our results conveniently as follows: We can now if for r=l the first column, and for r--?i, the last column is suppressed. But the sum of the determinants on the right hand is none other than the expansion of in terms of the elements of the last line. We have therefore proved that if fni can be expressed as a determinant of order?i--1, 4n can be similarly expressed as a determinant of order n. But the assumption is true when n--1 =1, and therefore for all greater values. Attention may be called to the fact that this determinant gives very beautifully the general term of jn in terms of the.s's. Let a be the number of cycles, involving?i, n, ua numbers respectively, where ns-fw8 +?iy=ii., into whose product any given substitution of the numbers 1, 2, 3 n can be decomposed. The general term of jH will consist of the products which correspond to the 6 cycles of the substitution. The modulus of the substitution will be the product of the moduli of the cycles, viz, (--1)"' x n't-14-na-1--(--)na. The coefficient of the term will be the number of ways in which the...show more

Product details

  • Paperback | 26 pages
  • 189 x 246 x 1mm | 68g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236570537
  • 9781236570536