On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quad-equations on decorated cubes.     

High-temperature expansion for nonequilibrium steady states in driven lattice gases

We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas, the "Ising model" for nonequilibrium physics. We represent the steady state as P(eta) alpha e(-betaH(eta)-psi(eta)) and evaluate the lowest order contribution to the nonequilibrium effective interaction psi(eta). We see that, in dimensions d > or = 2, all models with nonsingular transition rates yield the same summable psi(eta), suggesting the possibility of describing the state as a Gibbs state similar to equilibrium. The models with the Metropolis rule show exceptional behavior.     

Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector , which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.     

Perturbation of resonances in quantum mechanics

Schrödinger operators on L²($\text{R}$³) of the form ?Δ + V_λ with potentials V_λ real-analytic in λ are discussed. The analytic structure in V_λ and k (with k² the energy variable) of the resolvent kernel, the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions. The lower order expansion coefficients are computed explicitly. The resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions. This gives asymptotic low energy information about Schrödinger Hamiltonians with short range potentials. The perturbation theory of resonances, eigenvalues and of the corresponding functions for Hamiltonians describing n point interactions perturbed by a potential is also given.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.     

Photophysical properties of 4,4'-di-tert-butyl-2,2'-bipyridine supported 6-membered 2,2'-diphenyl-X platinacycles (X = CH2, NMe, O)

The six-membered platinacycle complex, Pt((t)Bu(2)bpy)(C(6)H(4)OC(6)H(4)) (6) ((t)Bu(2)bpy = 4,4'-di-tert-butyl-2,2'-bipyridine) has been prepared from Pt((t)Bu(2)bpy)Cl(2) and 2,2'-dilithio-diphenyl ether. Platinacycle 6 and its analogs with X = CH(2) (4) and NMe (5) exhibit intense solid-state photoluminescence and nearly identical crystal structures. The photophysical properties of 4-6 in the visible range are dominated by mixed metal-ligand-to-ligand charge transfer (MLL'CT) transitions involving high-lying filled mixed metal-ligand orbitals (ML), composed primarily of platinacyclic ring-based d- and π-orbitals, and a low lying vacant π* orbital (L') of the (t)Bu(2)bpy ligand. Lone pair donation from the bridging oxygen atom and especially the NMe group increases the energy of the mixed metal-ligand orbital (ML) without altering the energy of the (t)Bu(2)bpy π* orbital. As a result, the MLL'CT state energy decreases and the absorption and emission wavelengths are red-shifted. DFT and TD-DFT calculations support the experimental results. Additional calculations on the unknown platinacycles with X = CO (7) and SO(2) (8) predict a blue-shift for the MLL'CT absorption and emission. Two nearly equal energy triplet minima were located on the DFT triplet surface for 4-6. One of these (4T-6T) has a geometry very similar to the ground-state singlet (as represented by 4-6) and is associated with the emissive (3)MLL'CT excited state. The other triplet-state (4T'-6T') has a distorted structure where the platinacycle ring is twisted out of the Pt((t)Bu(2)bpy) plane. Thermal access to this distorted triplet may be responsible for the loss of photoluminescence in room temperature solutions of 4-6.