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We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over
-adic local fields with
. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.
This is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to On the Stabilization of the Trace Formula published in 2011. The books are intended primarily for two groups of readers: those interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information on multiplicities of automorphic representations; and those interested in the classification of I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap considerably. These volumes present convincing evidence supporting this, clearly and succinctly enough that readers can pass with minimal effort between the two points of view. Over a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, makes this series timely.
The thermal expansion coefficient (TEC) of nano-B4C having 50 nm mean particle size was measured as a function of applied direct current (DC) electric field strength varying from 0 to 12.7 V/mm and over a temperature range from 298 K up to 1273 K. The TEC exhibits a linear variation with temperature despite being measured over a range that is well below 50% of B4C’s normal melting temperature. The zeroth- and first-order TEC coefficients under zero-field condition are 4.8220 ± 0.009 × 10−6 K−1 and 1.462 ± 0.004 × 10−9 K−1, respectively. Both TECs exhibit applied DC electric field dependence. The higher the applied field strength, the steeper the linear thermal expansion response in nano-B4C, which suggests that the applied field affects the curvature of the interatomic potentials at the equilibrium bond length at a given temperature. No anisotropic thermal expansion with and without applied electric field was observed, although nano-B4C has a rhombohedral unit cell symmetry. The rhombohedral unit cell angle was determined as δR= 65.7046° (0.0007), and it remains unaffected by a change in temperature and applied electric field strength, which we attribute to B4C nanoparticle size and its carbon saturation.