Diffusion Matrix Research Articles

Long-time behaviour of a stochastic chemostat model with distributed delay

ABSTRACTIn this paper, we analyse a stochastic chemostat model with distributed delay and degenerate diffusion. We transform the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the positive solutions can converge in to an invariant density. The existence of a stable stationary distribution implies stochastic persistence of the microorganism.

Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems

The weak–strong uniqueness for renormalized solutions to reaction–cross-diffusion systems in a bounded domain with no-flux boundary conditions is proved. The system generalizes the Shigesada–Kawasaki–Teramoto population model to an arbitrary number of species. The diffusion matrix is neither symmetric nor positive definite, but the system possesses a formal gradient-flow or entropy structure. No growth conditions on the source terms are imposed. It is shown that any renormalized solution coincides with a strong solution with the same initial data, as long as the strong solution exists. The proof is based on the evolution of the relative entropy modified by suitable cutoff functions.

Enhanced adsorption and removal of urea from aqueous solutions using eco-friendly iron phosphate nanoparticles

This work reports of using iron phosphate nanoparticles (nano-FePh) as adsorbent for removing urea. Hence, the adsorption of urea on nano-FePh was investigated as a function of reaction time, temperature, catalyst weight, shaking speed and solution pH. Amorphous iron phosphate is synthesized by a cost-effective method reflux. The nano-FePh is characterized by different techniques including; X-ray diffraction (XRD), transmission electron micrograph (TEM), field-emission scanning electron microscopy (FE-SEM), energy dispersive X-Ray Spectroscopy (EDX), FT-IR and BET surface area determination. Highest adsorption extent obtained at pH range of 7–9 with uptake of urea equals to 1.2 × 102 mg/g (gram of nano-FePh). The reaction kinetics of the adsorption of urea follows a pseudo-first-order model. Application of homogenous film diffusion model and matrix diffusion model is carried out and analyzed. The model of the diffusion liquid film controls the process at the first adsorption step and then at the final stage is under control by matrix diffusion model. Frumkin adsorption isotherm is found to fit with the experimental data. The kinetics and thermodynamic parameters have been determined and analyzed. The nano-FePh adsorbent was stable and the adsorption capacity was remaining excellent after 10 cycles.

Strong well posedness of McKean–Vlasov stochastic differential equations with Hölder drift

Here, we prove strong well-posedness for stochastic systems of McKean–Vlasov type with Hölder drift, even in the measure argument, and uniformly non-degenerate Lipschitz diffusion matrix. The Hölder regularity of the drift with respect to the law argument being for the Wasserstein distance.Our proof is based on Zvonkin’s transformation (1974) and so on the regularization properties of the associated PDE, which is stated on the space [0,T]×Rd×P2(Rd), where T>0, d denotes the dimension equation and P2(Rd) is the space of probability measures on Rd with finite second order moment. Especially, a smoothing effect in the measure direction is exhibited. Our approach is based on a parametrix expansion of the transition density of the McKean–Vlasov process.

Cryptanalysis and Improvement in a Plaintext-Related Image Encryption Scheme Based on Hyper Chaos

Recently, a plaintext-related image encryption scheme based on a hyper-chaotic system has been proposed by Li et al . In their encryption scheme, the permutation process is related to the sum of plaintext pixel values, and the diffusion process is related to the values of nine special positions in the permuted image. This can resist the existing chosen plaintext attack due to the ever-changing permutation matrix and diffusion matrix when the plaintext is changed. Here, we show that Li’s encryption scheme contains two vulnerabilities. First, the gray values of the nine specific pixels do not change during diffusion. Second, the permutation process is reversible. Therefore, the diffusion process can be attacked and the permuted image can be obtained by constructing a special image, where the nine special position values in the special image’s permuted image are the same as those in the cipher image. Then, if the obtained permuted image is chosen as plaintext, we can get its permuted image by attacking the diffusion process. Since the permutation process is reversible, the permuted image is the original plaintext image. In addition, some improvements for Li’s encryption scheme are provided to enhance the security.

