Diffusion Matrix Research Articles

Weak-Strong Uniqueness for Maxwell--Stefan Systems

The weak-strong uniqueness for Maxwell--Stefan systems and some generalized systems is proved. The corresponding parabolic cross-diffusion equations are considered in a bounded domain with no-flux boundary conditions. The key points of the proofs are various inequalities for the relative entropy associated with the systems and the analysis of the spectrum of a quadratic form capturing the frictional dissipation. The latter task is complicated by the singular nature of the diffusion matrix. This difficulty is addressed by proving its positive definiteness on a subspace and using the Bott--Duffin matrix inverse. The generalized Maxwell--Stefan systems are shown to cover several known cross-diffusion systems for the description of tumor growth and physical vapor deposition processes.

Weighted p-Laplace approximation of linear and quasi-linear elliptic problems with measure data

We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p-Laplace term. A well chosen diffeomorphism between R and (−1,1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L1.

Coupled hydrodynamics in dipole-conserving quantum systems

We investigate the coupled dynamics of charge and energy in interacting lattice models with dipole conservation. We formulate a generic hydrodynamic theory for this combination of fractonic constraints and numerically verify its applicability to the late-time dynamics of a specific bosonic quantum system by developing a microscopic non-equilibrium quantum field theory. Employing a self-consistent $1/N$ approximation in the number of field components, we extract all entries of a generalized diffusion matrix and determine their dependence on microscopic model parameters. We discuss the relation of our results to experiments in ultracold atom quantum simulators.

Layered optimization strategy enables over 17.8% efficiency of layer-by-layer organic photovoltaics

In this work, layer-by-layer (LbL) type OPVs are constructed with wide bandgap polymer PNTB6-Cl as donor and nonfullerene material BTP-4F-12 as acceptor. The layered optimization strategy is employed via separately incorporating the diphenyl ether (DPE) and 1,8-diiodooctane (DIO) into PNTB6-Cl chlorobenzene solution and BTP-4F-12 chloroform solution. A power conversion efficiency (PCE) of 17.81% was achieved from the optimal LbL type OPVs with two solvent additives, which should be among the top level for LBL type binary OPVs. The incorporation of DPE and DIO can separately induce more ordered PNTB6-Cl and BTP-4F-12 orientation, which should contribute to charge transport with suppressed charge recombination in active layers. Meanwhile, BTP-4F-12 crystallization is strongly increased with the incorporation of DIO, which should facilitate more swell matrix for exciton diffusion in active layers. Over 13% PCE improvement can be realized in LbL type OPVs by incorporating two additives, benefiting from simultaneously improved short circuit current density (JSC) of 26.89 mA cm−2 and fill factor (FF) of 75.79%. Meanwhile, the PCE of optimized LbL type OPVs is higher than that of 17.33% for the optimized OPVs with bulk-heterojunction (BHJ) configuration, which reveals that layered optimization strategy should be a promising approach to achieve highly efficient LbL type OPVs.

Simulation of atrial fibrillation in a non-ohmic propagation model with dynamic gap junctions.

Gap junctions exhibit nonlinear electrical properties that have been hypothesized to be relevant to arrhythmogenicity in a structurally remodeled tissue. Large-scale implementation of gap junction dynamics in 3D propagation models remains challenging. We aim to quantify the impact of nonlinear diffusion during episodes of arrhythmias simulated in a left atrial model. Homogenization of conduction properties in the presence of nonlinear gap junctions was performed by generalizing a previously developed mathematical framework. A monodomain model was solved in which conductivities were time-varying and depended on transjunctional potentials. Gap junction conductances were derived from a simplified Vogel-Weingart model with first-order gating and adjustable time constant. A bilayer interconnected cable model of the left atrium with 100 μm resolution was used. The diffusion matrix was recomputed at each time step according to the state of the gap junctions. Sinus rhythm and atrial fibrillation episodes were simulated in remodeled tissue substrates. Slow conduction was induced by reduced coupling and by diffuse or stringy fibrosis. Simulations starting from the same initial conditions were repeated with linear and nonlinear gap junctions. The discrepancy in activation times between the linear and nonlinear diffusion models was quantified. The results largely validated the linear approximation for conduction velocities >20 cm/s. In very slow conduction substrates, the discrepancy accumulated over time during atrial fibrillation, eventually leading to qualitative differences in propagation patterns, while keeping the descriptive statistics, such as cycle lengths, unchanged. The discrepancy growth rate was increased by impaired conduction, fibrosis, conduction heterogeneity, lateral uncoupling, fast gap junction time constant, and steeper action potential duration restitution.

