Space Of Differential Operators Research Articles

Time-space fractional neutron point kinetics: Theory and simulations

In this work the mathematical derivation and numerical analysis of a fractional nuclear reactor point kinetics in time and space (TSFNPK) is presented. The TSFNPK model was derived considering a non-Fickian law for the neutron density current where the differential operators in space and time are of fractional order. The TSFNPK equations presented in this work constitutes a novel model for nuclear reactor kinetics, thus represent an extended model with respect to other fractional models and the classical neutron point kinetics equations. The model considers two diffusion exponents: one for the differential operator dependent in time and another for the dependent operator in space. Both exponents effect is numerically analyzed considering changes of reactivity step type, and temperature feedback reactivity. A first approach of the TSFNPK is presented, without temperature effects, and then a second approach considering temperature feedback effects is analyzed. In a following work, as a demonstration of application, a detailed analysis along with verification will be presented.

A fractional diffusion random laser

The goal of this letter is to introduce the concept of a non-resonant fractional random laser. This is achieved by extending the classical Letokhov model of photon diffusion through disordered gain media to fractional differential operators in space and time. Fractional transport equations effectively describe anomalous photon sub-diffusion phenomena in non-uniform random scattering media with memory and long-range spatial correlation effects. In particular, by analytically solving fractional transport equations in the one-dimensional slab geometry we obtain simple closed-form expressions for the critical amplification volumes required to initiate the laser action in both fractional-order (FO) and distributed-order (DO) space-time fractional reaction-diffusion equations. Our findings demonstrate the benefits of anomalous sub-diffusive photon transport in active media with correlated disorder and stimulate the engineering of novel non-resonant random lasers with significantly reduced footprint and amplification volumes beyond the limitations of uniform disorder and Markovian diffusion processes.

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a Galerkin method, uniformly stable in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert–Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov–Galerkin setting to evaluate the dual residual norm.

Cohomology and deformation of 𝔞𝔣𝔣(1|1) acting on differential operators

We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the second differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We classify formal deformations of the [Formula: see text]-module structure on the superspaces of symbols of differential operators. We prove that any formal deformation of a given infinitesimal deformation of this structure is equivalent to its infinitesimal part. This work is the simplest superization of a result by Basdouri [Deformation of [Formula: see text]-modules of pseudo-differential operators and symbols, J. Pseudo-differ. Oper. Appl. 7(2) (2016) 157–179] and application of work by Basdouri et al. [First cohomology of [Formula: see text] and [Formula: see text] acting on linear differential operators, Int. J. Geom. Methods Mod. Phys. 13(1) (2016)].

Quantization and injective submodules of differential operator modules

The Lie algebra of vector fields on Rm acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to slm+1, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor glm. We prove two results. First, we realize explicitly all injective objects of the parabolic category Oglm(slm+1) of glm-finite slm+1-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e.,slm+1-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.

First cohomology of 𝔞𝔣𝔣(1) and 𝔞𝔣𝔣(1|1) acting on linear differential operators

We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We study also the super analogue of this problem getting the same results.

Splines, lattice points, and arithmetic matroids

Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \{w \ge 0 : X w = u \}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^d$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this article we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate $T_X$) and the space of nice differential operators (i.e. operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix $X$. They are closely related to the $\mathcal{P}$-spaces studied by Ardila-Postnikov and Holtz-Ron in the context of zonotopal algebra and power ideals.

