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In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u = b e − a x 2 , a < 0 , a , b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u = b e − a x 2 2 , respectively, for the third-order PDE, u = A e i Φ , where the amplitude b and the phase function a are complex-valued functions, A > 0 , and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota's approach or the method of simplest equation. [ABSTRACT FROM AUTHOR]

This paper presents a review of the theory and applications of memristor cellular nonlinear networks. By mapping the physical processes to the memristive framework, all resistive switching devices can be modeled. The idea is to find a state variable that presents with high accuracy the important features of the system, its dynamics, and time evolution in response to the inputs of the memristor. In order to develop a new design of memristor-based cellular nonlinear networks (MCNN), new circuital and mathematical memristor models need to be introduced. In this way, implementation into new software packages for computer-aided integrated circuit realization can be achieved. Another challenging problem is studying the complex behavior of MCNN models by means of local activity theory and generalizing it for various test cases. An application of the hardware implementation of these models can be found in nanostructures. [ABSTRACT FROM AUTHOR]

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B 1 a Green function is constructed in the cases c > 0 , c ∉ − N , where c is the coefficient in front of u in the boundary condition ∂ u ∂ n + c u = f . To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λ u + c u = f . Elliptic boundary value problems for Δ m u = 0 in B 1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ u = 0 , Δ 2 u = 0 and Δ 3 u = 0 in B 1 as well as some additional results from the theory of spherical functions are proposed. [ABSTRACT FROM AUTHOR]

Information processing in the brain takes place in a dense network of neurons connected through synapses. The collaborative work between these two components (Synapses and Neurons) allows for basic brain functions such as learning and memorization. The so-called von Neumann bottleneck, which limits the information processing capability of conventional systems, can be overcome by the efficient emulation of these computational concepts. To this end, mimicking the neuronal architectures with silicon-based circuits, on which neuromorphic engineering is based, is accompanied by the development of new devices with neuromorphic functionalities. We shall study different memristor cellular nonlinear networks models. The rigorous mathematical analysis will be presented based on local activity theory, and the edge of chaos domain will be determined in the models under consideration. Simulations of these models working on the edge of chaos will show the generation of static and dynamic patterns. [ABSTRACT FROM AUTHOR]