The October Meeting in Washington

s of the papers presented at the meeting are listed below, those with the letter t" after their numbers having been read by title. Papers numbered 9 and 27 were read by Dr. Taussky and Dr. Carter respectively. Mr. Boone was introduced by Professor K. L. Chung, Mr. Leslie and Dr. Love by Professor L. C. Young. ALGEBRA AND THEORY OF NUMBERS It. A. A. Aucoin: Systems of Diophantine equations. Solutions of the following Diophantine systems are given: (l)H*_i ÜCLI »v*#* —fiiyù (* —If * • * » »)» where fiiyô^fiiyn, • • • , y%g) are homogeneous polynomials of degree m with integral coefficients and m and n are relatively prime. (2) fi(x) =giW, h(x)=g2(v)y where fi(x)=fi(xh • • • , xp), Mx)*=f%(xit • • • , * « ) are homogeneous polynomials of degree n and are such that integers Xi — ai exist for which all the partials of/i as well as those of ƒ2, of all orders less than n — 1, vanish, gi and g2 are homogeneous polynomials with integral coefficients of degree m where m and n are relatively prime. (3) fi(xit yi} Zi) =gi(xi, yit Zi), f2(xi, yit Zi) =#$(#*, y%, Zi), where the functions involved are polynomials, homogeneous in each of the sets of variables Xi, yit Zi. The solution depends upon the degree of homogeneity. The method applies to more than two equations. (Received August 15, 1951.) It. H. W. Becker: Planar rhyme schemes. Among forty odd other interpretations, Mn — (2n, n)/(n-{-\) is the number of planar rhyme schemes of n letters. (Nonplanar rhyme schemes, such as abab, have crossovers in their Puttenham diagrams.) A lexicon theorem ranks them individually. Known and new breakdowns of Mn have planar rhyme significations: iMnm — Mn-m -Mm-i (Euler), last "a" in mth position; nJIfn.»«(2w—m — 1, n—rn)rn/n (reverse Delannoy), m a's, initial ascending run of length mf m+1 progeny in Mn+i; mMn,m — 2~-(n — 1, 2m) Mm (Touchard), m inversions ; 1 vMn,m — (n,tn — l)(n — l,m — l)/in (W. H. Wise), m different letters, m ascending runs, w — 1 rising pairs; yMn,m = (n, m)AMo — (n, m)Nm, m singletons (unrhymed letters) ; yiMn,m = (n — 1, m)A Mif m couplets (consecutive-rhyme pairs); vnMn,m =iV~(iV+l)~, last singleton in mth. position. These lead to variations of the classic generating function for Mn. Thus Nn — (M— l) , the subset of Mn without singletons, has the generating function 1/(1 -tN) = [1 {(1 3 0 / ( 1 +t)} ]/2L (Received September 12, 1951.) 3t. H. W. Becker: Network severances. Given a net mXn subject to simultaneous failure of one knot in each row, what are 40 AMERICAN MATHEMATICAL SOCIETY [January the possibilities mSn of severances of the net? The difference equations through m =* 6 are set up, solution of the last one of the second degree being iSn — 2Fin (F the Fibonacci series). A more interesting approach is mSn-m • 3 _ 1 —2 F«, the subtrahend being an edge effect correction. Vn = 3 Vn-i + Wn-i, where Wn is the number of rhyme schemes of n letters, such that consecutive letters are of the form k, k or k, k± 1. In turn, Wn — SWn-i—Xn-h where Xn AMi—Wn, the membership of Wn terminating in "a". There are numerous relations between Vn, Wn, Xn, and Mn= (2n) n)/(n+l). For example,'if AUn ,s: Un+i~-mUn (J. Touchard, International Mathematical Congress, Toronto, pp. 465-472), the case m = 3 inverts to Wn+i — iS — M), Vn+i -Jlf-(3-Af)(-)~ . and the elegant dual Xn+i~M(3-M) , Mn+i~X(3-X) . Typical of their generating functions is 1/(1-*.£)[ 3 / 1 {l-t(2+3t)}i*]/2t. There are also numerous subset isomorphisms with planar rhyme schemes. (Received September 12, 1951.) 4. W. W. Boone: An extension of a result of Post. II. Magnus (J. Reine Angew. Math. vol. 103) terms a certain alteration in the form of a word of a group the elimination of a generator. The problem of determining when— for a given word—a similar elimination, described below, can be carried out is unsolvable. One can explicitly exhibit: a particular group, GT', with a finite number of defining relations Ri on a finite number of generators, gi, g2, • • • , gn, such that it is recursively unsolvable to determine for an arbitrary positive word W of GT> (i.e., a word in which all exponents are positive) whether or not W is expressible as a positive word W' not containing gi and made up only of generators occurring in W. Alternatively, regarding exponentiation as an abbreviation, words as made up of the distinct symbols, gt, g7 , and gj" g7 1 ~^r 1 ^ 1 = s T H E V 0 I D W 0 R D a s among thei?*, the foregoing obviously may be formulated as an (unsolvable) problem regarding the elimination of a symbol. The result follows by a comparison of proofs of (cf. abstract, An extension of a result of Post, Journal of Symbolic Logic vol. 16, no. 3) ]£r * * * » Mn of their characteristic roots such that any function (or linear function) f (A, B) has/(At-, m) as its characteristic roots. Pairs of matrices with property L are investigated, and it is shown in particular that hermitian matrices with property L have property P, are commutative, and can be transformed into diagonal matrices by the same similarity. Matrices with property L and w>2 do not in general have property P. (Received August 23, 1951.) 10. D. H. Wagner: On free products of groups. Preliminary report. It is proved that the homomorphisms on a free group F onto a free product are characterized as those homomorphisms (onto) which map some free basis of F into the union of the free factors. This result was, in effect, obtained by I. Gruschko (Rec. Math. (Mat. Sbornik) N. S. vol. 8 (1940) pp. 169-182) for the case where Fis finitely generated. The present proof uses a transfinite convergence procedure developed by H. Fédérer and B. Jónsson (Trans. Amer. Math. Soc. vol. 68 (1950) pp. 1-27) in conjunction with a procedure developed by the writer for reducing a finite subset of a free product. These reductions are motivated by J. Nielsen's reductions for free groups (Matematisk Tidsskrift (1921) pp. 77-94) and possess similar properties. An example is given of a free product which is a homomorphic image of a group which cannot be decomposed into a free product. (Received April 28, 1951.)