Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model

The two-dimensional Potts model can be studied either in terms of the original Q-component spins, or in the geometrical reformulation via Fortuin-Kasteleyn (FK) clusters. While the FK representation makes sense for arbitrary real values of Q by construction, it was only shown very recently that the spin representation can be promoted to the same level of generality. In this paper we show how to define the Potts model in terms of observables that simultaneously keep track of the spin and FK degrees of freedom. This is first done algebraically in terms of a transfer matrix that couples three different representations of a partition algebra. Using this, one can study correlation functions involving any given number of propagating spin clusters with prescribed colours, each of which contains any given number of distinct FK clusters. For 0 <= Q <= 4 the corresponding critical exponents are all of the Kac form h_{r,s}, with integer indices r,s that we determine exactly both in the bulk and in the boundary versions of the problem. In particular, we find that the set of points where an FK cluster touches the hull of its surrounding spin cluster has fractal dimension d_{2,1} = 2 - 2 h_{2,1}. If one constrains this set to points where the neighbouring spin cluster extends to infinity, we show that the dimension becomes d_{1,3} = 2 - 2 h_{1,3}. Our results are supported by extensive transfer matrix and Monte Carlo computations.Comment: 15 pages, 3 figures, 2 table

Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

Logarithmic observables in critical percolation

Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q = 1 limit of the Q-state Potts model, and analyzing the underlying S_Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q = 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d=2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by extensive Monte-Carlo simulations.Comment: 11 pages, 2 figures. V2: as publishe

Conductance of nano-systems with interactions coupled via conduction electrons: Effect of indirect exchange interactions

A nano-system in which electrons interact and in contact with Fermi leads gives rise to an effective one-body scattering which depends on the presence of other scatterers in the attached leads. This non local effect is a pure many-body effect that one neglects when one takes non interacting models for describing quantum transport. This enhances the non-local character of the quantum conductance by exchange interactions of a type similar to the RKKY-interaction between local magnetic moments. A theoretical study of this effect is given assuming the Hartree-Fock approximation for spinless fermions in an infinite chain embedding two scatterers separated by a segment of length L\_c. The fermions interact only inside the two scatterers. The dependence of one scatterer onto the other exhibits oscillations which decay as 1/L\_c and which are suppressed when L\_c exceeds the thermal length L\_T. The Hartree-Fock results are compared with exact numerical results obtained with the embedding method and the DMRG algorithm

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$

The one-dimensional Keller-Segel model with fractional diffusion of cells

We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent $0<\alpha\leq 2$. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when $\alpha<1$ and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for $\alpha\leq1$ if the initial density is small enough in the sense of the $L^{1/\alpha}$ norm.Comment: 12 page

Radiographic preoperative templating of extra-offset cemented THA implants: How reliable is it and how does it affect survival?

SummaryIntroductionSecuring femoral offset should in theory improve hip stability and abductor muscles moment arms. As problems arise mainly in case of originally increased offset (>40mm), a range of extra-offset stems is available; the exact impact in terms of fixation, however, is not known.HypothesisExtra-offset stems should more reliably reestablish original femoral offsets exceeding 40mm than standard femoral components, limiting instability risk without possible adverse effect on fixation.ObjectiveTo compare the ability of five commonly available femoral stem designs to restitute offset exceeding 40mm, and to assess function and cement fixation at a minimum 6 years’ follow-up in a stem conceived to reproduce such offset.Patients and methodsA continuous series of 74 total hip replacements (THR) in hips with increased (>40mm) femoral offset was studied. All underwent preoperative X-ray templating on Imagika™ software to assess offset reproduction by five models of stem: four standard, and one Lubinus SP2™ extra-offset stem. A retrospective clinical and X-ray study was conducted with a minimum 6 years’ follow-up on the Lubinus SP2™ 117° stems used to try to reproduce offset in the 74 THRs.ResultsApart from the increased (>40mm) offset, the cervicodiaphyseal angle was consistently <135°, <130° in 60 femurs (81%) and <125° in 45 (60%). Planning showed the four standard stems to induce (>5mm femoral offset reduction in 50–83% of cases, versus only 25% with the Lubinus SP2™ 117°). All 74 hips received Lubinus SP2™ 117° stems: at a mean 78 months FU (range, 70–94mo), their mean Postel-Merle d’Aubigné score was 17±1.8 (range, 13–18). Five of the 74 THRs underwent surgical revision: three cases of loosening, in which the stem was replaced, and two of instability, without change of stem. Loosening was not related to offset reproduction quality; two of the three cases were due to initial cementing defect, and the third occurred in a femur with previous history of two osteotomies. There were four cases of dislocation (5.4%: two primary, which were not operated on, and two recurrent, managed by acetabular revision), despite good reproduction of the preoperative offset in three of the four cases. Mean 7-year implant survivorship was 95.1% (±4.8).Discussion and conclusionThe anatomic form of the Lubinus™ SP2 117° should in theory provide a uniform cement mantle. Survivorship, however, is less good than for regular offset versions (126° or 135°). On the other hand, it does reproduce anatomy in case of >40mm offset, providing extra offset of more than 51mm. The slightly shorter survivorship requires more long-term surveillance.Level of evidenceLevel IV, retrospective study

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page