Showing papers on "Linear map published in 1985"
TL;DR: In this paper, it was shown that the scale map of the zero crossings of almost all signals filtered by a Gaussian filter of variable size determines the signal uniquely, up to a constant scaling.
Abstract: We prove that the scale map of the zero crossings of almost all signals filtered by a Gaussian filter of variable size determines the signal uniquely, up to a constant scaling. The proof assumes that the filtered signal can be represented as a polynomial of finite, albeit possibly high, order. The result applies to zero and level crossings of linear differential operators of Gaussian filters. In this case the signal is determined uniquely, modulus the null space of the linear operator. The theorem can be extended to two-dimensional functions. These results are reminiscent of Logan’s theorem [ Bell Syst. Tech. J.56, 487 ( 1977)]. They imply that extrema of derivatives at different scales are a complete representation of a signal. They are especially relevant for computational vision in the case of the Laplacian operator acting on image intensities, and they suggest rigorous foundations for the primal sketch.
105 citations
TL;DR: The LM N-inverse as mentioned in this paper is a generalized inverse of a linear transformation A: →, where and are arbitrary finite dimensional vector spaces, defined using only geometrical concepts of linear transformations.
81 citations
Book•
01 Jan 1985
76 citations
TL;DR: In this article, a linear model can be expressed as many different linear models, all of which yield identical first and second moments of the data vector, and the models of such a set are defined as linearly equivalent.
44 citations
TL;DR: The modeling of system and component noise and error sources in optical linear algebra processors (OLAP's) are considered, with attention to the frequency-multiplexed OLAP.
Abstract: The modeling of system and component noise and error sources in optical linear algebra processors (OLAP's) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.
42 citations
TL;DR: In this article, the authors show how certain invariants in the formal solution can be described and calculated through matrix-theoretic properties of the coefficients and at the same time show how they are related to ones for the differential operator.
36 citations
TL;DR: In this article, the authors considered the problem of defining the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set.
Abstract: In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.
36 citations
TL;DR: In this article, it was shown that if the underlying spaces possess a Schauder basis, or more generally, they have what the authors call property L' it is theoretically possible to establish a finite dimensional alternative problem for (13), where L: D(L) c X --+ Y is a linear map, N: X + Y is nonlinear and X and Y are Banach spaces
32 citations
TL;DR: In this article, the size of the operator C (A) where A is a linear operator in a Hilbert space with norm at most 1 was studied and an application to variable step integration of initial value problems using one-leg methods was given.
23 citations
TL;DR: In this article, the structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitians with fixed indefinite inertia is examined.
Abstract: The structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitian matrices with fixed indefinite inertia is examined. It turns out that such a map is either a unitary similarity or a unitary similarity followed by a transposition (the case when the fixed inertia has equal number of positive and negative eigenvalues is excluded).
20 citations
TL;DR: In this article, the spectral properties of cosine operator-valued functions and their representations are studied, and a generalization of the known spectral criterion of Loomis for almost periodicity is proposed.
Abstract: This article concerns: 1) theorems on the spectra of operators formed from cosine operator-valued functions and representations; 2) inequalities (of Bernstein type) connecting the norms of operators with their spectral radii; 3) applications to second-order differential equations; 4) generalizations of the known spectral criterion of Loomis for almost periodicity, and an application to the investigation of almost periodicity of cosine operator-valued functions, representations, and solutions of functional equations; and 5) linear methods for summation of Fourier series in eigenfunctions of a linear operator generating a bounded (one-parameter) cosine operator-valued function.Bibliography: 41 titles.
TL;DR: Gramsch and Lay as discussed by the authors studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space, and proved the spectral mapping results by means of a unified approach based on a factorization of the operators provided by the Dunford-Taylor calculus and well-known properties of products of operators present in Fredholm theory.
