1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9868.2011.00777.x

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

Barrier Lyapunov Functions for the control of output-constrained nonlinear systems

We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set ⌈0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

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Motion of Level Sets by Mean Curvature. I

: It is demonstrated the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

C1 + α local regularity of weak solutions of degenerate elliptic equations

Preface to the Second Edition * Preface to the First Edition * Elementary Properties * Multivariable Calculus of Real Analytic Functions * Classical Topics * Some Questions of Hard Analysis * Results Motivated by Partial Differential Equations * Topics in Geometry * Bibliography * Index

A Primer of Real Analytic Functions

Basics.- Caratheodory's Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: The Area Formula, the Coarea Formula, and Poincare Inequalities..- The Calculus of Differential Forms and Stokes's Theorem.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.

Geometric Integration Theory

This treatment of domains in space emphasizes the growing interaction between analysis and geometry. Geometric analysis is an important area of study for both pure and applied mathematicians and engineers.

/pdf/the-geometry-of-domains-in-space-3e2x6qaoz0.pdf

The geometry of domains in space

Explicit determination of area minimizing hypersurfaces, II

The dihedral angle of a regular n-dimensional simplex is the angle formed by a pair of intersecting faces. In the case n = 2 we are describing the angle at the vertex of an equilateral triangle, while in the case n = 3 we are describing the angle formed by the faces of the regular tetrahedron. (This angle is cos-'(1/3), or approximately 70.5?.) In the general n-dimensional case, the dihedral angle is known to equal cos-'(l/n) (see Coxeter [1]). This fact has been proved elegantly by R. Krasnodebski in [3]. Krasnodebski bases his work on a construction used by Coxeter in proving Gosset's theorem (see Coxeter [2]), so Krasnodebski's proof is neither self-contained nor elementary. As far as we know, there is no simple, elementary proof in print. om the sual generati g function fo the Stirling numbers of the second kind,

Harold R. Parks

Papers

A Primer of Real Analytic Functions

Geometric Integration Theory

The geometry of domains in space

Explicit determination of area minimizing hypersurfaces, II

An elementary calculation of the dihedral angle of the regular n-simplex