Showing papers on "Linear map published in 1989"

Linear Inverse and III-Posed Problems

Abstract: Publisher Summary This chapter considers linear inverse problems that have the following general structure. The first step is the definition of the direct problem, which must be linear. Then the solution of the direct problem defines a linear mapping L from the space X of all functions characterizing the properties of the physical sample (such as the density function in the case of a vibrating string or the refraction index in the case of a semi-transparent object, etc.) into the space Y of all corresponding measurable quantities (such as sequences of eigenvalues, scattering amplitudes, and so on). In the direct problem, the data are elements of X, while the solutions are elements of Y. In the corresponding inverse problem, the data and solutions are interchanged. The chapter also discusses the treatment of ill-posed problems. The basic idea in the treatment of ill-posed problem is the use of a priori information about the unknown object to constrict the class of approximate solutions. This means that additional information is needed. This information must be incorporated into the algorithm to produce a physically meaningful approximate solution. The additional information can consist of upper bounds on the solution and/or its derivatives, regularity properties of the solution (existence of derivatives up to a certain order, analyticity, etc.), localization properties of the solutions lower bounds on the solution and/or its derivative, and so on.

p-adic dynamics

Abstract: The quadratic map overp-adic numbers is studied in some detail. We prove that near almost all indifferent fixed points it is topologically conjugate to a quasiperiodic linear map. We also establish the existence of chaotic behavior and describe it using symbolic dynamics.

Linear and logarithmic capacities in associative neural networks

Abstract: A model of associate memory incorporating global linearity and pointwise nonlinearities in a state space of n-dimensional binary vectors is considered. Attention is focused on the ability to store a prescribed set of state vectors as attractors within the model. Within the framework of such associative nets, a specific strategy for information storage that utilizes the spectrum of a linear operator is considered in some detail. Comparisons are made between this spectral strategy and a prior scheme that utilizes the sum of Kronecker outer products of the prescribed set of state vectors, which are to function nominally as memories. The storage capacity of the spectral strategy is linear in n (the dimension of the state space under consideration), whereas an asymptotic result of n/4 log n holds for the storage capacity of the outer product scheme. Computer-simulated results show that the spectral strategy stores information more efficiently. The preprocessing costs incurred in the two algorithms are estimated, and recursive strategies are developed for their computation. >

Rank-preserving linear maps on b(x)

Abstract: Denote by B(X) the Banach algebra of all bounded linear operators on a complex Banach space X. In this paper, the representation of weakly continuous linear maps on B(X) which maps rank-1 operators to operators of rank at most 1 is given, and sequentially, some representation theorems for rank-preserving linear maps, spectrum-preserving linear maps and positivity-preserving linear maps on B(X) are obtained.

Discrete linear transformations of potential field data

Abstract: We present a new approach to perform any linear transformation of gridded potential field data using the equivalent‐layer principle. It is particularly efficient for processing areas with a large amount of data. An N × N data window is inverted using an M × M equivalent layer, with M greater than N so that the equivalent sources extend beyond the data window. Only the transformed field at the center of the data window is computed by premultiplying the equivalent source matrix by the row of the Green’s matrix (associated with the desired transformation) corresponding to the center of the data window. Since the inversion and the multiplication by the Green’s matrix are independent of the data, they are performed beforehand and just once for given values of N, M, and the depth of the equivalent layer. As a result, a grid operator for the desired transformation is obtained which is applied to the data by a procedure similar to discrete convolution. The application of this procedure in reducing synthetic anoma...

On interpolation of compact operators

Abstract: The authors extend a result of K. Hayakawa [J. Math. Soc. Jap. 21, 189-199 (1969; Zbl 0181.137)], and prove: If T is a linear operator such that T: A0 ! B0, is bounded,and T: A1 ! B1 is compact, and moreover, A1 A0, then T: ¯ A,q ! ¯B,q is compact for 0 < < 1, 0 < q 1.

Sequences of point transformations and linear canonical transformations in classical and quantum mechanics

Abstract: Sequences of point transformations and canonical linear transformations are considered in classical and quantum mechanics. It is shown that the unitary representations of such transformations can be obtained, in general, in the sense that in the limit, classical behavior is retrieved. In the particular case of one point transformation combined with two linear transformations, the results found in this way are exact. A new class of differential equations is thereby solved by quadratures.

