Favourite distances in 3-space

Let S be a set of n points in Euclidean 3-space. Assign to each x ∈ S a distance r(x) > 0, and let er(x,S) denote the number of points in S at distance r(x) from x. Avis, Erdo ̋s and Pach (1988) introduced the extremal quantity f3(n) = max ﰝx∈S er(x, S), where the maximum is taken over all n-point subsets S of 3-space and all assignments r: S → (0,∞) of distances. We show that if the pair (S,r) maximises f3(n) and n is sufficiently large, then, except for at most 2 points, S is contained in a circle C and the axis of symmetry L of C, and r(x) equals the distance from x to C for each x ∈ S ∩ L. This, together with a new construction, implies that f3(n) = n2/4 + 5n/2 + O(1)

Einstein's conversion from his static to an expanding universe

In 1917 Einstein initiated modern cosmology by postulating, based on general relativity, a homogeneous, static, spatially curved universe. To counteract gravitational contraction he introduced the cosmological constant. In 1922 Alexander Friedman showed that Einstein's fundamental equation also allowed dynamical worlds, and in 1927 Geroges Lemaitre, backed by observational evidence, concluded that our universe was expanding. Einstein impetuously rejected Friedman's as well as Lemaitre's findings. However, in 1931 he retracted his former static model in favour of a dynamic solution. This investigation follows Einstein on his hesitating path from a static to the expanding universe. Contrary to an often repeated belief the primary motive for his switch was not observational evidence, but the realisation that his static model was unstable.Comment: Published in EPJH (European Physics Journal - History) on February 4, 2014, 26 pages, 5 figures. A note on an unpublished draft by Einstein, found at the AEA, has been added in proof. Full biblio info was added on March 24, and a printing error was correcte

Cosmological Relativity: A General-Relativistic Theory for the Accelerating Expanding Universe

Recent observations of distant supernovae imply, in defiance of expectations, that the universe growth is accelerating, contrary to what has always been assumed that the expansion is slowing down due to gravity. In this paper a general-relativistic cosmological theory that gives a direct relationship between distances and redshifts in an expanding universe is presented. The theory is actually a generalization of Hubble's law taking gravity into account by means of Einstein's theory of general relativity. The theory predicts that the universe can have three phases of expansion, decelerating, constant and accelerating, but it is shown that at present the first two cases are excluded, although in the past it had experienced them. Our theory shows that the universe now is definitely in the stage of accelerating expansion, confirming the recent experimental results

On semi-classical pion production in heavy ion collisions

In high energy heavy ion collisions the pion multiplicity is large, and one might expect that pions are radiated semi-classically. The axial symmetry of the collision and approximately zero isotopic spin of the colliding nuclei result then in peculiar isotopic spin -- azimuth correlations of the produced pions. These correlations are easy to test -- and should be tested.Comment: Saclay-T93/06

High Energy Gamma--Radiation from the Galactic Center due to Neutralino Annihilation

We study the NGS (Non--dissipative Gravitational Singularity) model, which successfully describes the non--linear stage of evolution of perturbations (see [1], [2] and references therein). This model predicts DM density distribution $\rho(r) \sim r^{-\alpha}$ with $\alpha \simeq 1.8$ which holds from very small distances $r_{\rm min} \simeq 0.01~{\rm pc}$ up to very large distances $r_{\rm max} \simeq 5~{\rm Mpc}$. Assuming the neutralino to be a CDM particle, we calculate the annihilation of neutralinos in the vicinity of the singularity (Galactic Center). If neutralinos are the dominant component of DM in our Galaxy, the produced energy is enough to provide the whole observed activity of the GC. Neutralinos of the most general composition and of mass in the range 20~{\rm GeV} \leq m_\c \leq 1~{\rm TeV} are considered. We find the neutralino compositions which give the relic density needed for the Mixed Dark Matter (MDM) model and we evaluate for these compositions the high--energy ($E_{\gamma} > 100 ~{\rm MeV}$) gamma--ray flux under the constraint that the radio flux is lower than the observational limit. The compositions with the detectable gamma--ray flux which we found are provided by a set of almost pure gaugino states with the neutralino mass between $100$ and $500$ GeV. We demonstrate that a detectable high--energy gamma--ray flux is produced by the neutralino annihilation also in the case when neutralinos provide a small fraction (down to $0.1 \%$) of the DM in our Galaxy. The predicted flux is $F_\gamma \sim 10^{-7}-10^{-8}~{\rm cm}^{-2}~{\rm s}^{-1}$ for E_\gamma \gsim 300~{\rm MeV}Comment: Plain TeX 11 pages 4 figures available on request. Preprint numbers LNGS 94/90 - DFTT 5/9

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances