Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the Q- and T -systems based on A _r . The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.     

A Meinardus Theorem with Multiple Singularities

We solve the quantum version of the A ₁ T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A ₁ Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.     

The Bethe Equation at q = 0, The Möbius Inversion Formula,and Weight Multiplicities: III. The X

It is shown that the numbers of off-diagonal solutions to the U _q (X ^(r) _N ) Bethe equation at q = 0 coincide with the coefficients in the recently introduced canonical power series solution of the Q-system. Conjecturally, the canonical solutions are characters of KR (Kirillov–Reshetikhin) modules. This implies that the numbers of off-diagonal solutions agree with the weight multiplicities, which is interpreted as a formal completeness of the U _q (X ^(r) _N ) Bethe ansatz at q = 0.     

<Emphasis Type="Italic">Q</Emphasis>-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property

We define the cluster algebra associated with the Q-system for the Kirillov–Reshetikhin characters of the quantum affine algebra \({U_q(\widehat{\mathfrak {g}})}\) for any simple Lie algebra \({\mathfrak {g}}\), generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the “initial cluster seeds”, including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov–Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such “generalized T-systems” with appropriate boundary conditions.     

Symmetry of nuclear Hamiltonian as an origin of clustering

It is established that the set of solutions of the Hamiltonian of the generalized Elliott SU(3) model in the space of A-fermion wave functions contains a subset of multicluster solutions Ψ_Δ ^A which can be precisely represented in terms of solutions of the Hamiltonian of the same form acting in the reduced space of certain intercluster coordinates. The conditions imposed on a multicluster partition which exhibits such properties are determined by applying the group-theory methods.     

Boundary Chromatic Polynomial

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Q _s colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition function of a boundary loop model. Using results for the latter, the phase diagram of the coloring problem (with real Q and Q _s) is inferred, in the limits of two-dimensional or quasi one-dimensional infinite graphs. We find in particular that the special role played by Beraha numbers for the usual chromatic polynomial does not extend to the case Q≠Q _s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.     

Effect of alkali-earth ions on the local structure of LaAlO3-La0.67A0.33MnO3 (A = Ca, Sr, Ba) diluted solid solutions: 27Al NMR studies

²⁷Al magic angle spinning (MAS) nuclear magnetic resonance (NMR) was studied for diluted alkali earth metal-doped lanthanum manganite solid solutions in the lanthanum aluminate (1 − y)LaAlO₃-yLa_0.67 A _0.33MnO₃ (A = Ca, Sr, Ba) with y =0, 2, 3, and 5 mol %. The spectra depended on the dopant species and showed higher substitutional ordering for the Ba-containing mixed crystals. Magnetically shifted lines were observed in all solid solutions and were attributed to Al in the octahedral oxygen environment near manganese trivalent ions. Nonlinear dependences of their intensity were referred to the manganese-rich cluster formation. An additional MAS NMR line corresponding to aluminum at sites different from the octahedral site in pure LaAlO₃ was observed only in solutions doped with Ba. 3Q MAS NMR revealed that the broadening of this line is governed mainly by quadrupole coupling and made it possible to calculate the isotropic chemical shift. The article was submitted by the authors in English.     

The Canonical Solutions of the Q-Systems¶ and the Kirillov–Reshetikhin Conjecture

We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite $Q$-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems. Received: 2 August 2001 / Accepted: 27 December 2001     

The NLO Parton Distribution in the (x, Q 2)-Plane: A Relativistic Quark-Exchange Approach to A = 3 Mirror Nuclei

A next-to-leading order QCD analysis of polarized and unpolarized structure functions of the proton in the (x, Q²)-plane is discussed within the scheme of the radiation parton formalism. The valence quark distribution is obtained from the application of the relativistic quark-exchange model to A = 3 mirror nuclei, i.e., ³He and ³H. The sea quark and gluon distributions are calculated using the inverse Mellin technique in the NLO approximation. A comparison is made with the corresponding available experimental data. We find a good fit for F₂^p(x, Q²) to the data. It is shown that our new NLO calculation improves our previous works. We argue that the valence quark scenario at some μ₀² ≪ Q² is a reasonable assumption in the framework of the DGLAP evolution equation. In agreement with the data, it is demonstrated that the asymmetry A₁^p(x, Q²) has no significant Q²-dependence as we go to the small x, even at NLO limit. Finally we argue that for small x ≤ 0.2 it is a good approximation to consider ³He and ³H structure functions as those of neutrons and protons, respectively.     

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.     

The QKa, Branches of Carbodiimide, HNCNH, between 1.8 and 3.3 THz

The ^rQ_Ka branches of carbodiimide, HNCNH, have been recorded with a tunable far-infrared spectrometer. The ^rQ_Ka branches with K_a { 2, 3, 4 }, which occur at approximately 1.8, 2.5, and 3.2 THz, respectively, were measured rotationally resolved. The torsional doublet splitting observed in the ^rQ₀ branch and several ^rR branches by Birk et al. (J. Mol. Spectrosc.136, 402-445 (1989)) was also observed in all the Q branches reported here. In addition, anomalous K_a-type doubling was detected within the ^rQ₂ branch for rotational quantum numbers J > 18. Molecular constants were obtained by fitting the line positions, including older data, with Watson′s Hamiltonian in S-reduction. The anomalous K_a-type doubling could be described as a centrifugal distortion interaction with ΔK_a = 4. The splitting is called anomalous because it inverts the K_a = 2 energy levels of an accidentally nearly prolate symmetric top molecule (Ray′s asymmetry parameter for HNCNH κ = −0.999995343(32)).     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

From macroscopic to atomic motions in liquid metals

Summary In the present review of liquid dynamics studies on liquid metals are reported. Particularly the case of liquid lead is reviewed because this case was carefully studied by neutron scattering technique,S(Q,ω) being determined at two widely different temperaturesT=623 K andT=1170 K and therefore different densities. In addition extensive supplementary MD simulations were made using a 16 384-particle system. The simulations ranged from a determination of an effective pair potential for lead to simulation of the density correlation functionsF(Q,t) andF _s(Q,t), as well as the longitudinal and transversal current correlation functionsJ ₁(Q,t) andJ _T(Q,t). The MD simulation ?calibrated? via the experimentalS(Q) andS(Q,ω) was used to prolong the range of neutron data to draw conclusions regarding such quantities as dispersion relations for the current correlationsJ ₁(Q,t) andJ _T(Q,t), the generalized viscosity functions ν₁(Q,t), ν₁(Q) and ν_s(Q). Information regarding bulk viscosity ν_B(Q) is also gained. Conclusions are drawn regarding the relative importance of the derived pair potential form by comparison to corresponding hard-sphere data. The general framework of linearized hydrodynamic equations for the macroscopic situation transforming to visco-elastic equations of motion for finite wave-length and high frequency works well also for the case of a continuous potential. The region of transition from simple visco-elastic to hydrodynamic behaviour is occurring at wavelengths in the range (12÷20) ? for the cases studied. The spatial properties of the viscosity functions ν₁(r), ν_s(r) and ν_B(r) are found to correlate well with the range of the radial distribution function for the liquid. The general results for liquid lead probably have wide range of applicability to other simple liquids with similarS(Q) andg(r) properties. The authors have agreed not to receive proofs for correction.     

氧化亚氮3000—0200和1001—0110跃迁带在高温下的线强度

采用乘积近似法计算了氧化亚氮分子的总配分函数，其中转动配分函数考虑了离心扭曲修正，振动配分函数采用谐振子近似. 利用计算所得的配分函数和实验振动跃迁矩平方及Herman-Wallis因子系数，计算了氧化亚氮3000—0200和1001—0110跃迁带在常温和高温下的线强度. 结果显示，当温度高达3000K时，计算所得线强度与实验值及HITRAN数据库提供的结果仍符合较好. 这表明高温下的分子配分函数和线强度的计算是可靠的. 还进一步计算了氧化亚氮3000—0200和1001—0110跃迁带在更高温度(40 关键词： 氧化亚氮 配分函数 线强度 高温     

Controlled mixing enhancement in turbulent rectangular jets responding to periodically forced inflow conditions

