Symmetric Diffusion Research Articles

On symmetric linear diffusions

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.

Vector-valued Littlewood-Paley-Stein Theory for Semigroups II

Abstract Inspired by a recent work of Hytönen and Naor, we solve a problem left open in our previous work joint with Martínez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any $T$ which is the square of a symmetric diffusion Markovian operator on a measure space $(\Omega , \mu )$. Moreover, we show that $T\otimes{ \textrm{Id}}_X$ extends to an analytic contraction on $L_p(\Omega ; X)$ for any $1<p<\infty $ and any uniformly convex Banach space $X$.

Limit behaviour of diffusion in high-contrast periodic media and related Markov semigroups

ABSTRACTThe goal of the paper is to describe the large time behaviour of a symmetric diffusion in a high-contrast periodic environment and to characterize the limit process under the diffusive scaling. We consider separately the and settings.

Symmetrical information filtering via punishing superfluous diffusion

For niche recommendation, information filtering has attracted much attention from various fileds. Especially, mass diffusion based models behave prominently. Nevertheless, these models intrinsically suffer superfluous diffusion, transferring redundant preferences to the object and damaging the accuracy, diversity, and personalization of recommendation. Besides, we discover that the symmetrical diffusion can effectively improve recommendation performances. Thus, we assume that the superfluous diffusion should be symmetrically punished. Hence, we propose a symmetrical punishment model on superfluous diffusion for accurate information recommendation. Extensive experiments on two data sets Netflix and Movielens show that our proposed model outperforms mainstream indices remarkably in accuracy, diversity and personalization.

Reflections at infinity of time changed RBMs on a domain with Liouville branches

Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.

Singular and Calabi–Yau varieties linked with billiard trajectories and diffusion operators

The images under a certain map of periodic billiard trajectories inside the fundamental region of the affine Weyl group of the root system $$A_{2}$$ are closed curves and configurations of lines described by a one-parameter family of polynomials. The polynomials are related with eigenvectors of symmetric diffusion operators connected with the deltoid. Several associated singular varieties and Calabi–Yau threefolds defined over the rationals are constructed.

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the...

Noise-induced drift in two-dimensional anisotropic systems.

We study the isothermal Brownian dynamics of a particle in a system with spatially varying diffusivity. Due to the heterogeneity of the system, the particle's mean displacement does not vanish even if it does not experience any physical force. This phenomenon has been termed "noise-induced drift," and has been extensively studied for one-dimensional systems. Here, we examine the noise-induced drift in a two-dimensional anisotropic system, characterized by a symmetric diffusion tensor with unequal diagonal elements. A general expression for the mean displacement vector is derived and presented as a sum of two vectors, depicting two distinct drifting effects. The first vector describes the tendency of the particle to drift toward the high diffusivity side in each orthogonal principal diffusion direction. This is a generalization of the well-known expression for the noise-induced drift in one-dimensional systems. The second vector represents a novel drifting effect, not found in one-dimensional systems, originating from the spatial rotation in the directions of the principal axes. The validity of the derived expressions is verified by using Langevin dynamics simulations. As a specific example, we consider the relative diffusion of two transmembrane proteins, and demonstrate that the average distance between them increases at a surprisingly fast rate of several tens of micrometers per second.

Speckle reduction using deformable mirrors with diffusers in a laser pico-projector.

We propose a design for speckle reduction in a laser pico-projector adopting diffusers and deformable mirrors. This research focuses on speckle noise suppression by changing the angle of divergence of the diffuser. Moreover, the speckle contrast value can be further reduced by the addition of a deformable mirror. The speckle reduction ability obtained using diffusers with different divergence angles is compared. Three types of diffuser designs are compared in the experiments. For Type 1 which uses a circular symmetric diffuser the speckle contrast value can be decreased to 0.0264. For Type 2, the speckle contrast value can be reduced to 0.0267 because of the inclusion of an elliptical distribution diffuser. With Type 3 which includes a combination of the circular distribution diffuser and elliptical distribution diffuser, the speckle contrast value can be reduced to 0.0236. For all three types, the speckle contrast value is lower than 0.05. Under this speckle value, the speckle phenomenon is invisible to the human eye.

The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor

We present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with non-symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We present new approach for the construction that guarantees positive definiteness of the non-symmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, we observed super-convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor. The new scheme was also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics.

Improved tractography using asymmetric fibre orientation distributions

Diffusion MRI allows us to make inferences on the structural organisation of the brain by mapping water diffusion to white matter microstructure. However, such a mapping is generally ill-defined; for instance, diffusion measurements are antipodally symmetric (diffusion along x and –x are equal), whereas the distribution of fibre orientations within a voxel is generally not symmetric. Therefore, different sub-voxel patterns such as crossing, fanning, or sharp bending, cannot be distinguished by fitting a voxel-wise model to the signal. However, asymmetric fibre patterns can potentially be distinguished once spatial information from neighbouring voxels is taken into account. We propose a neighbourhood-constrained spherical deconvolution approach that is capable of inferring asymmetric fibre orientation distributions (A-fods). Importantly, we further design and implement a tractography algorithm that utilises the estimated A-fods, since the commonly used streamline tractography paradigm cannot directly take advantage of the new information. We assess performance using ultra-high resolution histology data where we can compare true orientation distributions against sub-voxel fibre patterns estimated from down-sampled data. Finally, we explore the benefits of A-fods-based tractography using in vivo data by evaluating agreement of tractography predictions with connectivity estimates made using different in-vivo modalities. The proposed approach can reliably estimate complex fibre patterns such as sharp bending and fanning, which voxel-wise approaches cannot estimate. Moreover, histology-based and in-vivo results show that the new framework allows more accurate tractography and reconstruction of maps quantifying (symmetric and asymmetric) fibre complexity.

A Conservative, Entropic Multispecies BGK Model

We derive a conservative multispecies BGK model that follows the spirit of the original, single species BGK model by making the specific choice to conserve species masses, total momentum, and total kinetic energy and to satisfy Boltzmann’s $$\mathcal {H}$$ -Theorem. The derivation emphasizes the connection to the Boltzmann operator which allows for direct inclusion of information from higher-fidelity collision physics models. We also develop a complete hydrodynamic closure via the Chapman-Enskog expansion, including a general procedure to generate symmetric diffusion coefficients based on this model. We numerically investigate velocity and temperature relaxation in dense plasmas and compare the model with previous multispecies BGK models and discuss the trade-offs that are made in defining and using them. In particular, we demonstrate that the BGK model in the NRL plasma formulary does not conserve momentum or energy in general.

Finite element method for a symmetric tempered fractional diffusion equation

A space fractional diffusion equation involving symmetric tempered fractional derivative of order 1<α<2 is considered. A Galerkin finite element method is implemented to obtain spatial semi-discrete scheme and first order centered difference in time is used to find a fully discrete scheme for tempered fractional diffusion equation. We construct a variational formulation and show its existence, uniqueness and regularity. Stability and error estimates of numerical scheme are discussed. The theoretical and computational study of accuracy and consistence of the numerical solutions are presented.

Symmetrical and overloaded effect of diffusion in information filtering

In physical dynamics, mass diffusion theory has been applied to design effective information filtering models on bipartite network. In previous works, researchers unilaterally believe objects’ similarities are determined by single directional mass diffusion from the collected object to the uncollected, meanwhile, inadvertently ignore adverse influence of diffusion overload. It in some extent veils the essence of diffusion in physical dynamics and hurts the recommendation accuracy and diversity. After delicate investigation, we argue that symmetrical diffusion effectively discloses essence of mass diffusion, and high diffusion overload should be published. Accordingly, in this paper, we propose an symmetrical and overload penalized diffusion based model (SOPD), which shows excellent performances in extensive experiments on benchmark datasets Movielens and Netflix.

Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data

This study examines the theoretical and empirical perspectives of the symmetric Hawkes model of the price tick structure. Combined with the maximum likelihood estimation, the model provides a proper method of volatility estimation specialized in ultra-high-frequency analysis. Empirical studies based on the model using the ultra-high-frequency data of stocks in the S&P 500 are performed. The performance of the volatility measure, intraday estimation, and the dynamics of the parameters are discussed. A new approach of diffusion analogy to the symmetric Hawkes model is proposed with the distributional properties very close to the Hawkes model. As a diffusion process, the model provides more analytical simplicity when computing the variance formula, incorporating skewness and examining the probabilistic property. An estimation of the diffusion model is performed using the simulated maximum likelihood method and shows similar patterns to the Hawkes model.

A linearity based comparison between symmetric and asymmetric lateral diffusion for a 22 nm Underlapped DG-MOSFET

The paper presents a comparative study between symmetric and asymmetric straggle architecture, where asymmetric nature is observed by considering straggle in source and drain side alternatively by using the data obtained from 2D numerical simulator Sentaurus TCAD. The analog parameters such as the transconductance (gm), the gain per unit current (gm/Ids)and the intrinsic gain (gmRo) has been analyzed. RF parameters such as intrinsic gate to source and gate to drain capacitances Cgs and Cgd, intrinsic gate to source and gate to drain resistances Rgs and Rgd, the transport delay τm, the unity gain cut off frequency (fT) and the maximum frequency of oscillation (fmax) are also computed using the Non Quasi Static(NQS) approach. The RF feasibility has also been scrutinized with the help of linearity analysis and Harmonic Distortion (HD). The Second and Third order HD (HD2 and HD3) and Total HD(THD) have been analyzed to evaluate distortion performance. The circuit performance is also analyzed by using the devices as a driver MOSFET in a single stage amplifier circuit whose dc performance as well as small signal gain is evaluated and Gain Bandwidth Product (GBW) is also calculated in this paper.

Expression, Functional Characterization, and Solid-State NMR Investigation of the G Protein-Coupled GHS Receptor in Bilayer Membranes

The expression, functional reconstitution and first NMR characterization of the human growth hormone secretagogue (GHS) receptor reconstituted into either DMPC or POPC membranes is described. The receptor was expressed in E. coli. refolded, and reconstituted into bilayer membranes. The molecule was characterized by 15N and 13C solid-state NMR spectroscopy in the absence and in the presence of its natural agonist ghrelin or an inverse agonist. Static 15N NMR spectra of the uniformly labeled receptor are indicative of axially symmetric rotational diffusion of the G protein-coupled receptor in the membrane. In addition, about 25% of the 15N sites undergo large amplitude motions giving rise to very narrow spectral components. For an initial quantitative assessment of the receptor mobility, 1H-13C dipolar coupling values, which are scaled by molecular motions, were determined quantitatively. From these values, average order parameters, reporting the motional amplitudes of the individual receptor segments can be derived. Average backbone order parameters were determined with values between 0.56 and 0.69, corresponding to average motional amplitudes of 40–50° of these segments. Differences between the receptor dynamics in DMPC or POPC membranes were within experimental error. Furthermore, agonist or inverse agonist binding only insignificantly influenced the average molecular dynamics of the receptor.

Er:Ti:LiNbO3 ridge waveguide optical amplifiers by optical grade dicing and three-side Er and Ti in-diffusion

Erbium-doped titanium in-diffused ridge waveguide optical amplifiers in x-cut congruent LiNbO3 substrates pumped by a 1486 nm laser diode are reported for the first time. High internal gain of 2.7 dB/cm has been demonstrated in 2.5 cm long waveguides for the coupled pump power 200 mW exceeding the best literature values reported for Er:Ti:LiNbO3 channel waveguides. For gain improvement, we have developed a novel technique which is comprised of ridge definition using diamond blade dicing followed by three-side erbium (Er) and titanium (Ti) layer deposition and high-temperature diffusion. Thereby both higher Er concentrations with more symmetric diffusion profiles and lower scattering losses due to the high-temperature smoothened waveguide side walls have been obtained.

Last Multipliers for Riemannian Geometries, Dirichlet Forms and Markov Diffusion Semigroups

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carre du champ associated to the infinitesimal generator of the semigroup.