I'm broadly interested in applicable mathematics. Below are some descriptions of the projects I've worked on.
Nonlinear dynamics in capillary networks
In vivo observation of blood flow in capillaries has turned up plenty of interesting dynamics. It can be difficult to tell, however, what exactly causes these behaviors: the boundary conditions of the in vivo are not constant, the physical characteristics (e.g., diameter) of each vessel can change over time, blood is distributed in a nonlinear fashion at diverging vessel junctions, and there are propagation effects related to the transit time of each vessel.
We have been working for several years on determining the simplest system that contains a given set of behaviors. We've found, for instance, that even a network with just two branches supports periodic oscillations and multiple steady state flow rates for a given set of network parameters. A network with three branches supports a richer set of dynamics. We've verified these predictions experimentally.
We have been working for several years on determining the simplest system that contains a given set of behaviors. We've found, for instance, that even a network with just two branches supports periodic oscillations and multiple steady state flow rates for a given set of network parameters. A network with three branches supports a richer set of dynamics. We've verified these predictions experimentally.
L(2,1) graph labelings
In the channel assignment problem, frequencies are assigned to transmitters in a network such that two transmitters receive sufficiently different frequencies to avoid interference due to transmitter proximity. In one model, each vertex of a graph $G$ represents a transmitter, edges connect vertices whose corresponding transmitters are operating in close proximity, and each frequency is represented by a non-negative integer so that adjacent vertices receive integers at least two apart, and vertices at distance two receive integers at least one apart. We call this an L(2,1) labeling of $G$.
L(2,1) labelings have turned out to be of theoretic interest as well. Griggs and Yeh conjectured that every $G$ with maximum degree $\Delta$ can be L(2,1) labeled with at most $\Delta^2$ distinct labels.
Sarah Adams (Olin College) and Denise Troxell (Babson College) have been advising an undergraduate research group working on problems in this space for quite a while. They have a really impressive publication record. Since my starting to work with the group in 2011, our undergraduates have verified the $\Delta^2$ conjecture for amalgamations of graphs, edge-path replacements of graphs, and amalgamations of Cartesian products.
L(2,1) labelings have turned out to be of theoretic interest as well. Griggs and Yeh conjectured that every $G$ with maximum degree $\Delta$ can be L(2,1) labeled with at most $\Delta^2$ distinct labels.
Sarah Adams (Olin College) and Denise Troxell (Babson College) have been advising an undergraduate research group working on problems in this space for quite a while. They have a really impressive publication record. Since my starting to work with the group in 2011, our undergraduates have verified the $\Delta^2$ conjecture for amalgamations of graphs, edge-path replacements of graphs, and amalgamations of Cartesian products.
Securing the smart grid
Networking the electrical distribution grid has tremendous potential to increase efficiency and decrease overall consumption. Providing users with fine grain information about system-wide electricity usage and pricing allows them to make more informed decisions about when (and if) to perform high energy tasks. And providing utility companies with detailed data about consumption allows for efficient utilization of generators.
Gathering the data on the scale necessary to make these changes isn't feasible using traditional wired transmission. Instead, many new smart meters come equipped with a wireless transmitter that broadcasts usage data. These data are aggregated and passed along to the utility provider. In return, the utility provider sends customers pricing information. Both directions of communication are subject to interception and spoofing. Interestingly, a similar system is being adopted at the home-wide level, in which individual appliances report their electricity consumption to a base station; this base station can then coordinate high load appliances like washing machines and dishwashers to run only when electricity prices are sufficiently low.
The abstraction for the neighborhood- or home-wide systems is the same: we have a single base station that is administrates a collection of smart meters. Each smart meter should be able to communicate to the base station, and the base station should be able to communicate to all group members in a single broadcast transmission. All communications must be secure, and we assume symmetric encryption due to the power and memory limitations on many smart metering devices. We investigated how to distribute cryptographic keys to each of the smart meters to allow for the ejections of multiple colluders from the network while at the same time respecting memory constraints on the devices. We've made connections between key distributions and set systems, and in particular between key distributions after the ejection of one or more member and residual set systems.
Gathering the data on the scale necessary to make these changes isn't feasible using traditional wired transmission. Instead, many new smart meters come equipped with a wireless transmitter that broadcasts usage data. These data are aggregated and passed along to the utility provider. In return, the utility provider sends customers pricing information. Both directions of communication are subject to interception and spoofing. Interestingly, a similar system is being adopted at the home-wide level, in which individual appliances report their electricity consumption to a base station; this base station can then coordinate high load appliances like washing machines and dishwashers to run only when electricity prices are sufficiently low.
