Cusp. - Seminar for Applied Mathematicsmhg/pub/mhg-published/68-Gut02... · 2014. 12. 5. · double...

T

TACNODE, point of osculation, osculation point, double cusp - The third in the series of Ak-curve singularities. The point (0,0) is a tacnode of the curve X 4 __ y 2 • 0 in R 2.

The first of the Ak-curve singularities are: an ordinary double point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.

They are exemplified by the curves X k+l - y2 = 0

for k = 1,2,3,4. The terms 'crunode' and 'spinode' are seldom used

nowadays (2000).

See also Node ; Cusp.

R e f e r e n c e s [1] ABHYANKAR, S.S.: Algebraic geometry for scientists and en-

gineers, Amer. Math . Soc., 1990, p. 3; 60. [2] DIMCA, A.: Topics on real and complex singularities, Vieweg,

1987. [3] GRIFFITHS, PH., AND HARRIS, J.: Principles of algebraic ge-

ometry, Wiley, 1978, p. 293; 507. [4] WALKER, R.J. : Algebraic curves, Pr ince ton Univ. Press, 1950,

Reprint : Dover 1962.

M. Hazewinkel

MSC 1991:14H20

TANGLE, relative link - A one-dimensional manifold properly embedded in a 3-ball, D a.

Two tangles are considered equivalent if they are am- bient isotopic with their boundary fixed. An n-tangle has 2n points on the boundary; a link is a 0-tangle. The term arcbody is used for a one-dimensional manifold properly embedded in a 3-dimensional manifold.

Tangles can be represented by their diagrams, i.e. regular projections into a 2-dimensional disc with additional over- and under-information at crossings. Two tangle diagrams represent equivalent tangles if they are related by Reidemeister moves (cf. R e i d e m e i s t e r theorem). The word 'tangle' is often used to mean a tangle diagram or part of a link diagram.

The set of n-tangles forms a mono id ; the identity tangle and composition of tangles is illustrated in Fig. 1.

o . o T 1 ~ T2

J I

TId T 1 * T 2

Fig. 1.

Several special families of tangles have been considered, including the r a t i o n a l t angles , the a lgebra ic t ang les and the periodic tangles (see Rotor ) . The n- braid group is a subgroup of the monoid of n-tangles (cf. also B r a i d e d group) . One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is important in developing quantum invariants of links and 3-manifolds (e.g. Reshetikhin-Turaev invariants). Tan- gles are also used to construct topological quantum field theories.

Re fe rences [1] BONAHON, P., AND SIEBENMANN, L.: Geometric splittings of

classical knots and the algebraic knots of Conway, Vol. 75 of Lecture Notes, London Math . Soc., to appear.

[2] CONWAY, J .H.: ' A n enumera t ion of knots and links' , in J. LEECH (ed.): Computational Problems in Abstract Alge- bra, Pergamon Press , 1969, pp. 329-358.

[3] LOZANO, M.: 'Arcbodies ' , Math. Proc. Cambridge Philos. Soe. 94 (1983), 253-260.

Jozef Przytycki

MSC 1991:57M25

T A N G L E M O V E - For given n-tangles 2/"1 and T2 (cf. also Tangle) , the tangle move, or more specifically the (T1,T2)-move, is substitution of the tangle T2 in the place of the tangle T1 in a link (or tangle). The simplest tangle 2-move is a crossing change. This can be generalized to n-moves (cf. M o n t e s i n o s - N a k a n i s h i con j ec tu r e or [5]), (m, q)-moves (cf. Fig. 1), and p/q- rational moves, where a rational 2-tangle is substituted in place of the identity tangle [6] (Fig. 2 illustrates a 13/5-rational move).

TANGLE MOVE

A p/q-rational move preserves the space of Fox p-

colourings of a link or tangle (cf. Fox n -co lou r ing ) .

For a fixed prime number p, there is a conjecture that

any link can be reduced to a trivial link by p/q-rational m o v e s (Iql _< p/2).

Kirby moves (cf. K i r b y ca lcu lus ) can be interpreted

as tangle moves on framed links.

J ~ " - J ' ~ " ~ " " " ~ ' ~ (m,q)-move

m half twists

Fig. 1.

... q half twists

13/5-move

Fig. 2.

Habiro Cn-moves [1] are prominent in the theory of

Vassiliev Gusarov invariants of links and 3-manifolds. The simplest and most extensively studied Habiro move

(beyond the crossing change) is the A-move on a 3-

tangle (cf. Fig. 3). One can reduce every knot into the

trivial knot by A-moves [4].

- m o v e

Fig. 3.

References [1] HABIRO, K.: 'Claspers and finite type invar iants of links' , Ge-

ometry and Topology 4 (2000), 1-83. [2] HARIKAE, T., AND UCHIDA, Y.: ' I r regular dihedral b ranched

coverings of knots ' , in M. BOZH/SY/)K (ed.): Topics in Knot Theory, Vol. 399 of NATO ASI Ser. C, Kluwer Acad. Publ . , 1993, pp. 269-276.

[3] KIRBY, R.: ' P rob l ems in low-dimensional topology' , in W. KAZEZ (ed.): Geometric Topology (Proc. Georgia Inter- nat. Topolo9y Conf., 1993), Vol. 2 of Studies in Adv. Math., Amer. Math . Soc. / IP, 1997, pp. 35-473.

[4] MURAKAMI, H., AND NAKANISHI, Y.: 'On a cer tain move gen- erat ing link homology ' , Math. Ann. 284 (1989), 75-89.

[5] PRZYTYCKI, J .H.: '3-coloring and o ther e lementary invar iants of knots ' : Knot Theory, Vol. 42, Banach Center Publ . , 1998, pp. 275-295.

[6] UCHTDA, Y., in S. SUZUKI (ed.): Knots '96, Proc. Fifth Inter- nat. Research Inst. of MS J, World Sei., 1997, pp. 109 113.

Jozef Przytycki MSC 1991:57M25

TAU METHOD, r method A method initially for-

mulated as a tool for the approximation of special func-

tions of mathematical physics (cf. also Special functions), which could be expressed in terms of simple dif-

ferential equations. It developed into a powerful and ac-

curate tool for the numerical solution of complex differ-

ential and functional equations. A main idea in it is to

approximate the solution of a given problem by solving

exactly an approximate problem.

L a n c z o s ~ f o r m u l a t i o n o f the tau m e t h o d . In [17], C. Lanczos remarked tha t t runcat ion of the series solution

of a differential equation is, in some way, equivalent to introducing a per turbat ion te rm in the right-hand side

of the equation. Conversely, a polynomial per turbat ion

te rm can be used to produce a t runcated series, that is,

a polynomial solution. Assume one wishes to solve by means of a power se-

ries expansion the simple linear differential equation (cf.

also Linear differential operator)

D y ( x ) : = y ' ( x ) + y ( x ) = 0 , O < x < l ,

v(0 ) = 1,

which defines y(x) = e x p ( - x ) . To find the coefficients

of a formal series expansion of y(x), one substitutes the

series in the equation and generates a system of alge-

braic equations for the coefficients: jaj + aj-1 = 0 for j = 1, 2 , . . . , solving it in terms of a0. The value of a0 is

fixed using the initial condition. To find a finite expan-

sion, say of order n, one needs to make all coefficients

aj with j > n equal to zero. This is achieved by adding

a term of the form r x n to the right-hand side of the

differential equation. One has (n + 1)an+l + an = % so

that a,,+l, and all the coefficients following it, will be

equal to zero if one chooses as = r . The same condi-

tion follows by substi tuting a segment of degree n of the series expansion of y(x) = e x p ( - x ) into the equation.

If the solution of the per turbed differential equation is

regarded as an approximation to that of the original

equation with, say, a r ight-hand side equal to zero, it

seems natural to replace it by the best u n i f o r m ap-

p r o x l m a t i o n of zero over the same interval J , which is

a Chebyshev polynomial T2 (x) of degree n, defined over

J (cf. also C h e b y s h e v p o l y n o m i a l s ) .

Therefore, to find an accurate polynomial approxima-

tion of y(x), Lanczos proposed solving exactly the more

complex per turbed problem (the tau problem):

Dye(x) = rT,~ (x),

with the same initial conditions as before. The polyno-

mial y*(x) is called the tau method approximation of

y(x) over the given interval J .

This tau problem can be solved for the unknown co-

efficients of y*(x) using several alternative procedures.

396

TAU METHOD

One of them is described above, that is, to set up and

solve a system of linear algebraic equations linking the

unknown coefficients of Dy* (x) with those of 7T~ (x). In this process one can assume that yn(X) itself can be expressed in either powers of x, or in Chebyshev, Legendre

or other polynomials. The first choice was Lanczos' orig-

inal choice, and he explicitly indicated the possibility of

choosing the others.

The second choice is a tau method, often [8] called the Chebyshev method (or Legendre method) and, also, the spectral method. This last formulation of the tau method has been extensively used and applied, since 1971, to complex problems in fluid dynamics by S.A. Orsag [11].

There are at least three other approaches to the tau method. One of them is to find the coefficients of the approximant through a process of interpolation at the

zeros of the per turbat ion term. This early form of collocation was termed the 'method of selected points' by

Lanczos [17]. When the per turbat ion term is an orthogonal polynomial (such as a Chebyshev, Legendre, or other polynomial), this process is called 'orthogonal collocation'. This is the name by which Lanczos' method of selected points is usually designated today (as of 2000); the name 'pseudo-spectral method' is also often applied to it. Algorithms for these methods have been well de-

veloped.

R e c u r s l v e f o r m u l a t i o n o f t h e t a u m e t h o d b a s e d on c a n o n i c a l p o l y n o m i a l s . In his classic [18], Lanc-

zos noted that if a sequence of polynomials Q~(x), n = 0, 1 , . . . , such that DQn(x) := x ~ for all n E N can be found for any linear differential operator with polynomial coefficients D, then, since Tg(x) := c~ + e~x + • .. + c~,x '~ (the coefficients of which are tabulated), the solution of the tau problem would be immediately given

by: n

k=0

where the parameter T is fixed using the initial condi-

tion.

An extension of this approach to a wider range of

differential operators than the trivial one, given in the

example, has several advantages: canonical polynomials are independent of the interval in which the solution is sought, allowing for easy segmentation of the domain; they are permanent, in the sense that if an approximation of a higher degree is required, the computation does not need to be repeated from scratch; they are also independent of the supplementary conditions of the problem, which can now equally be initial, boundary or multi- point conditions. Furthermore, the tau method does not require a stage of discretization of the given differential operator, as discrete-variable methods do.

A sequence of canonical polynomials defined as simply as DQn(x) := x n for all n = 0, 1 , . . . , need not always exist or need not be unique. An algebraic and algo- rithmic theory of the tau method, initially constructed for elements D of the class D of linear differential operators of arbitrary integer order, with polynomial or rational coefficients (essentially the tools a computer

handles) was discussed by E.L. Ortiz in [24]. In this work, canonical polynomials are defined as realizations

of classes of equivalence of polynomials, for which the algebraic kernel of the differential operator is the mod- ulus. These classes have gaps in their index sequence. Elements D E D are then uniquely associated with re- presentatives of such classes of canonical sequences. The codimension of the image of the space of polynomials under operators D C D is usually small, and bounded by the order of D plus the height h := maxncN{a~ - n } (where an is the degree of Dx n) of the differential operator.

For more general operators than the one used as an example, more than a single ~- term is usually required to satisfy the more elaborate supplementary conditions and, also, internal conditions of the method. In the case of a problem defined by a differential operator D in l?, of order # > i and with non-constant coefficients, the question of the number of 7 terms required for a tau method

approximation has been shown to be related to the size of the gap in the canonical sequence, and to the existence of a non-empty algebraic kernel in D. The number of ~- terms can be easily determined in this approach using information on the degree of polynomial (or rational) coefficients and the order of differentiation of the

function to which they apply. It was also shown in [24] that canonical sequences can be generated recursively. This approach was used to formulate the first recur-

sire algorithms for the automatic solution of differential equations using the tau method. The theory of canonical polynomials has been discussed and extended by several authors; see [10] and the references given therein.

Theoretical error analysis for the tau method [18], [30], [9], [22], [26] have shown that tau method approximations are of the order of best uniform approximations by polynomials defined over the same interval. This connection with best approximation is preserved when a

tau method based on rational approximation [18], [21] is used [5].

O p e r a t i o n a l f o r m u l a t i o n o f t h e t a u m e t h o d . There

is yet another way in which tau method approximations can be constructed. An operational formulation of the tau method was introduced by Ortiz and H. Samara in [27]. In this formulation, derivatives and polynomial coefficients of operators in 7? are represented in terms of

397

TAU M E T H O D

mul t ip l i ca t ive d iagona l mat r ices . Fu r the rmore , the dif-

ferent ia l o p e r a t o r and the s u p p l e m e n t a r y condi t ions are

decoupled . T h r o u g h a s imple and sys t ema t i c a lgor i thm,

which t r e a t s the different ial ope ra to r and supplemen-

t a r y condi t ions wi th s imi lar machinery, this technique

t r ans fo rms a given different ia l t au m e t h o d p rob lem into

one in l inear a lgebra . The a p p r o x i m a t e so lu t ion can be

genera ted , indis t inct ively , in t e rms of powers of the vari-

ables or in t e r m s of e lements of a more s tab le po lynomia l

basis , such as Chebyshev, Legendre or o the r po lynomi -

als. The o p e r a t i o n a l fo rmula t ion fur ther s implif ied the

deve lopmen t of sof tware for the t au me thod .

