China 
Africa 
Explore chapters and articles related to this topic
Introduction to Dynamics: Implications on the Design of Precision Machines
Published in Richard Leach, Stuart T. Smith, Basics of Precision Engineering, 2017
Patrick Baird, Stuart T. Smith
A rigid body, because of its extended shape, can exhibit modes of vibration that have directional characteristics beyond those of simple oscillations, such as those represented by ideal springs and pendulums. These modes of vibration can be solved as an eigensystem, which makes use of matrix representation and calculations. Eigensystems were originally used to investigate rigid body rotations, but now have many diverse applications, including stability and vibration analysis and, more generally, matrix algebra and differential operators. The whole subject of eigensystems was generalised into spectral theory by David Hilbert, using complex mathematical spaces for use in quantum theory (eigenstates and eigenfunctions).
Relative Morse index and multiple homoclinic orbits for a nonperiodic Hamiltonian system
Published in Applicable Analysis, 2023
We define an self-adjoint operator as where Id denotes the identity map on ( That is, on ). Moreover, by (16), we have Recall that is a compact operator and is finite dimensional, we have is a compact operator from to itself. Applying to the spectral theory of compact self-adjoint operator, we know there is a basis of , and a sequence in as such that Thus, for any , which can be expressed as , we have Since , all the coefficients are positive except a finite number. Thus, our Lemma follows by
Inverse problems of recovering first-order integro-differential operators from spectra
Published in Applicable Analysis, 2022
The majority of applications are concerned with the second-order integro-differential operators in the form or in the more general form with , where l is the interval length. Spectral theory of such operators was developed in connection with investigation of the behavior of particles on a crystalline surface [8, 9] and with the nonlocal theory of elasticity [10]. The first-order operator (1) plays a role of the ‘square root’ of the second-order operator (2) (see [11]), so the spectral theory of operator (1) also causes interest. In this connection, even if the kernel of (2) is real-valued, the investigation of the square root (1) leads to the need to study complex-valued solutions.
Recovery of a degenerate space-dependent heat capacity
Published in Inverse Problems in Science and Engineering, 2021
In the fifties, Krein used the inverse spectral theory of the string operator which is generated by the differential operator where is a Stieljtes measure defined by a nondecreasing function M, which represents the mass of a string, and is the right derivative at x, see [8,10] for the spectral theory of the string. When is an absolutely continuous function, the string operator reduces to (4) and is understood as and thus, when ρ vanishes on a subinterval, the solution f is a linear function.