Bipolar Orientations Research Articles

On the enumeration of plane bipolar posets and transversal structures

We show that plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection due to Kenyon, Miller, Sheffield and Wilson. We then derive exact and asymptotic counting results. In particular we prove (computationally and then bijectively) that the number of plane bipolar posets on n+2 vertices equals the number of plane permutations (i.e., avoiding the vincular pattern ▪ ) of size n. Regarding transversal structures, for each v≥0 we consider tn(v) the number of such structures with n+4 vertices and weight v per quadrangular inner face (the case v=0 corresponds to having only triangular inner faces). We obtain a recurrence to compute tn(v), and an asymptotic formula that for v=0 gives tn(0)∼c(27/2)nn−1−π/arccos(7/8) for some c>0, which also ensures that the associated generating function is not D-finite.

Modulation of water transport in carbon nanotubes by local charges

Carbon nanotubes (CNTs) enable extremely fast water transport that make them strong candidates for constructing artificial nanochannels and membranes. However, water transport properties in CNTs with defects along with local charges are rarely studied. In this work, with molecular dynamics simulations, it is found that water transport in CNTs can be greatly modulated by local charges at the CNT middle. The flux modulation factors show an increasing trend with CNT diameters and are slightly influenced by the axial positions of local charges and the CNT lengths. Analysis suggests that water transport is hindered by the confinement-enhanced concentration oscillations and bipolar orientations of water molecules induced by local charges. It is also interesting that the water flux of CNT (6, 6) with high local charges has a reversal behavior due to the overscreening effect. Simultaneously, ions are also impeded due to the water structuring and bipolarizations, which can improve the salt rejection of wide CNTs. In general, introducing local charges on CNT (10, 10) can reject 100% ions while maintaining a high water permeability of ∼90 L/cm2/day/Mpa in the reverse osmosis desalination. This study promotes the understanding of water transport in nanochannels with defects and guides the design of CNTs for high-performance desalination membranes.

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Baxter permutations, plane bipolar orientations, and a specific family of walks in the nonnegative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes and we relate it to the three families mentioned above. We prove joint Benjamini–Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure on the unit square, called the Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. In addition, we relate the limiting objects of the four families to each other, both in the local and scaling limit case. The scaling limit result is based on the convergence of the trajectories of the coalescent-walk process to the coalescing flow—in the terminology of Le Jan and Raimond (Ann. Probab. 32 (2004) 1247–1315)—of a perturbed version of the Tanaka stochastic differential equation. Our scaling result entails joint convergence of the tandem walks of a plane bipolar orientation and its dual, giving an alternative answer to Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (Ann. Probab. 47 (2019) 1240–1269) compared to the one of Gwynne, Holden, Sun ((2016), arXiv:1603.01194).

Random walk on random planar maps: Spectral dimension, resistance and displacement

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a “mating-of-trees” type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the γ-mated-CRT map for γ∈(0,2). For each of these maps, we obtain an upper bound for the Green’s function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after n steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors. When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after 2n steps is n−1+on(1). Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least n1/4−on(1) units of graph distance in n units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (Probab. Theory Related Fields 178 (2020) 567–611). These two works together resolve a conjecture of Benjamini and Curien (Geom. Funct. Anal. 23 (2013) 501–531) in the UIPT case. Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.

Utility of novel Omnipolar activation mapping for the detection of ventricular premature contraction origin

Abstract Background Bipolar electrograms are significantly influenced by direction of the propagating wavefront in relation to the recording bipole. Omnipolar voltage mapping may be superior to standard bipolar mapping since it obtains maximum voltage of all possible bipolar electrode orientations without the need for catheter rotation. Therefore, omnipolar maps can provide voltage maps with larger voltages as well as better defined boundaries. Purpose Whether omnipolar activation maps also describe better activation maps versus traditional bipolar maps during ventricular premature contraction (VPC) catheter ablation is unclear. Methods A high-density mapping catheter was advanced to the ventricular outflow tract and a high-resolution activation map was created. Each electrode along and across the splines of the catheter are 4mm apart. Bipoles were calculated along (MAP 2), across (MAP 3) and bidirectional (MAP 4) the splines while omnipoles (MAP 1) were derived from a right triangle clique. Within a square area, four omnipolar and two bipolar values along, across and bidirectional values were defined. Results Though the earliest activation site was vague by along and across maps (arrow), white color became evident by bidirectional map, and the VPC origin became distinct with omnipolar mapping. RF lesions were given via an open-irrigated ablation catheter targeting a lesion size index 5.0. The VPC was eliminated by radiofrequency ablation. Conclusion Omnipolar activation mapping may be more accurate than traditional bipolar mapping during ventricular premature contraction (VPC) catheter ablation. Figure 1 Funding Acknowledgement Type of funding source: None

