A Kind of Affine Weighted Moment Invariants

A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of local affine differential invariants and a framework of global integral, they can more effectively extract features of images and help to increase the number of low-order invariants and to decrease the calculating cost. The experimental results show that AWMIs have good stability and distinguishability and achieve better results in image retrieval than traditional moment invariants. An extension to 3D is straightforward

Group analysis of a class of nonlinear Kolmogorov equations

A class of (1+2)-dimensional diffusion-convection equations (nonlinear Kolmogorov equations) with time-dependent coefficients is studied with Lie symmetry point of view. The complete group classification is achieved using a gauging of arbitrary elements (i.e. via reducing the number of variable coefficients) with the application of equivalence transformations. Two possible gaugings are discussed in detail in order to show how equivalence groups serve in making the optimal choice.Comment: 12 pages, 4 table

Evolution of magnetic fields through cosmological perturbation theory

The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute the energy momentum tensor to second order in perturbation theory in terms of gauge invariant quantities. Two possible scenarios are discussed, first we consider a FLRW background without magnetic field and we study the perturbation theory introducing the magnetic field as a perturbation. The second scenario, we consider a magnetized FLRW and build up the perturbation theory from this background. We compare the cosmological dynamo like equation in both scenarios

Simulation of Harmonic Oscillators on the Lattice

[EN] This work deals with the simulation of a two¿dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second¿order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state¿of¿the¿art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two¿dimensional lattice and check its energy conservation.This work has been supported by the Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF) under grant TIN2017-89314-P, and the Programa de Apoyo a la Investigacion y Desarrollo 2018 (PAID-06-18) of the Universitat Politecnica de Valencia under grant SP20180016.Tung, MM.; Ibáñez González, JJ.; Defez Candel, E.; Sastre, J. (2020). Simulation of Harmonic Oscillators on the Lattice. Mathematical Methods in the Applied Sciences. 43(14):8237-8252. https://doi.org/10.1002/mma.6510S823782524314Dehghan, M., & Hajarian, M. (2009). Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite. Journal of Computational and Applied Mathematics, 231(1), 67-81. doi:10.1016/j.cam.2009.01.021Dehghan, M., & Hajarian, M. (2010). Computing matrix functions using mixed interpolation methods. Mathematical and Computer Modelling, 52(5-6), 826-836. doi:10.1016/j.mcm.2010.05.013Kazem, S., & Dehghan, M. (2017). Application of finite difference method of lines on the heat equation. Numerical Methods for Partial Differential Equations, 34(2), 626-660. doi:10.1002/num.22218Kazem, S., & Dehghan, M. (2018). Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL). Engineering with Computers, 35(1), 229-241. doi:10.1007/s00366-018-0595-5Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Efficient orthogonal matrix polynomial based method for computing matrix exponential. Applied Mathematics and Computation, 217(14), 6451-6463. doi:10.1016/j.amc.2011.01.004Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling, 54(7-8), 1835-1840. doi:10.1016/j.mcm.2010.12.049Serbin, S. M., & Blalock, S. A. (1980). An Algorithm for Computing the Matrix Cosine. SIAM Journal on Scientific and Statistical Computing, 1(2), 198-204. doi:10.1137/0901013Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Higham, N. J. (1988). FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation. ACM Transactions on Mathematical Software, 14(4), 381-396. doi:10.1145/50063.21438

On classification of Poisson vertex algebras

We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators)

Discrete structures in continuum descriptions of defective crystals

I discuss various mathematical constructions that combine together to provide a natural setting for discrete and continuum geometric models of defective crystals. In particular I provide a quite general list of `plastic strain variables', which quantifies inelastic behaviour, and exhibit rigorous connections between discrete and continuous mathematical structures associated with crystalline materials that have a correspond-ingly general constitutive specification

Discrete moving frames on lattice varieties and lattice based multispace

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalization of the jet bundle that also generalizes Olver’s one dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame

Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207), an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The form-preserving transformations for this class of systems are constructed and it is shown that this list contains only non-equivalent systems. The obtained symmetries permit to reduce the reaction-diffusion systems under study to two-dimensional systems of ordinary differential equations and to find exact solutions. As a non-trivial example, multiparameter families of exact solutions are explicitly constructed for two nonlinear reaction-diffusion systems. A possible interpretation to a biologically motivated model is presented

Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation

We apply multivariate Lagrange interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments

Group Analysis of the Novikov Equation

We find the Lie point symmetries of the Novikov equation and demonstrate that it is strictly self-adjoint. Using the self-adjointness and the recent technique for constructing conserved vectors associated with symmetries of differential equations, we find the conservation law corresponding to the dilations symmetry and show that other symmetries do not provide nontrivial conservation laws. Then we investigat the invariant solutions