Narrow Results

The propagation and reflection of thermo-elastic waves through diffusive nonlocal isotropic medium had been studied in this paper. The Green–Lindsay model of thermo-elasticity is incorporated in the context of Temperature Rate Dependent Theory. Using Helmholtz vector decomposition rule, the system of governing equations has been transformed to their respective components. The dispersion relation in frequency indicates the existence of three coupled waves and one independent wave propagating through the medium. The coupled waves are affected by non-locality, temperature field and diffusivity in the medium; anyhow, un-coupled shear vertical wave is only affected by the non-local parameter. The reflection of $P$-wave is also studied at the free boundary of the solid and their corresponding amplitude ratios are computed using set of suitable boundary conditions. The obtained results are further discussed graphically for significant physical parameters of interest. The results in the literature are obtained as a special case after ignoring the diffusivity in the solid.

Let be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form . In particular, we establish results on the distribution kernels and mapping properties of e^-tL and (μ + L)^s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the W^s,p-Sobolev spaces on M and E. We also prove that L satisfies maximal L^p–L^q-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂_tu + Lu = F(t, x, u, ∇ u).

In this short survey, we describe some recent developments on the modeling of propagation by reaction-differential equations with free boundaries, which involve local as well as nonlocal diffusion. After the pioneering works of Fisher, Kolmogorov–Petrovski–Piskunov (KPP) and Skellam, the use of reaction–diffusion equations to model propagation and spreading speed has been widely accepted, with remarkable progresses achieved in several directions, notably on propagation in heterogeneous media, models for interacting species including epidemic spreading, and propagation in shifting environment caused by climate change, to mention but a few. Such models involving a free boundary to represent the spreading front have been studied only recently, but fast progress has been made. Here, we will concentrate on these free boundary models, starting with those where spatial dispersal is represented by local diffusion. These include the Fisher–KPP model with free boundary and related problems, where both the one space dimension and high space dimension cases will be examined; they also include some two species population models with free boundaries, where we will show how the long-time dynamics of some competition models can be fully determined. We then consider the nonlocal Fisher–KPP model with free boundary, where the diffusion operator $\mathrm{\Delta }u$ is replaced by a nonlocal one involving a kernel function. We will show how a new phenomenon, known as accelerated spreading, can happen to such a model. After that, we will look at some epidemic models with nonlocal diffusion and free boundaries, and show how the long-time dynamics can be rather fully described. Some remarks and comments are made at the end of each section, where related problems and open questions will be briefly discussed.

Cortical spreading depression (CSD), which is a significant pathological phenomenon that correlates with migraines and cerebral ischemia, has been characterized by a wave of depolarization among neuronal cells and propagates across the cortex at a rate of 2–5 mm/min. Although the propagation pattern of CSD was well-investigated using high-resolution optical imaging technique, the variation of propagation speed of CSD across different regions of cortex was not well-concerned, partially because of the lack of ideal approach to visualize two-dimensional distribution of propagation speed of CSD over the whole imaged cortex. Here, we have presented a method to compute automatically the propagation speed of CSD throughout every spots in the imaged cortex. In this method, temporal clustering analysis (TCA) and least square estimation (LSE) were first used to detect origin site where CSD was induced. Taking the origin site of CSD as the origin of coordinates, the data matrix of each image was transformed into the corresponding points based on the polar-coordinate representation. Then, two fixed-distance regions of interest (ROIs) are sliding along with the radial coordinate at each polar angle within the image for calculating the time lag with correlating algorithm. Finally, we could draw a two-dimensional image, in which the value of each pixel represented the velocity of CSD when it spread through the corresponding area of the imaged cortex. The results demonstrated that the method can reveal the heterogeneity of propagation speed of CSD in the imaged cortex with high fidelity and intuition.

This paper considers several aspects of the relationship between size, structure, speed of propagation and the number of autonomous cognitive agents in a neural network. While memory and function generation capacities of neural networks with scale-invariant structure have been investigated extensively, the number of autonomous agents, associated with the simultaneous processing of the independent components of the data, has not received prior attention. We propose the emergence of the dichotomy of causal and noncausal regions created by the speed of propagation, in which the autonomous cognitive agents are not bound in a causal relationship with other agents. Arguments are presented for why the count of autonomous agents is best estimated with respect to the dimensionality of the underlying space. The number of autonomous agents obtained for the human brain equals 25, and it is significant that the number in the sub-system modules also turns out to be close to the same value. This number equals the number of enumerative categories of reality in at least one philosophical tradition, and this coincidence may be attributed to the capacity of consciousness to reflect on its own structure. It is possible that near equality across layers provides a special uniqueness to the human brain. We propose that the findings of this study will be useful in the design of neural-network-based AI systems that are designed to emulate human cognitive capacity.