N u m e r i c a l a p p l i c a t i o n s o f t h e t a u m e t h o d . The

recurs ive and ope ra t i ona l approaches to the t au m e t h o d

have been ex t ended in several direct ions. To sys tems of

l inear different ia l equa t ions [9], [4]; to non- l inear p rob-

lems [25], [23], [26]; to p a r t i a l differential equa t ions [28],

[29]; and, in pa r t i cu la r , to the numer ica l so lu t ion of non-

l inear sys tems of pa r t i a l different ial equa t ions the solu-

t ion of which has sha rp spikes, wi th high gradients , as

in the case of soliton interactions [14], [13]; to the ap-

proximate solu t ion of o r d i n a r y and pa r t i a l funct ional -

different ial equa t ions [25], [20], [15]; and to s ingular

p rob lems for pa r t i a l different ial equat ions re la ted to

crack p r o p a g a t i o n [7]. The t a u m e t h o d is well a d a p t e d

to p roduce accu ra t e a p p r o x i m a t i o n s in the numer ica l

t r e a t m e n t of differential eigenvalue p rob lems with one

or mul t ip le spec t ra l p a r a m e t e r s , enter ing e i ther l inear or

non- l inear ly into the equa t ion [2], [19]. T h e t a u m e t h o d

has been extens ive ly used for the h igh-precis ion approx-

ima t ion of real- [16] and complex-va lued funct ions. A

weak fo rmula t ion of the t a u m e t h o d has been p roposed

and appl ied to inverse p rob lems for pa r t i a l differential

equat ions [1].

A n a l y t i c a l a p p l i c a t i o n s o f t h e t a u m e t h o d . The

t a u m e t h o d has also been used in a t o t a l l y different di-

rect ion, as a tool in the discussion of p rob lems in m a t h -

emat i ca l analysis , for example , in complex funct ion the-

ory [12].

Possible connect ions be tween the t a u me thod , col-

locat ion, Ga le rk in ' s me thod , a lgebra ic kernel me thods ,

and o ther po lynomia l or d i sc re te -var iab le techniques

have also been explored [31], [13], [6].

The t au m e t h o d has also received some a t t en t i on as

an ana ly t ic tool in the discussion of equivalence resul ts

across nmner ica l me thods [6]. I t has been found tha t ,

wi th it, it is poss ible to cons t ruc t special ' t a u me thods ' ,

which recurs ively genera te solu t ions numer ica l ly ident i -

cal to those of col locat ion, Ga le rk in ' s and o ther weighted

res idual me thods , and to those of d i sc re te -var iab le me th -

ods, such as soph i s t i ca t ed forms of R u n g e - K u t t a meth-

ods. This work suggests a way of unifying a large group

of cont inuous- and d i sc re t e -va r i ab le a p p r o x i m a t i o n tech-

niques.

R e f e r e n c e s

[1] BANKS, H.T., AND WADE, J.G.: 'Weak tau approximations for distributed parameter systems in inverse problems', Nu- met. Funct. Anal. Optim. 12 (1991), 1-31.

[2] CHAVES, T., AND ORTIZ, E.L.: 'On the numerical solution of two point boundary value problems for linear differential equations', Z. Angew. Math. Mech. 48 (1968), 415 418.

[3] CRISCI, M.R., AND RUSSO, E.: 'A-stability of a class of methods for the numerical integration of certain linear systems of differential equations', Math. Comput. 41 (1982), 431-435.

[41 CRISCg M.R., AND RUSSO, E.: 'An extension of Ortiz's recursive formulation of the tau method to certain linear systems of ordinary differential equations', Math. Comput. 41 (1983), 27-42.

[5] EL DAOU, M., NAMASIVAYAM, S., AND ORTIZ, E.L.: 'Dif- ferential equations with piecewise approximate coefficients: discrete and continuous estimation for initial and boundary value problems', Computers Math. Appl. 24 (1992), 33-47.

[6] EL DAOU, M., AND ORTIZ, E.L.: 'The tau method as an analytic tool in the discussion of equivalence results across nu- mericaI methods', Computing 60 (1998), 365-376.

[7] EL MISlERY, A.E.M., AND ORTIZ, E.L.: 'Tau-lines: a new hy- brid approach to the numerical treatment of crack problems based on the tau method', Computer Methods in Applied Me- chanics and Engin. 56 (1986), 265 282.

[8] Fox, L., AND PARKER, I.B.: Chebyshev polynomials in numerical analysis, Oxford Univ. Press, 1968.

[9] FREILICH, J.G., AND ORTIZ, E.L.: 'Numerical solution of sys- terns of differential equations: an error analysis', Math. Com- put. 39 (1982), 467-479.

[10] FROES BUNCHAFT, M.E.: 'Some extensions of the Lanczos- Ortiz theory of canonical polynomials in the tau method', Math. Comput. 66, no. 218 (1997), 609 621.

[11] GOTLIEB, D., AND ORSZAG, S.A.: Numerical analysis of spectral methods: Theory and applications, Philadelphia, 1977.

[12] HAYMAN, W.K., AND ORTIZ, E.L.: 'An upper bound for the largest zero of Hermite's function with applications to subhar- monic functions', Proc. Royal Soc. Edinburgh 75A (1976), 183-197.

[13] HOSSEIM AH-ABAD% M., AND ORTm, E.L.: 'The algebraic kernel method', Namer. Funct. Anal. Optim. 12, no. 3-4 (1991), 339 360.

[14] HOSSEINI ALI-ABADI, M., AND ORTIZ, E.L.: 'A tau method based on non-uniform space-time elements for the numerical simulation of solitons', Computers Math. Appl. 22 (1991), 7-19.

[15] KHAJAH, H.G., AND ORTIZ, E.L.: 'Numerical approximation of solutions of functional equations using the tau method', Appl. Namer. Anal. 9 (1992), 461-474.

[16] KHAJAH, H.G., AND ORTIZ, E.L.: 'Ultra-high precision com- putations', Computers Math. Appl. 27, no. 7 (1993), 41-57.

[17] LANeZOS, C.: 'Trigonometric interpolation of empirical and analytic functions', J. Math. and Physics iT (1938), 123-199.

[18] LANCZOS, C.: Applied analysis, New Jersey, 1956. [19] LIu, K.M., AND ORTIZ, E.L.: 'Tau method approximation of

differential eigenvalue problems where the spectral parameter enters nonlinearly', Y. Comput. Phys. 72 (1987), 299-310. LIU, K.M., AND ORTIZ, E.L.: 'Numerical solution of ordinary and partial functional-differential eigenvalue problems with the tau method', Computing 41 (1989), 205-217.

[2o]

398

T A Y L O R J O I N T S P E C T R U M

[21] LUKE, Y.L.: The special functions and their approximations l-II, New York, 1969.

[22] NAVASIMAYAN, S., AND ORTIZ, E.L.: 'Best approximation and the numerical solution of partial differential equations with the tau method', Portugal. Math. 41 (1985), 97-119.

[23] ONUMANYI, P., AND ORTIZ, E.L.: 'Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method', Math. Comput. 43 (1984), 189-203.

[24] ORTIZ, E.L.: 'The tau method', SIAM J. Numer. Anal. 6 (1969), 480--492.

[25] ORTm, E.L.: 'On the numerical solution of nonlinear and functional differential equations with the tau method', in R. ANSORGE AND W. ThRmC (eds.): Numerical Treatment of Differential Equations in Applications, Berlin, 1978, pp. 127- 139.

[26] ORTIZ, E.L., AND PHAM NGOC DINH, A.: 'Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the tau method', SIAM J. Math. Anal. 18 (1987), 452-464.

[27] ORTIZ, E.L., AND SAMARA, H.: 'An operational approach to the tau method for the numerical solution of nonlinear differential equations', Computing 27 (1981), 15-25.

[28] OaTm, E.L., AND SAMARA, H.: 'Numerical solution of partial differential equations with variable coefficients with an operational approach to the tau method', Computers Math. Appl. 10, no. 1 (1984), 5-13.

[29] PUN, K.S., AND ORTm, E.L.: 'A bidimensional tau-elements method for the numerical solution of nonlinear partial differential equations, with an application to Burgers equation', Computers Math. Appl. 12B (1986), 1225-1240.

[30] RIVLIN, T.J.: The Chebyshev polynomials, New York, 1974, 2nd. ed. 1990.

[31] WRICHT, K.: 'Some relationships between implicit Runge- Kutta, collocation and Lanczos tau methods', BIT 10 (1970), 218-227.

Eduardo L. Ortiz M S C 1991: 65Lxx

T A Y L O R J O I N T S P E C T R U M - Let A = A[e] =

An[e] be the e x t e r i o r a l g e b r a on n generators

e l , . . . , e m with ident i ty e0 - 1. A is the algebra of

forms in e l , . . . , en with complex coefficients, subject to

the collapsing proper ty eiej + ejei = 0 (1 _< i, j < n).

Let E~: A --+ A denote the creat ion operator , given

by Ei~ := ei~ (~ • A, 1 _< i < n). If one de-

clares { e q , . . . , e i ~ : 1 < il < . . . < ik < n} to be an

or thonormal basis, the exterior algebra A becomes a

H i l b e r t space , admi t t ing an or thogonal decomposi t ion

A = ~Jk=lZ'~n A k, where dim A k = ( ; ) . Thus, each ~ • A ad- e t ~ t t 1 m r s a unique or thogonal decomposi t ion ~ = ~ +

where ~1 and ~" have no ei contribution. It then read-

ily follows tha t E*~ = ~ . Indeed, each Ei is a part ial

isometry, satisfying E~Ej + E j E [ = 5ij (1 _< i , j < n).

Let X be a n o r m e d s p a c e , let A =- ( A 1 , . . . , An) be

a commut ing n- tuple of bounded operators on X" and

set A(X) := X ® c A. One defines DA: A(X) --+ A(X)

by DA := E lL1 Ai ® El. Clearly, D~ = 0, so Ran DA C_ Ker DA.

The commut ing n- tuple A is said to be non-singular

on X if R a n D A = K e r D A . The Taylor joint spectrum,

or simply the Taylor spectrum, of A on X is the set

aT (A, X) := {A • C ~ : A - A is s ingular}.

The decomposi t ion A = O~=1Ak gives rise to a

cochain complex K ( A , X) , the so-called K o s z u l c o m -

p l e x associated to A on A/, as follows:

on-1 z ) : 0 A°(X) 4 AN(X) -+ 0,

where D k denotes the restr ict ion of DA to the subspace

Ak(X). Thus,

a T ( A , X ) = {A • C ~ : K ( A - A,X) is not exac t} .

J.L. Taylor showed in [18] t ha t if X is a B a n a c h

s p a c e , then aT(A, 2() is compact , non-empty, and con-

tained in a t(A), the (joint) algebraic spec t rum of A (cf.

also S p e c t r u m o f a n o p e r a t o r ) with respect to the

commutant of A, (A)' := {B • £ ( X ) : B A = A B } .

Moreover, aT carries an analyt ic f u n c t i o n a l c a l c u l u s

with values in the double c o m m u t a n t of A, so that , in

part icular , aT possesses the projec t ion property.

Example: n = 1. For n = 1, DA admits the following

(2 x 2)-matr ix relative to the direct sum decomposi t ion

(z ® e0) • (x ®

00) Then Ker D A / R a n DA = Ker A® ( X / R a n A). It follows

at once tha t aT agrees with ~, the spec t rum of A.

Example: n = 2. For n = 2,

DA = A1 0 0 2 0 0 '

- A 2 A1

so K e r D A / R a n D A = (KerA1 N KerA2) ®

{ (x l , x2 ) : A2xl = A l x 2 } / { ( A l x o , A 2 x o ) : x0 e X} (9

( X / ( R a n A1 + Ran A2)).

Note tha t since aT is defined in terms of the actions

of the opera tors Ai on vectors of X, it is intrinsically

'spat ial ' , as opposed to a I, a " and other algebraic joint

spectra, aT contains o ther well-known spatial spectra,

like ap (the point spect rum), a~ (the approximate point

spect rum) and a5 (the defect spectrum). Moreover, if

/3 is a commuta t ive Banach algebra, a -= ( a l , . . . , a , 0 ,

with each ai E /3, and L~ denotes the n- tuple of left

mult ipl ications by the ais, it is not hard to show tha t

aT (L~,/3) = a• (a). As a mat te r of fact, the same result

holds w h e n / 3 is not commuta t ive , provided all the ais

come from the centre of/3.

Spectral permanence. W h e n / 3 is a C*-algebra, s ay /3 C

£(7-0, then aT(La, B) = aT(a, 7- 0 [5]. This fact, known as spectral permanence for the Taylor spectrum, shows

399

TAYLOR JOINT SPECTRUM

that for C*-algebra elements (and also for Hilbert space operators), the non-singularity of La is equivalent to the invertibility of the associated Dirac operator Da + D t . .

Finite-dimensional ease. When dim A" < oc,

0.p = 0" 1 = 0-7r -= 0-5 = G-r = 0"T = 0-1 = 0-H = ~ ,

where 0-1, 0"r and ~ denote the left, right and polyno- mially convex spectra, respectively. As a mat ter of fact,

in this case the commuting n-tuple A can be simultane- / (k) ~dim W ously triangularized as Ak = ~ai, j h,j=l , and

0-T (A, X) ~" (a!~) -(~)' } = (~ ~ , ' " , u i i J: l < i < d i m X .

Case of compact operators. If A is a commuting n-tuple of compact operators acting on a Banach space 32, then 0-T(A, 2() is countable, with (0 , . . . , 0 ) as the only ac- cumulation point. Moreover, a . (A, 2() = 0.5(A,X) = 0-T (A, X). Invariant subspaces. If 2( is a Banach space, Y is a closed subspace of X and A is a commuting n-tuple leaving y invariant, then the union of any two of the sets (7 T (A, ,-32'),

0-T(A,Y) and aT(A, X / y ) contains the third [18]. This can be seen by looking at the long cohomology sequence associated to the Koszul complex and the canonical

short exact sequence 0 --+ J; --~ 2( --+ 2(/32 -+ O.

Additional properties. In addition to the above- mentioned properties of O-T, the following facts can be found in the survey article [6] and the references therein:

i) 0-T gives rise to a compact non-empty subset

M~ T (B, W) of the maximal ideal space of any commutative Banach algebra B containing A, in such a way that

0.T(A, Z) = .4(M~T (~ , W)) [18]; ii) for n = 2, 00.T(A,7/) C c90.H(A,7/), where ~H :=

0-10 0"r denotes the Harte spectrum; iii) the upper semi-continuity of separate parts holds

for the Taylor spectrum;

iv) every isolated point in 0-B(A) is an isolated point of 0-T (A, 7/) (and, afort iori , an isolated point of al (A, 7/) N

O'r (A, 7 / ) ) ;

v) if 0 C 0.T(A, 7-t), up to approximate unitary equivalence one can always assume that Ran DA ~ Ker DA [7];

vi) the functional calculus introduced by Taylor in [17] admits a concrete realization in terms of the Bochner- Martinelli kernel (cf. B o e h n e r - M a r t i n e l l i represen-

t a t i o n fo rmula ) in case A acts on a Hilbert space or on a C*-a lgebra [20];

vii) M. Putinar established in [13] the uniqueness of

the functional calculus, provided it extends the polynomial calculus.

Fredholm n-tuples. In a way entirely similar to the development of Fredholm theory, one can define the notion of Fredholm n-tuple: a commuting n-tuple A is said

to be Fredholm on X if the associated Koszul complex K(A, 2() has finite-dimensional cohomology spaces. The Taylor essential spectrum of A on A ~ is then

0-Te(A, 2() := {A C C n : A - A is not Fredholm}.

The Fredholm index of A is defined as the E u l e r c h a r a c t e r i s t i c of K ( A , X ) . For example, if n = 2, index(A) = d i m K e r D ° - dim(Ker D1A/RanD °) +

d i m ( X / R a n D y ) . In a Hilbert space, o-we(A,7/) = 0"T(La, Q(7/)), where a := 7r(A) is the coset of A in the Calkin algebra for 7/.

Example. If 7/ = H2(S 3) and Ai := Mz, (i = 1,2),

then 0-1(A) = 0-1e(A) = 0-re(A) = 0"Te(A) = S 3, 0"r(A) =

0.T(A) = B4, and index(A - A) = 1 (A ¢ B4). The Taylor spectral and Fredholm theories of multi-

plication operators acting on Bergman spaces over Rein-

hardt domains or bounded pseudo-convex domains, or acting on the Hardy spaces over the Shilov boundary of bounded symmetric domains on several complex variables, have been described in [3], [4], [8], [9], [10], [15], [16], [19], and [21]; for Toeplitz operators with H °° symbols acting on bounded pseudo-convex domains, concrete descriptions appear in [11].

Spectral inclusion. If S is a subnormal n-tuple acting on 7/ with minimal normal extension N acting on ]C (cf. also N o r m a l o p e r a t o r ) , 0.T(N,]C) _C 0-T(S, 7/) C_ ~(N, K)[14].

Left and right multiplications. For A and B two commuting n-tuples of operators on a Hilbert space 7/, and

LA and RB the associated n-tuples of left and right multiplication operators [7],

0-T((LA, RB ), £(7/)) = 0.T( A, 7/) X 0.T( B, 7/),

and

0-Te((LA, RB), £(7/)) =

= [aTe(A,7/) X 0.T(B,7/)] U [0-T(A,7/) X 0"Te(B,7/)].

During the 1980s and 1990s, Taylor spectral theory has received considerable attention; for further details and information, see [2], [11], [20], [6], [1]. There is also a parallel 'local spectral theory' , described in [11], [12] and [20]. R e f e r e n c e s

[1] ALBaEeHT, E., aND VASILESCU, F.-H.: 'Semi-Fredholm complexes' , Oper. Th. Adv. Appl. 11 (1983), 15-39.

[2] AMBROZIE, C.-C-., AND VASILESCU, F.-H.: Banach space com-

plexes, Kluwer Acad. Publ . , 1995. [3] BERGER, C., AND COBURN, L.: 'Wiener Hopf operators on

U2', Integral Eq. Oper. Th. 2 (1979), 139 173. [4] BERGER, C., COBURN, L., AND KORANYI, A.: 'Opfirateurs

de Wiene r -Hopf sur les spheres de Lie', C.R. Acad. Sci.

Paris Sdr. A 290 (1980), 989-991. [5] CURTO, R.: 'Spectral permanence for joint spectra ' ,

Trans. Amer. Math. Soc. 270 (1982), 659-665.

400

T H E O D O R S E N I N T E G R A L E Q U A T I O N

[6] CURTO, R.: 'Applications of several complex variables to multiparameter spectral theory', in J.B. CONWAY AND B.B. MORREL (eds.): Surveys of Some Recent Results in Operator Theory II, Vol. 192 of Pitman Res. Notes in Math., Longman Sci. Tech., 1988, pp. 25-90.

[7] CURTO, R., AND FIALKOW, L.: 'The spectral picture of (LA ,RB) ' , J. Funct. Anal. 71 (1987), 371-392.

[8] CURTO, R., AND MUHLY, P.: 'C*-algebras of multiplication operators on Bergman spaces', J. Funct. Anal. 64 (1985), 315-329.

[9] CURTO, R., AND SALINAS, N.: 'Spectral properties of cyclic subnormal m-tuples', Amer. J. Math. 107 (1985), 113-138.

[10] CURTO, R., AND VAN, K.: 'The spectral picture of Reinhardt measures', J. Funct. Anal. 131 (1995), 279-301.

[11] ESCHMEIER, J., AND PUTINAR, M.: Spectral decompositions and analytic sheaves, London Math. Soc. Monographs. Ox- ford Sci. Publ., 1996.

[12] LAURSEN, K., AND NEUMANN, M.: Introduction to local spectral theory, London Math. Soc. Monographs. Oxford Univ. Press, 2000.

[13] PUTINAR, M.: 'Uniqueness of Taylor's functional calculus', Proc. Amer. Math. Soc. 89 (1983), 647-650.

[14] PUTINAR, M.: 'Spectral inclusion for subnormal n-tuples', Proc. Amer. Math. Soc. 90 (1984), 405 406.

[15] SALINAS, N.: 'The cg-formalism and the C*-algebra of the Bergman n-tuple', J. Oper. Th. 22 (1989), 325 343.

[16] SALINAS, N., SHEU~ A., AND UPMEIER, H.: 'Toeplitz operators on pseudoconvex domains and foliation C*-algebras', Ann. of Math. 130 (1989), 531 565.

[17] TAYLOR, J.L.: 'The analytic functional calculus for several commuting operators', Acta Math. 125 (1970), 1-48.

[18] TAYLOR, J.L.: 'A joint spectrum for several commuting operators', g. Funct. Anal. 6 (1970), 172-191.

[19] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric domains', Ann. of Math. 119 (1984), 549-576.

[20] VASlLESOU, F.-H.: Analytic functional calculus and spectral decompositions, Reidel, 1982.

[21] VENUGOPALKRISHNA, U.: 'Fredholm operators associated with strongly pseudoconvex domains in C '~', Y. Funct. Anal. 9 (1972), 349 373.

Ragl E. Curto

M S C 1991: 47Dxx

T A Y L O R T H E O R E M - One of several results,

of which the most impor tan t is the T a y l o r f o r m u l a

and its various generalizations, e.g., to wider function

classes, to a s tochast ic sett ing or to multiple centres (in

which case one deals with interpolat ion- type formulas).

M S C 1991: 41A05, 41A58

T H E O D O R S E N I N T E G R A L E Q U A T I O N -

T h e o d o r s e n ' s integral equat ion [7] is a well-known tool

for comput ing numerical ly the c o n f o r m a l m a p p i n g g

of the unit disc D onto a star-like region A given by

the polar coordinates r , p( r ) of its boundary F. The

mapping g is assumed to be normalized by g(0) = 0,

g '(0) > 0. It is uniquely determined by its boundary

correspondence funct ion 0, which is implicitly defined

by

g(e it) = p (0(t)) e (vt c R),

/o ~ O(t) = dt 2~ 2 "

Theodorsen ' s equat ion follows f rom the fact t h a t the

funct ion h ( w ) := l o g ( g ( w ) / w ) is analyt ic in D and can

b e extended to a h o m e o m o r p h i s m of the closure D

o n t o the closure A. I t s imply s tates t ha t the 2~r-periodic

funct ion y: t ~-~ 0 - t is the conjugate periodic funct ion

of x: t ~ logp(O( t ) ) , t ha t is, y = K x , where I4 is the

conjugat ion opera tor defined on L[0, 21r] by the principal

value integral

( K x ) ( t ) := P.V. x ( s ) cot t - s ds (a.e.).

W h e n restr icted to L2[0, 2rF], K is a skew-symmetr ic en-

domorph i sm of no rm 1 with a very simple diagonal rep-

resentat ion in Fourier space: when x has the real Fourier

coefficients a o , a l , . . , b l , b 2 , . . . , then y has the coem-

cients 0, - b l , -b2 , .., a l , a2 , . . . .

Hence, while Theodorsen ' s integral equation is nor-

mal ly wri t ten as

_~ /o 21r t - - 8 o(t) - t = P.V. logp(0(s) ) cot - g - d, ,

for pract ical purposes the conjugat ion is executed by

t rans format ion to Fourier space: x is approximated

by a t r i g o n o m e t r i c p o l y n o m i a l of degree N , whose

Fourier coefficients are quickly found by the fast Fourier

t ransform, which then can also be applied to determine

values at 2 N equi-spaced points of the t r igonometr ic

polynomial tha t approximates y = K x (cf. also F o u r i e r

se r ies ) . Before the fast Fourier t ransform became the

s t anda rd tool for this discrete conjugat ion process, the

t ransi t ion from the values of z to those of y was based

on mult ipl icat ion by a matr ix, called the Wit t ich mat r ix

in [1]. The fast Fourier t rans form meant a cost reduct ion

f rom O ( N 2) to O ( N log N) operat ions per i teration.

Until the end of the 1970s the recommendat ion was

to solve a so-obta ined discrete version of Theodorsen ' s

equat ion by fixed-point (Picard) iteration, an approach

tha t is limited to Jo rdan regions with piecewise differ-

entiable bounda ry satisfying IP'/Pl < 1, and is very slow

when the bound 1 is nearly at tained. Other regions, like

those from airfoil design, which was the s tandard ap-

plication ta rge ted by T. Theodorsen, could be handled

by using first a suitable prel iminary conformal mapping,

which turned the exterior of the wing cross-section into

the exterior of a Jo rdan curve tha t is close to a circle;

see [6, Chapt . 10]. Moreover, for this application, the

equat ion has to be modified slightly to map the exterior

of the disc onto the exterior of a Jo rdan curve.

401

T H E O D O R S E N I N T E G R A L E Q U A T I O N

M. Gutknech t [2], [3] extended the applicability of

Theodorsen ' s equat ion by applying more refined itera-

tive me thods and discretizations, and O. H/ibner [5] im-

proved the convergence order from linear to quadrat ic by

adap t ing R. Wegmann ' s t r ea tment of a similar equat ion

obta ined by choosing h(w) := g ( w ) / w instead. Weg-

mann ' s me thod [9], [10] applies the N e w t o n m e t h o d

and solves the linear equat ion for the corrections by in-

terpre t ing it as a R i e m a n n - H i l b e r t p r o b l e m tha t

can be solved with four fast Fourier t ransforms.

