Showing papers on "Linear map published in 1982"
Book•
01 Jan 1982
TL;DR: Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.
Abstract: A unique introduction to the theory of linear operators on Hilbert space. The author presents the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.
693 citations
TL;DR: In this paper, the Ulam problem of finding conditions in order for a linear mapping near an approximately linear mapping to exist is solved. But it is not shown that the conditions in this problem can be expressed in terms of a Banach space.
532 citations
TL;DR: In this paper, the modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described.
Abstract: The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII1 factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.
309 citations
Book•
02 Dec 1982
TL;DR: In this article, first-order autonomous systems and second-order Hamiltonian systems of one degree of freedom are studied. But they do not consider the non-linear transformations of the plane.
Abstract: 1. First-order autonomous systems 2. Linear transformations of the plane 3. Second-order autonomous systems 4. Conservative Hamiltonian systems of one degree of freedom 5. Legrangians 6. Transformation theory 7. Angle-action variables 8. Perturbation theory 9. Adiabatic and rapidly oscillating conditions 10. Linear systems 11. Chaotic motion and non-linear maps Appendixes Index.
215 citations
TL;DR: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0.
Abstract: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...
83 citations
Book•
26 Apr 1982
TL;DR: In this article, a linear transformation and its matrix representation of a vector space is used for rigid-body guidance in three-dimensional kinematics, where the objective is to find a plane linkage for a rigid body.
Abstract: 1 Mathematical Preliminaries.- 1.0 Introduction.- 1.1 Vector space, linear dependence and basis of a vector space.- 1.2 Linear transformation and its matrix representation.- 1.3 Range and null space of a linear transformation.- 1.4 Eigenvalues and eigenvectors of a linear transformation.- 1.5 Change of basis.- 1.6 Diagonalization of matrices.- 1.7 Bilinear forms and sign definition of matrices.- 1.8 Norms, isometries, orthogonal and unitary matrices.- 1.9 Properties of unitary and orthogonal matrices.- 1.10 Stationary points of scalar functions of a vector argument.- 1.11 Linear algebraic systems.- 1.12 Numerical solution of linear algebraic systems.- 1.13 Numerical solution of nonlinear algebraic systems.- References.- 2. Fundamentals of Rigid-Body Three-Dimensional Kinematics.- 2.1 Introduction.- 2.2 Motion of a rigid body.- 2.3 The Theorem of Euler and the revolute matrix.- 2.4 Groups of rotations.- 2.5 Rodrigues' formula and the Cartesian decomposition of the rotation matrix.- 2.6 General motion of a rigid body and Chasles' Theorem.- 2.7 Velocity of a point of a rigid body rotating about a fixed point.- 2.8 Velocity of a moving point referred to a moving observer.- 2.9 General motion of a rigid body.- 2.10 Theorems related to the velocity distribution in a moving rigid body.- 2.11 Acceleration distribution in a rigid body moving about a fixed point.- 2.12 Acceleration distribution in a rigid body under general motion.- 2.13 Acceleration of a moving point referred to a moving observer.- References.- 3. Generalities on Lower-Pair Kinematic Chains.- 3.1 Introduction.- 3.2 Kinematic pairs.- 3.3 Degree of freedom.- 3.4 Classification of lower pairs.- 3.5 Classification of kinematic chains.- 3.6 Linkage problems in the Theory of Machines and Mechanisms.- References.- 4. Analysis of Motions of Kinematic Chains.- 4.1 Introduction.- 4.2 The method of Denavit and Hartenberg.- 4.3 An alternate method of analysis.- 4.4 Applications to open kinematic chains.- References.- 5. Synthesis of Linkages.- 5.1 Introduction.- 5.2 Synthesis for function generation.- 5.3 Mechanism synthesis for rigid-body guidance.- 5.4 A different approach to the synthesis problem for rigid-body guidance.- 5.5 Linkage synthesis for path generation.- 5.6 Epilogue.- References.- 6. An Introduction to the Optimal Synthesis of Linkages.- 6.1 Introduction.- 6.2 The optimisation problem.- 6.3 Overdetermined problems of linkage synthesis.- 6.4 Underdetermined problems of linkage synthesis subject to no inequality constraints.- 6.5 Linkage optimisation subject to inequality constraints. Penalty function methods.- 6.6 Linkage optimisation subject to inequality constraints. Direct methods.- References.- Appendix 1 Algebra of dyadics.- Appendix 2 Derivative of a determinant with respect to a scalar argument.- Appendix 4 Synthesis of plane linkages for rigid-body guidance.
49 citations
TL;DR: In this article, it was shown that the set SA(k) of k-dimensional A-invariant subspaces is a compact subvariety of the Grassmann manifold Gk(V), but it need not be a Schubert variety.
Abstract: If V is a finite-dimensional vector space over R or C and A E Hom( V), the set SA(k) of k-dimensional A-invariant subspaces is a compact subvariety of the Grassmann manifold Gk(V), but it need not be a Schubert variety. We study the topology of SA(k). We reduce to the case where A is nilpotent. In this case we prove that SA(k) is connected but need not be a manifold. However, the subset of SA(k) consisting of those subspaces with a fixed cyclic structure is a regular submanifold of Gk(V).
30 citations
TL;DR: In this paper, the set of all m × n matrices over an algebraically closed field whose ranks lie in the set E, where E is a subset of {1,2,…, m }.
19 citations
TL;DR: In this paper, it was shown that if L is a linear map from the set of n × n complex matrices into itself such that L(adj A) = adj L(A) for all A, and if n ≠ 2, then L has one of the forms, a linear combination of maps of the form PA(adj P) and/or QAT (adj Q) need not be nonsingular when n = 2
Abstract: It is shown that if L is a linear map from the set of n × n complex matrices into itself such that L(adj A) = adj L(A) for all A, and if n ≠ 2, then L has one of the forms is a linear combination of maps of the form PA(adj P) and/or QAT (adj Q) The P's and/or Q's need not be nonsingular when n = 2
17 citations
01 May 1982
TL;DR: A grouping of phoneme is proposed so that one adaptation parameter set is used for all phonemes that belong to any one group, and the cost of phoneme class-specific adaptation is very high, but the method needs a large learning set.
Abstract: Speaker dependence of automatic speech recognition systems can be reduced by applying speaker-specific transformations to adapt the speech signal of a new speaker to that of the reference speaker. Initial investigations showed that speaker adaptation can be performed by transformations using spectral weighting and spectral warping. These heuristic methods can be substituted by a general linear matrix transformation, the parameters of which are determined by mean square error optimisation. The improvement of the recognition rate achievable by this matrix transformation is very high, but the method needs a large learning set. This can be reduced by restriction of the matrix to a band including the main diagonal in the middle. This banded matrix yields results close to those of the general matrix. Adaptation can be performed speaker-specifically as well as speaker- and class-specifically. As the cost of phoneme class-specific adaptation is very high, a grouping of phonemes is proposed so that one adaptation parameter set is used for all phonemes that belong to any one group.
15 citations
TL;DR: In this paper, the existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.
Abstract: A known existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is being proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.
TL;DR: In this paper, a linear map between central simple algebras preserves reduced norms, and it is an isomorphism or antiisomorphism followed by multiplication by an element of reduced norm 1.
TL;DR: In this paper, the similarity class of an n×n diagonal matrix A with distinct eigenvalues and nonzero trace is defined and the form of a linear transformation on the n × n matrices that preserves S{A] is determined.
Abstract: Let S(A) be the similarity class of an n×n diagonal matrix A with distinct eigenvalues and nonzero trace. The form of a linear transformation on the n×n matrices that preserves S{A) is determined.
TL;DR: In this article, the authors make use of appropriate forms of sector conditions in conjunction with the decomposition of a linear operator into two linear operators φ and G - φ, where φ has a normal transfer function matrix, and propose a useful stability criterion for a class of multivariable nonlinear feedback systems.
Abstract: The definition and properties of sector conditions introduced by Zames for scalar systems can be extended to the multivariable case. The present paper makes use of appropriate forms of sector conditions in conjunction with the decomposition of a linear operator into two linear operators φ and G - φ, where φ has a normal transfer function matrix ; and proposes a useful stability criterion for a class of multivariable non-linear feedback systems. The relevant stability result is developed further by a suitable interpretation of sector conditions in the frequency domain. A particular choice for φ leads to a generalization of the circle criterion in which the Nyquist plot of the frequency response of scalar systems is replaced by bands swept by circles whose centres and radii are related in a direct manner to the numerical range of G(jω). The result has a simple graphical interpretation and lends itself to a computer implementation which can be shown to be numerically stable. The approach is also suitable for...
TL;DR: In this article, the alternating powers of a Hilbert space were studied and it was proved that the norm of the linear map D ∧k(A) depends only upon |A| and is assumed at the identity.
TL;DR: In this paper, the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds) were determined for i = 1,2 and σ was a lattice isomorphism from the invariant subspace lattice of A 1 onto A 2.
TL;DR: In this article, the Floquet theory for the quasi-periodic system was established for the first time, and the authors established the FLOFT theory for all quasiperiodic systems.
