Bounds on Dimension Reduction in the Nuclear Norm

$\newcommand{\schs}{\scriptstyle{\mathsf{S}}_1}$For all $n \ge 1$, we give an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices $A'_1,\ldots,A'_n$ that satisfy \|A'_i-A'_j\|_{\schs} \,\leq\, \|A_i-A_j\|_{\schs}\,\leq\, (1+\delta) \|A'_i-A'_j\|_{\schs} for all $i,j\in\{1,\ldots,n\}$ and small enough $\delta = O(n^{-c})$, where $c> 0$ is a universal constant, it must be the case that $d \ge 2^{\lfloor n/2\rfloor -1}$. This stands in contrast to the metric theory of commutative $\ell_p$ spaces, as it is known that for any $p\geq 1$, any $n$ points in $\ell_p$ embed exactly in $\ell_p^d$ for $d=n(n-1)/2$. Our proof is based on matrices derived from a representation of the Clifford algebra generated by $n$ anti-commuting Hermitian matrices that square to identity, and borrows ideas from the analysis of nonlocal games in quantum information theory.Comment: 16 page

Bounds on Dimension Reduction in the Nuclear Norm

For all n ≥ 1, we give an explicit construction of m × m matrices A_1,…,A_n with m = 2^([n/2]) such that for any d and d × d matrices A′_1,…,A′_n that satisfy ∥A_′i−A′_j∥S_1 ≤ ∥A_i−A_j∥S_1 ≤ (1+δ)∥A′_i−A′_j∥S_1 for all i,j∈{1,…,n} and small enough δ = O(n^(−c)), where c > 0 is a universal constant, it must be the case that d ≥ 2^([n/2]−1). This stands in contrast to the metric theory of commutative ℓ_p spaces, as it is known that for any p ≥ 1, any n points in ℓ_p embed exactly in ℓ^d_p for d = n(n−1)/2. Our proof is based on matrices derived from a representation of the Clifford algebra generated by n anti-commuting Hermitian matrices that square to identity, and borrows ideas from the analysis of nonlocal games in quantum information theory

Protocadherin 15 suppresses oligodendrocyte progenitor cell proliferation and promotes motility through distinct signalling pathways

Oligodendrocyte progenitor cells (OPCs) express protocadherin 15 (Pcdh15), a member of the cadherin superfamily of transmembrane proteins. Little is known about the function of Pcdh15 in the central nervous system (CNS), however, Pcdh15 expression can predict glioma aggression and promote the separation of embryonic human OPCs immediately following a cell division. Herein, we show that Pcdh15 knockdown significantly increases extracellular signal-related kinase (ERK) phosphorylation and activation to enhance OPC proliferation in vitro. Furthermore, Pcdh15 knockdown elevates Cdc42-Arp2/3 signalling and impairs actin kinetics, reducing the frequency of lamellipodial extrusion and slowing filopodial withdrawal. Pcdh15 knockdown also reduces the number of processes supported by each OPC and new process generation. Our data indicate that Pcdh15 is a critical regulator of OPC proliferation and process motility, behaviours that characterise the function of these cells in the healthy CNS, and provide mechanistic insight into the role that Pcdh15 might play in glioma progression

The split property for quantum field theories in flat and curved spacetimes

The split property expresses a strong form of independence of spacelike separated regions in algebraic quantum field theory. In Minkowski spacetime, it can be proved under hypotheses of nuclearity. An expository account is given of nuclearity and the split property, and connections are drawn to the theory of quantum energy inequalities. In addition, a recent proof of the split property for quantum field theory in curved spacetimes is outlined, emphasising the essential ideas

Somatic ‘Soluble’ Adenylyl Cyclase Isoforms Are Unaffected in Sacytm1Lex/Sacytm1Lex ‘Knockout’ Mice