A Novel and Fast Encryption System Based on Improved Josephus Scrambling and Chaotic Mapping.

To address the shortcomings of weak confusion and high time complexity of the existing permutation algorithms, including the traditional Josephus ring permutation (TJRP), an improved Josephus ring-based permutation (IJRBP) algorithm is developed. The proposed IJRBP replaces the remove operation used in TJRP with the position exchange operation and employs random permutation steps instead of fixed steps, which can offer a better scrambling effect and a higher permutation efficiency, compared with various scrambling methods. Then, a new encryption algorithm based on the IJRBP and chaotic system is developed. In our scheme, the plaintext feature parameter, which is related to the plaintext and a random sequence generated by a chaotic system, is used as the shift step of the circular shift operation to generate the diffusion matrix, which means that a minor change in the source image will generate a totally different encrypted image. Such a strategy strikes a balance between plaintext sensitivity and ciphertext sensitivity to obtain the ability to resist chosen-plaintext attacks (CPAs) and the high robustness of resisting noise attacks and data loss. Simulation results demonstrate that the proposed image cryptosystem has the advantages of great encryption efficiency and the ability to resist various common attacks.

Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the L^2-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.

Emergence of Turing patterns and dynamic visualization in excitable neuron model

This article is focused on studying the spatially extended reaction-diffusion system with a diagonal diffusion matrix in a bounded domain for a biophysically motivated excitable model. Diffusion induces spontaneous stationary patterns in the spatially extended homogeneous medium. We investigate the dynamics of the diffusively coupled network modulated by a Hindmarsh–Rose prototype model that describes the emergence of self-excited spiking activities with certain parameters and a constant injected stimulus. The linear stability analysis in this framework around the homogeneous steady states illustrates the emergence of stationary patterns. Turing domains are reported in the parameter space where Hopf bifurcation is determined. The bifurcation diagram helps us in understanding the transition mechanism in the spatial system. We have investigated the existence of Turing–Hopf bifurcation and established how Hopf and Turing curves divide the parameter space into three different dynamically significant regions. We have also studied the existence of Hopf bifurcation in the spatiotemporal system. Theoretically, the amplitude equations are derived by means of nonlinear multiple-scale analysis method and analyzed near the Hopf and Turing instabilities in the system. In particular, we observe asymptotic expressions for a wide range of various patterns (stationary, hexagonal, mixed-state) sustained by the spatial system. We obtain the explicit conditions to establish the structural transitions and stability of the diverse forms of these Turing patterns. These results reveal how the diffusive network evolves. To establish the results, the analytical derivations are demonstrated that are corroborated by numerical simulations of the corresponding diffusion induced system. Finally, we observe that the coupled excitable systems participate in a collective behavior that may contribute significantly to irregular neural dynamics associated with certain brain pathologies.

Image Encryption Based on Hopfield Neural Network and Bidirectional Flipping.

Many encryption systems face two problems: the key has nothing to do with the plaintext; only a single chaotic sequence is adopted during the encryption. To solve the problems, this paper proposes an image encryption method based on Hopfield neural network and bidirectional flipping. Firstly, the plaintext image was segmented into blocks, the resulting image matrix was block scrambled, and each block was bidirectionally flipped to complete the scrambling process. After that, the plaintext image was processed by the hash algorithm to obtain the initial values and control parameters of the chaotic system, producing a pseudo-random sequence. Then, a diffusion matrix was generated through the optimization by Hopfield neural network and used to derive a ciphertext image through diffusion transformation. Experimental results show that our algorithm is highly sensitive to plaintext, strongly resistant to common attacks, and very efficient in encryption.

