Haag Duality and the Distal Split Property for Cones in the Toric Code

We prove that Haag duality holds for cones in the toric code model. That is, for a cone ??, the algebra ${\mathcal{R}_{\Lambda}}$ of observables localized in ?? and the algebra ${\mathcal{R}_{\Lambda^c}}$ of observables localized in the complement ?? ^c generate each other??s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if ${\Lambda_1 \subset \Lambda_2}$ are two cones whose boundaries are well separated, there is a Type I factor ${\mathcal{N}}$ such that ${\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}$ . We demonstrate this by explicitly constructing ${\mathcal{N}}$ .     

Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra ${S(\mathfrak{g})}$ over a Lie algebra ${\mathfrak{g}}$ has the structure of a Poisson algebra. Assume ${\mathfrak{g}}$ is complex semisimple. Then results of Fomenko–Mischenko (translation of invariants) and Tarasov construct a polynomial subalgebra ${{\mathcal {H}} = {\mathbb C}[q_1,\ldots,q_b]}$ of ${S(\mathfrak{g})}$ which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of ${\mathfrak{g}}$ . Let G be the adjoint group of ${\mathfrak{g}}$ and let ? = rank ${\mathfrak{g}}$ . Using the Killing form, identify ${\mathfrak{g}}$ with its dual so that any G-orbit O in ${\mathfrak{g}}$ has the structure (KKS) of a symplectic manifold and ${S(\mathfrak{g})}$ can be identified with the affine algebra of ${\mathfrak{g}}$ . An element ${x\in \mathfrak{g}}$ will be called strongly regular if ${\{({\rm d}q_i)_x\},\,i=1,\ldots,b}$ , are linearly independent. Then the set ${\mathfrak{g}^{\rm{sreg}}}$ of all strongly regular elements is Zariski open and dense in ${\mathfrak{g}}$ and also ${\mathfrak{g}^{\rm{sreg}}\subset \mathfrak{g}^{\rm{ reg}}}$ where ${\mathfrak{g}^{\rm{reg}}}$ is the set of all regular elements in ${\mathfrak{g}}$ . A Hessenberg variety is the b-dimensional affine plane in ${\mathfrak{g}}$ , obtained by translating a Borel subalgebra by a suitable principal nilpotent element. Such a variety was introduced in Kostant (Am J Math 85:327–404, 1963). Defining Hess to be a particular Hessenberg variety, Tarasov has shown that ${{\rm{Hess}}\subset \mathfrak{g}^{\rm{sreg}}}$ . Let R be the set of all regular G-orbits in ${\mathfrak{g}}$ . Thus if ${O\in R}$ , then O is a symplectic manifold of dimension 2n where n = b ? ?. For any ${O\in R}$ let ${O^{\rm{sreg}} = \mathfrak{g}^{\rm{sreg}} \cap O}$ . One shows that O ^sreg is Zariski open and dense in O so that O ^sreg is again a symplectic manifold of dimension 2n. For any ${O\in R}$ let ${{\rm{Hess}}(O) = {\rm{Hess}}\cap O}$ . One proves that Hess(O) is a Lagrangian submanifold of O ^sreg and that $${\rm{Hess}} = \sqcup_{O\in R}{\rm{Hess}}(O).$$ The main result of this paper is to show that there exists simultaneously over all ${O\in R}$ , an explicit polarization (i.e., a “fibration” by Lagrangian submanifolds) of O ^sreg which makes O ^sreg simulate, in some sense, the cotangent bundle of Hess(O).     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Theoretical Investigation into Production of Double-Λ Hypernuclei from Stopped Ξ − on 6Li

