Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise

Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model??the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points $\mathcal{N}_{M}(x)$ above a level $\frac{x}{2}V_{m}$ , with V _m =2lnM standing for the leading-order typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of $\mathcal{N}_{M}(x)$ for 0<x<2. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level x, results in an important difference between the mean and the typical values of $\mathcal{N}_{M}(x)$ . This can be further used to determine the typical threshold x _m of extreme values in the pattern which turns out to be given by $x_{m}^{(\mathit{typ})}=2-c\ln\ln M /\ln M $ with $c=\frac{3}{2}$ . Such observation provides a rather compelling explanation of the mechanism behind universality of c. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum p _max of intensity is to be given by $-\ln p_{\mathit{max}}=\alpha_{-}\ln M +\frac{3}{2f'(\alpha_{-})}\ln\ln M+O(1)$ , where f(??) is the corresponding singularity spectrum positive in the interval ????(?? _?,?? ₊) and vanishing at ??=?? _?>0. For the 1/f noise case we further study asymptotic values of the prefactors in scaling laws for the moments of the counting function. Our numerics shows however that one needs prohibitively large sample sizes to reach such asymptotics even with a moderate precision. This motivates us to derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.     

Non-normal tracer diffusion from stirring by swimming microorganisms

The experimentally observed non-Gaussian form of passive tracer distributions in media stirred by active swimmers (Leptos et al., Phys. Rev. Lett. 103, 198103 (2009)) are analyzed in terms of continuous time random walks. The walks are characterized by a trapping time distribution ??(??) with long time behaviour ??(??) ?? ?? ^?1??? and a step size distribution p(??x) ?? (??x)^?2??? . The experimentally observed behaviour that ??x ²?? ?? t is obtained for a one-parameter family of exponents with ?? = 2??. However, the distribution function for this case is non-Gaussian and shows exponential tails. The shape of the distributions agrees rather well with the experimental observations from Leptos et al. and allows for the determination of the exponents.      

Chiral SU(3) dynamics and the quasibound K – pp cluster

The prototype of a $\bar{K}$ nuclear cluster, K ^???pp, has been investigated using effective $\bar{K}N$ potentials based on chiral SU(3) dynamics. Variational calculation shows a bound state solution with shallow binding energy B(K ^???pp)?=?20±3 MeV and broad mesonic decay width $\Gamma(\bar{K}NN \rightarrow \pi Y N)=40$ –70 MeV. The $\bar{K}N(I=0)$ pair in the K ^???pp system exhibits a similar structure as the Λ(1405). We have also estimated the dispersive correction, p-wave $\bar{K}N$ interaction, and two-nucleon absorption width.     

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant ??, which is related to energy density of cosmological vacuum by $ \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda $ . In the Veneziano ghost theory, the vacuum energy density ?? _?? is expressed by absolute value of the product of quark vacuum condensate and quark current mass: $ \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q < 0|:\bar qq:|0 > | $ . We calculate the quark local vacuum condensates ??0|: $ \bar q $ q: |0?? by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator S _f (p) with an effective gluon propagator G _???? ^ab (q). The quark current mass m _q is predicted by use of chiral perturbation theory. Our theoretical result of ??, with the resulting ??0|: 471-4 q: |0?? = ?(235 MeV)³ and light quark current mass m _q ? 3.29?C6.15 MeV, is in a good agreement with the observable of the ?? ?? 10^?52 m^?2 used widely in a great amount of literatures.     

Integrable and Conformal Boundary Conditions for \widehat{s\ell}(2) A–D–E Lattice Models and Unitary Minimal Conformal Field Theories

Integrable boundary conditions are studied for critical A–D–E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley–Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang–Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A–D–E cases are naturally in one-to-one correspondence with the conformal boundary conditions of $\widehat{s\ell }$ (2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories.     

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation ? _t ?? $\mathcal{D}$ Δ?=λS ²?, where S(x, t) is a Gaussian stochastic field with covariance C(x?x′, t, t′), and x∈ $\mathbb{R}$ ^d . It is shown that the coupling λ _cN (t) at which the N-th moment <? ^N (x, t)> diverges at time t, is always less or equal for $\mathcal{D}$ >0 than for $\mathcal{D}$ =0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫ ^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$ =0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$ →i $\mathcal{D}$ , the case of interest for backscattering instabilities in laser-plasma interaction.     

Cosmological perturbations in SFT inspired non-local scalar field models

We clearly and consistently supersymmetrize the celebrated horizontality condition to derive the off-shell nilpotent and absolutely anticommuting Becchi?CRouet?CStora?CTyutin (BRST) and anti-BRST symmetry transformations for the supersymmetric system of a free spinning relativistic particle within the framework of superfield approach to BRST formalism. For the precise determination of the proper (anti-)BRST symmetry transformations for all the bosonic and fermionic dynamical variables of our system, we consider the present theory on a (1,2)-dimensional supermanifold parameterized by an even (bosonic) variable (??) and a pair of odd (fermionic) variables ?? and $\bar{\theta}$ (with $\theta^{2} = \bar{\theta}^{2} = 0$ , $\theta\bar{\theta}+ \bar{\theta}\theta= 0$ ) of the Grassmann algebra. One of the most important and novel features of our present investigation is the derivation of (anti-)BRST invariant Curci?CFerrari type restriction which turns out to be responsible for the absolute anticommutativity of the (anti-)BRST transformations and existence of the coupled (but equivalent) Lagrangians for the present theory of a supersymmetric system. These observations are completely new results for this model.     

Inelastic Character of Solitons of Slowly Varying gKdV Equations

In this paper we study soliton-like solutions of the variable coefficients, the subcritical gKdV equation $$u_t + (u_{xx} -\lambda u + a(\varepsilon x) u^m )_x =0,\quad {\rm in} \quad \mathbb{R}_t\times\mathbb{R}_x, \quad m=2,3\,\, { \rm and }\,\, 4,$$ with ${\lambda\geq 0, a(\cdot ) \in (1,2)}$ a strictly increasing, positive and asymptotically flat potential, and ${\varepsilon}$ small enough. In previous works (Mu?oz in Anal PDE 4:573?C638, 2011; On the soliton dynamics under slowly varying medium for generalized KdV equations: refraction vs. reflection, SIAM J. Math. Anal. 44(1):1?C60, 2012) the existence of a pure, global in time, soliton u(t) of the above equation was proved, satisfying $$\lim_{t\to -\infty}\|u(t) - Q_1(\cdot -(1-\lambda)t) \|_{H^1(\mathbb{R})} =0,\quad 0\leq \lambda<1,$$ provided ${\varepsilon}$ is small enough. Here R(t, x) := Q _c (x ? (c ? ??)t) is the soliton of R _t +? (R _xx ??? R + R ^m ) _x =?0. In addition, there exists ${\tilde \lambda \in (0,1)}$ such that, for all 0?<??? <?1 with ${\lambda\neq \tilde \lambda}$ , the solution u(t) satisfies $$\sup_{t\gg \frac{1}{\varepsilon}}\|u(t) - \kappa(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}\lesssim \varepsilon^{1/2}.$$ Here ${{\rho'(t) \sim (c_\infty(\lambda) -\lambda)}}$ , with ${{\kappa(\lambda)=2^{-1/(m-1)}}}$ and ${{c_\infty(\lambda)>\lambda}}$ in the case ${0<\lambda<\tilde\lambda}$ (refraction), and ${\kappa(\lambda) =1}$ and c _??(??)?<??? in the case ${\tilde \lambda<\lambda<1}$ (reflection). In this paper we improve our preceding results by proving that the soliton is far from being pure as t ?? +???. Indeed, we give a lower bound on the defect induced by the potential a(·), for all ${{0<\lambda<1, \lambda\neq \tilde \lambda}}$ . More precisely, one has $$\liminf_{t\to +\infty}\| u(t) - \kappa_m(\lambda)Q_{c_\infty}(\cdot-\rho(t)) \|_{H^1(\mathbb{R})}>rsim \varepsilon^{1 +\delta},$$ for any ${{\delta>0}}$ fixed. This bound clarifies the existence of a dispersive tail and the difference with the standard solitons of the constant coefficients, gKdV equation.     

The Dual Stream Function Representation of an Ideal Steady Fluid Flow and its Local Geometric Structure

Using the methodology of Lie groups and Lie algebras we determine new symmetry and equivalence classes of the stationary three-dimensional Euler equations by introducing potential functions that are based on the so-called dual stream function representation of the steady state velocity field u(x, y, z) = ?λ(x, y, z) × ?μ(x, y, z), which itself can only be defined locally. In particular an infinite dimensional Lie algebra for Beltrami fields is gained. We show that this Lie algebra generates canonical transformations of a Hamiltonian flow for the dual pair of variables \(\lambda \) and \(\mu \) . It enables us to make the classification of a two-dimensional Riemannian manifold \(M^{2}\) wherein \((\lambda ,\mu )\) presents the local coordinates of \(M^{2}\) . Furthermore the local geometry of this manifold is explored in detail. As a result an infinite set of locally conserved currents and charges in the context of a conformal field theory is finally observed.     

Lie Triple Derivations of CSL Algebras

Let be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space with , and the associated CSL algebra. It is proved that every Lie triple derivation from into any σ-weakly closed algebra containing is of the form X→XT?TX+h(X)I, where and h is a linear mapping from into ? such that h([[A,B],C])=0 for all .     

Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

We define cylindric versions of skew Macdonald functions P _λ/μ (q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, viewed as a subset of the two-dimensional integer lattice, by the period vector (n, ?k). Imposing a periodicity condition one defines cylindric skew tableaux and associated weight functions. The resulting weighted sums over these cylindric tableaux are symmetric functions. They appear in the coproduct of a commutative Frobenius algebra which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a ${U_{q}\widehat{\mathfrak{sl}}(n)}$ Kirillov-Reshetikhin module. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig’s canonical basis. In the limit q = t = 0, this Frobenius algebra is isomorphic to the ${\widehat{\mathfrak{sl}}(n)}$ Verlinde algebra at level k, i.e. the structure constants become the ${\widehat{\mathfrak{sl}}(n)_{k}}$ Wess-Zumino-Novikov-Witten fusion coefficients. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions discussed here arise as partition functions of so-called vertex models obtained from solutions to the Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

The Taylor Expansion at Past Time-like Infinity

We construct initial data for the conformal vacuum field equations on a cone ${{\mathcal{N}}_p}$ with vertex p so that for the prospective vacuum solution, the point p will represent past time-like infinity i ^?, the set ${{\mathcal{N}}_p {\setminus}\{p\}}$ will represent past null infinity ${{\mathcal{J}}^-}$ , and the freely prescribed (suitably smooth) data will acquire the meaning of the incoming radiation field. It is shown that: (i) On some coordinate neighbourhood of p there exist smooth fields which satisfy at the point p the conformal vacuum field equations at all orders and induce the given data at all orders. The Taylor coefficients of these fields at p are uniquely determined by the free data. (ii) On the cone ${{\mathcal{N}}_p}$ there exists a unique set of fields which induce the given free data and satisfy the transport equations and the inner constraints induced on ${{\mathcal{N}}_p}$ by the conformal field equations. These fields are smooth at p in the sense that they coincide there at all orders with the fields which are obtained by restricting to ${{\mathcal{N}}_p}$ the functions considered in (i) and they are smooth on the smooth three-manifold ${{\mathcal{N}}_p {\setminus}\{p\}}$ in the standard sense.     

A Triviality Result in the AdS/CFT Correspondence for Euclidean Quantum Fields with Exponential Interaction

We consider scalar quantum fields with exponential interaction on Euclidean hyperbolic space ${\mathbb{H}^2}$ in two dimensions. Using decoupling inequalities for Neumann boundary conditions on a tessellation of ${\mathbb{H}^2}$ , we are able to show that the infra-red limit for the generating functional of the conformal boundary field becomes trivial.     

Contribution of vector resonances to the \bar{B}_{d}^{0}\to\bar {K}^{*0}\mu^{+}\mu^{-} decay

The fully differential angular distribution for the rare flavor-changing neutral current decay $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) \mu^{+}\mu^{-} $ is studied. The emphasis is placed on accurate treatment of the contribution from the processes $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) V $ with intermediate vector resonances V=??(770),??(782),?(1020),J/??,??(2S),?? decaying into the ?? ⁺ ?? ^? pair. The dilepton invariant-mass dependence of the branching ratio, longitudinal polarization fraction f _L of the $\bar{K}^{*0}$ meson, and forward?Cbackward asymmetry A _FB is calculated and compared with data from Belle, CDF and LHCb. It is shown that inclusion of the resonance contribution may considerably modify the branching ratio, calculated in the SM without resonances, even in the invariant-mass region far from the so-called charmonia cuts applied in the experimental analyses. This conclusion crucially depends on values of the unknown phases of the B ⁰??K ^?0 J/?? and B ⁰??K ^?0 ??(2S) decay amplitudes with zero helicity.     

