Let E be a closed nonempty subset of a general real Hilbert space H. Its singular set ΣE consists of those points in H∖E at which the distance function dE fails to be Fréchet differentiable. In particular, this paper demonstrates in full generality that ΣE is of the same homotopy type as the open set GE={x∈H:dco−−E(x)<dE(x)} consisting of the points whose distance to the closed convex hull of E is strictly smaller than to E itself. Moreover, it is shown that GE is intimately connected to the Aubry set of E. In the literature, the singular set ΣE is also known as the medial axis of E when dimH<∞.

For a given closed nonempty subset E of a Hilbert space H,the singular set ΣE consists of the points in H \ E where the distancefunction dE is not Fr´echet differentiable. It is known that ΣE is a weakdeformation retract of the open set GE = {x ∈ H : dco E(x) < dE(x)}.This short paper sheds light on the relationship between the connectedcomponents of the three sets ΣE ⊂ GE ⊆ H\E.

The Moreau envelopes of nonconvex functions on a real Hilbert space have points ofnondifferentiability called singularities. For the first time in an infinite-dimensionalspace, the propagation of singularities along intrinsic characteristic curves x(t) is studied.The properties of the intrinsic characteristics are investigated in some detail. It isproved that singularities propagate along intrinsic characteristics not only locally butglobally in t. Actually, this paper distinguishes between “singularities” and “strongsingularities,” two concepts that agree for distance functions but disagree for Moreauenvelopes, in general. Along intrinsic characteristics, propagating singularities instantaneouslytransform into strong singularities. Concerning the distance function to aclosed nonempty set E, it is demonstrated that the singular set and C E are homotopyequivalent provided E satisfies a certain condition which is weaker than the boundednessof C E. The notion of intrinsic characteristics was introduced by Cannarsa andCheng (Calc Var Part Differ Equ 56, 2017).

We study propagation of singularities for the Hamilton–Jacobi equation S t +H(∇S)=0,(t,x)∈(0,T)×R n , St+H(∇S)=0,(t,x)∈(0,T)×Rn,where H(p)=12 ⟨p,Ap⟩ H(p)=12⟨p,Ap⟩ is a positive definite quadratic form. Each viscosity solution S S is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015), namely that the singularities propagate along generalized characteristics indefinitely forward in time. 

We study propagation of singularities for Hamilton–Jacobi equations View the MathML sourceSt+H(t,x,∇S)=0,(t,x)∈(0,∞)×Rn,Turn MathJax onby means of the excess Lagrangian action and a related class of characteristics. In a sense, the excess action gauges how far a curve View the MathML sourceX(t) is from being action minimizing for a given viscosity solution S(t,x)S(t,x) of the Hamilton–Jacobi equation. Broken characteristics are defined as curves along which the excess action grows at the slowest pace possible. In particular, we demonstrate that broken characteristics carry the singularities of the viscosity solution.

Whenever there is an out-of-control signal in process parameter control charts, maintenance engineers try to diagnose the cause near the time of the signal which is not always lead to prompt identification of the source(s) of the out-of-control condition and this in some cases yields to extremely high monetary loses for manufacture owner. This paper applies multivariate exponentially weighted moving average (MEWMA) control charts and neural networks to make the signal identification more effective. The simulation of this procedure shows that this new control chart can be very effective in detecting the actual change point for all process dimension and all shift magnitudes considered. This methodology can be used in manufacturing and process industries to predict change points and expedite the search for failure causing parameters, resulting in improved quality at reduced overall cost. This research shows development of MEWMA by usage of neural network for identifying the step change point and the variable responsible for the change in the process mean vector.

This paper contributes to the analysis of propagation of singularities for semiconcave solutions of Hamilton–Jacobi equations. Under certain conditions, we establish the existence and uniqueness of certain broken characteristics termed strong characteristics. If u is a viscosity solution of a Hamilton–Jacobi equation, then strong characteristics carry the singularities of u.

The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton{Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.

We study the following system of the viscous Hamilton-Jacobi and the continuity equations in the limit as epsilon down arrow 0: S-t(epsilon) + 1/2 vertical bar DS epsilon vertical bar(2) + V(x) - epsilon Delta S-epsilon = 0 in Q(T), S-epsilon(0, x) = S-0(x) in R-n; rho(epsilon)(t) + div(rho(epsilon) DS epsilon) = 0 in Q(T), rho(epsilon)(0, x) = rho(0)(x) in R-n. Here Q(T) = (0, T] x R-n. The potential V and the initial function S-0 are allowed to grow quadratically while rho(0) is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity-measure solution (S, rho).

This paper revisits the duality between differentiability and strict or strong convexity under the Legendre-Fenchel transform {Mathematical expression}. Functions f defined on a Banach space X are considered. For a lower semicontinuous but not necessarily convex function f we relate essential Fréchet differentiability of the conjugate function f* to essential strong convexity of f