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.     

Radiative and semileptonic B decays involving higher K-resonances in the final states

We study the radiative and semileptonic B decays involving a spin-J resonant $K_{J}^{(*)}$ with parity (?1) ^J for $K_{J}^{*}$ and (?1) ^J+1 for K _J in the final state. Using large energy effective theory (LEET) techniques, we formulate $B\to K_{J}^{(*)}$ transition form factors in the large recoil region in terms of two independent LEET functions $\zeta_{\perp}^{K_{J}^{(*)}}$ and $\zeta_{\parallel}^{K_{J}^{(*)}}$ , the values of which at zero momentum transfer are estimated in the BSW model. According to the QCD counting rules, $\zeta_{\perp,\parallel}^{K_{J}^{(*)}}$ exhibit a dipole dependence in q ². We predict the decay rates for $B\to K_{J}^{(*)}\gamma$ , $B\to K_{J}^{(*)}\ell^{+}\ell^{-}$ and $B\to K_{J}^{(*)}\nu \bar{\nu}$ . The branching fractions for these decays with higher K-resonances in the final state are suppressed due to the smaller phase spaces and the smaller values of $\zeta^{K_{J}^{(*)}}_{\perp,\parallel}$ . Furthermore, if the spin of $K_{J}^{(*)}$ becomes larger, the branching fractions will be further suppressed due to the smaller Clebsch–Gordan coefficients defined by the polarization tensors of the $K_{J}^{(*)}$ . We also calculate the forward–backward asymmetry of the $B\to K_{J}^{(*)}\ell^{+}\ell^{-}$ decay, for which the zero is highly insensitive to the K-resonances in the LEET parametrization.     

Analysis of the \frac{1} {2}^ - and \frac{3} {2}^ - heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the $\frac{1} {2}^ -$ and $\frac{3} {2}^ -$ heavy and doubly heavy baryon states $\Sigma _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi '_Q \left( {\frac{1} {2}^ - } \right)$ , $\Omega _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Omega _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Sigma _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Omega _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ and $\Omega _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ by subtracting the contributions from the corresponding $\frac{1} {2}^ +$ and $\frac{3} {2}^ +$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.     

Analysis of the {3\over 2}^{+} heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the ${3\over 2}^{+}$ heavy and doubly heavy baryon states $\varXi^{*}_{cc}$ , $\varOmega^{*}_{cc}$ , $\varXi^{*}_{bb}$ , $\varOmega^{*}_{bb}$ , $\varSigma_{c}^{*}$ , $\varXi_{c}^{*}$ , $\varOmega_{c}^{*}$ , $\varSigma_{b}^{*}$ , $\varXi_{b}^{*}$ and $\varOmega_{b}^{*}$ by subtracting the contributions from the corresponding ${3\over 2}^{-}$ heavy and doubly heavy baryon states with the QCD sum rules, and we make reasonable predictions for their masses.     

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 ${\mathcal{W}_e}$ .     

Analysis of the vertexes \Xi_{Q}^{*}′QV , \Sigma_{Q}^{*} \Sigma_{Q}^{}V and radiative decays \Xi_{Q}^{*} \rightarrow ′Q \gamma , \Sigma_{Q}^{*} \rightarrow \Sigma_{Q}^{} \gamma

In this article, we study the vertexes $ \Xi_{Q}^{*}$ ′_Q V and $ \Sigma_{Q}^{*}$ $ \Sigma_{Q}^{}$ V with the light-cone QCD sum rules, then assume the vector meson dominance of the intermediate $ \phi$ (1020) , $ \rho$ (770) and $ \omega$ (782) , and calculate the radiative decays $ \Xi_{Q}^{*}$ $ \rightarrow$ ′_Q $ \gamma$ and $ \Sigma_{Q}^{*}$ $ \rightarrow$ $ \Sigma_{Q}^{}$ $ \gamma$ .     