Data-driven modeling of a four-dimensional stochastic projectile system

The dynamical modeling of projectile systems with sufficient accuracy is of great difficulty due to high-dimensional space and various perturbations. With the rapid development of data science and scientific tools of measurement recently, there are numerous data-driven methods devoted to discovering governing laws from data. In this work, a data-driven method is employed to perform the modeling of the projectile based on the Kramers–Moyal formulas. More specifically, the four-dimensional projectile system is assumed as an Itô stochastic differential equation. Then the least square method and sparse learning are applied to identify the drift coefficient and diffusion matrix from sample path data, which agree well with the real system. The effectiveness of the data-driven method demonstrates that it will become a powerful tool in extracting governing equations and predicting complex dynamical behaviors of the projectile.

A discrete boundedness-by-entropy method for finite-volume approximations of cross-diffusion systems

Abstract An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

A new order from the combination of exact coupling and the Euler scheme

<abstract><p>Davie defined a Levy variant and the combination of single random variables to ensure that the diffusion matrix did not degenerate. The use of the method proposed by Davie, which is a combination of the Euler method and the exact combination, was investigated for applying the degenerate Levy diffusion approach to $ \big(B_{ik}(Y)\big) $. We use certain degenerate conditions of diffusion which contribute to order convergence. We also show MATLAB codes to apply the integrated solution to an SDE and observe a convergence behavior. We also evaluate the agreement between the theoretical values and the MATLAB numerical example.</p></abstract>

Modeling effects of matrix heterogeneity on population persistence at the patch-level.

Habitat loss and fragmentation is the largest contributing factor to species extinction and declining biodiversity. Landscapes are becoming highly spatially heterogeneous with varying degrees of human modification. Much theoretical study of habitat fragmentation has historically focused on a simple theoretical landscape with patches of habitat surrounded by a spatially homogeneous hostile matrix. However, terrestrial habitat patches are often surrounded by complex mosaics of many different land cover types, which are rarely ecologically neutral or completely inhospitable environments. We employ an extension of a reaction diffusion model to explore effects of heterogeneity in the matrix immediately surrounding a patch in a one-dimensional theoretical landscape. Exact dynamics of a population exhibiting logistic growth, an unbiased random walk in the patch and matrix, habitat preference at the patch/matrix interface, and two functionally different matrix types for the one-dimensional landscape is obtained. These results show existence of a minimum patch size (MPS), below which population persistence is not possible. This MPS can be estimated via empirically derived estimates of patch intrinsic growth rate and diffusion rate, habitat preference, and matrix death and diffusion rates. We conclude that local matrix heterogeneity can greatly change model predictions, and argue that conservation strategies should not only consider patch size, configuration, and quality, but also quality and spatial structure of the surrounding matrix.

Stochastic perturbation of a cubic anharmonic oscillator

<p style='text-indent:20px;'>We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion.</p>

Innovative Dynamic Matrix-Type Learning Environment Based on the Example of the St. Petersburg School of Conference Interpreting and Translation, Herzen University, Russia

The paper examines an innovative cluster-type matrix model for training interpreters and translators, based on recent developments in text and pragmatics theory, as well as on the scenario approach in education. The aim is to create a single, multifaceted space for open interpreter and translator training with an anthropological focus, drawing on the achievements in Russian teaching culture and best world practices, including a system of feedback from employers, to enable a flexible response to the needs of the society. To achieve the aim, an innovative interactive system of intensive interpreter and translator training has been developed based on a cluster principle and continuous education quality monitoring. At the heart of the system are a scenario strategy and a text typology approach. What is unique about the St Petersburg School of Conference Interpreting and Translation (SCIT)’s project is the fact that it is not just an educational programme or a system or an algorithm, but a dynamic matrix-type learning environment.It is a flexible model for professional retraining that presupposes a range of basic skills and can fill any gaps in the general translators’ competence and current performance by applying the professional environment matrix. The cluster approach provides a combination of in-person, distant and independent learning components, as well as increased transparency of the educational and professional spaces, opening up future employment prospects.The ‘diffuse matrix’ version of the innovative matrix principle in education underlies the diversity of methods employed by SCIT: methodological pluralism, dominant in the process of training, implies a combination of diverse techniques developed by different authors for the matrix, making it most flexible and the whole approach – customer tailored. It should be pointed out that the external diversity is strictly regulated and targeted by the project’s creators by means of a monitoring mechanism.The future of the model is seen in its use in Russia and other countries to train highly professional interpreters and translators in other spheres (for example, audiovisual translation). It could also be used at the final stage of higher vocational training; as a standard mechanism for improving professional skills and retraining; as well as a social harmonising element (creating new links among academic, educational, and professional activities).

