Explicit Kernels Research Articles

Polyanalytic Neumann-n problem in the unit disk

An iterated Neumann boundary value problem for the polyanalytic operator is investigated in the unit disc of the complex plane. Although this problem can be treated for domains in the complex plane having a harmonic Green function, the unit disk is particularly interesting as it is the origin of complex boundary value problems and the prototype of simply connected domains. Moreover, the resulting formulas for solutions and for solvability conditions are expressed by integral formulas with explicit kernels. By the way, the result verifies the findings for domains with harmonic Green function.

Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.

Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

Let u α , d = u α , d t , x , t ∈ 0 , T , x ∈ ℝ d be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process u α , d , in time, having infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for SHEs with spatially colored noise. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.

Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable.

Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

Explicit heat kernels of a model of distorted Brownian motion on spaces with varying dimension

In this paper, we study a particular model of distorted Brownian motion (dBM) on state spaces with varying dimension. Roughly speaking, the state space of such a process consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the 3- or 1-dimensional component receives a strong “push” toward the origin. On each component, the “magnitude” of a “push” can be parametrized by a constant γ>0. In this article, using the probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any 0<t<∞.

Diffusions on a space of interval partitions: Poisson–Dirichlet stationary distributions

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson–Dirichlet laws with parameters α,0 and α,α. The construction has two steps. The first is a general construction of interval partition processes obtained previously by decorating the jumps of a Lévy process with independent excursions. Here, we focus on the second step which requires explicit transition kernels and, what we call, pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (Adv. in Appl. Probab. 13 (1981) 429–452) and Petrov (Funktsional. Anal. i Prilozhen. 43 (2009) 45–66). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step toward resolving longstanding conjectures by Feng and Sun on measure-valued Poisson–Dirichlet diffusions and by Aldous on a continuum-tree-valued diffusion.

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.

The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators

This paper deals with the static Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$ The system is reformulated in quaternion analysis by Kravchenko in the form $$\mathcal {L}F=0$$ with $$\mathcal {L}F=DF+F\alpha $$ . We consider special cases of the coefficient function $$\Phi =\Phi _0(x_0)\Phi _1(x_1)\Phi _2(x_2)$$ and prove the existence of four generalized Cauchy kernels of the operator $$\mathcal {L}$$ . We construct four explicit generalized Cauchy kernels in the case $$\Phi =x_0^{2p}x_1^{2m}x_2^{2n}$$ .

Simple bisection procedure in quickly convergent explicit ODE solver to numerically analyze FRCM strengthening systems

A fast bisection procedure embedded in a standard ODE45 solver is proposed for three different kinds of FRCM experimental tests, namely (1) coupon tensile test, (2) pull-out test and (3) single-lap shear test. In order to predict global and local behavior of FRCM in such kind of experiments, a mono-axial differential equations model can be derived, requiring in the most general case the numerical solution of Boundary Value Problems BVPs, involving as independent variables mortar and fiber displacement fields and their first derivatives. Assuming a nonlinear behavior of both matrix and reinforcement-matrix interface, implicit and general purpose BVPs exhibit poor numerical stability and slow convergence. Furthermore, the initial guess of the solution is an integral part of solving a BVP, making the approach even less appealing, especially to reproduce FRCM behavior in the whole possible range of deformations, where severe softening and snap back phenomena are encountered. In the paper, the BVP is transformed in an Ordinary Differential Equation ODE system with initial values, i.e. into an Initial Value Problem IVP, by means of a classic shooting method where guess values of the matrix displacement on the opposite side of the loaded edge are assumed and varied until the desired boundary condition on the loaded edge are obtained. A classic bisection procedure allows to converge quickly to the exact value of matrix displacement to assign in order to recover quickly the boundary condition of the original BVP problem imposed on the opposite side. The advantage -common to all shooting methods available- is that a BVP is transformed in a sequence of ODE problems with initial conditions, which can be solved by means of explicit Runge Kutta kernels. Any kind of non-linearity can be assumed for matrix –for which a particular saw tooth approximation of the real behavior in tension is proposed to strengthen further convergence- and matrix-reinforcement interface, for which both a newly proposed exponential law and a stepped softening relationship are adopted. The numerical procedure is tested on the aforementioned three different experimental setups, showing excellent fitting capability of both the experimental behavior and previously presented theoretical models, where available.

