Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page

Gordon's inequality and condition numbers in conic optimization

The probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry

Commensurate Nb2Zr5O15: Accessible Within the Field Nb2ZrxO2x+5 After All

Doped niobium zirconium oxides are applied in field-effect transistors and as special-purpose coatings. Whereas their material properties are sufficiently known, their crystal structures remain widely uncharacterized. Herein, we report on the comparably mild sol–gel synthesis of Nb2Zr5O15 and the elucidation of its commensurately modulated structure via neutron diffraction. We describe the structure using the most appropriate superspace as well as the convenient supercell approach. It is part of an α-PbO2-homeotypic field with the formula Nb2ZrxO2x+5, which has previously been reported only for x ≥ 5.1, and is closely related to the structure of Hf3Ta2O11. The results, supported by X-ray diffraction and additional synthesis experiments, are contextualized within the existing literature. Via the sol–gel route, metastable Nb–Zr–O compounds and their heavier congeners are accessible that shed light on possible structures of these commercially utilized materials.DFG, 198634447, SPP 1613: Regenerativ erzeugte Brennstoffe durch lichtgetriebene Wasserspaltung: Aufklärung der Elementarprozesse und Umsetzungsperspektiven auf technologische KonzepteTU Berlin, Open-Access-Mittel - 201

Data structures for retrieval on integer grids

A family of data structures is presented for retrieval of the sum of values of points within a half-plane or polygon, given that the points are on integer coordinates in the plane. Fredman has shown that the problem has a lower bound of Ω(N^2/3) for intermixed updates and retrievals. Willard has shown an upper bound of O(N^2log6^4) for the case where the points are not restricted to integer coordinates.We have developed families of related data structures for retrievals of half-planes or polygons. One of the data structures permits intermixed updates and half-plane retrievals in O(N^2/3log N) time, where N is the size of the grid.We use a technique we call "Rotation" to permit a better match of a portion of the data structure to the particular problem. Rotations appear to be an effective method for trading-off storage redundancy against retrieval time for certain classes of problems

Effective Condition Number Bounds for Convex Regularization

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with arXiv:1408.301

Reference data for phase diagrams of triangular and hexagonal bosonic lattices

We investigate systems of bosonic particles at zero temperature in triangular and hexagonal optical lattice potentials in the framework of the Bose-Hubbard model. Employing the process-chain approach, we obtain accurate values for the boundaries between the Mott insulating phase and the superfluid phase. These results can serve as reference data for both other approximation schemes and upcoming experiments. Since arbitrary integer filling factors g are amenable to our technique, we are able to monitor the behavior of the critical hopping parameters with increasing filling. We also demonstrate that the g-dependence of these exact parameters is described almost perfectly by a scaling relation inferred from the mean-field approximation.Comment: 6 pages, 5 figures, accepted for publication in EP