In this paper we generalize the famous Kariya theorem on the perspectivity of a given triangle with the homothetic images of the intouch triangle from the incenter to the touchpoints triangles of the excircles, leading to the triad of ex-Feuerbach hyperbolas. We also study in some details the triangle formed by the orthocenters of the touchpoints triangles. An elegant construction is given for ...

and we know that this ring is isomorphic to Z or an order O in an imaginary quadratic field K; in fact, the ring on the right is equal to Z or O (viewed as a subring of C).1 To simplify the discussion, we shall treat the isomorphism in (1) as an equality and view elements of End(EL) as elements of Z or O. How might we construct an elliptic curve with endomorphism ring O? An obvious way is to us...

Abstract. We introduce the mixed convolution bodies of two convex symmetric bodies. We prove that if the boundary of a body K is smooth enough then as δ tends to 1 the δ–M∗–convolution body of K with itself tends to a multiple of the Euclidean ball after proper normalization. On the other hand we show that the δ–M∗–convolution body of the n–dimensional cube is homothetic to the unit ball of ln 1 .

A brief summary of results on kinematic self-similarities in general relativity is given. Attention is focussed on locally rotationally symmetric models admitting kinematic self-similar vectors. Coordinate expressions for the metric and the kinematic self-similar vector are provided. Einstein’s field equations for perfect fluid models are investigated and all the homothetic perfect fluid soluti...

We estimate the number and ratio of negative homothetic copies of a d-dimensional convex body C sufficient for the covering of C. If the number of those copies is not very large, then our estimates are better than recent estimates of Rogers and Zong. Particular attention is paid to the 2-dimensional case. It is proved that every planar convex body can be covered by two copies of ratio −4 3 (thi...

We introduce the mixed convolution bodies of two convex symmetric bodies. We prove that if the boundary of a body K is smooth enough then as δ tends to 1 the δ-M∗-convolution body of K with itself tends to a multiple of the Euclidean ball after proper normalization. On the other hand we show that the δ-M∗-convolution body of the n-dimensional cube is homothetic to the unit ball of `1 .

We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to t...

A general method for solving the Cosmological Einstein’s Field Equations (EFE) in vacuum is presented, and is applied to the Bianchi Type III vacuum cosmology. As a result, the entire space of solutions for this model is found. This space is composed of two distinct, disconnected ,three-parameter families. Both of them admit four Killing vector Fields and no homothetic vector. One family repres...

We review the similarity solutions proposed by Waylen for a regular time-dependent axisymmetric vacuum space-time, and show that the key equation introduced to solve the invariant surface conditions is related by a Bäcklund transform to a restriction on the similarity variables. We further show that the vacuum space-times produced via this path automatically possess a (possibly homothetic) Kill...

We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a YangMills field, with or without mass term, if the spatial geometry of the underlying spacetime is homothetic to R. The energy levels of the resulting quantum model, i.e., the eigenvalues of the corresponding self-adjoint Hamiltonian with a pure point spectrum, are strictly posi...