When Lanchester met Richardson,the outcome was stalemate: A parable for mathematical models of insurgency

Many authors have used dynamical systems to model asymmetric war. We explore this approach more broadly, first returning to the prototypical models such as Richardson’s arms race, Lanchester’s attrition models and Deitchman’s guerrilla model. We investigate combinations of these and their generalizations, understanding how they relate to assumptions about asymmetric conflict. Our main result is that the typical long-term outcome is neither annihilation nor escalation but a stable fixed point, a stalemate. The state cannot defeat the insurgency by force alone, but must alter the underlying parameters. We show how our models relate to or subsume other recent models. This paper is a self-contained introduction to 2D continuous dynamical models of war, and we intend that, by laying bare their assumptions, it should enable the reader to critically evaluate such models and serve as a reminder of their limitations.     

What causes the hyperintense T2-weighting and increased short T2 signal in the corticospinal tract?

The corticospinal tract (CST) appears hyperintense on both T₂-weighted images and myelin water maps. Here, an extended multiecho T₂ relaxation sequence with echoes out to 1120 ms was used to characterize the longer T₂ times present in the CST. The T₂ distribution from the CST was compared to other white matter structures in 14 healthy subjects. The intra-/extracellular T₂ peak of the CST was broadened relative to other white matter structures and often split into two distinct peaks. In the CST, it appeared that the intracellular and extracellular water environments had unique T₂ times, causing the intracellular water peak to be pushed down into the myelin water T₂ regime and the extracellular peak to be pushed up to longer T₂ times. The conventional myelin water T₂ limits of 5-40 ms resulted in an artificial increase in myelin water fraction (MWF), causing the CST to be bright on myelin water images. When the upper limit for MWF was decreased to 25 ms, the CST regions exhibited MWF values similar to those found for adjacent anterior and posterior regions. The CST has unique magnetic resonance characteristics, which should be taken into consideration when being examined, especially when compared to pathological tissue.     

Optimization of small deviation for mixed fractional Brownian motion with trend

In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of this optimal split. The asymptotic behaviour of the probability as the ball becomes small is analysed for zero trend function and for the particular choice of the upper limiting function.     

A mild halogenation of pyrazoles using sodium halide salts and Oxone

A mild, inexpensive, and operationally simple pyrazole halogenation method utilizing Oxone and sodium halide salts is reported. This work documents 17 examples of alkyl, aryl, allyl, and benzyl substituted 4-chloro and 4-bromopyrazoles, obtained in up to 93% yield. Reactions are performed in water under ambient conditions and generation of organic byproducts is avoided.     

Conserved charges and supersymmetry in principal chiral and WZW models

Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess–Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess–Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension.