Physical Sciences and Mathematics Commons^™

Viscosity Of Bacterial Suspensions: Hydrodynamic Interactions And Self-Induced Noise, Shawn D. Ryan, Brian M. Haines, Leonid Berlyand, Falko Ziebert, Igor S. Aranson

Shawn Ryan

The viscosity of a suspension of swimming bacteria is investigated analytically and numerically. We propose a simple model that allows for efficient computation for a large number of bacteria. Our calculations show that long-range hydrodynamic interactions, intrinsic to self-locomoting objects in a viscous fluid, result in a dramatic reduction of the effective viscosity. In agreement with experiments on suspensions of Bacillus subtilis, we show that the viscosity reduction is related to the onset of large-scale collective motion due to interactions between the swimmers. The simulations reveal that the viscosity reduction occurs only for relatively low concentrations of swimmers: Further …

An Elastica Model Of The Buckling Of A Nanoscale Sheet Perpendicular To A Rigid Substrate, Shawn D. Ryan, Dmitry Golovaty, Patrick Wilber

Shawn Ryan

We study a variation on the classical problem of the buckling of an elastica. The elastica models a nanoscale sheet that interacts with a rigid substrate by intermolecular forces. We formulate a buckling problem in which the sheet is perpendicular to the substrate and a load is applied to the edge of the sheet further from the substrate. Our study is motivated by problems in nanomechanics such as the bending of a graphene sheet interacting with a rigid substrate by van der Waals forces. After identifying a trivial branch, we combine computation and analysis to determine the stability and bifurcations …

A Kinetic Model For Semidilute Bacterial Suspensions, Shawn D. Ryan, Leonid Berlyand, Brian M. Haines, D. A. Karpeev

Shawn Ryan

Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated with the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal …

Nonsupereulerian Graphs With Large Size, Paul A. Catlin, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

On The Edge Arboricity Of A Random Graph, P. A. Catlin, Zhi-Hong Chen, E. M. Palmer

Zhi-Hong Chen

No abstract provided.

On Hamiltonian Line Graphs, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

Supereuleriaun Graphs And The Petersen Graph, Zhi-Hong Chen

Zhi-Hong Chen

Using a contraction method, we find some best-possible sufficient condi­tions for 3-edge-connected simple graphs such that either the graphs have spanning eulerian subgraphs or the graphs are contractible to the Petersen graph.

Supereulerian Graphs And The Petersen Graph, Ii, Zhi-Hong Chen, Hong-Jian Lai

Zhi-Hong Chen

In this note, we verify two conjectures of Catlin in [J. Graph Theory 13 (1989) 465 - 483] for graphs with at most 11 vertices. These are used to prove the following theorem which improves prior results in [10] and [13]:

Let G be a 3-edge-connected simple graph with order n. If n is large and if for every edge 11.v E E(G), d(u) + d(v) 2 % - 2, then either G has a spanning eulerian subgraph or G can be contracted to the Petersen graph.

Fan-Type Conditions For Collapsible Graphs, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

Connectivity Of Cycle Matroids And Bicircular Matroids, Zhi-Hong Chen, Kuang Ying-Qiang, Hong-Jian Lai

Zhi-Hong Chen

A unified approach to prove former connectivity results of Tutte, Cunningham, Inukai and Weinberg, Oxley and Wagner.

Reinforcing The Number Of Disjoint Spanning Trees, Damin Liu, Hong-Jian Lai, Zhi-Hong Chen

Zhi-Hong Chen

The spanning tree packing number of a connected graph G, denoted by _T(G), is the maximum number of edge-disjoint spanning trees of G. In this paper, we determine the minimum number of edges that must be added to G so that the resulting graph has spanning tree packing number at least k, for a given value of k.

Even Subgraphs Of A Graph, Hong-Jian Lai, Zhi-Hong Chen

Zhi-Hong Chen

No abstract provided.

Closed-Form Probability Distribution Of Number Of Infections At A Given Time In A Stochastic Sis Epidemic Model, Olusegun M. Otunuga

Olusegun Michael Otunuga

We study the effects of external fluctuations in the transmission rate of certain diseases and how these affect the distribution of the number of infected individuals over time. To do this, we introduce random noise in the transmission rate in a deterministic SIS model and study how the number of infections changes over time. The objective of this work is to derive and analyze the closed form probability distribution of the number of infections at a given time in the resulting stochastic SIS epidemic model. Using the Fokker-Planck equation, we reduce the differential equation governing the number of infections to …

Spanning Eulerian Subgraphs And Catlin’S Reduced Graphs, Wei-Guo Chen, Zhi-Hong Chen

Zhi-Hong Chen

A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph H_R of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n …

Properties Of Catlin’S Reduced Graphs And Supereulerian Graphs, Wei-Guo Chen, Zhi-Hong Chen, Mei Lu

Zhi-Hong Chen

A graph G is called collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is the reduction of G if it is obtained from G by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs G of order n with d(u) + d(v) ≥ 2(n/p − …

Lai’S Conditions For Spanning And Dominating Closed Trails, Wei-Guo Chen, Zhi-Hong Chen, Mei Lu

Zhi-Hong Chen

No abstract provided.

Closed-Form Probability Distribution Of Number Of Infections At A Given Time In A Stochastic Sis Epidemic Model.Pdf, Michael Otunuga

Olusegun Michael Otunuga

Reverse Mathematics Of Matroids, Jeffry L. Hirst, Carl Mummert

Carl Mummert

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \Sigma^0_2 formulas.

Acid And Base Stress And Transcriptomic Responses In Bacillus Subtilis., Brian Jones, Jessica C. Wilks, Ryan D. Kitko, Sarah H. Cleeton, Grace E. Lee, Chinagozi S. Ugwu, Sandra S. Bondurant, Joan L. Slonczewski

Brian Jones

Acid and base environmental stress responses were investigated in Bacillus subtilis. B. subtilis AG174 cultures in buffered potassium-modified Luria broth were switched from pH 8.5 to pH 6.0 and recovered growth rapidly, whereas cultures switched from pH 6.0 to pH 8.5 showed a long lag time. Log-phase cultures at pH 6.0 survived 60 to 100% at pH 4.5, whereas cells grown at pH 7.0 survived <15%. Cells grown at pH 9.0 survived 40 to 100% at pH 10, whereas cells grown at pH 7.0 survived <5%. Thus, growth in a moderate acid or base induced adaptation to a more extreme acid or base, respectively. Expression indices from Affymetrix chip hybridization were obtained for 4,095 protein-encoding open reading frames of B. subtilis grown at external pH 6, pH 7, and pH 9. Growth at pH 6 upregulated acetoin production (alsDS), dehydrogenases (adhA, ald, fdhD, and gabD), and decarboxylases (psd and speA). Acid upregulated malate metabolism (maeN), metal export (czcDO and cadA), oxidative stress (catalase katA; OYE family namA), and the SigX extracytoplasmic stress regulon. Growth at pH 9 upregulated arginine catabolism (roc), which generates organic acids, glutamate synthase (gltAB), polyamine acetylation and transport (blt), the K(+)/H(+) antiporter (yhaTU), and cytochrome oxidoreductases (cyd, ctaACE, and qcrC). The SigH, SigL, and SigW regulons were upregulated at high pH. Overall, greater genetic adaptation was seen at pH 9 than at pH 6, which may explain the lag time required for growth shift to high pH. Low external pH favored dehydrogenases and decarboxylases that may consume acids and generate basic amines, whereas high external pH favored catabolism-generating acids.

Reverse Mathematics And Uniformity In Proofs Without Excluded Middle, Jeffry L. Hirst, Carl Mummert

Carl Mummert

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a \Pi^1_2 sentence of a certain form is provable using E-HA^ω along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles …

Reverse Mathematics And Properties Of Finite Character, Damir D. Dzhafarov, Carl Mummert

Carl Mummert

We study the reverse mathematics of the principle stating that,for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey’s lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these …

The Multilinear Structure Of Relu Networks, Thomas Laurent

Thomas Laurent

We study the loss surface of neural networks equipped with a hinge loss criterion and ReLU or leaky ReLU nonlinearities. Any such network defines a piecewise multilinear form in parameter space. By appealing to harmonic analysis we show that all local minima of such network are non-differentiable, except for those minima that occur in a region of parameter space where the loss surface is perfectly flat. Non-differentiable minima are therefore not technicalities or pathologies; they are heart of the problem when investigating the loss of ReLU networks. As a consequence, we must employ techniques from nonsmooth analysis to study these …

Deep Linear Networks With Arbitrary Loss: All Local Minima Are Global, Thomas Laurent

Thomas Laurent

We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possible in the following sense: If the loss is convex and Lipschitz but not differentiable then deep linear networks can have sub-optimal local minima.

Degree And Neighborhood Conditions For Hamiltonicity Of Claw-Free Graphs, Zhi-Hong Chen

Zhi-Hong Chen

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers …

Circumferences Of 3-Connected Claw-Free Graphs, Ii, Zhi-Hong Chen

Zhi-Hong Chen

For a graph H , the circumference of H , denoted by c ( H ) , is the length of a longest cycle in H . It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ ≥ 8 , then c ( H ) ≥ min { 9 δ − 3 , n } . In Li (2006), Li conjectured that every 3-connected k -regular claw-free graph H of order n has c ( H ) ≥ min { 10 k − 4 , n } . Later, Li posed …

A Geometric Generalizaion Of The Planar Gale-Nikaidô Theorem, Eduardo C. Balreira

Eduardo Cabral Balreira

The Gale-Nikaidô Theorem establishes global injectivity of maps defined over rectangular regions provided the Jacobian matrix is a P-matrix. We provide a purely geometric generalization of this result in the plane by showing that if the image of each edge of the rectangular domain is realized as a graph of a function over the appropriate axis, then the map is injective. We also show that the hypothesis that the Jacobian matrix is a P-matrix is simply one way to analytically check this geometric condition.

Global Stability Of Higher Dimensional Monotone Maps, Eduardo C. Balreira, Saber Elaydi, Rafael Luis

Eduardo Cabral Balreira

We develop a notion of monotonicity for maps defined on Euclidean spaces R^k₊, of arbitrary dimension k. This is a geometric approach that extends the classical notion of planar monotone maps or planar competitive difference equations. For planar maps, we show that our notion and the classical notion of monotonicity are equivalent. In higher dimensions, we establish certain verifiable conditions under which Kolmogorov monotone maps on R^k₊ have a globally asymptotically stable fixed point. We apply our results to two competition population models, the Leslie–Gower and the Ricker models of two- and three-species. It …

Two Games Displayed By Butler’S 2017 Celebration Of Mind, Jeremiah Farrell

Jeremiah Farrell

Jeremiah's two games displayed by Butler's 2017 Celebration of Mind.

Flying Saucer, Jeremiah Farrell, Karen Farrell

Jeremiah Farrell

Jeremiah's puzzle "Flying Saucer", which was exchanged at the 2013 International Puzzle Party in Washington, DC. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exchange them. This puzzle is also manufactured by Walter Hoppe as "Flying Saucer".

The French Resolution, Jeremiah Farrell, Karen Farrell

Jeremiah Farrell

Jeremiah's puzzle "The French Resolution", which was exchanged at the 2017 International Puzzle Party in Paris, France. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exchange them. This puzzle is also manufactured by Kadon Enterprises, Inc. as "The French Resolution".