We present numerical studies of active flow control applied to jet flow. We focus on rectangular jets, which are more unstable than their circular counterparts. The higher level of instability is expressed mainly by an increased intensity of mixing of the main flow with its surroundings. We analyse jets with aspect ratio A_r = 1, A_r = 2 and A_r = 3 at Re = 10,000. It is shown that the application of control with a suitable excitation (forcing) at the jet nozzle can amplify the mixing and qualitatively alter the character of the flow. This can result in an increased spreading rate of the jet or even splitting into nearly separate streams. The excitations studied are obtained from a superposition of axial and flapping forcing terms. We consider the effect of varying parameters such as the frequency of the excitations and phase shift between forcing components. The amplitude of the forcing is 10% of the inlet centreline jet velocity and the forcing frequencies correspond to Strouhal numbers in a range St = 0.3–0.7. It is shown that qualitatively different flow regimes and a rich variety of possible flow behaviours can be achieved simply by changing aspect ratio and forcing parameters. The numerical results are obtained applying large eddy simulation in combination with a high-order compact difference code for incompressible flows. The solutions are validated based on experimental data from literature for non-excited jets for A_r = 1 at Re = 1.84 × 10⁵ and A_r = 2 at Re = 1.28 × 10⁵. Both the mean velocities as well as their fluctuations are predicted with good accuracy.     

13C−1H heteronuclear dipolar correlation studies of the hydrogen bonding of the quinones inRhodobacter sphaeroides R26 reaction centers

The photosynthetic reaction center (RC) of the photosynthetic bacteriumRhodobacter sphaeroides R26 contains two quinones, Q_A and Q_B. Solid-state heteronuclear (¹H?¹³C) dipolar correlation spectroscopy has been used to study the binding of the quinones in the ground state for RCs reconstituted with l-¹³C ubiquinone-10. Lee-Goldburg cross-polarization buildup curves are recorded to determine distancesr _CH between the l-¹³C carbon labels and the protons involved in the polarization transfer. The l-¹³C of both Q_A and Q_B have intermolecular correlations with protons that resonate downfield, in the region of hydrogen-bonding protons. The distances between the carbon labels and the correlated protons are short, 0.21±0.01 nm. Hence the nuclear magnetic resonance provides evidence for strong hydrogen-bonding interactions at the l-C=O of both Q_A and Q_B for RCs in the ground state. The environment of the l-¹³C of the Q_B is structurally heterogeneous compared to that of the Q_A. The data can be reconciled with a strong H-bonding interaction of the l-C=O of Q_A with Ala M260 NH, and with complex hydrogen bonding involving NH of Ile-L224 and of Gly-L225, and possibly the Ser-L223 hydroxyl group of the l-C=O of the Q_B, in the proximal site.     

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

The structure of concentrated NiCl2 aqueous solutions: what is molecular simulation revealing about the neutron scattering methodologies?

An analysis is made of adequacy and limitations of the neutron weighted ion distribution function for the determination of the hydration ion structure in hydrothermal solutions. Our analysis indicates that the coordination number based on the O—Ni²⁺ interactions is unambiguously defined by the first peak of the neutron weighted cation distribution function G _Ni(r), but that the corresponding H—Ni⁺² and H—Cl^? coordination numbers may be ill-defined due to the occurrence of Ni⁺²-Cl^? ion pairing. For the system considered in this work, this effect contributes about 1.5 units to the H—Ni^?2 and 0.85 units to the H—Cl^? coordination numbers, respectively, for a 3.9 M NiCl₂ aqueous solution under ambient conditions. A comparison under ambient conditions between the most reliable NDIS data on Ni²⁺ hydration and our simulation results suggests that the present intermolecular potential models underestimate the O-Ni⁺² coordination numbers by about 1.5 units, and it might indicate the need for a reparametrization of the current ion-water intermolecular potentials. The hydration structure exhibits practically no temperature dependence for the isochore studied (1.356 g cm^?3) and composition. The Ni⁺²-Cl^? ion pair formation appears to affect the location of the shoulder in the neutron weighted distribution functions G _Ni(r) and G _Cl(r), although it does not affect the magnitude of the ion—water coordination.     

Quantum Yang-Mills on a Riemann surface

We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/G _m of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path^2g G consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path^2g G, each with respect to the push-forward toA/G _m of the measuree ^–S(A) D A. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/G _m to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path^2g G is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the caseg=1.Research supported in part by DOE grant DE-FGO2-88ER25066