On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1

Imaging geometry through dynamics: the observable representation

For many stochastic processes there is an underlying coordinate space, $V$, with the process moving from point to point in $V$ or on variables (such as spin configurations) defined with respect to $V$. There is a matrix of transition probabilities (whether between points in $V$ or between variables defined on $V$) and we focus on its ``slow'' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the ``observables,'' and they can be used to recover geometrical features of $V$

Long-Range Order in Electronic Transport through Disordered Metal Films

Ultracold atom magnetic field microscopy enables the probing of current flow patterns in planar structures with unprecedented sensitivity. In polycrystalline metal (gold) films we observe long-range correlations forming organized patterns oriented at +/- 45 deg relative to the mean current flow, even at room temperature and at length scales orders of magnitude larger than the diffusion length or the grain size. The preference to form patterns at these angles is a direct consequence of universal scattering properties at defects. The observed amplitude of the current direction fluctuations scales inversely to that expected from the relative thickness variations, the grain size and the defect concentration, all determined independently by standard methods. This indicates that ultracold atom magnetometry enables new insight into the interplay between disorder and transport

Calculation of mechanical and thermal influences during coiling of hot strip

Coiled steel strip is the final product from flat hot rolling processes. With increasing demand for higher quality of hot rolled strips, especially the evolution of strip flatness during and after coiling becomes a crucial aspect. The main impacts on the flatness properties of hot rolled strips result from residual stresses and “eigen-strains” induced by the last hot rolling passes, by strip cooling at the run-out table, and finally, by the mechanical and thermal conditions during and after the coiling process itself. In this paper, a mathematical model is presented, which takes into account the mechanical and thermal effects on hot rolled strip during and after the coiling process. To improve the prediction quality of the underlying process, a customized self-developed 3D finite-element model has been developed and programmed in C++, leading to a software prototype, which is highly superior to commercial FEM-packages with respect to calculation time and storage capacities. The model is based on a dynamic implicit total Lagrangian formulation. All relevant devices directly interacting with the strip, such as pinch rolls, coiler rolls and mandrel are incorporated in the calculation model. Well known and established methods in the solid-shell theory, like the EAS- and ANS-method, were applied to prevent the occurrence of locking phenomena resulting from low order interpolation functions. Selected benchmark tests were performed to evaluate the accuracy of these novel solid-shell elements in comparison to the results attained by the FEM- package ABAQUS©. The results obtained so far agree very satisfactorily. A further important topic is the contact and friction algorithm, where Coulomb’s friction law is applied. The accurate and reliable determination of the contact between strip and interacting devices as well as the aspect of self-contact was treated by applying a sophisticated two dimensional contact search algorithm, leading to a significantly reduced calculation time. The highly non-linear time-dependent system of equations is integrated by utilizing the (implicit) Newmark time-integration scheme. The developed calculation model serves as an effective tool to predict the interesting stress-distributions and local plastic deformations inside the strip, which induce residual stresses and strip unflatness (latent or even manifest waviness). Furthermore, this tool p ovides the basis for further improvements and investigations on hot rolling production lines

Increase in degraded collagen type II in synovial fluid early in the rabbit meniscectomy model of osteoarthritis

SummaryObjectiveThe objective of this study was to determine whether collagen type II breakdown products in synovial fluid (SF), detected by an enzyme-linked immunoassay, represent a useful marker for early events in osteoarthritis (OA) in the rabbit medial meniscectomy model.DesignComplete medial meniscectomy was performed on the right knee joints of 32 rabbits. Balanced groups of rabbits were then sacrificed at 2, 4, 8, and 12 weeks post-surgery. An additional 8 unoperated and 11 sham-operated animals served as controls. SF lavages were performed on right and left knee joints of the same animals at sacrifice. The proteolytic epitope of type II collagen was monitored using an enzyme-linked immunoassay.ResultsMacroscopically visible surface fibrillation and focal erosions appeared as early as 2 weeks after meniscectomy in the femorotibial joint (P<0.01). OA developed gradually during the later observation period, and then predominantly on the medial tibial plateau and medial femur. Significant histological alterations in cartilage, including a loss of proteoglycans, surface irregularities, and clefts, were detected at 2 weeks after meniscectomy (P<0.01). Collagen type II epitope levels in SF lavage samples were elevated peaking at 2 weeks after meniscectomy (P<0.02). Levels decreased at later time points, but they were still raised at 12 weeks (P≤0.05). Highly significant correlations were found between the SF collagen type II epitope levels and the macroscopic and microscopic scoring results (Spearman rho correlation coefficient, macroscopy—collagen type II epitope r=0.222, P=0.025; microscopy—collagen type II epitope r=0.436, P≤0.01).ConclusionIn this rabbit model of medial meniscectomy, levels of type II collagen fragments in SF appear to provide a useful marker of the early degenerative changes

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Designing potentials by sculpturing wires

Magnetic trapping potentials for atoms on atom chips are determined by the current flow in the chip wires. By modifying the shape of the conductor we can realize specialized current flow patterns and therefore micro-design the trapping potentials. We have demonstrated this by nano-machining an atom chip using the focused ion beam technique. We built a trap, a barrier and using a BEC as a probe we showed that by polishing the conductor edge the potential roughness on the selected wire can be reduced. Furthermore we give different other designs and discuss the creation of a 1D magnetic lattice on an atom chip.Comment: 6 pages, 8 figure