A mathematical model of cartilage regeneration after chondrocyte and stem cell implantation – II: the effects of co-implantation

We present a mathematical model of cartilage regeneration after cell therapy, to show how co-implantation of stem cells (mesenchymal stem cells) and chondrocytes into a cartilage defect can impact chondral healing. The key mechanisms involved in the regeneration process are simulated by modelling cell proliferation, migration and differentiation, nutrient diffusion and Extracellular Matrix (ECM) synthesis at the defect site, both spatially and temporally. In addition, we model the interaction between mesenchymal stem cells and chondrocytes by including growth factors. In Part I of this work, we have shown that matrix formation was enhanced at early times when mesenchymal stem cell-to-chondrocyte interactions due to the effects of growth factors were considered. In this article, we show that the additional effect of co-implanting mesenchymal stem cells and chondrocytes further enhances matrix production within the first year in comparison to implanting only chondrocytes or only mesenchymal stem cells. This could potentially reduce healing time allowing the patient to become mobile sooner after surgery.

"Teaching old dogs new tricks": targeting neural extracellular matrix for normal and pathological aging-related cognitive decline.

Cognitive decline is a feature of normal and pathological aging. As the proportion of the global aged population continues to grow, it is imperative to understand the molecular and cellular substrates of cognitive aging for therapeutic discovery. This review focuses on the critical role of neural extracellular matrix in the regulation of neuroplasticity underlying learning and memory in another under-investigated “critical period”: the aging process. The fascinating ideas of neural extracellular matrix forming a synaptic cradle in the tetrapartite synapse and possibly serving as a substrate for storage of very long-term memories will be introduced. We emphasize the distinct functional roles of diffusive neural extracellular matrix and perineuronal nets and the advantage of the coexistence of two structures for the adaptation to the ever-changing external and internal environments. Our study of striatal neural extracellular matrix supports the idea that chondroitin sulfate proteoglycan-associated extracellular matrix is restrictive on synaptic neuroplasticity, which plays important functional and pathogenic roles in early postnatal synaptic consolidation and aging-related cognitive decline. Therefore, the chondroitin sulfate proteoglycan-associated neural extracellular matrix can be targeted for normal and pathological aging. Future studies should focus on the cell-type specificity of neural extracellular matrix to identify the endogenous, druggable targets to restore juvenile neuroplasticity and confer a therapeutic benefit to neural circuits affected by aging.

A Simultaneous Scrambling and Diffusion Color Image Encryption Algorithm Based on Hopfield Chaotic Neural Network

The classic scrambling-diffusion structure for image encryption is vulnerable to be attacked. The scrambling and diffusion process can be cracked separately due to the two processes are independent. In this paper, we proposed a novel color image encryption algorithm which can overcome the problem. In the proposed algorithm, firstly, the first diffusion is performed simultaneously with the first scrambling which can resist the attacker to separately crack the scrambling or diffusion. Then, the second diffusion matrix generated from the Hopfield chaotic neural network (HCNN) which is related to the plaintext is performed which can not only resist common chosen-plaintext attack, but also improved the key sensitivity. Finally, a second scrambling is performed on some special pixels to further improve the security. Experiments and security analysis indicate that the proposed algorithm has an excellent performance in image encryption and various attacks.

A chaotic image encryption scheme with complex diffusion matrix for plain image sensitivity

A chaotic image encryption scheme with complex diffusion matrix for plain image sensitivity

Influence of temperature on multicomponent diffusion in calcium and sodium aluminosilicate melts

The effect of temperature on multicomponent chemical diffusion in liquids of the quaternary system CaO – Na2O – Al2O3 – SiO2 has been studied. Diffusion-couple experiments were performed around a central composition of 64.5 wt % SiO2, 13.3 wt % Na2O, 10.8 wt % CaO, 11.4 wt % Al2O3. Experiments were performed for three temperatures far above the glass transition (1200, 1280 and 1360 °C), as well as 30°C above the glass transition. Strong multidiffusive effects were observed for all temperatures, with significant uphill diffusion of calcium, demonstrating that uphill diffusion happens close to the glass transition as well as at high temperature. For each temperature, we determined the diffusion matrix of the system and quantified its eigenvectors and eigenvalues, which correspond formally to (respectively) exchanges between ions, and associated diffusion coefficients. Little variation of the eigenvectors of the diffusion matrix was observed as a function of temperature, with a dominant eigenvector corresponding to the exchange of sodium with calcium, the two other eigenvectors corresponding to the exchange of calcium with network formers. For the temperature range 1200–1360 °C, the eigenvalues of the diffusion matrix have an Arrhenian temperature dependence, with an activation energy consistent with electrical conductivity for the exchange of sodium and calcium, and an activation energy consistent with viscosity for eigenvectors involving network formers. Multicomponent diffusion close to the glass transition is characterized by the same eigenvectors as at higher temperature, but some diffusion profiles are asymmetric due to strong viscosity contrasts resulting in concentration-dependent eigenvalues. Moreover, we observe some departure from Eyring relation close to the glass transition, with diffusion eigenvalues several orders of magnitude greater than the Eyring prediction.

Super Diffusion for Salient Object Detection.

One major branch of saliency object detection methods are diffusion-based which construct a graph model on a given image and diffuse seed saliency values to the whole graph by a diffusion matrix. While their performance is sensitive to specific feature spaces and scales used for the diffusion matrix definition, little work has been published to systematically promote the robustness and accuracy of salient object detection under the generic mechanism of diffusion. In this work, we firstly present a novel view of the working mechanism of the diffusion process based on mathematical analysis, which reveals that the diffusion process is actually computing the similarity of nodes with respect to the seeds based on diffusion maps. Following this analysis, we propose super diffusion, a novel inclusive learning-based framework for salient object detection, which makes the optimum and robust performance by integrating a large pool of feature spaces, scales and even features originally computed for non-diffusion-based salient object detection. A closed-form solution of the optimal parameters for the integration is determined through supervised learning. At the local level, we propose to promote each individual diffusion before the integration. Our mathematical analysis reveals the close relationship between saliency diffusion and spectral clustering. Based on this, we propose to re-synthesize each individual diffusion matrix from the most discriminative eigenvectors and the constant eigenvector (for saliency normalization). The proposed framework is implemented and experimented on prevalently used benchmark datasets, consistently leading to state-of-the-art performance.

Dendrite tip selection during isothermal free growth in multi-component alloys: Marginal stability theories and insights from phase-field simulations

The principal length scales associated with dendritic solidification, which being the dendritic tip-radius (Rtip) and the primary dendritic arm spacing, are functions of the relative solutal interdiffusivities of the different components in a multi-component alloy. In this paper, we firstly derive marginal stability based theories to predict Rtip during isothermal free growth in any generic multi-component alloy for any generic diffusivity matrix. Here, we extend the Ivanstov solution of the diffusion problem in the liquid with a parabolic solid-liquid interface and propose the closure conditions between the Rtip and the selected dendrite tip velocity (V) based on modification of the classical marginal stability criteria for multi-component situations. Additionally, we derive the constant Rtip2V from phase-field simulations and using the Ivantsov parabola as the approximate interface shape compute the Rtip and the V, which we term as “Approximate microsolvability prediction (AMP)”. Thereafter, we compare our predictions of Rtip,V as well as phase compositions with independent phase-field simulation results for different choices of the diffusivity matrix, where we show that the analytical theories based upon the marginal stability theory are unable to accurately predict variations of the dendrite tip velocity V and the Rtip. We thereby utilize the phase-field simulation results to investigate the tip selection in multi-component alloys and derive an appropriate solvability constant σ∗. We find that the σ∗ in a multi-component alloy is a function of the diffusivity ratios of the components for the case of a diagonal diffusivity matrix.

Global martingale solutions for a stochastic population cross-diffusion system

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski’s generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia’s truncation method due to Chekroun, Park, and Temam.

Functional integrals method for systems of stochastic differential equations

Systems of stochastic differential equations, for which the Riemannian manifold generated by a diffusion matrix has zero curvature, are considered in this article. The method for approximate evaluation of characteristics of the solution of the systems of stochastic differential equations is proposed. This method is based on the representation of the probability density function through the functional integral. To compute functional integrals we use the expansion of action with respect to a classical trajectory, for which the action takes an extreme value. The classical trajectory is found as the solution of the multidimensional Euler – Lagrange equation.

Fick and Fokker–Planck Diffusion Law in Inhomogeneous Media

We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick's law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non-homogeneous Einstein relation and the possibility of uphill diffusion.

Estimation of γ/γ′ interfacial energy in Ni-Co base superalloy TMW-4M3

The interfacial energy between the γ and γ′ phase in Ni-Co base superalloy TMW-4M3 was investigated using both a kinetic analysis of experimental data and a thermodynamic analysis. The coarsening rate of γ′ precipitates during the aging treatment process was analyzed using extended Lifshitz-Slyozov-Wagner theory, including the contribution of a full diffusion matrix with a thermodynamic calculation. The interfacial energies calculated from the coarsening kinetics using only the diagonal components were half as large as those using the off-diagonal contributions as well. The interfacial energy with full diffusion matrix contributions is about 75 mJ/m2 between 1373 and 1413 K. In addition, the interfacial energy was also analyzed on the basis of the extended Becker’s model (EBM), and the results, which ranged from 73.9 to 76.3 mJ/m2, showed good agreements with those from the coarsening kinetics. A comparison of the interfacial energy between TMW-4M3 and several Ni base superalloys indicated an identical tendency of higher interfacial energy with a larger Ti/Al ratio. The off-diagonal terms of a diffusion matrix are very important in analyzing the coarsening kinetics of γ′, and the γ/γ′ interfacial energy estimation using the EBM is a suitable method for TMW-4M3.

Robust Fuzzy Filter Design for Nonlinear Parabolic Partial Differential Systems with Continuous Wiener Noise and Discontinuous Poisson Noise

This paper addresses a robust \(H_{\infty }\) fuzzy filter design problem for nonlinear stochastic partial differential systems (NSPDSs) with continuous random fluctuation, discontinuous Poisson jumping noise, random external disturbance and measurement noise in the spatiotemporal domain. For NSPDSs, the robust \(H_{\infty }\) filter design problem through a measurement output needs to solve a complex second-order Hamilton Jacobi integral inequality. In order to simplify the design procedure, a fuzzy stochastic partial differential system based on a fuzzy interpolation approach is proposed to approximate the NSPDS. Then, the robust \(H_{\infty }\) fuzzy filter design problem can be reformulated as a diffusion matrix inequality (DMI) problem. Since the DMI problem is difficult to be solved via traditional algebraic techniques, we utilize the divergence theorem and Poincare inequality to transform the DMIs to a set of linear matrix inequalities (LMIs) which could be easily solved with the help of MATLAB LMI Toolbox. Finally, a robust state estimation problem of an ecology system with intrinsic spatiotemporal continuous Wiener noise and discontinuous Poisson jump fluctuation is provided as an example to illustrate the design procedure and to confirm the \(H_{\infty }\) filtering performance of the proposed \(H_{\infty }\) fuzzy filter design method.

A color image cryptosystem based on dynamic DNA encryption and chaos

• A color image cryptosystem based on dynamic DNA encryption and a four-wing hyperchaotic system is presented. • Dynamic DNA encoding and decoding rules for all the pixels are generated and dependent on the plain image. • A novel diffusion mechanism based on random numbers related to plaintext (DMRNRP) is introduced to diffuse the DNA sequences of the plain image. • The image encryption method is highly sensitive to the plain image. • Experimental results demonstrate that the proposed encryption algorithm has good security performance. This paper presents a color image cryptosystem based on dynamic DNA encryption and chaos. The color plain image is firstly decomposed into red, green and blue components, and then a simultaneous intra-inter-component permutation mechanism dependent on the plaintext (SCPMDP) is introduced to shuffle them. Secondly, convert the recombined permutated components into a DNA matrix by a DNA encoding rule, and subsequently, different from the traditional DNA sequence operations according to the law of binary calculation, a diffusion mechanism based on random numbers related to plaintext (DMRNRP) is presented to diffuse it. Further, transform the diffused DNA matrix into a decimal one according to a DNA decoding rule, and divide it into three equal images. Finally, to enhance security of the image cryptosystem, the second confusion scheme is applied to respectively scramble images, and the color cipher image is gotten. A four-wing hyperchaotic system is used to provide pseudo-random chaotic sequences, SHA 384 hash function of the plain image and external parameters are combined to compute its initial values, and one-time-pad encryption policy makes the proposed encryption effectively resist plaintext attacks. Experimental results and security analysis demonstrate that our encryption algorithm has good performance and may resist against various typical attacks.

Upscaling of a parabolic system with a large nonlinear surface reaction term

Motivated by the study of the dynamics of calcium ions in biological cells, the authors derived in [33], via periodic homogenization, a macroscopic bidomain model, by considering in the corresponding microscopic two-component problem a properly scaled nonlinear exchange term. We study here, at the microscopic scale, a similar parabolic system, with a large nonlinear interfacial reaction term. At the macroscopic scale, the nonlinear effect of this reaction term is recovered in the homogenized diffusion matrix, which is not anymore constant. This nonstandard phenomenon shows the fine interplay between reaction and diffusion in such processes.

Stationary distribution of a stochastic predator–prey model with distributed delay and general functional response

In this paper, we study a stochastic predator–prey model with distributed delay and general functional response. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition does not hold. The Markov semigroup theory is used to derive the existence of a unique stable stationary distribution. We verify the densities of the distributions of the positive solutions can converge in L1 to an invariant density.