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Matrices and linear transformations
Published in Alan Jeffrey, Mathematics, 2004
The notion of a determinant, when first introduced in Chapter 4, was that of a single number associated with a square array of numbers. In its subsequent application in that chapter it was used in a subsidiary role to simplify the manipulation of the vector product, and in that capacity it gave rise to a vector. The notion of a determinant is, in fact, implicit in Theorem 5.22 when, in connection with the change of variable in partial differentiation, it can be introduced with functions as elements. It is then called a Jacobian, and in this role it is often called a functional determinant and gives rise to a function that is closely related to the one–one nature of the change of variables involved.
Supersymmetric Theory of Stochastics:Demystification of Self-Organized Criticality
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Here, in the first line, the functional δ-function that limits the path integration only to closed solutions of SDE follows after integrating out the Lagrange multiplier, B, whereas the functional determinant of the infinite-dimensional matrix of the functional derivatives of the SDE is provided by out-integration of the fermionic fields, χ, X¯. The second line is the pathintegral analog of the Lefschetz index in Equation 17.88.
Ultralong-range Rydberg molecules
Published in Molecular Physics, 2020
Christian Fey, Frederic Hummel, Peter Schmelcher
An alternative fully quantum mechanical method that captures both, few-body and many-body features of polyatomic s-state ULRMs, is the functional determinant approach [19,21,89]. The resulting spectra reproduce resonance energies and the lineshape of experimental signals very accurately. Furthermore, they explain beyond mean-field effects related to the collective polaron character of the system, which are not included in the semiclassical sampling approach [19,21,89].
Dynamic caustics in unidirectional fiber-reinforced composites with mode I cracks: Experiment and numerical simulation
Published in Mechanics of Advanced Materials and Structures, 2018
Wenfeng Hao, Xixuan Sheng, Guangping Guo, Xinwen Chen, Jianguo Zhu
From the formation principle of caustics, it is concluded that the caustic is a strongly illuminated singular curve. The sufficient and necessary condition of existence of such a singularity is the zeroing of the Jacobian functional determinant J, i.e.:
An inverse spectral analysis of transmission problem in thermo- and photo-acoustic tomography
Published in Waves in Random and Complex Media, 2019
Secondly, by linear algebra the existence of and are equivalent to finding the zeros of the following functional determinant: