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Assessing the origin of Pluto and Triton has profound implications for the bigger picture of Solar System formation and evolution. In such a context, this chapter reviews our current knowledge of the formation conditions of Pluto and Triton's constitutive building blocks in the protosolar nebula, which can be derived from their known or estimated volatile contents. Assuming that the ultravolatiles carbon monoxide and dinitrogen detected in Pluto and Triton are primordial, the presence of these molecules suggest that the two bodies accreted material originating from the vicinity of the carbon monoxide and dinitrogen icelines. Dinitrogen--rich and water--poor comets such as comet C/2016 R2 (PanSTARRS) obviously present a compositional link with Pluto and Triton, indicating that their building blocks formed in nearby regions of the protosolar nebula, despite of the variation of the water abundance among those bodies. Also, the assumption of Triton's growth in Neptune's circumplanetary disk requires that its building blocks formed at earlier epochs in the protosolar nebula, to remain consistent with its estimated composition., Comment: Accepted for publication as Chapter 1 in the book "Triton and Pluto: The long lost twins of active worlds"

Ocean worlds, or icy bodies in the outer solar system that have or once had subsurface liquid water oceans, are among the most compelling topics of astrobiology. Typically, confirming the existence of a subsurface ocean requires close spacecraft observations. However, combining our understanding of the chemistry that takes place in a subsurface ocean with our knowledge of the building blocks that formed potential ocean worlds provides an opportunity to identify tracers of endogenic activity in the surface volatiles of Pluto and Triton. We show here that the current composition of the volatiles on the surfaces and in the atmospheres of Pluto and Triton are deficient in carbon, which can only be explained by the loss of CH4 through a combination of aqueous chemistry and atmospheric processes. Furthermore, we find that the relative nitrogen and water abundances are within the range observed in building block analogs, comets, and chondrites. A lower limit for N/Ar in Pluto's atmosphere also suggests source building blocks that have a cometary or chondritic composition, all pointing to an origin for their nitrogen as NH3 or organics. Triton's lower abundance of CH4 compared to Pluto, and the detection of CO2 at Triton but not at Pluto points to aqueous chemistry in a subsurface ocean that was more efficient at Triton than Pluto. These results have applications to other large Kuiper Belt objects as well as the assessment of formation locations and times for the four giant planets given future probe measurements of noble gas abundances and isotope ratios.

Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when $q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377].

The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $\gamma_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $\gamma_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $\gamma_P(G) \le n/6$, and this bound is sharp. We also provide new bounds on $\gamma_P(G \Box H)$ where $G\Box H$ is the Cartesian product of graphs $G$ and $H$. In the specific case that $G$ and $H$ are trees whose power domination number and domination number are equal, we show the Vizing-like inequality holds and $\gamma_P(G \Box H) \ge \gamma_P(G)\gamma_P(H)$.

Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$ using this process is known as a power dominating set of $G$, and the power domination number of $G$, $\gamma_P(G)$, is the minimum number of vertices in a power dominating set. We introduce a new partition on the vertices of a graph to provide a lower bound for the power domination number. We also consider the power domination number of the Cartesian product of two graphs, $G \Box H$, and show certain graphs satisfy a Vizing-like bound with regards to the power domination number. In particular, we prove that for any two trees $T_1$ and $T_2$, $\gamma_P(T_1)\gamma_P(T_2) \leq \gamma_P(T_1 \Box T_2)$.

Observations of comet C/2016 R2 (PanSTARRS) have revealed exceptionally bright emission bands of N$_2^+$, the strongest ever observed in a comet spectrum. Alternatively, it appears to be poor in CN compared to other comets, and remarkably depleted in H$_2$O. Here we quantify the N$_2$ production rate from N$_2^+$ emission lines using the Haser model. We derived effective parent and daughter scalelengths for N2 producing N2+. This is the first direct measurement of such parameters. Using a revised fluorescence efficiency for N2+, the resulting production rate of molecular nitrogen is inferred to be Q(N$_2$) ~ 1 $\times 10^{28}$ molecules.s-1 on average for 11, 12, and 13 Feb. 2018, the highest for any known comet. Based on a CO production rate of Q(CO) ~ 1.1 $\times 10^{29}$ molecules.s-1, we find Q(N$-2$)/Q(CO)~0.09, which is consistent with the N$_2^+$/CO$^+$ ratio derived from the observed intensities of N$_2^+$ and CO$^+$ emission lines. We also measure significant variations in this production rate between our three observing nights, with Q(N$_2$) varying by plus or minus 20% according to the average value, Comment: 8 pages, 6 figures

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph $G$ is also equimatchable, meaning that every maximal matching in $G$ is maximum. In this paper, we show that if $G$ is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then $G$ is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product $G\Box H$ is well-edge-dominated, where $G$ and $H$ have order at least $2$, if and only if $G\Box H = K_2 \Box K_2$., Comment: 18 pages, 2 figures, 18 references

We develop a formalism for photoionization (PI) and potential energy curves (PECs) of Rydberg atoms in ponderomotive optical lattices and apply it to examples covering several regimes of the optical-lattice depth. The effect of lattice-induced PI on Rydberg-atom lifetime ranges from noticeable to highly dominant when compared with natural decay. The PI behavior is governed by the generally rapid decrease of the PI cross sections as a function of angular-momentum ($\ell$), and by lattice-induced $\ell$-mixing across the optical-lattice PECs. In GHz-deep lattices, $\ell$-mixing leads to a rich PEC structure, and the significant low-$\ell$ PI cross sections are distributed over many lattice-mixed Rydberg states. In lattices less than several tens-of-MHz deep, atoms on low-$\ell$ PECs are essentially $\ell$-mixing-free and maintain large PI cross sections, while atoms on high-$\ell$ PECs trend towards being PI-free. Characterization of PI in GHz-deep Rydberg-atom lattices may be beneficial for optical control and quantum-state manipulation of Rydberg atoms, while data on PI in shallower lattices are potentially useful in high-precision spectroscopy and quantum-computing applications of lattice-confined Rydberg atoms.

The building blocks of Titan and Enceladus are believed to have formed in a late-stage circumplanetary disk around Saturn. Evaluating the evolution of the abundances of volatile species in this disk as a function of the migration, growth, and evaporation of icy grains is then of primary importance to assess the origin of the material that eventually formed these two moons. Here we use a simple prescription of Saturn's circumplanetary disk in which the location of the centrifugal radius is varied, to investigate the time evolution of the icelines of water ice, ammonia hydrate, methane clathrate, carbon monoxide and dinitrogen pure condensates. To match their compositional data, the building blocks of both moons would have had to form in a region of the circumplanetary disk situated between the icelines of carbon monoxide and dinitrogen at their outer limit, and the iceline of methane clathrate as their inner limit. We find that a source of dust at the location of centrifugal radius does not guarantee the replenishment of the disk in the volatiles assumed to be primordial in Titan and Enceladus. Only simulations assuming a centrifugal radius in the 66--100 Saturnian radii range allow for the formation and growth of solids with compositions consistent with those measured in Enceladus and Titan. The species are then able to evolve in solid forms in the system for longer periods of time, even reaching an equilibrium, thus favoring the formation of Titan and Enceladus building blocks in this region of the disk., Comment: Accepted for publication in The Planetary Science Journal

Throttling addresses the question of minimizing the sum or the product of the resources used to accomplish a task and the time needed to complete that task for various graph searching processes. Graph parameters of interest include various types of zero forcing, power domination, and Cops and Robbers. We provide a survey of product throttling for these parameters.