T
TACNODE, point of osculation, osculation point, double cusp - The third in the series of Ak-curve sin- o.o T 1 ~ T2 gularities. The point (0,0) is a tacnode of the curve X 4 __ y2 • 0 in R 2. J I The first of the Ak-curve singularities are: an ordi- TId T 1 * T 2 nary double point, also called a node or crunode; the Fig. 1. cusp, or spinode; the tacnode; and the ramphoid cusp. They are exemplified by the curves X k+l - y2 = 0 Several special families of tangles have been consid- for k = 1,2,3,4. ered, including the rational tangles, the algebraic The terms 'crunode' and 'spinode' are seldom used tangles and the periodic tangles (see Rotor). The n- nowadays (2000). braid group is a subgroup of the monoid of n-tangles (cf. See also Node; Cusp. also Braided group). One has also considered framed tangles and graph tangles. The category of tangles, with References boundary points as objects and tangles as morphisms, is [1] ABHYANKAR, S.S.: Algebraic geometry for scientists and en- gineers, Amer. Math. Soc., 1990, p. 3; 60. important in developing quantum invariants of links and [2] DIMCA, A.: Topics on real and complex singularities, Vieweg, 3-manifolds (e.g. Reshetikhin-Turaev invariants). Tan- 1987. gles are also used to construct topological quantum field [3] GRIFFITHS, PH., AND HARRIS, J.: Principles of algebraic ge- theories. ometry, Wiley, 1978, p. 293; 507. [4] WALKER,R.J.: Algebraic curves, Princeton Univ. Press, 1950, References Reprint: Dover 1962. [1] BONAHON, P., AND SIEBENMANN, L.: Geometric splittings of M. Hazewinkel classical knots and the algebraic knots of Conway, Vol. 75 of MSC 1991:14H20 Lecture Notes, London Math. Soc., to appear. [2] CONWAY, J.H.: 'An enumeration of knots and links', in J. LEECH (ed.): Computational Problems in Abstract Alge- TANGLE, relative link - A one-dimensional manifold bra, Pergamon Press, 1969, pp. 329-358. properly embedded in a 3-ball, D a. [3] LOZANO, M.: 'Arcbodies', Math. Proc. Cambridge Philos. Soe. 94 (1983), 253-260. Two tangles are considered equivalent if they are am- bient isotopic with their boundary fixed. An n-tangle Jozef Przytycki has 2n points on the boundary; a link is a 0-tangle. MSC 1991:57M25 The term arcbody is used for a one-dimensional mani- fold properly embedded in a 3-dimensional manifold. TANGLE MOVE - For given n-tangles 2/"1 and T2 Tangles can be represented by their diagrams, i.e. (cf. also Tangle), the tangle move, or more specifically regular projections into a 2-dimensional disc with ad- the (T1,T2)-move, is substitution of the tangle T2 in ditional over- and under-information at crossings. Two the place of the tangle T1 in a link (or tangle). The tangle diagrams represent equivalent tangles if they are simplest tangle 2-move is a crossing change. This can related by Reidemeister moves (cf. Reidemeister the- be generalized to n-moves (cf. Montesinos-Nakanishi orem). The word 'tangle' is often used to mean a tangle conjecture or [5]), (m, q)-moves (cf. Fig. 1), and p/q- diagram or part of a link diagram. rational moves, where a rational 2-tangle is substituted The set of n-tangles forms a monoid; the identity in place of the identity tangle [6] (Fig. 2 illustrates a tangle and composition of tangles is illustrated in Fig. 1. 13/5-rational move). TANGLE MOVE
A p/q-rational move preserves the space of Fox p- TAU METHOD, r method A method initially for- colourings of a link or tangle (cf. Fox n-colouring). mulated as a tool for the approximation of special func- For a fixed prime number p, there is a conjecture that tions of mathematical physics (cf. also Special func- any link can be reduced to a trivial link by p/q-rational tions), which could be expressed in terms of simple dif- moves (Iql _< p/2). ferential equations. It developed into a powerful and ac- Kirby moves (cf. Kirby calculus) can be interpreted curate tool for the numerical solution of complex differ- as tangle moves on framed links. ential and functional equations. A main idea in it is to approximate the solution of a given problem by solving exactly an approximate problem.
... q half twists Lanczos ~ formulation of the tau method. In [17], C. J~"-J'~"~"" "~'~ (m,q)-move Lanczos remarked that truncation of the series solution m half twists of a differential equation is, in some way, equivalent to introducing a perturbation term in the right-hand side Fig. 1. of the equation. Conversely, a polynomial perturbation term can be used to produce a truncated series, that is, a polynomial solution. 13/5-move Assume one wishes to solve by means of a power se- ries expansion the simple linear differential equation (cf. also Linear differential operator) Dy(x):=y'(x)+y(x)=0, O
396 TAU METHOD
One of them is described above, that is, to set up and A sequence of canonical polynomials defined as sim- solve a system of linear algebraic equations linking the ply as DQn(x) := x n for all n = 0, 1,..., need not al- unknown coefficients of Dy* (x) with those of 7T~ (x). In ways exist or need not be unique. An algebraic and algo- this process one can assume that yn(X) itself can be ex- rithmic theory of the tau method, initially constructed pressed in either powers of x, or in Chebyshev, Legendre for elements D of the class D of linear differential op- or other polynomials. The first choice was Lanczos' orig- erators of arbitrary integer order, with polynomial or inal choice, and he explicitly indicated the possibility of rational coefficients (essentially the tools a computer choosing the others. handles) was discussed by E.L. Ortiz in [24]. In this The second choice is a tau method, often [8] called the work, canonical polynomials are defined as realizations Chebyshev method (or Legendre method) and, also, the of classes of equivalence of polynomials, for which the spectral method. This last formulation of the tau method algebraic kernel of the differential operator is the mod- has been extensively used and applied, since 1971, to ulus. These classes have gaps in their index sequence. complex problems in fluid dynamics by S.A. Orsag [11]. Elements D E D are then uniquely associated with re- There are at least three other approaches to the tau presentatives of such classes of canonical sequences. The method. One of them is to find the coefficients of the codimension of the image of the space of polynomials approximant through a process of interpolation at the under operators D C D is usually small, and bounded zeros of the perturbation term. This early form of col- by the order of D plus the height h := maxncN{a~ -n} location was termed the 'method of selected points' by (where an is the degree of Dx n) of the differential oper- Lanczos [17]. When the perturbation term is an orthogo- ator. nal polynomial (such as a Chebyshev, Legendre, or other For more general operators than the one used as an polynomial), this process is called 'orthogonal colloca- example, more than a single ~- term is usually required tion'. This is the name by which Lanczos' method of to satisfy the more elaborate supplementary conditions selected points is usually designated today (as of 2000); and, also, internal conditions of the method. In the case the name 'pseudo-spectral method' is also often applied of a problem defined by a differential operator D in l?, of to it. Algorithms for these methods have been well de- order # > i and with non-constant coefficients, the ques- veloped. tion of the number of 7 terms required for a tau method Recurslve formulation of the tau method based approximation has been shown to be related to the size on canonical polynomials. In his classic [18], Lanc- of the gap in the canonical sequence, and to the exis- zos noted that if a sequence of polynomials Q~(x), tence of a non-empty algebraic kernel in D. The num- n = 0, 1,..., such that DQn(x) := x ~ for all n E N can ber of ~- terms can be easily determined in this approach be found for any linear differential operator with poly- using information on the degree of polynomial (or ratio- nomial coefficients D, then, since Tg(x) := c~ + e~x + nal) coefficients and the order of differentiation of the • .. + c~,x '~ (the coefficients of which are tabulated), the function to which they apply. It was also shown in [24] solution of the tau problem would be immediately given that canonical sequences can be generated recursively. by: This approach was used to formulate the first recur- n sire algorithms for the automatic solution of differential equations using the tau method. The theory of canonical k=0 polynomials has been discussed and extended by several where the parameter T is fixed using the initial condi- authors; see [10] and the references given therein. tion. Theoretical error analysis for the tau method [18], An extension of this approach to a wider range of [30], [9], [22], [26] have shown that tau method approxi- differential operators than the trivial one, given in the mations are of the order of best uniform approximations example, has several advantages: canonical polynomials by polynomials defined over the same interval. This con- are independent of the interval in which the solution is nection with best approximation is preserved when a sought, allowing for easy segmentation of the domain; tau method based on rational approximation [18], [21] they are permanent, in the sense that if an approxima- is used [5]. tion of a higher degree is required, the computation does not need to be repeated from scratch; they are also inde- Operational formulation of the tau method. There pendent of the supplementary conditions of the problem, is yet another way in which tau method approximations which can now equally be initial, boundary or multi- can be constructed. An operational formulation of the point conditions. Furthermore, the tau method does not tau method was introduced by Ortiz and H. Samara in require a stage of discretization of the given differential [27]. In this formulation, derivatives and polynomial co- operator, as discrete-variable methods do. efficients of operators in 7? are represented in terms of
397 TAU METHOD multiplicative diagonal matrices. Furthermore, the dif- of continuous- and discrete-variable approximation tech- ferential operator and the supplementary conditions are niques. decoupled. Through a simple and systematic algorithm, which treats the differential operator and supplemen- References tary conditions with similar machinery, this technique [1] BANKS, H.T., AND WADE, J.G.: 'Weak tau approximations transforms a given differential tau method problem into for distributed parameter systems in inverse problems', Nu- met. Funct. Anal. Optim. 12 (1991), 1-31. one in linear algebra. The approximate solution can be [2] CHAVES, T., AND ORTIZ, E.L.: 'On the numerical solution generated, indistinctively, in terms of powers of the vari- of two point boundary value problems for linear differential ables or in terms of elements of a more stable polynomial equations', Z. Angew. Math. Mech. 48 (1968), 415 418. basis, such as Chebyshev, Legendre or other polynomi- [3] CRISCI, M.R., AND RUSSO, E.: 'A-stability of a class of meth- als. The operational formulation further simplified the ods for the numerical integration of certain linear systems of differential equations', Math. Comput. 41 (1982), 431-435. development of software for the tau method. [41 CRISCg M.R., AND RUSSO, E.: 'An extension of Ortiz's recur- Numerical applications of the tau method. The sive formulation of the tau method to certain linear systems recursive and operational approaches to the tau method of ordinary differential equations', Math. Comput. 41 (1983), 27-42. have been extended in several directions. To systems of [5] EL DAOU, M., NAMASIVAYAM,S., AND ORTIZ, E.L.: 'Dif- linear differential equations [9], [4]; to non-linear prob- ferential equations with piecewise approximate coefficients: lems [25], [23], [26]; to partial differential equations [28], discrete and continuous estimation for initial and boundary [29]; and, in particular, to the numerical solution of non- value problems', Computers Math. Appl. 24 (1992), 33-47. linear systems of partial differential equations the solu- [6] EL DAOU, M., AND ORTIZ, E.L.: 'The tau method as an an- alytic tool in the discussion of equivalence results across nu- tion of which has sharp spikes, with high gradients, as mericaI methods', Computing 60 (1998), 365-376. in the case of soliton interactions [14], [13]; to the ap- [7] EL MISlERY, A.E.M., AND ORTIZ, E.L.: 'Tau-lines: a new hy- proximate solution of ordinary and partial functional- brid approach to the numerical treatment of crack problems differential equations [25], [20], [15]; and to singular based on the tau method', Computer Methods in Applied Me- problems for partial differential equations related to chanics and Engin. 56 (1986), 265 282. [8] Fox, L., AND PARKER, I.B.: Chebyshev polynomials in nu- crack propagation [7]. The tau method is well adapted merical analysis, Oxford Univ. Press, 1968. to produce accurate approximations in the numerical [9] FREILICH, J.G., AND ORTIZ, E.L.: 'Numerical solution of sys- treatment of differential eigenvalue problems with one terns of differential equations: an error analysis', Math. Com- or multiple spectral parameters, entering either linear or put. 39 (1982), 467-479. non-linearly into the equation [2], [19]. The tau method [10] FROES BUNCHAFT, M.E.: 'Some extensions of the Lanczos- Ortiz theory of canonical polynomials in the tau method', has been extensively used for the high-precision approx- Math. Comput. 66, no. 218 (1997), 609 621. imation of real- [16] and complex-valued functions. A [11] GOTLIEB, D., AND ORSZAG, S.A.: Numerical analysis of spec- weak formulation of the tau method has been proposed tral methods: Theory and applications, Philadelphia, 1977. and applied to inverse problems for partial differential [12] HAYMAN, W.K., AND ORTIZ, E.L.: 'An upper bound for the equations [1]. largest zero of Hermite's function with applications to subhar- monic functions', Proc. Royal Soc. Edinburgh 75A (1976), Analytical applications of the tau method. The 183-197. tau method has also been used in a totally different di- [13] HOSSEIM AH-ABAD% M., AND ORTm, E.L.: 'The algebraic kernel method', Namer. Funct. Anal. Optim. 12, no. 3-4 rection, as a tool in the discussion of problems in math- (1991), 339 360. ematical analysis, for example, in complex function the- [14] HOSSEINI ALI-ABADI, M., AND ORTIZ, E.L.: 'A tau method ory [12]. based on non-uniform space-time elements for the numerical Possible connections between the tau method, col- simulation of solitons', Computers Math. Appl. 22 (1991), 7-19. location, Galerkin's method, algebraic kernel methods, [15] KHAJAH, H.G., AND ORTIZ, E.L.: 'Numerical approximation and other polynomial or discrete-variable techniques of solutions of functional equations using the tau method', have also been explored [31], [13], [6]. Appl. Namer. Anal. 9 (1992), 461-474. The tau method has also received some attention as [16] KHAJAH, H.G., AND ORTIZ, E.L.: 'Ultra-high precision com- putations', Computers Math. Appl. 27, no. 7 (1993), 41-57. an analytic tool in the discussion of equivalence results [17] LANeZOS, C.: 'Trigonometric interpolation of empirical and across nmnerical methods [6]. It has been found that, analytic functions', J. Math. and Physics iT (1938), 123-199. with it, it is possible to construct special 'tau methods', [18] LANCZOS, C.: Applied analysis, New Jersey, 1956. which recursively generate solutions numerically identi- [19] LIu, K.M., AND ORTIZ, E.L.: 'Tau method approximation of cal to those of collocation, Galerkin's and other weighted differential eigenvalue problems where the spectral parameter enters nonlinearly', Y. Comput. Phys. 72 (1987), 299-310. residual methods, and to those of discrete-variable meth- [2o] LIU, K.M., AND ORTIZ, E.L.: 'Numerical solution of ordinary ods, such as sophisticated forms of Runge-Kutta meth- and partial functional-differential eigenvalue problems with ods. This work suggests a way of unifying a large group the tau method', Computing 41 (1989), 205-217.
398 TAYLOR JOINT SPECTRUM
[21] LUKE, Y.L.: The special functions and their approximations The commuting n-tuple A is said to be non-singular l-II, New York, 1969. on X if RanDA = KerDA. The Taylor joint spectrum, [22] NAVASIMAYAN,S., AND ORTIZ, E.L.: 'Best approximation and or simply the Taylor spectrum, of A on X is the set the numerical solution of partial differential equations with the tau method', Portugal. Math. 41 (1985), 97-119. aT (A, X) := {A • C ~ : A - A is singular}. [23] ONUMANYI, P., AND ORTIZ, E.L.: 'Numerical solution of stiff and singularly perturbed boundary value problems with a The decomposition A = O~=1Ak gives rise to a segmented-adaptive formulation of the tau method', Math. cochain complex K(A, X), the so-called Koszul com- Comput. 43 (1984), 189-203. plex associated to A on A/, as follows: [24] ORTIZ, E.L.: 'The tau method', SIAM J. Numer. Anal. 6 on-1 (1969), 480--492. z): 0 A°(X) 4 AN(X) -+ 0, [25] ORTm, E.L.: 'On the numerical solution of nonlinear and functional differential equations with the tau method', in where D k denotes the restriction of DA to the subspace R. ANSORGEAND W. ThRmC (eds.): Numerical Treatment of Ak(X). Thus, Differential Equations in Applications, Berlin, 1978, pp. 127- 139. aT(A,X) = {A • C~: K(A- A,X) is not exact}. [26] ORTIZ, E.L., AND PHAM NGOC DINH, A.: 'Linear recursive J.L. Taylor showed in [18] that if X is a Banach schemes associated with some nonlinear partial differential equations in one dimension and the tau method', SIAM J. space, then aT(A, 2() is compact, non-empty, and con- Math. Anal. 18 (1987), 452-464. tained in a t(A), the (joint) algebraic spectrum of A (cf. [27] ORTIZ, E.L., AND SAMARA, H.: 'An operational approach to also Spectrum of an operator) with respect to the the tau method for the numerical solution of nonlinear dif- commutant of A, (A)' := {B • £(X): BA = AB}. ferential equations', Computing 27 (1981), 15-25. Moreover, aT carries an analytic functional calculus [28] OaTm, E.L., AND SAMARA,H.: 'Numerical solution of partial differential equations with variable coefficients with an oper- with values in the double commutant of A, so that, in ational approach to the tau method', Computers Math. Appl. particular, aT possesses the projection property. 10, no. 1 (1984), 5-13. Example: n = 1. For n = 1, DA admits the following [29] PUN, K.S., AND ORTm, E.L.: 'A bidimensional tau-elements (2 x 2)-matrix relative to the direct sum decomposition method for the numerical solution of nonlinear partial dif- ferential equations, with an application to Burgers equation', (z ® e0) • (x ® Computers Math. Appl. 12B (1986), 1225-1240. [30] RIVLIN, T.J.: The Chebyshev polynomials, New York, 1974, 00) 2nd. ed. 1990. [31] WRICHT, K.: 'Some relationships between implicit Runge- Then Ker DA/Ran DA = Ker A® (X/Ran A). It follows Kutta, collocation and Lanczos tau methods', BIT 10 (1970), at once that aT agrees with ~, the spectrum of A. 218-227. Example: n = 2. For n = 2, Eduardo L. Ortiz MSC 1991: 65Lxx DA = A1 0 0 2 0 0 ' TAYLOR JOINT SPECTRUM - Let A = A[e] = An[e] be the exterior algebra on n generators -A2 A1 el,...,em with identity e0 - 1. A is the algebra of so KerDA/RanDA = (KerA1 N KerA2) ® forms in el,..., en with complex coefficients, subject to {(xl,x2): A2xl = Alx2}/{(Alxo,A2xo): x0 e X} (9 the collapsing property eiej + ejei = 0 (1 _< i, j < n). (X/(Ran A1 + Ran A2)). Let E~: A --+ A denote the creation operator, given Note that since aT is defined in terms of the actions by Ei~ := ei~ (~ • A, 1 _< i < n). If one de- of the operators Ai on vectors of X, it is intrinsically clares {eq,...,ei~: 1 < il < ... < ik < n} to be an 'spatial', as opposed to a I, a" and other algebraic joint orthonormal basis, the exterior algebra A becomes a spectra, aT contains other well-known spatial spectra, Hilbert space, admitting an orthogonal decomposition like ap (the point spectrum), a~ (the approximate point A = ~Jk=lZ'~nA k, where dim A k = (;). Thus, each ~ • A ad- spectrum) and a5 (the defect spectrum). Moreover, if mrs a unique orthogonal decomposition ~ = e ~ t + ~tt 1 /3 is a commutative Banach algebra, a -= (al,...,a,0, where ~1 and ~" have no ei contribution. It then read- with each ai E /3, and L~ denotes the n-tuple of left ily follows that E*~ = ~. Indeed, each Ei is a partial multiplications by the ais, it is not hard to show that isometry, satisfying E~Ej + EjE[ = 5ij (1 _< i,j < n). aT (L~,/3) = a• (a). As a matter of fact, the same result Let X be a normed space, let A =- (A1,..., An) be holds when/3 is not commutative, provided all the ais a commuting n-tuple of bounded operators on X" and come from the centre of/3. set A(X) := X ®c A. One defines DA: A(X) --+ A(X) Spectral permanence. When/3 is a C*-algebra, say/3 C by DA := ElL1 Ai ® El. Clearly, D~ = 0, so Ran DA C_ £(7-0, then aT(La, B) = aT(a, 7-0 [5]. This fact, known Ker DA. as spectral permanence for the Taylor spectrum, shows
399 TAYLOR JOINT SPECTRUM
that for C*-algebra elements (and also for Hilbert space to be Fredholm on X if the associated Koszul complex operators), the non-singularity of La is equivalent to the K(A, 2() has finite-dimensional cohomology spaces. The invertibility of the associated Dirac operator Da + Dt.. Taylor essential spectrum of A on A~ is then
Finite-dimensional ease. When dim A" < oc, 0-Te(A, 2() := {A C C n : A - A is not Fredholm}.
