On divisible weighted Dynkin diagrams and reachable elements

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra \g. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in \g determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.Comment: 17 pages; v2 minor corrrections; final version, to appear in Transformation Groups (2010

There are no magnetically charged particle-like solutions of the Einstein Yang-Mills equations for Abelian models

We prove that there are no magnetically charged particle-like solutions for Abelian models in Einstein Yang-Mills, but for non-Abelian models the possibility remains open. An analysis of the Lie algebraic structure of the Yang-Mills fields is essential to our results. In one key step of our analysis we use invariant polynomials to determine which orbits of the gauge group contain the possible asymptotic Yang-Mills field configurations. Together with a new horizontal/vertical space decomposition of the Yang-Mills fields this enables us to overcome some obstacles and complete a dynamical system existence theorem for asymptotic solutions with nonzero total magnetic charge. We then prove that these solutions cannot be extended globally for Abelian models and begin an investigation of the details for non-Abelian models.Comment: 48 pages, 1 figur

Non-Supersymmetric Attractor Flow in Symmetric Spaces

We derive extremal black hole solutions for a variety of four dimensional models which, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupled to a sigma model with symmetric target space. The solutions are in correspondence with certain nilpotent generators of the isometry group. In particular, we provide the exact solution for a non-BPS black hole with generic charges and asymptotic moduli in N=2 supergravity coupled to one vector multiplet. Multi-centered solutions can also be generated with this technique. It is shown that the non-supersymmetric solutions lack the intricate moduli space of bound configurations that are typical of the supersymmetric case.Comment: 50 pages, 4 figures; v2: Reference added. To appear in JHE

On all possible static spherically symmetric EYM solitons and black holes

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for any action of the rotation group (or SU(2)) by automorphisms of a principal bundle over space-time whose structure group is a compact semisimple Lie group G. These actions are characterized by a vector in the Cartan subalgebra of g and are called regular if the vector lies in the interior of a Weyl chamber. In the irregular cases (the majority for larger gauge groups) the boundary value problem that results for possible asymptotically flat soliton or black hole solutions is more complicated than in the previously discussed regular cases. In particular, there is no longer a gauge choice possible in general so that the Yang-Mills potential can be given by just real-valued functions. We prove the local existence of regular solutions near the singularities of the system at the center, the black hole horizon, and at infinity, establish the parameters that characterize these local solutions, and discuss the set of possible actions and the numerical methods necessary to search for global solutions. That some special global solutions exist is easily derived from the fact that su(2) is a subalgebra of any compact semisimple Lie algebra. But the set of less trivial global solutions remains to be explored.Comment: 26 pages, 2 figures, LaTeX, misprints corrected, 1 reference adde

Two dimensional computational fluid dynamics model of pollutant transport in an open pit mine under Arctic inversion

Thesis (M.S.) University of Alaska Fairbanks, 2012A better understanding of the microscale meteorology of deep, open pit mines is important for mineral exploitation in arctic and subarctic regions. During strong temperature inversions in the atmospheric boundary layer--which are common in arctic regions during the winter--the concentrations of gaseous pollutants in open pit mines can reach dangerous levels. In this research, a two dimensional computational fluid dynamics (CFD) model was used to study the atmosphere of an open pit mine. The natural airflow patterns in an open pit mine are strongly dependent on the geometry of the mine. Generally, mechanical turbulence created by the mine topography results in a recirculatory region at the bottom of the mine that is detached from the freestream. The presence of a temperature inversion further inhibits natural ventilation in open pit mines, and the air can quickly become contaminated if a source of pollution is present. Several different exhaust fan configurations were modeled to see if the pollution problem could be mitigated. The two dimensional model suggests that mitigation is possible, but the large quantity of ventilating air required would most likely beimpractical in an industrial setting.1. Introduction -- 1.1. Scientific rationale -- 1.2. Air inversion -- 1.3. Previous modeling approaches -- 1.4. Solution approaches -- 1.5. Proposed remediation measures -- 1.6. Scope of this research -- 1.7. Work plan -- 2. Data collection -- 3. Model development -- 3.1. Fundamental transport equations -- 3.2. Cell zone and boundary conditions -- 3.3. Meshing -- 3.4. Discretization -- 3.5. TurbulenceModeling -- 3.6. Geometry and mesh creation -- 3.7. Wind flow in open pit mines -- 3.8. Development of an atmospheric inversion -- 4. Pollutant transport in an open pit mine under Arctic air inversion -- 5. Mitigation of pollutants -- 5.1. Helicopter -- 5.2. Exhaust fan: 142 m³/s -- 5.3. Exhaust fan: 556 m³/s -- 5.4. Exhaust fans: multiple fans, multiple sources (142 m³/s) -- 5.5. Exhaust fans: multiple fans, multiple sources (284 m³/s) -- 6. Summary, conclusions, and recommendations for future work -- 6.1. Summary and conclusions -- 6.2. Future work -- 7. References

Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi--Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.Comment: 65 pages, 67 reference

New Probation Law of Michigan

Nothing connected with the work of a circuit judge demands more thoughtful consideration or occasions him more anxiety than the punishment to be meted out to the men and women who have violated the laws of the state. In almost every other matter there is an opportunity for review by an appellate court. Where litigants differ widely from the decision of the circuit court it is altogether likely that there will be an appeal and the matter will be finally adjudicated by another court. But from the sentence given to one who has plead guilty, or has been found guilty after a jury trial, there is practically no appeal except to the pardoning power of the Executive

Metal ion binding to the amyloid beta monomer studied by native top-down FTICR mass spectrometry

Native top-down mass spectrometry is a fast, robust biophysical technique that can provide molecular-scale information on the interaction between proteins or peptides and ligands, including metal cations. Here we have analyzed complexes of the full-length amyloid β (1-42) monomer with a range of (patho)physiologically relevant metal cations using native Fourier transform ion cyclotron resonance mass spectrometry and three different fragmentation methods—collision-induced dissociation, electron capture dissociation, and infrared multiphoton dissociation—all yielding consistent results. Amyloid β is of particular interest as its oligomerization and aggregation are major events in the etiology of Alzheimer’s disease, and it is known that interactions between the peptide and bioavailable metal cations have the potential to significantly damage neurons. Those metals which exhibited the strongest binding to the peptide (Cu2+, Co2+, Ni2+) all shared a very similar binding region containing two of the histidine residues near the N-terminus (His6, His13). Notably, Fe3+ bound to the peptide only when stabilized toward hydrolysis, aggregation, and precipitation by a chelating ligand, binding in the region between Ser8 and Gly25. We also identified two additional binding regions near the flexible, hydrophobic C-terminus, where other metals (Mg2+, Ca2+, Mn2+, Na+, and K+) bound more weakly—one centered on Leu34, and one on Gly38. Unexpectedly, collisional activation of the complex formed between the peptide and [CoIII(NH3)6]3+ induced gas-phase reduction of the metal to CoII, allowing the peptide to fragment via radical-based dissociation pathways. This work demonstrates how native mass spectrometry can provide new insights into the interactions between amyloid β and metal cations

Global behavior of solutions to the static spherically symmetric EYM equations

The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a black hole horizon as well as those that are analytic and bounded near infinity were shown to exist. Some globally bounded solutions are also known to exist because they can be obtained by embedding solutions for the $G=SU(2)$ case which is well understood. Here we derive some asymptotic properties of an arbitrary global solution, namely one that exists locally near a radial value $r_{0}$, has positive mass $m(r)$ at $r_{0}$ and develops no horizon for all $r>r_{0}$. The set of asymptotic values of the Yang-Mills potential (in a suitable well defined gauge) is shown to be finite in the so-called regular case, but may form a more complicated real variety for models obtained from irregular rotation group actions.Comment: 43 page

Extremal Multicenter Black Holes: Nilpotent Orbits and Tits Satake Universality Classes

Four dimensional supergravity theories whose scalar manifold is a symmetric coset manifold U[D=4]/Hc are arranged into a finite list of Tits Satake universality classes. Stationary solutions of these theories, spherically symmetric or not, are identified with those of an euclidian three-dimensional sigma-model, whose target manifold is a Lorentzian coset U[D=3]/H* and the extremal ones are associated with H* nilpotent orbits in the K* representation emerging from the orthogonal decomposition of the algebra U[D=3] with respect to H*. It is shown that the classification of such orbits can always be reduced to the Tits-Satake projection and it is a class property of the Tits Satake universality classes. The construction procedure of Bossard et al of extremal multicenter solutions by means of a triangular hierarchy of integrable equations is completed and converted into a closed algorithm by means of a general formula that provides the transition from the symmetric to the solvable gauge. The question of the relation between H* orbits and charge orbits W of the corresponding black holes is addressed and also reduced to the corresponding question within the Tits Satake projection. It is conjectured that on the vanishing locus of the Taub-NUT current the relation between H*-orbit and W-orbit is rigid and one-to-one. All black holes emerging from multicenter solutions associated with a given H* orbit have the same W-type. For the S^3 model we provide a complete survey of its multicenter solutions associated with all of the previously classified nilpotent orbits of sl(2) x sl(2) within g[2,2]. We find a new intrinsic classification of the W-orbits of this model that might provide a paradigm for the analogous classification in all the other Tits Satake universality classes.Comment: 83 pages, LaTeX; v2: few misprints corrected and references adde