3D weak-dispersion reverse time migration using a stereo-modeling operator

Reliable 3D imaging is a required tool for developing models of complex geologic structures. Reverse time migration (RTM), as the most powerful depth imaging method, has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts. Finite-difference methods are among the most popular numerical approaches used for RTM. However, these methods often encounter a serious issue of numerical dispersion, which is typically suppressed by reducing the grid interval of the propagation model, resulting in large computation and memory requirements. In addition, even with small grid spacing, numerical anisotropy may degrade images or, worse, provide images that appear to be focused but position events incorrectly. Recently, stereo-operators have been developed to approximate the partial differential operator in space. These operators have been used to develop several weak-dispersion and efficient stereo-modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy. We generalized one stereo-modeling method, fourth-order nearly analytic central difference (NACD), from 2D to 3D and applied it to 3D RTM. The RTM results for the 3D SEG/EAGE phase A classic data set 1 and the SEG Advanced Modeling project model demonstrated that, even when using a large grid size, the NACD method can handle very complex velocity models and produced better images than can be obtained using the fourth-order and eighth-order Lax-Wendroff correction (LWC) schemes. We also applied 3D NACD and fourth-order LWC to a field data set and illustrated significant improvements in terms of structure imaging, horizon/layer continuity and positioning. We also investigated numerical dispersion and found that not only does the NACD method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for LWC.

Integral Cauchy Type Representations of Infinite Order Differential Operators in Spaces of Exponential Type Entire Functions Bounded on the Real Axis

We obtain an integral Cauchy type representation of a vector differential operator of infinite order in the space of entire functions of exponential type that are bounded on the real axis. This representation is used for obtaining integral representations of entire solutions to explicit and implicit linear inhomogeneous differential equations in a Banach space. Bibliography: 5 titles.

Conformally invariant quantization – towards the complete classification

Let M be a smooth manifold equipped with a conformal structure, E[w] the space of densities with the conformal weight w and Dw,w+δ the space of differential operators from E[w] to E[w+δ]. Conformal quantization Q is a right inverse of the principle symbol map on Dw,w+δ such that Q is conformally invariant and exists for all w. There are partial results concerning the dependence on δ: the existence of Q is known for generic values of δ∈R and the nonexistence is known for a set of critical weights δ∈Σ⊆R. We complete these results, i.e. we show that Q exists for all noncritical weights δ∉Σ. We give explicit formulae for Q for all such δ in terms of the conformal tractor calculus.

Spo(2|2) -Equivariant Quantizations on the Supercircle S1|2

We consider the space of differential operators $\mathcal{D}_{\lambda\mu}$ acting between $\lambda$- and $\mu$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{\lambda\mu}$ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space $\mathcal{D}_{\lambda\mu}$ and the associated graded space of symbols $\mathcal{S}_{\delta}$ ($\delta=\mu-\lambda$) can be considered as $\mathfrak{spo}(2|2)$-modules, where $\mathfrak{spo}(2|2)$ is the Lie superalgebra of contact projective vector fields on $S^{1|2}$. We show in this paper that there is a unique isomorphism of $\mathfrak{spo}(2|2)$-modules between $\mathcal{S}_{\delta}$ and $\mathcal{D}_{\lambda\mu}$ that preserves the principal symbol (i.e. an $\mathfrak{spo}(2|2)$-equivariant quantization) for some values of $\delta$ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages, arXiv:0912.5190] which were established for second-order differential operators. The method used here to build the $\mathfrak{spo}(2|2)$-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331, arXiv:1003.3320] to prove the existence of a $\mathfrak{pgl}(p+1|q)$-equivariant quantization on $\mathbb{R}^{p|q}$.

Deformation of [formula omitted]-modules of symbols

We consider the action of the Lie algebra of polynomial vector fields, v e c t ( 1 ) , by the Lie derivative on the space of symbols S β n = ⨁ j = 0 n F β − j . We study the deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H diff 2 ( v e c t ( 1 ) , D ν , μ ) where D ν , μ is the space of differential operators from F ν to F μ . Necessary second-order integrability conditions of any infinitesimal deformations of S β n are given. We describe completely the formal deformations for some spaces S β n and we give concrete examples of nontrivial deformations.

Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

We analyze explicit Runge-Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs type. For the time discretization, we consider explicit second- and third-order Runge-Kutta schemes. We identify a general set of properties on the space stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then we establish $L^2$-norm error estimates with quasi-optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge-Kutta schemes and any polynomial degree in space and for second-order Runge-Kutta schemes and first-order polynomials in space. For second-order Runge-Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for smooth and rough solutions. The case of finite volumes is briefly discussed.