0.p = 0"1 = 0-7r -= 0-5 = G-r = 0"T = 0-1 = 0-H = ~, The Fredholm index of A is defined as the Euler where 0-1, 0"r and ~ denote the left, right and polyno- characteristic of K(A,X). For example, if n = 2, mially convex spectra, respectively. As a matter of fact, index(A) = dimKerD ° - dim(Ker D1A/RanD °) + in this case the commuting n-tuple A can be simultane- dim(X/RanDy). In a Hilbert space, o-we(A,7/) = ously triangularized as Ak = ~ai,/ (k)j h,j=l~dim W , and 0"T(La, Q(7/)), where a := 7r(A) is the coset of A in the Calkin algebra for 7/. 0-T (A, X) = (~~"(a!~) ~ ,'",uii-(~)' J: l
400 THEODORSEN INTEGRAL EQUATION
[6] CURTO, R.: 'Applications of several complex variables to by multiparameter spectral theory', in J.B. CONWAYAND B.B. MORREL (eds.): Surveys of Some Recent Results in Operator g(e it) = p (0(t)) e (vt c R), Theory II, Vol. 192 of Pitman Res. Notes in Math., Longman Sci. Tech., 1988, pp. 25-90. dt 2~ 2 " [7] CURTO, R., AND FIALKOW, L.: 'The spectral picture of /o ~ O(t) = (LA,RB)' , J. Funct. Anal. 71 (1987), 371-392. Theodorsen's equation follows from the fact that the [8] CURTO, R., AND MUHLY, P.: 'C*-algebras of multiplication operators on Bergman spaces', J. Funct. Anal. 64 (1985), function h(w) := log(g(w)/w) is analytic in D and can 315-329. be extended to a homeomorphism of the closure D [9] CURTO, R., AND SALINAS, N.: 'Spectral properties of cyclic onto the closure A. It simply states that the 2~r-periodic subnormal m-tuples', Amer. J. Math. 107 (1985), 113-138. function y: t ~-~ 0 - t is the conjugate periodic function [10] CURTO, R., AND VAN, K.: 'The spectral picture of Reinhardt of x: t ~ logp(O(t)), that is, y = Kx, where I4 is the measures', J. Funct. Anal. 131 (1995), 279-301. [11] ESCHMEIER, J., AND PUTINAR, M.: Spectral decompositions conjugation operator defined on L[0, 21r] by the principal and analytic sheaves, London Math. Soc. Monographs. Ox- value integral ford Sci. Publ., 1996. [12] LAURSEN,K., AND NEUMANN, M.: Introduction to local spec- (Kx)(t) := P.V. x(s) cot t - s ds (a.e.). tral theory, London Math. Soc. Monographs. Oxford Univ. Press, 2000. [13] PUTINAR, M.: 'Uniqueness of Taylor's functional calculus', When restricted to L2[0, 2rF], K is a skew-symmetric en- Proc. Amer. Math. Soc. 89 (1983), 647-650. domorphism of norm 1 with a very simple diagonal rep- [14] PUTINAR, M.: 'Spectral inclusion for subnormal n-tuples', resentation in Fourier space: when x has the real Fourier Proc. Amer. Math. Soc. 90 (1984), 405 406. coefficients ao,al, .., bl,b2,..., then y has the coem- [15] SALINAS, N.: 'The cg-formalism and the C*-algebra of the cients 0, -bl, -b2, .., al, a2,.... Bergman n-tuple', J. Oper. Th. 22 (1989), 325 343. [16] SALINAS,N., SHEU~ A., AND UPMEIER, H.: 'Toeplitz operators Hence, while Theodorsen's integral equation is nor- on pseudoconvex domains and foliation C*-algebras', Ann. of mally written as Math. 130 (1989), 531 565. [17] TAYLOR, J.L.: 'The analytic functional calculus for several o(t) - t = _~ P.V. /o 21r logp(0(s)) cot -g-t -- 8 d,, commuting operators', Acta Math. 125 (1970), 1-48. [18] TAYLOR, J.L.: 'A joint spectrum for several commuting op- for practical purposes the conjugation is executed by erators', g. Funct. Anal. 6 (1970), 172-191. [19] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric transformation to Fourier space: x is approximated domains', Ann. of Math. 119 (1984), 549-576. by a trigonometric polynomial of degree N, whose [20] VASlLESOU, F.-H.: Analytic functional calculus and spectral Fourier coefficients are quickly found by the fast Fourier decompositions, Reidel, 1982. transform, which then can also be applied to determine [21] VENUGOPALKRISHNA, U.: 'Fredholm operators associated values at 2N equi-spaced points of the trigonometric with strongly pseudoconvex domains in C '~', Y. Funct. Anal. 9 (1972), 349 373. polynomial that approximates y = Kx (cf. also Fourier Ragl E. Curto series). Before the fast Fourier transform became the MSC 1991: 47Dxx standard tool for this discrete conjugation process, the transition from the values of z to those of y was based on multiplication by a matrix, called the Wittich matrix TAYLOR THEOREM - One of several results, in [1]. The fast Fourier transform meant a cost reduction of which the most important is the Taylor formula from O(N 2) to O(N log N) operations per iteration. and its various generalizations, e.g., to wider function Until the end of the 1970s the recommendation was classes, to a stochastic setting or to multiple centres (in to solve a so-obtained discrete version of Theodorsen's which case one deals with interpolation-type formulas). equation by fixed-point (Picard) iteration, an approach that is limited to Jordan regions with piecewise differ- MSC 1991: 41A05, 41A58 entiable boundary satisfying IP'/Pl < 1, and is very slow when the bound 1 is nearly attained. Other regions, like THEODORSEN INTEGRAL EQUATION - those from airfoil design, which was the standard ap- Theodorsen's integral equation [7] is a well-known tool plication targeted by T. Theodorsen, could be handled for computing numerically the conformal mapping g by using first a suitable preliminary conformal mapping, of the unit disc D onto a star-like region A given by which turned the exterior of the wing cross-section into the polar coordinates r, p(r) of its boundary F. The the exterior of a Jordan curve that is close to a circle; mapping g is assumed to be normalized by g(0) = 0, see [6, Chapt. 10]. Moreover, for this application, the g'(0) > 0. It is uniquely determined by its boundary equation has to be modified slightly to map the exterior correspondence function 0, which is implicitly defined of the disc onto the exterior of a Jordan curve.
401 THEODORSEN INTEGRAL EQUATION
M. Gutknecht [2], [3] extended the applicability of payments before and after time t + dt leads to Theodorsen's equation by applying more refined itera- Vt = #x+t dt S - P dt+ (1) tive methods and discretizations, and O. H/ibner [5] im- proved the convergence order from linear to quadratic by +(1 - #x+t dt)e -~ atvt+at + o(dt), adapting R. Wegmann's treatment of a similar equation from which one obtains that Vt is the solution to obtained by choosing h(w) := g(w)/w instead. Weg- dv, mann's method [9], [10] applies the Newton method ~/ ~ = P + ~vt - ~x+~(s - vd, (2) and solves the linear equation for the corrections by in- subject to the condition V~ = 0. terpreting it as a Riemann-Hilbert problem that can be solved with four fast Fourier transforms. This is the celebrated Thiele differential equation, proclaimed 'the fundament of modern life insurance A common framework for conformal mapping meth- mathematics' in the authoritative textbook [1], and ods based on function conjugation is given in [4]; named after its inventor Th.N. Thiele (1838-1910). It Theodorsen's restriction to regions given in polar co- dates back to 1875, but was published only in 1910 in ordinates can be lifted. Both Theodorsen's [8] and Weg- the obituary on Thiele by J.P. Gram [2], and appeared mann's [11] equations and methods can be extended to in a scientific text [7] only in 1913. the doubly connected case. As is apparent from the proof sketched in [1], Thiele's References differential equation is a simple example of a Kol- [1] GAIER, D.: Konstruktive Methoden der konformen Abbil- dung, Springer, 1964. mogorov backward equation (cf. Kolmogorov equa- [2] GUTKNECHT, M.H.: 'Solving Theodorsen's integral equation tion), which is a basic tool for determining conditional for conformal maps with the fast Fourier transform and var- expected values in intensity-driven Markov processes. ious nonlinear iterative methods', Numer. Math. 36 (1981), Thus, today there exist Thiele differential equations for 405-429. a variety of life insurance products described by multi- [3] GUTKNECHT, M.H.: 'Numerical experiments on solving Theodorsen's integral equation for conformal maps with the state Markov processes and for various aspects of the fast Fourier transform and various nonlinear iterative meth- discounted payments, e.g. higher order moments and ods', SIAM a. Sci. Statist. Comput. 4 (1983), 1 30. probability distributions. The technique is an indispens- [4] GUTKNECHT,M.H.: 'Numerical conformal mapping methods able constructive device in theoretical and practical life based on function conjugation', J. Comput. Appl. Math. 14 insurance mathematics and also forms the basis for nu- (1986), 31-77. [5] H/iBNEa, O.: 'The Newton method for solving the merical procedures, see [8]. Theodorsen equation', a. Comput. Appl. Math. 14 (1986), Thiele was Professor of Astronomy at the University 19-30. of Copenhagen from 1875, cofounder and Director (ac- [6] KYTHE, P.K.: Computational conformal mapping, tuary) of the Danish life insurance company Hafnia from Birkhguser, 1998. 1872, and first president of the Danish Actuarial Soci- [7] THEODORSEN, T.: 'Theory of wing sections of arbitrary shape', Rept. NACA 411 (1931). ety founded in 1901. In 52 written works (three mono- [8] THEODORSEN,T., AND GARRICK, I.E.: 'General potential the- graphs; [11], [12], [13]) he made contributions (a number ory of arbitrary wing sections', Rept. NACA 452 (1933). of them fundamental) to astronomy, mathematical sta- [9] WEGMANN, R.: 'Ein Iterationsverfahren zur konformen Ab- tistics, numerical analysis, and actuarial mathematics. bildung', Numer. Math. 30 (1978), 453-466. Biographical/bibliographical accounts are given in [3], [10] WEGMANN,R.: 'An iterative method for conformal mapping', J. Comput. Appl. Math. 14 (1986), 7-18, English translation [4], [51, [6], [9], [10]. of [9]. (In German.) References [11] WEGMANN, R.: 'An iterative method for the conformal map- [1] BERCER, A.: Mathematik der Lebensversicherung, Springer ping of doubly connected regions', J. Comput. Appl. Math. Wien, 1939. 14 (1986), 79-98. [2] GRAM, J.P.: 'Professor Thiele sore aktuar', Dansk For- Martin H. Gutknecht sikringsdrbog (1910), 26-37. MSC 1991: 30C20, 30C30 [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 THIELE DIFFERENTIAL EQUATION- Consider 1930, Wiley, 1998. an n year term life insurance, with sum insured S and [5] HOEM, J.M.: 'The reticent trio: Some little-known early dis- level premium P per time unit, issued at time 0 to an coveries in insurance mathematics by L.H.F. Oppermann, x years old person. Denote by py the force of mortality T.N. Thiele, and J.P. Gram', Internat. Statist. Bey. 51 at age y and by d the force of interest. If the insured is (1983), 213-221. [6] JOHNSON, N.L., AND KOTZ, S. (eds.): Leading personalities still alive at time t E [0, n), then the insurer must pro- in statistical science, Wiley, 1997. vide a reserve, Vt, which by statute is the mean value of [7] JORGENSEN, N.R.: Grundz@e einer Theorie der Lebensver- future discounted benefits less premiums. Splitting into sicherung, G. Fischer, 1913.
402 TILTED ALGEBRA
[8] NORBERG, R.: 'Reserves in life and pension insurance', Scan& 1) For every surjective stratified morphism f: M Actuarial d. (1991), 1-22. N, the restriction of f to the inverse image f-1 (S) of a [9] NORBERG, R.: Thorvald Nicolai Thiele, statisticians of the stratum S is a fibration. centuries, Internat. Statist. Inst., 2001. [10] SCHWEDER, W.: 'Scandinavian statistics, some early lines of 2) If there is a sequence of stratified morphisms M development', Scan& J. Statist. 7" (1980), 113-129. N 2~ I, where f is a Thorn mapping (an 'application [11] THIELE, T.N.: Element~er Iagttagelseslaere, Gyldendal, sans ficlatement') and I is a segment, then the map- Copenhagen, 1897. pings fa and fb over two points a and b in I have the [12] THIELE, T.N.: Theory of observations, Layton, London, 1903, same topological type, i.e. there are homeomorphisms h Reprinted in: Ann. Statist. 2 (1931), 165-308. (Translated from the Danish edition 1897.) and h' such that the following diagram commutes: [13] THIELE, T.N.: Interpolationsrechnung, Teubner, 1909. M~ h M6 Ragnar Norberg
MSC 1991:62P05 N~ -~ Nb h'
THOM-MATIIER STRATIFICATION - A stratifi- The importance of Thom-Mather stratifications is cation of a space such that each stratum has a neigh- emphasized by their applications to stability and topo- bourhood which fibres over that stratum, with levels logical triviality theorems. Among other important re- defined by a tubular function (called 'fonction tapis' in sults in singularity theory is the fact that any Whitney Thorn's and 'distance function' in Mather's terminoI- stratification (see Stratification) is a Thom-Mather ogy), and the fibrations and tubular functions associated stratification. Hence, a Whitney stratification satisfies to the strata are compatible with each other. Thorn topological triviality. The converse is false [1]; in fact, Mather stratifications satisfy the Thorn first and sec- being a Whitney stratification is equivalent to topolog- ond isotopy lemmas (see below), providing results such ical triviality for all sections by a generic flag [3]. as local topological triviality of the stratification, lo- References cal topological triviality along the strata of a morphism [1] BRIAN~ON, J., AND SPEDER, J.P.: 'La trivialit~ topologique and topological stability of generic smooth mappings n'implique pas les conditions de Whitney', Note C.R. Acad. Sci. Paris Ser. A 280 (1975), 365. ('generic' meaning transverse to the natural stratifies- [2] GORESKY, M., AND MACPHEHSON, R.: Stratified Morse the- tion of the jet space). ory, Springer, 1988. The word 'stratification' has been introduced by R. [3] Lg, D.T., AND TEISSIER, B.: 'Cycles fivanescents, sections Thorn in [5]. He proposed regularity conditions on how planes et conditions de Whitney II': Proe. Syrup. Pure Math., the strata of a stratification should fit together and Vol. 40, Amer. Math. Soc., 1983, pp. 65-103. [4] MATHER, J.: Notes on topological stability, Harvard Univ., stated the isotopy lemmas. The notes [4] of J. Mather 1970. provide a detailed proof, with improvements and sire- [5] THOU, R.: 'La stabilit~ topologique des applications polyno- plifications (cf. [2], which contains an excellent history miales', Enseign. Math. 8, no. 2 (1962), 24 33. of stratification theory). [6] THOU, R.: 'Ensembles et morphismes stratifies', Bull. Amer. Math. Soc. 75 (1969), 240-284. A Thom-Mathcr stratification of a space M consists [7] WHITNEY, H.: 'Local properties of analytic varieties', in of a tube system (Tx, 7Cx, px) associated to the strata S. CAIRNS (ed.): Differential and Combinatorial Topology, X of M, such that Tx is a tubular neighbourhood Princeton Univ. Press, 1965, pp. 205-244. of X in M, 7rx : Tx --+ X is the fibre projection associ- [8] WHITNEY, H.: 'Tangents to an analytic variety', Ann. of ated to Tx and the tubular function Px : Tx -+ R is a Math. 81 (1965), 496-549. continuous mapping satisfying p} 1 (0) = X. These data Jean-Paul Brasselet MSC 1991:57N80 are controlled in the following sense: If X and Y are two strata such that X is in the frontier of Y, then TILTED ALGEBRA - The endomorphism ring of a i) the restriction mapping (zcx, Px) : Tx n Y --+ X x tilting module over a finite-dimensional hereditary al- ]0, ec[ is a smooth submersion; gebra (cf. also Algebra; Endomorphism). ii) for a E Tx N Ty such that Try (a) C Tx, there are Let H be a finite-dimensional hereditary K-algebra, commutation relations K some field, for example the path-algebra of some finite C1) ~rx o Try(a) = rex(a), quiver without oriented cycles. A finite-dimensional H- 02) px o ~y (a) = ~x (a) module HT is called a tilting module if whenever both sides of the formulas are defined. i) p. dimT < 1, which always is satisfied in this con- Thom-Mather stratifications satisfy the isotopy lem- text; mas (as proposed by Thom): ii) Ext~(T,T) = 0; and
403 TILTED ALGEBRA
iii) there exists a short exact sequence 0 --+ H -+ If the hereditary algebra H is representation-finite Tz --+ T.2 -+ 0 with rl and T~ in add T, the category of (cf. also Algebra of finite representation type), finite direct sums of direct summands of T. Here, p. dim then any tilted algebra of type H also is representation- is projective dimension. finite. If H is tame (cf. also Representation of an associative algebra), then a tilted algebra of type H The third condition also says that T is maximal with re- either is tame and one-parametric, or representation- spect to the property Ext,(T, T) = 0. Note further, that finite. The latter case is equivalent to the fact that the a tilting module T over a hereditary algebra is uniquely tilting module contains non-zero pre-projective and pre- determined by its composition factors. Cf. also Tilting injective direct summands simultaneously. If H is wild module. (cf. also Representation of an associative algebra), The algebra B = EndH(T) is called a tilted algebra then a tilted algebra of type H may be wild, or tame of type H, and T becomes an (H, B)-bimodule (cf. also domestic, or representation-finite. Bimodule). See also Tilting theory. In H-mod, the category of finite-dimensional H- References modules, the module T defines a torsion pair (G,$-) [1] ASSEM, I.: 'Tilting theory - an introduction', in N. BAL- with torsion class G consisting of modules, generated by CERZYK ET AL. (eds.): Topics in Algebra, Vol. 26, Banach T and torsion-free class • = {Y: HomH(T,Y) = 0}. Center Publ., 1990, pp. 127-180. In B-mod it defines the torsion pair (X,3;) with tor- [2] AUSLANDER, M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter functors without diagrams', Trans. Amer. Math. Soc. 250 sion class 2( = {Y: T ®B Y = 0} and torsion-free (1979), 1-46. class ~2 = {Y: TorB(T,Y) = 0}. The Brenner-Butler [3] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAROW, V.A.: theorem says that the functors HomH(T,-), respec- 'Coxeter functors and Gabriel's theorem', Russian Math. tively T ®B --, induce equivalences between G and J;, Surveys 28 (1973), 17-32. whereas Extf/(T,-), respectively TorB(T,-), induce [4] BONGARTZ, K.: 'Tilted algebras', in M. AUSLANDER AND E. LLUIS (eds.): Representations of Algebras. Proc. ICRA III, equivalences between )c and X. In B-rood the torsion Vol. 903 of Lecture Notes in Mathematics, Springer, 1981, pair is splitting, that is, any indecomposable B-module pp. 26 38. is either torsion or torsion-free. In this sense, B-mod has [5] BRENNER, S., AND BUTLER, M.: 'Generalizations of 'less' indecomposable modules, and information on the the Bernstein-Gelfand-Ponomarev reflection functors', in category H-mod can be transferred to B-mod. V. DLAB AND P. GABRIEL (eds.): Representation Theory II. Proc. ICRA II, Vol. 832 of Lecture Notes in Mathematics, For example, B has global dimension at most 2 and Springer, 1980, pp. 103 169. any indecomposable B-module has projective dimension [6] HAPPEL, D.: Triangulated categories in the representation or injective dimension at most 1 (cf. also Dimension theory of finite dimensional algebras, Vol. 119 of London for dimension notions). These condition characterize the Math. Soc. Lecture Notes, Cambridge Univ. Press, 1988. [7] HAPPEL, D., REITEN, I., AND SMAL0, S.O.: 'Tilting in abelian more general class of quasi-tilted algebras. categories and quasitilted algebras', Memoirs Amer. Math. The indecomposable injective H-modules are in the Soc. 575 (1996). torsion class ~ and their images under the tilting func- [8] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras', tor HomH (T, -) are contained in one connected compo- Trans. Amer. Math. Soc. 274 (1982), 399-443. nent of the Auslander Reiten quiver F(B) of B-rood (cf. [9] KERNER, O.: 'Tilting wild algebras', J. London Math. Soc. 39, no. 2 (1989), 29-47. also Quiver; Riedtmann classification), and they [10] KERNER, O.: 'Wild tilted algebras revisited', Colloq. Math. form a complete slice in this component. Moreover, the 73 (1997), 67-81. existence of such a complete slice in a connected com- [11] LIu, S.: 'The connected components of the Auslander-Reiten ponent of F(B) characterizes tilted algebras. Moreover, quiver of a tilted algebra', J. Algebra 161 (1993), 505-523. the Auslander-Reiten quiver of a tilted algebra always [12] RINGEL, C.M.: Tame algebras and integral quadratic forms, Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984. contains pre-projective and pre-injective components. [13] RINGEL, C.M.: 'The regular components of the Auslander- If H is a basic hereditary algebra and He is a sim- Reiten Quiver of a tilted algebra', Chinese Ann. Math. Set. ple projective module, then T = H(1 - e) ® TrD He, B. 9 (1988), 1-18. where TrD denotes the Auslander-Reiten translation [14] STRAUSS, H.: 'On the perpendicular category of a partial tilt- ing module', J. Algebra 144 (1991), 43-66. (cf. Riedtmann classification), is a tilting module, sometimes called APR-tilting module. The induced tor- O. Kerner sion pair (G,¢-) in H-rood is splitting and He is the MSC 1991: 16G10, 16G20, 16G60, 16G70 unique indecomposable module in F. The tilting func- tor HomH(T,-) corresponds to the reflection functor TILTING FUNCTOR - When studying an algebra introduced by I.N. BernshteYn, I.M. Gel'land and V.A. A, it is sometimes convenient to consider another al- Ponomarev for their proof of the Gabriel theorem [3]. gebra, given for instance by the endomorphism of an
404 TILTING THEORY appropriate A-module, and functors between the two ii) Ext (T,T) = 0; and module categories. For instance, this is the basis of the iii) the number of non-isomorphic indecomposable Morita equivalence or the construction of the so- summands of T equals the number of simple A-modules. called Auslander algebras. An important example of this The fundamental work by S. Brenner and M.C.R. But- strategy is given by the tilting theory and the tilting ler, and D. Happel and C.M. Ringel, on tilting theory functors, as now described. have established the relations between the module cate- Let A be a finite-dimensional k-algebra, where k is gories rood A and rood B, where B = EndA(T), through a field, T a tilting (finitely-generated) A-module (cf. the tilting functors HOmA (T, -) and Ext~ (T, -) (cf. also Tilting module) and B = EndA(T). One can then as- Tilting functor). The particular case where A is a sign to T the functors HomA (T,-), -®B T, Ext~ (T,-), hereditary algebra gives rise to the notion of a tilted and TOrlB(-, T), which are called tilting functors. The algebra, which nowadays (as of 2000) plays a very im- importance of considering such functors is that they give portant role in the representation theory of algebras. equivalences between subcategories of the module cate- One can also consider the dual notion of eotiltin9 rood- gories mod A and rood B, results first established by S. ules. Brenner and M.C.R. Butler. Namely, HomA(T,-) and Tilting theory goes back to the work of I.N. its adjoint - ®B T give an equivalence between the sub- Bernshtein, I.M. Gel'fand and V.A. Ponomarev on the categories characterization of representation-finite hereditary alge- T(TA) = {MA: Extl(T, M) = 0} bras through their ordinary quivers (cf. also Quiver). and Their reflection functors on quivers has led to a Y(TA) = {NB: TorB(N,T) = 0}, module-theoretical interpretation by M. Auslander, M.I. Platzeck and I. Reiten. Next steps in this theory are the while Ext}4(T ,-) and TorB(-,T) give an equivalence work by Brenner and Butler and Happel and Ringel, between the subcategories which gave the basis for all its further development. Y(TA) = {NB: Tor~(N,T) = O} Worthwhile mentioning is the connection of tilting the- ory with derived categories established by Happel (cf. and also Derived category). X(TA) = {NB: N ®, T = 0}. The success of this strategy to study a bigger class of It is not difficult to see that (T(TA),5(TA)) and algebras through tilting theory has led to several gener- (X(TA), Y(TA)) are torsion pairs in rood A and rood B, alizations. On one hand, one can relax the condition on respectively. Clearly, one can now transfer information the projective dimension and consider tilting modules from rood A to rood B. One of the most interesting cases of finite projective dimension. In this way it was possi- occurs when A is a hereditary algebra and so the tot- ble to show the connection between tilting theory and sion pair (X(TA), Y(TA)) splits, giving in particular that some other homological problems in the representation each indecomposable B-module is the image of an in- theory of algebras. On the other hand, this concept can decomposable A-module either by HomA(T,-) or by be generalized to a so-called tilting object in more gen- Ext,(T,-) (in this case, the algebra B is called tilted, eral Abelian categories. For instance, this has led to the cf. also Tilted algebra). notion of a quasPtilted algebra. Recently (as of 2000), This procedure has been generalized in several ways there has been much work also on exploring such no- and it is worthwhile mentioning, for instance, its con- tions in categories of (not necessarily finitely-generated) nection with derived categories (cf. also Derived cate- modules over arbitrary rings. gory), or the notion of quasi-tilted algebras. It has also For references, see also Tilting theory; Tilted al- been considered for infinitely-generated modules over ar- gebra. bitrary rings. Fldvio Ulhoa Coelho For referenes, see also Tilting theory; Tilted alge- MSC 1991: 16Gxx bra. Fldvio Ulhoa CoeIho MSC 1991: 16Gxx TILTING THEORY- Artin algebras. A finitely-generated module T over TILTING MODULE - A (classical) tilting module an Artin algebra A (cf. also Artinian module) is called over a finite-dimensional k-algebra A (cf. also Algebra), a tilting module if p. dim A T _< 1 and Ext,(T, T) = 0 where k is a field, is a (finitely-generated) A-module T and there is a short exact sequence 0 -+ A --+ To --> satisfying: T1 ~ 0 with To,T1 C addT. Here, p. dimAT denotes i) the projective dimension of T is at most one; the projective dimension of T and add T is the category
405 TILTING THEORY of finite direct sums of direct summands of T (see Tilt- An important theoretical development of tilt- ing module). Dually, a A-module T is called a cotilt- ing theory was the connection with derived cat- ing module if the A°P-module D(T) is a tilting module, egories established by Happel [10]. The functor where D denotes the usual duality. If T is a tilting rood- HomA(T, .) : rood A --+ rood F when T is a tilting module ule and F = Endr(T) °p, then T is a tilting module over induces an equivalence RHomA(T,-): Db(A) -+ Db(F), F °p. Hence D(T) is a cotilting F-module. where Db(A) denotes the derived category whose ob- Let T be a tilting module, and let T = Fact be jects are the bounded complexes of A-modules. the category of finitely-generated A-modules gener- The set of all tilting modules (up to isomorphism) ated by T. The category T is a torsion class in the over a k-algebra A, k an algebraically closed field, category modA of finitely-generated A-modules. This has an interesting combinatorial structure: It is a count- yields an associated torsion pair (T,~C), where • = able simplicial complex E. This complex has been in- {C: HomA(T, C) = 0}. Dually, there is associated with vestigated by L. Unger in [21] and [22], where it was a cotilting module T the subcategory y = Sub T of A- proved that E is a shellable simplicial complex provided modules cogenerated by T. The category 3; is a torsion- it is finite, and that certain representation-theoretical free class and there is an associated torsion pair (2(, y) invariants are reflected by its structure. where 2( = {C: HomA(C,Y) = 0}. Analogues and generalizations. There is an analo- An important feature of tilting theory is the follow- gous concept of a tilting sheaf T for the category coh X ing connection between modA and modF when F = of coherent sheaves of a weighted projective line X (cf. EndA(T) °p for a tilting module T: If (T, 2r) denotes the also Coherent sheaf) as studied in [9]. The canonical torsion pair in mod A associated with T and (2(, y) the algebras introduced in [19] can be realized as endomor- torsion pair associated with D(T), then there are equiv- phism algebras of certain tilting sheaves. alences of categories: To obtain a common treatment of both the class of HomA(T, .): T --+ 32 tilted algebras and the canonical algebras, in [12] tilt- and ing theory was generalized to hereditary categories 7{, Ext ,(T, .): f 2(. that is, 7{ is a connected Abelian k-category with van- ishing Yoneda functor Ext2( ., .) and finite-dimensional (Cf. also Tilting functor.) In the special case where homomorphism and extension spaces. Here, k denotes T is a projective generator one recovers the Morita an algebraically closed field. An object T in 7{ with equivalence HOmA(T, .) : rood A --+ mod F, where T Ext~(T,T) = 0 such that Hom~(T,X) = 0 = is a projective generator of mod A. For a general mod- Ext~t(T,X ) implies X = 0, is called a tilting object in ule T, the Artin algebras A and F may be quite dif- 7{. The endomorphism algebra End~ T of a tilting ob- ferent, but they share many homological properties; in ject T is called a quasi-tilted algebra. Tilted algebras particular, one uses the tilting functors Homa(T, .) and and canonical algebras furnish examples for quasi-tilted Ext~ (T, .) in order to transfer properties between mod A algebras. and mod F. The transfer of information is especially use- ful when one already knows a lot about mod A and when There are two types of hereditary categories 7-/with the torsion pair (2(, y) splits, that is, when each inde- tilting objects: those derived equivalent to modH for composable F-module is in 2( or in y. This is the case some finite-dimensional hereditary k-algebra H and when A is hereditary. In this case, F is called a tilted those derived equivalent to some category coh X of co- algebra (cf. also Tilted algebra). Tilted algebras have herent sheaves on a weighted projective line X. Two played an important part in representation theory, since categories are called derived equivalent if their derived many questions can be reduced to this class of algebras. categories are equivalent as triangulated categories. In Tilting theory goes back to the reflection functors in- 2000, Happel [11] proved that these are the only possible troduced by I.N. BernshteYn, I.M. Gel'fand and V.A. hereditary categories with tilting object. This proved a Ponomarev [5] in the early 1970s. A module-theoretic conjecture stated, for example, in [17]. interpretation of these functors was given by M. Aus- Generalizations and applications of tilting modules. A lander, M.I. Platzeck and I. Reiten [3]. Further general- A-module T is called a generalized tilting module if izations where given by S. Brenner and M.C.R. Butler pd A T = n < ec and Ext~ (T, T) = 0 for i > 0 and there [6], where the equivalence Homa(T, .): T -~ 3; was es- is an exact sequence 0 ~ A --+ T1 --+ '.. --+ Tn --+ 0 tablished. The above definitions where given by D. Hap- with Ti C add T. Generalized tilting modules were in- pel and C.M. Ringel [13], who developed an extensive troduced in [16]. This concept was generalized to the theory of tilted algebras. A good reference for the early notion of tilting complexes by J. Rickard [18], who es- work in tilting theory is [2]. tablished some :Morita theory for derived categories'.
406 TITS QUADRATIC FORM
Let R be a ring and let PA be the category of finitely- [6] BRENNER, S., AND BUTLER, M.C.R.: 'Generalization of generated projective A-modules. Denote by Kb(PA) the Bernstein-Gelfand-Ponomarev reflection functors': Proc. Ot- category of bounded complexes over PA modulo homo- tawa Conf. on Representation Theory, 1979, Vol. 832 of Lec- ture Notes in Mathematics, Springer, 1980, pp. 103-169. topy. A complex is called a T E Kb(pA) tilting complex [7] CRAWLEY-BOEVEY, W., AND KERNER, O.: 'A functor between if Homgb(pA)(T,T[i]) = 0 for all/# 0 (here, [.] denotes categories of regular modules for wild hereditary algebras', the shift functor) and if addT generates Kb(pA) as a Math. Ann. 298 (1994), 481-487. triangulated category. Rickard proved that two rings R [8] DONKIN, S.: 'On tilting modules for algebraic groups', Math. and R ~ are derived equivalent (i.e. their module cate- Z. 212, no. 1 (1993), 39-60. [9] GEIGLE, W., AND LENZING, H.: 'Perpendicular categories gories are derived equivalent) if and only if R ~ is the with applications to representations and sheaves', J. Algebra endomorphism ring of a tilting complex T E Kb(PA). 144 (1991), 273 343. [10] HAPPEL, D.: 'Triangulated categories in the representa- The results mentioned above uses tilting mod- tion theory of finite dimensional algebras', London Math. ules/objects mainly to compare modA and modF, Soc. Lecture Notes 119 (1988). [11] HAPPEL, D.: 'A characterization of hereditary categories with where F = EndA T for some tilting module/object. tilting object', preprint (2000). There are other approaches, which use tilting modules [12] HAPPEL, D., REITEN, R., AND SMALO, S.O.: 'Tilting in to describe subcategories of mod A. Kerner [15] and W. abelian categories and quasitilted algebras', Memoirs Amer. Crawley-Boevey and Kerner [7] used tilting modules to Math. Soc. 575 (1996). investigate subcategories of regular modules over wild [13] HAPPEL, D., AND RINGEL, C.M.: 'Tilted algebras', hereditary 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. Quasi-hereditary algebras. Auslander and Reiten [4] [15] KERNER, O.: 'Tilting wild algebras', J. London Math. Soc. proved that there is a one-to-one correspondence be- 39, no. 2 (1989), 29-47. tween basic generalized tilting modules and certain co- [16] MIYASHITA, Y.: 'Tilting modules of finite projective dimen- sion', Math. Z. 193 (1986), 113-146. variantly finite subcategories of rood A. This correspon- [17] REITEN, I.: 'Tilting theory and quasitilted algebras': Proc. dence was further investigated [14]. The Auslander- Internat. Congress Math. Berlin, Vol. II, 1998, pp. 109-120. Reiten correspondence was applied to quasi-hereditary [18] RICKARD, J.: 'Morita theory for derived categories', J. Lon- algebras by Ringel [20] and his results served as a basis don Math. Soc. 39, no. 2 (1989), 436-456. for applications to Schur algebras by S. Donkin [8] and [19] RINGEL, C.M.: 'The canonical algebras': Topics in Algebra, Vol. 26:1 of Banach Center Publ., PWN, 1990, pp. 407-432. to quantum groups by H.H. Andersen [1]. In dealing [20] RINGEL, C.M.: 'The category of modules with good filtra- with quasi-hereditary algebras and highest-weight cat- tion over a quasi-hereditary algebra has alost split sequences', egories, the notion of a tilting module is now (2000) Math. Z. 208 (1991), 209-224. used in a related but deviating way, namely for all the [21] UNGER, L.: 'The simplicial complex of tilting modules over objects or modules that have both a A-filtration and quiver algebras', Proc. London Math. Soc. 73, no. 3 (1996), 27-46. a V-filtration. The isomorphism classes of the indecom- [22] UNGER, L.: 'Shellability ofsimplicial complexes arising in rep- posables that have both a A-filtration and a V-filtration resentation theory', Adv. Math. 144 (1999), 221-246. correspond bijectively to the elements of the weight L. Unger poset, and their direct sum is a tilting module in the MSC 1991: 16Gxx sense considered above. TITS QUADRATIC FORM- Let Q = (Qo, Q1) be References a finite quiver (see [8]), that is, an oriented graph with vertex set Q0 and set Q1 of arrows (oriented edges; cf. [i] ANDERSEN, H.H.: 'Tensor products of quantized tilting mod- ules', Commun. Math. Phys. 149, no. 1 (1992), 149-159. also Graph, oriented; Quiver). Following P. Gabriel [2] ASSEM, I.: 'Tilting theory - an introduction': Topics in Alge- [8], [9], the Tits quadratic form qQ: Z Qo -+ Z of Q is bra, Vol. 26 of Banach Center Publ., PWN, 1990, pp. 127- defined by the formula 180. [3] AUSLANDER, M., PLATZECK, M.I., AND REITEN, I.: 'Coxeter 2 functors without diagrams', Trans. Amer. Math. Soe. 250 jCQo i,jCQo (1979), 1-12. [4] AUSLANDER,M., AND REITEN, I.: 'Applications of contravari- where x = (xi)icQo E Z Q° and diy is the number of antly finite subcategories', Adv. Math. 86, no. 1 (1991), 111- arrows from i to j in Q1. 152. There are important applications of the Tits form [5] BERNSTEIN, I.N., CELFAND, I.M., AND PONOMAREV, V.A.: 'Coxeter functors and Gabriel's theorem', Russian Math. in representation theory. One easily proves that if Q is Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973), connected, then qQ is positive definite if and only if Q 19-33.) (viewed as a non-oriented graph) is any of the Dynkin
407 TITS QUADRATIC FORM diagrams An, D~, E6, ET, or Es (cf. also Dynkin dia- (see also [18]). In particular, he showed [6] that if 13 gram). On the other hand, the Gabriel theorem [8] as- is of tame representation type, then q~ is weakly non- serts that this is the case if and only if Q has only finitely negative, that is, q~(v) > 0 for all v C N ~. many isomorphism classes of indeeomposable K-linear K. Bongartz [3] associated with any finite- representations, where K is an algebraically closed dimensional basic K-algebra R a Tits quadratic form field (see also [2]). Let rePK(Q ) be the Abelian cate- as follows. Let {el,..., en} be a complete set of prim- gory of finite-dimensional K-linear representations of itive pairwise non-isomorphic orthogonal idempotents Q formed by the systems X = (Xi,¢9)jeQo,9~Q~ of of the algebra R. Fix a finite quiver Q = (Qo,Q1) finite-dimensional vector/(-spaces Xj, connected by K- with Qo = {1,...,n} and a K-algebra isomorphism linear mappings CZ : Xi --+ Xj corresponding to arrows R ~- KQ/I, where KQ is the path K-algebra of the /3: i --+ j of Q. By a theorem of L.A. Nazarova [12], quiver O (see [1], [10], [19]) and I is an ideal of R con- given a connected quiver Q the category rePK(Q) is of tained in the square of the Jacobson radical rad R of tame representation type (see [7], [10], [19] and Quiver) R and containing a power of rad R. Assume that Q has if and only if qQ is positive semi-definite, or equivalently, no oriented cycles (and hence the global dimension of R if and only if Q (viewed as a non-oriented graph) is any is finite). The Tits quadratic form qn : Z n -+ Z of R is of the extended Dynkin diagrams A~, Dn, E6, ET, or defined by the formula Es (see [1], [10], [19]; and [4] for a generalization). 2 Let Ko(Q) = K0(reptc(Q)) be the Grothendieck qR(x) = Z Z xixj+ Z r ,j i j, jCQo (~: i--+j)cQ1 (fl: i--+j)cQ1 group of the category repK(Q ). By the Jordan- HSlder theorem, the correspondence X where ri,j = IL M ejIei[, for a minimal set L of genera- dimX = (dimKXj)jeQo defines a group isomor- tors of I contained in ~i,j~Qo ejIei. One checks that phism dim: Ko(Q) -+ Z Q°. One shows that the rid = dimK Ext~(Sj,Si), where St is the simple R- Tits form qQ coincides with the Euler charac- module associated to the vertex t E Q0. Then the def- teristic XQ: Ko(Q) --+ Z, IX] ~ XQ([X]) = inition of qR depends only on R, and when R is of dimK EndQ(X) - dimN Ext~(X,X), along the iso- global dimension at most two, the form qR coincides morphism dim: Ko(Q) --+ Z Q°, that is, qQ(dimX) = with the Euler characteristic XR: K0(modR) -+ Z, XQ([X]) for any X in repg(Q) (see [10], [17]). [X] ~ Xn([X]) = Y2~=0(-1) m dimg Ext,(X, X), un- The Tits quadratic form qQ is related with an al- der a group isomorphism dim: K0(modR) -+ Z Q°, gebraic geometry context defined as follows (see [9], where K0(mod R) is the Grothendieck group of the [10], [19]). category mod R of finite-dimensional right R-modules For any vector v = (vj)jcQo E N Q°, con- (see [17]). Note that qR = qQ if R = KQ. sider the att:ine irreducible K-variety AQ(V) = By applying a Tits-type equality as above, Bongartz [3] proved that if R is of finite representation type, then I-Ii,jEQo H(j3:j--~i)CQ1 M~ ×~j (K)z of K-representations of Q of the dimension type v (in the Zariski topol- qn is weakly positive, that is, qn(v) > 0 for all non-zero ogy), where M~ ×vj (K)Z = M~ x~j (K) is the space of vectors v E N ~. The converse implication does not hold (vi x vj)-matrices for any arrow fl : j -+ i of Q. Consider in general, but it has been established if the Auslander- the algebraic group GgQ(d) = [IjcQo GI(vj,K) and Reiten quiver of R (see Riedtmann classification) has the algebraic group action *: ~gQ(d) x AQ(d) --+ AQ(d) a post-projective component (see [10]), by applying an defined by the formula (hi) * (M~) = (h~-lM~hj), idea of Drozd [5]. J.A. de la Pefia [14] proved that if R is of tame representation type, then qR is weakly non- where fl: j ~ i is an arrow of Q, M~ C Mvj×v~(K)9, negative. The converse implication does not hold in gen- hj E GI(vj,K), and hi C Gl(vi,K). An important eral, but it has been proved under a suitable assumption role in applications is played by the Tits-type equality on R (see [13] and [16] for a discussion of this problem qQ(v) = dimGfQ(V) - dimAQ(v), v C N Q°, where dim and relations between the Tits quadratic form and the denotes the dimension of the algebraic variety (see Euler quadratic form of R). [8]). Let (I, ~) be a partially ordered set with partial or- Following the above ideas, Yu.A. Drozd [5] introduced der relation -< and let max I be the set of all maximal and successfully applied a Tits quadratic form in the elements of (I, __). Following [5] and [15], D. Simson [20] study of finite representation type of the Krull-Schmidt defined the Tits quadratic form q~: Z I --+ Z of (I, _-3) by category Mats of matrix K-representations of partially the formula ordered sets ([, -<) with a unique maximal element (see [10], [19]). In [6] and [7] he also studied bimodule ma- trix problems and the representation type of boxes 13 by iEI i~j pCmax I means of an associated Tits quadratic form qB : Z ~ ~ Z jEI\max I
408 TOEPLITZ C*-ALGEBRA and applied it in the study of prinjective KI-modules, [10] GABRIEL, P., AND ROITER, A.V.: 'Representations of finite that is, finite-dimensional right modules X over the in- dimensional algebras': Algebra VIII, Vol. 73 of Encycl. Math. Stud., Springer, 1992. cidence K-algebra KI = K(I, ~_) of (I,__) such that [11] KASJAN, S., AND SIMSON, D.: 'Tame prinjective type and Tits there is an exact sequence 0 -+ P1 -+ P0 ~ X -+ 0, form of two-peak posers II', J. Algebra 187 (1997), 71-96. where P0 is a projective KI-module and P1 is a direct [12] NAZAROVA, L.A.: 'Representations of quivers of infinite type', sum of simple projectives. The additive Krull-Schmidt Izv. Akad. Nauk. SSSR 37 (1973), 752-791. (In Russian.) category prinKI of prinjective KI-modules is equiv- [13] PEI~A, J.A. DE LA: 'Algebras with hypercritical Tits form': Topics in Algebra, Vol. 26:1 of Banaeh Center Publ., PWN, alent to the category of matrix K-representations of 1990, pp. 353-369. (I, _) [20]. Under an identification Ko(prinKI) ~- Z I, [14] PEI~A, J.A. DE LA: 'On the dimension of the module-varieties the Tits form qI is equal to the Euler characteristic of tame and wild algebras', Commun. Algebra 19 (1991), XKZ: Ko(prin KI) -+ Z. A Tits-type equality is also 1795-1807. valid for qr [15]. It has been proved in [20] that q1 is [15] PE~A, J.A. DE LA, AND S~MSON, D.: 'Prinjective modules, re- flection functors, quadratic forms and Auslander-Reiten se- weakly positive if and only if prin KI has only a finite quences', Trans. Amer. Math. Soe. 329 (1992), 733-753. number of iso-classes of indecomposable modules. By [16] PE~A, J.A. DE LA, AND SKOWROr@KI, A.: 'The Euler and Tits [15], if prinK/is of tame representation type, then qI forms of a tame algebra', Math. Ann. 315 (2000), 37-59. is weakly non-negative. The converse implication does [17] RINGEL, C.M.: Tame algebras and integral quadratic forms, not hold in general, but it has been proved under some Vol. 1099 of Lecture Notes in Mathematics, Springer, 1984. [18] ROITER, A.V., AND I~[LEINER, M.M.: Representations of dif- assumption on (I, _) (see [11]). ferential graded categories, Vol. 488 of Lecture Notes in Math- A Tits quadratic form qA : Z n --+ Z for a class of clas- ematics, Springer, 1975, pp. 316-339. [19] SIMSON, D.: Linear representations of partially ordered sets sical D-orders A, where D is a complete discrete valu- and vector space categories, Vol. 4 of Algebra, Logic Appl., ation domain, has been defined in [21]. Criteria for the Gordon & Breach, 1992. finite lattice type and tame lattice type of A are given [20] SIMSON, D.: 'Posets of finite prinjective type and a class of in [21] by means of qA. orders', J. Pure Appl. Algebra 90 (1993), 77-103. [21] SIMSON, D.: 'Representation types, Tits reduced quadratic For a class of K-co-algebras C, a Tits quadratic form forms and orbit problems for lattices over orders', Contemp. qc:Z (Iv) --+ Z is defined in [22], and the co-module Math. 229 (1998), 307-342. [22] SIMSON,D.: 'Coalgebras, comodules, pseudoeompact algebras types of C are studied by means of qc, where Ic is a and tame comodule type', Colloq. Math. in press (2001). complete set of pairwise non-isomorphic simple left C- Daniel Simson co-modules and Z (Ic) is a free Abelian group of rank MSC 1991: 16Gxx Ircl.
References TOEPLITZ C*-ALGEBRA - A uniformly closed *- algebra of operators on a Hilbert space (a uniformly [1] AUSLANDER, V.I., REITEN, I., AND SMAL0, S.: Representa- closed C*-algebra). Such algebras are closely connected tion theory of Artin algebras, Vol. 36 of Studies Adv. Math., Cambridge Univ. Press, 1995. to important fields of geometric analysis, e.g., index the- [2] BERNSTEIN, I.N., GELFAND, I.M., AND PONOMAREV, V.A.: ory, geometric quantization and several complex vari- 'Coxeter functors and Gabriel's theorem', Russian Math. ables. Surveys 28 (1973), 17-32. (Uspekhi Mat. Nauk. 28 (1973), In the one-dimensional case one considers the Hardy 19 33.) [3] BONGARTZ, N.: 'Algebras and quadratic forms', J. London space H 2 (T) over the one-dimensional torus T (cf. also Math. Soe. 28 (1983), 461-469. Hardy spaces), and defines the Toeplitz operator [4] DLAB, V., AND RINGEL, C.M.: Indecomposable representa- Tf with 'symbol' function f E L°°(T) by Tfh := P(fh) tions of graphs and algebras, Vol. 173 of Memoirs, Amer. for all h E H2(T), where P: L2(T) --+ H2(T) is the Math. Soc., 1976. orthogonal projection given by the Cauchy integral [5] DROZD, Yu.A.: 'Coxeter transformations and representations of partially ordered sets', Funkts. Anal. Prilozhen. 8 (1974), theorem. The C*-algebra T(T) := C* (Tf : f E C(T)) 34 42. (In Russian.) generated by all operators Tf with continuous symbol f [6] DROZD, Yu.A.: 'On tame and wild matrix problems': Matrix is not commutative, but defines a C*-algebra extension Problems, Akad. Nauk. Ukr. SSR., Inst. Mat. Kiev, 1977, pp. 104-114. (In Russian.) 0 --+ K:(H2(T)) ~ T(T) --+ C(T) --+ 0 [7] DROZD, Yu.A.: 'Tame and wild matrix problems': Represen- tations and Quadratic Forms, 1979, pp. 39-74. (In Russian.) of the C*-algebra ~ of all compact operators; in fact, [8] GABRIEL, P.: 'Unzerlegbare Darstellungen 1', Manuscripta this 'Toeplitz extension' is the generator of the Abelian Math. 6 (1972), 71-103, Also: Berichtigungen 6 (1972), 309. group Ext(C(T)) ~ Z. [9] GABRIEL, P.: 'Reprfisentations ind~composables': Sdminaire Bourbaki (1973/73), Vol. 431 of Lecture Notes in Mathemat- C*-algebra extensions are the building blocks of K- ics, Springer, 1975, pp. 143-169. theory and index theory; in our case a Toeplitz
409 TOEPLITZ C*-ALGEBRA operator Tf is Fredholm (cf. also Fredhohn opera- such as a non-compact solution operator of the Neu- tor) if f C C(T) has no zeros, and then the index mann 0-problem [6]. Index(Tf) = dim Ker T/- dim Coker T/is the (negative) References winding number of f. [i] BOUTET DE MONVEL, L.: 'On the index of Toeplitz opera- In the multi-variable case, Toeplitz C*-algebras have tors of several complex variables', Invent. Math. 50 (1979), been studied in several important cases, e.g. for strictly 249-272. pseudo-convex domains D C C ~ [1], including the unit [2] COBURN, L.: 'Singular integral operators and Toeplitz oper- ators on odd spheres', Indiana Univ. Math. Y. 23 (1973), ball D = {z 6 Cn: Izll 2 +... + Iznl 2 < 1} [2], [10], for 433-439. tube domains and Siegel domains over convex 'symmet- [3] DOUGLAS,R., AND HOWE, R.: 'On the C*-algebra of Toeplitz ric' cones [5], [8], and for general bounded symmetric do- operators on the quarter-plane', Trans. Amer. Math. Soe. mains in C n having a transitive semi-simple Lie group 158 (1971), 203-217. of holomorphic automorphisms [7]. Here, the principal [4] LANDSTAD,M., PHILLIPS, J., RAEBURN, [., AND SUTHERLAND, C.: 'Representations of crossed products by coactions and new feature is the fact that Toeplitz operators Tf (say, principal bundles', Trans. Amer. Math. Soe. 299(1987), 747- on the Hardy space H2(S) over the Shilov boundary S 784. of a pseudo-convex domain D C C ~) with continuous [5] MUHLY, P., AND RENAULT, J.: 'C*-algebras of multivari- symbols f E Co(S) are not essentially commuting, i.e. able Wiener-Hopf operators', Trans. Amer. Math. Soc. 274 (1982), 1-44. [T/1,T/=] f~ K.(H2(S)), [6] SALINAS, N., SHEU, A., AND UPMEIER, H.: 'Toeplitz opera- tors on pseudoconvex domains and foliation algebras', Ann. in general. Thus, the corresponding Toeplitz C*-algebra Math. 130 (1989), 531-565. T(S) is not a (one-step) extension of K:; instead one [7] UPMEIER, H.: 'Toeplitz C*-algebras on bounded symmetric obtains a multi-step C*-filtration domains', Ann. Math. 119 (1984), 549-576. [8] UPMEIER, H.: 'Toeplitz operators on symmetric Siegel do- = ~i ~"" ~ I~ ~ T(S) mains', Math. Ann. 271 (1985), 401-414. [9] UPMmER, H.: Toeplitz operators and index theory in several of C*-ideals, with essentially commutative subquotients complex variables, Birkh£user, 1996. Zk+l/27k, whose maximal ideal space (its spectrum) re- [10] VENUGOPALKRISHNA, W.: 'Fredholm operators associated with strongly pseudoconvex domains in C n', J. Funct. Anal. fleets the boundary strata of the underlying domain. 9 (1972), 349 373. The length r of the composition series is an important [11] WASSERMANN, A.: 'Alg~bres d'op~rateurs de Toeplitz sur les geometric invariant, called the rank of D. The index the- groupes unitaires', C.R. Acad. Sci. Paris 299 (1984), 871- ory and K-theory of these multi-variable Toeplitz C*- 874. algebras is more difficult to study; on the other hand H. Upmeier one obtains interesting classes of operators arising by MSC 1991: 46Lxx geometric quantization of the underlying domain D, re- garded as a complex Kiihler manifold. TOEPLITZ SYSTEM - A system of linear equations A general method for studying the structure and rep- Tx = a with T a Toeplitz matrix. resentations of Toeplitz C*-algebras, at least for Shilov boundaries S arising as a symmetric space (not nec- essarily Riemannian), is the so-called C*-duality [11], MSC1991:15A57 [9]. For example, if S is a Lie group with (reduced) group C*-algebra C*(S), then the so-called co-crossed TRAVELLING SALESMAN PROBLEM A generic product C*-algebra C*(S) ®5 Co(S) induced by a natu- name for a number of very different problems. For in- ral co-action 5 can be identified with 1~(L2(S)). Now stance, suppose that a facility (a 'machine') starting the Cauehy-Szeg5 orthogonal projection E: L2(S) -+ from an 'idle' position is assigned to process a finite H2(S) (cf. also Cauchy operator) defines a certain set of 'jobs' (say n, n > 3 jobs). If the machine has to C*-completion C~(S) D C*(S), and the corresponding be 'calibrated' (or %et-up') for processing each of these Toeplitz C*-algebra T(S) can be realized as (a corner jobs and if the machine's 'calibration time' (the distance of) C~(S) ®5 Co(S). In this way the well-developed rep- metric) between processing of a pair of jobs in succession resentation theory of (co-) crossed product C*-algebras is dependent on the particular pair, then a reasonable [4] can be applied to obtain Toeplitz C*-representations objective is to organize this job assignment so it will related to the boundary cgD. For example, the two- minimize the total machine calibration time. One might dimensional torus S = T 2 gives rise to non-type-I want to assume that after the last job is processed the C*-algebras (for cones with irrational slopes), and the machine returns to its idle position. underlying 'Reinhardt' domains (cf. also Reinhardt A very similar problem exists when the 'machine' cor- domain) have interesting complex-analytic properties, responds to a computer centre which has n programs to
410 TRIANGLE CENTRE run, and each program requires resources such as a com- possible order magnitude of the required number of com- piler, a certain portion of the main memory, and perhaps puter operations. This casts these problems (the trav- some other 'devices'. I.e., each program requires a spe- elling salesman and the Hamiltonian circuit problems) cific configuration of devices. Conversion cost (or time) as being 'hard' (cf. also A/P). Essentially, for this sort from one configuration to another, say from the config- of problems, one does not presently (2000) know of any uration of program i to that of program j is denoted by solution scheme which does not require some sort of enu- cij (>_ 0). Thus, the question becomes that of determin- meration of all possible 'configuration' sequences. ing the cost minimizing order in which all the programs See [2], [1] for recent overviews of the problem. ought to be run. References If at the end of running all the programs by the com- [1] FLEISHNER, H.: '~IYaversing graphs: The Eulerian and Hamil- puter centre the system returns to an 'idle' configura- tonian theme', in M. DROR (ed.): Arc Routing: Theory, So- tion, then the number of possible ways to run these pro- lutions, and Applications, Kluwer Acad. Publ., 2000. [2] LAWLER, E.L., LENSTRA, J.K., RINNOY KAN, A.H.G., AND grams one after the other equals n! (for n + 1 configu- SHMOYS, D.B. (eds.): The traveling salesman problem, Wi- rat!ons). ley, 1985. This is the same problem as that in the story about Moshe Dror the lonely salesman who has to visit n sales outlets MSC 1991:90C08 (starting from his home) and wishes to travel the short- est total distance in the process. It is the salesman's TRIANGLE CENTRE - Given a triangle A1A2A3, a problem to select a distance-minimizing travel order of triangle centre is a point dependent on the three vertices outlet visits. Thus, the name travelling salesman prob- lem. of the triangle in a symmetric way. Classical examples are: In graph terminology terms, the problem is presented as that of a graph G = (V, E), where V is a finite set • the centroid (i.e. the centre of mass), the common of nodes ('cities') and E C_ V x V is the set of edges intersection point of the three medians (see Median (of connecting the node pairs in V. If one associates a real- a triangle)); valued 'cost' matrix (cij), i,j = 1,..., IVI, with the set • the incentre, the common intersection point of the of edges E, the travelling salesman dilemma becomes three bisectrices (see Bisectrix) and hence the centre that of constructing a cost-minimizing circuit on G that of the ineirele (see Plane trigonometry); visits all the nodes in V exactly once, if such a circuit • the circumcentre, the centre of the circumcircle exists (eft also Graph circuit). (see Plane trigonometry); If the requirement is that all the nodes in V are vis- • the orthocentre, the common intersection point of ited in a cost-minimizing fashion but without necessarily the three altitude lines (see Plane trigonometry); forming a circuit, then the problem is referred to as a • the Gergonne point, the common intersection travelling salesman path problem, or travelling salesman point of the lines joining the vertices with the opposite walk problem. Again, the question of the existence of tangent points of the incircle; such a path has to be addressed first. • the Format point (also called the Torrieelli point or first isogonic centre), the point X that minimizes the If the graph G = (V, A), A _C V x V, assigns a 'direc- sum of the distances IAIX] + IA2XI + IA3XI; tion' to each element in A (a subset of arcs), then the • the Grebe point (also called the Lemoine point or corresponding travelling salesman problem is of the 'di- symmedean point), the common intersection point of the rected' variety. Clearly, there is the option of the mixed three symmedeans (the symmedean through Ai is the problem, where some of the node pairs are connected by isogonal line of the median through Ai, see Isogonal); arcs and some by edges. • the Nagel point, the common intersection point The question of whether a circuit exists in a graph G of the lines joining the vertices with the centre points of which visits each node in V exactly once is commonly re- the corresponding excircles (see Plane trigonometry). ferred to as that of determining the existence of a Hamil- tonian circuit (or path; cf. also Hamiltonian tour). In [2], 400 different triangle centres are described. Graphs for which such a circuit (path) is guaranteed to The Nagel point is the isotomic conjugate of the Ger- exist are called Hamiltonian graphs. gonne point, and the symmedean point is the isogonal The difficulty of determining the existence of a Hamil- conjugate of the centroid (see Isogonal for both notions tonian circuit for a graph G and that of constructing a of 'conjugacy'). cost-minimizing travelling salesman circuit on a graph References G are very much the same when measured by the worst [1] JOHNSON, R.A.: Modern geometry, Houghton-Mifflin, 1929.
