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- Dynamic equation (2)
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Articles 1 - 14 of 14
Full-Text Articles in Physical Sciences and Mathematics
Periodic Functions Related To The Gompertz Difference Equation, Tom Cuchta, Nick Wintz
Periodic Functions Related To The Gompertz Difference Equation, Tom Cuchta, Nick Wintz
Faculty Scholarship
We investigate periodicity of functions related to the Gompertz difference equation. In particular, we derive difference equations that must be satisfied to guarantee periodicity of the solution.
Spectral Properties Of The Hierarchical Product Of Graphs, Per Sebastian Skardal, Kirsti Wash
Spectral Properties Of The Hierarchical Product Of Graphs, Per Sebastian Skardal, Kirsti Wash
Faculty Scholarship
The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large …
Erosion Of Synchronization: Coupling Heterogeneity And Network Structure [Post-Print], Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
Erosion Of Synchronization: Coupling Heterogeneity And Network Structure [Post-Print], Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
Faculty Scholarship
We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, …
Collective Frequency Variation In Network Synchronization And Reverse Pagerank, Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
Collective Frequency Variation In Network Synchronization And Reverse Pagerank, Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
Faculty Scholarship
A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals’ natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an outflow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the …
Synchronization Of Heterogeneous Oscillators Under Network Modifications: Perturbation And Optimization Of The Synchrony Alignment Function, Dane Taylor, Per Sebastian Skardal, Jie Sun
Synchronization Of Heterogeneous Oscillators Under Network Modifications: Perturbation And Optimization Of The Synchrony Alignment Function, Dane Taylor, Per Sebastian Skardal, Jie Sun
Faculty Scholarship
Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications---for which proper functionality depends sensitively on the extent of synchronization---there remains a lack of understanding for how systems evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies …
What The Integral Does: Physics Students' Efforts At Making Sense Of Integration, Fr. Joseph Wagner S.J.
What The Integral Does: Physics Students' Efforts At Making Sense Of Integration, Fr. Joseph Wagner S.J.
Faculty Scholarship
Students use a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts. This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discomforts and skepticism some students maintain even after constructing useful conceptions of the integral. In particular, many students attempt to explain what integration does by trying to interpret the algebraic manipulations and computations involved in finding antiderivatives. This …
An Extended Theoretical Framework For The Concept Of The Derivative, Joseph F. Wagner, David Roundy, Tevian Dray, Corinne Monogue, Eric Weber
An Extended Theoretical Framework For The Concept Of The Derivative, Joseph F. Wagner, David Roundy, Tevian Dray, Corinne Monogue, Eric Weber
Faculty Scholarship
No abstract provided.
The Kalman Filter For Linear Systems On Time Scales, Martin Bohner, Nick Wintz
The Kalman Filter For Linear Systems On Time Scales, Martin Bohner, Nick Wintz
Faculty Scholarship
We introduce the Kalman filter for linear systems on time scales, which includes the discrete and continuous versions as special cases. When the system is also stochastic, we show that the Kalman filter is an observer that estimates the system when the state is corrupted by noisy measurements. Finally, we show that the duality of the Kalman filter and the Linear Quadratic Regulator (LQR) is preserved in their unification on time scales. A numerical example is provided.
Strict Topologies On Spaces Of Vector-Valued Functions, Terje Hõim, David A. Robbins
Strict Topologies On Spaces Of Vector-Valued Functions, Terje Hõim, David A. Robbins
Faculty Scholarship
No abstract provided.
The Linear Quadratic Tracker On Time Scales, Martin Bohner, Nick Wintz
The Linear Quadratic Tracker On Time Scales, Martin Bohner, Nick Wintz
Faculty Scholarship
In this work, we study a natural extension of the Linear Quadratic Regulator (LQR) on time scales. Here, we unify and extend the Linear Quadratic Tracker (LQT). We seek to find an affine optimal control that minimizes a cost functional associated with a completely observable linear system. We then find an affine optimal control for the fixed final state case in terms of the current state. Finally we include an example in disturbance/rejection modelling. A numerical example is also included.
Sharp Weighted Estimates For Classical Operators [Post-Print], David Cruz-Uribe Sfo, José María Martell, Carlos Perez
Sharp Weighted Estimates For Classical Operators [Post-Print], David Cruz-Uribe Sfo, José María Martell, Carlos Perez
Faculty Scholarship
See abstract at: http://www.sciencedirect.com/science/article/pii/S0001870811003136
On The Structure Of Graphs With Non-Surjective L(2,1)-Labelings, John P. Georges, David W. Mauro
On The Structure Of Graphs With Non-Surjective L(2,1)-Labelings, John P. Georges, David W. Mauro
Faculty Scholarship
For a graph G, an L(2,1)-labeling of G with span k is a mapping $L \right arrow \{0, 1, 2, \ldots, k\}$ such that adjacent vertices are assigned integers which differ by at least 2, vertices at distance two are assigned integers which differ by at least 1, and the image of L includes 0 and k. The minimum span over all L(2,1)-labelings of G is denoted $\lambda(G)$, and each L(2,1)-labeling with span $\lambda(G)$ is called a $\lambda$-labeling. For $h \in \{1, \ldots, k-1\}$, h is a hole of Lif and only if h …
Infinitely Many Hyperbolic 3-Manifolds Which Contain No Reebless Foliation, R. Roberts, J. Shareshian, Melanie Stein
Infinitely Many Hyperbolic 3-Manifolds Which Contain No Reebless Foliation, R. Roberts, J. Shareshian, Melanie Stein
Faculty Scholarship
We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented -order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.
On Regular Graphs Optimally Labeled With A Condition At Distance Two, John P. Georges, David W. Mauro
On Regular Graphs Optimally Labeled With A Condition At Distance Two, John P. Georges, David W. Mauro
Faculty Scholarship
For positive integers $j \geq k$, the $\lambda_{j,k}$-number of graph Gis the smallest span among all integer labelings of V(G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the $\lambda_{j,k}$-number of any r-regular graph is no less than the $\lambda_{j,k}$-number of the infinite r-regular tree $T_{\infty}(r)$. Defining an r-regular graph G to be $(j,k,r)$-optimal if and only if $\lambda_{j,k}(G) = \lambda_{j,k}(T_{\infty}(r))$, we establish the equivalence between $(j,k,r)$-optimal graphs and r-regular bipartite …