Trajectories in Random Monads

Let us have a non-empty finite set S with n>1 elements which we call points and a map M:S→S. After V.I. Arnold, we call such pairs (S, M) monads, but we consider random monads in which all the values of M(?) are random, independent and uniformly distributed in S. We fix some ⊙∈S and consider the infinite sequence M ^t (⊙), t=0,1,2,…?. A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis, Rec, Tra the numbers of visited, recurrent and transient points respectively and study their distributions. The distributions of Vis, Rec, Tra are unimodal. The modes of Rec and Tra equal their minimal values, that is 1 and 0 respectively. The mode of Vis is approximated by \(\sqrt{n}\), plus-minus a constant. The mathematical expectations: \(\mathbb{E}(\mathit{Vis})\) is approximated by \(2 \sqrt{\pi\, n/8}\) plus-minus a constant; \(\mathbb{E}(\mathit{Rec})\) and \(\mathbb{E}(\mathit{Tra})\) are approximated by \(\sqrt{\pi\, n/8}\) plus-minus a constant. For the standard deviations σ(Vis) and σ(Rec)=σ(Tra) respectively we present the approximations

$\sqrt{\frac{4-\pi}{2} \cdot n} \quad\mbox{and}\quad \sqrt{\frac{16-3\pi}{24} \cdot n},$

from which they also deviate at most by a constant. We prove that when n tends to infinity, the correlations Corr(Rec,Tra) and Corr(Rec,Vis)=Corr(Tra,Vis) converge to

$\frac{8-3\pi}{16-3\pi}\quad \mbox{and}\quad \sqrt{\frac{12-3\pi}{16-3\pi}}.$

     

Phase Transitions in the Dynamics of Slow Random Monads

Let us have a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B. Following Vladimir Arnold, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter in [0,1].     

Bottomonium γ(5<Emphasis Type="Italic">S</Emphasis>) decays into <Emphasis Type="Italic">BB</Emphasis> and <Emphasis Type="Italic">BB</Emphasis>π

Two-and three-body decays of γ(5S) into BB, BB*, B*B*, B _s B _s , B _s B _s ^*, and BB*π, B*B*π are evaluated using the theory developed earlier for dipion-bottomonium transitions. The theory contains only two parameters—vertex masses M _br and M _ω—known from the dipion spectra and width. Predicted values of Γ_tot(5S) and six partial widths Γ _k (5S), k = BB, BB*, ... are in agreement with the experiment. The decay widths Γ_5S (πBB*) and Γ_5S (πB*B*) are also calculated and found to be on the order of 10 keV. The text was submitted by the authors in English.     

Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries L _kk  = k ², and the matrix B is off-diagonal, with nonzero entries B _k,k+1 = B _k+1,k  = k ^α , 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the nth eigenvalue E _n (z), E _n (0) = n ², is a well-defined analytic function. Let R _n be the convergence radius of its Taylor’s series about z = 0. It is proved that

Comment: “On the computation of molecular auxiliary functions<Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>and<Emphasis Type="Italic">B</Emphasis><Subscript>n</Subscript>”

Guseinov, Mamedov, Kara and Orbay (Pramana - J. Phys. 56, 691 (2001)) propose methods for evaluating the molecular auxiliary functionsA _n(p) andB _n(pt) for the range 17 ≤n ≤ 60 and 25 ≤pt ≤ 60. However, their procedure forA _n(p) is not new, and that forB _n(pt) is less efficient for their target range than another well-known method. Their approach does have merit for smaller non-zero values ofpt. Two minor errors in table 1 of their paper are also identified.     

Sum-of-Squares Results for Polynomials Related to the Bessis–Moussa–Villani Conjecture

We show that the polynomial S _m,k (A,B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R〈X,Y〉, where X ²=A and Y ²=B, for all even values of m and k with 6≤k≤m−10, and also for (m,k)=(12,6). This leaves only the case (m,k)=(16,8) open. This topic is of interest in connection with the Lieb–Seiringer formulation of the Bessis–Moussa–Villani conjecture, which asks whether Tr (S _m,k (A,B))≥0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using “descent + sum-of-squares” to prove the BMV conjecture.     

Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φ _jk (M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φ _jk (M) is O(1/n), and the CLT is valid for . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.     

The hyperfine splittings in heavy-light mesons and quarkonia

Hyperfine splittings (HFS) are calculated within the Field Correlator Method, taking into account relativistic corrections. The HFS in bottomonium and the B _q (q = n, s) mesons are shown to be in full agreement with experiment if a universal coupling α _HF = 0.310 is taken in perturbative spinspin potential. It gives M(B*) −M(B) = 45.7(3) MeV, M(B _s ^* ) − M(B _s ) = 46.7(3) MeV (n _f = 4), while in bottomonium Δ_HF(b $ \bar b $ \bar b ) = M(Υ(9460)) − M(η _b (1S)) = 63.4 MeV for n _f = 4 and 71.1 MeV for n _f = 5 are obtained; just the latter agrees with recent BaBar data. For unobserved excited states we predict M(Υ(2S))−M(η _b (2S)) = 36(2)MeV,M(Υ(3S))−M(η _b (3S)) = 28(2)MeV, and also M(B _c ^*) = 6334(4) MeV, M(B _c (2S)) = 6868(4) MeV, M(B _c ^* (2S)) = 6905(4) MeV. The mass splittings between D(2³ S ₁) − D(2¹ S ₀), D _s (2³ S ₁) − D _s (2¹ S ₀) are predicted to be ∼75 MeV, which are significantly smaller than in several other studies but agree with the mass splitting between recently observed D(2533) and D*(2610).     