A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; gamma =sqrt{8/3}); and planar maps weighted by the number of different spanning trees (gamma =sqrt{2}), bipolar orientations (gamma =sqrt{4/3}), or Schnyder woods (gamma =1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of gamma -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when gamma =sqrt{8/3}, we instead deduce estimates for the sqrt{8/3}-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

The active bijection for graphs

The active bijection forms a package of results studied by the authors in a series of papers in oriented matroids. The present paper is intended to state the main results in the particular case, and more widespread language, of graphs. We study fundamental properties of (directed) graphs on a linearly ordered set of edges. The central result is that, for a directed graph on a linearly ordered set of edges, we determine in a canonical way one particular spanning tree, which we call the active spanning tree and which has important properties. For any graph on a linearly ordered set of edges, this yields a surjective mapping from orientations onto spanning trees, which preserves activities (for orientations in the sense of Las Vergnas, for spanning trees in the sense of Tutte), as well as some partitions (or filtrations) of the edge set associated with orientations and spanning trees. This yields a canonical bijection between classes of orientations and spanning trees, as well as a refined bijection between all orientations and edge subsets, containing various notable bijections, for instance: between orientations in which smallest edges of cycles and cocycles have a fixed orientation and spanning trees; or between acyclic orientations and no-broken-circuit subsets. Several constructions of independent interest are involved. The basic case concerns bipolar orientations, which are in bijection with their fully optimal spanning trees, as proved in a previous paper. We give a canonical decomposition of a directed graph on a linearly ordered set of edges into acyclic/cyclic bipolar directed graphs. Considering all orientations of a graph, we obtain an expression of the Tutte polynomial in terms of products of beta invariants of minors, an important partition of the set of orientations into activity classes, and a simple expression of the Tutte polynomial using four orientation-activity parameters. We derive a similar decomposition theorem for spanning trees. We also provide a general deletion/contraction framework for these bijections and relatives.

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, \begin{document}$u_t = u_{xx} + f(x, u, u_x)\, , $ \end{document} on the unit interval \begin{document}$0 with Neumann boundary conditions. Equilibria \begin{document}$v_t = 0$\end{document} are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} as 3-cell templates \begin{document}$\mathcal{C}$\end{document} . The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries \begin{document}$x = 0$\end{document} and \begin{document}$x = 1$\end{document} , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.

Co-segregation of sex chromosomes in the male black widow spider Latrodectus mactans (Araneae, Theridiidae).

During meiosis I, homologous chromosomes join together to form bivalents. Through trial and error, bivalents achieve stable bipolar orientations (attachments) on the spindle that eventually allow the segregation of homologous chromosomes to opposite poles. Bipolar orientations are stable through tension generated by poleward forces to opposite poles. Unipolar orientations lack tension and are stereotypically not stable. The behavior of sex chromosomes during meiosis I in the male black widow spider Latrodectus mactans (Araneae, Theridiidae) challenges the principles governing such a scenario. We found that male L. mactans has two distinct X chromosomes, X1 and X2. The X chromosomes join together to form a connection that is present in prometaphase I but is lost during metaphase I, before the autosomes disjoin at anaphase I. We found that both X chromosomes form stable unipolar orientations to the same pole that assure their co-segregation at anaphase I. Using micromanipulation, immunofluorescence microscopy, and electron microscopy, we studied this unusual chromosome behavior to explain how it may fit the current dogma of chromosome distribution during cell division.

A Generic Method for Bijections between Blossoming Trees and Planar Maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of application so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face.The bijective construction presented here relies deeply on the theory of $\alpha$-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases many previously known bijections involving blossoming trees: for example Eulerian maps, $m$-Eulerian maps, non-separable maps and simple triangulations and quadrangulations of a $k$-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and $d$-angulations of girth $d$ of a $k$-gon.As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode or sample planar maps.In a recent work, Bernardi and Fusy introduced another unified bijective scheme; we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.

Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes

We study global attractors $\mathcal{A}_f$ of scalar partial differential equations $u_t=u_{xx}+f(x,u,u_x)$ on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call $\mathcal{A}_f$ a Sturm global attractor. We assume all equilibria $v$ to be hyperbolic. Due to a gradient-like structure we can then write \begin{equation} \mathcal{A}_f = \bigcup\limits_{v}\, W^u(v) (*) \end{equation} as a dynamic decomposition into finitely many disjoint invariant sets: the unstable manifolds $W^u(v)$ of the equilibria $v$. Based on our previous Schoenflies result [17], we prove that the dynamic decomposition $(*)$ is in fact a regular finite CW-complex with cells $W^u(v)$, in the Sturm case. We call this complex the regular dynamic complex or Sturm complex of the Sturm attractor $\mathcal{A}_f$. We characterize the planar Sturm complexes by bipolar orientations of their 1-skeletons. We also show that any regular finite CW-complex which is the closure of a single 3-cell arises as a Sturm complex. We include a preliminary discussion of the tetrahedron and the octahedron as Sturm complexes.

Motor-mediated bidirectional transport along an antipolar microtubule bundle: A mathematical model

Long-distance bidirectional transport of organelles depends on the coordinated motion of various motor proteins on the cytoskeleton. Recent quantitative live cell imaging in the elongated hyphal cells of Ustilago maydis has demonstrated that long-range motility of motors and their endosomal cargo occurs on unipolar microtubules (MTs) near the extremities of the cell. These MTs are bundled into antipolar bundles within the central part of the cell. Dynein and kinesin-3 motors coordinate their activity to move early endosomes (EEs) in a bidirectional fashion where dynein drives motility towards MT minus ends and kinesin towards MT plus ends. Although this means that one can easily assign the drivers of bidirectional motion in the unipolar section, the bipolar orientations in the bundle mean that it is possible for either motor to drive motion in either direction. In this paper we use a multilane asymmetric simple exclusion process modeling approach to simulate and investigate phases of bidirectional motility in a minimal model of an antipolar MT bundle. In our model, EE cargos (particles) change direction on each MT with a turning rate Ω and there is switching between MTs in the bundle at the minus ends. At these ends, particles can hop between MTs with rate q(1) on passing from a unipolar to a bipolar section (the obstacle-induced switching rate) or q(2) on passing in the other direction (the end-induced switching rate). By a combination of numerical simulations and mean-field approximations, we investigate the distribution of particles along the MTs for different values of these parameters and of Θ, the overall density of particles within this closed system. We find that even if Θ is low, the system can exhibit a variety of phases with shocks in the density profiles near plus and minus ends caused by queuing of particles. We discuss how the parameters influence the type of particle that dominates active transport in the bundle.

Molecular dynamics simulation of water conduction within carbon nanotube

Water molecule transportation through a single-walled carbon nanotube under a charge influence on tube was simulated. As the charge located at middle tube increases from 0.0e to 0.7e, water molecular orientations develop from a consistent orientation (0⩽q⩽0.4e) to bipolar orientations (L defect, when q⩾0.5e). Water molecule permeation through the channel is almost the same as in charge-free nanotube when q⩽0.2e. As q increases further the permeation decreases sharply. When q=0.6e the permeation becomes negligible and the tube channel effectively closes.

Local On/Off Currents in BiFeO3 Thin Films Modulated by Bipolar Polarization Orientations

Nonvolatile memories on the basis of tunneling junctions of ferroelectric ultrathin-film barriers make use of resistance switching between high and low conductance states upon polarization reversal, which facilitates the nondestructive readout of the binary information within a tiny memory cell. The apparent electroresistance effect for the generation of a large on/off current ratio depends on the modulation of the tunneling barrier height in two opposite polarization orientations due to asymmetric finite screening lengths of top and bottom electrodes, where the direct tunneling current attenuates quickly with enhanced film thickness. To break through the atomistic thickness requirement of the tunneling junction, we separately observed the same electroresistance effect in a semiconducting BiFeO3 film with the thickness of 120 nm. Its working mechanism depends on the formation of a ferroelectric diode for the film in contacts with top and bottom electrodes, where the polarity of diode junction can be switched by polarization reversal.

Partial truths in the pursuit of patient safety

The paper explores several issues in the form of partial truths that dominate current thinking as investigators continue their pursuit of patient safety. Among the partial truths examined—cast as bipolar...

Bijections for Baxter families and related objects

The Baxter number B n can be written as B n = ∑ k = 0 n Θ k , n − k − 1 with Θ k , ℓ = 2 ( k + 1 ) 2 ( k + 2 ) ( k + ℓ k ) ( k + ℓ + 1 k ) ( k + ℓ + 2 k ) . These numbers have first appeared in the enumeration of so-called Baxter permutations; B n is the number of Baxter permutations of size n, and Θ k , ℓ is the number of Baxter permutations with k descents and ℓ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers Θ k , ℓ . Apart from Baxter permutations, these include plane bipolar orientations with k + 2 vertices and ℓ + 2 faces, 2-orientations of planar quadrangulations with k + 2 white and ℓ + 2 black vertices, certain pairs of binary trees with k + 1 left and ℓ + 1 right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of Θ k , ℓ as an application of the lemma of Lindström Gessel–Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plane bipolar orientations, also Schnyder woods of triangulations. Most of the enumerative results and some of the bijections are not new. Our contribution is mainly in the simplified and unifying presentation of this beautiful piece of combinatorics.

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results. Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.

New bijective links on planar maps via orientations

This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific “transversal structures” on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are respectively non-separable maps and irreducible triangulations. This gives another bijective proof (after Wormald’s construction published in 1980) of the fact that rooted loopless maps with n edges are equinumerous to rooted triangulations with n inner vertices.

Bijective counting of plane bipolar orientations and Schnyder woods

A bijection Φ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to Baxter for the number Θ i j of plane bipolar orientations with i non-polar vertices and j inner faces: Θ i j = 2 ( i + j ) ! ( i + j + 1 ) ! ( i + j + 2 ) ! i ! ( i + 1 ) ! ( i + 2 ) ! j ! ( j + 1 ) ! ( j + 2 ) ! . In addition, it is shown that Φ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words. This is the extended and revised journal version of a conference paper with the title “Bijective counting of plane bipolar orientations”, which appeared in Electr. Notes in Discr. Math. pp. 283–287 (Proceedings of Eurocomb’07, 11–15 September 2007, Sevilla).

The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

The present paper is the first in a series of four dealing with a mapping, introduced by the present authors, from orientations to spanning trees in graphs, from regions to simplices in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). This mapping is actually defined for ordered oriented matroids. We call it the active orientation-to-basis mapping, in reference to an extensive use of activities, a notion depending on a linear ordering, first introduced by W.T. Tutte for spanning trees in graphs. The active mapping, which preserves activities, can be considered as a bijective generalization of a polynomial identity relating two expressions–one in terms of activities of reorientations, and the other in terms of activities of bases–of the Tutte polynomial of a graph, a hyperplane arrangement or an oriented matroid. Specializations include bijective versions of well-known enumerative results related to the counting of acyclic orientations in graphs or of regions in hyperplane arrangements. Other interesting features of the active mapping are links established between linear programming and the Tutte polynomial. We consider here the bounded case of the active mapping, where bounded corresponds to bipolar orientations in the case of graphs, and refers to bounded regions in the case of real hyperplane arrangements, or of oriented matroids. In terms of activities, this is the uniactive internal case. We introduce fully optimal bases, simply defined in terms of signs, strengthening optimal bases of linear programming. An optimal basis is associated with one flat with a maximality property, whereas a fully optimal basis is equivalent to a complete flag of flats, each with a maximality property. The main results of the paper are that a bounded region has a unique fully optimal basis, and that, up to negating all signs, fully optimal bases provide a bijection between bounded regions and uniactive internal bases. In the bounded case, up to negating all signs, the active mapping is a bijection.