On the inverse of the Caputo matrix exponential

[EN] Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index a > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with constant coefficients. This paper analyzes the most significant properties of the Caputo matrix exponential, in particular those related to its inverse. Several numerical test examples are discussed throughout this exposition in order to outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in general is not another Caputo matrix exponential.This work has been partially supported by Spanish Ministerio de Economia y Competitividad and European Regional Development Fund (ERDF) grants TIN2017-89314-P and by the Programa de Apoyo a la Investigacion y Desarrollo 2018 of the Universitat Politecnica de Valencia (PAID-06-18) grant SP20180016.Defez Candel, E.; Tung, MM.; Chen-Charpentier, BM.; Alonso Abalos, JM. (2019). On the inverse of the Caputo matrix exponential. Mathematics. 7(12):1-11. https://doi.org/10.3390/math7121137S111712Moler, C., & Van Loan, C. (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, 45(1), 3-49. doi:10.1137/s00361445024180Ortigueira, M. D., & Tenreiro Machado, J. A. (2015). What is a fractional derivative? Journal of Computational Physics, 293, 4-13. doi:10.1016/j.jcp.2014.07.019Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent--II. Geophysical Journal International, 13(5), 529-539. doi:10.1111/j.1365-246x.1967.tb02303.xRodrigo, M. R. (2016). On fractional matrix exponentials and their explicit calculation. Journal of Differential Equations, 261(7), 4223-4243. doi:10.1016/j.jde.2016.06.023Garrappa, R., & Popolizio, M. (2018). Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus. Journal of Scientific Computing, 77(1), 129-153. doi:10.1007/s10915-018-0699-

On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.Comment: 19 page

Spectrality of ordinary differential operators

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford's spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein's conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on a new direct method, exploiting \textit{almost orthogonality} of Birkhoff's solutions of the equation $l(y)=\lambda y$, which was discovered earlier by the author.Comment: AmsLaTeX, 26 pages, added section on dissipative operators and reference

Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Comment: 38 pages, LaTex file. Proceedings of the Vth International Conference on Mathematical Physics, Strings and Quantum gravity, Alushta, Ukraine 199

Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure U_q(sl(2)) \odot U_{\qhat}(sl(2)) is a non trivial extension of $U_q(sl(2))$ and U_{\qhat}(sl(2)). We construct the corresponding generalized exponentials and the generalized Liouville field.Comment: 36 pages, LaTex, LPTENS 93/4

Simulating Quantum Dynamics On A Quantum Computer

We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are significant because they are the first to have time-complexities that are comparable to the best known methods for simulating time-independent Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian are satisfied. We provide a thorough cost analysis of these algorithms that considers discretizion errors in both the time and the representation of the Hamiltonian. In addition, we provide the first upper bounds for the error in Lie-Trotter-Suzuki approximations to unitary evolution operators, that use adaptively chosen time steps.Comment: Paper modified from previous version to enhance clarity. Comments are welcom