Embeddings and separable differential operators in spaces of Sobolev-Lions type

We study embedding theorems for anisotropic spaces of Bessel-Lions type H ,γ (Ω; E 0, E), where E 0 and E are Banach spaces. We obtain the most regular spaces E α for which mixed differentiation operators D α from H ,γ (Ω; E 0, E) to L p,γ(Ω; E α ) are bounded. The spaces E α are interpolation spaces between E 0 and E, depending on α = (α 1, α 2, …, α n ) and l = (l 1, l 2, …, l n ). The results obtained are applied to prove the separability of anisotropic differential operator equations with variable coefficients.

Antiplectic Structure, Diffeomorphism and Generalized KdV Family

In this paper we show that the generalized KdV, generalized Camassa–Holm equations and the corresponding Mobius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by \(Vect(S^1)\) on the space of differential operators and on the space of immersion curves, respectively. These are Euler–Poincare type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler–Poincare flow with respect to \(H^1\) metric, and this induces generalized Camassa–Holm equation. In the final section we discuss the Antiplectic pair in \(2+1\) dimensions.

Time discretization of an evolution equation via Laplace transforms

Following earlier work by Sheen, Sloan, and Thomée concerning parabolic equations we study the discretization in time of a Volterra type integro‐differential equation in which the integral operator is a convolution of a weakly singular function and an elliptic differential operator in space. The time discretization is accomplished by using a modified Laplace transform in time to represent the solution as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. Stability and error bounds of high order are derived for two different choices of the quadrature rule. The method is combined with finite‐element discretization in the spatial variables.

On the domain of some ordinary differential operators in spaces of continuous functions

We characterize the domain of some ordinary differential operators with unbounded coefficients in spaces of continuous functions.

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and \(\mathcal{F}_{\lambda } (M)\) the space of densities of degree λ on M. Denote \(\mathcal{D}_{{\lambda ,}\mu }^k (M)\) the space of differential operators from \(\mathcal{F}_{\lambda } (M)\) to \(\mathcal{F}_\mu (M)\) of order k and Skδ with δ = μ − λ the corresponding space of symbols. We construct (the unique) conformally invariant quantization map \(Q_{{\lambda ,}\mu }^3 :S_\delta ^3 \to \mathcal{D}_{{\lambda ,}\mu }^3 \). This result generalizes that of Duval and Ovsienko.

Infinite order differential operators in spaces of entire functions

Differential operators ϕ( Δ θ, ω ), where ϕ is an exponential type entire function of a single complex variable and Δ θ, ω =( θ+ ωz) D+ zD 2, D= ∂/ ∂z, z∈ C , θ⩾0, ω∈ R , acting in the spaces of exponential type entire function are studied. It is shown that, for ω⩾0, such operators preserve the set of Laguerre entire functions provided the function ϕ also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane C . The operator exp( aΔ θ, ω ), a⩾0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all ω∈ R . An integral representation of exp( aΔ θ, ω ), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ θ, ω .

Projective and affine connections on S1 and integrable systems

It is known that the Korteweg–de Vries (KdV) equation is a geodesic flow of an L 2 metric on the Bott–Virasoro group. This can also be interpreted as a flow on the space of projective connections on S 1. The space of differential operators Δ ( n) = ∂ n + u 2 ∂ n−2 +⋯+ u n form the space of extended or generalized projective connections. If a projective connection is factorizable Δ ( n) =( ∂−(( n+1)/2−1) p 1)⋯( ∂+( n−1)/2 p n ) with respect to quasi primary fields p i ’s, then these fields satisfy ∑ i=1 n (( n+1)/2− i) p i =0. In this paper we discuss the factorization of projective connection in terms of affine connections. It is shown that the Burgers equation and derivative non-linear Schrödinger (DNLS) equation or the Kaup–Newell equation is the Euler–Arnold flow on the space of affine connections.