Analysis of the {1\over2}^{\pm} flavor antitriplet heavy baryon states with QCD sum rules

In this article, we study the masses and pole residues of the ${1\over2}^{\pm}$ flavor antitriplet heavy baryon states ( $\varLambda _{c}^{+}$ , $\varXi _{c}^{+},\varXi _{c}^{0})$ and ( $\varLambda _{b}^{0}$ , $\varXi _{b}^{0},\varXi _{b}^{-})$ by subtracting the contributions from the corresponding ${1\over2}^{\mp}$ heavy baryon states with the QCD sum rules, and observe that the masses are in good agreement with the experimental data and make reasonable predictions for the unobserved ${1\over2}^{-}$ bottom baryon states. Once reasonable values of the pole residues λ _Λ and λ _Ξ are obtained, we can take them as basic parameters to study the relevant hadronic processes with the QCD sum rules.     

Higher harmonics of azimuthal anisotropy in relativistic heavy-ion collisions in HYDJET++ model

The LHC data on azimuthal anisotropy harmonics from PbPb collisions at center-of-mass energy 2.76 TeV per nucleon pair are analyzed and interpreted in the framework of the HYDJET++ model. The cross-talk of elliptic $v_2$ and triangular $v_3$ flow in the model generates both even and odd harmonics of higher order. Comparison with the experimental data shows that this mechanism is able to reproduce the $p_\mathrm{T}$ and centrality dependencies of quadrangular flow $v_4$ , and also the basic trends for pentagonal $v_5$ and hexagonal $v_6$ flows.     

Spectroscopic properties of Ho 3 + -doped K–Sr–Al phosphate glasses

Trivalent holmium-doped K–Sr–Al phosphate glasses ( $\mathrm{P}_{2}\mathrm{O}_{5}$ – $\mathrm{K}_{2}\mathrm{O}$ –SrO– $\mathrm{Al}_{2}\mathrm{O}_{3}$ – $\mathrm{Ho}_{2}\mathrm{O}_{3}$ ) were prepared, and their spectroscopic properties have been evaluated using absorption, emission, and excitation measurements. The Judd–Ofelt theory has been used to derive spectral intensities of various absorption bands from measured absorption spectrum of 1.0 mol% $\mathrm{Ho}_{2}\mathrm{O}_{3}$ -doped K–Sr–Al phosphate glass. The Judd–Ofelt intensity parameters ( $\varOmega_{\lambda}$ , $\times10^{-20}~\mathrm{cm}^{2}$ ) have been determined of the order of $\varOmega_{2} = 11.39$ , $\varOmega_{4} = 3.59$ , and $\varOmega_{6} = 2.92$ , which in turn used to derive radiative properties such as radiative transition probability, radiative lifetime, branching ratios, etc. for excited states of $\mathrm{Ho}^{3+}$ ions. The radiative lifetimes for the ${}^{5}F_{4}$ , ${}^{5}S_{2}$ , and ${}^{5}F_{5}$ levels of $\mathrm{Ho}^{3+}$ ions are found to be 169, 296, and 317 μs, respectively. The stimulated emission cross-section for 2.05-μm emission was calculated by the McCumber theory and found to be $9.3\times10^{-2 1}~\mathrm{cm}^{2}$ . The wavelength-dependent gain coefficient with population inversion rate has been evaluated. The results obtained in the titled glasses are discussed systematically and compared with other $\mathrm{Ho}^{3+}$ -doped systems to assess the possibility for visible and infrared device applications.     

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.     

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).     

Local structure of Fe-doped In 2 O 3 films investigated by X-ray absorption fine structure spectroscopy

$(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ $(x=0.07, 0.09, 0.16, 0.22, 0.31)$ films were deposited on Si (100) substrates by RF-magnetron sputtering technique. The influence of Fe doping on the local structure of films was investigated by X-ray absorption spectroscopy (XAS) at Fe K-edge and L-edge. For the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.07, 0.09 \mbox{ and } 0.16$ , Fe ions dissolve into $\mathrm{In}_{2}\mathrm{O}_{3}$ and substitute for $\mathrm{In}^{3+}$ sites with a mixed-valence state ( $\mathrm{Fe}^{2+}/\mathrm{Fe}^{3+}$ ) of Fe ions. However, a secondary phase of Fe metal clusters is formed in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.22 \mbox{ and } 0.31$ . The qualitative analyses of Fe-K edge extended X-ray absorption fine structure (EXAFS) reveal that the Fe–O bond length shortens and the corresponding Debye–Waller factor ( $\sigma^{2}$ ) increases with the increase of Fe concentration, indicating the relaxation of oxygen environment of Fe ions upon substitution. The anomalously large structural disorder and very short Fe–O distance are also observed in the films with high Fe concentration. Linear combination fittings at Fe L-edge further confirm the coexistence of $\mathrm{Fe}^{2+}$ and $\mathrm{Fe}^{3+}$ with a ratio of ${\sim}3:2$ ( $\mathrm{Fe}^{2+}: \mathrm{Fe}^{3+}$ ) for the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.16$ . However, a significant fraction ( ${\sim}40~\mbox{at\%}$ ) of the Fe metal clusters is found in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.31$ .     

