Polynomial functors and combinatorial Dyson-Schwinger equations

We present a general abstract framework for combinatorial Dyson-Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson-Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild $1$-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structure. Precisely, for any finitary polynomial endofunctor $P$ defined over groupoids, the system of combinatorial Dyson-Schwinger equations $X=1+P(X)$ has a universal solution, namely the groupoid of $P$-trees. The isoclasses of $P$-trees generate naturally a Connes-Kreimer-like bialgebra, in which the abstract Dyson-Schwinger equation can be internalised in terms of canonical $B_+$-operators. The solution to this equation is a series (the Green function) which always enjoys a Fa\`a di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying $P$ yields different bialgebras, and cartesian natural transformations between various $P$ yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson-Schwinger equations. Finally, all constructions can be pushed inside the classical Connes-Kreimer Hopf algebra of trees by the operation of taking core of $P$-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy

Kinky Behavior in Josephson Junctions

We analyze nonperturbatively the behavior of a Josephson junction in which two BCS superconductors are coupled through an Anderson impurity. We recover earlier perturbative results which found that a $\delta=\pi$ phase difference is preferred when the impurity is singly occupied and the on-site Coulomb interaction is large. We find a novel intermediate phase in which one of $\delta=0$ and $\delta=\pi$ is stable while the other is metastable, with the energy $E(\delta)$ having a kink somewhere in between. As a consequence of the kink, the $I-V$ characteristics of the junction are modified at low voltages.Comment: 7 pages, 7 encapsulated PostScript figures; figure 3 correcte

Stereotypical risks and threats in the youth’s opinion (diachronic comparative aspect)

The paper reveals the structure of associative fields of words-stimuli "danger", "risk", "threat", fixed in 1988-90 (the materials of "Russian Association Dictionary") and in 2015 (the results of authors’ associative experiment). The obtained results demonstrate the structural stability of these fields diachronically on the one hand and explicit redistribution of "association vectors" within them on the other on

Theory of quantum metal to superconductor transitions in highly conducting systems

We derive the theory of the quantum (zero temperature) superconductor to metal transition in disordered materials when the resistance of the normal metal near criticality is small compared to the quantum of resistivity. This can occur most readily in situations in which ``Anderson's theorem'' does not apply. We explicitly study the transition in superconductor-metal composites, in an s-wave superconducting film in the presence of a magnetic field, and in a low temperature disordered d-wave superconductor. Near the point of the transition, the distribution of the superconducting order parameter is highly inhomogeneous. To describe this situation we employ a procedure which is similar to that introduced by Mott for description of the temperature dependence of the variable range hopping conduction. As the system approaches the point of the transition from the metal to the superconductor, the conductivity of the system diverges, and the Wiedemann-Franz law is violated. In the case of d-wave (or other exotic) superconductors we predict the existence of (at least) two sequential transitions as a function of increasing disorder: a d-wave to s-wave, and then an s-wave to metal transition

Generalized Paraxial Ray Trace Procedure Derived from Geodesic Deviation

Paraxial ray tracing procedures have become widely accepted techniques for acoustic models in seismology and underwater acoustics. To date a generic form of these procedures including fluid motion and time dependence has not appeared in the literature. A detailed investigation of the characteristic curves of the equations of hydrodynamics allows for an immediate generalization of the procedure to be extracted from the equation form geodesic deviation. The general paraxial ray trace equations serve as an ideal supplement to ordinary ray tracing in predicting the deformation of acoustic beams in random environments. The general procedure is derived in terms of affine parameterization and in a coordinate time parameterization ideal for application to physical acoustic ray propagation. The formalism is applied to layered media, where the deviation equation reduces to a second order differential equation for a single field with a general solution in terms of a depth integral along the ray path. Some features are illustrated through special cases which lead to exact solutions in terms of either ordinary or special functions.Comment: Original; 40 pages (double spaced), 1 figure Replaced version; 36 pages single spaced, 7 figures. Expanded content; Complete derivation of the equations from the equations of hydrodynamics, introduction of an auxiliary basis for three dimensional wave-front modeling. Typos in text and equations correcte

Hydrodynamic description of transport in strongly correlated electron systems

We develop a hydrodynamic description of the resistivity and magnetoresistance of an electron liquid in a smooth disorder potential. This approach is valid when the electron-electron scattering length is sufficiently short. In a broad range of temperatures, the dissipation is dominated by heat fluxes in the electron fluid, and the resistivity is inversely proportional to the thermal conductivity, $\kappa$. This is in striking contrast with the Stokes flow, in which the resistance is independent of $\kappa$ and proportional to the fluid viscosity. We also identify a new hydrodynamic mechanism of spin magnetoresistance

Mesoscopic mechanism of adiabatic charge transport

We consider adiabatic charge transport through mesoscopic metallic samples caused by a periodically changing external potential. We find that both the amplitude and the sign of the charge transferred through a sample per period are random sample specific quantities. The characteristic magnitude of the charge is determined by the quantum interference.Comment: 4 pages, 2 figure

On the Nature of Infrared Singularities in d≤2d\leq 2 Disordered Interacting Systems

We address the problem of infrared singularities in the perturbation theory for disordered interacting systems in $d\leq 2$. We show that a typical, sufficiently large interacting system exhibits a linear instability in the spin triplet channel. In a density-density channel, although stability is preserved, a large number of soft modes is accumulated. These phenomena are responsible for the instability of the weak-interacting fixed point. Although generic, the unstable direction and soft modes are highly sample specific and can not be effectively captured by conventional techniques based on an averaging procedure. Rather, the instability is determined by the largest eigenvalues of the polarization operator. We propose to employ the optimal fluctuation method for evaluating the probability of such events.Comment: 4 pages, RevTeX. References added, minor change

Signature of the electron-electron interaction in the magnetic field dependence of nonlinear I-V characteristics in mesoscopic systems

We show that the nonlinear I-V characteristics of mesoscopic samples with metallic conductivity should contain parts which are linear in the magnetic field and quadratic in the electric field. These contributions to the current are entirely due to the electron-electron interaction and consequently they are proportional to the electron-electron interaction constant. We also note that both the amplitude and the sign of the current exhibit random oscillations as a function of temperature

Counting statistics for arbitrary cycles in quantum pumps

Statistics of charge transport in an adiabatic pump are determined by the dynamics of the scattering matrix S(t). We show that, up to an integer offset, the statistics depend only on the corresponding path N(t)=S^\dagger\sigma_3 S in the coset space (the sphere for a single channel). For a general loop S(t) we solve for the noise-minimizing pumping strategy. The average current is given by the area enclosed by N(t) in the coset space; its minimal noise by the area of a minimal surface (soap film) spanned by N(t) in the space of all matrices. We formulate conditions for quantization of the pumped charge.Comment: 4 pages, 2 figure