Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.Comment: 24 page

Control Barrier Function Based Quadratic Programs for Safety Critical Systems

Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds

Mixed N‑Heterocyclic Carbene−Bis(oxazolinyl)borato Rhodium and Iridium Complexes in Photochemical and Thermal Oxidative Addition Reactions

In order to facilitate oxidative addition chemistry of fac-coordinated rhodium(I) and iridium(I) compounds, carbene–bis(oxazolinyl)phenylborate proligands have been synthesized and reacted with organometallic precursors. Two proligands, PhB(OxMe2)2(ImtBuH) (H[1]; OxMe2 = 4,4-dimethyl-2-oxazoline; ImtBuH = 1-tert-butylimidazole) and PhB(OxMe2)2(ImMesH) (H[2]; ImMesH = 1-mesitylimidazole), are deprotonated with potassium benzyl to generate K[1] and K[2], and these potassium compounds serve as reagents for the synthesis of a series of rhodium and iridium complexes. Cyclooctadiene and dicarbonyl compounds {PhB(OxMe2)2ImtBu}Rh(η4-C8H12) (3), {PhB(OxMe2)2ImMes}Rh(η4-C8H12) (4), {PhB(OxMe2)2ImMes}Rh(CO)2 (5), {PhB(OxMe2)2ImMes}Ir(η4-C8H12) (6), and {PhB(OxMe2)2ImMes}Ir(CO)2 (7) are synthesized along with ToMM(η4-C8H12) (M = Rh (8); M = Ir (9); ToM = tris(4,4-dimethyl-2-oxazolinyl)phenylborate). The spectroscopic and structural properties and reactivity of this series of compounds show electronic and steric effects of substituents on the imidazole (tert-butyl vs mesityl), effects of replacing an oxazoline in ToMwith a carbene donor, and the influence of the donor ligand (CO vs C8H12). The reactions of K[2] and [M(μ-Cl)(η2-C8H14)2]2 (M = Rh, Ir) provide {κ4-PhB(OxMe2)2ImMes′CH2}Rh(μ-H)(μ-Cl)Rh(η2-C8H14)2 (10) and {PhB(OxMe2)2ImMes}IrH(η3-C8H13) (11). In the former compound, a spontaneous oxidative addition of a mesityl ortho-methyl to give a mixed-valent dirhodium species is observed, while the iridium compound forms a monometallic allyl hydride. Photochemical reactions of dicarbonyl compounds 5 and 7 result in C–H bond oxidative addition providing the compounds {κ4-PhB(OxMe2)2ImMes′CH2}RhH(CO) (12) and {PhB(OxMe2)2ImMes}IrH(Ph)CO (13). In 12, oxidative addition results in cyclometalation of the mesityl ortho-methyl similar to 10, whereas the iridium compound reacts with the benzene solvent to give a rare crystallographically characterized cis-[Ir](H)(Ph) complex. Alternatively, the rhodium carbonyl 5 or iridium isocyanide {PhB(OxMe2)2ImMes}Ir(CO)CNtBu (15) reacts with PhSiH3 in the dark to form the silyl compound {PhB(OxMe2)2ImMes}RhH(SiH2Ph)CO (14) or {PhB(OxMe2)2ImMes}IrH(SiH2Ph)CNtBu (17). These examples demonstrate the enhanced thermal reactivity of {PhB(OxMe2)2ImMes}-supported iridium and rhodium carbonyl compounds in comparison to tris(oxazolinyl)borate, tris(pyrazolyl)borate, and cyclopentadienyl-supported compounds