Dirac multimode ket-bra operators’ \mathfrak{Q}-ordered and \mathfrak{P}-ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$ -ordering or $\mathfrak{P}$ -ordering to multimode case, where $\mathfrak{Q}$ -ordering means all Qs are to the left of all Ps and $\mathfrak{P}$ -ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k } $ . Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle $ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.     

Reaction pp \rightarrow pp \pi \pi \pi as a background for hadronic decays of the \eta^{{\prime}}_{} meson

Isospin violating hadronic decays of the $ \eta$ and $ \eta{^\prime}$ mesons into 3 $ \pi$ mesons are driven by a term in the QCD Lagrangian proportional to the mass difference of the d and u quarks. The source giving large yield of the mesons for such decay studies are pp interactions close to the respective kinematical thresholds. The most important physics background for $ \eta$ , $ \eta{^\prime}$ $ \rightarrow$ $ \pi$ $ \pi$ $ \pi$ is coming from direct three-pion production reactions. In case of the $ \eta$ meson the background for the decays is relatively low ( $ \approx$ 10% . The purpose of this article is to provide an estimate of the direct pion production background for the $ \eta{^\prime}$ $ \rightarrow$ 3 $ \pi$ decays. Using the inclusive data from the COSY-11 experiment we have extracted the differential cross-section for the pp $ \rightarrow$ pp -multipion production reactions with the invariant mass of the pions equal to the $ \eta{^\prime}$ meson mass and estimated an upper limit for the signal to background ratio for studies of the $ \eta{^\prime}$ $ \rightarrow$ $ \pi^{+}_{}$ $ \pi^{-}_{}$ $ \pi^{0}_{}$ decay.     

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).     

On q-Deformed {\mathfrak{gl}_{\ell+1}}-Whittaker Function II

A representation of a specialization of a q-deformed class one lattice ${\mathfrak{gl}_{\ell+1}}$ -Whittaker function in terms of cohomology groups of line bundles on the space ${\mathcal{QM}_d(\mathbb{P}^{\ell})}$ of quasi-maps ${\mathbb{P}^1 \to \mathbb{P}^{\ell}}$ of degree d is proposed. For ? = 1, this provides an interpretation of the non-specialized q-deformed ${\mathfrak{gl}_{2}}$ -Whittaker function in terms of ${\mathcal{QM}_d(\mathbb{P}^1)}$ . In particular the (q-version of the) Mellin-Barnes representation of the ${\mathfrak{gl}_2}$ -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed ${\mathfrak{gl}_2}$ -Toda chain is also discussed.     

Study of the ion exchange equilibrium of Cl−, NO_3^{ - } , and SO_4^{{2 - }} ions on the AMX membrane

