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To reach sub-picometer sensitivity in the millihertz range, displacement sensors based on laser interferometry require suppression of laser-frequency noise by several orders of magnitude. Many optical frequency stabilization methods exist with varying levels of complexity, size, and performance. In this paper, we describe the performance of a compact Mach-Zehnder interferometer based on a monolithic optic. The setup consists of a commercial fiber injector, a custom-designed pentaprism used to split and recombine the laser beam, and two photoreceivers placed at the complementary output ports of the interferometer. The structural stability of the prism is transferred to the laser frequency via amplification, integration, and feedback of the balanced-detection signal, achieving a fractional frequency instability better than 6 parts in $10^{13}$, corresponding to an interferometer pathlength stability better than $10^{-12}$ m$/\sqrt{\mathrm{Hz}}$., Comment: 10 pages, 4 figures

Bacteria meticulously regulate their intracellular ion concentrations and create ionic concentration gradients across the bacterial membrane. These ionic concentration gradients provide free energy for many cellular processes and are maintained by transmembrane transport. Given the physical dimensions of a bacterium and the stochasticity in transmembrane transport, intracellular ion concentrations and hence the charge state of a bacterium are bound to fluctuate. Here, we investigate the charge noise of 100s of non-motile bacteria by combining electrical measurement techniques from condensed matter physics with microfluidics. In our experiments, bacteria in a microchannel generate charge density fluctuations in the embedding electrolyte due to random influx and efflux of ions. Detected as electrical resistance noise, these charge density fluctuations display a power spectral density proportional to $1/f^2$ for frequencies $0.05~{\rm Hz} \leq f \leq 1 ~{\rm Hz}$. Fits to a simple noise model suggest that the steady-state charge of a bacterium fluctuates by $\pm 1.30 \times 10^6 {e}~({e} \approx 1.60 \times 10^{-19}~{\rm C})$, indicating that bacterial ion homeostasis is highly dynamic and dominated by strong charge noise. The rms charge noise can then be used to estimate the fluctuations in the membrane potential; however, the estimates are unreliable due to our limited understanding of the intracellular concentration gradients., Comment: Main article 24 pages, 12 figures, 1 table; supplementary 6 pages, 6 figure, 1 table

The Laser Ranging Interferometer onboard the Gravity Recovery and Climate Experiment Follow-On mission proved the feasibility of an interferometric sensor for inter-satellite length tracking with sub-nanometer precision, establishing an important milestone for space laser interferometry and the general expectation that future gravity missions will employ heterodyne laser interferometry for satellite-to-satellite ranging. In this paper we present the design of an on-axis optical bench for next-generation laser ranging which enhances the received optical power and the transmit beam divergence, enabling longer interferometer arms and relaxing the optical power requirement of the laser assembly. All design functionalities and requirements are verified by means of computer simulations. A thermal analysis is carried out to investigate the robustness of the proposed optical bench to the temperature fluctuations found in orbit., Comment: 18 pages, 8 figures

Tracking moving masses in several degrees of freedom with high precision and large dynamic range is a central aspect in many current and future gravitational physics experiments. Laser interferometers have been established as one of the tools of choice for such measurement schemes. Using sinusoidal phase modulation homodyne interferometry allows a drastic reduction of the complexity of the optical setup, a key limitation of multi-channel interferometry. By shifting the complexity of the setup to the signal processing stage, these methods enable devices with a size and weight not feasible using conventional techniques. In this paper we present the design of a novel sensor topology based on deep frequency modulation interferometry: the self-referenced single-element dual-interferometer (SEDI) inertial sensor, which takes simplification one step further by accommodating two interferometers in one optic. Using a combination of computer models and analytical methods we show that an inertial sensor with sub-picometer precision for frequencies above 10 mHz, in a package of a few cubic inches, seems feasible with our approach. Moreover we show that by combining two of these devices it is possible to reach sub-picometer precision down to 2 mHz. In combination with the given compactness, this makes the SEDI sensor a promising approach for applications in high precision inertial sensing for both next-generation space-based gravity missions employing drag-free control, and ground-based experiments employing inertial isolation systems with optical readout., Comment: 18 pages, 9 figures

