Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

Linear combinations x ₁ a ₁ ⋯ x _n a _n of n given natural numbers a _s (with nonnegative integral coefficients x _s ) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers a _s have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta ₁ ⋯ a _n depends peculiarly from the direction α of the vector a = σα. The article proves the lower bound of order and the upper bound of order σ ². Both orders are reached from some directions α. The averaging of N(a) along all directions, performed for σ = 7, 19, 41 and 97, provides the values, confirming the rate σ ^p for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large σ. One gets check p ≈ 1, 66 for σ between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures. ^★ Partially supported by RFBR grant 05-01-00104.     

Conservation laws for classical and relativistic collisions. II

On the basis of elementary symmetry arguments it is shown that (1) if in classical mechanics there exists a quantity λ+Σ_iμ_iυ_i+1/2νυ ² that is conserved, where λ,μ _i, andν are particle parameters, then theμ _i andν are all proportional to a single parameterμ and the quantityAμ+Σ_iB_iμυ_i+C(λ+ 1/2Dμυ ²), whereD ≡ν/μ, is conserved for all values ofA, B _i, andC; (2) if in relativistic mechanics there exists a quantity λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc[1−(υ ²/c ²)]^−1/2 that is conserved, then theμ _i andν are all proportional to a single parameterμ and the quantityAλ+Σ_iB_iμν_i[1−(υ ²/c ²)]^−1/2+Cμc [1−(υ ²/c ²)]^−1/2 is conserved for all values ofA, B _i, andC.     

Constant-cutoff approach to Λ (1405) resonance in the bound-state soliton model

We suggest a quantum stabilization method for theSU(2) σ-model, based on the constant-cutoff limit of the cutoff quantization method developed by Balakrishnaet al., which avoids the difficulties with the usual soliton boundary conditions pointed out by Iwasaki and Ohyama. We investigate the baryon numberB = 1 sector of the model and show that after the collective coordinate quantization it admits a stable soliton solution which depends on a single dimensional arbitrary constant. We then study strong and electromagnetic properties of the Λ(1405) hyperon in the bound-state approach to theSU(3)-soliton model for the hyperons, withSU(3)-symmetry breaking. We calculate the strong coupling constantg _Λ*NK; , the magnetic moment of Λ*, the mean square radii, and the radiative decay amplitudes. Finally we compare the present results with those obtained using other models and with the available empirical data. We show that there is a general qualitative agreement between our results and the results of other models and available empirical data, except for the Λ*πΣ coupling, which, as in the case of the complete Skyrme model, vanishes in the second-order approximation of the kaon fluctuations used in this work.     

New data for the 197Au(γ, nX) and 197Au(γ, 2nX) reaction cross sections

The results from experimental and theoretical studies of the total and partial cross sections of photoneutron reactions on the ¹⁹⁷Au isotope were analyzed. The cross sections for reactions σ(γ, nX) = σ(γ, n) + σ(γ, np) + … + σ(γ, 2nX) = σ(γ, 2n) + σ(γ, 2np) + … were evaluated in the energy range 7 ≤ E _γ ≤ 30 MeV using an approach free of the shortcomings of experimental photoneutron multiplicity sorting methods. The total photoneutron reaction cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) + … = σ^exp(γ, n) + σ^exp(γ, np) + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + … were used as the initial experimental data. The contributions from the cross sections σ(γ, nX) and σ(γ, 2nX) to the cross sections σ^exp(γ, xn) were separated using the multiplicity transition functions F ₁ ^theor = σ^theor(γ, 1nX)/σ^theor(γ, xn) and F ₂ ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn), calculated within an updated version of the pre-equilibrium model of photonuclear reactions. New evaluated data for both partial reaction cross sections, i.e., σ^eval (γ, 1nX) = F ₁ ^theorσ^exp(γ, xn) and σ^eval(γ, 2nX) = F ₂ ^theorσ^exp(γ, xn), were obtained. The cross sections σeval(γ, nX) and σ^eval.(γ, 2nX) evaluated using the theoretically calculated functions F _1,2^theor are consistent with the Livermore data, but substantially contradict the Saclay data.     

New cw FIR laser lines from optically pumped ammonia analogues

Two new cw submillimetre lines optically pumped by the 9 μR(30) CO₂ laser have been observed for¹⁴NH₃ and assigned to thea(6,0)→s(5,0) transition and to its cascade transitions(5,0)→a(4,0) in thev ₂=1 state. In addition, six new emission lines have been observed for NH₂D, and one tentatively assigned to the (11,1)→(10,1) transition of ND₃. Methylamine has also been investigated and twenty-eight new emission lines have been discovered.     

