In his proof of Andreev’s theorem, Thurston in [1] introduced a conformal geometric structure on two dimensional simplicial complexes which is an analogue of a Riemannian metric. He then used a version of curvature to prove the existence of circle-packings (see also Marden-Rodin [2] for more exposition). Techniques very similar to elliptic partial differential equation techniques were used by Y...

subject to (tf , x(tf )) ∈ S = [t0,∞)× S1 where S1 is a k dimensional manifold in Rn S1 = {x ∈ R : h1(x) = h2(x) = · · · = hn−k(x) = 0} where hi are C1 functions from Rn to R subject to ẋ = f(x, u), x(t0) = x0 for u ∈ C[t0, T ] and u(t) ∈ U ⊂ Rm with f and L being C1 functions. Let u∗ : [t0, tf ] → R be an optimal control with state trajectory x∗ : [t0, tf ] → Rn and a constant. Then there exis...

In many practical situations, the Maximum Entropy (MaxEnt) approach leads to reasonable distributions. However, in an important case when all we know is that the value of a random variable is somewhere within the interval, this approach leads to a uniform distribution on this interval – while our intuition says that we should have a distribution whose probability density tends to 0 when we appr...

We discuss an innnite-dimensional version of Pontrya-gin's maximum principle as a uniied variational method in many familiar classes of analytic functions, and its interrelation with classical variational methods in geometric function theory.

We study the strong maximum principle for the heat equation associated with the Dirichlet form on an infinite network. We prove that the strong maximum principle is equivalent to the underlying graph being connected after deletion of the nodes with infinite degree. Using this result, we prove that the multiplicity of the eigenvalue 0 of a generalization of the Laplace matrix equals the number o...

The concept of uncertain entropy is used to provide a quantitative measurement of the uncertainty associated with uncertain variables. After introducing the definition, this paper gives some examples of entropy of uncertain variables. Furthermore this paper proposes the maximum entropy principle for uncertain variables, that is, out of all the uncertainty distributions satisfying given constrai...

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow’s numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, ...

In this paper, the authors study an optimal control problem for quasilinear elliptic PDEs with pointwise state constraints. Weak and strong optimality conditions of Pontryagin maximum principle type are derived. In proving these results, we penalized the state constraints and respectively use the Ekeland variational principle and an exact penalization method. In this paper, our aim is to prove ...

Some generalizations of the maximum principle for harmonic functions are discussed. §

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow’s numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, ...