Assume submodels are in exponential family: pm(x, z; θm) = exp { θ m ( φm(x)φ Z(z) ) −Am(θm) } for x ∈ X , z ∈ Zm and 0 otherwise Reformulation of Product EM: Aggregate parameters: b = ∑ m bm, bm = φ X m(x) θm E-step: compute expected sufficient statistics μ = E(b,∩mZm) def = Eq(z;b)φ(z) with support ∩mZm M-step: set θm to match moments φ X m(x)μ Exponential family formulation Two sources of in...

We study the regularized, oscillatory, multiscale, quasilinear, elliptic problem in divergence form − div(Qη(x, x , u)∇u) = f, in Ω, (0.1) u = g, on ∂Ω, (0.2) assuming that the conductivity has the form Qη(x, y, r) = a−(x) + (a+(x)− a−(x))βη(r − ψ(x, y)), (0.3) for some domain Ω ⊂ Rn, n ≥ 1, a regularization βη of the Heaviside step function and a locally periodic function ψ(x, y), Y -periodic ...

Any quasismooth function f (x) in a finite interval [0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e., f (...

We study semi-linear elliptic free boundary problems in which the forcing term is discontinuous; i.e., a Poisson’s equation where the forcing term is the Heaviside function applied to the unknown minus a constant. This approach uses level sets to construct a monotonic sequence of iterates which converge to a class of solutions to the free boundary problem. The monotonicity of the construction b...

It is well-known that distributional solutions to the Cauchy problem for ut +(b(t, x)u)x = 0 with b(t, x) = 2H(x− t), where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied.

Abstract An extended meshfree method is employed in this paper for investigating the dynamic behaviour of cracked plates based on first-order shear deformation theory (FSDT). The FSDT a straightforward formulation with assumption as its name implies, which appropriate relatively thick plates. In study, chosen an alternative to conventional mesh-based methods model plate structures. Among variou...

This paper is devoted to the study of inhomogeneous wave equation with singular (less than continuous) time dependent coefficients. Particular attention given role lower order terms and suitable Levi conditions are formulated in obtain a very weak solution as introduced [15]. Very solutions for this kind equations also investigated from numerical point view two toy models: Heaviside function de...

In this work, we study a quasilinear elliptic problem involving the 1-Laplacian operator, with discontinuous, superlinear and subcritical nonlinearity Heaviside function $H(\cdot - \beta )$ . Our approach is based on an analysis of associated p-Laplacian problem, followed by thorough asymptotic behaviour or such solutions as $p \to 1^+$ We also solutions, $\beta 0^+$ prove that it converges to ...

We analyze a time independent integral equation defined on a spatially extended domain which arises in the modelling of neuronal networks. In this paper, the coupling function is oscillatory and the firing rate is a smooth “heaviside-like” function. We will derive an associated fourth order ODE and establish that any bounded solution of the ODE is also a solution of the integral equation. We wi...