The Quantization of Gravity: Quantization of the Hamilton Equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form −Δu=0 in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided n≠4. Using then separation of variables, the solutions u can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space SL(n,R)/SO(n). Since one can define a Schwartz space and tempered distributions in SL(n,R)/SO(n) as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

Numerical analysis of sparse initial data identification for parabolic problems

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations ∫Sf(xyt)dµ(t)+∫Sf(xσ(y)t)dµ(t)= 2f(x)f(y), x,y ∈ S; ∫Sf(xσ(y)t)dµ(t)-∫Sf(xyt)dµ(t)= 2f(x)f(y), x,y ∈ S; where S is a semigroup, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.

Constructing equivalence-preserving Dirac variational integrators with forces

The dynamical motion of mechanical systems possesses underlying geometric structures and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction (Marsden, J. & West, M. (2001) Discrete mechanics and variational integrators. Acta Numer., 10, 357–514). This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics (Leok, M. & Ohsawa, T. (2011) Variational and geometric structures of discrete Dirac measures. Found. Comput. Math., 11, 529–562). Forced Lagrange–Dirac systems are described by a Lagrangian and an external force pair, and two pairs of Lagrangians and external forces are said to be equivalent if they yield the same equations of motion. However, the variational discretization of a forced Lagrange–Dirac system discretizes the Lagrangian and forces separately, and will generally depend on the choice of representation. In this paper we derive a class of Dirac variational integrators with forces that yield well-defined numerical methods that are independent of the choice of representation. We present a numerical simulation to demonstrate how such equivalence-preserving discretizations avoid spurious solutions that otherwise arise from poorly chosen representations.

Detecting the filamentary structure of 3D point clouds: Proposal of a wavelet-based method

The analysis of the filamentary structure of the cosmo as well as that of the internal structure of the polar ice suggests the development of models based on three-dimensional (3D) point processes. A point process, regarded as a random measure, can be expressed as a sum of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads the continuous wavelet transform of the process itself. As possible mother wavelets, we propose the application of the Mexican hat and the Morlet wavelet in order to implement the scale-angle energy density of the process, depending on the dilation parameter and on the three angles which define the direction in the Euclidean space. Such indicator proves to be a sensitive detector of any variation in the direction and it can be successfully implemented to study the isotropy or the filamentary structure in 3D point patterns.

Structural and hidden properties of 1D point processes: A wavelet-based study

A one-dimensional (1D) point process, if considered as a random measure, can be represented by a sum, at most countable, of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads to the definition of the continuous wavelet transform of the process itself. As a possible choice of the mother wavelet, we propose the Mexican hat and the Morlet wavelet in order to implement the energy density of the process as a function of two wavelet parameters. Such mathematical tool works as a microscope to process an in-depth analysis of some classes of processes, in particular homogeneous, cluster, and locally scaled processes.