The Circuit Ideal of a Vector Configuration

The circuit ideal, \ica, of a configuration \A = \{\a_1, ..., \a_n\} \subset \Z^d is the ideal generated by the binomials {\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n] as \cc = \cc^+ - \cc^- \in \Z^n varies over the circuits of \A. This ideal is contained in the toric ideal, \ia, of \A which has numerous applications and is nontrivial to compute. Since circuits can be computed using linear algebra and the two ideals often coincide, it is worthwhile to understand when equality occurs. In this paper we study \ica in relation to \ia from various algebraic and combinatorial perspectives. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of \ica and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that \ia is the unique minimal prime of \ica and that the embedded primes of \ica are indexed by certain faces of the cone spanned by \A. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to \ia and \ica. The Gr\"obner fan of \ica is shown to refine that of \ia when the codimension of the ideals is at most two.Comment: 25 page

Scaling of resistivities and guided vortex motion in MgB2 thin films

Longitudinal and transverse voltages have been measured on thin films of MgB2 with different superconducting transition widths. The study has been performed in zero and non-zero external magnetic fields. The non-zero transverse voltage has been observed in close vicinity of the critical temperature in zero external magnetic field, while further away from Tc this voltage becomes zero. In magnetic field it becomes a transverse voltage which is an even function with respect to the direction of the field. The usual Hall voltage starts to appear with increasing magnetic field and in large fields the even voltage disappears and only the Hall voltage is measurable (i.e. the transverse even voltage is suppressed with increasing magnetic field and increasing transport current). New scaling between transverse and longitudinal resistivities has been observed. This correlation is valid not only in the zero magnetic field but also in nonzero magnetic field where transverse even voltage is detected. Several models trying to explain observed results are discussed. The most promising one seems to be guided motion of the vortices, though further theoretical work will be required to confirm this

From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of mutually interacting subsystem and heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the "hypothesis of equal weights for eigenstates", we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.Comment: LaTeX, 8 pages, no figures. The title, abstract and some discussions are modified to stress physical motivation of the work. References are added to [2]. This version will appear in Phys. Rev. Lett. There is an accompanying unpublished note (cond-mat/9707255

Computing Tropical Varieties

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.Comment: 22 pages, 2 figure

Transverse voltage in zero external magnetic fields, its scaling and violation of the time reversal symmetry in MgB2

The longitudinal and transverse voltages (resistances) have been measured for MgB$_2$ in zero external magnetic fields. Samples were prepared in the form of thin film and patterned into the usual Hall bar shape. In close vicinity of the critical temperature T$_c$ non-zero transverse resistance has been observed. Its dependence on the transport current has been also studied. New scaling between transverse and longitudinal resistivities has been observed in the form $\rho{_{xy}}\sim d\rho{_{xx}}/dT$. Several models for explanation of the observed transverse resistances and breaking of reciprocity theorem are discussed. One of the most promising explanation is based on the idea of time-reversal symmetry violation

Spin-polarized transport through carbon nanotubes

Carbon nanotubes (CNT) belong to the most promising new materials which can in the near future revolutionize the conventional electronics. When sandwiched between ferromagnetic electrodes, the CNT behaves like a spacer in conventional spin-valves, leading quite often to a considerable giant magneto-resistance effect (GMR). This paper is devoted to reviewing some topics related to electron correlations in CNT. The main attention however is directed to the following effects essential for electron transport through nanotubes: (i) nanotube/electrode coupling and (ii) inter-tube interactions.It is shown that these effects may account for some recent experimental reports on GMR, including those on negative (inverse) GMR.Comment: 7 pages, 3 figure

A numerical study of the spectrum and eigenfunctions on a tubular arc

The Hamiltonian for a particle constrained to move on the surface of a curved nanotube is derived using the methods of differential forms. A two-dimensional Gram-Schmidt orthonormalization procedure is employed to calculate basis functions for determining the eigenvalues and eigenstates of a tubular arc (a nanotube in the shape of a hyperbolic cosine) with several hundred scattering centers. The curvature of the tube is shown to induce bound states that are dependent on the curvature parameters and bend location of the tube.Comment: 14 pages, 5 tables, 6 figure

Interacting Random Walkers and Non-Equilibrium Fluctuations

We introduce a model of interacting Random Walk, whose hopping amplitude depends on the number of walkers/particles on the link. The mesoscopic counterpart of such a microscopic dynamics is a diffusing system whose diffusivity depends on the particle density. A non-equilibrium stationary flux can be induced by suitable boundary conditions, and we show indeed that it is mesoscopically described by a Fourier equation with a density dependent diffusivity. A simple mean-field description predicts a critical diffusivity if the hopping amplitude vanishes for a certain walker density. Actually, we evidence that, even if the density equals this pseudo-critical value, the system does not present any criticality but only a dynamical slowing down. This property is confirmed by the fact that, in spite of interaction, the particle distribution at equilibrium is simply described in terms of a product of Poissonians. For mesoscopic systems with a stationary flux, a very effect of interaction among particles consists in the amplification of fluctuations, which is especially relevant close to the pseudo-critical density. This agrees with analogous results obtained for Ising models, clarifying that larger fluctuations are induced by the dynamical slowing down and not by a genuine criticality. The consistency of this amplification effect with altered coloured noise in time series is also proved.Comment: 8 pages, 7 figure