Creation and Growth of Components in a Random Hypergraph Process

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend

Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for $1 \leq q n(1/2-1/p)$, if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^s(\R^n)$ then $f \in L^p(\R^n)$ and there exists a constant $c_{p,q,s}$ such that $$\|f\|_{L^p} \leq c_{p,q,s} \|f\|_{L^{q,\infty}}^\theta \|f\|_{\dot H^s}^{1-\theta},$$ where $1/p = \theta/q + (1-\theta)(1/2-s/n)$. In particular, in $\R^2$ we obtain the generalised Ladyzhenskaya inequality $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|f\|_{\dot H^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\|f\|_{\dot H^{n/2}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon-Zygmund decompositions

Destruction of long range magnetic order in an external magnetic field and the associated spin dynamics in Cu2GaBO5 and Cu2AlBO5 ludwigites

The quantum spin systems Cu2M BO5 M Al,Ga with the ludwigite crystal structure consist of a structurally ordered Cu2 sublattice in the form of three leg ladders, interpenetrated by a structurally disordered sublattice with a statistically random site occupation by magnetic Cu2 and nonmagnetic Ga3 or Al3 ions. A microscopic analysis based on density functional theory calculations for Cu2GaBO5 reveals a frustrated quasi two dimensional spin model featuring five inequivalent antiferromagnetic exchanges. A broad low temperature 11B nuclear magnetic resonance points to a considerable spin disorder in the system. In zero magnetic field, antiferromagnetic order sets in below TN approximation 4.1 K and 2.4 K for the Ga and Al compounds, respectively. From neutron diffraction, we find that the magnetic propagation vector in Cu2GaBO5 is commensurate and lies on the Brillouin zone boundary in the H0L plane, qm 0.45, 0, 0.7 , corresponding to a complex noncollinear long range ordered structure with a large magnetic unit cell. Muon spin relaxation is monotonic, consisting of a fast static component typical for complex noncollinear spin systems and a slow dynamic component originating from the relaxation on low energy spin fluctuations. Gapless spin dynamics in the form of a diffuse quasielastic peak is also evidenced by inelastic neutron scattering. Most remarkably, application of a magnetic field above 1 T destroys the static long range order, which is manifested in the gradual broadening of the magnetic Bragg peaks. We argue that such a crossover from a magnetically long range ordered state to a spin glass regime may result from orphan spins on the structurally disordered magnetic sublattice, which are polarized in magnetic field and thus act as a tuning knob for field controlled magnetic disorde

A Re-Evaluation of the nuclear Structure Function Ratios for D, He, Li, C and Ca

We present a re-evaluation of the structure function ratios F2(He)/F2(D), F2(C)/F2(D) and F2(Ca)/F2(D) measured in deep inelastic muon-nucleus scattering at an incident muon momentum of 200 GeV. We also present the ratios F2(C)/F2(Li), F2(Ca)/F2(Li) and F2(Ca)/F2(C) measured at 90 GeV. The results are based on data already published by NMC; the main difference in the analysis is a correction for the masses of the deuterium targets and an improvement in the radiative corrections. The kinematic range covered is 0.0035 < x < 0.65, 0.5 < Q^2 <90 GeV^2 for the He/D, C/D and Ca/D data and 0.0085 < x < 0.6, 0.84 < Q^2 < 17 GeV^2 for the Li/C/Ca ones.Comment: 6 pages, Latex, 3 figures as uuencoded compressed tar file included at the end, in case of problems contact [email protected] (Antje Bruell

The Structure Function Ratios F2(Li)/F2(D) and F2(C)/F2(D) at small x

We present the structure function ratios F2(Li)/F2(D) and F2(C)/F2(D) measured in deep inelastic muon-nucleus scattering at a nominal incident muon energy of 200 GeV. The kinematic range 0.0001 < x < 0.7 and 0.01< Q^2 < 70 GeV^2 is covered. For values of $x$ less than $0.002$ both ratios indicate saturation of shadowing at values compatible with photoabsorption results

Exocyst subunit Sec6 is positioned by microtubule overlaps in the moss phragmoplast prior to cell plate membrane arrival

During plant cytokinesis a radially expanding membrane-enclosed cell plate is formed from fusing vesicles that compartmentalizes the cell in two. How fusion is spatially restricted to the site of cell plate formation is unknown. Aggregation of cell-plate membrane starts near regions of microtubule overlap within the bipolar phragmoplast apparatus of the moss Physcomitrella patens Since vesicle fusion generally requires coordination of vesicle tethering and subsequent fusion activity, we analyzed the subcellular localization of several subunits of the exocyst, a tethering complex active during plant cytokinesis. We found that the exocyst complex subunit Sec6 but not the Sec3 or Sec5 subunits localized to microtubule overlap regions in advance of cell plate construction in moss. Moreover, Sec6 exhibited a conserved physical interaction with an ortholog of the Sec1/Munc18 protein KEULE, an important regulator for cell-plate membrane vesicle fusion in Arabidopsis Recruitment of the P. patens protein KEULE and vesicles to the early cell plate was delayed upon Sec6 gene silencing. Our findings, thus, suggest that vesicle-vesicle fusion is, in part, enabled by a pool of exocyst subunits at microtubule overlaps, which is recruited independently of vesicle delivery.</p