Saturable absorption dynamics of mode-locking dye DODCI

0 –S₁–S_m–S_n multilevel system. An S_m→S₁ absorption recovery time of τ^S1 _A=(600±100) fs is determined. The picosecond and femtosecond pulse excitation leads to S₀→S₁ ground-state absorption bleaching and S₁→S_m first excited-state absorption bleaching. The excited-state absorption cross-sections σ^S1 _ex(S₁→S_m) and σ^Sm _ex(S_m→S_n) are determined. Received: 3 June 1996     

Complex Harmonic-Oscillator Basis for the Relativistic Three-Body Problem

A complex harmonic-oscillator basis is employed for the three-body problem obeying S ₃-symmetry. Unlike a real basis it generates an additional quantum number (N _a ), in addition to the standard principal quantum number (N), and thus facilitates a more quantitative S ₃-classification of the various states than is usually possible. It is shown that certain bilinear forms with definite S ₃-symmetry properties, which can be constructed out of the linear harmonic-oscillator operators (a, a ^†) satisfy several uncoupled sets of SO(2, 1) algebras with spectra bounded from below. It is also briefly indicated how this S ₃-formalism can be adapted to the core structure of a more general relativistic three-particle system with unequal-mass kinematics through an appropriate choice of internal variables. Received May 11, 1994; revised November 3, 1994; accepted for publication November 23, 1994     

Perturbations in the rotational structure of the 4<Emphasis Type="Italic">p</Emphasis> complex of H<Subscript>2</Subscript> molecule terms

A statistical analysis of all the available data on the wave numbers of spectral lines related to triplet-triplet electronic-vibrational-rotational (rovibronic) radiation transitions into the H₂ molecule (1sσ2sσ) a ³Σ _g ⁺ electronic state was performed for the first time. This allowed us to check and refine the controversial identification of several spectral lines. Optimum rovibronic term values were found for 15 electronic states, including the (4pσ)f ³Σ _u ⁺, (4pπ)k ³Π _u ⁺, and (4pπ)k ³Π _u ⁻ states studied in this work. The ratios between the oscillator strengths of R- and P-branch lines with common upper levels (branching coefficients) for the f ³Σ _u ⁺ → a ³Σ _g ⁺ and k ³Π _u ⁺ → a ³Σ _g ⁺ systems of H₂ molecule bands were measured for the first time. Substantial deviations of the measured branching coefficients from the corresponding ratios between the Henl-London factors were observed. The deviations monotonically increased as the rotational quantum number N grew, which, in combination with substantial Λ-doubling in the k ³Π _u state, was evidence of an important role played by electronicrotational interaction in the 4pσ³Σ _u ⁺ and 4pπ³Π _u ⁺ adiabatic electronic states. A strong correlation was observed between the N dependences of branching coefficients for transitions from the mutually perturbed f ³Σ _u ⁺ and k ³Π _u ⁺ electronic states. The results of this work show that the measured branching coefficients are a much more sensitive and capacious channel of information about perturbation effects than rovibronic term values.     

Kinetics of formation of O2(1Σ) molecules in reactions involving excited O2(1Δ) oxygen molecules and I(2P1/2) iodine atoms

The results of our experimental study of the kinetics of formation of O₂(¹Σ) molecules in energy-exchange reactions O₂(¹Δ) + I(⁵ p,² P _1/2) and O₂(a,¹Δ) + O₂(a,¹Δ) are presented. The ratio of rate constants was obtained for these reactions (4800 ± 300). Setting the rate constant of the deactivation of O₂(¹Σ) molecules on CO₂ molecules at 4.1 · 10^–13 cm³/s, we evaluated the rate constants for these reactions at a temperature of approximately 330 K: (1.7 ± 0.2) · 10⁻¹³ and (3.6 ± 0.5) · 10⁻¹⁷ cm³/s, respectively.     

Observation of a resonance in the KSp decay channel at a mass of 1765?MeV/c2

We report on the observation of a K_Sp resonance signal at a mass of 1765 ± 5 MeV/c ², with intrinsic width Γ = 108 ± 22 MeV/c ², produced inclusively in Σ^--nucleus interactions at 340 GeV/c in the hyperon beam experiment WA89 at CERN. The signal was observed in the kinematic region x_F>0.7, in this region its production cross section rises approximately linearly with (1-x_F), reaching BR(X→K_Sp)dσ/dx_F=(5.2±2.3)μb per nucleon at x_F=0.8. The hard x_F spectrum suggests the presence of a strong leading particle effect in the production and hence the identification as a Σ^*+ state. No corresponding peaks were observed in the K^-p and Λπ^± mass spectra. PACS 25.80.-e; 13.60.Rj; 14.20.Jn     

Evaluated cross sections of the σ(γ, nX) and σ(γ, 2nX) reactions on 112, 114, 116, 117, 118, 119, 120, 122, 124Sn isotopes

A combined analysis of experimental data on total and partial photoneutron reaction cross sections, obtained using bremsstrahlung γ-radiation and quasi-monoenergetic annihilation photon beams, was performed for nine Sn isotopes. The partial reactions σ^eval(γ, nX) and σ^eval(γ, 2nX) cross sections were evaluated using an approach free of the shortcomings of experimental neutron multiplicity sorting methods. This approach involves calculations within the photonuclear reaction model, based on Fermi gas densities and considering the effects of nucleus deformation, the isospin splitting of its giant dipole resonance (GDR), and experimental data on the total photoneutron cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) = σ^exp(γ, n) + σ^exp(γ, np) + … + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + …. The evaluated σ^eval(γ, nX) and σ^eval(γ, 2nX) reactions cross sections were obtained using the introduced transition multiplicity functions F ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn) = σ^theor(γ, 2nX)/[σ^theor(γ, nX) + 2σ^theor(γ, 2nX) + …]; and σ^eval.(γ, 2nX) = F ^theor.σ^exp.(γ, xn) = σ^eval(γ, nX) = (1 − 2F ^theor)σ^exp(γ, xn). The evaluated partial reaction cross sections were used to assess the total photoneutron reaction cross sections σ^eval(γ, sn) = σ^eval(γ, nX) + σ^eval(γ, 2nX) + … as functions of the mass number A. The GDR features of ^{112, 114, 116, 117, 118, 119, 120, 122, 124}Sn isotopes were studied and are discussed here.     

Tails in harnesses

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa _s ^t , wheres ∈ Z ^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z _d ⁺ define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a _s ^t =a _s ^t . Then by induction, wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a _s ^t diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a _s ^t diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a _s ^t , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a _s ^t converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima _s ^t )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a _s ^t )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a _s ^t )=E-decay(ν).     

Conservation laws for classical and relativistic collisions. I

On the basis of simple kinematic arguments it is shown that any quantity, depending only on the nature and velocity of a particle, that is conserved in a collision must, in classical mechanics, be of the form λ+Σ_iμ_iυ_i+1/2vυ ² or in relativistic mechanics of the form λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc [1−(υ ²/c ²)]^−1/2 where λ,μ _i, andν are particle parameters.     

Total scattering cross-section and extraction of low energy parameters of Σ±p scattering

The lowest momentum at which the total scattering cross-section data are available for Σ⁺ p and Σ⁻ p scattering is 145 MeV/c and 142.5 MeV/c respectively. Thus extracting low energy parameters amounts to extrapolating the data to still lower energies. Using the analytic structure of foward scattering amplitude to advantage a parameterization of theσ _T is presented which is hoped to be more reliable and stable for deriving results through extrapolation. The scattering lengths and effective ranges for the Σ⁺ p and Σ⁻ p are also estimated.     

Temperature dependences of the high-field magnetization of dilute frustrated ferrimagnetic spinels

Results are presented of an investigation of the magnetic properties of dilute frustrated ferrimagnetic spinels Li_0.5Fe_2.5−x Ga_xO₄ (x=0.8–1.2), which characterize the main parameters of the ferrimagnetic state and provide evidence of local violation of collinear spin ordering and frustrations. In particular, measurements were made of the concentration dependences of the magnetic moment n ₀(x) and the Curie point T _c (x), the magnetization isotherms σ _T (H) at T=4.2 K and H⩽10 kOe, and also the low-and high-field magnetization polytherms σ _H (T). It was established that for x⩾0.8 in fields exceeding the technical saturation field H _s ∼2 kOe, the temperature dependences of the high-field magnetization σ _H (T) between 4.2 and 230 K cannot be described by the Bloch T ^3/2 law whereas this law is satisfied for undiluted Li spinel (x=0). Over the entire temperature range (4.2–230 K) the experimental curves σ _H (T) may be approximated by σ _H (T)=σ ₀(1−AT ^3/2 − BT ^5/2) for x=0.8–1.0 and σ _H (T)=σ ₀[1−CT ^3/2exp(μ(H−H ₀)/k _B T)] for x=1.1, 1.2, where μH ₀∼15 K is the internal field produced by competition between exchange interactions and frustrations. Fiz. Tverd. Tela (St. Petersburg) 40, 1075–1079 (June 1998)     