Equilibrium between the ion exchange membrane and solutions of anions at various valences has been the subject of this investigation. Competitive ion exchange reactions were studied on a strong base anion exchange membrane AMX manufactured by Tokuyama, commercialized by Eurodia, involving Cl^?, $ {\text NO}_3^{ - } $ and $ {\text SO}_4^{{2 - }} $ ions. Solution concentrations studied were 0.05 and 0.1 M for all the systems reported. Experiments were performed with sodium as the counter ion, and the temperature was kept constant (T?=?298 K). Ionic exchange isotherms for the binary systems— $ {{\text Cl}^{ - }}/{\text NO}_3^{ - } $ , $ {{\text Cl}^{ - }}/{\text SO}_4^{{2 - }} $ , and $ {\text NO}_3^{ - }/{\text SO}_4^{{2 - }} $ —were established. The obtained results show that the sulfate was the most strongly sorbed, and the selectivity order is $ {\text SO}_4^{{2 - }} > {\text NO}_3^{ - } > {{\text Cl}^{ - }} $ at 0.05 M and $ {\text NO}_3^{ - } > {\text SO}_4^{{2 - }} > {{\text Cl}^{ - }} $ at 0.1 M under the experimental conditions. Selectivity coefficients $ K_{{{{{\text Cl} }^{ - }}}}^{{{\text NO}_3^{ - }}} $ , $ K_{{2{{{\text Cl} }^{ - }}}}^{{{\text SO}_4^{{2 - }}}} $ , and $ K_{{2{\text NO}_3^{ - }}}^{{{\text SO}_4^{{2 - }}}} $ for the three binary systems were determined. All the results given by this membrane were compared with those obtained, in the same conditions, with the RPA membrane (produced by RHONE POULENC). Ternary equilibrium data were taken for $ {{\text Cl}^{ - }}/{\text NO}_3^{ - }/{\text SO}_4^{{2 - }} $ . The prediction of the ternary system based only on the binary data was consistent with the experimental data obtained for this system. The good agreement between the experimental and the predicted data showed that the proposed framework can be considered as an effective method to predict many ternary systems from binary systems.     

Annihilation-type charmless radiative decays of B meson in non-universal Z′ model

We study charmless pure annihilation type radiative B decays within the QCD factorization approach. After adding the vertex corrections to the naive factorization approach, we find that the branching ratios of $\overline{B}^{0}_{d}\to\phi\gamma$ , $\overline{B}^{0}_{s}\to\rho^{0}\gamma$ and $\overline{B}^{0}_{s}\to\omega\gamma$ within the standard model are at the order of $\mathcal{O}(10^{-12})$ , $\mathcal{O}(10^{-10})$ and $\mathcal{O}(10^{-11})$ , respectively. The smallness of these decays in the standard model makes them sensitive probes of flavor physics beyond the standard model. To explore their physics potential, we have estimated the contribution of Z′ boson in the decays. Within the allowed parameter space, the branching ratios of these decay modes can be enhanced remarkably in the non-universal Z′ model: The branching ratios can reach to $\mathcal{O}(10^{-8})$ for $\overline{B}_{s}^{0}\to \rho^{0}(\omega)\gamma$ and $\mathcal{O}(10^{-10})$ for the $\overline{B}_{d}^{0}\to \phi \gamma$ , which are large enough for LHC-b and/or Super B-factories to detect those channels in near future. Moreover, we also predict large CP asymmetries in suitable parameter space. The observation of these modes could in turn help us to constrain the Z′ mass within the model.     

On the Quasi-linear Elliptic PDE {-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_k a_k \delta_{s_k}} in Physics and Geometry

It is shown that for each finite number N of Dirac measures ${\delta_{s_n}}$ supported at points ${s_n \in {\mathbb R}^3}$ with given amplitudes ${a_n \in {\mathbb R} \backslash\{0\}}$ there exists a unique real-valued function ${u \in C^{0, 1}({\mathbb R}^3)}$ , vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form ${-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_{n=1}^N a_n \delta_{s_n}}$ . Moreover, ${u \in C^{\omega}({\mathbb R}^3\backslash \{s_n\}_{n=1}^N)}$ . The result can be interpreted in at least two ways: (a) for any number N of point charges of arbitrary magnitude and sign at prescribed locations s _n in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as |s| ?? ??; (b) for any number N of integral mean curvatures assigned to locations ${s_n \in {\mathbb R}^3 \subset{\mathbb R}^{1, 3}}$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime ${{\mathbb R}^{1, 3}}$ , having lightcone singularities over the s _n but being smooth otherwise, and whose height function vanishes as |s| ?? ??. No struts between the point singularities ever occur.     

Liouville-Type Theorems for the Forced Euler Equations and the Navier–Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in ${\mathbb {R}^N}$ . If we assume “single signedness condition” on the force, then we can show that a ${C^1 (\mathbb {R}^N)}$ solution (v, p) with ${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$ is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying ${v(x) \to 0}$ as ${|x| \to \infty}$ , the condition ${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$ , which is stronger than the important D-condition, ${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$ , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203–215, 2007), using the self-similar Euler equations directly.