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1.
Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.  相似文献   

2.
In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) –1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.  相似文献   

3.
4.
We study the regularity of the minimizer u for the functional F (u,f)=|u|2 + |u–f{2 over all maps uH 1(, S 2). We prove that for some suitable functions f every minimizer u is smooth in if 0 and for the same functions f, u has singularities when is large enough.
Résumé On étudie la régularité des minimiseurs u du problème de minimisation minueH 1(,S2)(|u|2 + |u–f{2. On montre que pour certaines fonctions f, u est régulière lorsque 0 et pour les mêmes f, si est assez grand, alors u possède des singularités.
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5.
In this paper, we deal with the following stability problem: given a differential inclusion of the formx'F(t,x,), where is a parameter varying in a topological space , find conditions under which the set of all , such that the differential inclusion is controllable, is open in . Applying Theorem 3.1 of Ref. 3, we get a result in this direction, assuming, as leader hypotheses, thatF(t,·,) is a convex process, from into itself, and thatF(t,x,·) is lower semicontinuous.  相似文献   

6.
In this paper the relation betweenEP--matrices andE k P--matrices over an arbitrary filedF is studied. Further, conditions for the product ofE k P--matrices to be anE k P--matrix and for the reverse order law to hold for the polynomial Moore-Penrose inverse of the product ofE k P--matrices are determined  相似文献   

7.
Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL –1() into the sume of its principal and regular parts.  相似文献   

8.
The solution of the eigenvalue problem is examined for the polyomial matrixD()=Aos+A1s–1+...+As when the matricesA 0 andA 2 (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD() and to the zero eigenvalue of matrixA 0. The computation of the other eigenvalues ofD() is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 80–92, 1976.  相似文献   

9.
The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by n , a representation by polynomialsP n (), 1, connected with a Jacobi matrix, appears; 2) in the representation, instead of n , the expression n figures, where is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).The work is partially supported by the DFG, Project 436 UKR 113/39/0 and by the CRDF, Project UM1-2090.  相似文献   

10.
This paper describes a method for computing all the eigenvalues in a user supplied interval [a,b], and their associated eigenvectors, of the symmetric definite quadratic-matrix problem (M 2+C+K)x=0, where the matrices ar sufficiently sparse that methods based on similarity transformations are inappropriate. Only the triangular factorization of onen ×n matrix need be computed.Research sponsored in part by the Mathematics Department, University of Linköping, Sweden, and the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.  相似文献   

11.
Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim n n , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .  相似文献   

12.
A model in which strongness of is indestructible under + -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.  相似文献   

13.
14.
This paper examines the solving of the eigenvalue problem for a matrix M () with a nonlinear occurrence of the spectral parameter. Two methods are suggested for replacing the equation dat M()=0 by a scalar equationf()=0. Here the functionf() is not written formally, but a rule for computingf() at a fixed point of the domain in which the desired roots lie is indicated. Müller's method is used to solve the equationf()=0. The eigenvalue found is refined by Newton's method based on the normalized expansion of matrix M(), and the linearly independent vectors corresponding to it are computed. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 54–66, 1976.  相似文献   

15.
LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze 2i )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH 0=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a i (), 0) at 0 k , whereH (z)–z=a i ()z i . Then one can find a parameter solutionS (z) of (1) which has at each pointz 0 belonging to the domain of definition ofS 0, an expansion in seriesS (z)=z+b i ()(z–z 0) i with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 2 such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX 0, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé  相似文献   

16.
LetT B(H) be a bounded linear operator on a complex Hilbert spaceH. Let 0 (T) be an isolated point of (T) and let be the Riesz idempotent for 0. In this paper, we prove that ifT isp-hyponormal or log-hyponormal, thenE is self-adjoint andE H=ker(H0)=ker(H0 *.This research was supported by Grant-in-Aid Research 1 No. 12640187.  相似文献   

17.
Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf 1 –1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].  相似文献   

18.
We show here that by modifying the eigenvalues 2 < 3 < 0 < 1 of the geometric Lorenz attractor, replacing the usualexpanding condition 3+1 > 0 by acontracting condition 3+1 < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.  相似文献   

19.
Cauchy's problem for the equationu xx +x –1 u x =u t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.  相似文献   

20.
We consider generalized ruled surfaces in euclidean n-space n with k-dimensional generators and central ruled surface of dimension k–m+1 (O < m < k). Every orthogonal trajectoryy of the generators of defines a principal ruled surface y with generators totally orthogonal to the generators of . In each generator of y there exists an ellipsoid — called the indicatrix of the distribution parameters — which is defined by the distribution parameters of the tangent spaces to or y. Formulars will be given for the distribution parameters of and y .

Herrn Prof. Dr. H.R. Müller zum 70. Geburtstag  相似文献   

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