The papers [11], [13], [1], [15], [2], [8], [9], [16], [5], [12], [10], [4], [6], [7], and [14] provide the notation and terminology for this paper. In this paper x denotes a real number and Z denotes an open subset of R. One can prove the following propositions: (1) Suppose Z ⊆ dom((the function tan) ·(the function cot)). Then (i) (the function tan) ·(the function cot) is differentiable on Z, ...

The new curve contains all of the information about the original curve’s tangent lines. For example, it crosses the y-axis whenever the original curve has a horizontal tangent line. The straight line y = mx + b, or (x(t), y(t)) = (t,mt + b), has only one tangent line and thus a trivial transformation to a point, (x∗(t), y∗(t)) = (m, b). Some conic sections transform to conic sections. For examp...

Now, let (u, v) = Rα(θ, φ′). Here, Rα = Ry(α). We omit the z rotation since that does not affect Yl0 which has no azimuthal dependence. The vector corresponding to coordinates (u, v) is then given by   sinu cos v sinu sin v cos u   =   cosα 0 sinα 0 1 0 − sinα 0 cosα     sin θ′ cosφ′ sin θ′ sinφ′ cos θ′   =   cosα sin θ′ cosφ′ + sinα cos θ′ sin θ′ sinφ′ cosα cos θ′ + sinα sin θ′ (...

(b) Give a general real solution of the homogeneous equation Ly = 0. Solution. A general solution is y = c1e −2t cos(3t) + c2e −2t sin(3t) + c3 cos(7t) + c4 sin(7t) + c5t cos(7t) + c6t sin(7t) + c7e 5t + c8t e 5t + c9t e + c10e −3t + c11 + c12t . The reasoning is as follows: • the single conjugate pair −2± i3 yields e cos(3t) and e sin(3t); • the double conjugate pair ±i7 yields cos(7t) , sin(7...

Background: Breast cancer is a common cause of brain metastasis. Although breast has relatively high survival rates, its rate after metastasis to the lower. Conventional two-dimensional cell culture models and animal are widely used in metastatic research, these have tremendously contributed understanding this disease. However, some limitations, such as different physiological features genetic ...

We estimate the contribution of millisecond pulsars to the diffuse gamma-ray background of the Galaxy, and show that a significant fraction of the Galactic background may originate from them. A small number of the unidentified COS-B point sources may, in fact, be millisecond pulsars. It is argued that several hundred millisecond pulsars may be detectable as point sources by the GRO satellite.

as the price relative which is how much a stock goes up or down in a single day. St denotes the amount of wealth we have at the start of day t and we assume S1 = 1. We denote wt(i) to be the fraction of our wealth that we have in stock i at the beginning of day t which can be viewed as a probability distribution as ∀i, wt(i) ≥ 0 and ∑ iwt(i) = 1. We can then derive the total wealth in stock i a...