A Derivative Formula for the Free Energy Function

We consider bond percolation on the Z ^d lattice. Let M _n be the number of open clusters in B(n)=[−n,n] ^d . It is well known that E _p M _n /(2n+1) ^d converges to the free energy function κ(p) at the zero field. In this paper, we show that s²_p(M_n)/(2n+1)^d\sigma^{2}_{p}(M_{n})/(2n+1)^{d} converges to −p(1−p)κ′(p).     

D-dimensional massless particle with extended gauge invariance

We propose the model ofD-dimensional massless particle whose Lagrangian is given by theN-th extrinsic curvature of world-line. The system hasN+1 gauge degrees of freedom constitutingW-like algebra; the classical trajectories of the model are space-like curves which obey the conditionsk _N+a=k_N−a, k_2N =0,a=1, ...,N−1,N≤[(D−2)/2], while the firstN curvaturesk _i remain arbitrary. We show that the model admits consistent formulation on the anti-DeSitter space. The solutions of the system are the massless irreducible representations of Poincaré group withN nonzero helicities, which are equal to each other. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice

A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ ₁=Q,l=1,2,..., on the Sierpiski-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM _k-spin correlation functions, whereM _k=3 ^k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM _k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM _k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.     

Integrability ofSU(N) spin chains with elliptic exchange

For some one-parameter setH _N of linear combinations ofN(N−1)/2 elementary transpositions {P _jk} (1≤j<k≤N) at arbitrary naturalN≥3 one can construct a variety {I _m} (3≤m≤N) of operators which commute withH _N. Being applied toSU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

On the computation of molecular auxiliary functions A n and B n

Molecular auxiliary functions A _n (p) and B _n (pt), arising in the Hartree-Fock-Roothaan (HFR) approximation for molecules, Ewald’s crystal lattice theory, electromagnetic stopping theory, and other approximate methods, are evaluated and analysed in the range of 17≤n≤60 and 25≤pt≤60.     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

The 2009 world average of <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Bold">s</Emphasis></Subscript>

Measurements of α _s, the coupling strength of the Strong Interaction between quarks and gluons, are summarised and an updated value of the world average of a_s(M_Z⁰)\alpha_{\mathrm{s}}(M_{\mathrm{Z}^{0}}) is derived. Special emphasis is laid on the most recent determinations of α _s. These are obtained from τ-decays, from global fits of electroweak precision data and from measurements of the proton structure function F₂, which are based on perturbative QCD calculations up to O(a_s⁴)\mathcal{O}(\alpha_{\mathrm{s}}^{4}); from hadronic event shapes and jet production in e⁺e⁻ annihilation, based on O(a_s³)\mathcal{O}(\alpha_{\mathrm{s}}^{3}) QCD; from jet production in deep inelastic scattering and from ϒ decays, based on O(a_s²)\mathcal{O}(\alpha_{\mathrm{s}}^{2}) QCD; and from heavy quarkonia based on unquenched QCD lattice calculations. A pragmatic method is chosen to obtain the world average and an estimate of its overall uncertainty, resulting in

Full chains of twists for orthogonal algebras

We show that for some Hopf subalgebras inU _F(so(M)) nontrivially deformed by a twistF it is possible to find the nonlinear primitive copies. This enlarges the possibilities to construct chains of twists. For orthogonal algebraU(so(M)) we present a method to compose the full chains with carrier space as large as the Borel subalgebraB(so(M)). These chains can be used to construct the new deformed Yangians. Presented at the 9-th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. This work was partially supported by the Russian Foundation for Basic Research under the grant 00-01-00500 (VDL) and 98-01-00310 (PPK).     

A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation

In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces \mathringB^1-a_￥,q{\mathring{B}^{1-\alpha}_{\infty,q}}, and global well-posedness of the critical quasi-geostrophic equation in \mathringB⁰_￥,q{\mathring{B}^{0}_{\infty,q}} for all 1 ≤ q < ∞. Here \mathringB^s_￥,q {\mathring{B}^{s}_{\infty,q} } is the closure of the Schwartz functions in the norm of B^s_￥,q{B^{s}_{\infty,q}}.     

Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property

This paper continues the analysis of the quantum states introduced in previous works and determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state λ _M , of a massless conformally coupled scalar field propagating in M. λ _M is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in particular whenever the spacetime M and the associated unphysical one, M͂, are globally hyperbolic and M admits future time infinity i ⁺. λ _M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state λ _M enjoys the following further remarkable properties:

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.