411 TRIANGLE CENTRE
[2] KIMBERLING, C.: 'Triangle centres and central triangles', References Congr. Numer. 129 (1998), 1-285. [1] ATANASSOV, K., HLEBAROVA, J., AND MIHOV, S.: 'Recur- M. Hazewinkel rent formulas of the generalized Fibonacci and Tribonacci MSC 1991:51M04 sequences', The Fibonacci Quart. 30, no. 1 (1992), 77-79. [2] BRUCE, I.: 'A modified Tribonacci sequence', The Fibonacci TRIBONACCI NUMBER A member of the Tri- Quart. 22, no. 3 (1984), 244-246. [3] FEINBERG, M.: 'Fibonacci-Tribonacci', The Fibonacci Quart. bonacci sequence. The formula for the nth number is 1, no. 3 (1963), 71-74. given by A. Shannon in [1]: [41 LEE, J.-Z., AND LEE, J.-S.: 'Some properties of the general- [~/2] In/a] ization of the Fibonacci sequence', The Fibonacci Quart. 25, no. 2 (1987), 111 117. m=0 r=0 m -I- r r [5] SCOTT, A., DELANEY, T., AND HOGGATT JR., V.: 'The Tri- bonacci sequence', The Fibonacci Quart. 15, no. 3 (1977), Binet's formula for the nth number is given by W. 193-200. Spickerman in [2]: [6] SHANNON, A.: 'Tribonacci numbers and Pascal's pyramid', fin+2 G-n+2 The Fibonacci Quart. 15, no. 3 (1977), 268; 275. T~= + t- [71 VALAVIGI, C.: 'Properties of Tribonacci numbers', The Fi- bonacci Quart. 10, no. 3 (1972), 231-246. ~n+2 Krassimir Atanassov
- - M~,~er~ ! 91".; : 11B39 where TRIGOiX,,~v.~ETRIC PSEUDO-SPECTRAL METH- 1 ((19 -~- 3X/~) 1/3 -l- (19 -- 3V/~) 1/3 -~- 1) P=~ ODS - Trigonometric pseudo-spectral methods, and spectral methods in general, are methods for solving ~=~ 1 [2 - (19 + 3v/~) '/a - (19 - 3V~) 1/3] -~- differential and integral equations using trigonometric +-g-v~i [(19 + 3X/~) 1/3 - (19 - 3v/~) 1/3] functions as the basis. Suppose the boundary value problem Lu = f is to and ~ is the complex conjugate of ~. be solved for u(x) on the interval x = [a, b], where L is References a differential operator in x and f(x) is some given [1] SHANNON, A.: 'Tribonacci numbers and Pascal's pyramid', smooth function (cf. also Boundary value problem, The Fibonacci Quart. 15, no. 3 (1977), 268; 275. ordinary differential equations). Also, u must sat- [2] SPICKERMAN, W.: 'Binet's formula for the Tribonacci se- u(a) = u~ u(b) = u b. quence', The Fibonacci Quart. 15, no. 3 (1977), 268; 275. isfy given boundary conditions and Krassimir Atanassov As in most numerical methods, an approximate so- MSC 1991:11B39 lution, UN, is sought which is the sum of N + 1 ba- sis functions, ¢,(x), n = 0,...,N, in the form UN = N TRIBONACCI SEQUENCE - An extension of the se- ~=0 a,¢~(x), where the coefficients an are the finite quence of Fibonacci numbers having the form (with set of unknowns for the approximate solution. A 'resid- a, b, c given constants): ual equation', formed by plugging the approximate so- lution into the differential equation and subtracting the to = a, tl = b, t2 = c, right-hand side, R( x; ao, . . . , aN) =- L[u g( x) ] -- f , is then tn+3 = tn+2 + t~+l + tn (n ~ 0). minimized over the interval to find the coefficients. The The concept was introduced by the fourteen-year- difference between methods boils down to the choice old student M. Feinberg in 1963 in [3] for the case: of basis and how R is minimized. The basis functions a = b = 1, c = 2. The basic properties are introduced in should be easy to compute, be complete or represent [2], [5], [6], [7]. the class of desired functions in a highly accurate man- The Tribonacci sequence was generalized in [1], [4] to ner, and be orthogonal (cf. also Complete system of the form of two sequences: functions; Orthogonal system). In spectral methods, trigonometric functions and their relatives as well as an÷3 = ttn+2 -~- Wn+l ~- Yn, other orthogonal polynomials are used. bn+3 = Vn+2 -}- Xn+l ~- Zn, If the basis functions are trigonometric functions such where u,v,w,x,y,z E {a,b} and each of the tuples as sines or cosines, the method is said to be a Fourier (u, v), (w, x), (y, z) contains the two symbols a and b. spectral method. If, instead, Chebyshev polynomials There are eight different such schemes. An open problem are used, the method is a Chebyshev spectral method. (as of 2000) is the construction of an explicit formula for The method of mean weighted residuals is used to each of them. minimize R and find the unknowns coefficients a~. An See also Tribonacci number. inner product (.,-) and weight function p(x) are defined,
412 TRIGONOMETRIC PSEUDO-SPECTRAL METHODS as well as N + 1 test functions wi such that (wi, R) = 0 smooth solutions. A simple finite difference approxi- for i = 0,...,N and (u,v) = f:u(x)v(x)p(x) dx. This mation has a convergence lu- UNI = O(h~), where yields N + 1 equations for the N + 1 unknowns. Pseudo- h = (b - a)/N and c~ is an integer; double the num- spectral methods, including Fourier pseudo-spectral and ber of points in the interval and the error goes down Chebyshev pseudo-spectral methods, have Dirac delta- by a factor 2% The convergence rate of spectral meth- functions (cf. also Dirae distribution) as their test ods is O(hh), sometimes called exponential, infinite oi" functions: wi(z) = 6(x -zi), where xi are interpolation spectral, stemming from the fact that convergence of a or collocation points. The residual equation becomes trigonometric series is geometric. If the solution is not R(xi;ao,...,aN) = 0 for i = 0,...,N. smooth, however, spectral methods will have an alge- The Galerkin method uses the basis functions as braic convergence linked to the continuity of the solu- the test functions. If L is linear, the following ma- tion. trix equation can be formed: L~n,~a~ = fro, where Rapid convergence allows fewer unknowns to be used, L,~,~ = (4,~, L¢~) and f,~ = (f, ¢,~). An alternative but more computational processing per unknown. Hence Galerkin formulation can be found by transforming the spectral methods are particularly attractive for prob- residual equation into spectral space R(x; ao,..., aN) = lem/computer matches in which memory and not com- ~-~.nrn(ao,...,aN)¢n(x) and setting r~ = 0 for n = puting power is the critical factor. O,..., N. In using Gauss-Jacobi integration to evaluate Multi-dimensional problems are handled by tensor- the inner products of the Galerkin method, the inte- product basis functions, which are basis functions that grands are interpolated at the zeros of the iV + 1st basis are products of 1-dimensional basis functions. Other or- function. By using the same set of points as collocation thogonal polynomials can be used in pseudo-spectral points for a pseudo-spectral method, the two methods methods, such as Legendre and Hermite and spherical are made equivalent. Problems can be cast in either grid- harmonics for spherical geometries. point or spectral coefficient representation. For trigono- A disadvantage of spectral methods is that only rel- metric bases, this result allows the complexity of com- atively simple domains and boundaries can be handled. putation to be reduced in many problems through the Spectral element methods, a combination of spectral and use of fast transforms. finite element methods, have in many cases overcome A main difference between spectral and other meth- this difficulty. Another difficulty is that spectral meth- ods, such as finite difference or finite element methods, ods are, in general, more complicated to code and re- is that in the latter the domain is divided into smaller quire more analysis to be done prior to coding than subdomains in which local basis functions of low order simpler methods. are used. With the basis functions frozen, more accu- Aliasin 9 is a phenomenon in which modes of degree racy is gained by decreasing the size of the subdomains. higher than in the expansion are interpreted as modes In spectral methods, the domain is not subdivided, but that are within the range of the expansion. This occurs global basis functions of high order are used. Accuracy is in, say, a problem with quadratic non-linearity where gained by increasing the number and order of the basis twice the range is created. If the coefficients near the up- functions. per limit are sufficiently large in magnitude, there may The lower-order methods produce algebraic systems be a significant error associated with aliased modes. For which can be represented as sparse matrices. Spectral a Fourier pseudo-spectral method, the coefficient aN/2+k methods usually produce full matrices. The solution is interpreted as a coefficient aN/2-k. By zeroing the then involves finding the inverse. Through the use of upper 1/3 of the coefficients, the quadratic nonlinearity orthogonality and fast transforms, full matrix inversion will only fill a range from 2/(3N/2) to 4/(3N/2). This can usually be accomplished with a complexity similar will only produce aliasing errors for modes 2/(3N/2) to to the sparse matrices. N/2, but these are to be zeroed anyway. This '2/3' rule removes errors for one-dimensional problems with qua- Boundary conditions are handled in a reasonably dratic non-linearity. It is debatable, however, whether straightforward manner. Sometimes the boundary con- in a 'well-resolved' simulation there is need to address ditions are satisfied automatically, such as with period- aliasing errors. icity and a Fourier method. With other types of con- ditions, an extra equation may be added to the system References to satisfy it, or the basis functions may be modified to [1] BOYD, J.P.: Chebyshev and Fourier spectral methods, automatically satisfy the conditions. second ed., Dover, 2000, pdf version: http://www- personal.engin.umich.edu/~jpboyd/book_spectral2OOO.html. The attractiveness of spectral methods is that they ['2] CANUTO, C., HUSSAINI, M.Y., QUARTERONI, A., AND gANG, have a greater than algebraic convergence rate for T.A.: Spectral methods in fluid dynamics, Springer, 1987.
413 TRIGONOMETRIC PSEUDO-SPECTRAL METHODS
[3] FORNBERG, B.: A practical guide to pseudospectral methods, D. GOTTLIEB, AND M.Y. HUSSAINI (eds.): Spectral Methods Vol. 1 of Cambridge Monographs Appl. Comput. Math., Cam- for Partial Differential Equations, SIAM, 1984. bridge Univ. Press, 1996. [5] GOTTLIEB, D., AND ORSZAG, S.A.: Numerical analysis of [4] GOTTLmB, D., HUSSAINI, M.Y., AND ORSZAC, S.A.: 'The- spectral methods: Theory and applications, SIAM, 1977. ory and application of spectral methods', in R.G. VOIGT, Richard B. Pelz MSC 1991: 65M70, 65Lxx
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