We have investigated theoretically a feasible nuclear reaction to produce light double-Λ hypernuclei by choosing a suitable target. In the reaction from stopped Ξ ^? on ⁶Li target light doubly-strange nuclei, ${^5_{\Lambda\Lambda}{\rm H}}$ and ${^6_{\Lambda\Lambda}{\rm He}}$ , are produced: we have calculated the formation ratio of ${^5_{\Lambda\Lambda}{\rm H}}$ to ${^6_{\Lambda\Lambda}{\rm He}}$ for Ξ ^? absorptions from 2S, 2P and 3D orbitals of Ξ ^?–⁶Li atom by assuming a d?α cluster model for ⁶Li. From this cluster model the d?α relative wave functions has a node due to Pauli exclusion among nucleons belonging to d and α clusters. Two kinds of d?α wave functions, namely 1s relative wave function with a phenomenological one-range Gaussian (ORG) potential and that of an orthogonality-condition model (OCM) are used. It is found that the probability of ${^5_{\Lambda\Lambda}{\rm H}}$ formation is larger than that of ${^6_{\Lambda\Lambda}{\rm He}}$ for all absorption orbitals: in the case of the major 3D absorption their ratio is 1.08 for ORG and 1.96 for OCM. The dominant low momentum component of the d?α relative wave function favors the ${^5_{\Lambda\Lambda}{\rm H}}$ formation with a low Q value compared to the ${^6_{\Lambda\Lambda}{\rm He}}$ formation with a high Q value. We have also calculated momentum distributions of emitted particles, d and n, displaying continuum spectra for single-Λ hypernuclei, ${^4_{\Lambda}{\rm H}}$ and ${^5_{\Lambda}{\rm He}}$ , and line spectra for the ${^5_{\Lambda\Lambda}{\rm H}}$ and ${^6_{\Lambda\Lambda}{\rm He}}$ nuclei. Thus, our present theoretical analysis would be a significant contribution to experiments in the strangeness ?2 sector of hypernuclear physics.     

Joint System Quantum Descriptions Arising from Local Quantumness

We study the entropy flux in the stationary state of a finite one-dimensional sample ${\mathcal{S}}$ connected at its left and right ends to two infinitely extended reservoirs ${\mathcal{R}_{l/r}}$ at distinct (inverse) temperatures ${\beta_{l/r}}$ and chemical potentials ${\mu_{l/r}}$ . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator ${h_{\mathcal{S}, L}= - \Delta + v}$ . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system ${\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}$ in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit ${L \to \infty}$ and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator ${h_\mathcal{S}}$ .     

Selectivity of anion exchange membrane modified with polyethyleneimine

Previous works have been made on the improvement of selectivity of ion exchange membranes using adsorption of polyelectrolyte on the surface of the materials. The modification of the surface material in the case of an anion exchange membrane concerns the hydrophilic/hydrophobic balance properties and its relationship with the hydration state. Starting from this goal, the AMX membrane has been modified, in this work, by adsorption of polyethyleneimine on its surface. Many conditions of modification of the AMX membrane surface were studied. A factorial experimental design was used for determining the influent parameters on the AMX membrane modification. The results obtained have shown that the initial concentration of polyethyleneimine and the pH of solution were the main influent parameters on the adsorption of polyethyleneimine on the membrane surface. Competitive ion exchange reactions were studied for the modified and the unmodified membrane involving $ {\text{C}}{{\text{l}}^{ - }} $ , $ {\text{NO}}_3^{ - } $ and $ {\text{SO}}_4^{{2 - }} $ ions. All experiments were carried out at constant concentration of 0.3?mol?L^?1 and at 25?°C. Ion exchange isotherms for the binary systems $ \left( {{\text{C}}{{\text{l}}^{ - }}/{\text{NO}}_3^{ - }} \right) $ , $ \left( {{\text{C}}{{\text{l}}^{ - }}/{\text{SO}}_4^{{2 - }}} \right) $ and $ \left( {{\text{NO}}_3^{ - }/{\text{SO}}_4^{{2 - }}} \right) $ were studied. The obtained results show that chloride was the most sorbed and the selectivity order both for the modified membrane and the unmodified one is: $ {\text{Cl}} > {\text{NO}}_3^{ - } > {\text{SO}}_4^{{2 - }} $ , under the experimental conditions. Selectivity coefficients $ {\text{K}}_{{{\text{C}}{{\text{l}}^{ - }}}}^{{{\text{NO}}_3^{ - }}} $ , $ {\text{K}}_{{2{\text{C}}{{\text{l}}^{ - }}}}^{{{\text{SO}}_4^{{2 - }}}} $ and $ {\text{K}}_{{2{\text{NO}}_3^{ - }}}^{{{\text{SO}}_4^{{2 - }}}} $ for the three binary systems and for the two membranes were determined. It was also observed that for the modified membrane the selectivity towards sulfate ion decrease and the modified membrane became more selective towards monovalent anions.     

Smooth Crossed Products of Rieffel’s Deformations

Assume ${\mathcal{A}}$ is a Fréchet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel’s deformation ${\mathcal{A}_J}$ of ${\mathcal{A}}$ . We construct an explicit isomorphism between the smooth crossed products ${V\ltimes\mathcal{A}_J}$ and ${V\ltimes\mathcal{A}}$ . When combined with the Elliott–Natsume–Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C^*-algebras, we also get a simple proof of equivalence of Rieffel’s and Kasprzak’s approaches to deformation.     

How to Add a Boundary Condition

Given a conformal QFT local net of von Neumann algebras ${\mathcal {B}_2}$ on the two-dimensional Minkowski spacetime with irreducible subnet ${\mathcal {A} \otimes \mathcal {A}}$ , where ${\mathcal {A}}$ is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to ${\mathcal {B}_2}$ : we provide a procedure to construct a Boundary CFT net ${\mathcal {B}}$ of von Neumann algebras on the half-plane x >  0, associated with ${\mathcal {A}}$ , and locally isomorphic to ${\mathcal {B}_2}$ . All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of ${\mathcal {B}_2}$ to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S ¹.     

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

Femtosecond relaxation kinetics of highly excited \mathrm{M}_{\mathrm{Na}}^{**} states in CaF2 at 3.2 eV and 4.7 eV

Femtosecond (fs) laser pulses at variable delay times allowed us to track the fast non-radiative transitions between the manifold of highly excited $\mathrm{M}_{\mathrm{Na}}^{**}$ states to the lower lying fluorescent $\mathrm{M}_{\mathrm{Na}}^{*}$ state in CaF₂. Two distinct $\mathrm{M}_{\mathrm{Na}}^{**}$ states of the manifold at 3.16?eV ( $\mathrm{M}_{\mathrm{Na}2}^{**}$ ) and 4.73?eV ( $\mathrm{M}_{\mathrm{Na}3}^{**}$ ) were populated using the second (SH) and third harmonics (TH) of fs laser light at 785?nm. The population kinetics of the fluorescent $\mathrm{M}_{\mathrm{Na}}^{*}$ state in the 2?eV excitation energy range was revealed by depleting its fluorescence centered at 740?nm using fundamental near infrared (NIR) fs laser pulses. The related time constants for $\mathrm{M}_{\mathrm{Na}2,3}^{**}{\sim}{>} \mathrm{M}_{\mathrm{Na}}^{*}$ relaxation amounted to 1.0±0.14?ps and 3.0±0.3?ps upon SH and TH excitation, respectively.     

Equivariant GW Theory of Stacky Curves

We extend Okounkov and Pandharipande’s work on the equivariant Gromov–Witten theory of ${\mathbb{P}^1}$ to a class of stacky curves ${\mathcal{X}}$ . Our main result uses virtual localization and the orbifold ELSV formula to express the tau function ${\tau_\mathcal{X}}$ as a vacuum expectation on a Fock space. As corollaries, we prove the decomposition conjecture for these ${\mathcal{X}}$ , and prove that ${\tau_\mathcal{X}}$ satisfies a version of the 2-Toda hierarchy. Coupled with degeneration techniques, the result should lead to treatment of general orbifold curves.     

Complex Vector Formalism of Harmonic Oscillator in Geometric Algebra: Particle Mass,Spin and Dynamics in Complex Vector Space

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed.     

Some Properties of Generalized Quantum Operations

Let $\mathcal{B}(\mathcal{H})$ be the set of all bounded linear operators on the separable Hilbert space  $\mathcal{H}$ . A (generalized) quantum operation is a bounded linear operator defined on  $\mathcal{B}(\mathcal{H})$ , which has the form $\varPhi_{\mathcal{A}}(X)=\sum_{i=1}^{\infty}A_{i}XA_{i}^{*}$ , where $A_{i}\in\mathcal{B}(\mathcal{H})$ (i=1,2,…) satisfy $\sum_{i=1}^{\infty}A_{i}A_{i}^{*}\leq \nobreak I$ in the strong operator topology. In this paper, we establish the relationship between the (generalized) quantum operation $\varPhi_{\mathcal{A}}$ and its dual $\varPhi_{\mathcal {A}}^{\dag}$ with respect to the set of fixed points and the noiseless subspace. In particular, we also partially characterize the extreme points of the set of all (generalized) quantum operations and give some equivalent conditions for the correctable quantum channel.     

Level-Rank Duality via Tensor Categories

We give a new way to derive branching rules for the conformal embedding $$(\hat{\mathfrak{sl}}_n)_m\oplus(\hat{\mathfrak{sl}}_m)_n\subset(\hat{\mathfrak{sl}}_{nm})_1. $$ In addition, we show that the category ${\mathcal{C}(\hat{\mathfrak{sl}}_n)_m^0}$ of degree zero integrable highest weight ${(\hat{\mathfrak{sl}}_n)_m}$ -representations is braided equivalent to ${\mathcal{C}(\hat{\mathfrak{sl}}_m)_n^0}$ with the reversed braiding.     

Stability of {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} and {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} under H2 atmospheres

Previous H₂ permeation tests showed a degradation of H₂ permeation flux with time. To understand the cause of degradation and develop a solution, the stability of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ and $ {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ samples were studied under dry and wet H₂ atmospheres. Total conductivity of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ increased with time in dry H₂. The X-ray diffraction pattern of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ after dry hydrogen atmosphere heat treatments show CeO₂ peaks indicating that $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ decomposes under dry H₂ atmospheres; scanning electron microscopy and energy dispersive X-ray spectroscopy analyses prove that decomposition proceeded along the grain boundaries. $ {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ was investigated and demonstrated greater stability under dry hydrogen atmospheres. However, Zr substitution results in a tradeoff with electrical properties.     

Study of Light Hypernuclei by the (e,e′K +) Reaction

We have been performing Λ hypernuclear spectroscopic experiments by the (e,e′K ⁺) reaction since 2000 at Thomas Jefferson National Accelerator Facility (JLab). The (e,e′K ⁺) experiment can achieve a few 100 keV (FWHM) energy resolution compared to a few MeV (FWHM) by the (K ^?, π ^?) and (π ⁺, K ⁺) experiments. Therefore, more precise Λ hypernuclear structures can be investigated by the (e,e′K ⁺) experiment. ${^{7}_{\Lambda}{\rm He}}$ , ${^{9}_{\Lambda}{\rm Li}}$ , ${^{10}_{\Lambda}{\rm Be}}$ , ${^{12}_{\Lambda}{\rm B}}$ , ${^{28}_{\Lambda}{\rm Al}}$ , and ${^{52}_{\Lambda}{\rm V}}$ were measured in the experiment at JLab Hall-C. In addition, ${^{9}_{\Lambda}{\rm Li}}$ , ${^{12}_{\Lambda}{\rm B}}$ , and ${^{16}_{\Lambda}{\rm N}}$ were measured in the experiment at JLab Hall-A.     

