Multivariate density estimation using dimension reducing information and tail flattening transformations for truncated or censored data

This paper introduces a multivariate density estimator for truncated and censored data with special emphasis on extreme values based on survival analysis. A local constant density estimator is considered. We extend this estimator by means of tail flattening transformation, dimension reducing prior knowledge and a combination of both. The asymptotic theory is derived for the proposed estimators. It shows that the extensions might improve the performance of the density estimator when the transformation and the prior knowledge is not too far away from the true distribution. A simulation study shows that the density estimator based on tail flattening transformation and prior knowledge substantially outperforms the one without prior knowledge, and therefore confirms the asymptotic results. The proposed estimators are illustrated and compared in a data study of fire insurance claims.     

Rational and Heron tetrahedra

Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353-368] began a systematic search for tetrahedra having integer edges and volume by restricting his attention to those with two or three different edge lengths. Of the fifteen configurations identified for such tetrahedra, Buchholz leaves six unsolved. In this paper we examine these remaining cases for integer volume, completely solving all but one of them. Buchholz also considered Heron tetrahedra, which are tetrahedra with integral edges, faces and volume. Buchholz described an infinite family of Heron tetrahedra for one of the configurations. Another of the cases yields a new infinite family of Heron tetrahedra which correspond to the rational points on a two-parameter elliptic curve.     

Calculus of generalized hyperbolic tetrahedra

We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.     

The kissing numbers of tetrahedra

We determine the lattice kissing numbers of tetrahedra, by which we disprove a conjecture by Grünbaum. At the same time, we present a strange phenomenon concerning kissing numbers and packing densities of tetrahedra. This article is based upon part of the author's Ph.D. thesis which was supported by the Austrian Academic Exchange Service.     

An isoperimetric problem for tetrahedra

It is proved that a regular tetrahedron has the maximal possible surface area among all tetrahedra having surface with unit geodesic diameter. An independent proof of O’Rourke-Schevon’s theorem about polar points on a convex polyhedron is given. A. D. Aleksandrov’s general problem on the area of a convex surface with fixed geodesic diameter is discussed. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 28–55.     

The Fermat-Torricelli point and isosceles tetrahedra

An analytical, unifying approach to geometric properties of the Fermat-Torricelli point of four affinely independent points yields several characterizations of isosceles tetrahedra and, in particular, a characterization or regular tetrahedra within the set of isosceles tetrahedra by means of the solid angle sum.Dedicated to Professor N. Stephanidis on the occasion of his sixtyfifth birthday     

Geometry of Euclidean tetrahedra and knot invariants

We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of the sphere S ³, where the knot lies. Edges of the triangulation along which the knot goes are distinguished by a nonzero deficit angle, in the terminology of the Regge calculus. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 105–117, 2005.     

Reconstruction of tetrahedra from sets of edge lengths

Mathematical Notes - The problemof reconstructing tetrahedra fromgiven sets of edge lengths is studied. This is a special case of the problem of determining, up to isometry, the position of a...     

Tiling space and slabs with acute tetrahedra

We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than 74.21°, and another which tiles a slab in space.     

Constructing spherical CR manifolds by gluing tetrahedra

We propose a general method of constructing spherical CR manifolds by gluing tetrahedra adapted to CR geometry. We obtain spherical CR structures on the complement of the figure eight knot and the Whitehead link complement with holonomy in $\mathrm{PU}(2,1,\mathbb{Z}[\omega ])$ and $\mathrm{PU}(2,1,\mathbb{Z}[\mathrm{i}])$ respectively (the same integer rings appearing in real hyperbolic geometry). To cite this article: E. Falbel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Rational tetrahedra with edges in geometric progression

This paper discusses tetrahedra with rational edges forming a geometric progression, focussing on whether they can have rational volume or rational face areas. We examine the 30 possible configurations of such tetrahedra and show that no face of any of these has rational area. We show that 28 of these configurations cannot have rational volume, and in the remaining two cases there are at most six possible examples, and none have been found.     

Rational tetrahedra with edges in arithmetic progression

This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families are found which have rational volume, a face can have rational area only if its edges are themselves in arithmetic progression, and a tetrahedron can have at most one such rational face area.     

On the translative packing densities of tetrahedra and cuboctahedra

In 1900, as a part of his 18th problem, Hilbert asked the question to determine the density of the densest tetrahedron packings. However, up to now no mathematician knows the density δ^t(T)${\delta }^t(T)$ of the densest translative tetrahedron packings and the density δ^c(T)${\delta }^c(T)$ of the densest congruent tetrahedron packings. This paper presents a local method to estimate the density of the densest translative packings of a general convex solid. As an application, we obtain the upper bound in

0.3673469?≤δ^t(T)≤0.3840610?,$0.3673469?\le {\delta }^t(T)\le 0.3840610?,$

where the lower bound was established by Groemer in 1962, which corrected a mistake of Minkowski. For the density δ^t(C)${\delta }^t(C)$ of the densest translative cuboctahedron packings, we obtain the upper bound in

0.9183673?≤δ^t(C)≤0.9601527?.$0.9183673?\le {\delta }^t(C)\le 0.9601527?.$

In both cases we conjecture the lower bounds to be the correct answer.     

On the neutron flux flattening problem

In this paper, we present the critical neutron flux flattening problem governed by the critical transport equation in a nonuniform slab with periodic boundary conditions. Existence and uniqueness theorem of the optimal solution is shown in continuous function space.     

On truncated variation,upward truncated variation and downward truncated variation for diffusions

The truncated variation, TV^c${\text{TV}}^c$, is a fairly new concept introduced in ?ochowski (2008) [5]. Roughly speaking, given a càdlàg function f$f$, its truncated variation is “the total variation which does not pay attention to small changes of f$f$, below some threshold c>0$c>0$”. The very basic consequence of such approach is that contrary to the total variation, TV^c${\text{TV}}^c$ is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in ?ochowski (2011) [6], another characterization of TV^c${\text{TV}}^c$ has been found. Namely TV^c${\text{TV}}^c$ is the smallest possible total variation of a function which approximates f$f$ uniformly with accuracy c/2$c/2$. Due to these properties we envisage that TV^c${\text{TV}}^c$ might be a useful concept both in the theory and applications of stochastic processes.