Let K be a fixed number field, assumed to be Galois over Q. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that “on average,” the number of such prime ideals with norm less than or equal to x satisfies an as...

This note continues the study initiated in 2006 by P.L. Papini, R. R. Phelps and the author on some classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let H be the family of all closed, convex and bounded subsets of a Banach space endowed with the Hausdorff metric. A completion of A ∈ H is a diametrically maximal set D ∈ H satisfying A ⊂ D and diamA = d...

Our understanding of animal flight benefits greatly from specialized wind tunnels designed for flying animals. Existing facilities can simulate laminar flow during straight, ascending and descending flight, as well as at different altitudes. However, the atmosphere in which animals fly is even more complex. Flow can be laminar and quiet at high altitudes but highly turbulent near the ground, an...

The lifetimes of the unitary currents from ion channels, as revealed from single-channel recording, are traditionally thought to follow exponential or multiexponential distributions. The interpretation of these event-time distributions is that the gating process follows Markov kinetics among a small number of states. There is recent evidence, however, that certain systems exhibit distributions ...

It is well known that a spherically symmetric constant density static star, modeled as a perfect fluid, possesses a bound on its mass m by its radial size R given by 2m/R ≤ 8/9 and that this bound continues to hold when the energy density decreases monotonically. The existence of such a bound is intriguing because it occurs well before the appearance of an apparent horizon at m = R/2. However, ...

Let K be a fixed number field, assumed to be Galois over Q. Let r and f be fixed integers with f positive. Given an elliptic curve E , defined over K , we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r . Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an...

Let K be a fixed number field, assumed to be Galois over Q. Let r and f be fixed integers with f positive. Given an elliptic curve E , defined over K , we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r . Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an...

Let K be a number field and r an integer. Given an elliptic curve E , defined over K , we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r . Under certain restrictions on K , we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard...