A sharp upper bound on the signless Laplacian spectral radius of graphs

Let G be a simple connected graph of order n   with degree sequence _d1,_d2,…,_dn$d_1,d_2,\dots ,d_n$ in non-increasing order. The signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ of G   is the largest eigenvalue of its signless Laplacian matrix Q(G)$Q(G)$. In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ in terms of _di$d_i$, which improves and generalizes some known results.     

The inverse inertia problem for the complements of partial k-trees

Let F$\mathbb{F}$ be an infinite field with characteristic not equal to two. For a graph G=(V,E)$G=(V,E)$ with V={1,…,n}$V=\{1,\dots ,n\}$, let S(G;F)$S(G;\mathbb{F})$ be the set of all symmetric n×n$n\times n$ matrices A=[_ai,j]$A=[a_{i,j}]$ over F$\mathbb{F}$ with _ai,j≠0$a_{i,j}\ne 0$, i≠j$i\ne j$ if and only if ij∈E$ij\in E$. We show that if G is the complement of a partial k  -tree and m?k+2$m?k+2$, then for all nonsingular symmetric m×m$m\times m$ matrices K   over F$\mathbb{F}$, there exists an m×n$m\times n$ matrix U   such that U^TKU∈S(G;F)$U^{T}KU\in S(G;\mathbb{F})$. As a corollary we obtain that, if k+2?m?n$k+2?m?n$ and G is the complement of a partial k-tree, then for any two nonnegative integers p and q   with p+q=m$p+q=m$, there exists a matrix in S(G;R)$S(G;\mathbb{R})$ with p positive and q negative eigenvalues.     

Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by Brownian ratchets  . In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ with a changing reflection point _(Rt)t≥0${\left(R_t\right)}_{t\ge 0}$. The rate of change of _Rt$R_t$ is γ(_Xt−_Rt)$\gamma (X_t-R_t)$ and the new reflection boundary is distributed uniformly between _Rt−$R_{t-}$ and _Xt$X_t$. The asymptotic speed of the ratchet scales with γ^1/3${\gamma }^{1/3}$ and the asymptotic variance is independent of γ$\gamma$.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a definition, in the ring language, of _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ and of _Fp[[t]]${\mathbb{F}}_p[[t]]$ inside _Fp((t))${\mathbb{F}}_p((t))$, which works uniformly for all p   and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ uniformly for all p  . For any fixed finite extension of _Qp${\mathbb{Q}}_p$, we give an existential formula and a universal formula in the ring language which define the valuation ring.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

On the least distance eigenvalue of a graph

Denote by D(G)=_(di,j)n×n$D(G)={(d_{i,j})}_{n\times n}$ the distance matrix of a connected graph G with n   vertices, where _dij$d_{ij}$ is equal to the distance between vertices _vi$v_i$ and _vj$v_j$ in G  . The least eigenvalue of D(G)$D(G)$ is called the least distance eigenvalue of G  , denoted by _λn${\lambda }_n$. In this paper, we determine all the graphs with _λn∈[−2.383,0]${\lambda }_n\in [-2.383,0]$.     

Sharp upper bounds on the distance spectral radius of a graph

Let M=(_mij)$M=(m_{ij})$ be a nonnegative irreducible n×n$n\times n$ matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M  , denoted by ρ(M)$\rho (M)$. In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph.     

Inequivalent representations of matroids over prime fields

It is proved that for each prime field GF(p)$\mathit{GF}(p)$, there is an integer _np$n_p$ such that a 4-connected matroid has at most _np$n_p$ inequivalent representations over GF(p)$\mathit{GF}(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.