Partial Isometries of a Sub-Riemannian Manifold

In this paper, we obtain the following generalisation of isometric $C^1$-immersion theorem of Nash and Kuiper. Let $M$ be a smooth manifold of dimension $m$ and $H$ a rank $k$ subbundle of the tangent bundle $TM$ with a Riemannian metric $g_H$. Then the pair $(H,g_H)$ defines a sub-Riemannian structure on $M$. We call a $C^1$-map $f:(M,H,g_H)\to (N,h)$ into a Riemannian manifold $(N,h)$ a {\em partial isometry} if the derivative map $df$ restricted to $H$ is isometric; in other words, $f^*h|_H=g_H$. The main result states that if $\dim N>k$ then a smooth $H$-immersion $f_0:M\to N$ satisfying $f^*h|_H<g_H$ can be homotoped to a partial isometry $f:(M,g_H)\to (N,h)$ which is $C^0$-close to $f_0$. In particular we prove that every sub-Riemannian manifold $(M,H,g_H)$ admits a partial isometry in $\R^n$ provided $n\geq m+k$.Comment: 13 pages. This is a revised version of an earlier submission (minor revision

Scaling of load in communications networks

We show that the load at each node in a preferential attachment network scales as a power of the degree of the node. For a network whose degree distribution is p(k) ~ k^(-gamma), we show that the load is l(k) ~ k^eta with eta = gamma - 1, implying that the probability distribution for the load is p(l) ~ 1/l^2 independent of gamma. The results are obtained through scaling arguments supported by finite size scaling studies. They contradict earlier claims, but are in agreement with the exact solution for the special case of tree graphs. Results are also presented for real communications networks at the IP layer, using the latest available data. Our analysis of the data shows relatively poor power-law degree distributions as compared to the scaling of the load versus degree. This emphasizes the importance of the load in network analysis.Comment: 4 pages, 5 figure

Movement-Related Cortical Potential Amplitude Reduction after Cycling Exercise Relates to the Extent of Neuromuscular Fatigue.

Exercise-induced fatigue affects the motor control and the ability to generate a given force or power. Surface electroencephalography allows researchers to investigate movement-related cortical potentials (MRCP), which reflect preparatory brain activity 1.5 s before movement onset. Although the MRCP amplitude appears to increase after repetitive single-joint contractions, the effects of large-muscle group dynamic exercise on such pre-motor potential remain to be described. Sixteen volunteers exercised 30 min at 60% of the maximal aerobic power on a cycle ergometer, followed by a 10-km all-out time trial. Before and after each of these tasks, knee extensor neuromuscular function was investigated using maximal voluntary contractions (MVC) combined with electrical stimulations of the femoral nerve. MRCP was recorded during 60 knee extensions after each neuromuscular sequence. The exercise resulted in a significant decrease in the knee extensor MVC force after the 30-min exercise (-10 ± 8%) and the time trial (-21 ± 9%). The voluntary activation level (VAL; -6 ± 8 and -12 ± 10%), peak twitch (Pt; -21 ± 16 and -32 ± 17%), and paired stimuli (P100 Hz; -7 ± 11 and -12 ± 13%) were also significantly reduced after the 30-min exercise and the time trial. The first exercise was followed by a decrease in the MRCP, mainly above the mean activity measured at electrodes FC1-FC2, whereas the reduction observed after the time trial was related to the FC1-FC2 and C2 electrodes. After both exercises, the reduction in the late MRCP component above FC1-FC2 was significantly correlated with the reduction in P100 Hz (r = 0.61), and the reduction in the same component above C2 was significantly correlated with the reduction in VAL (r = 0.64). In conclusion, large-muscle group exercise induced a reduction in pre-motor potential, which was related to muscle alterations and resulted in the inability to produce a maximal voluntary contraction

The Large Scale Curvature of Networks

Understanding key structural properties of large scale networks are crucial for analyzing and optimizing their performance, and improving their reliability and security. Here we show that these networks possess a previously unnoticed feature, global curvature, which we argue has a major impact on core congestion: the load at the core of a network with N nodes scales as N^2 as compared to N^1.5 for a flat network. We substantiate this claim through analysis of a collection of real data networks across the globe as measured and documented by previous researchers.Comment: 4 pages, 5 figure

A computational framework for identifying design guidelines to increase the penetration of targeted nanoparticles into tumors

Targeted nanoparticles are increasingly being engineered for the treatment of cancer. By design, they can passively accumulate in tumors, selectively bind to targets in their environment, and deliver localized treatments. However, the penetration of targeted nanoparticles deep into tissue can be hindered by their slow diffusion and a high binding affinity. As a result, they often localize to areas around the vessels from which they extravasate, never reaching the deep-seeded tumor cells, thereby limiting their efficacy. To increase tissue penetration and cellular accumulation, we propose generalizable guidelines for nanoparticle design and validate them using two different computer models that capture the potency, motion, binding kinetics, and cellular internalization of targeted nanoparticles in a section of tumor tissue. One strategy that emerged from the models was delaying nanoparticle binding until after the nanoparticles have had time to diffuse deep into the tissue. Results show that nanoparticles that are designed according to these guidelines do not require fine-tuning of their kinetics or size and can be administered in lower doses than classical targeted nanoparticles for a desired tissue penetration in a large variety of tumor scenarios. In the future, similar models could serve as a testbed to explore engineered tissue-distributions that arise when large numbers of nanoparticles interact in a tumor environment.Human Frontier Science Program (Strasbourg, France)David H. Koch Institute for Integrative Cancer Research at MIT (Marie D. and Pierre Casimir-Lambert Fund)National Institutes of Health (U.S.) (Grant U54 CA151884)National Cancer Institute (U.S.) (Koch Institute Support (Core) Grant P30-CA14051

Multidimensional cut-off technique, odd-dimensional Epstein zeta functions and Casimir energy of massless scalar fields

Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose-Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement - due to quantum fluctuations - in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in $d$-dimensional rectangular spaces with $q$ large dimensions and $d-q$ dimensions of length $L$ and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulas we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization.Comment: 42 pages, 3 figures. v.2: typos corrected to match published versio

Self-adjustment mechanisms and their application for orthosis design

Medical orthoses aim at guiding anatomical joints along their natural trajectories while preventing pathological movements, especially in case of trauma or injuries. The motions that take place between bone surfaces have complex kinematics. These so-called arthrokinematic motions exhibit axes that move both in translation and rotation. Traditionally, orthoses are carefully adjusted and positioned such that their kinematics approximate the arthrokinematic movements as closely as possible in order to protect the joint. Adjustment procedures are typically long and tedious. We suggest in this paper another approach. We propose mechanisms having intrinsic self-aligning properties. They are designed such that their main axis self-adjusts with respect to the joint’s physiological axis during motion. When connected to a limb, their movement becomes homokinetic and they have the property of automatically minimizing internal stresses. The study is performed here in the planar case focusing on the most important component of the arthrokinematic motions of a knee joint

β-Catenin is a pH sensor with decreased stability at higher intracellular pH.

β-Catenin functions as an adherens junction protein for cell-cell adhesion and as a signaling protein. β-catenin function is dependent on its stability, which is regulated by protein-protein interactions that stabilize β-catenin or target it for proteasome-mediated degradation. In this study, we show that β-catenin stability is regulated by intracellular pH (pHi) dynamics, with decreased stability at higher pHi in both mammalian cells and Drosophila melanogaster β-Catenin degradation requires phosphorylation of N-terminal residues for recognition by the E3 ligase β-TrCP. While β-catenin phosphorylation was pH independent, higher pHi induced increased β-TrCP binding and decreased β-catenin stability. An evolutionarily conserved histidine in β-catenin (found in the β-TrCP DSGIHS destruction motif) is required for pH-dependent binding to β-TrCP. Expressing a cancer-associated H36R-β-catenin mutant in the Drosophila eye was sufficient to induce Wnt signaling and produced pronounced tumors not seen with other oncogenic β-catenin alleles. We identify pHi dynamics as a previously unrecognized regulator of β-catenin stability, functioning in coincidence with phosphorylation

Genomic Expansion of Magnetotactic Bacteria Reveals an Early Common Origin of Magnetotaxis with Lineage-specific Evolution

The origin and evolution of magnetoreception, which in diverse prokaryotes and protozoa is known as magnetotaxis and enables these microorganisms to detect Earth’s magnetic field for orientation and navigation, is not well understood in evolutionary biology. The only known prokaryotes capable of sensing the geomagnetic field are magnetotactic bacteria (MTB), motile microorganisms that biomineralize intracellular, membrane-bounded magnetic single-domain crystals of either magnetite (Fe3O4) or greigite (Fe3S4) called magnetosomes. Magnetosomes are responsible for magnetotaxis in MTB. Here we report the first large-scale metagenomic survey of MTB from both northern and southern hemispheres combined with 28 genomes from uncultivated MTB. These genomes expand greatly the coverage of MTB in the Proteobacteria, Nitrospirae, and Omnitrophica phyla, and provide the first genomic evidence of MTB belonging to the Zetaproteobacteria and “Candidatus Lambdaproteobacteria” classes. The gene content and organization of magnetosome gene clusters, which are physically grouped genes that encode proteins for magnetosome biosynthesis and organization, are more conserved within phylogenetically similar groups than between different taxonomic lineages. Moreover, the phylogenies of core magnetosome proteins form monophyletic clades. Together, these results suggest a common ancient origin of iron-based (Fe3O4 and Fe3S4) magnetotaxis in the domain Bacteria that underwent lineage-specific evolution, shedding new light on the origin and evolution of biomineralization and magnetotaxis, and expanding significantly the phylogenomic representation of MTB

On the experimental verification of quantum complexity in linear optics

The first quantum technologies to solve computational problems that are beyond the capabilities of classical computers are likely to be devices that exploit characteristics inherent to a particular physical system, to tackle a bespoke problem suited to those characteristics. Evidence implies that the detection of ensembles of photons, which have propagated through a linear optical circuit, is equivalent to sampling from a probability distribution that is intractable to classical simulation. However, it is probable that the complexity of this type of sampling problem means that its solution is classically unverifiable within a feasible number of trials, and the task of establishing correct operation becomes one of gathering sufficiently convincing circumstantial evidence. Here, we develop scalable methods to experimentally establish correct operation for this class of sampling algorithm, which we implement with two different types of optical circuits for 3, 4, and 5 photons, on Hilbert spaces of up to 50,000 dimensions. With only a small number of trials, we establish a confidence >99% that we are not sampling from a uniform distribution or a classical distribution, and we demonstrate a unitary specific witness that functions robustly for small amounts of data. Like the algorithmic operations they endorse, our methods exploit the characteristics native to the quantum system in question. Here we observe and make an application of a "bosonic clouding" phenomenon, interesting in its own right, where photons are found in local groups of modes superposed across two locations. Our broad approach is likely to be practical for all architectures for quantum technologies where formal verification methods for quantum algorithms are either intractable or unknown.Comment: Comments welcom