7,626 research outputs found
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
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 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
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