Muonic Anti-hydrogen {(\bar{\rm H}_{\mu}}) Formation in Low-energy Three-body Reactions

A few-body type computation is performed for a three-charge-particle collision with participation of a slow antiproton ${\bar{\rm{p}}}$ and a muonic muonium atom (true muonium), i.e. a bound state of two muons ${(\mu^{+}\mu^{-})}$ in its ground state. The total cross section of the following reaction ${\bar{\rm p}+(\mu^{+}\mu^{-}) \rightarrow \bar{\rm{H}}_{\mu} + \mu^{-}}$ , where muonic anti-hydrogen ${\bar{\rm{H}}_{\mu}=(\bar{\rm p}\mu^{+})}$ is a bound state of an antiproton and positive muon, is computed in the framework of a set of coupled two-component Faddeev-Hahn-type equation. A better known negative muon transfer low energy three-body reaction: ${{\rm t}^{+} + ({\rm d}^{+}\mu^{-})\rightarrow ({\rm t}^{+}\mu^{-}) + {\rm d}^{+}}$ is also computed as a test system. Here, t⁺ is triton and d⁺ is deuterium.     

Optimal Paths for Symmetric Actions in the Unitary Group

Given a positive and unitarily invariant Lagrangian ${\mathcal{L}}$ defined in the algebra of matrices, and a fixed time interval ${[0,t_0]\subset\mathbb R}$ , we study the action defined in the Lie group of ${n\times n}$ unitary matrices ${\mathcal{U}(n)}$ by $$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$ where ${\alpha:[0,t_0]\to\mathcal{U}(n)}$ is a rectifiable curve. We prove that the one-parameter subgroups of ${\mathcal{U}(n)}$ are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if ${\mathcal{L}}$ is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in ${\mathcal{U}(n)}$ as well as angular metrics in the Grassmann manifold.     

Structure of Neutron-Rich Be Hyper Isotopes

The deformation change of ${^{9}_\Lambda}$ Be and the low-lying states of ${^{12}_{\Lambda}}$ Be are studied by using the antisymmetrized molecular dynamics for hypernuclei (HyperAMD). In ${^{9}_{\Lambda}}$ Be, the Λ hyperon in p orbit enhances nuclear quadrupole deformation, while the Λ hyperon in s orbit reduces it. In ${^{12}_{\Lambda}}$ Be, the ground state parity inverted in ¹¹Be is reverted in ${^{12}_{\Lambda}}$ Be by adding a Λ hyperon as an impurity (impurity effect).     

A DFG-based cavity ring-down spectrometer for trace gas sensing in the mid-infrared

Continuing studies into an all-diode laser-based 3.3 μm difference frequency generation cavity ring-down spectroscopy system are presented. Light from a 1,560 nm diode laser, amplified by an erbium-doped fibre amplifier, was mixed with 1,064 nm diode laser radiation in a bulk periodically poled lithium niobate crystal to generate 16 μW of mid-IR light at 3,346 nm with a conversion efficiency of $0.05\,\%\,{\text{W}}^{-1}\,{\text{cm}}^{-1}$ . This radiation was coupled into a 77 cm long linear cavity with average mirror reflectivities of 0.9996, and a measured baseline ring-down time of $6.07\pm 0.03\,\upmu{\rm s}$ . The potential of such a spectrometer was illustrated by investigating the $P(3)$ transition in the fundamental $\nu_{3}(F_{2})$ band of ${\text{CH}}_4$ both in a 7.5 ppmv calibrated mixture of ${\text{CH}}_4$ in air and in breath samples from methane and non-methane producers under conditions where the minimum detectable absorption coefficient ( $\alpha_{\rm min}$ ) was $2.8 \times 10^{-8}\,{\rm cm}^{-1}$ over 6 s using a ring-down time acquisition rate of 20 Hz. Allan variance measurements indicated an optimum $\alpha_{\rm min}$ of $2.9\times 10^{-9}\,{\rm cm}^{-1}$ over 44 s.