Solving For The General Linear Transformation Relating 3-D Objects From The Minimum Moments

Abstract: This paper describes an extension of previous moment based techniques for the solution of the general linear transformation that describes the relationship between two 3D objects given either a 3D density map or a sufficient number of projections of the object. It is shown that values of the second and third order 3D moments are sufficient to solve this problem. The solution technique exploits the tensor nature of the moment set to enable solution of the object transformation matrix by direct (non-iterative) means. An implementation is also described that uses a fast moment method to generate the required data from object descriptions. Experimental results are shown.

Formula for Multiplicity of Contractions with Finite Defect Indices

Abstract: The multiplicity of a linear operator is invariant under various equivalence relations, such us unitary equivalence, linear similarity,. quasisimilarity. Moreover, under some restrictions on a given operator (e.g. for normal, or Dunford scalar operators), one can define a local version of the multiplicity, so called multiplicity function, which uniquely determins the equivalent class of such an operator.

Compact endomorphisms of Banach algebras. II

Abstract: In this note we exhibit examples which show that several conjectures concerning compact endomorphisms of commutative semisimple Banach algebras prove to be false. In this sequel to an earlier paper [4], we answer several open questions about compact endomorphisms of commutative semi-simple Banach algebras. In so doing, we show that their behavior is quite diverse. We recall that a compact endomorphism of a Banach algebra B is a compact linear map of B -* B which preserves multiplication. Further, if T is a non-zero endomorphism of B, and X is the maximal ideal space of B, then there exists a map q: X -* X such that Tf(x) = f(q$(x)) for all f E B, x E X. We will denote the nth iterate of q by q$, The main theorem in [4] states that if T is a non-zero compact endomnorphism of a commutative semi-simple Banach algebra B with maximal ideal space X, and if Tf = f o q$, thenn lq$ (x) is finite. As a consequence, if X is connected and B has an identity then l q$(X) is a singleton, while if B has no identity then 0 is the only compact endomorphism. Clearly, if B and X are as described above, and if B has an identity 1, then T: f -* f(xo) 1 is a compact endomorphism for each x0 E X. Much of what follows is concerned with the question of when such endomorphisms are the only non-zero compact endomorphisms. Two natural examples are C(X) for compact connected X and the disc algebra A. For the former, every non-zero compact endomorphism has the form T: f -* f(xo)1 for some xo E X, while for the disc algebra A, if Tf = f o q, then T is compact if, and only if, either q(z) = constant for all z, Izl 0, kk(z)l < I1 for all z, Izl < 1 [3]. In the case of the disc algebra, the unit circle F is both the Silov boundary and the strong boundary, and thus T: f -* f o q$ is a compact endomorphism of A if, and only if, q is constant or F n (range q) = 0 . Received by the editors October 18, 1988 and, in revised form, February 16, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B38, 46J99, 47B05. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

Optical computing using optical flip-flops in Fourier processors: use in matrix multiplication and discrete linear transforms.

Abstract: An architecture and the algorithms for matrix multiplication using optical flip-flops (OFFs) in optical processors are proposed based on residue arithmetic. The proposed system is capable of processing all elements of matrices in parallel utilizing the information retrieving ability of optical Fourier processors. The employment of OFFs enables bidirectional data flow leading to a simpler architecture and the burden of residue-to-decimal (or residue-to-binary) conversion to operation time can be largely reduced by processing all elements in parallel. The calculated characteristics of operation time suggest a promising use of the system in a real time 2-D linear transform.

