Frobenius distributions of elliptic curves in short intervals on average

The Ramanujan Journal - For an elliptic curve $$E/{\mathbb {Q}}$$ , let $$a_p$$ denote the trace of its Frobenius endomorphism over $${\mathbb {F}}_p$$ , where p is a prime of good reduction for E....     

Synthesis,spectroscopic studies,and x-ray crystallographic analysis of the organotin carbohydrate: 1,2∶3,4-di-O-isopropylidene-6-O-triphenylstannylmethyl-α-D-galactopyranose

The title organotin carbohydrate, C₃₁H₃₆O₆Sn, has been synthesized and its molecular structure has been determined in solution and in the solid state. NMR, infrared, mass and X-ray crystallographic techniques were used. The chiral molecules crystallize in the monoclinic space group P2₁ withZ=2. The triphenyltin and carbohydrate moieties are linked by a trans methylene-oxygen-methylene arrangement. The pyranosyl ring adopts a twist-boat conformation and the isopropylidene rings adopt different (half-chair and envelope) forms. Solution and solid-state conformations are similar as only three ¹³C shift values are greater than 2ppm; the ¹¹⁹Sn value is 12 ppm.     

Low temperature magnetic ordering of a series of structurally characterized chloroferrates

Iron-57 Mössbauer spectroscopic results for a new series of related chloroferrate salts of complex ammonium cations are presented. In particular, spectra of materials based on FeCl ₄ ^– , FeCl ₅ ^2– , FeCl ₆ ^– , [FeCl₅(H₂O)]^2–, and [FeCl₅(CH₃OH)]^2– anions are discussed relative to the systematics of their isomer shifts, coordination numbers, and low temperature 3D critical ordering temperatures. The following specific systems have been studied by Mössbauer spectroscopy and definitively characterized by single-crystal X-ray analysis: [CH₃NH ₃ ⁺ ]₄[FeCl₆]^3–[Cl^–](A), [Me₂N(-CH₂CH₂)₂NMe ₂ ⁺ ][FeCl ₅ ^2– ] (B), [H₃NCH₂CH₂NH ₃ ²⁺ ][FeCl₅·H₂O^2–], (C), and [NH₃(CH₂)₆NH ₃ ²⁺ ]₄[(FeCl ₆ ^3– )(FeCl ₄ ^– )ClCl ₄ ^– ] (D). The spectral data for these complexes are considered in the light of results of previous studies for systems such as [K₂FeCl₅·H₂O], [Co(NH₃)₆][FeCl₆], diamethyltriethylenediammoniumpentachloroferrate (E), and the related [FeCl₅·CH₃OH]^2– salt (F). The critical 3D ordering temperatures are 1.45, 6.80, 3.45, and 1.22K forA, B, C, andD, respectively.     

Mechanical properties of twist grain boundaries in Cu

In a previous paper we studied the Young's and shear modulus of a high-angle twist grain boundary (5) in Cu, using the EAM, and related it to the uniaxial strain derivatives of single crystals. In this paper, we discuss elastic properties of ten additional twist grain boundaries, from 8.8–43.6°. The monolayer Young's modulus at each boundary was calculated and found to be 20–50% higher than the bulk value for all eleven boundaries for both csl and type1 structures. The monolayer shear modulus at each boundary was calculated and found to be 93–98% lower than the bulk value for six grain boundaries with csl structure and found to decrease with increasing twist angle. The critical shear stress was also calculated for eleven boundaries with csl structure and found to roughly decrease with increasing twist angle.     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.