A ne w a l~orilhm is ~ive n fur c umpulin ~ Ih e soluliun of a n y secu nd-ord e r lin ea r diffe re nce e qu a li on which is a ppli c able when s impl e rec urrence proc e dures ca nnol be u se d bec au se uf in sla bilil Y. Co mpare d wilh Ih e we ll -knuwn lV1ill e r a l ~o rilhm Ih e ne w m e lhod has Ih e advanla~es of (i) a Ul umal ic ally d eLe rminin g th e currec t number of rec urre nce st eps.(ii ) app lying to illllOlllo ~e n eo u s diffe re nce e qlla ~ li uns, (iii ) e nab lin g mure powerful e rrur a n a lyses lu be co ns lru c le d.The me lhud is illu Slra le d by num e ri ca l comp ulalion s, ineludin~ e rror ana lyses.uf A n ~e r-W e b e r .
Th e c lass of a ll continuou s se lf•mappings uf a me tri zab le space whi c h ca n become contract ion s (in th e sen se of Banac h) und e r me tri cs co mpatibl e with th e to po logy u n th e space is c harac terized .Th e c haracte ri za ti o n amo unt s to a co n ve rse tu th e Co ntrac ti on Mapp ing Prin c iple.Key Words: Co ntrac ti ons, fun c ti on a l a na lysi s, me tri c spaces .topu lugy.:I This co ro ll ary is a reSlale l.n e n.' or a Ih eo re.m due lu J a nos [31.The prour of Th eo re m 1 uses ideas first used by Janos m hIS proof of t hiS theore m: the proof has just been published.
A class of Newton form s P (x )= ao + a,(x-xo)+ . . .+ a,, (x-xo) . . .(x-x ,,_ .)are di sc ussed whic h admit a stable e valu a tion algorithm in an inte rval [A , B l. Stability is defined in th e pape r.The estim a te wh ere L: = B -A and M(x): = laol + ladx + .. .+ la"lx", is shown to hold for the rela tiv e error of e valuati on of P (x) in fA, Bl.
So me purposes of thi s pa per are: (1) To ta ke se riously the term , " term ra nk." (2) T o ma ke a n iss ue of not " rea rra nging rows a nd co lu mn s" by not " a rran ging" th e m in th e firs t place.(3) To pro mote the nu me rica l use of C ra mer 's rul e. (4) To ill us tra te that the re le va nce of " numbe r of s te ps" to " a mount of wo rk " de pen d s on t he a mou nt of wo rk in a step.(5) To ca ll a tt e nti on to the com puta tional as pec t of SDR's, a n as pe ct wher e th e subjec t di ffe rs fro m be in g an insta nce of fa milia r li nea r alge bra.(6) To desc rib e a n SDR in s ta nce of a th eory on e xtre mal co m bi nato rics tha t uses lin ea r alge b ra in ve ry dif• fe rent ways tha n does to tall y un imod ul ar t heo ry.(The preceding pape r, O ptimum Branc hin gs, de• sc rib es a nothe r in sta nc e of tha t theory.)
The following class of inte gral transform pairs is established(2)The kernel in the tra nsform ( 1) is Mac Robert's E•func tion and integration is performed with res pect to the arg ume nt of thi s fun ction.In the inversion formula (2), the kernel is like wise a n E-fun ction, but the integration is performed with respect to its parameters.Known spec ial cases of thi s ge neral transform pair is the Kantorovich-Lebedev transform s pair:r tKixfy)ffy)dy , 7r 0 fIx) = f o oo Ki y(X)g(y)dy , and the ge neralized Mehler trans form pair g(x) = -;sinh (7rx)r(!-k+ iX)r(!-k -iX) 1. 00 pkix _t/2 fy)ffy)dy , fIx) = 10 00 P~y-t/2 (x)g(y) dy.
Given a set S of card in al ity tn , we determine the minimum cardinality/(m) for a family F of s ub sets of S s uch that each SES can be expressed as the intersection of some subfamily of F. Th e problem is solv ed in the following inverse form.For a given numbe r II of subsets of S, find g(Il): the maximum number of elements of S which can be written as the intersec tion of so m e of th ese s ubsets.We s how that g(ll) is the largest binomial coefficient for com binations of II things.
Any matrix 8 s uc h that A8A = A is ca ll ed a C,•inve rse of A and a C,-inverse of A such that BA8 = B is ca ll e d a C,-inverse of A. Some properties of s uc h inve rses are es tabli s hed.It is shown that if A is p-square of ra nk q < p and P is any pos itiv e se mid e finit e matrix, whose rank is the nullity of A, such th at U = A + Pi s non sin gular , th e n B = U-IAU-I is a C,•inverse of A with th e prop e rty that null s pace 8 = null s pace 8 *.That s uc h a P ex is ts for a rbitrar y squ are A is s how n.The relation be twee n thi s res ult a nd th e work of Go ldm an a nd Zelen is disc ussed.
An arborescence T is a tree whose edges a re directe d so that eac h is directed toward a differe nt node.Exactly one node of T, called the root , has no edge of T directed toward it.Le t C be any directe d grap h with a real numerical weight on eac h edge.A good algorithm is described for find ing in C (if there is one) a s panning arborescence, with prescribed root, whose ed ges have maximum (o r minimum) total weight.
j(x) = II " {x (xy) -kE j " I .I 1; +-1) + I : 1 -<1, +-, I ;:,1y) dy.II n wh e re II. is an y pos itive integer a nd E is Mac Ro be rt's fun ction a nd th e ge ne ra li zed Ma c Ro be rt's fun cti o n, res pec t ive ly.S pec ia l c ho ices of t.h e para me t.e rs in th e las t tra ns form lead in turn to th e de riva ti o n of Ha n ke l tra ns form , V-tra ns form , K-tra ns form , Fo uri e r tran s fo rm , Lap lace tra ns fo rm a nd ot.h e r int egra l tran sform s with ta bl es to illu strate t.h ese ne w tra ns form s.
Simple proofs are given of the following classical theorems: (1) An arbitrary set of commuting matrices may be simultaneously brought to triangular form by a unitary similarity.(2) An arbitrary set of co mmuting normal matrices may be simultaneously brought to diagonal form by a unitary similarity.
The problem of s tudying the growth of the error is most important for the numerical so lution of. differential equations. In thi s paper the Wilrs crite rion is generaJized to be applied for sys tems of diffe re ntial eq uations. A general th eore m is inves tigated and regions of s tability have to be determined. The use of an electronic computer is more essential for s uch region s to be c haracterized. These regions of s tability have the property that , the error introduced at any stage te nds to decay. The regions of stabi lity for partic ular numerical methods are exp licitl y determined.
The system of equations a;+a; + _ __ +a~_ l = b;+b;+ __ _ + b~_ l' r=2,3, ___ , n; has no nontrivial solutions in positive integers_
It is shown that if fez) is a polynomial with no zeroes inside the unit c ircle and if r is any positive number, then the coe ffi c ien ts of f'(z) tend to zero like n -,r, and thi s is best possibl e.
A fram e of a co ne C is a minimal se t of generators, and the lin eaJity space L of C is th e grea tes t lin ea r s ubs pace co ntain ed in C. Algorithm s are desc ribed for determinin g a fram e and th e lin ea lit y sp ace of a c one C(S) s panned by a finit e se t S .Th ese a lgorit hm s ca n be used for determ inin g th e ve rti ces, ed ges, and oth e r fa ces of low dim e ns ion of th e co nv ex hull of a finit e set H (S) .All algo• rit hill s are based on th e s impl ex me th od of lin ea r programm in g.Th e problem of findin g th e lin ea lit y s pac e ca n be success ive ly redu ced to prob le ms in s paces of lowe r dim e ns ions.
Th e complex, not necessaril y square matrix A is called a partial isom etry if the vectors x a nd A x ha ve the sam e Euclidean norm whe never x is in the orthogo nal co mple me nt of the null s pace of A .The main result s of th e paper give necessary and sufficient conditions for a matrix to be a partial isometry, for a partial iso metry to be normal and for the produ ct of two partial isometri es to be a partial isometry.A factorization for an arbitrary ma trix involving partial isometries is giv e n.The conce pt of a ge ne ralized inverse is used in establishing the primary results.
Expressions are derived for the indefinite integrals, J,!( r ) Co(ar )dr Irf(r ) q (ar )elr I ,!(r) C,,(ar ) Co(f3r)elr a "" f3 y,!(r) Co(ar ) Z,,(Ar )dr where Co(ar) are zero order Bessel fun c tions, ZoO...,.) are zero orde r modified Bessel fun ction s and f( r ) is a polynomial in r.In general , the expressions given for the integrals are given in te rms of prescribed fun c tions of the Bessel fun c tions, and th e coefficients of these func tions are determined from a finite series, the te rm s of which are found from recurre nce relationships that involve only the polynomial fIr) _ Coeffic ients of the term s of the fin ite series are given in tabular form for up to an eleventh degree polynomial.
It is proved that if C is a finite group of ord er n , th e n the ge nerator rank of C does not exceed the total numb er of prin1es div iding n a nd is equal to thi s number for in finite ly many gro up s C.
se ve ral oth e rs have deve lo pe d a fun cti un th eor y fo r " di sc re te anal yti c fun c ti o ns" de fi ne d u n " di sc rete regiuns" in th e " di sc re te co mpl ex pl a ne."In thi s pa per we brin g to lig ht s ome cu mbin a to ri al• topologica l pro pe rti es of "sim ple d isc re te regiuns," a nd we stud y som e bas ic pru pe rti es of d isc rete a nal ytic fun c ti o ns th a t a re de fi ned on s im ple di sc rete regions _ Th ese co m b in a tor ial-to po logica l p ro perti es a nd bas ic pro pe rti es a re th e n used to es ta bli s h a n ex ist e n ce a nd uni q ue ness th eo re m for d isc re te co mp lex fun c tio ns with presc rib ed " bound a ry va lues" a nd " residu es" 0 11 a n a r b itrary s im ple d isc re te regiun.Key Wo rd s : A na lyti c fu nc ti uns, co mpl ex a na lysis , Diri c hl et pro bl e m , d isc re te a na lyti c fun ct io ns .C f!lls .In this case R 1 = 5 I.CASE II: 5~ and 5 a are contained in Rs but 54 is not.This means that Zi precedes Zt -l in C (i.e., C:
