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In Evans and Francis (2022) and Hendel (2021) the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced s times provided the original grid has at least 4 s rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously. [ABSTRACT FROM AUTHOR]

We examine whether social data can be used to predict how members of Major League Baseball (MLB) and members of the National Basketball Association (NBA) transition between teams during their career. We find that incorporating social data into various machine learning algorithms substantially improves the algorithms' ability to correctly determine these transitions in the NBA but only marginally in MLB. We also measure the extent to which player performance and team fitness data can be used to predict transitions between teams. This data, however, only slightly improves our predictions for players for both basketball and baseball players. We also consider whether social, performance, and team fitness data can be used to infer past transitions. Here we find that social data significantly improves our inference accuracy in both the NBA and MLB but player performance and team fitness data again does little to improve this score. [ABSTRACT FROM AUTHOR]

In [3,4] the authors found formulas for the number of spanning 2-forests (equivalently for the resistance distance) between any pair of vertices in a "straight linear 2-tree" and in a "linear 2-tree with a single bend." The main result of this paper generalizes these formulas to linear 2-trees with any number of bends. We also give alternative and perhaps more transparent versions of these formulas in terms of block matrices. As an application we find the resistance distance between the extremal vertices of the "corrugated linear 2-tree" and use this result to determine its asymptotic rate of growth as the number of vertices goes to infinity. Finally we note that our formulas give a different way to determine all the resistance distances in a "fan graph." [ABSTRACT FROM AUTHOR]

A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices u and v in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating u and v divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood. For any connected graph G with a 2-separator separating vertices u and v , we show that the number of spanning trees and spanning 2-forests separating u and v can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if u and v are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend. [ABSTRACT FROM AUTHOR]

Numerical simulations suggest that average velocity of a biological cell depends largely on attachment dynamics and less on the forces exerted by the cell. We determine the relationship between two models of cell motion, one based on finite spring constants modeling attachment properties (a randomly switched differential equation) and a limiting case (a centroid model-a generalized random walk) where spring constants are infinite. We prove the main result of this paper, the Expected Velocity Relationship theorem. This result shows that the expected value of the difference between cell locations in the differential equation model at the initial time and at some elapsed time is proportional to the elapsed time. We also show that the relationship is time invariant. Numerical results show the model is consistent with experimental data. [ABSTRACT FROM AUTHOR]

Abstract We consider the graph G n with vertex set V (G n) = { 1 , 2 , ... , n } and { i , j } ∈ E (G n) if and only if 0 < | i − j | ≤ 2. We call G n the straight linear 2-tree on n vertices. Using Δ –Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance r G n (i , j) between any two vertices i and j of G n. To our knowledge { G n } n = 3 ∞ is the first nontrivial family with diameter going to ∞ for which all resistance distances have been explicitly calculated. Our result also gives formulae for the number of spanning trees and 2-forests in a straight linear 2-tree. We show that the maximal resistance distance in G n occurs between vertices 1 and n and the minimal resistance distance occurs between vertices n ∕ 2 and n ∕ 2 + 1 for n even (with a similar result for n odd). It follows that r n (1 , n) → ∞ as n → ∞. Moreover, our explicit formula makes it possible to order the non-edges of G n exactly according to resistance distance, and this ordering agrees with the intuitive notion of distance on a graph. Consequently, G n is a geometric graph with entirely different properties than the random geometric graphs investigated in Luxburg, Radl and Hein (2010). These results for straight linear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we consider a transmission problem across highly conductive layers of the pre-fractal Sierpinski gasket type. We apply a homogenization approach to obtain a convergence of solutions to the transmission problem for a fixed iteration of the Sierpinski gasket. We show Mosco convergence of a sequence of energy functionals to an energy functional, supported on the highly conductive layer that incorporates a singular term on this layer. Moreover, from this convergence we can obtain results related to the convergence of approximating solutions to the limit solutions. [Copyright &y& Elsevier]

Fluorescence Recovery After Photobleaching (FRAP) has been extensively used to understand molecular dynamics in cells. This technique when applied to soluble, globular molecules driven by diffusion is easily interpreted and well understood. However, the classical methods of analysis cannot be applied to anisotropic structures subjected to directed transport, such as cytoskeletal filaments or elongated organelles transported along microtubule tracks. A new mathematical approach is needed to analyze FRAP data in this context and determine what information can be obtain from such experiments. To address these questions, we analyze fluorescence intensity profile curves after photobleaching of fluorescently labelled intermediate filaments anterogradely transported along microtubules. We apply the analysis to intermediate filament data to determine information about the filament motion. Our analysis consists of deriving equations for fluorescence intensity profiles and developing a mathematical model for the motion of filaments and simulating the model. Two closed forms for profile curves were derived, one for filaments of constant length and one for filaments with constant velocity, and three types of simulation were carried out. In the first type of simulation, the filaments have random velocities which are constant for the duration of the simulation. In the second type, filaments have random velocities which instantaneously change at random times. In the third type, filaments have random velocities and exhibit pausing between velocity changes. Our analysis shows: the most important distribution governing the shape of the intensity profile curves obtained from filaments is the distribution of the filament velocity. Furthermore, filament length which is constant during the experiment, had little impact on intensity profile curves. Finally, gamma distributions for the filament velocity with pauses give the best fit to asymmetric fluorescence intensity profiles of intermediate filaments observed in FRAP experiments performed in polarized migrating astrocytes. Our analysis also shows that the majority of filaments are stationary. Overall, our data give new insight into the regulation of intermediate filament dynamics during cell migration. Author summary: Fluorescence Recovery After Photobleaching (FRAP) is a commonly-used technique to analyze the dynamics of fluorescently-tagged proteins or structures in biology. After photochemical altering the fluorophor in a specific region, fluorescent material from the surrounding region moves into the photobleached region. Usually applied to the diffusion of soluble or membrane associate proteins, the existing models of analysis are not suitable for the elucidation of directional transport of elongated structures. Different modes of motions for the elongated structures with distributed lengths and velocities in cells are considered. First, we observe that filament lengths can be inferred from the level of noisiness. We further show the characteristics of fluorescence profile curves mainly depend on the occurrence of changes in velocities and distributions of velocities; whereas length distributions have negligible impact. Analysis of experimental data using this new framework indicates intermediate filaments transported by kinesins along microtubules in polarized migrating cells have gamma distributed velocities changing over time between pausing. Most filaments are found to be very slow or stationary with a few moving fast. This new computational approach should permit the interpretation of FRAP experimental data obtained with any directionally moving elongated structures of various lengths. [ABSTRACT FROM AUTHOR]