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We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational…
In this paper we consider an acoustic problem of wave propagation through a discontinuous medium. The problem is reduced to the dissipative wave equation with distributional dissipation. We show that this problem has a so-called very weak…
Finding hidden layers in complex networks is an important and a non-trivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multi-layer system exist and if so then what is…
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of…
Adaptive networks consist of a collection of nodes with adaptation and learning abilities. The nodes interact with each other on a local level and diffuse information across the network to solve estimation and inference tasks in a…
In this paper, we study the partial differential equation models of neural networks. Neural network can be viewed as a map from a simple base model to a complicate function. Based on solid analysis, we show that this map can be formulated…
We consider a class of linear differential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear.…
Wave propagation through waveguides, quantum wires or films with a modest amount of disorder is in the semi-ballistic regime when in the transversal direction(s) almost no scattering occurs, while in the long direction(s) there is so much…
In this work we include the elastic scattering of longitudinal electromagnetic waves in transport theory using a medium filled with point-like, electric dipoles. The interference between longitudinal and transverse waves creates two new…
Partial differential equation (PDE) models and their associated variational energy formulations are often rotationally invariant by design. This ensures that a rotation of the input results in a corresponding rotation of the output, which…
We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
This work considers the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Rigorous numerical analysis for this…
In this work, we introduce a compartmental advection-diffusion network model to describe the propagation of stress in a population situated in two interconnected spatial zones during a disaster situation. The model accounts for interactions…
The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type…
Nodes residing in different parts of a graph can have similar structural roles within their local network topology. The identification of such roles provides key insight into the organization of networks and can be used for a variety of…
In this paper, we analyze the effects of random sampling on adaptive diffusion networks. These networks consist in a collection of nodes that can measure and process data, and that can communicate with each other to pursue a common goal of…
A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by…
This chapter is a pedagogical review of methods and results for studying wave propagation in one-dimensional complex structures. We describe and compare the tight-binding, scattering matrix, transfer matrix and Riccati formalisms. We…
The linear semigroup associated with age-structured diffusive populations is investigated in the $L_1$-setting. A complete determination of its generator is given along with detailed spectral information that imply, in particular, an…
The structure of real-world networks is usually difficult to characterize owing to the variation of topological scales, the nondyadic complex interactions, and the fluctuations in the network. We aim to address these problems by introducing…