Combinatorial proofs on the joint distribution of descents and inverse descents

Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $i-1$ descents and $j-1$ inverse descents. Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial and arithmetic properties of $A_{n,i,j}$, which contain a recurrence of $A_{n,i,j}$. Using the idea of balls in boxes, Petersen gave a combinatorial interpretation for the generating function of $A_{n,i,j}$, and obtained the same recurrence of $A_{n,i,j}$ from its generating function. Subsequently, Petersen asked whether there is a visual way to understand this recurrence. In this paper, after observing the internal structures of permutation grids, we present a combinatorial proof of the recurrence of $A_{n,i,j}$. Let $I_{n,k}$ and $J_{n,k}$ count the number of involutions and fixed-point free involutions on $[n]$ with $k$ descents, respectively. With the help of generating functions, Guo and Zeng derived two recurrences of $I_{n,k}$ and $J_{2n,k}$ that play an essential role in the proof of their unimodal properties. Unexpectedly, the constructive approach to the recurrence of $A_{n,i,j}$ is found to fuel the combinatorial interpretations of these two recurrences of $I_{n,k}$ and $J_{2n,k}$

Uniqueness and Nondegeneracy of Ground States for (−Δ)sQ+Q−Qα+1=0(-\Delta)^s Q + Q - Q^{\alpha+1} = 0 in R\mathbb{R}

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page

Double Twist in Helical Polymer Soft Crystals

In natural and synthetic materials having non-racemic chiral centers, chirality and structural ordering each play a distinct role in the formation of ordered states. Configurational chirality can be extended to morphological chirality when the phase, structures possess low liquid crystalline order. In the crystalline states the crystallization process suppresses the chiral helical morphology due to strong ordering interactions, In this Letter, we report the first observation of helical single lamellar crystals of synthetic non-racemic chiral polymers. Experimental evidence shows that the molecular chains twist along both the long and short axes of the helical lamellar crystals, which is the first time a double-twist molecular orientation in a helical crystal has been observed

The lower mass function of the young open cluster Blanco 1: from 30 Mjup to 3 Mo

We performed a deep wide field optical survey of the young (~100-150 Myr) open cluster Blanco1 to study its low mass population well down into the brown dwarf regime and estimate its mass function over the whole cluster mass range.The survey covers 2.3 square degrees in the I and z-bands down to I ~ z ~ 24 with the CFH12K camera. Considering two different cluster ages (100 and 150 Myr), we selected cluster member candidates on the basis of their location in the (I,I-z) CMD relative to the isochrones, and estimated the contamination by foreground late-type field dwarfs using statistical arguments, infrared photometry and low-resolution optical spectroscopy. We find that our survey should contain about 57% of the cluster members in the 0.03-0.6 Mo mass range, including 30-40 brown dwarfs. The candidate's radial distribution presents evidence that mass segregation has already occured in the cluster. We took it into account to estimate the cluster mass function across the stellar/substellar boundary. We find that, between 0.03Mo and 0.6Mo, the cluster mass distribution does not depend much on its exact age, and is well represented by a single power-law, with an index alpha=0.69 +/- 0.15. Over the whole mass domain, from 0.03Mo to 3Mo, the mass function is better fitted by a log-normal function with m0=0.36 +/- 0.07Mo and sigma=0.58 +/- 0.06. Comparison between the Blanco1 mass function, other young open clusters' MF, and the galactic disc MF suggests that the IMF, from the substellar domain to the higher mass part, does not depend much on initial conditions. We discuss the implications of this result on theories developed to date to explain the origin of the mass distribution.Comment: 18 pages, 15 figures and 5 tables accepted in A&

Polytetrahedral Clusters

By studying the structures of clusters bound by a model potential that favours polytetrahedral order, we find a previously unknown series of `magic numbers' (i.e. sizes of special stability) whose polytetrahedral structures are characterized by disclination networks that are analogous to hydrocarbons.Comment: 4 pages, 4 figure

Graphene plasmonics: A platform for strong light-matter interaction

Graphene plasmons provide a suitable alternative to noble-metal plasmons because they exhibit much larger confinement and relatively long propagation distances, with the advantage of being highly tunable via electrostatic gating. We report strong light- matter interaction assisted by graphene plasmons, and in particular, we predict unprecedented high decay rates of quantum emitters in the proximity of a carbon sheet, large vacuum Rabi splitting and Purcell factors, and extinction cross sections exceeding the geometrical area in graphene ribbons and nanometer-sized disks. Our results provide the basis for the emerging and potentially far-reaching field of graphene plasmonics, offering an ideal platform for cavity quantum electrodynamics and supporting the possibility of single-molecule, single-plasmon devices.Comment: 39 pages, 15 figure

Virtual Effects of Split SUSY in Higgs Productions at Linear Colliders

In split supersymmetry the gauginos and higgsinos are the only supersymmetric particles possibly accessible at foreseeable colliders like the CERN Large Hadron Collider (LHC) and the International Linear Collider (ILC). In order to account for the cosmic dark matter measured by WMAP, these gauginos and higgsinos are stringently constrained and could be explored at the colliders through their direct productions and/or virtual effects in some processes. The clean environment and high luminosity of the ILC render the virtual effects of percent level meaningful in unraveling the new physics effects. In this work we assume split supersymmetry and calculate the virtual effects of the WMAP-allowed gauginos and higgsinos in Higgs productions e+e- -> Z h and e+e- -> \nu_e \bar_\nu_e h through WW fusion at the ILC. We find that the production cross section of e+e- -> Zh can be altered by a few percent in some part of the WMAP-allowed parameter space, while the correction to the WW-fusion process e+e- -> \nu_e \bar_\nu_e h is below 1%. Such virtual effects are correlated with the cross sections of chargino pair productions and can offer complementary information in probing split supersymmetry at the colliders.Comment: more discussions added (7 pages, 10 figs

The helicase HAGE prevents interferon-a-induced PML expression in ABCB5+ malignant melanoma-initiating cells by promoting the expression of SOCS1

The tumour suppressor PML (promyelocytic leukaemia protein) regulates several cellular pathways involving cell growth, apoptosis, differentiation and senescence. PML also has an important role in the regulation of stem cell proliferation and differentiation. Here, we show the involvement of the helicase HAGE in the transcriptional repression of PML expression in ABCB5 + malignant melanoma-initiating cells (ABCB5 + MMICs), a population of cancer stem cells which are responsible for melanoma growth, progression and resistance to drug-based therapy. HAGE prevents PML gene expression by inhibiting the activation of the JAK-STAT (janus kinase-signal transducers and activators of transcription) pathway in a mechanism which implicates the suppressor of cytokine signalling 1 (SOCS1). Knockdown of HAGE led to a significant decrease in SOCS1 protein expression, activation of the JAK-STAT signalling cascade and a consequent increase of PML expression. To confirm that the reduction in SOCS1 expression was dependent on the HAGE helicase activity, we showed that SOCS1, effectively silenced by small interfering RNA, could be rescued by re-introduction of HAGE into cells lacking HAGE. Furthermore, we provide a mechanism by which HAGE promotes SOCS1 mRNA unwinding and protein expression in vitro