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1.
To gain understanding of the deformations of determinants andPfaffians resulting from deformations of matrices, the deformationtheory of composites f F with isolated singularities is studied,where f : YC is a function with (possibly non-isolated) singularityand F : XY is a map into the domain of f, and F only is deformed.The corresponding T1(F) is identified as (something like) thecohomology of a derived functor, and a canonical long exactsequence is constructed from which it follows that = µ(f F) – ß0 + ß1, where is the length of T1(F) and ßi is the lengthof ToriOY(OY/Jf, OX). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulaysingular locus (for example when f is the determinant function),relations between and the rank of the vanishing homology ofthe zero locus of f F are obtained.  相似文献   

2.
Exceptional Functions and Normality   总被引:1,自引:0,他引:1  
Yang proved in [10] that if f and f(k) have no fix-points forevery fF, where F is a family of meromorphic functions in adomain G and k a fixed integer, then F is normal in G. In thispaper we prove normality for families F for which every fF omits1 and f(k) omits 2, where 1 and 2 are analytic functions with. 1991 Mathematics SubjectClassification 30D35, 30D45.  相似文献   

3.
Given a measurable function f on (0, ) with Mellin transformF(s), let |f|p denote the Lp-norm of f with respect to the measuredx/x. We prove that under certain assumptions, for instanceif f is real and non-negative and F() converges for in an openinterval and F() 0, then wherecp (2e)–1. We derive similar inequalities for complex-valuedf, for the Lp-norm of the derivative of f, and for the supremumof real-valued f and of its derivative. The lower bounds areeminently applicable when f is a convolution product.  相似文献   

4.
Generalized Steffensen methods are nonderivative algorithmsfor the computation of fixed points of a function f. They replacethe functional iteration Zm+1=f(Zm) with Zm+1=Fn(Zm, where Fnis explicitly provided for every n 1 as a quotient of two Hankeldeterminants. In this paper we derive rules pertaining to thelocal behaviour of these methods. Specifically, and subjectto analyticity, given that is a bounded fixed point of f, thenit is also a fixed point of Fn. Moreover, unless f'() vanishesor is a root of unity, becomes a superattractive fixed pointof Fn of degree n; if f'() is a root of unity of minimal degreeq2, then is (as a fixed point of Fn) superattractive of degreemin {q-1, n}; if f'()=1, then is attractive for Fn; and, finally,if is superattractive of degree s (as a fixed point of f),then it becomes superattractive of degree (s + 1)n–1(ns+ s + 1)–1. Attractivity rules change at infinity (providedthat f()=). Broadly speaking, infinity becomes less attractivefor Fn, Since one is interested in convergence to finite fixedpoints, this further enhances the appeal of generalized Steffensenmethods.  相似文献   

5.
Minimal complex surfaces of general type with pg = 0 and K2= 7 or 8 whose bicanonical map is not birational are studied.It is shown that if S is such a surface, then the bicanonicalmap has degree 2 (see Bulletin of the London Mathematical Society33 (2001) 1–10) and there is a fibration f: S P1 suchthat (i) the general fibre F of f is a genus 3 hyperellipticcurve; (ii) the involution induced by the bicanonical map ofS restricts to the hyperelliptic involution of F. Furthermore, if , then f isan isotrivial fibration with six double fibres, and if , then f has five double fibres andit has precisely one fibre with reducible support, consistingof two components. 2000 Mathematics Subject Classification 14J29.  相似文献   

6.
Let Hilb6t–3(P3) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P3. Let H bethe component of Hilb6t–3(P3) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P3. Let F be the vector spaceof linear forms in the variables z1, z2, z3, z4. Denote by Fdthe vector space of homogeneous forms of degree d. Set X = (f2,f3)where f2 P(F2) is a quadric surface, and f3 P(F3/f2 ·F) is a cubic modulo f2. Wehave a rational map, : X ... Hdefined by (f2,f3) f2 f3. It fails to be regular along thelocus where f2 and f3 acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.  相似文献   

7.
Consider an analytic germ f:(Cm, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C4, 0)(C, 0), defined by f(x1, x2, x3, x4) =x1x2x3x4 has the homotopy type of S1xS1xS1. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [912, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF0Sn...Sn (whereF0 is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase FiFi, where the Fi are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].  相似文献   

