Planar 3-dimensional assignment problems with Monge-like cost arrays

Given an $n\times n\times p$ cost array $C$ we consider the problem $p$-P3AP which consists in finding $p$ pairwise disjoint permutations $\varphi_1,\varphi_2,\ldots,\varphi_p$ of $\{1,\ldots,n\}$ such that $\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k}$ is minimized. For the case $p=n$ the planar 3-dimensional assignment problem P3AP results. Our main result concerns the $p$-P3AP on cost arrays $C$ that are layered Monge arrays. In a layered Monge array all $n\times n$ matrices that result from fixing the third index $k$ are Monge matrices. We prove that the $p$-P3AP and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that in the layered Monge case there always exists an optimal solution of the $p$-3PAP which can be represented as matrix with bandwidth $\le 4p-3$. This structural result allows us to provide a dynamic programming algorithm that solves the $p$-P3AP in polynomial time on layered Monge arrays when $p$ is fixed.Comment: 16 pages, appendix will follow in v

Geometric versions of the 3-dimensional assignment problem under general norms

We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles measured according to a certain norm. (All our results also carry over to the corresponding special cases of the 3-dimensional matching problem.) The minimization version is NP-hard for every norm, even if the underlying Cartesian space is 2-dimensional. The maximization version is polynomially solvable, if the dimension of the Cartesian space is fixed and if the considered norm has a polyhedral unit ball. If the dimension of the Cartesian space is part of the input, the maximization version is NP-hard for every $L_p$ norm; in particular the problem is NP-hard for the Manhattan norm $L_1$ and the Maximum norm $L_{\infty}$ which both have polyhedral unit balls.Comment: 21 pages, 9 figure

Generation of specific antibodies against the rap1A, rap1B and rap2 small GTP-binding proteins. Analysis of rap and ras proteins in membranes from mammalian cells

Specific antibodies against rap1A and rap1B small GTP-binding proteins were generated by immunization of rabbits with peptides derived from the C-terminus of the processed proteins. Immunoblot analysis of membranes from several mammalian cell lines and human thrombocytes with affinity-purified antibodies against rap1A or rap1B demonstrated the presence of multiple immunoreactive proteins in the 22-23 kDa range, although at strongly varying levels. Whereas both proteins were present in substantial amounts in membranes from myelocytic HL-60, K-562 and HEL cells, they were hardly detectable in membranes from lymphoma U-937 and S49.1 cyc- cells. Membranes from human thrombocytes and 3T3-Swiss Albino fibroblasts showed strong rap1B immunoreactivity, whereas rap1A protein was present in much lower amounts. In the cytosol of HL-60 cells, only small amounts of rap1A and rap1B proteins were detected, unless the cells were treated with lovastatin, an inhibitor of hydroxymethylglutaryl-coenzyme A reductase, suggesting that both proteins are isoprenylated. By comparison with recombinant proteins, the ratio of rap1A/ras proteins in membranes from HL-60 cells was estimated to be about 4:1. An antiserum directed against the C-terminus of rap2 reacted strongly with recombinant rap2, but not with membranes from tested mammalian cells. In conclusion, rap1A and rap1B proteins are distributed differentially among membranes from various mammalian cell types and are isoprenylated in HL-60 cells

Polygons with inscribed circles and prescribed side lengths

AbstractWe prove NP-completeness of the following problem: For n given input numbers, decide whether there exists an n-sided, plane, convex polygon that has an inscribed circle and that has the input numbers as side lengths

Graph Orientation and Flows Over Time

Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems -- streets, tracks, etc. -- are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost in any orientation of the network if the time horizon remains fixed? We prove that there is always an orientation where we can still send $\frac{1}{3}$ of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is $\frac12$ which is again tight. We present more results of similar flavor and also show non-approximability results for finding the best orientation for single and multicommodity maximum flows over time

The Masticatory Contractile Load Induced Expression and Activation of Akt1/PKBα in Muscle Fibers at the Myotendinous Junction within Muscle-Tendon-Bone Unit

The cell specific detection of enzyme activation in response to the physiological contractile load within muscle-tendon-bone unit is essential for understanding of the mechanical forces transmission from muscle cells via tendon to the bone. The hypothesis that the physiological mechanical loading regulates activation of Akt1/PKBα at Thr308 and at Ser473 in muscle fibers within muscle-tendon-bone unit was tested using quantitative immunohistochemistry, confocal double fluorescence analysis, and immunoblot analysis. In comparison to the staining intensities in peripheral regions of the muscle fibers, Akt1/PKBα was detected with a higher staining intensity in muscle fibers at the myotendinous junction (MTJ) areas. In muscle fibers at the MTJ areas, Akt1/PKBα is dually phosphorylated at Thr308 and Ser473. The immunohistochemical results were confirmed by immunoblot analysis. We conclude that contractile load generated by masticatory muscles induces local domain-dependent expression of Akt1/PKBα as well as activation by dually phosphorylation at Thr308 and Ser473 in muscle fibers at the MTJ areas within muscle-tendon-bone unit

The Molecular Assembly of Amyloid Aβ Controls Its Neurotoxicity and Binding to Cellular Proteins

Accumulation of β-sheet-rich peptide (Aβ) is strongly associated with Alzheimer's disease, characterized by reduction in synapse density, structural alterations of dendritic spines, modification of synaptic protein expression, loss of long-term potentiation and neuronal cell death. Aβ species are potent neurotoxins, however the molecular mechanism responsible for Aβ toxicity is still unknown. Numerous mechanisms of toxicity were proposed, although there is no agreement about their relative importance in disease pathogenesis. Here, the toxicity of Aβ 1–40 and Aβ 1–42 monomers, oligomers or fibrils, was evaluated using the N2a cell line. A structure-function relationship between peptide aggregation state and toxic properties was established. Moreover, we demonstrated that Aβ toxic species cross the plasma membrane, accumulate in cells and bind to a variety of internal proteins, especially on the cytoskeleton and in the endoplasmatic reticulum (ER). Based on these data we suggest that numerous proteins act as Aβ receptors in N2a cells, triggering a multi factorial toxicity