Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

The one-dimensional Dirac operator with periodic potential $V=\begin{pmatrix} 0 & \mathcal{P}(x) \\ \mathcal{Q}(x) & 0 \end{pmatrix}$, where $\mathcal{P},\mathcal{Q}\in L^2([0,\pi])$ subject to periodic, antiperiodic or a general strictly regular boundary condition $(bc)$ has discrete spectrums. It is known that, for large enough $|n|$ in the disc centered at $n$ of radius 1/4, the operator has exactly two (periodic if $n$ is even or antiperiodic if $n$ is odd) eigenvalues $\lambda_n^+$ and $\lambda_n^-$ (counted according to multiplicity) and one eigenvalue $\mu_n^{bc}$ corresponding to the boundary condition $(bc)$. We prove that the smoothness of the potential could be characterized by the decay rate of the sequence $|\delta_n^{bc}|+|\gamma_n|$, where $\delta_n^{bc}=\mu_n^{bc}-\lambda_n^+$ and $\gamma_n=\lambda_n^+-\lambda_n^-.$ Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if $\sup\limits_{\gamma_n\neq0} \frac{|\delta_n^{bc}|}{|\gamma_n|}$ is finite.Comment: 29 pages, no figur

arrostii roots

A new acylated and triterpenoidal saponin, named GS1, was isolated from the roots of Gypsophila arrostii Guss. On the basis of acid hydrolysis, comprehensive spectroscopic analyses and comparison with spectral data of known compounds, its structure was established as 3-O--D-xylopyranosyl-(12)-[-D-xylopyranosyl-(13)]--D-glucopyranosyl-{21-O-[(E)-3,4,5trimethoxycinnamoyl]}21-hydroxygypsogenin 28-O--D-glucopyranosyl-(12)- [-D-arabinopyranosyl-(13)]--D-xylopyranosyl-(13]--L-rhamnopyranosyl ester. This article deals with the isolation and structural elucidation of new acylated and oleanane-type saponin

Semantics through Reference to the Unknown

In this paper, I dwell on a particular distinction introduced by Ilhan Inan—the distinction between ostensible and inostensible use of our language. The distinction applies to singular terms, such as proper names and definite descriptions, or to general terms like concepts and to the ways in which we refer to objects in the world by using such terms. Inan introduces the distinction primarily as an epistemic one but in his earlier writings (1997: 49) he leaves some room for it to have some semantic significance i.e., the view that in certain intensional de re contexts whether a term occurring in a sentence is ostensible or inostensible may have a bearing on the semantic content of the sentence. However, in his later writings e.g., The Philosophy of Curiosity, he appears to abandon his earlier thoughts regarding the semantic significance of his distinction. He says: “the ostensible/inostensible distinction is basically an epistemic one.... It is an epistemic distinction that has no semantic significance” (2012: 65). I argue that there are indeed such intensional contexts in which the distinction has some semantic significance, i.e., whether a term is ostensible or inostensible has in fact a bearing on what proposition is expressed by the sentence in which the term occurs