For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence of a G. An equivalent definition exists for non-orientable surfaces. In 1983, Jozef Širáň proved that for every decreasing, convex sequence of non-negative integers, there is a graph G such that this sequence is the crossing sequence of G. This main result of this thesis proves the existence of a graph with non-convex crossing sequence of arbitrary length.

Highway/railroad grade crossings present a danger to vehicular traffic. According to the USDOT Bureau of Transportation Statistics, in 2009, there were 1,887 crashes at highway/rail crossings resulting in 248 fatalities (FRA, 2009). The installation of presignals at grade crossings decreases crashes and fatalities at highway/rail crossings. There are no Federal standards that provide guidance for the installation of presignals. Therefore, current practices do not conform to any set of consistent nationwide standards except for guidelines specified in the MUTCD. These guidelines state that a presignal should be considered where the at-grade highway/rail crossing is located within 50 feet of a signalized intersection. The MUTCD also gives the option of installing a presignal at a distance greater than 50 feet, if an engineering study determines a need; however, no specific guidelines are provided for such studies.This work uses a case study to determine which measures are needed to warrant a presignal and examines if the distance criterion of 50 feet between signalized intersections and highway/rail crossings is adequate. It also explores the need for consistent national standards to provide guidance to practitioners in determining the needs for the installation of such signals.The study finds that distance criterion should not be used as the sole indicator for the installation of a presignal and therefore, engineering studies must be performed in all cases to determine presignal needs. Furthermore, the work concluded that the MUTCD must be modified to provide standards and guidelines that can be used nationwide for systematic quantitative assessment in determining when presignals are warranted near railroad crossings. This study proposes that presignals be installed based on warrants that consist of crash data, queue distance and no gates at the crossing. The proposed modifications include describing presignal types, defining their purpose, developing presignal warrants, and creating guidelines that can be used by practitioners.The changes and revisions recommended by this research work include queue length analysis, signal phasing and timing modifications, and existing intersection infrastructure needs. The resulting warrants and guidelines for presignal installation can be used nationally to provide uniform guidance and recommendations in performing presignal studies.

The crossing number of a graph is the minimum number of pairwise edge crossings in a drawing of a graph. A graph $G$ is $k$-crossing-critical if it has crossing number at least $k$, and any subgraph of $G$ has crossing number less than $k$. A consequence of Kuratowski's theorem is that 1-critical graphs are subdivisions of $K_{3,3}$ and $K_{5}$. The graph $V_{2n}$ is a $2n$-cycle with $n$ diameters. Bokal, Oporowski, Richter and Salazar found in \cite{bigpaper} all the critical graphs except the ones that contain a $V_{8}$ minor and no $V_{10}$ minor. We show that a 4-connected graph $G$ has crossing number at least 2 if and only if for each pair of disjoint edges there are two disjoint cycles containing them. Using a generalization of this result we found limitations for the 2-crossing-critical graphs remaining to classify. We showed that peripherally 4-connected 2-crossing-critical graphs have at most 4001 vertices. Furthermore, most 3-connected 2-crossing-critical graphs are obtainable by small modifications of the peripherally 4-connected ones.

For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence of a G. An equivalent definition exists for non-orientable surfaces. In 1983, Jozef Širáň proved that for every decreasing, convex sequence of non-negative integers, there is a graph G such that this sequence is the crossing sequence of G. This main result of this thesis proves the existence of a graph with non-convex crossing sequence of arbitrary length.

This thesis is concerned with boundary crossing probabilities and first crossing time densities for stochastic processes. This is a classical problem in probability that goes back to the famous ballot problem (first studied by W. A. Whitworth (1878) and J. Bertrand (1887)) and has numerous applications in diverse areas including mathematical statistics and financial mathematics. Our main objective is the study of approximation methods and control of the resulting approximation error for boundary crossing probabilities where a closed-form solution is unavailable. This leads to the study of bounds for the density of the first crossing time of the boundary, which in turn leads to the derivation of some analytic properties of the densities. This thesis presents a whole suite of closely related new results obtained when working on the outlined research program. (For complete abstract open document).

Charged particle accelerators often have strict requirements on the beam energy, and timing to calibrate, or control background processes. Longitudinal Phase Space Tomography is a technique developed in 1987 to visualize the time, and energy coordinates of a beam. With non-invasive detectors, the beam can be visualized at any point during operation of a synchrotron. With the progress of computing power over the last 27 years, it is now possible to compute tomographic reconstructions in real time accelerator operations for many bunches around the accelerator ring. This thesis describes a real-time, multi-bunch tomography system developed and implemented in Fermilab's Main Injector and Recycler Rings, and a study of bunch growth when crossing transition. Implications of these studies for high intensity operation of the Fermilab accelerators are presented. / text

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions. We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8. We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution.

The effect of different treatments on crossing over between and within genes at the tip of the X chromosome of Drosophila melanogaster was studied to determine whether exchange in the two regions occurs by different mechanisms. In response to autosomal inversions, ɣ -radiation, and heat shock, crossing over of both types was altered in the same direction and to a comparable extent. This would be expected if there were only one crossover mechanism involved. There was some difference in response of interlocus and intralocus crossing over after mitomycin C injection but, since the effect on interlocus crossing over in two separate regions was not consistent, the significance of this result is questionable. Although double crossing over involving the two interlocus regions was very rare doubles involving an inter and an intralocus region were recovered. The association of exchanges within a gene with a crossover between genes might indicate that there are two noninterfering mechanisms, or that multiple exchanges occur in short effectively paired regions. A further experiment designed to detect switch regions is outlined. / Science, Faculty of / Zoology, Department of / Graduate