A c o m m o n framework for conformal mapping meth-

ods based on function conjugat ion is given in [4];

Theodorsen ' s restr ict ion to regions given in polar co-

ordinates can be lifted. Both Theodorsen ' s [8] and Weg-

mann ' s [11] equat ions and methods can be extended to

the doubly connected case.

R e f e r e n c e s [1] GAIER, D.: Konstruktive Methoden der konformen Abbil-

dung, Springer, 1964. [2] GUTKNECHT, M.H.: 'Solving Theodorsen's integral equation

for conformal maps with the fast Fourier transform and various nonlinear iterative methods', Numer. Math. 36 (1981), 405-429.

[3] GUTKNECHT, M.H.: 'Numerical experiments on solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods', SIAM a. Sci. Statist. Comput. 4 (1983), 1 30.

[4] GUTKNECHT, M.H.: 'Numerical conformal mapping methods based on function conjugation', J. Comput. Appl. Math. 14 (1986), 31-77.

[5] H/iBNEa, O.: 'The Newton method for solving the Theodorsen equation', a. Comput. Appl. Math. 14 (1986), 19-30.

[6] KYTHE, P.K.: Computational conformal mapping, Birkhguser, 1998.

[7] THEODORSEN, T.: 'Theory of wing sections of arbitrary shape', Rept. NACA 411 (1931).

[8] THEODORSEN, T., AND GARRICK, I.E.: 'General potential theory of arbitrary wing sections', Rept. NACA 452 (1933).

[9] WEGMANN, R.: 'Ein Iterationsverfahren zur konformen Ab- bildung', Numer. Math. 30 (1978), 453-466.

[10] WEGMANN, R.: 'An iterative method for conformal mapping', J. Comput. Appl. Math. 14 (1986), 7-18, English translation of [9]. (In German.)

[11] WEGMANN, R.: 'An iterative method for the conformal mapping of doubly connected regions', J. Comput. Appl. Math. 14 (1986), 79-98.

Martin H. Gutknecht M S C 1991: 30C20, 30C30

THIELE DIFFERENTIAL E Q U A T I O N - Consider

an n year te rm life insurance, with sum insured S and

level premium P per t ime unit, issued at t ime 0 to an

x years old person. Denote by py the force of mor ta l i ty

at age y and by d the force of interest. If the insured is

still alive at t ime t E [0, n), then the insurer mus t pro-

vide a reserve, Vt, which by s ta tu te is the mean value of

future discounted benefits less premiums. Split t ing into

payments before and after t ime t + dt leads to

Vt = #x+t dt S - P dt+ (1)

+(1 - #x+t dt)e -~ atvt+at + o(dt),

from which one obtains t h a t Vt is the solution to

d v , ~/ ~ = P + ~ v t - ~ x + ~ ( s - v d , (2)

subject to the condit ion V~ = 0.

This is the celebrated Thiele differential equation,

proclaimed ' the fundament of modern life insurance

mathemat ics ' in the au thor i t a t ive t e x t b o o k [1], and

named after its inventor Th.N. Thiele (1838-1910). It

dates back to 1875, bu t was published only in 1910 in

the obi tuary on Thiele by J.P. G r a m [2], and appeared

in a scientific text [7] only in 1913.

As is apparent f rom the p roof sketched in [1], Thiele 's

differential equat ion is a simple example of a Kol-

mogorov backward equat ion (cf. K o l m o g o r o v e q u a -

t i on ) , which is a basic tool for determining condit ional

expected values in intensi ty-driven Markov processes.

Thus, today there exist Thiele differential equations for

a variety of life insurance p roduc t s described by multi-

s tate Markov processes and for various aspects of the

discounted payments , e.g. higher order moments and

probabi l i ty distr ibutions. The technique is an indispens-

able construct ive device in theoret ical and practical life

insurance mathemat ics and also forms the basis for numerical procedures, see [8].

Thiele was Professor of As t ronomy at the Universi ty

of Copenhagen from 1875, cofounder and Director (ac-

tuary) of the Danish life insurance company Hafnia from

1872, and first president of the Danish Actuarial Soci-

ety founded in 1901. In 52 wri t ten works (three mono-

graphs; [11], [12], [13]) he made contr ibut ions (a number

of them fundamenta l ) to astronomy, mathemat ica l sta-

tistics, numerical analysis, and actuarial mathemat ics .

Biographical /b ibl iographical accounts are given in [3],

[4], [51, [6], [9], [10].

R e f e r e n c e s [1] BERCER, A.: Mathematik der Lebensversicherung, Springer

Wien, 1939. [2] GRAM, J.P.: 'Professor Thiele sore aktuar', Dansk For-

sikringsdrbog (1910), 26-37. [3] HALD, A.: 'T.N. Thiele's contributions to statistics', Internat.

Statist. Rev. 49 (1981), 1-20. [4] HALD, A.: A history of mathematical statistics from 1750 to

1930, Wiley, 1998. [5] HOEM, J.M.: 'The reticent trio: Some little-known early dis-

coveries in insurance mathematics by L.H.F. Oppermann, T.N. Thiele, and J.P. Gram', Internat. Statist. Bey. 51 (1983), 213-221.

[6] JOHNSON, N.L., AND KOTZ, S. (eds.): Leading personalities in statistical science, Wiley, 1997.

[7] JORGENSEN, N.R.: Grundz@e einer Theorie der Lebensver- sicherung, G. Fischer, 1913.

402

TILTED ALGEBRA

[8] NORBERG, R.: 'Reserves in life and pension insurance', Scan& Actuarial d. (1991), 1-22.

[9] NORBERG, R.: Thorvald Nicolai Thiele, statisticians of the centuries, Internat. Statist. Inst., 2001.

[10] SCHWEDER, W.: 'Scandinavian statistics, some early lines of development', Scan& J. Statist. 7" (1980), 113-129.

[11] THIELE, T.N.: Element~er Iagttagelseslaere, Gyldendal, Copenhagen, 1897.

[12] THIELE, T.N.: Theory of observations, Layton, London, 1903, Reprinted in: Ann. Statist. 2 (1931), 165-308. (Translated from the Danish edition 1897.)

[13] THIELE, T.N.: Interpolationsrechnung, Teubner, 1909.

Ragnar Norberg

MSC 1991:62P05

T H O M - M A T I I E R STRATIFICATION - A s t r a t i f i -

c a t i o n of a space such that each stratum has a neigh-

bourhood which fibres over that stratum, with levels defined by a tubular function (called 'fonction tapis' in Thorn's and 'distance function' in Mather's terminoI- ogy), and the fibrations and tubular functions associated to the s trata are compatible with each other. Thorn Mather stratifications satisfy the Thorn first and second isotopy lemmas (see below), providing results such as local topological triviality of the stratification, local topological triviality along the strata of a morphism and topological stability of generic smooth mappings ('generic' meaning transverse to the natural stratifies- tion of the jet space).

The word 'stratification' has been introduced by R. Thorn in [5]. He proposed regularity conditions on how the strata of a stratification should fit together and stated the isotopy lemmas. The notes [4] of J. Mather provide a detailed proof, with improvements and sire- plifications (cf. [2], which contains an excellent history of stratification theory).

A Thom-Mathcr stratification of a space M consists

of a tube system (Tx, 7Cx, px ) associated to the strata X of M, such that Tx is a t u b u l a r n e i g h b o u r h o o d

of X in M, 7rx : Tx --+ X is the fibre projection associated to Tx and the tubular function Px : Tx -+ R is a continuous mapping satisfying p} 1 (0) = X. These data are controlled in the following sense: If X and Y are two strata such that X is in the frontier of Y, then

i) the restriction mapping (zcx, Px) : Tx n Y --+ X x ]0, ec[ is a smooth s u b m e r s i o n ;

ii) for a E Tx N Ty such that Try (a) C Tx, there are commutation relations

C1) ~rx o Try(a) = rex(a),

02) px o ~ y (a) = ~ x (a) whenever both sides of the formulas are defined.

Thom-Mather stratifications satisfy the isotopy lem-

mas (as proposed by Thom):

1) For every surjective stratified morphism f : M N, the restriction of f to the inverse image f - 1 (S) of a s tratum S is a f i b r a t i o n .

2) If there is a sequence of stratified morphisms M N 2~ I, where f is a Thorn mapping (an 'application sans ficlatement') and I is a segment, then the map-

pings fa and fb over two points a and b in I have the same topological type, i.e. there are homeomorphisms h and h' such that the following diagram commutes:

M~ h M6

N~ -~ Nb h'

The importance of Thom-Mathe r stratifications is emphasized by their applications to stability and topological triviality theorems. Among other important results in singularity theory is the fact that any Whitney stratification (see S t r a t i f i c a t i o n ) is a Thom-Mather

stratification. Hence, a Whitney stratification satisfies topological triviality. The converse is false [1]; in fact, being a Whitney stratification is equivalent to topological triviality for all sections by a generic flag [3].

R e f e r e n c e s [1] BRIAN~ON, J., AND SPEDER, J.P.: 'La trivialit~ topologique

n'implique pas les conditions de Whitney', Note C.R. Acad. Sci. Paris Ser. A 280 (1975), 365.

[2] GORESKY, M., AND MACPHEHSON, R.: Stratified Morse theory, Springer, 1988.

[3] Lg, D.T., AND TEISSIER, B.: 'Cycles fivanescents, sections planes et conditions de Whitney II': Proe. Syrup. Pure Math., Vol. 40, Amer. Math. Soc., 1983, pp. 65-103.

[4] MATHER, J.: Notes on topological stability, Harvard Univ., 1970.

[5] THOU, R.: 'La stabilit~ topologique des applications polyno- miales', Enseign. Math. 8, no. 2 (1962), 24 33.

[6] THOU, R.: 'Ensembles et morphismes stratifies', Bull. Amer. Math. Soc. 75 (1969), 240-284.

[7] WHITNEY, H.: 'Local properties of analytic varieties', in

S. CAIRNS (ed.): Differential and Combinatorial Topology, Princeton Univ. Press, 1965, pp. 205-244.

[8] WHITNEY, H.: 'Tangents to an analytic variety', Ann. of Math. 81 (1965), 496-549.

Jean-Paul Brasselet MSC 1991:57N80

TILTED AL G E B RA - The endomorphism ring of a t i l t i n g m o d u l e over a finite-dimensional hereditary algebra (cf. also A lg eb ra ; E n d o m o r p h i s m ) .

Let H be a finite-dimensional hereditary K-algebra, K some field, for example the path-algebra of some finite q u i v e r without oriented cycles. A finite-dimensional H- module HT is called a tilting module if

i) p. d i m T < 1, which always is satisfied in this context;

ii) E x t ~ ( T , T ) = 0; and

403

TILTED ALGEBRA

iii) there exists a short exac t s e q u e n c e 0 --+ H -+ Tz --+ T.2 -+ 0 with r l and T~ in add T, the category of finite direct sums of direct summands of T. Here, p. dim is projective dimension.

The third condition also says that T is maximal with respect to the property E x t , ( T , T) = 0. Note further, that a tilting module T over a hereditary algebra is uniquely determined by its composition factors. Cf. also T i l t i n g

m o d u l e .

The algebra B = EndH(T) is called a tilted algebra of type H, and T becomes an (H, B)-bimodule (cf. also B i m o d u l e ) .

In H-mod, the c a t e g o r y of finite-dimensional H- modules, the module T defines a torsion pair (G,$-) with torsion class G consisting of modules, generated by T and torsion-free class • = {Y: HomH(T,Y) = 0}. In B-mod it defines the torsion pair (X,3;) with torsion class 2( = {Y: T ®B Y = 0} and torsion-free class ~2 = {Y: TorB(T,Y) = 0}. The Brenner-Butler theorem says that the functors H o m H ( T , - ) , respectively T ®B --, induce equivalences between G and J;, whereas E x t f / ( T , - ) , respectively T o r B ( T , - ) , induce

equivalences between )c and X. In B-rood the torsion pair is splitting, that is, any indecomposable B-module is either torsion or torsion-free. In this sense, B-mod has 'less' indecomposable modules, and information on the category H-mod can be transferred to B-mod.

For example, B has global dimension at most 2 and any indecomposable B-module has projective dimension or injective dimension at most 1 (cf. also D i m e n s i o n for dimension notions). These condition characterize the

more general class of quasi-tilted algebras. The indecomposable injective H-modules are in the

torsion class ~ and their images under the t i l t i ng f u n c -

t o r HomH (T, - ) are contained in one connected component of the Auslander Reiten quiver F(B) of B-rood (cf. also Quiver ; R i e d t m a n n c lass i f ica t ion) , and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of F(B) characterizes tilted algebras. Moreover, the Auslander-Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components.

If H is a basic hereditary algebra and He is a simple projective module, then T = H(1 - e) ® TrD He, where TrD denotes the Auslander-Reiten translation (cf. R i e d t m a n n class i f ica t ion) , is a tilting module, sometimes called APR-tilting module. The induced torsion pair (G,¢-) in H-rood is splitting and He is the unique indecomposable module in F. The tilting functor H o m H ( T , - ) corresponds to the reflection functor introduced by I.N. BernshteYn, I.M. Gel'land and V.A. Ponomarev for their proof of the Gabriel theorem [3].