Extremal positive and self-adjoint extensions of suboperators

Abstract: F. ollowing Halmos [1] a suboperator is a map from a subspace of a (complex) Hilbert space into the whole space which is a restriction of some bounded linear transformation, an operator on the space. A suboperator is said to be subpositive, subself-adjoint (and so on) if it is a restriction of a positive, self-adjoint (and so on) operator of the space. The characterization problem of subself-adjoint suboperators is solved by Krein, see [2]. Krein shows that a bounded and symmetric linear map from a subspace of a Hilbert space has a symmetric extension to the whole space (hence a self-adjoint extension) with the same bound. Krein's method of proof yields the normpreserving smallest, resp. largest self-adjoint extensions, in the usual ordering of self-adjoint operators on Hilbert space, too. The extension theorem for subpositive suboperators, due to one of the authors [3], led naturally to the reduction of Krein's theorem to the just mentioned positive extendibility problem. Another characterization of subpositive suboperators is given by Halmos [1]. The purpose of this note is first to show that the extension of a subpositive suboperator constructed in [3] is of minimal norm and smallest in the ordering of self-adjoint operators, between all the positive extensions. The existence of a largest positive extension (with the same norm) is then a simple consequence of the existence of the smallest positive extension. The extremal norm-preserving extension problem with respect to subself-adjoint suboperators is thus reduced to the corresponding, just mentioned question for subpositive suboperators.

On continuous preservation of norms and areas

Abstract: We give various properties, examples and equivalent conditions for mapsT of then-dimensional euclidean space into itself (n ⩾ 2) satisfying the generalised orthogonality equation|Tx ⋅ Ty| = |x ⋅ y| for allx, y inRn, where ⋅ stands for the usual dot product, and we prove that the only continuous maps verifying this condition are the orthogonal linear transformations.

Multiparameter bifurcation for some particular reaction–diffusion systems

Abstract: In some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

On the Vector-Valued Hilbert Transform

Abstract: Let H : Lp (R) Lp(R), 1 < p < ∞ be the real HILBERT transform. A bounded, linear operator u:E F (E, F BANACH spaces) is a HT-operator, if the mapping u ⊗ H : E ⊗ L2(R, E) L2(R, F) has a bounded, linear extension to L2(R) L2(R, F). For E = F and u = idE BOURGAIN [3] and BURKHOLDER [5] have shown that this holds if and only if E ϵ UMD. We study these HT-operators and, in particular, we construct a HT-operator which is not UMD-factorable. Furthermore, we show that a UMD-space E is a HILBERT space if and only if |idE ⊗ H| = 1.

The averaging method for weakly nonlinear operator equations

Abstract: A method asymptotic with respect to a small parameter is presented for solving Cauchy problems for the evolution equations where is a linear operator and is a nonlinear operator. It is assumed that the method of regular expansion in powers of leads to secular terms. Such terms can be removed by suitably defining how the terms of the asymptotic solution depend on the slow variable . The proposed method is modified for equations of second order in . The possibility of getting rid of the terms secular with respect to , and of applying the asymptotic methods in the case of problems with strong forced resonance, is indicated. Examples are given which illustrate possibilities for the proposed methods.Bibliography: 16 titles.

Scale-space for discrete images

Abstract: This article addresses the formulation of a scale-space theory for one-dimensional discrete images. Two main subjects are treated:Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output image does not exceed the number of local extrema (or zero-crossings) in the original image?How should one create a multi-resolution family of representations with the property that an image at a coarser level of scale never contains more structure than an image at a finer level of scale?We propose that there is only one reasonable way to define a scale-space for discrete images comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T(n; t) = e^{-t} I_n(t),, where $I_n$ are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.Some obvious discretizations of the continuous scale-space theory are discussed in view of the results presented. An important result is that scale-space violations might occur in the family of representations generated by discrete convolution with the sampled Gaussian kernel.

Linear operators commuting with translations on () are continuous

Abstract: Let (R) denote the Schwartz space of all C""-functions f: R C with compact supports in the real line R. An earlier result of the author on the automatic continuity of translation-invariant linear functionals on 9?(R) is combined with a general version of the Closed-Graph Theorem due to A. P. Robertson and W. J. Robertson in order to prove that every linear mapping S of 227(R) into itself, which commutes with translations, is automatically continuous.

Generalized inverses of linear operators in Hilbert spaces

Abstract: On etudie l'inverse generalise pour des operateurs lineaires entre des espaces de Hilbert et en particulier pour des operateurs lineaires clos densement definis

Linear interpolation with binary neurons

Abstract: A two-layer network of binary neurons is considered. After learning a finite number of input-output combinations, the network performs linear interpolation between these combinations at the macroscopic level of correlations. It is not necessary to separate learning phase and testing phase. The network can also be taught linear transformations. It is shown that by introducing a special interpretation of the Hebb rule it is possible to construct the model with neurons which are either strictly excitatory or strictly inhibitory. >