Mathematics
http://hdl.handle.net/1903/2261
2017-04-13T16:08:33ZOn The Number of Unlabeled Bipartite Graphs
http://hdl.handle.net/1903/19186
On The Number of Unlabeled Bipartite Graphs
Atmaca, Abdullah; Oruc, Yavuz A
Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\Phi$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. It is shown that the following two-sided equality holds.
$\displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!} \le |B_u(n,r)| \le 2\frac{\binom{r+2^{n}-1}{r}}{n!} $
This paper describes a result that has been obtained in joint work with Abdullah Atmaca of Bilkent University, Ankara, Turkey
2016-01-01T00:00:00ZMathematical Models of Quorum Sensing
http://hdl.handle.net/1903/19008
Mathematical Models of Quorum Sensing
Ueda, Hana
Mathematical models of biological phenomena are constructed in order to further the understanding of the known and unknown interactions that result in the behaviors of dynamical systems. We present mathematical models dealing with quorum sensing, which is the biological process of communication in bacteria. The density-dependent means of communication are mediated by molecules called autoinducers that are both synthesized and collected by quorum sensing bacteria.
We employ differential equations to investigate and understand the dynamics of underlying signaling processes. The first two models of this study were constructed with the idea of relating flocking movements observed in birds to gene expression in quorum sensing. To this end, modified Cucker-Smale flocking equations which do not require detailed knowledge of signal transductions mechanisms or regulatory proteins are used to represent quorum sensing and chemotaxis. The dynamical behaviors of these models are analyzed and approximated using asymptotic analysis and simulations. The coupling of quorum sensing and chemotaxis systems results in the formation of two groups of cells during the migration towards the attractant, which is similar to behavior observed in experiments of chemotaxing E. coli. This consequence of density influencing the velocity of bacteria suggests the possibility that density (or a density-dependent system such as quorum sensing) affects the chemotaxis system. We also show an application of this coupled model that produces qualitatively similar results with experimental data.
To further analyze collective behavior emerging from the interactions of quorum sensing and chemotaxis, this study uses statistical physics to derive a partial differential equation that tracks the time evolution in phase space of the distribution of these cells. Lastly, this study combines theory and experimental data to present a compartmental model that predicts p-aminophenol (PAP) response to various autoinducer concentrations in quorum sensing cells. The use of compartments allows for the model to be customized for constructs that do not use autoinducer-mediated production of the beta-galactosidase enzyme.
2016-01-01T00:00:00ZPROCESSING INFORMATION ON INTERMEDIATE TIMESCALES WITHIN RECURRENT NEURAL NETWORKS
http://hdl.handle.net/1903/19005
PROCESSING INFORMATION ON INTERMEDIATE TIMESCALES WITHIN RECURRENT NEURAL NETWORKS
Rourke, Oliver
The cerebral cortex has remarkable computational abilities; it is able to solve prob- lems which remain beyond the most advanced man-made systems. The complexity arises due to the structure of the neural network which controls how the neurons interact. One surprising fact about this network is the dominance of ‘recurrent’ and ‘feedback’ connections. For example, only 5-10% of connections into the earliest stage of visual processing are ‘feedforward’, in that they carry information from the eyes (via the Lateral Geniculate Nucleus). One possible reason for these connec- tions is that they allow for information to be preserved within the network; the underlying ‘causes’ of sensory stimuli usually persist for much longer than the time scales of neural processing, and so understanding them requires continued aggrega- tion of information within the sensory cortices. In this dissertation, I investigate several models of such sensory processing via recurrent connections. I introduce the transient attractor network, which depends on recurrent plastic connectivity, and demonstrate in simulations how it might be involved in the processes of short term memory, signal de-noising, and temporal coherence analysis. I then show how a certain recurrent network structure might allow for transient associative learning to occur on the timescales of seconds using presynaptic facilitation. Finally, I consider how auditory scene analysis might occur through ‘gamma partitioning’. This process uses recurrent excitatory and inhibitory connections to preserve information within the neural network about its recent state, allowing for the separation of auditory sources into different perceptual cycles.
2016-01-01T00:00:00ZConic Economics
http://hdl.handle.net/1903/18973
Conic Economics
Raissi, Maziar
Modern general equilibria under uncertainty are modeled based on the recognition that all risks cannot be eliminated, perfect hedging is not possible, and some risk exposures must be tolerated. Therefore, we need to define the set of acceptable risks as a primitive of the financial economy. This set will be a cone, hence the word conic. Such a conic perspective challenges classical economics by introducing finance into the economic models and enables us to rewrite major chapters of classical micro- and macro-economics textbooks.
2016-01-01T00:00:00Z