Physics beyond the Standard Model through b → s μ ?+? μ ? transition

A comprehensive study of the impact of new-physics operators with different Lorentz structures on decays involving the b ?? s ?? ^?+? ?? ^? transition is performed. The effects of new vector?Caxial vector (VA), scalar?Cpseudoscalar (SP) and tensor (T) interactions on the differential branching ratios, forward?Cbackward asymmetries (A _FB??s), and direct CP asymmetries of ${\bar B}_{\rm s}^0 \to \mu^+ \mu^-$ , ${\bar B}_{\rm d}^0 \to$ $ X_{\rm s} \mu^+ \mu^-$ , ${\bar B}_{\rm s}^0 \to \mu^+ \mu^- \gamma$ , ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ , and ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ are examined. In ${\bar B}_{\rm d}^0\to {\bar{K}^*} \mu^+ \mu^-$ , we also explore the longitudinal polarization fraction f _L and the angular asymmetries $A_{\rm T}^{(2)}$ and A _LT, the direct CP asymmetries in them, as well as the triple-product CP asymmetries $A_{\rm T}^{\rm (im)}$ and $A^{\rm (im)}_{\rm LT}$ . While the new VA operators can significantly enhance most of the observables beyond the Standard Model predictions, the SP and T operators can do this only for A _FB in ${\bar B}_{\rm d}^0 \to {\bar K} \mu^+ \mu^-$ .     

New Optimal Asymmetric Quantum Codes Derived from Negacyclic Codes

The construction of quantum maximum-distance-separable (MDS) codes have been studied by many researchers for many years. Here, by using negacyclic codes, we construct two families of asymmetric quantum codes. The first family is the asymmetric quantum codes with parameters $[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$ , where 0≤t≤k≤(q?1)/2, $q \equiv1(\operatorname{mod} 4)$ , and k, t are positive integers. The second one is the asymmetric quantum codes with parameters $[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$ , where 1≤t≤k≤(q?1)/2, and k, t are positive integers. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.     

Galaxy cluster number count data constraints on cosmological parameters

We use data on massive galaxy clusters (M _cluster>8×10¹⁴ h ^?1 M _?? within a comoving radius of R _cluster=1.5h ^?1?Mpc) in the redshift range 0.05?z?0.83 to place constraints, simultaneously, on the nonrelativistic matter density parameter ?? _m , on the amplitude of mass fluctuations ?? ₈, on the index n of the power-law spectrum of the density perturbations, and on the Hubble constant H ₀, as well as on the equation-of-state parameters (w ₀,w _a ) of a smooth dark energy component. For the first time, we properly take into account the dependence on redshift and cosmology of the quantities related to cluster physics: the critical density contrast, the growth factor, the mass conversion factor, the virial overdensity, the virial radius and, most importantly, the cluster number count derived from the observational temperature data. We show that, contrary to previous analyses, cluster data alone prefer low values of the amplitude of mass fluctuations, ?? ₈??0.69 (1?? C.L.), and large amounts of nonrelativistic matter, ?? _m ??0.38 (1?? C.L.), in slight tension with the ??CDM concordance cosmological model, though the results are compatible with ??CDM at 2??. In addition, we derive a ?? ₈ normalization relation, $\sigma_{8} \varOmega_{m}^{1/3} = 0.49 \pm 0.06$ (2?? C.L.). Combining cluster data with ?? ₈-independent baryon acoustic oscillation observations, cosmic microwave background data, Hubble constant measurements, Hubble parameter determination from passively evolving red galaxies, and magnitude?Credshift data of type Ia supernovae, we find $\varOmega_{m} = 0.28^{+0.03}_{-0.02}$ and $\sigma_{8} = 0.73^{+0.03}_{-0.03}$ , the former in agreement and the latter being slightly lower than the corresponding values in the concordance cosmological model. We also find $H_{0} = 69.1^{+1.3}_{-1.5}~\mbox {km}/\mbox {s}/\mbox {Mpc}$ , the fit to the data being almost independent on n in the adopted range [0.90,1.05]. Concerning the dark energy equation-of-state parameters, we show that the present data on massive clusters weakly constrain (w ₀,w _a ) around the values corresponding to a cosmological constant, i.e. (w ₀,w _a )=(?1,0). The global analysis gives $w_{0} = -1.14^{+0.14}_{-0.16}$ and $w_{a} = 0.85^{+0.42}_{-0.60}$ (1?? C.L. errors). Very similar results are found in the case of time-evolving dark energy with a constant equation-of-state parameter w=const (the XCDM parametrization). Finally, we show that the impact of bounds on (w ₀,w _a ) is to favor top-down phantom models of evolving dark energy.