Coordinate time and proper time in the GPS

The Global Positioning System (GPS) provides an excellent educational example as to how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS (see \cite{ashby}) uses the Schwarzschild spacetime to deduce the {\it approximate} formula, ds/dt\approx 1+V-\frac{|\vv|^2}{2}, for the relation between the proper time rate $s$ of a satellite clock and the coordinate time rate $t$. Here $V$ is the gravitational potential at the position of the satellite and \vv is its velocity (with light-speed being normalized as $c=1$). In this note we give a different derivation of this formula, {\it without using approximations}, to arrive at ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}, where \n is the normal vector pointing outward from the center of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to ds/dt=\sqrt{1+2V-|\vv|^2}. We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice.Comment: 5 pages, revised, over-over-simplified... Does anyone care that the GPS uses an approximate formula, while a precise one is available in just a few lines??? Physicists don'

Fuglede's conjecture fails in dimension 4

In this note we give an example of a set \W\subset \R^4 such that L^2(\W) admits an orthonormal basis of exponentials \{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L} for some set \L\subset\R^4, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.Comment: 6 page

The linear polarization constant of R^n

The present work contributes to the determination of the n-th linear polarization constant cn(H) of an n-dimensional real Hilbert space H. We provide some new lower bounds on the value of supkyk=1 | hx1, yi · · · hxn, yi |, where x1, . . . , xn are unit vectors in H. In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture cn(H) = nn/2 remains open

On quasi-contractivity of C 0-semigroups on Banach spaces

A basic result in semigroup theory states that every C-0-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula (e(t/n) (A) p)(n) (where P denotes a bounded projection), we prove that whenever the generator A is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is not quasi-contractive

A Fourier analytic approach to the problem of mutually unbiased bases

We give an entirely new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique in additive combinatorics. The method provides a short and elegant generalization of the fact that there are at most $d+1$ MUBs in \Co^d. It may also yield a proof that no complete system of MUBs exists in some composite dimensions -- a long standing open problem.Comment: 11 page

An improvement on the Delsarte-type LP-bound with application to MUBs

The linear programming (LP) bound of Delsarte can be applied to several problems in various branches of mathematics. We describe a general Fourier analytic method to get a slight improvement on this bound. We then apply our method to the problem of mutually unbiased bases (MUBs) to prove that the Fourier family $F(a,b)$ in dimension 6 cannot be extended to a full system of MUBs.Comment: 10 page