If the hereditary algebra H is representation-finite (cf. also A l g e b r a o f f in i te r e p r e s e n t a t i o n t y p e ) , then any tilted algebra of type H also is representation- finite. If H is tame (cf. also R e p r e s e n t a t i o n o f an a s soc ia t ive a lgebra ) , then a tilted algebra of type H either is tame and one-parametric, or representation- finite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre- injective direct summands simultaneously. If H is wild (cf. also R e p r e s e n t a t i o n o f an a s so c i a t i v e a lgebra ) , then a tilted algebra of type H may be wild, or tame

domestic, or representation-finite.

See also T i l t i n g t h e o r y .

R e f e r e n c e s [1] ASSEM, I.: 'Tilt ing theory - an introduction' , in N. BAL-

CERZYK ET AL. (eds.): Topics in Algebra, Vol. 26, Banach Center Publ., 1990, pp. 127-180.

[2] AUSLANDER, M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter functors without diagrams' , Trans. Amer. Math. Soc. 250 (1979), 1-46.

[3] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAROW, V.A.: 'Coxeter functors and Gabriel 's theorem', Russian Math. Surveys 28 (1973), 17-32.

[4] BONGARTZ, K.: 'Tilted algebras', in M. AUSLANDER AND E. LLUIS (eds.): Representations of Algebras. Proc. ICRA III, Vol. 903 of Lecture Notes in Mathematics, Springer, 1981, pp. 26 38.

[5] BRENNER, S., AND BUTLER, M.: 'Generalizations of the Bernste in-Gelfand-Ponomarev reflection functors', in V. DLAB AND P. GABRIEL (eds.): Representation Theory II. Proc. ICRA II, Vol. 832 of Lecture Notes in Mathematics,

Springer, 1980, pp. 103 169. [6] HAPPEL, D.: Triangulated categories in the representation

theory of finite dimensional algebras, Vol. 119 of London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1988.

[7] HAPPEL, D., REITEN, I., AND SMAL0, S.O.: 'Tilting in abelian categories and quasitilted algebras', Memoirs Amer. Math.

Soc. 575 (1996). [8] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras',

Trans. Amer. Math. Soc. 274 (1982), 399-443. [9] KERNER, O.: 'Til t ing wild algebras', J. London Math. Soc.

39, no. 2 (1989), 29-47. [10] KERNER, O.: 'Wild tilted algebras revisited', Colloq. Math.

73 (1997), 67-81. [11] LIu, S.: 'The connected components of the Auslander-Reiten

quiver of a tilted algebra', J. Algebra 161 (1993), 505-523. [12] RINGEL, C.M.: Tame algebras and integral quadratic forms,

Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984. [13] RINGEL, C.M.: 'The regular components of the Auslander-

Reiten Quiver of a til ted algebra', Chinese Ann. Math. Set. B. 9 (1988), 1-18.

[14] STRAUSS, H.: 'On the perpendicular category of a partial tilting module', J. Algebra 144 (1991), 43-66.

O. K e r n e r

MSC 1991: 16G10, 16G20, 16G60, 16G70

TILTING F U N C T O R - When studying an a l g e b r a A, it is sometimes convenient to consider another algebra, given for instance by the endomorphism of an

404

TILTING THEORY

appropriate A-module, and functors between the two module categories. For instance, this is the basis of the M o r i t a equ iva lence or the construction of the so- called Auslander algebras. An important example of this strategy is given by the t i l t ing t h e o r y and the tilting functors, as now described.

Let A be a finite-dimensional k-algebra, where k is a field, T a tilting (finitely-generated) A-module (cf. Til t ing m o d u l e ) and B = EndA(T). One can then as- sign to T the functors HomA ( T , - ) , - ® B T, Ext~ ( T , - ) , and TOrl B ( - , T), which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categories mod A and rood B, results first established by S. Brenner and M.C.R. Butler. Namely, HomA(T, - ) and its adjoint - ®B T give an equivalence between the subcategories

T(TA) = {MA: Ext l (T , M) = 0}

and Y(TA) = {NB: TorB(N,T) = 0},

while Ext}4(T , - ) and TorB( - ,T ) give an equivalence between the subcategories

Y(TA) = {NB: Tor~(N,T) = O}

and

X(TA) = {NB: N ® , T = 0}.

It is not difficult to see that (T(TA),5(TA)) and (X(TA), Y(TA)) are torsion pairs in rood A and rood B, respectively. Clearly, one can now transfer information from rood A to rood B. One of the most interesting cases occurs when A is a hereditary algebra and so the tot- sion pair (X(TA), Y(TA)) splits, giving in particular that each indecomposable B-module is the image of an indecomposable A-module either by HomA(T, - ) or by E x t , ( T , - ) (in this case, the algebra B is called tilted, cf. also T i l t e d algebra) .

This procedure has been generalized in several ways and it is worthwhile mentioning, for instance, its connection with derived categories (cf. also Der ived category), or the notion of quasi-tilted algebras. It has also been considered for infinitely-generated modules over arbitrary rings.

For referenes, see also T i l t i ng theo ry ; T i l t ed algebra.

Fldvio Ulhoa CoeIho MSC 1991: 16Gxx

TILTING MODULE - A (classical) tilting module over a finite-dimensional k-algebra A (cf. also Algebra) , where k is a field, is a (finitely-generated) A-module T satisfying:

i) the projective d i m e n s i o n of T is at most one;

ii) Ext (T,T) = 0; and iii) the number of non-isomorphic indecomposable

summands of T equals the number of simple A-modules.

The fundamental work by S. Brenner and M.C.R. But- ler, and D. Happel and C.M. Ringel, on tilting theory have established the relations between the module categories rood A and rood B, where B = EndA(T), through the tilting functors HOmA (T, - ) and Ext~ (T, - ) (cf. also Til t ing func to r ) . The particular case where A is a hereditary algebra gives rise to the notion of a t i l t e d a lgebra , which nowadays (as of 2000) plays a very important role in the representation theory of algebras. One can also consider the dual notion of eotiltin9 rood- ules.

Til t ing t h e o r y goes back to the work of I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev on the characterization of representation-finite hereditary algebras through their ordinary quivers (cf. also Quiver) . Their reflection functors on quivers has led to a module-theoretical interpretation by M. Auslander, M.I. Platzeck and I. Reiten. Next steps in this theory are the work by Brenner and Butler and Happel and Ringel, which gave the basis for all its further development. Worthwhile mentioning is the connection of tilting theory with derived categories established by Happel (cf. also De r i v e d ca tegory) .

The success of this strategy to study a bigger class of algebras through tilting theory has led to several generalizations. On one hand, one can relax the condition on the projective dimension and consider tilting modules of finite projective dimension. In this way it was possible to show the connection between tilting theory and some other homological problems in the representation theory of algebras. On the other hand, this concept can be generalized to a so-called tilting object in more general Abelian categories. For instance, this has led to the notion of a quasPtilted algebra. Recently (as of 2000), there has been much work also on exploring such notions in categories of (not necessarily finitely-generated) modules over arbitrary rings.

For references, see also T i l t i ng theory ; T i l t ed algebra .

Fldvio Ulhoa Coelho MSC 1991: 16Gxx

TILTING THEORY-

Art in algebras . A finitely-generated m o d u l e T over an Artin algebra A (cf. also A r t i n i a n modu le ) is called a tilting module if p. dim A T _< 1 and Ext , (T, T) = 0 and there is a short exac t sequence 0 -+ A --+ To --> T1 ~ 0 with To,T1 C addT. Here, p. dimAT denotes the projective dimension of T and add T is the category

405

TILTING THEORY

of finite direct sums of direct summands of T (see T i l t -

ing m o d u l e ) . Dually, a A-module T is called a cotilting module if the A°P-module D(T) is a tilting module, where D denotes the usual duality. If T is a tilting rood- ule and F = End r (T ) °p, then T is a tilting module over F °p. Hence D(T) is a cotilting F-module.

Let T be a tilting module, and let T = F a c t be the c a t e g o r y of finitely-generated A-modules generated by T. The category T is a torsion class in the category modA of finitely-generated A-modules. This yields an associated torsion pair (T,~C), where • =

{C: HomA(T, C) = 0}. Dually, there is associated with a cotilting module T the subcategory y = Sub T of A- modules cogenerated by T. The category 3; is a torsion- free class and there is an associated torsion pair (2(, y )

where 2( = {C: HomA(C,Y) = 0}. An important feature of tilting theory is the follow-

ing connection between modA and modF when F = EndA(T) °p for a tilting module T: If (T, 2 r) denotes the

torsion pair in mod A associated with T and (2(, y ) the torsion pair associated with D(T), then there are equivalences of categories:

HomA(T, .): T --+ 32

and

Ext ,(T, .): f 2(.

(Cf. also T i l t i n g func to r . ) In the special case where T is a projective generator one recovers the M o r i t a

equ iva l e nc e HOmA (T, .) : rood A --+ mod F, where T is a projective generator of mod A. For a general module T, the Artin algebras A and F may be quite different, but they share many homological properties; in particular, one uses the tilting functors Homa(T, .) and Ext~ (T, .) in order to transfer properties between mod A and mod F. The transfer of information is especially use- ful when one already knows a lot about mod A and when the torsion pair (2(, y ) splits, that is, when each indecomposable F-module is in 2( or in y . This is the case when A is hereditary. In this case, F is called a tilted algebra (cf. also T i l t e d a lgebra ) . Tilted algebras have played an important part in representation theory, since many questions can be reduced to this class of algebras.

Tilting theory goes back to the reflection functors in-

troduced by I.N. BernshteYn, I.M. Gel'fand and V.A. Ponomarev [5] in the early 1970s. A module-theoretic

interpretation of these functors was given by M. Aus- lander, M.I. Platzeck and I. Reiten [3]. Further generalizations where given by S. Brenner and M.C.R. Butler [6], where the equivalence Homa(T, .): T -~ 3; was established. The above definitions where given by D. Hap- pel and C.M. Ringel [13], who developed an extensive theory of tilted algebras. A good reference for the early

work in tilting theory is [2].

An important theoretical development of tilting theory was the connection with derived categories established by Happel [10]. The functor HomA(T, .) : rood A --+ rood F when T is a tilting module induces an equivalence RHomA(T, - ) : Db(A) -+ Db(F), where Db(A) denotes the d e r i v e d c a t e g o r y whose objects are the bounded complexes of A-modules.

The set of all tilting modules (up to isomorphism) over a k-algebra A, k an a l g e b r a i c a l l y c losed field, has an interesting combinatorial structure: It is a countable s impl ic ia l c o m p l e x E. This complex has been investigated by L. Unger in [21] and [22], where it was proved that E is a shellable simplicial complex provided it is finite, and that certain representation-theoretical invariants are reflected by its structure.

A n a l o g u e s a n d g e n e r a l i z a t i o n s . There is an analo- gous concept of a tilting sheaf T for the category coh X of coherent sheaves of a weighted projective line X (cf. also C o h e r e n t sheaf ) as studied in [9]. The canonical algebras introduced in [19] can be realized as endomorphism algebras of certain tilting sheaves.

To obtain a common t reatment of both the class of tilted algebras and the canonical algebras, in [12] tilting theory was generalized to hereditary categories 7{, that is, 7{ is a connected Abelian k-category with van- ishing Yoneda functor Ext2( ., .) and finite-dimensional

homomorphism and extension spaces. Here, k denotes an algebraically closed field. An object T in 7{ with

E x t ~ ( T , T ) = 0 such that H o m ~ ( T , X ) = 0 = Ext~t(T,X ) implies X = 0, is called a tilting object in 7{. The endomorphism algebra End~ T of a tilting object T is called a quasi-tilted algebra. Tilted algebras and canonical algebras furnish examples for quasi-tilted algebras.

There are two types of hereditary categories 7-/with tilting objects: those derived equivalent to m o d H for some finite-dimensional hereditary k-algebra H and those derived equivalent to some category coh X of coherent sheaves on a weighted projective line X. Two categories are called derived equivalent if their derived categories are equivalent as triangulated categories. In 2000, Happel [11] proved that these are the only possible hereditary categories with tilting object. This proved a conjecture stated, for example, in [17].

Generalizations and applications of tilting modules. A A-module T is called a generalized tilting module if pd A T = n < ec and Ext~ (T, T) = 0 for i > 0 and there is an exact sequence 0 ~ A --+ T1 --+ ' . . --+ Tn --+ 0 with Ti C add T. Generalized tilting modules were introduced in [16]. This concept was generalized to the notion of tilting complexes by J. Rickard [18], who established some :Morita theory for derived categories'.

406

TITS QUADRATIC FORM

Let R be a ring and let PA be the category of finitely- generated projective A-modules. Denote by Kb(PA) the category of bounded complexes over PA modulo homo- topy. A complex T E Kb(pA) is called a tilting complex if Homgb(pA)(T,T[i]) = 0 for a l l / # 0 (here, [.] denotes the shift functor) and if addT generates Kb(pA) as a triangulated category. Rickard proved that two rings R and R ~ are derived equivalent (i.e. their module categories are derived equivalent) if and only if R ~ is the endomorphism ring of a tilting complex T E Kb(PA).

The results mentioned above uses tilting mod-

ules/objects mainly to compare modA and modF,

where F = EndA T for some tilting module/object. There are other approaches, which use tilting modules

to describe subcategories of mod A. Kerner [15] and W.

Crawley-Boevey and Kerner [7] used tilting modules to

investigate subcategories of regular modules over wild hereditary algebras.

Q u a s i - h e r e d i t a r y a lgebras . Auslander and Reiten [4] proved that there is a one-to-one correspondence between basic generalized tilting modules and certain co- variantly finite subcategories of rood A. This correspondence was further investigated [14]. The Auslander- Reiten correspondence was applied to quasi-hereditary algebras by Ringel [20] and his results served as a basis for applications to Schur algebras by S. Donkin [8] and to q u a n t u m groups by H.H. Andersen [1]. In dealing with quasi-hereditary algebras and highest-weight categories, the notion of a tilting module is now (2000) used in a related but deviating way, namely for all the objects or modules that have both a A-filtration and a V-filtration. The isomorphism classes of the indecom- posables that have both a A-filtration and a V-filtration correspond bijectively to the elements of the weight poset, and their direct sum is a tilting module in the sense considered above.

References

[i] ANDERSEN, H.H.: 'Tensor products of quantized tilting mod-

ules', Commun. Math. Phys. 149, no. 1 (1992), 149-159.

[2] ASSEM, I.: 'Tilting theory - an introduction': Topics in Alge- bra, Vol. 26 of Banach Center Publ., PWN, 1990, pp. 127-

180. [3] AUSLANDER, M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter

functors without diagrams', Trans. Amer. Math. Soe. 250 (1979), 1-12.

[4] AUSLANDER, M., AND REITEN, I.: 'Applications of contravari- antly finite subcategories', Adv. Math. 86, no. 1 (1991), 111- 152.

[5] BERNSTEIN, I.N., CELFAND, I.M., AND PONOMAREV, V.A.: 'Coxeter functors and Gabriel's theorem', Russian Math.

Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973), 19-33.)

[6] BRENNER, S., AND BUTLER, M.C.R.: 'Generalization of Bernstein-Gelfand-Ponomarev reflection functors': Proc. Ot- tawa Conf. on Representation Theory, 1979, Vol. 832 of Lec-

ture Notes in Mathematics, Springer, 1980, pp. 103-169. [7] CRAWLEY-BOEVEY, W., AND KERNER, O.: 'A functor between

categories of regular modules for wild hereditary algebras',

Math. Ann. 298 (1994), 481-487. [8] DONKIN, S.: 'On tilting modules for algebraic groups', Math.

Z. 212, no. 1 (1993), 39-60. [9] GEIGLE, W., AND LENZING, H.: 'Perpendicular categories

with applications to representations and sheaves', J. Algebra 144 (1991), 273 343.

[10] HAPPEL, D.: 'Triangulated categories in the representation theory of finite dimensional algebras', London Math. Soc. Lecture Notes 119 (1988).

[11] HAPPEL, D.: 'A characterization of hereditary categories with tilting object', preprint (2000).

[12] HAPPEL, D., REITEN, R., AND SMALO, S.O.: 'Tilting in abelian categories and quasitilted algebras', Memoirs Amer.

Math. Soc. 575 (1996). [13] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras',

Trans. Amer. Math. Soc. 274 (1982), 399-443. [14] HAPPEL, D., AND UNGER, L.: 'Modules of finite projective

dimension and cocovers', Math. Ann. 306 (1996), 445-457. [15] KERNER, O.: 'Tilting wild algebras', J. London Math. Soc.

39, no. 2 (1989), 29-47. [16] MIYASHITA, Y.: 'Tilting modules of finite projective dimen-

sion', Math. Z. 193 (1986), 113-146. [17] REITEN, I.: 'Tilting theory and quasitilted algebras': Proc.

Internat. Congress Math. Berlin, Vol. II, 1998, pp. 109-120. [18] RICKARD, J.: 'Morita theory for derived categories', J. Lon-

don Math. Soc. 39, no. 2 (1989), 436-456. [19] RINGEL, C.M.: 'The canonical algebras': Topics in Algebra,

Vol. 26:1 of Banach Center Publ., PWN, 1990, pp. 407-432. [20] RINGEL, C.M.: 'The category of modules with good filtra-

tion over a quasi-hereditary algebra has alost split sequences', Math. Z. 208 (1991), 209-224.

[21] UNGER, L.: 'The simplicial complex of tilting modules over quiver algebras', Proc. London Math. Soc. 73, no. 3 (1996), 27-46.

[22] UNGER, L.: 'Shellability ofsimplicial complexes arising in representation theory', Adv. Math. 144 (1999), 221-246.

L. Unger

MSC 1991: 16Gxx

TITS QUADRATIC F O R M - Let Q = (Qo, Q1) be a finite quiver (see [8]), that is, an oriented graph with vertex set Q0 and set Q1 of arrows (oriented edges; cf. also G r a p h , o r ien ted ; Quiver) . Following P. Gabriel [8], [9], the Tits quadratic form qQ: Z Qo -+ Z of Q is defined by the formula

2

jCQo i,jCQo

where x = (xi)icQo E Z Q° and diy is the number of arrows from i to j in Q1.

There are important applications of the Tits form in representation theory. One easily proves that if Q is connected, then qQ is positive definite if and only if Q (viewed as a non-oriented graph) is any of the Dynkin

407

TITS QUADRATIC FORM

diagrams An, D~, E6, ET, or Es (cf. also D y n k i n dia- g ram) . On the other hand, the Gabriel theorem [8] as- serts that this is the case if and only if Q has only finitely many isomorphism classes of indeeomposable K-linear representations, where K is an a lgebra ica l ly c losed field (see also [2]). Let rePK(Q ) be the A b e l i a n c a t e - g o r y of finite-dimensional K-linear representations of Q formed by the systems X = (Xi,¢9)jeQo,9~Q~ of finite-dimensional vector/(-spaces Xj, connected by K- linear mappings CZ : Xi --+ Xj corresponding to arrows /3: i --+ j of Q. By a theorem of L.A. Nazarova [12], given a connected quiver Q the category rePK(Q) is of tame representation type (see [7], [10], [19] and Quiver) if and only if qQ is positive semi-definite, or equivalently, if and only if Q (viewed as a non-oriented graph) is any of the extended Dynkin diagrams A~, Dn, E6, ET, or Es (see [1], [10], [19]; and [4] for a generalization).

Let Ko(Q) = K0(reptc(Q)) be the G r o t h e n d i e c k g roup of the category repK(Q ). By the J o r d a n - HSlde r t h e o r e m , the correspondence X dimX = (dimKXj)jeQo defines a group isomorphism dim: Ko(Q) -+ Z Q°. One shows that the Tits form qQ coincides with the E u l e r charac- t e r i s t i c XQ: Ko(Q) --+ Z, IX] ~ XQ([X]) = dimK EndQ(X) - dimN Ext~(X,X) , along the isomorphism dim: Ko(Q) --+ Z Q°, that is, qQ(dimX) = XQ([X]) for any X in repg(Q) (see [10], [17]).

The Tits quadratic form qQ is related with an algebra ic g e o m e t r y context defined as follows (see [9], [10], [19]).

For any vector v = (vj)jcQo E N Q°, consider the att:ine irreducible K-variety AQ(V) = I-Ii,jEQo H(j3:j--~i)CQ1 M ~ ×~j (K)z of K-representations of Q of the dimension type v (in the Zar iski topology), where M ~ ×vj (K)Z = M ~ x~j (K) is the space of (vi x vj)-matrices for any arrow fl : j -+ i of Q. Consider the a lgebraic g roup GgQ(d) = [IjcQo GI(vj,K) and the algebraic group action *: ~gQ(d) x AQ(d) --+ AQ(d) defined by the formula (hi) * (M~) = (h~-lM~hj),

where fl: j ~ i is an arrow of Q, M~ C Mvj×v~(K)9, hj E GI(vj,K), and hi C Gl(vi,K). An important role in applications is played by the Tits-type equality qQ(v) = dimGfQ(V) - dimAQ(v), v C N Q°, where dim denotes the d i m e n s i o n of the a lgebra ic va r i e ty (see [8]).

Following the above ideas, Yu.A. Drozd [5] introduced and successfully applied a Tits quadratic form in the study of finite representation type of the Krull-Schmidt category Mats of matrix K-representations of partially ordered sets ([, -<) with a unique maximal element (see [10], [19]). In [6] and [7] he also studied bimodule matrix problems and the representation type of boxes 13 by means of an associated Tits quadratic form qB : Z ~ ~ Z

(see also [18]). In particular, he showed [6] that if 13 is of tame representation type, then q~ is weakly non- negative, that is, q~(v) > 0 for all v C N ~.

K. Bongartz [3] associated with any finite- dimensional basic K-algebra R a Tits quadratic form as follows. Let { e l , . . . , en} be a complete set of prim- itive pairwise non-isomorphic orthogonal idempotents of the algebra R. Fix a finite quiver Q = (Qo,Q1) with Qo = { 1 , . . . , n } and a K-algebra isomorphism R ~- KQ/ I , where KQ is the path K-algebra of the quiver O (see [1], [10], [19]) and I is an ideal of R contained in the square of the J a c o b s o n rad ica l rad R of R and containing a power of rad R. Assume that Q has no oriented cycles (and hence the global dimension of R is finite). The Tits quadratic form qn : Z n -+ Z of R is defined by the formula

2 qR(x) = Z Z xixj+ Z r ,j i j, jCQo (~: i--+j)cQ1 (fl: i--+j)cQ1

where ri,j = IL M ejIei[, for a minimal set L of generators of I contained in ~i,j~Qo ejIei. One checks that

rid = dimK Ext~(Sj,Si), where St is the simple R- module associated to the vertex t E Q0. Then the def- inition of qR depends only on R, and when R is of global dimension at most two, the form qR coincides with the Euler characteristic XR: K0(modR) -+ Z, [X] ~ Xn([X]) = Y2~=0(-1) m dimg E x t , ( X , X), under a group isomorphism dim: K0(modR) -+ Z Q°, where K0(mod R) is the G r o t h e n d i e c k g roup of the category mod R of finite-dimensional right R-modules (see [17]). Note that qR = qQ if R = KQ.

By applying a Tits-type equality as above, Bongartz [3] proved that if R is of finite representation type, then qn is weakly positive, that is, qn(v) > 0 for all non-zero vectors v E N ~. The converse implication does not hold in general, but it has been established if the Auslander- Reiten quiver of R (see R i e d t m a n n classif icat ion) has a post-projective component (see [10]), by applying an idea of Drozd [5]. J.A. de la Pefia [14] proved that if R is of tame representation type, then qR is weakly non- negative. The converse implication does not hold in general, but it has been proved under a suitable assumption on R (see [13] and [16] for a discussion of this problem and relations between the Tits quadratic form and the Euler quadratic form of R).

Let (I, ~) be a partially ordered set with partial order relation -< and let max I be the set of all maximal elements of (I, __). Following [5] and [15], D. Simson [20] defined the Tits quadratic form q~: Z I --+ Z of (I, _-3) by the formula

iEI i~ j pCmax I jEI\max I

408

T O E P L I T Z C*-ALGEBRA

and applied it in the study of prinjective KI-modules, that is, finite-dimensional right modules X over the in-

cidence K-algebra K I = K(I , ~_) of (I,__) such that there is an exact sequence 0 -+ P1 -+ P0 ~ X -+ 0, where P0 is a projective KI-module and P1 is a direct sum of simple projectives. The additive Krull-Schmidt category p r i n K I of prinjective KI-modules is equivalent to the category of matrix K-representations of (I, _) [20]. Under an identification Ko(prinKI) ~- Z I, the Tits form qI is equal to the Euler characteristic XKZ: Ko(prin KI) -+ Z. A Tits-type equality is also valid for qr [15]. It has been proved in [20] that q1 is weakly positive if and only if prin K I has only a finite

number of iso-classes of indecomposable modules. By [15], if p r i n K / i s of tame representation type, then qI is weakly non-negative. The converse implication does not hold in general, but it has been proved under some assumption on (I, _) (see [11]).

A Tits quadratic form qA : Z n --+ Z for a class of classical D-orders A, where D is a complete discrete valu- ation domain, has been defined in [21]. Criteria for the finite lattice type and tame lattice type of A are given

in [21] by means of qA.

For a class of K-co-algebras C, a Tits quadratic form q c : Z (Iv) --+ Z is defined in [22], and the co-module

types of C are studied by means of qc, where Ic is a complete set of pairwise non-isomorphic simple left C- co-modules and Z (Ic) is a free Abelian group of rank Ircl.

