Abstract:
Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitz continuous then f(t,x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t,x) will be convex in x and decreasing in t. We also define the marginal support of a process and show that it almost surely contains the paths of the process. Although f need not be jointly continuous, we show that it will be continuous on the marginal support of X. We prove these results for a generalization of diffusion processes that we call `almost-continuous diffusions', and includes all continuous and strong Markov processes.

Abstract:
In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.

Abstract:
We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it is shown that requiring $f$ to be locally Lipschitz continuous in $x$ is enough for an It\^{o}-style decomposition to exist. In particular, $Y$ will be a Dirichlet process. We also look at the case where $f(t,x)$ can depend on $t$, possibly discontinuously. It is shown, under some additional mild constraints on $f$, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes. Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.

Abstract:
Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X. The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions.

Abstract:
We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant mean. We show that, under the extra condition that they are weakly continuous, the marginals can always be fitted in a unique way by a martingale which lies in a particular class of strong Markov processes. It is also shown that the map that this gives from the sets of marginal distributions to the martingale measures is continuous. Furthermore, we prove that it is the unique continuous method of fitting martingale measures to the marginal distributions.

Abstract:
The helminth parasite Fasciola hepatica secretes cathepsin L cysteine proteases to invade its host, migrate through tissues and digest haemoglobin, its main source of amino acids. Here we investigated the importance of pH in regulating the activity and functions of the major cathepsin L protease FheCL1. The slightly acidic pH of the parasite gut facilitates the auto-catalytic activation of FheCL1 from its inactive proFheCL1 zymogen; this process was ~40-fold faster at pH 4.5 than at pH 7.0. Active mature FheCL1 is very stable at acidic and neutral conditions (the enzyme retained ~45% activity when incubated at 37°C and pH 4.5 for 10 days) and displayed a broad pH range for activity peptide substrates and the protein ovalbumin, peaking between pH 5.5 and pH 7.0. This pH profile likely reflects the need for FheCL1 to function both in the parasite gut and in the host tissues. FheCL1, however, could not cleave its natural substrate Hb in the pH range pH 5.5 and pH 7.0; digestion occurred only at pH≤4.5, which coincided with pH-induced dissociation of the Hb tetramer. Our studies indicate that the acidic pH of the parasite relaxes the Hb structure, making it susceptible to proteolysis by FheCL1. This process is enhanced by glutathione (GSH), the main reducing agent contained in red blood cells. Using mass spectrometry, we show that FheCL1 can degrade Hb to small peptides, predominantly of 4–14 residues, but cannot release free amino acids. Therefore, we suggest that Hb degradation is not completed in the gut lumen but that the resulting peptides are absorbed by the gut epithelial cells for further processing by intracellular di- and amino-peptidases to free amino acids that are distributed through the parasite tissue for protein anabolism.

Abstract:
In the last decade, computation has played a valuable role in the understanding of materials. Hard materials, in particular, are only part of the application. Although materials involving B, C, N or O remain the most valued atomic component of hard materials, with diamond retaining its distinct superiority as the hardest, other materials involving a wide variety of metals are proving important. In the present work the importance of both ab-initio approaches and molecular dynamics aspects will be discussed with application to quite different systems. On one hand, ab-initio methods are applied to lightweight systems and advanced nitrides. Following, the use of molecular dynamics will be considered with application to strong metals that are used for high temperature applications.

Abstract:
When the Quadrennial Defense Review (QDR) was released in February of 2006, the United States was in the middle of a multi-front Global War on Terror (GWOT) that had been underway for more than four years. Beginning with the initial response to the 9/11 attacks in October of 2001, the US Navy began to play a significant part in the unconventional operations that characterised the early days of OPERATION ENDURING FREEDOM. While the Navy carried out its mission admirably supporting Special Operations Forces (SOF) by providing the USS Kitty Hawk (CV 63) as a “lily pad”, which enabled Rangers, Delta operators, Green Berets, and SEALs to move in and out of Afghanistan from a maritime staging area, it was a role well outside the norm of American naval operations and one the Navy is yet to fully embrace.

Abstract:
We report a long campaign to track the 1.8 hr photometric wave in the recurrent nova T Pyxidis, using the global telescope network of the Center for Backyard Astrophysics. During 1996-2011, that wave was highly stable in amplitude and waveform, resembling the orbital wave commonly seen in supersoft binaries. The period, however, was found to increase on a timescale P/P-dot=3x10^5 yr. This suggests a mass transfer rate of ~10^-7 M_sol/yr in quiescence. The orbital signal became vanishingly weak (<0.003 mag) near maximum light of the 2011 eruption. After it returned to visibility near V=11, the orbital period had increased by 0.0054(6) %. This is a measure of the mass ejected in the nova outburst. For a plausible choice of binary parameters, that mass is at least 3x10^-5 M_sol, and probably more. This represents >300 yr of accretion at the pre-outburst rate, but the time between outbursts was only 45 yr. Thus the erupting white dwarf seems to have ejected at least 6x more mass than it accreted. If this eruption is typical, the white dwarf must be eroding, rather than growing, in mass -- dashing the star's hopes of ever becoming famous via a supernova explosion. Instead, it seems likely that the binary dynamics are basically a suicide pact between the eroding white dwarf and the low-mass secondary, excited and rapidly whittled down, probably by the white dwarf's EUV radiation.

Abstract:
The formation of magnetic moment due to the dopants with p-orbital (d-orbital) is named d0 (d ) magnetism, where the ion without (with) partially filled d states is found to be responsible for the observed magnetic properties. To study the origin of magnetism at a fundamental electronic level in such materials, as a representative case, we theoretically investigate ferromagnetism in MgO doped with transition metal (Mn) and non-metal (C). The generalized gradient approximation based first-principles calculations are used to investigate substitutional doping of metal (Mn) and non-metal (C), both with and without the presence of neighboring oxygen vacancy sites. Furthermore, the case of co-doping of (Mn, C) in MgO system is also investigated. It is observed that the oxygen vacancies do not play a role in tuning the ferromagnetism in presence of Mn dopants, but have a significant influence on total magnetism of the C doped system. In fact, we find that in MgO the d0 magnetism through C doping is curtailed by pairing of the substitutional dopant with naturally occurring O vacancies. On the other hand, in case of (Mn, C) co-doped MgO the strong hybridization between the C (2p) and the Mn(3d) states suggests that co-doping is a promising approach to enhance the ferromagnetic coupling between the nearest-neighboring dopant and host atoms. Therefore, (Mn,C) co-doped MgO is expected to be a ferromagnetic semiconductor with long ranged ferromagnetism and high Curie temperature.