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Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq \kappa E[X])$, where $\kappa$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions., Comment: 10 pages

Type I CRISPR-Cas systems are the most common among six types of CRISPR-Cas systems, however, non-self-targeting genome editing based on a single Cas3 of type I CRISPR-Cas systems has not been reported. Here, we present the subtype I-B-Svi CRISPR-Cas system (with three confirmed CRISPRs and a cas gene cluster) and genome editing based on this system found in Streptomyces virginiae IBL14. Importantly, like the animal-derived bacterial protein SpCas9 (1368 amino-acids), the single, compact, non-animal-derived bacterial protein SviCas3 (771 amino-acids) can also direct template-based microbial genome editing through the target cell's own homology-directed repair system, which breaks the view that the genome editing based on type I CRISPR-Cas systems requires a full Cascade. Notably, no off-target changes or indel-formation were detected in the analysis of potential off-target sites. This discovery broadens our understanding of the diversity of type I CRISPR-Cas systems and will facilitate new developments in genome editing tools., Comment: 113 pages, 10 figures, and 6 tables

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a

Lifetimes for the lowest three transitions in the nucleus $^{120}$Xe have been measured using the Recoil Distance Technique. Our data indicate that the lifetime for the $2_{1}^{+} \to 0_{1}^{+}$ transition is more than a factor of two lower than the previously adopted value and is in keeping with more recent measurements performed on this nucleus. The theoretical implications of this discrepancy and the possible reason for the erroneous earlier results are discussed. All measured lifetimes in $^{120}$Xe, as well as the systematics of the lifetimes of the 2$_{1}^{+}$ states in Xe isotopes, are compared with predictions of various models. The available data are best described by the Fermion Dynamic Symmetry Model (FDSM)., Comment: 9 pages, RevTeX, 3 figures with Postscript file available on request at Umesh.Garg.1@nd.edu, pan@einstein.physics.drexel.edu. Submitted to Phys. Rev. C

It is shown that the SU(3) symmetry of the fermion dynamical symmetry model is essentially preserved even for highly nondegenerate spherical single-particle energies. The breaking of SU(3) symmetry by single-particle energy terms for either normal deformation or superdeformation occurs only through an indirect Pauli effect and is significant only when the spherical single-particle splitting within shells is artificially large relative to that observed experimentally.

According to the analysis based on the fermion dynamical symmetry model, nuclei previously regarded as SO(6)-like (e.g. $^{128}$Xe and $^{196}$Pt) are shown to be more akin to the transitional nuclei between SO(7) and SO(6) symmetries.