Recently we have created a multitime maximum principle gathering together some concepts in Mechanics, Field Theory, Differential Geometry, and Control Theory. The basic tools of our theory are variational PDE systems, adjoint PDE systems, Hamiltonian PDE systems, duality, multitime maximum principle, incavity on manifolds etc. Now we justify the multitime maximum principle for curvilinear integ...

Objective Bayesian epistemology invokes three norms: the strengths of our beliefs should be probabilities, they should be calibrated to our evidence of physical probabilities, and they should otherwise equivocate sufficiently between the basic propositions that we can express. The three norms are sometimes explicated by appealing to the maximum entropy principle, which says that a belief functi...

Let ÍÍ be a polygonal domain in the plane and Sy(£l) denote the finite element space of continuous piecewise polynomials of degree < r — 1 (r > 2) defined on a quasi-uniform triangulation of ii (with triangles roughly of size h). It is shown that if un e Sy(Sl) is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form ""ftHi^n) < CII"/illL„.(3iï) holds. ...

The maximum principle for SPDEs is established in multidimensional C domains. An application is given to proving the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs. The maximum principle is one of the most powerful tools in the theory of second-order elliptic and parabolic partial differential equations. However, until now it did not play any significant role in the ...

Since the second half of the 20th century, Pontryagin’s Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer underst...

Many science and engineering problems can be formulated as optimization problems that are governed by contact distributions (multitime Pfaff evolution systems) and by cost functionals expressed as multiple integrals or curvilinear integrals. Our paper discuss the contact distribution constrained optimization problems, focussing on a nonholonomic approach of multitime maximum principle. This pri...

Define the following parabolic operator in divergence form Lu ≡ ∂u ∂t − ∂ ∂x j a ij (x, t) ∂u ∂x i for (x, t) ∈ Ω T = Ω × (0, T ], where Ω is a bounded simply connected subset of R m for m ≥ 2 and T > 0 is a fixed but arbitrary positive number. We also assume that the a ij are bounded and measurable in ¯ Ω T and satisfy the uniform parabolicity condition: 1/ν |z| 2 ≤ m i,j=1 a ij (x, t)z i z j ...

A method, which we suggest to call the Empirical Maximum Entropy method, is implicitly present at Maximum Entropy Empirical Likelihood method, as its special, non-parametric case. From this vantage point the entropy-based empirical approach to estimation is surveyed.