8.
A function f: Rn R is a connectivity function if the graphof its restriction f|C to any connected C Rn is connected inRn x R. The main goal of this paper is to prove that every functionf: Rn R is a sum of n + 1 connectivity functions (Corollary2.2). We will also show that if n > 1, then every functiong: Rn R which is a sum of n connectivity functions is continuouson some perfect set (see Theorem 2.5) which implies that thenumber n + 1 in our theorem is best possible (Corollary 2.6). Toprove the above results, we establish and then apply the followingtheorems which are of interest on their own. For every dense G-subset G of Rn there are homeomorphisms h1,..., hn of Rn such that Rn = G h1(G) ... hn(G) (Proposition2.4). For every n > 1 and any connectivity function f: Rn R, ifx Rn and > 0 then there exists an open set U Rn such thatx U Bn(x, ), f|bd(U) is continuous, and |(x) – f(y)|< for every y bd(U) (Proposition 2.7). 1991 MathematicsSubject Classification: 26B40, 54C30, 54F45.  相似文献   

9.
We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tnx = 0 for somepositive integer n or the sequence {Tnx/||Tnx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f Lp [0,1] then the limit limn (n!||Vnf||p)1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace Lp[0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.  相似文献   

10.
Let K be a kernel on Rn, that is, K is a non-negative, unboundedL1 function that is radially symmetric and decreasing. We definethe convolution K * F by and note from Lp-capacity theory [11, Theorem 3] that, if F Lp, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rn with CK,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rn, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=Py*(K*F)=Py*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.  相似文献   

11.
Let G1 and G2 be locally compact groups. If T is an algebraisomorphism of L1(G1) onto L1(G2) with ||T|| (1+3), then G1and G2 are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.  相似文献   

12.
Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C*-algebraC*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w*-strongly exposed point of P(G)0, where P(G)0={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w*-strongly exposed point of P(G)0, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w*-strongly exposed point of P(G)0. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w*-stronglyexposed point of P(G)0, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F2 on two generators), there isno closed point in its reduced dual space .  相似文献   

13.
Let A be an order integral over a valuation ring V in a centralsimple F-algebra, where F is the fraction field of V. We showthat (a) if (Vh, Fh) is the Henselization of (V, F), then Ais a semihereditary maximal order if and only if AVVh is a semihereditarymaximal order, generalizing the result by Haile, Morandi andWadsworth, and (b) if J(V) is a principal ideal of V, then asemihereditary V-order is an intersection of finitely many conjugatesemihereditary maximal orders; if not, then there is only onemaximal order containing the V-order. 1991 Mathematics SubjectClassification 16H05.  相似文献   

14.
We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L1(0,a) is a given function. The number k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Letp p x/a) denote the eigenfunction associated with 1; then p(kp x/a) is the eigenfunction associated with k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and ft L1(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t* [0,1] such that (P) has a solutionfor f = ft*. Moreover, an appropriate choice of t* yields asolution u with an arbitrarily large L1(0,a)-norm which meansthat such f cannot be orthogonal to pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t* [0,1]such that (P) has a solution for f = ft*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.  相似文献   

15.
A family of transcendental meromorphic functions, fp(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of fp satisfies dim J(fp) 1/p, for0 < < 1/6p, and dim J(fp) 1–(30 ln ln p/ln p),for p4p–1/105 ln p < < p4p–1/104 ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.  相似文献   

16.
Normal Families and Shared Values   总被引:57,自引:0,他引:57  
For f a meromorphic function on the plane domain D and a C,let f(a) = {z D: f(z) = a}. Let F be a family of meromorphicfunctions on D, all of whose zeros are of multiplicity at leastk. If there exist b 0 and h > 0 such that for every f F,f(0) = f(k)(b) and 0 < |f(k+1)(z)| h whenever z f(0), thenF is a normal family on D. The case f(0) = Ø is a celebratedresult of Gu [5]. 1991 Mathematics Subject Classification 30D45,30D35.  相似文献   

17.
** Email: anil{at}math.iitb.ac.in*** Email: mcj{at}math.iitb.ac.in**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u*,y*) is then obtained as the limit of the optimal pair sequence{(un*, yn*)}, where un* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and yn* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.  相似文献   

18.
Consider the bounded linear operator, L: F Z, where Z RN andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, Li:F R. By the Riesz representation theorem, there exist functionsk(xi, ·) F : Lif = f, k(xi, ·)F. The functions,k(xi, ·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.  相似文献   

19.
In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of Fp and U be the formal powerseries algebra F[T], with group of units Ux. Let Q be a finitelygenerated abelian pro-p group. We write Zp[Q] for the completedgroup algebra of Q over Zp. Let T(Q) be the abelian group Hom(Q,Ux) of continuous homomorphisms from Q to Ux. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Zp[Q]to U.  相似文献   

20.
Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a1, a2,..., am}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L1, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.  相似文献   

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