Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials

For an infinitely renormalizable quadratic map \({f_c: z\mapsto z^2+c}\) with the sequence of renormalization periods {k _m } and rotation numbers {t _m  = p _m /q _m }, we prove that if \({\limsup k_m^{-1} \log |p_m| >0 }\), then the Mandelbrot set is locally connected at c. We prove also that if \({\limsup |t_{m+1}|^{1/q_m} <1 }\) and q _m → ∞, then the Julia set of f _c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.     

Signal velocity in oscillator arrays

We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. After describing necessary and sufficient conditions for asymptotic stability, we derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocitiesc₊ > 0 and c_? < 0 such that low frequency disturbances travel through the flock as f₊(x ? c₊t) in the direction of increasing agent numbers and f_?(x ? c_?t) in the other.     

Fermi Surface Topology in the Case of Spontaneously Broken Rotational Symmetry

The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f(p, p') having a sharp peak at |p ? p'| = q₀. It is shown that, in the case of attraction and q₀ = 2p_F the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q₀, the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ? π(p_F/q₀ ? 1) small hole pockets at the distance p_F ? q₀ from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L. At q₀ → 0, when the model under study transforms to the two-dimensional Nozières model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.     

A model of classical thermodynamics based on the partition theory of integers,Earth gravitation,and semiclassical asymptotics. I

In the paper, a new construction of the theory of partitions of integers is proposed. The author defines entropy as the natural logarithm of the number of partitions of a number M into natural summands with repetitions allowed p(M) and repetitions forbidden q(M). The passage from ln p(M) to lnq(M) through the mesoscopic values M → 0 is studied. The topological transition from the mesoscopic lower levels of the Bohr–Kalckar construction to the macroscopic levels corresponding to the critical number of neutrons according to the consequence of Einstein’s inequality M ≤ c N _c , where c is determined for the particles of the given atomic nucleus. The role of quantum mechanics in establishing the new world outlook in physics is analyzed. It is pointed out that the main equations of thermodynamics in the volume “Statistical Physics” of the Landau–Lifshits treatise are obtained without appealing to the so-called “three main principles of thermodynamics”. It is also pointed out that Niels Bohr’s liquid model of the nucleus does not involve any interaction of particles in the form of attraction and is based on the presence of a common potential trough for all elements of the nucleus. The author constructs a new approach to thermodynamics, using quantum mechanics and the Earth’s gravitational attraction as a common potential trough.     

Noncompact <Emphasis Type="Italic">u</Emphasis><Subscript>q</Subscript>(2, 1) quantum algebra: Discrete series of highest weight irreducible representations

The structure of unitary irreducible representations of the noncompact u_q(2, 1) quantum algebra that are related to a negative discrete series is examined. With the aid of projection operators for the su_q(2) subalgebra, a q analog of the Gelfand-Graev formulas is derived in the basis corresponding to the reduction u_q(2, 1) → su_q(2)×u(1). Projection operators for the su_q(1, 1) subalgebra are employed to study the same representations for the reduction u_q(2, 1) → u(1)×su_q(1, 1). The matrix elements of the generators of the u_q(2, 1) algebra are computed in this new basis. A general analytic expression for an element of the transformation brackets <U∣T>_q between the bases associated with the above two reductions (the elements of this matrix are referred to as q Weyl coefficients) is obtained for a general case where the deformation parameter q is not equal to a root of unity. It is shown explicitly that, apart from a phase, the q Weyl coefficients coincide with the q Racah coefficients for the su_q(2) quantum algebra.     

Using a chiral anomaly to determine the number of colors

The N_c dependence of PPPγ vertices, where P is a pseudoscalar meson and N_c is the number of colors, is analyzed with allowance for the N_c dependence of the quark charges. It is shown that the reactions Kγ → Kπ and π^±γ → π^±η and the decay η → π⁺π^?γ are the best processes for determining N_c. The cross section σ(π^?γ → π^?η) as measured by using the VES facility at IHEP agrees with the value of N_c=3.     

Structure,ionic conduction,and phase transformations in lithium titanate Li<Subscript>4</Subscript>Ti<Subscript>5</Subscript>O<Subscript>12</Subscript>

The spinel structure of lithium titanate Li₄Ti₅O₁₂ is refined by the Rietveld full-profile analysis with the use of x-ray and neutron powder diffraction data. The distribution and coordinates of atoms are determined. The Li₄Ti₅O₁₂ compound is studied at high temperatures by differential scanning calorimetry and Raman spectroscopy. The electrical conductivity is measured in the high-temperature range. It is shown that the Li₄Ti₅O₁₂ compound with a spinel structure undergoes two successive order-disorder phase transitions due to different distributions of lithium atoms and cation vacancies (□, V) in a defect structure of the NaCl type: (Li)_8a[Li_0.33Ti_1.67]_16dO₄ → [Li□]_16c[Li_1.33Ti_1.67]_16dO₄ → [Li_1.33□_0.67]_16c[Ti_1.67□_0.33]_16dO₄. The low-temperature diffusion of lithium predominantly occurs either through the mechanism ... → Li(8a) → V(16c) → V(8a) → ... in the spinel phase or through the mechanism ... → Li(16c) → V(8a) → V(16c) → ... in an intermediate phase. In the high-temperature phase, the lithium cations also migrate over 48f vacancies: ... Li(16c) → V(8a, 48f) → V(16c) → ....     

Fundamental Entangling Operators in Quantum Mechanics and Their Properties

For the first time, we introduce so-called fundamental entangling operators \(e^{iQ_{1} P_{2}}\) and \(e^{iP_{1} Q_{2} }\) for composing bipartite entangled states of continuum variables, where Q_i and P_i (i = 1, 2) are coordinate and momentum operator, respectively. We then analyze how these entangling operators naturally appear in the quantum image of classical quadratic coordinate transformation (q₁, q₂) → (Aq₁ + Bq₂, Cq₁ + Dq₂), where AD?BC = 1, which means even the basic coordinate transformation (Q₁, Q₂) → (AQ₁ + BQ₂, CQ₁ + DQ₂) involves entangling mechanism. We also analyse their Lie algebraic properties and use the integration technique within an ordered product of operators to show they are also one- and two- mode combinatorial squeezing operators.     

Investigation of tunnel-diode oscillators activated on the internal capacitance of their coils

Comparison of two tunnel-diode oscillators is performed, one of them having a solenoid pick-up coil, while another one has a single-layer flat-geometry coil. Oscillators operate in the frequency ranges, respectively, 6–25 MHz and 8–24 MHz and the coils had approximately equal inductances. These oscillators have been activated both without and with an external capacitance C ₀ in their resonant circuit, i.e., the studied oscillators could be activated also with their internal (own) capacitances C _c . It was succeeded to obtain more stable oscillations in case of flat coils. The analysis of data indicates that this fact is a consequence of a relatively high value of the own capacitance C _c of the flat coils as compared to their parasitic capacitance C _par with respect to the environment. This causes the advantages of flat-coil based oscillators. Also the inductance and the quantity C _c + C _par have been determined for both coils from the data of measurements.     

Corona effect in <Emphasis Type="Italic">AA</Emphasis> collisions at the LHC

Following our earlier finding based on RHIC data on the dominant jet production from nucleus corona region, we reconsider this effect in nucleus–nucleus collisions at the LHC energies. Our hypothesis was based on experimental data, which raised the idea of a finite formation time for the produced medium. At the RHIC energy and in low-density corona region, this time reaches about 2 fm/c. Following this hypothesis, the nuclear modification factor R _AA at high p _t should be independent on particle momentum, and the azimuthal anisotropy of high p _t particles, v ₂, should be finite. A separate prediction held that, at the LHC energy, the formation time in the corona region should be about 1 fm/c. New LHC data show that R _AA is not flat and is rising with p _t . We add to our original hypothesis an assumption that a fast parton traversing the produced medium loses the fixed portion of its energy. A shift of about 7 GeV from the original power law p ^?6 production cross section in pp explains well all the observed R _AA dependencies. The shift of about 7 GeV is also valid at the RHIC energy. We also show that the observed at the LHC dependence of v ₂ at high p _t and our previous predictions agree.     

On the possibility of revealing the transition of a baryon pair state to a six-quark confinement state

Proton-proton collisions are considered to find favourable conditions for searching for the transition of a baryon pair state to a hexa-quark confinement state (3q) + (3q) → (6q)_cnf. It is admitted that central pp collisions in a definite range of the initial energy can lead to creation of an intermediate compound system where the hexa-quark dibaryon can be formed. Criteria for selection of central collision events and for manifestation of the quark-structure dibaryon production are proposed.     

