Odd Correlation Functions Research Articles

The BEG Model in the Disordered Region and at the Antiquadrupolar-Disordered Line of Parameters

We analyse the d-dimensional BEG model with $$d\ge 2$$ and parameters in the disordered region and at the antiquadrupolar-disordered line. We obtain a subset of these regions for which the odd correlation functions are zero and the even ones decay exponentially fast, at all non-zero temperatures, for a wide class of boundary conditions.

Anomalous scaling of the magnetic field in the Kazantsev–Kraichnan model

Using the field theoretic renormalization group technique and the operator-product expansion, the problem of anomalous scaling in magnetohydrodynamic turbulence is investigated in the framework of the Kazantsev–Kraichnan model, where the magnetic field is passively advected by the δ-correlated in time Gaussian velocity field, up to the second order of the perturbation theory (two-loop approximation). The model is studied in the isotropic case as well as in the case of large-scale anisotropy. It is shown that the two-loop corrections to the critical dimensions of relevant composite operators are much more important in the model of the passively advected vector field studied in the present paper, i.e. they lead to a more pronounced anomalous scaling behavior than in the corresponding problem of a passively advected scalar field in the framework of the Kraichnan model. Explicit two-loop expressions for the anomalous exponents of the single-time correlation functions of the magnetic field of arbitrary order are present. Their hierarchies with respect to the degree of anisotropy are investigated and it is shown that the hierarchies which are valid at the one-loop level of approximation are not destroyed by two-loop corrections. Thus, the leading contributions to the even correlation functions are given by the exponents from the isotropic shell in accordance with the predictions given at the one-loop level of approximation. The persistence of anisotropy in the inertial interval is studied by the behavior of the odd correlation functions. It is shown that the two-loop corrections to the scaling behavior of the odd dimensionless ratios of the correlation functions confirm the persistence of anisotropy in the inertial interval.

A Ternary Zero-Correlation Zone Sequence Set Having Wide Inter-Subset Zero-Correlation Zone

The present paper introduces a novel construction of ternary sequences having a zero-correlation zone. The cross-correlation function and the side-lobe of the auto-correlation function of the proposed sequence set is zero for the phase shifts within the zero-correlation zone. The proposed sequence set consists of more than one subset having the same member size. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a wider zero-correlation zone than that of the correlation function of sequences of the same subset (intra-subset correlation function). The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set. The proposed sequence set has a zero-correlation zone for periodic, aperiodic, and odd correlation functions.

"Toy models" of turbulent convection and the hypothesis of the local isotropy restoration

A brief review is given of recent results devoted to the effects of large-scale anisotropy on the inertial-range statistics of the passive scalar quantity $\theta(t,{\bf x})$, advected by the synthetic turbulent velocity field with the covariance $\propto\delta(t-t')|{\bf x}-{\bf x'}|^{\eps}$. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions $ S_n (\r) \equiv $ are obtained; they are represented by superpositions of power laws with universal (independent of the anisotropy parameters) anomalous exponents, calculated to the first order in $\eps$ in any space dimension. The exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The small-scale anisotropy reveals itself in odd correlation functions: for the incompressible velocity field, $S_{3}/S_{2}^{3/2}$ decreases going down towards to the depth of the inertial range, while the higher-order odd ratios increase; if the compressibility is strong enough, the skewness factor also becomes increasing.

Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence

The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. The velocity field is Gaussian, delta-correlated in time, and scales with a positive exponent xi. Explicit inertial-range expressions for the magnetic correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy and forcing) anomalous exponents. The complete set of anomalous exponents for the pair correlation function is found nonperturbatively, in any space dimension d, using the zero-mode technique. For higher-order correlation functions, the anomalous exponents are calculated to O(xi) using the renormalization group. The exponents exhibit a hierarchy related to the degree of anisotropy; the leading contributions to the even correlation functions are given by the exponents from the isotropic shell, in agreement with the idea of restored small-scale isotropy. Conversely, the small-scale anisotropy reveals itself in the odd correlation functions: the skewness factor is slowly decreasing going down to small scales and higher odd dimensionless ratios (hyperskewness, etc.) dramatically increase, thus diverging in the r-->0 limit.

Even-odd transformation with application to multiuser CW radars

An even odd transformation is a linear phase transformation which translates a set of sequences into another set, all of whose absolute even (also called periodic) and odd correlation functions become the respective absolute odd and even correlation functions of the original set. This transformation enables the choice of periodic or odd-periodic sequence sets to simplify the implementation of a multiuser CW radar without sacrificing its pulse compression performance. As an application example, the odd-perfect counterparts of the famous Zadoff-Chu sequence sets are derived. Of particular interest is a pair of odd perfect sequences with optimal odd cross-correlation levels, whose phase alphabet size is only half of that of the well-known P3 code for even lengths.

Correlations in disordered crystals and diffuse scattering of x rays or neutrons.

The scattering cross section for diffuse scattering of x rays or neutrons from disordered crystals is expressed systematically in terms of density-correlation functions by using a cumulant expansion. This approach allows one to distinguish between contributions owing to pair-, three-, and higher-order correlations of the defects; the latter ones enter even within the first Born approximation due to the nonlinear dependence of the scattering amplitude on the lattice displacements. The terms involving even and odd correlation functions, respectively, can be classified according to their inversion symmetry properties. This analysis is applied to x-ray scattering data obtained from niobium-molybdenum alloys loaded with hydrogen. Furthermore, we propose how to test, on the basis of the scattering data alone, approximation schemes for the three-point correlation function, such as, e.g., the Kirkwood superposition principle.

Correlation properties of dual-BCH, Kasami and other sequences for spread-spectrum, multiple-access systems

Odd correlation functions and average interference parameters for dual-BCH, small sets of Kasami sequences and a new set of sequences are investigated. Results are presented for periods up to 255, and an investigation into the effects of choice of initial phase is included. Results indicate that the new sequences provide a level of multiple-access performance which is on average similar to other sequences, but is insensitive to variations in the statistics of the data used to modulate the sequences.

Coexistence of phases in Ising ferromagnets

We derive a new inequality for ferromagnetic Ising spin systems and then use it to obtain information about the number of phases which can coexist in such systems. We show in particular that for even interactions only two phases (up and down magnetization) can coexist below the critical temperature at zero magnetic field (h=0) whenever the energy is a continuous function of the temperature. We also prove that the derivatives with respect toh ath=0 of the odd correlation functions (triplet,...) diverge like the susceptibility in the vicinity of the critical temperature (at least for pair interactions). Our results also apply to higher order Ising spins (not just spin 1/2).