Abstract: "B. Gramsch and D. Lay have studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space. Firstly they consider operators belonging to the Banach algebra of all bounded linear operators on the space, and later they derive the theorems for unbounded closed linear operators with non-empty resolvent from the aboye case ; but bounded closed linear operators with domain a proper subspace are not included. In this note we introduce a notion of extended essential spectra for any closed linear operator with non-empty resolvent, which covers the above cases. Then, in this more general context, we are able to prove the spectral mapping theorems by means of a more unified approach based on a factorization of the operators provided by the Dunford-Taylor calculus and well-known properties of products of operators present in Fredholm theory. "
TL;DR: In this article, for each S ⊆ R n, n, n, those linear transformations L : R n,n → R n/n,n which map S onto S are characterized.
01 Jan 1985
TL;DR: In this article, the authors identify subgroups of the group of affine linear transformations of finite fields of order p and n (for certainp andn) as groups of typeF a,b,−c for certain (not unique) choices ofa, b andc.
Abstract: For integersa, b andc, the groupF
a,b,−c is defined to be the group 〈R, S : R
2=RS
aRSbRS−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp
n (for certainp andn) as groups of typeF
a,b,−c for certain (not unique) choices ofa, b andc.
TL;DR: The finite Fourier-transform is considered as a linear transformation on a certain space of theta functions and thereby is seen to induce an invertible morphism of Abelian varieties.
Abstract: The finite Fourier-transform is considered as a linear transformation on a certain space of theta functions and thereby is seen to induce an invertible morphism of Abelian varieties. This is explained in the context of the representation theory of the finite symplectic group. Finally the MacWilliams identities in coding theory are discussed in the light of the theory of theta functions.
TL;DR: It is shown that over any cummutative ring R, the combinations, of 2 × 2 minors are the only quadratic forms vanishing on the matrices of rank 1, so any invertible linear transformation on matrices that preserves the rank-1 set over R will automatically do the same over all extensions of R.
Abstract: We show that over any cummutative ring R,the combinations, of 2 × 2 minors are the only quadratic forms vanishing on the matrices of rank 1. Hence any invertible linear transformation on matrices that preserves the rank-1 set over R will automatically do the same over all extensions of R. Similarly, the linear combinations of 4 × 4 Paffians are the only quadratic forms vanishing on the alternating matrices of rank 2. Hence again any invertible transformation preserving that set over R will do so formally. This fact allows us to determine the collection of such transformations
TL;DR: In this paper, the authors considered the n-dimensional real space R n, n > 3, provided with a closed solid cone K and thus being a Kantorovich space ("K-space").
Abstract: We consider the n-dimensional real space R n, n > 3, provided with a closed solid cone K and thus being a Kantorovich space ("K-space"). Let A be a nonnegative linear operator: A ~0r c K. The multiplication of A by a positive constant does not lead out from the set of nonnegative linear operators; therefore, for the investigation of the character of the spectrum it is sufficient to restrict ourselves to the class of nonnegative operators with spectral radius equal to unity. Everywhere in the sequel, by the term "nonnegative operator" we shall mean a nonnegative linear operator with spectral radius equal to unity.
TL;DR: In this paper, the evolution of physical systems in which two separate scales are present can be described in terms of slow and fast variables, by a linear transformation the system is decomposed into a slow and a fast subsystem.
Abstract: The evolution of physical systems in which two separate scales are present can be described in terms of slow and fast variables. In the present communication linear equations are studied. By a linear transformation the system is decomposed into a slow and a fast subsystem. This method has some advantages over previous methods. One simplifying feature is that initial conditions need not be taken into account. As an application it is shown that in the case that only even variables are involved the Onsager symmetry remains present at the reduced level.
TL;DR: In this paper, it is shown that Gα achieves a minimum at a unique point xα in, and by using the alternative method the xα are shown to converge to a least square solution of Lx = y with rate of convergence of order α2.
Abstract: The method of regularization is used in a general setting to obtain least squares solutions of thelinear equation Lx = y, permitting applications to linear boundary value problems of ordinary and partial differential equations. In regularization the functional is minimized over the domain , where L is assumed to be aclosed densely defined linear operator from a Hilbert space X into a Hilbert space Y, α is a nonzero parameter, and T is a linearoperator from X into a Hilbert space Z with . Under suitable conditions on L and T, it is shown that Gα achieves a minimum at a unique point xα in , and by using the alternative method the xα are shown to converge to a least squares solution of Lx = y with rate of convergence of order α2 . The regularization method is also recast as a least squares process. Finally, the important special case Z = X and T = I is examined in detail, andthe method is applied to the numerical solution of some model boundary value problems.
TL;DR: In this paper, the so-called "perturbative error correction" is analyzed on the pattern of the correlation problem, and the second quantization formalism expands under a polynomial form both the cler and the correlation operator.
Abstract: We reexamine an error potential method. The so-called "perturbative error correction" is analysed on the pattern of the correlation problem. In the whole C.I. space the idea of a perturbation correction of each w.f., cannot lead to a linear operator. But a restriction of this process to the uncorrelated q-configurations only permits proposing a q-operator which is of linear form in the whole C.I. space. Summing over q we obtain a Cler operator adapted to an r-repetitive method which performs the whole C.I. without diagonalization. On the other hand Cler is determined by a rapidly convergent, recurrent process. Its r-times product has as a limit a correlation operator adapted to all the electronic configurations. Now, the second quantization formalism expands under a polynomial form both the cler and the correlation operator. Both these operators can be expanded in clusters. Then the excitation matrix appears to be a sum of an infinite set of r-matrices which can be determined by an ordinary perturbation.
01 Mar 1985
TL;DR: The necessary and sufficient conditions for S φ to be a subspace were given in this paper, where the set consisting of the vectors whose minimal polynomial φ(x) together with the zero vector was considered.
Abstract: Let V be a finite dimensional vector space over the field Fand φ (x)∊F[x].Letx V → V be a linear operator. Let Sφ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S φ to be a subspace.
01 Jan 1985
TL;DR: In this paper, mutually equivalent sets of necessary and sufficient conditions are derived that there exists a completely positive, unity preserving linear map Ttransforming 0-1,..., an simultaneously into o i.., o)n: a)k = ak° T,k = l,...9n.
Abstract: Let {o^,..., a)n] and {
01 Jan 1985
TL;DR: In this paper, eight new CAL operations are presented which can be combined with general matrix operations to construct element matrices from any conceivable variational principle, covering one and two dimensions.
Dissertation•
01 Jan 1985
TL;DR: In this paper, the authors give sufficient conditions for global bifurcation of solutions to the nonlinear eigenvalue problem: F(X, lambda) = 0, where F : X x IR→Y, with X X IR, Y Banach spaces and (x,lambda) ∈ Xx IR.
Abstract: The objective of this thesis is to give sufficient conditions for global bifurcation of solutions to the nonlinear eigenvalue problem:F(X,lambda) = 0, where F : X x IR→Y, with X x IR, Y Banach spaces and (x,lambda) ∈ X x IR. F(.,lambda) is assumed to belong to the class of A-proper maps and to be of the non-standard form, an A-proper, linear operator A - lambdaB : X → Y plus a nonlinear mapping R(.,lambda) : X → Y. R(X,lambda) is taken to satisfy a smallness condition in x at the origin in X. Our analysis is based on an extension of known methods, for obtaining global bifurcation results, which have been used successfully when the mappings involved are compact or k-set contractive. Chapter One is an introduction to the concepts used throughout the thesis, including Fredholm maps of index zero, A-proper maps and generalised topological degree. In Chapter Two we state and prove our main global bifurcation theorem in terms of the generalised degree; this result forms the basis for the proofs of all the main theorems in the thesis. Chapters Three and Four contain various global bifurcation theorems, for different sets of hypotheses imposed on the mapping F and the underlying spaces X x IR and Y. Finally, in Chapter Five we apply our results to certain classes of ordinary differential equations and obtain existence results, for periodic solutions in one case and not necessarily periodic solutions in another. The main results are: Theorem 2.10; Theorems 3.3 and 3.13; Theorems 4.7, 4.12, 4.15 and 4.18.