Abstract: In this paper we establish the Floquet theory for the quasi-periodic system
$$\frac{{dx}}{{dt}} = A(\omega _1 t,\omega _2 t, \cdots ,\omega _m t)x$$
(0.1)
TL;DR: In this paper, the authors give the criterion about solution existence of the nonlinear operator equation and the convergence criterion of Galerkin approximate solutions, and give a convergence criterion for the solution of this problem.
TL;DR: In this paper, a new definition of generalized separability for a linear operator L of arbitrary order is given which, when specialized to the Helmholtz operator, includes ordinary separability in nonorthogonal and heat type coordinates.
Abstract: A new definition of (generalized) separability for a linear operator L of arbitrary order is given which, when specialized to the Helmholtz operator, includes ordinary separability in nonorthogonal and “heat type” coordinates. On the basis of this definition, a set of commuting operators is explicitly constructed and various relationships between eigenvalue problems for these operators and separated equations corresponding to $Lu = 0$ are derived. Moreover, a transformation of Backlund type for separable operators is obtained.
TL;DR: In this paper, the problem of generating the state sensitivity functions of linear time-invariant continuous systems using only one additional cascaded model having the same dimension as that of the system is treated.
Abstract: The problem of generating the state sensitivity functions of linear time-invariant continuous systems using only one additional cascaded model having the same dimension as that of the system is treated in this paper. Necessary and sufficient conditions for the state sensitivity functions to be expressed as linear transformations of the combined state of the system and the cascaded model are derived. Explicit expressions for these linear transformations are obtained via the necessary and sufficient conditions for the controllable and alpha canonical forms, both with single input.
TL;DR: An efficient algorithm is developed to solve linear equation systems which can be partitioned into symmetric and unsymmetric parts and the software allows partial factorization thus allowing even more efficient solution of those systems whose symmetric part is derived from a linear operator.
01 Dec 1982
TL;DR: In this article, the problem of minimax state estimation of a linear time invariant system which is driven by and observed in the presence of noise processes with uncertain second order statistics is considered.
Abstract: This paper considers the problem of minimax state estimation of the states of a linear time invariant system which is driven by and observed in the presence of noise processes with uncertain second order statistics. When the process noise and observations are scalars, the problem is shown to be equivalent to a scalar minimax estimation problem. The existence of a minimax solution is thereby established, and the minimax filter is shown to be a linear transformation of the minimax filter for the scalar problem.
TL;DR: In this article, the tensor virial theorem is analyzed in relation to orthogonal linear transformations, and the physical implications are discussed in terms of orthogonality and tensor curvature.
Abstract: The tensor virial theorem is analyzed in relation to orthogonal linear transformations. The physical implications are discussed.
TL;DR: In this paper, an iterative method is described for improving the accuracy of the solution of linear operator equations when there are errors in computing the operator and errors in reading the right-hand side (with cut-off or rounding) in computing systems with contracted place mesh.
Abstract: An iterative method is described for improving the accuracy of the solution of linear operator equations when there are errors in computing the operator and errors in reading the right-hand side (with cut-off or rounding) in computing systems with contracted place mesh.
TL;DR: In this paper, the authors generalize Wendel's theorem to cover an arbitrary closed linear map in L1(G), G a locally compact group, and generalize it to cover any locally compact linear map.
01 Jan 1982
TL;DR: In this paper, the authors consider linear spaces L over the fields R and C, and define the set of elements z ∈ L of the form λx -I-μy, where x ∈ A, y ∈ B. In cases when a statement does not depend on the choice of field, they write K instead of R or C. If A and B are two subsets of L and λ and μ are two numbers in K, then
Abstract: We shall consider linear spaces L over the fields R and C. In cases when a statement does not depend on the choice of field, we write K instead of R or C. If A and B are two subsets of L and λ and μ are two numbers in K, then λA + μB denotes the set of elements z ∈ L of the form λx -I- μy, where x ∈ A, y ∈ B.
01 Feb 1982
TL;DR: In this paper, the spectral theory of elliptic operators has been studied in Hulbert spaces, where the spectral properties of linear operators have been investigated in the context of spectral spectral theory.
Abstract: Let A be a compact linear operator on a Hulbert space H, Sn-(A) = 1/2(A*A), Q be a linear operator, IQf I 0, c > 0, Vf E H. Let Sn(A) = cn-r{1 + 0(n-q)}, r, q > 0, B = A(I + Q). Then Sn(B) = sn(A){l + O(n-')}, -y = min1 qra) Some applications of this result to the spectral theory of elliptic operators are given.
01 Feb 1982
TL;DR: In this article, the authors consider the converse problem and find conditions on the range S(X) which guarantee that a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous.
Abstract: Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].