Abstract:
Luttinger liquid theory accounts for the low energy boson excitations of one-dimensional quantum liquids, but disregards the high energy excitations. The most important high energy excitations are holes which have infinite lifetime at zero temperature. At finite temperatures they can be scattered by thermally excited bosons. We describe the interaction of the hole with the bosons by treating it as a mobile impurity in a Luttinger liquid. This approach enables us to evaluate the scattering probability at arbitrary interaction strength. In general, the result is expressed in terms of the hole spectrum, its dependence on the density and momentum of the fluid, and the parameters of the Luttinger liquid Hamiltonian. In the special case of Galilean invariant systems the scattering probability is expressed in terms of only the hole spectrum and its dependence on the fluid density. We apply our results to the problem of equilibration of one-dimensional quantum liquids.

Abstract:
We study Coulomb blockade oscillations of thermoelectric coefficients of a single electron transistor based on a quantum dot strongly coupled to one of the leads by a quantum point contact. At temperatures below the charging energy E_C the transport of electrons is dominated by strong inelastic cotunneling. In this regime we find analytic expressions for the thermopower as a function of temperature T and the reflection amplitude $r$ in the contact. In the case when the electron spins are polarized by a strong external magnetic field, the thermopower shows sinusoidal oscillations as a function of the gate voltage with the amplitude of the order of $e^{-1}|r|\frac{T}{E_C}$. We obtain qualitatively different results in the absence of the magnetic field. At temperatures between $E_C$ and $E_C|r|^2$ the thermopower oscillations are sinusoidal with the amplitude of order $e^{-1}|r|^2 \ln \frac{E_C}{T}$. On the other hand, at $T\ll E_C|r|^2$ we find non-sinusoidal oscillations of the thermopower with the amplitude $\sim e^{-1} |r| \sqrt{T/E_C} \ln(E_C/T)$.

Abstract:
We study how a Luttinger liquid of spinless particles in one dimension approaches thermal equilibrium. Full equilibration requires processes of backscattering of excitations which occur at energies of order of the bandwidth. Such processes are not accounted for by the Luttinger liquid theory. We treat the high-energy excitations as mobile impurities and derive an expression for the equilibration rate in terms of their spectrum. Our results apply at any interaction strength.

Abstract:
We consider Coulomb blockade oscillations of thermoelectric coefficients of a single electron transistor based on a quantum dot strongly coupled to one of the leads. Analytic expression for the thermopower as a function of temperature $T$ and the reflection amplitude $r$ in the quantum point contact is obtained. Two regimes can be identified: $T \ll E_C|r|^2$ and $T \gg E_C |r|^2$, where $E_C$ is the charging energy of the dot. The former regime is characterized by weak logarithmic dependence of the thermopower on the reflection coefficient, in the latter the thermopower is linear in the reflection coefficient $|r|^2$ but depends on temperature only logarithmically.

Abstract:
Luttinger liquid theory describes one-dimensional electron systems in terms of non-interacting bosonic excitations. In this approximation thermal excitations are decoupled from the current flowing through a quantum wire, and the conductance is quantized. We show that relaxation processes not captured by the Luttinger liquid theory lead to equilibration of the excitations with the current and give rise to a temperature-dependent correction to the conductance. In long wires, the magnitude of the correction is expressed in terms of the velocities of bosonic excitations. In shorter wires it is controlled by the relaxation rate.

Abstract:
We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph we enumerate all corresponding knot projections, and after that we construct the corresponding minimal diagrams. Several known and new tricks made it possible to keep the process within reasonable limits and offer a rigorous theoretical proof of the completeness of the table. For proving that all knots are different we use a generalized version of the Kauffman polynomial.

Abstract:
Mechanisms of acoustic energy dissipation in heterogeneous solids attract much attention in view of their importance for material characterization, nondestructive testing, and geophysics. Due to the progress in measurement techniques in recent years it has been revealed that rocks can demonstrate extremely high strain sensitivity of seismo-acoustic loss. In particular, it has been found that strains of order $10^{-8}$ produced by lunar and solar tides are capable to cause variations in the seismoacoustic decrement on the order of several percents. Some laboratory data (although obtained for higher frequencies) also indicate the presence of very high dissipative nonlinearity. Conventionally discussed dissipation mechanisms (thermoelastic loss in dry solids, Biot and squirt-type loss in fluid-saturated ones) do not suffice to interpret such data. Here, the dissipation at individual cracks is revised taking into account the influence of wavy asperities of their surfaces quite typical of real cracks, which can drastically change the values of the relaxation frequencies and can result in giant strain sensitivity of the dissipation without the necessity to assume the presence of unrealistically thin (and, therefore, unrealistically soft) cracks. In particular, these mechanisms suggest interpretation for observations of pronounced amplitude modulation of seismo-acoustic waves by tidal strains.

Abstract:
We show that a well known for NLS equation scaling invariance and Galilean invariance property of its solutions can be extended with appropriate modifications to the whole AKNS hierahchy and its reduced and deformed versions.