The decay of theT=1 isospin triplet inA=12 systems: IV. The energy dependence of the asymmetry coefficients\alpha _{\beta ^ \mp } of the beta-ray angular distribution in aligned12B and12N and the induced pseudotensor interaction

The asymmetry parameters \(\alpha _{\beta ^ \mp } \) of the beta-ray emitted from aligned¹²B and¹²N are evaluated as a function of the energy. The agreement with experimental differential data is excellent for both \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W). This work confirms, using available nuclear model information, that no induced pseudotensor (IPT) interaction is required for a correct theoretical interpretation of the data. An upper limit for the IPT coupling constantf _T is determined from a simultaneous fit of \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W).     

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.     

Alternative technique for Standard Model estimation of the rare kaon decay branchings \left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM}

We estimate $BR(K \to \pi \nu \bar \nu )$ in the context of the Standard Model by fitting for λ _t≡V _tdV _ts ^* of the “kaon unitarity triangle” relation. To find the vertex of this triangle, we fit data from |? _K|, the CP-violating parameter describing K mixing, and a _ψ,K , the CP-violating asymmetry in B _d ⁰ → J/ψK ⁰ decays, and obtain the values $\left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM} = (7.07 \pm 1.03) \times 10^{ - 11} $ and $\left. {BR(K_L^0 \to \pi ^0 \nu \bar \nu )} \right|_{SM} = (2.60 \pm 0.52) \times 10^{ - 11} $ . Our estimate is independent of the CKM matrix element V _cb and of the ratio of B-mixing frequencies ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ . We also use the constraint estimation of λ _t with additional data from $\Delta m_{B_d } $ and |V _ub|. This combined analysis slightly increases the precision of the rate estimation of $K^ + \to \pi ^ + \nu \bar \nu $ and $K_L^0 \to \pi ^0 \nu \bar \nu $ (by ?10 and ?20%, respectively). The measured value of $BR(K^ + \to \pi ^ + \nu \bar \nu )$ can be compared both to this estimate and to predictions made from ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ .     

Three-loop relation of quark\overline {MS} and pole masses

We calculate, exactly, the next-to-leading correction to the relation between the \(\overline {MS} \) quark mass, \(\bar m\) , and the scheme-independent pole mass,M, and obtain $$\begin{gathered} \frac{M}{{\bar m(M)}} \approx 1 + \frac{4}{3}\frac{{\bar \alpha _s (M)}}{\pi } + \left[ {16.11 - 1.04\sum\limits_{i = 1}^{N_F - 1} {(1 - M_i /M)} } \right] \hfill \\ \cdot \left( {\frac{{\bar \alpha _s (M)}}{\pi }} \right)^2 + 0(\bar \alpha _s^3 (M)), \hfill \\ \end{gathered} $$ as an accurate approximation forN _F?1 light quarks of massesM _i <M. Combining this new result with known three-loop results for \(\overline {MS} \) coupling constant and mass renormalization, we relate the pole mass to the \(\overline {MS} \) mass, \(\bar m\) (μ), renormalized at arbitrary μ. The dominant next-to-leading correction comes from the finite part of on-shell two-loop mass renormalization, evaluated using integration by parts and checked by gauge invariance and infrared finiteness. Numerical results are given for charm and bottom \(\overline {MS} \) masses at μ=1 GeV. The next-to-leading corrections are comparable to the leading corrections.     

Singular $$ \mathcal{R}$$ -Matrices and Drinfeld's Comultiplication

We compute the $\mathcal{R}$ -matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for ${\text{U}}_q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ . This $\mathcal{R}$ -matrix contains terms proportional to the δ-function. We construct the algebra $A\left( \mathcal{R} \right)$ generated by the elements of the matrices L^±(z) with relations determined by $\mathcal{R}$ . In the category of highest-weight representations, there is a Hopf algebra isomorphism between $A\left( \mathcal{R} \right)$ and an extension $\overline {\text{U}} _q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ of Drinfeld's algebra.     

A Remark on the Unconditionally Convergent Series

For every unconditionally convergent series $\sum_{j=1}^{\infty}x_{j}$ in sequentially complete Abelian topological group, we show that the sum $\sum_{j=1}^{\infty}x_{\theta(j)}$ is same for all permutations θ:?→?. This result justify the measures defined on quantum structures.     

Jack Superpolynomials with Negative Fractional Parameter: Clustering Properties and Super-Virasoro Ideals

The Jack polynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible partitions are known to span an ideal ${I_{N}^{(k,r)}}$ of the space of symmetric functions in N variables. The ideal ${I_{N}^{(k,r)}}$ is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in ${I_{N}^{(k,r)}}$ admit clusters of size at most k: they vanish when k?+?1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible superpartitions span an ideal ${\mathcal{I}_{N}^{(k,r)}}$ of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal ${\mathcal{I}_{N}^{(k,r)}}$ is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in ${\mathcal {I}_{N}^{(k,r)}}$ vanish when k?+?1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of ${\mathcal{I}_{N}^{(k,2)}}$ provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k?+?1 commuting variables are set equal to each other.     

Relative x-ray transition probabilities to theK-shell forZ=81, 92, 94, and 96

Accurate intensity measurements of the majorK x-ray groups have been performed with high resolution Ge(Li) detectors in singles and coincidence arrangements and with a high-purity Ge detector of the intrinsic type. Previously reportedK x-ray intensities forZ=96 are in error due to the presence of a 121.5 keV γ-ray in the decay of²⁴⁹Cf. The present results are as follows: forZ=81,K _α2/K _α1=0.589±0.008, \(K_{\beta _1^\prime } /K_{\alpha ^1 } = 0.344 \pm 0.008, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.102 \pm 0.004\) , andK _β/K _α=0.281±0.006; forZ=92 \(K_{\alpha _2 } /K_{\alpha _1 } = 0.611 \pm 0.008,K_{\beta _1^\prime } /K_{\alpha _1 } = 0.365 \pm 0.008, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.125 \pm 0.004\) , andK _β/K _α=0.300±0.006; forZ=94, \(K_{\alpha _2 } /K_{\alpha _1 } = 0.610 \pm 0.008, K_{\beta _1^\prime } /K_{\alpha _1 } = 0.369 \pm 0.010, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.127 \pm 0.004\) , andK _β/K _α=0.308±0.008; and forZ=96, \(K_{\alpha _2 } /K_{\alpha _1 } = 0.627 \pm 0.008, K_{\beta _1^\prime } /K_{\alpha _1 } = 0.372 \pm 0.009, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.133 \pm 0.005\) , andK _β/K _α=0.310±0.008. The error limits are the 2σ statistical errors to which a systematic error in the detector efficiencies has been added linearly. The present results are compared with recent theoretical calculations.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

QCD calculation ofB \to \pi l\bar v_l and the matrix element |V bu|

Using QCD sum rules for a two-point function involving beauty vector currents, together with current algebra-PCAC sum rules, we estimate the hadronic matrix element in \(B \to \pi l\bar v_l \) . We find \(\Gamma \left( {\bar {\rm B}^0 \to \pi ^ + l\bar v_l } \right) = \left( {1.45 \pm 0.59} \right) \times 10^{13} \left| {V_{bu} } \right|^2 s^{ - 1} \) . As a byproduct, the vector current contribution to the decay \(B \to \rho l\bar v_l \) is also estimated.     

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.     

Invariant Algebraic Surfaces for Generalized Raychaudhuri Equations

We consider a generalized Raychaudhuri equation, $$\begin{array}{llll} \dot x = -\frac 1 2 x^2 -\alpha x -2(y^2 +z^2 -w^2)-2 \beta,\\ \dot y = -(\alpha +x) y -\gamma,\\ \dot z = -(\alpha +x) z -\delta,\\ \dot w = -(\alpha +x) w, \end{array}$$ where ??, ??, ??, ?? are real parameters. This model has appeared in modern string cosmology. We study the algebraic invariants of this model for all values of the parameters ${\alpha,\beta,\gamma,\delta\in \mathbb{R}}$ . We prove that when ???=????=?0 the system is integrable and for any other values of the parameters ??, ??, ??, ?? we characterize all the invariant surfaces of this system. In particular we characterize all the polynomial and proper rational first integrals.     

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: $$ i\frac{\rm d}{{\rm d}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t)\quad \varepsilon > 0\,\quad V\in L^1(\mathbb{R},(1+|x|){\rm d}x) \cap L^\infty(\mathbb{R}).$$ This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: $$ i\frac{{\rm d}}{{\rm d}t} \psi(t) =H_{\alpha} \psi(t) .$$ where H _α is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\rm d}x}\) . The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if V ≥  0 and for every  \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.     

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices $H_N$ , $N=|\Lambda |W$ with elements $H_{jk,\alpha \beta }$ , where $j,k \in \Lambda =[1,m]^d\cap \mathbb {Z}^d$ (they parameterize the lattice sites) and $\alpha , \beta = 1,\ldots , W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha \beta }$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha _1\beta _1}H_{j_2k_2,\alpha _2\beta _2}\rangle =\delta _{j_1k_2}\delta _{j_2k_1} \delta _{\alpha _1\beta _2}\delta _{\beta _1\alpha _2} J_{j_1k_1},$ where $J=1/W+\alpha \Delta /W$ , $\alpha < 1/4d$ . This matrices are the special case of Wegner’s $W$ -orbital models. Assuming that the number of sites $|\Lambda |$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda _0|< \sqrt{2}$ .     

Surface tension with normal curvature in curved space-time

With an aim to include the contribution of surface tension in the action of the boundary, we define the tangential pressure in terms of surface tension and Normal curvature in a more naturally geometric way. For a thin shell approximation of a static spherically symmetric surface and for weak and slowly varying fields, the negative tangential pressure $\tau _{\alpha \beta }$ is chosen to be analogous to $S_{\alpha \beta },$ where $S_{\alpha \beta }$ is the classical surface tension. First, by a suitable choice of the enveloping surfaces, we show that the negative tangential pressure is independent of the four-velocity of a very thin hyper-surface. Second, using suitable definition of the normal curvature for such a surface layer, we relate the 3-pressure of a surface layer to the normal curvature and the surface tension. Third, using the fact that the tangential pressure on the surface layer is independent of the four-velocity and a central force interaction, we relate the surface tension $S_{\alpha \beta }$ to the energy of the surface layer. Four, we show that the delta like energy flows across the hypersurface will be zero for such a representation of intrinsic 3-pressure. Five, for the weak field approximation and for static spherically symmetric configuration, we deduce the classical Kelvin’s relation between surface tension, pressure difference and mean curvature from this sort of representation of negative tangential pressure $\tau _{\alpha \beta }$ in terms of surface tension $S_{\alpha \beta }$ and the normal curvature. Six, using the representation of tangential pressure in terms of surface tension and normal curvature, we write a modified action for the boundary having contributions both from surface tension and normal curvature of the surface layer. Also we propose a method to find the physical action assuming a reference background, where the background is not flat. (The $g_{\mu \nu }^{+}$ or just $g_{\mu \nu }$ has been chosen to represent the metric coefficent of the hypersurface of $V_{+}$ space which is time-like surface layer here. The $g_{\mu \nu }^{-}$ represents the metric coefficient of the space like hypersurface of $V_{-}$ space.)     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

Estimates of general Mayer graphs III: Upper bounds obtained by means of spanningn-trees

We obtain computable upper bounds for any given Mayer graph withn root-points (orn-graph). These are products of integrals of the type \(\left( {\int {\left| {f_L } \right|^{z_{iL} y_i^{ - 1} } dx} } \right)^{yi} \) , where thez _iL andy _i are nonnegative real numbers whose sum overi is equal to 1. As a particular case, we obtain the canonical bounds (see their definition in Section 2.2): $$\left| {\int {\prod\limits_L {f_L \left( {x_i ,x_j } \right)dx_{n + 1} \cdot \cdot \cdot dx_{n + k} } } } \right| \leqslant \prod\limits_L {\left( {\int {\left| {f_L } \right|^{\alpha _L } dx} } \right)^{\alpha _L^{ - 1} } } $$ where theα _L 's satisfy the conditionα _L ≥1 for anyL, and ∑ _L α _L ^?1 =k (k is the number of variables that are integrated over). These bounds are finite for alln-graphs of neutral systems. We obtain also finite bounds for all irreduciblen-graphs of polar systems, and for certainn-graphs occurring in the theory of ionized systems. Finally, we give a sufficient condition for an arbitraryn-graph to be finite.