The abstraction for the neighborhood- or home-wide systems is the same: we have a single base station that is administrates a collection of smart meters. Each smart meter should be able to communicate to the base station, and the base station should be able to communicate to all group members in a single broadcast transmission. All communications must be secure, and we assume symmetric encryption due to the power and memory limitations on many smart metering devices. We investigated how to distribute cryptographic keys to each of the smart meters to allow for the ejections of multiple colluders from the network while at the same time respecting memory constraints on the devices. We've made connections between key distributions and set systems, and in particular between key distributions after the ejection of one or more member and residual set systems.
Cell allocation in honey bee colonies
Honey bees use their hexagonal cells for one of three purposes: store honey, store pollen or house their offspring (brood). Interestingly enough, these cells are not dispersed randomly around the come, but instead occur in a predictable pattern: a concentrated group of brood cells in the middle surrounded by a ring of pollen cells, with the remainder of cells filled by honey.
If we assume that each bee doesn't have some sort of master plan of this pattern, then we must somehow confront the idea that the collective action of thousands of bees acting individually results in the formation of the cell allocation we observe.
Cellular automata models have been put forward to show that even a small collection of rules followed by every bee can result in the formation of the observed pattern. We found that pattern retention over multiple brood gestational cycles requires additional constraints and provide several biologically inspired rules that could be tested in the field. This work was published in 2013.
If we assume that each bee doesn't have some sort of master plan of this pattern, then we must somehow confront the idea that the collective action of thousands of bees acting individually results in the formation of the cell allocation we observe.
Cellular automata models have been put forward to show that even a small collection of rules followed by every bee can result in the formation of the observed pattern. We found that pattern retention over multiple brood gestational cycles requires additional constraints and provide several biologically inspired rules that could be tested in the field. This work was published in 2013.
Wireless communications
Efficient wireless communications is becoming all the more important as users demand richer mobile experiences. Unfortunately, the urban cellular environment is pretty tough on radio signals. Alamouti introduced an interesting solution, in which a single signal is sent from multiple spatially distributed antennas with the hope that the multiple corrupted versions that are picked up at the receiver can be pieced back together to form the original. We call this scheme space-time block coding.
The two most important characteristics of a space-time block code on $m$ transmit antennas are its rate $R$, defined as the number of information symbols sent per unit time, and the decoding delay $n$, defined as the total number of time units necessary to send the entire code. Liang showed the the maximum rate for a space-time block code on $m$ antennas is $R_{max} = {m+1 \over 2m}$. We showed in two papers that the minimum decoding delay for a maximum rate code was $2{2m \choose m-1}$ when $m \equiv 2$ modulo 4, and ${2m \choose m-1}$ otherwise.
We also considered other technical traits of these codes, including transceiver linearization, peak-to-average power ratio (PAPR), and signal polarization.
The two most important characteristics of a space-time block code on $m$ transmit antennas are its rate $R$, defined as the number of information symbols sent per unit time, and the decoding delay $n$, defined as the total number of time units necessary to send the entire code. Liang showed the the maximum rate for a space-time block code on $m$ antennas is $R_{max} = {m+1 \over 2m}$. We showed in two papers that the minimum decoding delay for a maximum rate code was $2{2m \choose m-1}$ when $m \equiv 2$ modulo 4, and ${2m \choose m-1}$ otherwise.
We also considered other technical traits of these codes, including transceiver linearization, peak-to-average power ratio (PAPR), and signal polarization.
Publication list
25. N. Karst, J. Langowitz, J. Oehrlein, D. Troxell, “Radio k-Chromatic Number of Cycles for Large k,” accepted at Discrete Mathematics, Applications and Algorithms.
24. M. Beaudouin-Lafon, S. Chen, N. Karst, J. Oehrlein, D. Troxell, “Labeling crossed prisms with a condition at distance two,” accepted at Involve.
23. N. Karst, J. Geddes, R. Carr, “Model microvascular networks can have many equilibria,” Bulletin of Mathematical Biology, doi:10.1007/s11538-017-0251-z, 2017.
22. D. Dralle, N. Karst, K. Charalampous, S. Thompson, “Event-scale power law recession analysis: quantifying methodological uncertainty,” Hydrology and Earth System Sciences, doi:10.5194/hess-21-65-2017, 2017.
21. D. Khachatryan, N. Karst, “V for voice: strategies for bolstering communication skills in undergraduate and graduate classrooms,” accepted at Journal of Statistics Education.
20. N. Karst, R. Slegers, “Cryptography in context: co-teaching ethics and mathematics,” under review at Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS).
19. N. Karst, D. Dralle, S. Thompson, “Spiral and rotor patterns produced by fairy ring fungi,” PLoS ONE, doi:10.1371/journal.pone.0149254, 2016.
18. D. Dralle, N. Karst, S. Thompson, “Dry season streamflow persistence in seasonal climates,” Water Resources Research, doi:10.1002/2015WR017752, 2016.
17. “a, b careful! Consequences of scale invariance in power-law models of the streamflow recession,” with D. Dralle and S. Thompson, Geophysical Research Letters, doi:10.1002/2015GL066007, 2015.
16. "Oscillations and multiple equilibria in microvascular networks", with B. Storey, J. Geddes, Bulletin of Mathematical Biology, doi:10.1007/s11538-015-0089-1, 2015.
15. "Observations of spontaneous oscillations in simple two-fluid networks", with B. Storey, D. Hellen, J. Geddes, Physical Review E, doi:10.1103/PhysRevE.91.023004, 2015.
14. "On distance labelings of amalgamations and injective labelings of general graphs", with J. Oehrlein, D. Troxell, J. Zhu, Involve, doi:10.2140/involve.2015.8.535, 2015.
13. "The minimum span of L(2, 1)-labelings of generalized flowers", with J. Oehrlein, D. Troxell, J. Zhu, Discrete Applied Mathematics, doi:10.1016/j.dam.2014.10.010, 2015.
12. "Labeling amalgamations of Cartesian products of complete graphs with a condition at distance two", with J. Oehrlein, D. Troxell, J. Zhu, Discrete Applied
Mathematics, doi:10.1016/j.dam.2014.06.022, 2014.
11. “Spontaneous oscillations simple fluid networks”, with J. Geddes and B. Storey, SIAM Journal on Applied Dynamical Systems, doi:10.1137/130926304, 2014.
10. “Individual behavioral rules sustain the cell allocation pattern in the comb of honey bees (Apis mellifera)”, with K. Montovan, T. Seeley and L. Jones. Journal of Theoretical Biology, doi:10.1016/j.jtbi.2013.07.010, 2013.
9. "L(d,1)-labeling of the edge-path-replacement by factorization of graphs", with J. Oehrlein, D. Troxell, and J. Zhu. Journal of Combinatorial Optimization, doi:10.1007/s10878-013-9632-x, 2013.
8. “On the L(2,1)-labelings of amalgamations of graphs" with S. Adams, N. Howell, D. Troxell, and J. Zhu. Discrete Applied Mathematics, doi:10.1016/j.dam.2012.11.007, 2012.
7. “On the rekeying load in group key distributions using cover-free families” with S. Wicker. IEEE Transactions on Information Theory, doi:10.1109/TIT.2012.2204542, 2012.
6. “On transceiver signal linearization and the decoding delay of maximum rate complex orthogonal space-time block codes” with S. Adams, M. Murugan, and
T. Wysocki. IEEE Transactions on Information Theory, doi:10.1109/TIT.2011.2137050, 2011.
5. “Novel classes of minimal delay and low PAPR rate-1/2 complex orthogonal designs” with S. Adams, J. Davis, M. Murugan, B. Lee, M. Crawford, and C. Greeley. IEEE Transactions on Information Theory, doi:10.1109/TIT.2011.2110730, 2011.
4. “The final case of the decoding delay problem for maximum rate complex orthogonal designs” with S. Adams and M. Murugan. IEEE Transactions on Information Theory, doi:10.1109/TIT.2009.2034818, 2010.
3. “Quaternion orthogonal designs from complex companion designs” with S. Adams, J. Seberry, and J. Pollack. Linear Algebra and Its Applications, doi:10.1016/j.laa.2007.09.013, 2008.
2. “The minimum decoding delay of maximum rate complex orthogonal designs” with S. Adams, and J. Pollack. IEEE Transactions on Information Theory, doi:10.1109/TIT.2007.901174, 2007.
1. “The onset of oscillations in microvascular blood flow” with J. Geddes, R. Carr, and F. Wu. SIAM Journal on Applied Dynamical Systems, doi:10.1137/060670699, 2007.
