Belinda Trotta, Robert Johnson, Catherine de Burgh-Day, Debra Hudson, Esteban Abellan, James Canvin, Andrew Kelly, Daniel Mentiplay, Benjamin Owen, Jennifer Whelan
Artificial Intelligence (AI) weather models are now reaching operational-grade performance for some variables, but like traditional Numerical Weather Prediction (NWP) models, they exhibit systematic biases and reliability issues. We test the application of the Bureau of Meteorology's existing statistical post-processing system, IMPROVER, to ECMWF's deterministic Artificial Intelligence Forecasting System (AIFS), and compare results against post-processed outputs from the ECMWF HRES and ENS models. Without any modification to processing workflows, post-processing yields comparable accuracy improvements for AIFS as for traditional NWP forecasts, in both expected value and probabilistic outputs. We show that blending AIFS with NWP models improves overall forecast skill, even when AIFS alone is not the most accurate component. These findings show that statistical post-processing methods developed for NWP are directly applicable to AI models, enabling national meteorological centres to incorporate AI forecasts into existing workflows in a low-risk, incremental fashion.
Sarah Mechbal, Pierre-Simon Mangeard, John M. Clem, Paul A. Evenson, Robert P. Johnson, Brian Lucas, James Roth
We report on a new measurement of the cosmic ray (CR) electron and positron spectra in the energy range of 20 MeV -- 1 GeV. The data were taken during the first flight of the balloon-borne spectrometer AESOP-Lite (Anti Electron Sub Orbital Payload), which was flown from Esrange, Sweden, to Ellesmere Island, Canada, in May 2018. The instrument accumulated over 130 hours of exposure at an average altitude of 3 g.cm$^{-2}$ of residual atmosphere. The experiment uses a gas Cherenkov detector and a magnetic spectrometer, consisting of a permanent dipole magnet and silicon strip detectors (SSDs), to identify particle type and measure the rigidity. Electrons and positrons were detected against a background of protons and atmospheric secondary particles. The primary cosmic ray spectra of electrons and positrons, as well as the re-entrant albedo fluxes, were extracted between 20 MeV -- 1 GeV during a positive solar magnetic polarity epoch. The positron fraction below 100 MeV appears flat, suggesting diffusion dominated solar modulation at low rigidity. The all-electron spectrum is presented and compared with models from a heliospheric numerical transport code.
J. Robert Johnson, Imre Leader, Eoin Long
In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris--Kleitman sense). Thus, for example, for a (uniformly) random permutation $π$, the event that no point is displaced by more than a fixed distance $d$ and the event that $π$ is the product of at most $k$ adjacent transpositions are positively correlated. In contrast, under the weak Bruhat order we show that this completely fails: surprisingly, there are two up-sets each of measure $1/2$ whose intersection has arbitrarily small measure. We also prove analogous correlation results for a class of non-uniform measures, which includes the Mallows measures. Some applications and open problems are discussed.
J. Robert Johnson, Trevor Pinto
The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.
J. Robert Johnson
A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.
Robert W. Johnson
Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real and complex quadratic spaces with dimension 4 or less. A model for the representations of these Heisenberg groups and automorphism groups is constructed. A pseudo-differential operator enables a parallel treatment of spaces defined over finite and real fields.
J. Robert Johnson, Imre Leader, Mark Walters
Positional games are a well-studied class of combinatorial game. In their usual form, two players take turns to play moves in a set (`the board'), and certain subsets are designated as `winning': the first person to occupy such a set wins the game. For these games, it is well known that (with correct play) the game cannot be a second-player win. In the avoidance (or misère) form, the first person to occupy such a set \emph{loses} the game. Here it would be natural to expect that the game cannot be a first-player win, at least if the game is transitive, meaning that all points of the board look the same. Our main result is that, contrary to this expectation, there are transitive games that are first-player wins, for all board sizes which are not prime or a power of 2. Further, we show that such games can have additional properties such as stronger transitivity conditions, fast winning times, and `small' winning sets.
J. Robert Johnson, Imre Leader, Paul A. Russell
The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $α>0$ there is a family of $α2^n$ sets containing $(α+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.
Justin Deighan, Robert E Johnson
Mar 15, 2013·astro-ph.EP·PDF All three terrestrial planets with atmospheres support O3 layers of some thickness. While currently only that of Earth is substantial enough to be climatically significant, we hypothesize that ancient Mars may also have supported a thick O3 layer during volcanically quiescent periods whenthe atmosphere was oxidizing. To characterize such an O3 layer and determine the significance of its fedback on the Martian climate, we apply a 1D line-by-line radiative-convective model under clear sky conditions coupled to a simple photochemical model. The parameter space of atmospheric pressure, insolation, and O2 mixing fraction are explored to find conditions favorable to O3 formation. We find that a substantial O3 layer is most likely for surface pressures of 0.3-1.0 bar, and could produce an O3 column comparable to that of modern Earth for O2 mixing fractions approaching 1%. However, even for thinner O3 layers, significant UV shielding of the surface occurs along with feedback on both the energy budget and photochemistry of the atmosphere. In particular, CO2 condensation in the middle atmosphere is inhibited and the characteristics of H2O dissociation are modified, shifting from a direct photolysis dominated state similar to modern Mars to a more Earth-like state controlled by O(1D) attack.
Natalie C. Behague, J. Robert Johnson
An automaton is synchronizing if there is a word that maps all states onto the same state. Černý's conjecture on the length of the shortest such word is probably the most famous open problem in automata theory. We consider the closely related question of determining the minimum length of a word that maps $k$ states onto a single state. For synchronizing automata, we improve the upper bound on the minimum length of a word that sends some triple to a a single state from $0.5n^2$ to $\approx 0.19n^2$. We further extend this to an improved bound on the length of such a word for 4 states and 5 states. In the case of non-synchronizing automata, we give an example to show that the minimum length of a word that sends $k$ states to a single state can be as large as $Θ\left(n^{k-1}\right)$.
J. Robert Johnson, John Talbot
Let $\mathcal{Q}_n$ be the $n$-dimensional hypercube: the graph with vertex set $\{0,1\}^n$ and edges between vertices that differ in exactly one coordinate. For $1\leq d\leq n$ and $F\subseteq \{0,1\}^d$ we say that $S\subseteq \{0,1\}^n$ is \emph{$F$-free} if every embedding $i:\{0,1\}^d\to \{0,1\}^n$ satisfies $i(F)\not\subseteq S$. We consider the question of how large $S\subseteq \{0,1\}^n$ can be if it is $F$-free. In particular we generalise the main prior result in this area, for $F=\{0,1\}^2$, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal sets. We also show that the density required to guarantee an embedded copy of at least one of a family of forbidden configurations may be significantly lower than that required to ensure an embedded copy of any individual member of the family. Finally we show that any subset of the $n$-dimensional hypercube of positive density will contain exponentially many points from some embedded $d$-dimensional subcube if $n$ is sufficiently large.
J. Robert Johnson
The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a family of sets of edges of our graph and are not allowed to use all the edges of any member of this family. We consider two natural ways of expressing constraints of this kind using graphs and using set systems. For the first version we ask for conditions on regular bipartite graphs $G$ and $H$ for there to exist a perfect matching in $G$, no two edges of which form a $4$-cycle with two edges of $H$. In the second, we ask for conditions under which a Hamilton cycle in the complete graph (or equivalently a cyclic permutation) exists, with the property that it has no collection of intervals of prescribed lengths whose union is an element of a given family of sets. For instance we prove that the smallest family of $4$-sets with the property that every cyclic permutation of an $n$-set contains two adjacent pairs of points has size between $(1/9+o(1))n^2$ and $(1/2-o(1))n^2$. We also give bounds on the general version of this problem and on other natural special cases. We finish by raising numerous open problems and directions for further study.
Robert W. Johnson
I construct an algebraic model for a typical fiber on a 1+1 dimensional spacetime. The vector space comprising the fiber is composed of elements formed from the direct product of two copies of an element x in the D2=C2xC2 finite group algebra over the real numbers. The fiber contains subspaces whose elements are associated with the tangent and momentum vectors of trajectories in the manifold. The fiber also contains a subspace whose elements are associated with the local flow of action of each trajectory. The condition of minimum action translates into a constraint on the original vector x in the direct product structure.
Robert E Johnson, Wei-Lin Tseng, Meredith K Elrod, Ann M Persoon
The observed disparity between the radial dependence of the ion and electron densities measured by the Cassini plasma and radio science instruments are used to show that the region between the outer edge of Saturn's main rings and its tenuous G-ring is permeated with small charged grains (nanograins). These grains emanate from the edge of the A-ring and from the tenuous F-ring and G-ring. This is a region of Saturn's magnetosphere that is relatively unexplored, but will be a focus of Cassini's F-ring orbits prior to the end of mission in September 2017. Confirmation of the grain densities predicted here will enhance our ability to describe the formation and destruction of material in this important region of Saturn's magnetosphere.
J. Robert Johnson, Mark Walters
Given a graph on n vertices with m edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions -- graphs like a star, and graphs which are close to regular -- with the transition between them occuring when the average degree is 3. However, one of our main aims in this paper is to show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs -- namely `which rooted graph minimises the average resistance to the root?'. The rooted case is much simpler to analyse than the unrooted, and one of the main results of this paper is that the two cases are asymptotically equivalent.
Rahil Baber, J. Robert Johnson, John Talbot
We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from $K_{3,3,3}$ has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.
Meredith K. Elrod, Wei-Ling Tseng, Adam K. Woodson, Robert E. Johnson
Dec 14, 2013·astro-ph.EP·PDF A goal of Cassini's extended mission has been to examine the seasonal variations of Saturn's magnetosphere, moons, and rings. Recently we showed that the magnetospheric plasma between the main rings and Enceladus exhibited a time dependence that we attributed to a seasonally variable source of oxygen from the main rings (Elrod et al., 2012). Such a temporal variation was subsequently seen in the energetic ion composition (Christon et al., 2013). Here we include the most recent measurements by the Cassini Plasma Spectrometer (CAPS) in our analysis (Elrod et al., 2012) and modeling (Tseng et al., 2013a) of the temporal and radial dependence of the thermal plasma in the region between the main rings and the orbit of Enceladus. Data taken in 2012, well past equinox for which the northern side of the main rings were illuminated, appear consistent with a seasonal variation. Although the thermal plasma in this region comes from two sources, the extended ring atmosphere and the Enceladus torus that have very different radial and temporal trends, the heavy ion density is found to exhibit a steep radial dependence that is similar for all years examined. Using our chemical model, we show that this dependence requires a radial dependence for Enceladus torus than differs from recent models or, more likely, enhanced heavy ion quenching with decreasing distance from the edge of the main rings. We examine the possible physical processes and suggest that the precipitation of the inward diffusing high energy background radiation onto the edge of the main rings could play an important role.
Robert E. Johnson, Alexey N. Volkov, Justin T. Erwin
Feb 26, 2013·astro-ph.EP·PDF The equations of gas dynamics are extensively used to describe atmospheric loss from solar system bodies and exoplanets even though the boundary conditions at infinity are not uniquely defined. Using molecular-kinetic simulations that correctly treat the transition from the continuum to the rarefied region, we confirm that the energy-limited escape approximation is valid when adiabatic expansion is the dominant cooling process. However, this does not imply that the outflow goes sonic. In fact in the sonic regime, the energy limited approximation can significantly under estimate the escape rate. Rather large escape rates and concomitant adiabatic cooling can produce atmospheres with subsonic flow that are highly extended. Since this affects the heating rate of the upper atmosphere and the interaction with external fields and plasmas, we give a criterion for estimating when the outflow goes transonic in the continuum region. This is applied to early terrestrial atmospheres, exoplanet atmospheres, and the atmosphere of the ex-planet, Pluto, all of which have large escape rates. The paper and its erratum, combined here, are published: ApJL 768, L4 (2013); ApJ, 779, L30 (2013).
Alexey N. Volkov, Robert E. Johnson, Orenthal J. Tucker, Justin T. Erwin
Sep 26, 2010·astro-ph.EP·PDF Thermally-driven atmospheric escape evolves from an organized outflow (hydrodynamic escape) to escape on a molecule by molecules basis (Jeans escape) with increasing Jeans parameter, the ratio of the gravitational to thermal energy of molecules in a planet's atmosphere. This transition is described here using the direct simulation Monte Carlo method for a single component spherically symmetric atmosphere. When the heating is predominantly below the lower boundary of the simulation region, R0, and well below the exobase, this transition is shown to occur over a surprisingly narrow range of Jeans parameters evaluated at R0: λ0 ~ 2-3. The Jeans parameter λ0 ~ 2.1 roughly corresponds to the upper limit for isentropic, supersonic outflow and for λ0 >3 escape occurs on a molecule by molecule basis. For λ0 > ~6, it is shown that the escape rate does not deviate significantly from the familiar Jeans rate evaluated at the nominal exobase, contrary to what has been suggested. Scaling by the Jeans parameter and the Knudsen number, escape calculations for Pluto and an early Earth's atmosphere are evaluated, and the results presented here can be applied to thermally-induced escape from a number of solar and extrasolar planetary bodies.
J. Robert Johnson, Klas Markström
The discrete cube $\{0,1\}^d$ is a fundamental combinatorial structure. A subcube of $\{0,1\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no $k$ which have non-empty intersection and no $l$ which are pairwise disjoint? These questions are naturally expressed as Turán and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Turán and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of $n$ separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Turán and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.