37,947 research outputs found
SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
We propose a group-theoretical approach to the generalized oscillator algebra
Ak recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The case
k > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonic
oscillator and the Poeschl-Teller systems) while the case k < 0 is described by
the compact group SU(2) (as for the Morse system). We construct the phase
operators and the corresponding temporally stable phase eigenstates for Ak in
this group-theoretical context. The SU(2) case is exploited for deriving
families of mutually unbiased bases used in quantum information. Along this
vein, we examine some characteristics of a quadratic discrete Fourier transform
in connection with generalized quadratic Gauss sums and generalized Hadamard
matrices
Fractional biorthogonal partners in channel equalization and signal interpolation
The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners
Log-correlated Gaussian fields: an overview
We survey the properties of the log-correlated Gaussian field (LGF), which is
a centered Gaussian random distribution (generalized function) on , defined up to a global additive constant. Its law is determined by the
covariance formula
which holds for mean-zero test functions . The LGF belongs to
the larger family of fractional Gaussian fields obtained by applying fractional
powers of the Laplacian to a white noise on . It takes the
form . By comparison, the Gaussian free field (GFF)
takes the form in any dimension. The LGFs with coincide with the 2D GFF and its restriction to a line. These objects
arise in the study of conformal field theory and SLE, random surfaces, random
matrices, Liouville quantum gravity, and (when ) finance. Higher
dimensional LGFs appear in models of turbulence and early-universe cosmology.
LGFs are closely related to cascade models and Gaussian branching random walks.
We review LGF approximation schemes, restriction properties, Markov properties,
conformal symmetries, and multiplicative chaos applications.Comment: 24 pages, 2 figure
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