Uniqueness of a Probability Solution to the Kolmogorov Equation with a Diffusion Matrix Satisfying Dini’s Condition

Uniqueness of a Probability Solution to the Kolmogorov Equation with a Diffusion Matrix Satisfying Dini’s Condition

Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and ℝ, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H1 (Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L1. This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p-Laplace vanishing stabilisation term.

Image Processing Method Based on Chaotic Encryption and Wavelet Transform for Planar Design

This paper provides an in-depth study and analysis of image processing for graphic design through chaotic encryption combined with a wavelet transform algorithm. Firstly, the traditional Mallat algorithm is optimized; since the mean value of the transform coefficients generated after the wavelet transform of the image is used as the initial value of the chaotic system to iterate, when the image is modified, then the mean value of the wavelet coefficients will also change, and the final iteration comes out as two different sequences using the property that the chaotic system is extremely sensitive to the initial conditions, so the algorithm has a certain sensitivity to tampering and localization effect. The image of the encrypted graphic design is decrypted by the chaos decryption system, and the final image information of the graphic design is obtained. In terms of the security of the graphic design image itself, the complex dynamical properties of chaos are fully utilized to encrypt it, and the algorithm has a good encryption effect after statistical characteristic analysis, attack complexity, difference analysis, adjacent pixel correlation analysis, and key sensitivity analysis. The plaintext image is decomposed in odd-even sequence using the boosting algorithm to get the sequence with an even index and the sequence with an odd index; then, the diffusion algorithm is applied to the two sequences by the prediction and update algorithm, and this process is repeated many times to get the two ciphertext sequences after scrambling, merging these two sequences, and matrixing them to get the ciphertext image. Finally, the testing of the embedded planar designed image revolves around four aspects, namely, image processing, tamper sensitivity, robustness, and imperceptibility, to examine the designed planar designed image system. In the diffusion manipulation, the key matrix for diffusion is obtained using the threshold processing method of the Local Binary Pattern (LBP) algorithm for the matrices generated by the Logistic Chaos system, which improves the randomness of the algorithm. The experimental results verify the effectiveness and security of the algorithm.

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify on one hand the regularity of the solution with respect to the random domain mapping required for many multilevel quadrature methods, such as the multilevel quasi-Monte Carlo quadrature using Halton points, the multilevel sparse anisotropic Gauss–Legendre and Clenshaw–Curtis quadratures and multilevel interlaced polynomial lattice rules. On the other hand, we derive the coupling formulation and show by numerical results that the approach is feasible.

Structure and Dynamics in Mg2+-Stabilized γ-Na3PO4.

In parallel with advances in the synthesis of solid-state ionic conductors, there is a need to understand the underlying mechanisms behind their improved ionic conductivities. This can be achieved by obtaining an atomic level picture of the interplay between the structure of materials and the resultant ionic diffusion processes. To this end, the structure and dynamics of Mg2+-stabilized rotor phase material γ-Na3PO4, characterized by neutron scattering, are detailed in this work. The Mg2+-stabilized rotor phase is found to be thermally stable from 4 to 650 K. However, signatures of orientational disorder of the phosphate anions are also evident in the average structure. Long-range Na+ self-diffusion was probed by quasi-elastic neutron scattering and subsequently modeled via a jump diffusion matrix with consideration of the phosphate anion rotations. The resultant diffusion model points directly to coupled anion-cation dynamics. Our approach highlights the importance of considering the whole system when developing an atomic level picture of structure and dynamics, which is critical in the rational design and optimization of energy materials.