BACKGROUND: Mammalian Soluble adenylyl cyclase (sAC, Adcy10, or Sacy) represents a source of the second messenger cAMP distinct from the widely studied, G protein-regulated transmembrane adenylyl cyclases. Genetic deletion of the second through fourth coding exons in Sacy(tm1Lex)/Sacy(tm1Lex) knockout mice results in a male sterile phenotype. The absence of any major somatic phenotype is inconsistent with the variety of somatic functions identified for sAC using pharmacological inhibitors and RNA interference. PRINCIPAL FINDINGS: We now use immunological and molecular biological methods to demonstrate that somatic tissues express a previously unknown isoform of sAC, which utilizes a unique start site, and which 'escapes' the design of the Sacy(tm1Lex) knockout allele. CONCLUSIONS/SIGNIFICANCE: These studies reveal increased complexity at the sAC locus, and they suggest that the known isoforms of sAC play a unique function in male germ cells

Quantum fields and local measurements

The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the "system", using another, the "probe". The measurement process involves a dynamical coupling of "system" and "probe" within a bounded spacetime region. The resulting "coupled theory" determines a scattering map on the uncoupled combination of the "system" and "probe" by reference to natural "in" and "out" spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the "in" region, the scattering map determines a completely positive map from "probe" observables in the "out" region to "induced system observables", thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies-Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both "system" and "probe" are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory

Not the End of the World? Post-Classical Decline and Recovery in Rural Anatolia

Between the foundation of Constantinople as capital of the eastern half of the Roman Empire in 330 CE and its sack by the Fourth Crusade in 1204 CE, the Byzantine Empire underwent a full cycle from political-economic stability, through rural insecurity and agrarian decline, and back to renewed prosperity. These stages plausibly correspond to the phases of over-extension (K), subsequent release (Ω) and recovery (α) of the Adaptive Cycle in Socio-Ecological Systems. Here we track and partly quantify the consequences of those changes in different regions of Anatolia, firstly for rural settlement (via regional archaeological surveys) and secondly for land cover (via pollen analysis). We also examine the impact of climate changes on the agrarian system. While individual histories vary, the archaeological record shows a major demographic decline between ca .650 and ca. 900 CE in central and southwestern Anatolia, which was then a frontier zone between Byzantine and Arab armies. In these regions, and also in northwest Anatolia, century-scale trends in pollen indicate a substantial decline in the production of cereal and tree crops, and a smaller decline in pastoral activity. During the subsequent recovery (α) phase after 900 CE there was strong regional differentiation, with central Anatolia moving to a new economic system based on agro-pastoralism, while lowland areas of northern and western Anatolia returned to the cultivation of commercial crops such as olive trees. The extent of recovery in the agrarian economy was broadly predictable by the magnitude of its preceding decline, but the trajectories of recovery varied between different regions

A SELEX-Screened Aptamer of Human Hepatitis B Virus RNA Encapsidation Signal Suppresses Viral Replication

Background: The specific interaction between hepatitis B virus (HBV) polymerase (P protein) and the e RNA stem-loop on pregenomic (pg) RNA is crucial for viral replication. It triggers both pgRNA packaging and reverse transcription and thus represents an attractive antiviral target. RNA decoys mimicking e in P protein binding but not supporting replication might represent novel HBV inhibitors. However, because generation of recombinant enzymatically active HBV polymerase is notoriously difficult, such decoys have as yet not been identified. Methodology/Principal Findings: Here we used a SELEX approach, based on a new in vitro reconstitution system exploiting a recombinant truncated HBV P protein (miniP), to identify potential e decoys in two large e RNA pools with randomized upper stem. Selection of strongly P protein binding RNAs correlated with an unexpected strong enrichment of A residues. Two aptamers, S6 and S9, displayed particularly high affinity and specificity for miniP in vitro, yet did not support viral replication when part of a complete HBV genome. Introducing S9 RNA into transiently HBV producing HepG2 cells strongly suppressed pgRNA packaging and DNA synthesis, indicating the S9 RNA can indeed act as an e decoy that competitively inhibits P protein binding to the authentic e signal on pgRNA. Conclusions/Significance: This study demonstrates the first successful identification of human HBV e aptamers by an in vitro SELEX approach. Effective suppression of HBV replication by the S9 aptamer provides proof-of-principle for the abilit