We reflect real spectra of new logarithmic model PT-symmetry operators with singular and non-singular in nature. also notice the iso-spectral nature between inverted noninverted potentials. Present numerical results give good agreement previous results.

INTRODUCTION SUMMARY We present an exact and analytical expression for the Fourier transform of a function that has been sampled logarithmically. The procedure is significantly more efficient computationally than the fast Fourier transformation (FFT) for transforming functions or measured responses which decay slowly with increasing abscissa value. We illustrate the proposed method with an exam...

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential λ SW , are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c = c 2,1 = −2. The fields of this theory precisely simulate the branched do...

We present a new method to determine contraction kernels for the construction of graph pyramids. The new method works with undirected graphs and yields a reduction factor of at least 2.0. This means that with our method the number of vertices in the subgraph induced by any set of contractible edges is reduced to half or less by a single parallel contraction. Our method yields better reduction f...

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for ...

We analyse the SU(2) k WZNW models beyond the integrable representations and in particular the case of SU(2) 0. We find that these are good examples of logarithmic conformal field theories as indecomposable representations are naturally produced in the fusion of discrete irreducible representations. We also find extra, chiral and non-chiral, multiplet structure in the theory. The chiral fields,...

We present a complete solution of the WZW model on the supergroup GL(1|1). Our analysis begins with a careful study of its minisuperspace limit (“harmonic analysis on the supergroup”). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the represe...

Logarithmic general error distribution, an extension of the log-normal distribution, is proposed. Some interesting properties of the logarithmic general error distribution are derived. These properties are applied to establish the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the logarithmic general error and log-normal distributions, and to derive the ...

We use generalized power series to construct algebraically a nonstandard model of the theory of the real eld with exponentiation. This model enables us to show the undeenability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic...