Surgery on links with unknotted components and three-manifolds

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M.Comment: 10 pages, 8 figure

The CHSH-type inequalities for infinite-dimensional quantum systems

By establishing CHSH operators and CHSH-type inequalities, we show that any entangled pure state in infinite-dimensional systems is entangled in a $2\otimes2$ subspace. We find that, for infinite-dimensional systems, the corresponding properties are similar to that of the two-qubit case: (i) The CHSH-type inequalities provide a sufficient and necessary condition for separability of pure states; (ii) The CHSH operators satisfy the Cirel'son inequalities; (iii) Any state which violates one of these Bell inequalities is distillable.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1006.3557 by other author

Any entanglement of assistance is polygamous

We propose a condition for a measure of quantum correlation to be polygamous without the traditional polygamy inequality. It is shown to be equivalent to the standard polygamy inequalities for any continuous measure of quantum correlation with the polygamy power. We then show that any entanglement of assistance is polygamous but not monogamous and any faithful entanglement measure is not polygamous.Comment: 5 page

Unified view of monogamy relations for different entanglement measures

A particularly interesting feature of nonrelativistic quantum mechanics is the monogamy laws of entanglement. Although the monogamy relation has been explored extensively in the last decade, it is still not clear to what extent a given entanglement measure is monogamous. We give here a conjecture on the amount of entanglement contained in the reduced states by observing all the known related results at first. Consequently, we propose the monogamy power of an entanglement measure and the polygamy power for its dual quantity, the assisted entanglement, and show that both the monogamy power and the polygamy power exist in any multipartite systems with any dimension, from which we formalize exactly for the first time when an entanglement measure and an assisted entanglement obey the monogamy relation and the polygamy relation respectively in a unified way. In addition, we show that any entanglement measure violates the polygamy relation, which is misstated in some recent papers. Only the existence of monogamy power is conditioned on the conjecture, all other results are strictly proved.Comment: 6 pages. Comments are welcom

Sharp capacity estimates in s-John domains

It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney cube $Q_0$ in terms of the Hausdorff $q$-content of $E$ in an $s$-John domain $\Omega$. Moreover, we construct several examples to show the essential sharpness of our estimates.Comment: 13 pages, 3 figure

Fractional Sobolev-Poincare inequalities in irregular domains

This paper is devoted to the study of fractional (q,p)-Sobolev-Poincare inequalities in irregular domains. In particular, we establish (essentially) sharp fractional (q,p)-Sobolev-Poincare inequality in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tends to the results for the usual derivative. Furthermore, we verified that those domains that support the fractional (q,p)-Sobolev-Poincare inequality together with a separation property are s-diam John domains for certain s, depending only on the associated data. We also point out an inaccurate statement in [2]

Rescaled range and transition matrix analysis of DNA sequences

In this paper we treat some fractal and statistical features of the DNA sequences. First, a fractal record model of DNA sequence is proposed by mapping DNA sequences to integer sequences, followed by R/S analysis of the model and computation of the Hurst exponents. Second, we consider transition between the four kinds of bases within DNA. The transition matrix analysis of DNA sequences shows that some measures of complexity based on transition proportion matrix are of interest. We use some measures of complexity to distinguish exon and intron. Regarding the evolution, we find that for species of higher grade, the transition rate among the four kinds of bases goes further from the equilibrium.Comment: 8 pages with one figure. Communication in Theoretical Physics (2000) (to appear

Fuzzy L languages

In this paper we introduce some families of fuzzy L-systems and investigate their properties. We further discuss the relationship between fuzzy L languages and the fuzzy languages generated by fuzzy grammar proposed in Ref.[3,5]. A measure of fuzziness for a string, called the fuzzy entropy of a string with respect to a given fuzzy L system, will be defined. The relationship between fuzzy L languages and the ordinary L languages is also discussed.Comment: 9 pages with no figure. International J. of Fuzzy sets and Systems (Accepted for publ ication

Nonlinear Optical Properties of Transition Metal Dichalcogenide MX2_2 (M = Mo, W; X = S, Se) Monolayers and Trilayers from First-principles Calculations

Due to the absence of interlayer coupling and inversion symmetry, transition metal dichalcogenide (MX$_2$) semiconductor monolayers exhibit novel properties that are distinctly different from their bulk crystals such as direct optical band gaps, large band spin splittings, spin-valley coupling, piezoelectric and nonlinear optical responses, and thus have promising applications in, e.g., opto-electronic and spintronic devices. Here we have performed a systematic first-principles study of the second-order nonlinear optical properties of MX$_2$ (M = Mo, W; X = S, Se) monolayers and trilayers within the density functional theory with the generalized gradient approximation plus scissors correction. We find that all the four MX$_2$ monolayers possess large second-order optical susceptibility $\chi^{(2)}$ in the optical frequency range and significant linear electro-optical coefficients in low frequency limit, thus indicating their potential applications in non-linear optical devices and electric optical switches. The $\chi^{(2)}$ spectra of the MX$_2$ trilayers are overall similar to the corresponding MX$_2$ monolayers, {\it albeit} with the magnitude reduced by roughly a factor of 3. The prominent features in the $\chi^{(2)}$ spectra of the MX$_2$ multilayers are analyzed in terms of the underlying band structures and optical dielectric function, and also compared with available experiments.Comment: references updated, new Figure 2, revised Figure 8 and text improve

Constructing positive maps from block matrices

Positive maps are useful for detecting entanglement in quantum information theory. Any entangled state can be detected by some positive map. In this paper, the relation between positive block matrices and completely positive trace-preserving maps is characterized. Consequently, a new method for constructing decomposable maps from positive block matrices is derived. In addition, a method for constructing positive but not completely positive maps from Hermitian block matrices is also obtained.Comment: 13 page