The lack of rapid antibiotic susceptibility tests adversely affects the treatment of bacterial infections and contributes to increased prevalence of multidrug resistant bacteria. Here, we describe an all-electrical approach that allows for ultra-sensitive measurement of growth signals from only tens of bacteria in a microfluidic device. Our device is essentially a set of microfluidic channels, each with a nano-constriction at one end and cross-sectional dimensions close to that of a single bacterium. Flowing a liquid bacteria sample (e.g., urine) through the microchannels rapidly traps the bacteria in the device, allowing for subsequent incubation in drugs. We measure the electrical resistance of the microchannels, which increases (or decreases) in proportion to the number of bacteria in the microchannels. The method and device allow for rapid antibiotic susceptibility tests in about two hours. Further, the short-time fluctuations in the electrical resistance during an antibiotic susceptibility test are correlated with the morphological changes of bacteria caused by the antibiotic. In contrast to other electrical approaches, the underlying geometric blockage effect provides a robust and sensitive signal, which is straightforward to interpret without electrical models. The approach also obviates the need for a high-resolution microscope and other complex equipment, making it potentially usable in resource-limited settings., Comment: Main article 8 pages, 4 figures; supplementary 15 pages, 6 figure, 4 tables, 2 movies

We study the canonical orbit category of the bounded derived category of finite dimensional representations of the quiver of type $D_{\infty}$. We prove that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure as defined in [\ref{BIRS}]., Comment: 21 pages, 5 figures. arXiv admin note: text overlap with arXiv:1505.06062 by other authors

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for (rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also, we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case., Comment: 41 pages

In this short paper we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every $\tau$-tilting module is tilting., Comment: 4 pages, 0 figure

In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $\tau$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$., Comment: 9 pages, 6 figures

In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the $11$ exceptional cases, the distribution of genuses is $0$ or $1$. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It is verified that the genus of an oriented surface is an upper bound for the genuses of cluster quivers from this surface. Furthermore, for any non-negative integer $n$ and a closed oriented surface of genus $n$, we show that there always exist a set of punctures and a triangulation of this surface such that the corresponding cluster quiver from this triangulation is exactly of genus $n$., Comment: 14 pages, 17 figures, Pacific Journal of Mathematics, 2014. arXiv admin note: text overlap with arXiv:math/0608367 by other authors

In general, representation rings are well-known as Green rings from module categories of Hopf algebras. In this paper, we study Green rings in the context of monoidal categories such that representations of Hopf algebras can be investigated through Green rings of various levels from module categories to derived categories in the unified view-point. Firstly, as analogue of representation rings of Hopf algebras, we set up the so-called Green rings of monoidal categories, and then list some such categories including module categories, complex categories, homotopy categories, derived categories and (derived) shift categories, etc. and the relationship among their corresponding Green rings. The main part of this paper is to characterize representation rings and derived rings of a class of inite dimensional Hopf algebras constructed from the Nakayama truncated algebras $KZ_{n}/J^{d}$ with certain constraints. For the representation ring $r(KZ_{n}/J^{d})$, we completely determine its generators and the relations of generators via the method of Pascal triangle. For the derived ring $dr(KZ_{n}/J^{2})$(i.e., $d=2$), we determine its generators and give the relations of generators. In these two aspects, the polynomial characterizations of the representation ring and the derived ring are both given., Comment: 34 pages, 7 figures

As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevisky \cite{[4]}, in this paper, we give the test method of positive definite matrices via the planar networks and the so-called cluster subalgebra respectively, introduced here originally. This work firstly gives a combinatorial realization of all matrices through planar network, and then sets up a test method for positive definite matrices by $LDU$-decompositions and the horizontal weightings of all lines in their planar networks. On the other hand, mainly the relationship is built between positive definite matrices and cluster subalgebras., Comment: 11 pages, 6 figures