Bound states of the muon-antimuon system: Lifetimes and hyperfine splitting

We examine the properties of an atomic system consisting of a muon and antimuon. Expressions are derived for the probability of decay and the hyperfine splitting of the lower levels with allowance for the leading radiative corrections, which are of relative order α. The results for the lifetimes and the ground-state energy are τ(1³ S ₁)=1.7907(8)×10⁻¹² s, τ(1¹ S ₀)=0.59547(33)×10⁻¹² s, and E _hfs (1s)=4.23284(35)×10⁷ MHz. The relative probabilities for the various decay channels are calculated; in particular, for the 1 ³ S ₁ level it is found that Γ (μμ→eeγ)/Γ(μμ→ee)≈15%. Finally, possible applications are discussed. Zh. éksp. Teor. Fiz. 113, 409–431 (February 1998)     

Electron-vibrational energy exchange in collisions of Ar atoms in the 3P2 metastable state with N2(X1S g+ ) molecules

Rate constants for electron-vibrational energy exchange Ar(³ P ₂) + N₂(X ¹Σ _g ⁺, ν = 0) → Ar(¹ S ₀) + N₂(C ³Π _u , ν′), where ν′ = 0, 1, 2, were calculated. Calculations were performed taking into account the presence of a resonance in electron scattering by N₂(X ¹Σ _g ⁺). As a result, the interaction of Ar(³ P ₂) with N₂(X ¹Σ _g ⁺, ν = 0) was characterized by attraction and, in the end, intersection of electron-vibrational potential surfaces correlating with Ar(³ P ₂) + N₂(X ¹Σ _g ⁺, ν = 0) and Ar(¹ S ₀) + N₂(C ³Π _u , ν′) at interparticle distances of 2.5–3.5 ?. Exchange interaction at which electron-vibrational transitions in the region of intersection of electron-vibrational transitions in the region of intersection of electron-vibrational potential surfaces accompanied by spin exchange were induced was calculated by the asymptotic method. The rate constants determined at 300–600 K were on the order of 10⁻¹¹−10⁻¹² cm³/s and weakly increased as the temperature grew. Mainly the C ³Π _u , ν′ = 0 state of the N₂ molecule was populated. The calculation results were in satisfactory agreement with the experimental data obtained at 300 K.     

Hyperfine interactions in molecules through laser spectroscopy

The infrared and millimeter wave spectroscopy, laser Stark spectroscopy, and beam maser spectroscopy of CH₃CN and its isotopic species will be discussed. The beam maser spectroscopy and hyperfine structure of molecules like NCCCD, ClCCD and CH₃CCH are reviewed. The laser magnetic resonance and hyperfine structure in CF, CH and CH₂ free radicals will be discussed. The Lamb dip spectroscopy and laser-induced fluorescence in I₂ involving theB ³Π(0 _u ⁺ ) state are reviewed with special reference to its hyperfine structure. The splitting of the rotational levels of N ₂ ⁺ in itsX ²Σ _g ⁺ andB ²Σ _u ⁺ states due to hyperfine interactions (along with the hyperfine structure) in laser-induced fluorescence in theB−X transition is discussed. Recent results obtained in the laser photo-acoustic spectrum of ICl in the transitionX ¹Σ⁺ −A ³Π₁ will be presented and the possibility of the use of this technique in studying the hyperfine structure will be discussed.     

Spin exchange during collisions of potassium atoms: Complex cross sections

Singlet (X ¹Σ⁺) and triplet (a ³Σ⁺) potentials of interaction of two potassium atoms residing in the ground state (4s ² S _1/2) are presented. Based on the given interaction potentials, the complex cross sections of spin exchange q = [`(q)] + i[`([`(q)])]q = \bar q + i\overline{\overline q} for the system under investigation are calculated. Obtained dependences of the real and imaginary parts of the spin-exchange cross section on temperature allow one to obtain information both on the broadening of a magnetic resonance line of K atoms and on the frequency shift of the magnetic resonance during collision.     

Masses of light tetraquarks and scalar mesons in the relativistic quark model

Masses of the ground-state light tetraquarks are dynamically calculated in the framework of the relativistic diquark–antidiquark picture. The internal structure of the diquark is taken into account by calculating the form factor of the diquark–gluon interaction in terms of the overlap integral of the diquark wave functions. It is found that scalar mesons with masses below 1 GeV, f ₀(600) (σ), K ₀^*(800) (κ), f ₀(980) and a ₀(980), agree well with the light-tetraquark interpretation.