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[en] We will describe four classes of photoionization matrix element zeros and their physical consequences. Three of these classes have been discussed previously. The fourth class of zeros, discovered more recently, occur at photon energies above 100 keV. We will compare this newer class of zeros, relativistic high energy zeros (RHEZ), with the other three classes. Dipole RHEZ were noted by Kim and later discussed by Tong et al. In the dipole case RHEZ are independent of n and Z, central potential and retardation. They occur only in (n,lb,jb) → (ε,lc=lb+1,jc=jb) transitions, at photon energies Elbp = mc2/(lb + 1). These properties, first observed numerically, are exact consequences of the Dirac equation. Although RHEZ occur at very high energies, they do have physical consequences. Most striking is a sign change in the photon-electron polarization correlation coefficient C23, the correlation of longitudinally polarized electrons with linearly polarized photons, at photon energies near the RHEZ. Low energy zeros that result in Cooper minima (CM) occur near threshold, in nonrelativistic dipole matrix elements of ground state atoms only for (n,lb) → (ε,lc=lb+1) transitions. There are simple rules regarding the number of zeros. Unlike RHEZ, low energy CM zeros are very dependent on atomic properties, including screening, and depend on n, l and Z. They result in major physical consequences, including minima in total cross sections and sign changes in angular distribution asymmetry parameters. Point Coulomb relativistic zeros (PCRZ) have been observed above threshold in ns1/2 → ε p3/2 transitions when Z ≥ 128 and in np1/2 → εd3/2 transitions when Z ≥ 133. Unlike RHEZ, PCRZ occur at near threshold energies. Higher energy nonrelativistic Coulomb zeros (HENRCZ) occur at 1-50 keV in beyond dipole matrix elements. Unlike RHEZ, positions of HENRCZ depend on n and Z. Like low energy CM, they affect angular distributions. (author)
[en] The matrix calculus in classical photoelasticity is used. Transfer functions for different polariscope arrangements are calculated. Linear polariscopes, circular polariscopes, double-exposure method to obtain isochromatics and Tardy and Senarmont method of measuring fractional relative retardations are analysed using coherency matrix formalism. (author)
[en] We suggest a new fast algorithm for the matrix generator of random numbers which has been earlier proposed. This algorithm reduces N2 operation of the matrix generator to NlnN and essentially reduces the generation time. It also clarifies the algebraic structure of this type of K system generators. This approach allows to suggest a K system matrix generator with almost empty matrix elements, so that only N of them are nonzero. (orig.)
[en] We calculate the unpolarized twist-2 three-loop splitting functions and and the associated anomalous dimensions using massive three-loop operator matrix elements. While we calculate directly, is computed from 1200 even moments, without any structural prejudice, using a hierarchy of recurrences obtained for the corresponding operator matrix element. The largest recurrence to be solved is of order 12 and degree 191. We confirm results in the foregoing literature.
[en] In this paper we study robustness of strong correlation between eigenvalues and diagonal matrix elements sorted from the smallest to the largest, in the presence of large single-particle splittings for both realistic and schematic systems. We also study the nearest level spacing distribution obtained by the linear correlation. (author)
[en] We show that given one Cl(a) improvement constant, bm, all the remaining quantities needed to define the renormalized and O(a) improved dimension-3 quark bilinears can be obtained by studying the matrix elements of these operators between external quark states in a fixed gauge.
[en] In this paper, we introduce a new Fibonacci Gp,m matrix for the m-extension of the Fibonacci p-numbers where p (≥0) is integer and m (>0). Thereby, we discuss various properties of Gp,m matrix and the coding theory followed from the Gp,m matrix. In this paper, we establish the relations among the code elements for all values of p (nonnegative integer) and m(>0). We also show that the relation, among the code matrix elements for all values of p and m=1, coincides with the relation among the code matrix elements for all values of p [Basu M, Prasad B. The generalized relations among the code elements for Fibonacci coding theory. Chaos, Solitons and Fractals (2008). doi: 10.1016/j.chaos.2008.09.030]. In general, correct ability of the method increases as p increases but it is independent of m.
[en] Using the Sklyanin-Kharchev-Lebedev method of separation of variables adapted to the cyclic Baxter-Bazhanov-Stroganov or the τ(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.