Dominant decays of scalar leptoquarks and scalar gluons in the minimal four-color symmetry model

Fermionic and weak decays of the scalar leptoquarks S = S ₁ ⁽⁺⁾ , S ₁ ^(?) , and S _m and the scalar gluons F = F ₁ and F ₂ predicted by the minimal model involving four-color symmetry and the Higgs mechanism of quark-and lepton-mass splitting are considered. The widths and the branching ratios are calculated for these decays, and the results are analyzed versus the couplings and masses of decaying particles. It is shown that, at relatively small mass splittings Δm within scalar doublets (Δm < m _W), the fermionic decays S ₁ ⁽⁺⁾ → tl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → t \(\tilde \nu \) _j, F ₁ → t \(\tilde b\), and F ₂ → t \(\tilde t\), which are characterized by few-GeV widths for m _S, m _F < 1 TeV and decay branching ratios close to unity, are dominant, but that, for Δm > m _W, the weak decays S → S′W and F → F’W compete with the above fermionic decays. In the case of m _S < m _t, the processes S ₁ ⁽⁺⁾ → cl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → bl _j ⁺ , and S _m → c \(\tilde \nu \) _j, whose total branching ratios are Br(S ₁ ⁽⁺⁾ → cl ⁺) ≈ Br(S ₁ ^(?) → v \(\tilde b\)) ≈ 1, Br(S _m → bl ⁺) ≈ 0.9, and Br(S _m → c \(\tilde \nu \)) ≈ 0.1, appear to be dominant decays of scalar leptoquarks. Searches for these decays at LHC and the Tevatron are of interest.     

An Analysis of Charged Anisotropic Star with de Sitter Spacetime

We propose a model for charged anisotropic star in de Sitter spacetime. We have taken Krori and Barua (J. Phys. A, Math. Gen. 8, 508, 1975) metric in de Sitter spacetime with non-zero cosmological constant. The model is free from singularity. We incorporate the existence of the cosmological constant on a small scale to study the structure of anisotropic charged star. To solve the Einstein-Maxwell field equations we assume the relation between the radial and transverse pressure as p _t ?p _r =g q(r)² r ² (where g is a non-zero positive constant). The physical conditions inside the stellar model are also discussed.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

Inclusive neutral-pion production in <Emphasis Type="Italic">d</Emphasis>C and <Emphasis Type="Italic">d</Emphasis>Cu interactions at a momentum of 4.5 GeV/. per nucleon

The cross sections for inclusive neutral-pion production in the reactions d + C → π⁰ + x and d + Cu → π⁰ + x at a momentum of 4.5 GeV/c per nucleon were measured over the kinematical region specified by the inequalities Θ_π≤16° and E_π≥2 GeV (in the laboratory frame). From the ratio of the cross sections for neutral-pion generation on carbon and copper nuclei, the exponent n in the parametrization Ed³σ/d³p ～ A _T ⁿ is obtained as a function of the cumulative number X in the range 0.6 ≤ X ≤ 1.8 and as a function of the square of the transverse momentum in the range 0.04 ≤ P _t ² ≤ 0.40 (GeV/c)². The probabilities of the formation of six-quark configurations in the D, ⁴He, and ¹²C nuclei are estimated. The double-differential cross section for the reaction d + C → π⁰ + x is determined for the first time by using a data sample containing more than 40 000 neutral pions.     

Inclusive Production Λc in the e+e− - Annihilation

In the framework of the quark-diquark model of baryons, Λ_c can be considered as constituent c quark and ud diquark. In hadronizasion process baryon Λ_c can be produced from ud scalar diquark and c quark fragmentation. So in this work, in the lowest order of perturbative QCD, fragmentation functions c → Λ_c and ud → Λ_c are calculated and finally in the electron-positron annihilation in LEP, the inclusive cross section production of Λ_c is calculated about pole of z⁰.     

Feynman-Diagrammatic Description of the Asymptotics of the Time Evolution Operator in Quantum Mechanics

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on \({\mathbb{R}^n}\), with magnetic and potential terms. In particular, for each classical path γ connecting points q ₀ and q ₁ in time t, we define a formal power series V _γ (t, q ₀, q ₁) in \({\hbar}\), given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V _γ ) satisfies Schrödinger’s equation, and explain in what sense the \({t \to 0}\) limit approaches the δ distribution. As such, our construction gives explicitly the full \({\hbar\to 0}\) asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.     

Why <Emphasis Type="Italic">X</Emphasis>(3872) is not a molecule

We discuss the scenario where the X(3872) resonance is the \(c\bar c\) = χ_c1(2P) charmonium which “sits on” the D*⁰\({\bar D^0}\) threshold. We explain the shift of the mass of the X(3872) resonance with respect to the prediction of a potential model for the mass of the χ_c1(2P) charmonium by the contribution of the virtual D*\(\bar D\) + c.c. intermediate states into the self energy of the X(3872) resonance. This allows us to estimate the coupling constant of the X(3872) resonance with the D*⁰\({\bar D^0}\) channel, the branching ratio of the X(3872) → D*⁰\({\bar D^0}\) + c.c. decay, and the branching ratio of the X(3872) decay into all non-D*⁰\({\bar D^0}\) + c.c. states. We predict a significant number of unknown decays of X(3872) via two gluon: X(3872) → gluongluon → hadrons. We suggest a physically clear program of experimental researches for verification of our assumption.