Upper and lower fast Khintchine spectra in continued fractions

For an irrational number \(x\in [0,1)\), let \(x=[a_1(x), a_2(x),\ldots ]\) be its continued fraction expansion. Let \(\psi : \mathbb {N} \rightarrow \mathbb {N}\) be a function with \(\psi (n)/n\rightarrow \infty \) as \(n\rightarrow \infty \). The (upper, lower) fast Khintchine spectrum for \(\psi \) is defined as the Hausdorff dimension of the set of numbers \(x\in (0,1)\) for which the (upper, lower) limit of \(\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)\) is equal to 1. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.     

Some variations on Tverberg’s theorem

Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in R^d into r pairwise disjoint subsets whose convex hulls have a common point. The numbers T(d, r) are known as Tverberg numbers. Reay added another parameter k (2 ≤ k ≤ r) and asked: what is the smallest number n, such that every set of n points in R^d admits an r-partition, in such a way that each k of the convex hulls of the r parts meet. Call this number T(d, r, k). Reay conjectured that T(d, r, k) = T(d, r) for all d, r and k. In this paper we prove Reay’s conjecture in the following cases: when k ≥ [d+3/2], and also when d < rk/r-k - 1. The conjecture also holds for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.     

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its continuum limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL _N , goes over in the continuum limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL _N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL _N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL _N image represented in the spin-chain. On the continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).     

Subtle Interactions and Electron Transfer between UIII,NpIII, or PuIII and Uranyl Mediated by the Oxo Group

A dramatic difference in the ability of the reducing An^III center in AnCp₃ (An=U, Np, Pu; Cp=C₅H₅) to oxo‐bind and reduce the uranyl(VI) dication in the complex [(UO₂)(THF)(H₂L)] (L=“Pacman” Schiff‐base polypyrrolic macrocycle), is found and explained. These are the first selective functionalizations of the uranyl oxo by another actinide cation. At‐first contradictory electronic structural data are explained by combining theory and experiment. Complete one‐electron transfer from Cp₃U forms the U^IV‐uranyl(V) compound that behaves as a U^V‐localized single molecule magnet below 4 K. The extent of reduction by the Cp₃Np group upon oxo‐coordination is much less, with a Np^III‐uranyl(VI) dative bond assigned. Solution NMR and NIR spectroscopy suggest Np^IVU^V but single‐crystal X‐ray diffraction and SQUID magnetometry suggest a Np^III‐U^VI assignment. DFT‐calculated Hirshfeld charge and spin density analyses suggest half an electron has transferred, and these explain the strongly shifted NMR spectra by spin density contributions at the hydrogen nuclei. The Pu^III–U^VI interaction is too weak to be observed in THF solvent, in agreement with calculated predictions.     

Exit Time of a Hyperbolic α-Stable Process from a Halfspace or a Ball

For a hyperbolic α-stable process in the hyperbolic space \(\mathbb {H}^{d}, d\ge 2\), we prove that the mean exit time from a halfspace \(H(a)=\{x_{d}>a\}\subset \mathbb {H}^{d} \) is equal to \(\mathbb {E}^{x} \tau _{H(a)} = c(\alpha , d) \delta ^{\alpha /2}_{H(a)} (x),\) where δ_D(x) is the (hyperbolic) distance of x to D^c. Based on this exact result we provide a sharp estimate of the mean exit time from a hyperbolic ball B(x₀,R) of radius R and center x₀: \(\mathbb {E}^{x}\tau _{B(x_{0},R)}\approx (\delta _{B(x_{0},R)}(x) \tanh R)^{\alpha /2}, x\in \mathbb {H}^{d}\). By usual isomorphism argument the same estimate holds in any other model of real hyperbolic space.     

Charged-particle multiplicity near midrapidity in central Au+Au collisions at sqrt[SNN]=56 and 130 GeV

We present the first measurement of pseudorapidity densities of primary charged particles near midrapidity in Au+Au collisions at sqrt[s(NN)] = 56 and 130 GeV. For the most central collisions, we find the charged-particle pseudorapidity density to be dN/deta|(|eta|<1) = 408+/-12(stat)+/-30(syst) at 56 GeV and 555+/-12(stat)+/-35(syst) at 130 GeV, values that are higher than any previously observed in nuclear collisions. Compared to proton-antiproton collisions, our data show an increase in the pseudorapidity density per participant by more than 40% at the higher energy.     

Non-invasive surface analysis of ancient bronze arrowheads with scanning electron microscopy and X-ray powder diffractometry

In this research, surfaces of eight ancient metal arrowheads were investigated regarding chemical composition, homogeneity, and products of corrosion. To perform that, two nondestructive techniques were applied: Scanning electron microscopy coupled with energy dispersive spectroscopy (SEM-EDS), and X-ray powder diffractometry (XRPD). Importantly, both methods did not require sampling, cutting, nor significant cleaning of the historical artifacts, which made the measurements not only nondestructive but noninvasive too. SEM-EDS measurements provided information on the morphology and elemental composition of the surfaces of the studied objects as well as the distribution of chemical elements on the surfaces and supported crystalline phase analysis. It was revealed that the arrowheads were cast of tin bronze, but some of them contained high amounts of lead and admixtures of antimony and arsenic while copper and tin oxides and lead carbonates were found as the major corrosion products. In some cases, distribution of elements in the surface exhibited serious nonhomogeneity, probably resulting from limited solubility of the casting metals and degradation processes. Based on the obtained results, authenticity and declared provenience of the arrowheads were assessed in reference to the characteristics of similar objects described in literature.