We study the Cauchy problem for quasilinear parabolic inequalities containing squares of first derivatives an unknown function (the so-called nonlinearities KPZ type). The coefficients leading nonlinear terms considered either can be continuous functions regular case) or admit power singularities singular degree no greater than 1. For case, we prove damping global nonnegative solutions to studi...

In this work, an effective numerical method is developed to solve a class of singular boundary value problems arising in various physical models by using the improved differential transform method (IDTM). The IDTM applies the Adomian polynomials to handle the differential transforms of the nonlinearities arising in the given differential equation. The relation between the Adomian polynomials of...

We study traveling waves for reaction diffusion equations on the spatially discrete domain Z2. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [1] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the saw...

In this paper, a composite approach to the adaptive neural network (NN) controller is proposed for a rigid link flexible joint (RLFJ) robot manipulator with unknown nonlinearities. Based on a singular perturbation formulation of robot motion dynamics, the RLFJ robot is described by a reduced-order flexible-joint model. The concept of integral manifold is used to decompose the model into fast an...

Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network’s input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spectra of Jacobians at initialization. To this end, we leverage powerful tools ...

We study the flat phase of nematic elastomer membranes with rotational symmetry spontaneously broken by in-plane nematic order. Such state is characterized by a vanishing elastic modulus for simple shear and soft transverse phonons. At harmonic level, in-plane orientational (nematic) order is stable to thermal fluctuations, that lead to short-range in-plane translational (phonon) correlations. ...

We consider the following superlinear elliptic equation on S ε∆Snu− u + u = 0 in S, u > 0 in S where ∆Sn is the Laplace-Beltrami operator on S. We prove that for any k = 1, ..., n−1, there exists pk > 1 such that for 1 < p < pk and ε sufficiently small, there exist at least n − k positive solutions concentrating on k−dimensional subset of the equator. We also discuss the problem on geodesic bal...

We propose an estimator for nonlinear systems with unmatched unknown inputs and under measurement noise. The estimator design is based on the combination of observer design for descriptor systems, sliding-modes theory and adaptive control. The estimation of the measurement noise is achieved thanks to the transformation of the original system into a singular form where the measurement noise make...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...