Hierarchical matrix approximations for space-fractional diffusion equations

Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N, full representation of a fractional diffusion matrix would require O(N2) memory storage requirement, with a similar estimate for matrix–vector products. In this work, we present H2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix–vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudo-transient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix–vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 105 – 106. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.

An extension problem related to the fractional Branson–Gover operators

The Branson–Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson–Gover operators. For Euclidean spaces we show that the fractional Branson–Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli–Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties.

Evolution equations on time-dependent intervals

Abstract We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l_1(t)&lt;x&lt;l_2(t)$, $0&lt;t&lt;T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

Integral representation of solutions to higher-order fractional Dirichlet problems on balls

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power [Formula: see text] of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if [Formula: see text]. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of [Formula: see text]-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.

Explicit Linear Kernels for Packing Problems

During the last years, several algorithmic meta-theorems have appeared (Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive from them constructive kernels with reasonably low explicit constants. To fill this gap, we recently presented [STACS 2014] a framework to obtain explicit linear kernels for some families of problems whose solutions can be certified by a subset of vertices. In this article we enhance our framework to deal with packing problems, that is, problems whose solutions can be certified by collections of subgraphs of the input graph satisfying certain properties. F-Packing is a typical example: for a family F of connected graphs that we assume to contain at least one planar graph, the task is to decide whether a graph G contains k vertex-disjoint sub-graphs such that each of them contains a graph in F as a minor. We provide explicit linear kernels on sparse graphs for the following two orthogonal generalizations of F-Packing: for an integer 1, one aims at finding either minor-models that are pairwise at distance at least in G (-F-Packing), or such that each vertex in G belongs to at most minors-models (F-Packing with-Membership). Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.

Examples of Interacting Particle Systems on mathbb {Z} as Pfaffian Point Processes: Annihilating and Coalescing Random Walks

A class of interacting particle systems on mathbb {Z}, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffian point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffian kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on mathbb {R}, depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffian point process for real eigenvalues for the real Ginibre ensemble of random matrices. For Brownian motions on mathbb {R}_{+} with absorbing or reflected boundary conditions at zero, new interesting Pfaffian kernels appear. We illustrate the utility of the Pfaffian structure by determining the extreme statistics of the rightmost particle for the purely annihilating Brownian motions, and also computing the probability of overcrowded regions for all models.

Overdamped large-eddy simulations of turbulent pipe flow up to Reτ = 1500

We present results from large-eddy simulations (LES) of turbulent pipe flow in a computational domain of 42 radii in length. Wide ranges of shear the Reynolds number and Smagorinsky model parameter are covered, 180 ≤ Reτ ≤ 1500 and 0.05 ≤ Cs ≤ 1.2, respectively. The aim is to asses the effect of Cs on the resolved flow field and turbulence statistics as well as to test whether very large scale motions (VLSM) in pipe flow can be isolated from the near-wall cycle by enhancing the dissipative character of the static Smagorinsky model with elevated Cs values. We found that the optimal Cs to achieve best agreement with reference data varies with Reτ and further depends on the wall normal location and the quantity of interest. Furthermore, for increasing Reτ, the optimal Cs for pipe flow LES seems to approach the theoretically optimal value for LES of isotropic turbulence. In agreement with previous studies, we found that for increasing Cs small-scale streaks in simple flow field visualisations are gradually quenched and replaced by much larger smooth streaks. Our analysis of low-order turbulence statistics suggests, that these structures originate from an effective reduction of the Reynolds number and thus represent modified low-Reynolds number near-wall streaks rather than VLSM. We argue that overdamped LES with the static Smagorinsky model cannot be used to unambiguously determine the origin and the dynamics of VLSM in pipe flow. The approach might be salvaged by e.g. using more sophisticated LES models accounting for energy flux towards large scales or explicit anisotropic filter kernels.

Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel

In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given.

L-Kuramoto–Sivashinsky SPDEs vs. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws

We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto–Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by the space–time white noise in one-to-three dimensional spaces. Both classes were introduced—with Brownian-time-type kernel formulations—by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in d=1,2,3. We show that both classes are (1/2)− Hölder continuously differentiable in space when d=1, and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by the space–time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order <1/2 and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba's earlier works on (1) and (2). Similar regularity results hold for the Allen–Cahn nonlinear members of (1) and (2) on compacts via change of measure.