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

Orthogonal Projection of an Infinite Round Cone in Real Hilbert Space

We fully characterize orthogonal projections of infinite right circular (round) cones in real Hilbert spaces. Another interpretation is that, given two vectors in a real Hilbert space, we establish the optimal estimate on the angle between the orthogonal projections of the two vectors. The estimate depends on the angle between the two vectors and the position of only one of the two vectors. Our results also make a contributions to Cauchy-Bunyakovsky-Schwarz type inequalities

Initial conditions, equations of state and final state in hydrodynamics

In this paper we present properties of relativistic and non-relativistic perfect hydrodynamical models. In particular we show illustrations of the fact that different initial conditions and equations of state can lead to the same hadronic final state. This means that alone from the hadronic observables one cannot determine either of the above, one needs for example penetrating probes that inherit their properties from each timeslice of the evolution of the fireball.Comment: Presented at the IV Workshop on Particle Correlations and Femtoscopy. 6 pages, 3 figure

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