References

[1] AUSLANDER, V.I., REITEN, I., AND SMAL0, S.: Representa- tion theory of Artin algebras, Vol. 36 of Studies Adv. Math., Cambridge Univ. Press, 1995.

[2] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAREV, V.A.: 'Coxeter functors and Gabriel 's theorem', Russian Math. Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973), 19 33.)

[3] BONGARTZ, N.: 'Algebras and quadratic forms', J. London Math. Soe. 28 (1983), 461-469.

[4] DLAB, V., AND RINGEL, C.M.: Indecomposable representations of graphs and algebras, Vol. 173 of Memoirs, Amer. Math. Soc., 1976.

[5] DROZD, Yu.A.: 'Coxeter transformations and representations of partially ordered sets', Funkts. Anal. Prilozhen. 8 (1974), 34 42. (In Russian.)

[6] DROZD, Yu.A.: 'On tame and wild matr ix problems': Matrix Problems, Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev, 1977, pp. 104-114. (In Russian.)

[7] DROZD, Yu.A.: 'Tame and wild matr ix problems': Represen- tations and Quadratic Forms, 1979, pp. 39-74. (In Russian.)

[8] GABRIEL, P.: 'Unzerlegbare Darstellungen 1', Manuscripta Math. 6 (1972), 71-103, Also: Berichtigungen 6 (1972), 309.

[9] GABRIEL, P.: 'Reprfisentations ind~composables': Sdminaire Bourbaki (1973/73), Vol. 431 of Lecture Notes in Mathemat- ics, Springer, 1975, pp. 143-169.

[10] GABRIEL, P., AND ROITER, A.V.: 'Representations of finite dimensional algebras': Algebra VIII, Vol. 73 of Encycl. Math. Stud., Springer, 1992.

[11] KASJAN, S., AND SIMSON, D.: 'Tame prinjective type and Tits form of two-peak posers II', J. Algebra 187 (1997), 71-96.

[12] NAZAROVA, L.A.: 'Representations of quivers of infinite type' , Izv. Akad. Nauk. SSSR 37 (1973), 752-791. (In Russian.)

[13] PEI~A, J.A. DE LA: 'Algebras with hypercritical Tits form': Topics in Algebra, Vol. 26:1 of Banaeh Center Publ., PWN, 1990, pp. 353-369.

[14] PEI~A, J.A. DE LA: 'On the dimension of the module-varieties of tame and wild algebras', Commun. Algebra 19 (1991), 1795-1807.

[15] PE~A, J.A. DE LA, AND S~MSON, D.: 'Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences', Trans. Amer. Math. Soe. 329 (1992), 733-753.

[16] PE~A, J.A. DE LA, AND SKOWROr@KI, A.: 'The Euler and Tits forms of a tame algebra', Math. Ann. 315 (2000), 37-59.

[17] RINGEL, C.M.: Tame algebras and integral quadratic forms, Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984.

[18] ROITER, A.V., AND I~[LEINER, M.M.: Representations of differential graded categories, Vol. 488 of Lecture Notes in Math- ematics, Springer, 1975, pp. 316-339.

[19] SIMSON, D.: Linear representations of partially ordered sets and vector space categories, Vol. 4 of Algebra, Logic Appl., Gordon & Breach, 1992.

[20] SIMSON, D.: 'Posets of finite prinjective type and a class of orders', J. Pure Appl. Algebra 90 (1993), 77-103.

[21] SIMSON, D.: 'Representation types, Tits reduced quadratic forms and orbit problems for lattices over orders', Contemp. Math. 229 (1998), 307-342.

[22] SIMSON, D.: 'Coalgebras, comodules, pseudoeompact algebras and tame comodule type', Colloq. Math. in p r e s s (2001).

Daniel S imson

MSC 1991: 16Gxx

TOEPLITZ C*-ALGEBRA - A uniformly closed *- algebra of operators on a Hilbert space (a uniformly

closed C*-a lgebra) . Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex vari-

ables.

In the one-dimensional case one considers the Hardy space H 2 (T) over the one-dimensional torus T (cf. also

Hardy spaces), and defines the T o e p l i t z operator Tf with 'symbol' function f E L°°(T) by Tfh := P(fh) for all h E H2(T) , where P : L2(T) --+ H2(T) is the orthogonal projection given by the C a u c h y i n t eg ra l

t h e o r e m . The C* -a lg eb ra T ( T ) := C* (Tf : f E C(T)) generated by all operators Tf with continuous symbol f is not commutative, but defines a C*-algebra extension

0 --+ K:(H2(T)) ~ T ( T ) --+ C(T) --+ 0

of the C*-algebra ~ of all compact operators; in fact, this 'Toeplitz extension' is the generator of the Abelian group Ext(C(T)) ~ Z.

C*-algebra extensions are the building blocks of K- t h e o r y and i n d e x t h e o r y ; in our case a Toeplitz

409

T O E P L I T Z C*-ALGEBRA

operator Tf is Fredholm (cf. also F r e d h o h n o p e ra - to r ) if f C C(T) has no zeros, and then the index Index(Tf) = dim Ker T / - dim Coker T / i s the (negative) winding n u m b e r of f .

In the multi-variable case, Toeplitz C*-algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains D C C ~ [1], including the unit ball D = {z 6 Cn: Izll 2 + . . . + Iznl 2 < 1} [2], [10], for

tube domains and Siegel domains over convex 'symmet-

ric' cones [5], [8], and for general bounded symmetric domains in C n having a transitive semi-simple Lie g r o u p

of holomorphic automorphisms [7]. Here, the principal new feature is the fact that Toeplitz operators Tf (say, on the Hardy space H2(S) over the Shilov boundary S of a pseudo-convex domain D C C ~) with continuous symbols f E Co(S) are not essentially commuting, i.e.

[T/1,T/=] f~ K.(H2(S)),

in general. Thus, the corresponding Toeplitz C*-algebra T(S) is not a (one-step) extension of K:; instead one obtains a multi-step C*-filtration

= ~i ~"" ~ I~ ~ T(S)

of C*-ideals, with essentially commutative subquotients

Zk+l/27k, whose maximal ideal space (its spectrum) re- fleets the boundary strata of the underlying domain. The length r of the composition series is an important geometric invariant, called the rank of D. The index the-

ory and K- theory of these multi-variable Toeplitz C*- algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain D, regarded as a complex Ki ih le r man i fo ld .

A general method for studying the structure and representations of Toeplitz C*-algebras, at least for Shilov boundaries S arising as a symmetric space (not necessarily Riemannian), is the so-called C*-duality [11], [9]. For example, if S is a Lie g r o u p with (reduced) group C*-algebra C*(S), then the so-called co-crossed product C*-algebra C*(S) ®5 Co(S) induced by a natural co-action 5 can be identified with 1~(L2(S)). Now the Cauehy-Szeg5 orthogonal projection E: L2(S) -+ H2(S) (cf. also C a u c h y o p e r a t o r ) defines a certain

C*-completion C~(S) D C*(S), and the corresponding Toeplitz C*-algebra T(S) can be realized as (a corner of) C~(S) ®5 Co(S). In this way the well-developed representation theory of (co-) crossed product C*-algebras [4] can be applied to obtain Toeplitz C*-representations related to the boundary cgD. For example, the two- dimensional torus S = T 2 gives rise to non-type-I C*-algebras (for cones with irrational slopes), and the underlying 'Reinhardt ' domains (cf. also R e i n h a r d t doma in ) have interesting complex-analytic properties,

such as a non-compact solution operator of the Neu-

mann 0-problem [6].

References [i] BOUTET DE MONVEL, L.: 'On the index of Toeplitz opera-

tors of several complex variables', Invent. Math. 50 (1979),

249-272.

[2] COBURN, L.: 'Singular integral operators and Toeplitz oper-

ators on odd spheres', Indiana Univ. Math. Y. 23 (1973),

433-439.

[3] DOUGLAS, R., AND HOWE, R.: 'On the C*-algebra of Toeplitz operators on the quarter-plane', Trans. Amer. Math. Soe.

158 (1971), 203-217. [4] LANDSTAD, M., PHILLIPS, J., RAEBURN, [., AND SUTHERLAND,

C.: 'Representations of crossed products by coactions and principal bundles', Trans. Amer. Math. Soe. 299(1987), 747- 784.

[5] MUHLY, P., AND RENAULT, J.: 'C*-algebras of multivari- able Wiener-Hopf operators', Trans. Amer. Math. Soc. 274 (1982), 1-44.

[6] SALINAS, N., SHEU, A., AND UPMEIER, H.: 'Toeplitz operators on pseudoconvex domains and foliation algebras', Ann.

Math. 130 (1989), 531-565. [7] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric

domains', Ann. Math. 119 (1984), 549-576. [8] UPMEIER, H.: 'Toeplitz operators on symmetric Siegel do-

mains', Math. Ann. 271 (1985), 401-414. [9] UPMmER, H.: Toeplitz operators and index theory in several

complex variables, Birkh£user, 1996. [10] VENUGOPALKRISHNA, W.: 'Fredholm operators associated

with strongly pseudoconvex domains in C n', J. Funct. Anal.

9 (1972), 349 373. [11] WASSERMANN, A.: 'Alg~bres d'op~rateurs de Toeplitz sur les

groupes unitaires', C.R. Acad. Sci. Paris 299 (1984), 871- 874.

H. Upmeier

MSC 1991: 46Lxx

TOEPLITZ SYSTEM - A system of linear equations Tx = a with T a T o e p l i t z m a t r i x .

MSC1991:15A57

TRAVELLING SALESMAN PROBLEM A generic name for a number of very different problems. For instance, suppose that a facility (a 'machine') starting from an 'idle' position is assigned to process a finite set of 'jobs' (say n, n > 3 jobs). If the machine has to be 'calibrated' (or %et-up') for processing each of these jobs and if the machine's 'calibration time' (the distance metric) between processing of a pair of jobs in succession is dependent on the particular pair, then a reasonable objective is to organize this job assignment so it will

minimize the total machine calibration time. One might want to assume that after the last job is processed the machine returns to its idle position.

A very similar problem exists when the 'machine' corresponds to a computer centre which has n programs to

410

TRIANGLE CENTRE

run, and each program requires resources such as a com- piler, a certain portion of the main memory, and perhaps some other 'devices'. I.e., each program requires a spe- cific configuration of devices. Conversion cost (or time) from one configuration to another, say from the configuration of program i to that of program j is denoted by cij (>_ 0). Thus, the question becomes that of determining the cost minimizing order in which all the programs ought to be run.

If at the end of running all the programs by the computer centre the system returns to an 'idle' configura-

tion, then the number of possible ways to run these programs one after the other equals n! (for n + 1 configu- rat!ons).

This is the same problem as that in the story about the lonely salesman who has to visit n sales outlets (starting from his home) and wishes to travel the short- est total distance in the process. It is the salesman's problem to select a distance-minimizing travel order of outlet visits. Thus, the name travelling salesman problem.

In graph terminology terms, the problem is presented as that of a g r a p h G = (V, E), where V is a finite set of nodes ('cities') and E C_ V x V is the set of edges connecting the node pairs in V. If one associates a real- valued 'cost' matrix (cij), i , j = 1 , . . . , IVI, with the set of edges E, the travelling salesman dilemma becomes that of constructing a cost-minimizing circuit on G that visits all the nodes in V exactly once, if such a circuit exists (eft also G r a p h c i rcui t ) .

If the requirement is that all the nodes in V are vis- ited in a cost-minimizing fashion but without necessarily forming a circuit, then the problem is referred to as a travelling salesman path problem, or travelling salesman walk problem. Again, the question of the existence of such a path has to be addressed first.

If the graph G = (V, A), A _C V x V, assigns a 'direc-

tion' to each element in A (a subset of arcs), then the corresponding travelling salesman problem is of the 'di- rected' variety. Clearly, there is the option of the mixed problem, where some of the node pairs are connected by arcs and some by edges.

The question of whether a circuit exists in a graph G which visits each node in V exactly once is commonly referred to as that of determining the existence of a Hamil- tonian circuit (or path; cf. also H a m i l t o n i a n tou r ) . Graphs for which such a circuit (path) is guaranteed to exist are called Hamiltonian graphs.

The difficulty of determining the existence of a Hamil- tonian circuit for a graph G and that of constructing a cost-minimizing travelling salesman circuit on a graph G are very much the same when measured by the worst

possible order magnitude of the required number of computer operations. This casts these problems (the travelling salesman and the Hamiltonian circuit problems) as being 'hard' (cf. also A/P). Essentially, for this sort of problems, one does not presently (2000) know of any solution scheme which does not require some sort of enu- meration of all possible 'configuration' sequences.

See [2], [1] for recent overviews of the problem.

R e f e r e n c e s [1] FLEISHNER, H.: '~IYaversing graphs: The Eulerian and Hamil-

tonian theme', in M. DROR (ed.): Arc Routing: Theory, So- lutions, and Applications, Kluwer Acad. Publ., 2000.

[2] LAWLER, E.L., LENSTRA, J.K., RINNOY KAN, A.H.G., AND SHMOYS, D.B. (eds.): The traveling salesman problem, Wi- ley, 1985.

Moshe Dror MSC 1991:90C08

T R I A N G L E C E N T R E - Given a triangle A1A2A3, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:

• the centroid (i.e. the centre of mass), the common intersection point of the three medians (see M e d i a n (of a t r i a n g l e ) ) ;

• the incentre, the common intersection point of the three bisectrices (see B i s e c t r i x ) and hence the centre of the ineirele (see P l a n e t r i g o n o m e t r y ) ;

• the circumcentre, the centre of the circumcircle (see P l a n e t r i g o n o m e t r y ) ;

• the orthocentre, the common intersection point of the three altitude lines (see P l a n e t r i g o n o m e t r y ) ;

• the G e r g o n n e p o in t , the common intersection

point of the lines joining the vertices with the opposite tangent points of the incircle;

• the Format point (also called the Torrieelli point or first isogonic centre), the point X that minimizes the

sum of the distances IAIX] + IA2XI + IA3XI; • the Grebe point (also called the Lemoine point or

symmedean point), the common intersection point of the three symmedeans (the symmedean through Ai is the isogonal line of the median through Ai, see Isogonal ) ;

• the N a g e l p o in t , the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see P l a n e t r i g o n o m e t r y ) .

In [2], 400 different triangle centres are described.

The Nagel point is the isotomic conjugate of the Ger- gonne point, and the symmedean point is the isogonal conjugate of the centroid (see I sogona l for both notions of 'conjugacy').

R e f e r e n c e s [1] JOHNSON, R.A.: Modern geometry, Houghton-Mifflin, 1929.

411

TRIANGLE CENTRE

[2] KIMBERLING, C.: 'Tr iangle centres and central t r iangles ' , Congr. Numer. 129 (1998), 1-285.

M. Hazewinkel MSC 1991:51M04

TRIBONACCI NUMBER A member of the Tri- b o n a c c i s equence . The formula for the nth number is given by A. Shannon in [1]:

[~/2] In/a]

m= 0 r =0 m -I- r r

Bine t ' s formula for the n th number is given by W. Spickerman in [2]:

fin+2 G-n+2 T ~ = + t-

~ n + 2

- -

where

1 ((19 -~- 3X /~ ) 1/3 -l- (19 -- 3V /~ ) 1/3 -~- 1) P=~ 1 [2 - (19 + 3 v / ~ ) ' /a - (19 - 3 V ~ ) 1/3] -~- ~ = ~

v~i [(19 + 3X/~) 1/3 - (19 - 3v /~ ) 1/3] + - g -

and ~ is the complex conjugate of ~.

R e f e r e n c e s [1] SHANNON, A.: 'Tr ibonacci number s and Pascal ' s pyramid ' ,

The Fibonacci Quart. 15, no. 3 (1977), 268; 275. [2] SPICKERMAN, W.: 'B ine t ' s formula for the Tribonacci se-

quence' , The Fibonacci Quart. 15, no. 3 (1977), 268; 275. Krassimir Atanassov

MSC 1991:11B39

T R I B O N A C C I S E Q U E N C E - An extension of the sequence of F i b o n a c c i n u m b e r s having the form (with a, b, c given constants):

to = a, tl = b, t2 = c,

tn+3 = tn+2 + t~+l + tn (n ~ 0). The concept was introduced by the fourteen-year-

old student M. Feinberg in 1963 in [3] for the case: a = b = 1, c = 2. The basic properties are introduced in [2], [5], [6], [7].

The Tribonacci sequence was generalized in [1], [4] to the form of two sequences:

an÷3 = t t n + 2 -~- Wn+l ~- Yn,

bn+3 = Vn+2 -}- Xn+l ~- Zn,

where u , v , w , x , y , z E {a ,b} and each of the tuples (u, v), (w, x), (y, z) contains the two symbols a and b. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.

See also T r i b o n a c c i n u m b e r .

R e f e r e n c e s [1] ATANASSOV, K., HLEBAROVA, J. , AND MIHOV, S.: 'Recur-

rent formulas of the general ized Fibonacci and Tribonacci sequences ' , The Fibonacci Quart. 30, no. 1 (1992), 77-79.

[2] BRUCE, I.: 'A modified Tr ibonacc i sequence ' , The Fibonacci Quart. 22, no. 3 (1984), 244-246.

[3] FEINBERG, M.: 'F ibonacc i -Tr ibonacc i ' , The Fibonacci Quart. 1, no. 3 (1963), 71-74.

[41 LEE, J.-Z., AND LEE, J.-S.: 'Some proper t ies of the general- izat ion of the Fibonacci sequence ' , The Fibonacci Quart. 25, no. 2 (1987), 111 117.

[5] SCOTT, A., DELANEY, T. , AND HOGGATT JR., V.: 'The Tri- bonacci sequence ' , The Fibonacci Quart. 15, no. 3 (1977), 193-200.

[6] SHANNON, A.: 'Tr ibonacci number s and Pasca l ' s pyramid ' , The Fibonacci Quart. 15, no. 3 (1977), 268; 275.

[71 VALAVIGI, C.: 'P roper t i e s of Tr ibonacci numbers ' , The Fi- bonacci Quart. 10, no. 3 (1972), 231-246.

Krass imir Atanassov er~ ! 91".; : M~,~ 11B39

TRIGOiX, ,~v .~ETRIC P S E U D O - S P E C T R A L M E T H -

O D S - Trigonometric pseudo-spectral methods, and spectral methods in general, are methods for solving

differential and integral equations using trigonometric functions as the basis.

Suppose the boundary value problem L u = f is to be solved for u(x ) on the interval x = [a, b], where L is

a d i f f e r en t i a l o p e r a t o r in x and f ( x ) is some given smooth function (cf. also B o u n d a r y va lue p r o b l e m , o r d i n a r y d i f f e r en t i a l eq u a t i o n s ) . Also, u must sat-

isfy given boundary conditions u(a) = u~ and u(b) = u b.

As in most numerical methods, an approximate solution, UN, is sought which is the sum of N + 1 basis functions, ¢ , (x ) , n = 0 , . . . , N , in the form UN =

N ~ = 0 a ,¢~(x) , where the coefficients an are the finite set of unknowns for the approximate solution. A 'residual equation', formed by plugging the approximate solution into the differential equation and subtracting the

right-hand side, R ( x; ao, . . . , aN) =- L[u g ( x) ] -- f , is then minimized over the interval to find the coefficients. The

difference between methods boils down to the choice of basis and how R is minimized. The basis functions should be easy to compute, be complete or represent the class of desired functions in a highly accurate manner, and be orthogonal (cf. also C o m p l e t e s y s t e m of func t ions ; O r t h o g o n a l s y s t e m ) . In spectral methods, t r i g o n o m e t r i c f u n c t i o n s and their relatives as well as other o r t h o g o n a l p o l y n o m i a l s are used.

If the basis functions are trigonometric functions such as sines or cosines, the method is said to be a Fourier

spectral method. If, instead, C h e b y s h e v p o l y n o m i a l s

are used, the method is a Chebyshev spectral method.

The method of mean weighted residuals is used to minimize R and find the unknowns coefficients a~. An inner product (.,-) and weight function p(x) are defined,

412

T R I G O N O M E T R I C P S E U D O - S P E C T R A L M E T H O D S

as well as N + 1 test functions wi such that (wi, R) = 0

for i = 0 , . . . , N and (u,v) = f : u ( x ) v ( x ) p ( x ) dx. This yields N + 1 equations for the N + 1 unknowns. Pseudo-

spectral methods, including Fourier pseudo-spectral and

Chebyshev pseudo-spectral methods, have Dirac delta- functions (cf. also D i r a e d i s t r i b u t i o n ) as their test

functions: wi(z) = 6(x - z i ) , where xi are interpolation or collocation points. The residual equation becomes

R ( x i ; a o , . . . , a N ) = 0 for i = 0 , . . . , N .

The G a l e r k i n m e t h o d uses the basis functions as

the test functions. If L is linear, the following ma-

trix equation can be formed: L~n,~a~ = fro, where

L,~,~ = (4,~, L¢~) and f,~ = (f, ¢,~). An alternative Galerkin formulation can be found by transforming the

residual equation into spectral space R(x; ao , . . . , aN) = ~-~.nrn(ao,. . . ,aN)¢n(x) and setting r~ = 0 for n = O,.. . , N. In using Gauss-Jacobi integration to evaluate the inner products of the Galerkin method, the inte-

grands are interpolated at the zeros of the iV + 1st basis

function. By using the same set of points as collocation

points for a pseudo-spectral method, the two methods

are made equivalent. Problems can be cast in either grid- point or spectral coefficient representation. For trigono-

metric bases, this result allows the complexity of com-

putat ion to be reduced in many problems through the

use of fast transforms.

A main difference between spectral and other meth-

ods, such as finite difference or finite element methods,

is that in the latter the domain is divided into smaller

subdomains in which local basis functions of low order

are used. With the basis functions frozen, more accu-

racy is gained by decreasing the size of the subdomains.

In spectral methods, the domain is not subdivided, but

global basis functions of high order are used. Accuracy is

gained by increasing the number and order of the basis

functions.

The lower-order methods produce algebraic systems

which can be represented as sparse matrices. Spectral

methods usually produce full matrices. The solution then involves finding the inverse. Through the use of

orthogonality and fast transforms, full matr ix inversion can usually be accomplished with a complexity similar

to the sparse matrices.

Boundary conditions are handled in a reasonably

straightforward manner. Sometimes the boundary con-

ditions are satisfied automatically, such as with period-

icity and a Fourier method. With other types of con-

ditions, an extra equation may be added to the system to satisfy it, or the basis functions may be modified to

automatically satisfy the conditions.

The attractiveness of spectral methods is that they have a greater than algebraic convergence rate for

smooth solutions. A simple finite difference approxi-

mat ion has a convergence l u - UNI = O(h~), where

h = (b - a ) / N and c~ is an integer; double the number of points in the interval and the error goes down

by a factor 2% The convergence rate of spectral methods is O(hh), sometimes called exponential, infinite oi"

spectral, s temming from the fact that convergence of a

tr igonometric series is geometric. If the solution is not smooth, however, spectral methods will have an alge-

braic convergence linked to the continuity of the solu-

tion.

Rapid convergence allows fewer unknowns to be used,

but more computat ional processing per unknown. Hence

spectral methods are part icularly at tract ive for prob-

l em/computer matches in which memory and not com-

puting power is the critical factor.

Multi-dimensional problems are handled by tensor-

product basis functions, which are basis functions that

are products of 1-dimensional basis functions. Other or -

t h o g o n a l p o l y n o m i a l s can be used in pseudo-spectral methods, such as Legendre and Hermite and spherical

harmonics for spherical geometries.

A disadvantage of spectral methods is that only rel-

atively simple domains and boundaries can be handled.

Spectral element methods, a combination of spectral and

finite element methods, have in many cases overcome

this difficulty. Another difficulty is that spectral meth-

ods are, in general, more complicated to code and re-

quire more analysis to be done prior to coding than

simpler methods.

Aliasin 9 is a phenomenon in which modes of degree

higher than in the expansion are interpreted as modes

tha t are within the range of the expansion. This occurs

in, say, a problem with quadratic non-linearity where

twice the range is created. If the coefficients near the up-

per limit are sufficiently large in magnitude, there may

be a significant error associated with aliased modes. For a Fourier pseudo-spectral method, the coefficient aN/2+k is interpreted as a coefficient aN/2-k. By zeroing the upper 1/3 of the coefficients, the quadratic nonlinearity will only fill a range from 2 / (3N/2 ) to 4/ (3N/2) . This

will only produce aliasing errors for modes 2/ (3N/2) to N/2, but these are to be zeroed anyway. This '2/3' rule

removes errors for one-dimensional problems with qua-

dratic non-linearity. It is debatable, however, whether

in a 'well-resolved' simulation there is need to address

aliasing errors.

References [1] BOYD, J.P.: Chebyshev and Fourier spectral methods,

second ed., Dover, 2000, pdf version: http://www- personal.engin.umich.edu/~jpboyd/book_spectral2OOO.html.

['2] CANUTO, C., HUSSAINI, M.Y., QUARTERONI, A., AND gANG, T.A.: Spectral methods in fluid dynamics, Springer, 1987.

413

TRIGONOMETRIC PSEUDO-SPECTRAL METHODS

[3] FORNBERG, B.: A practical guide to pseudospectral methods, Vol. 1 of Cambridge Monographs Appl. Comput. Math., Cam- bridge Univ. Press, 1996.

[4] GOTTLmB, D., HUSSAINI, M.Y., AND ORSZAC, S.A.: 'The- ory and application of spectral methods', in R.G. VOIGT,

D. GOTTLIEB, AND M.Y. HUSSAINI (eds.): Spectral Methods for Partial Differential Equations, SIAM, 1984.

[5] GOTTLIEB, D., AND ORSZAG, S.A.: Numerical analysis of spectral methods: Theory and applications, SIAM, 1977.

Richard B. Pelz MSC 1991: 65M70, 65Lxx

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