The papers [2], [7], [13], [3], [1], [6], [9], [4], [14], [8], [5], [15], [11], [12], and [10] provide the notation and terminology for this paper. We adopt the following rules: n denotes an element of N, h, k, x, x0, x1, x2, x3 denote real numbers, and f , g denote functions from R into R. Next we state a number of propositions: (1) If x0 > 0 and x1 > 0, then loge x0 − loge x1 = loge( x1 ). (2...

Znajdziemy teraz wzór na odleg lość punktu (x0, y0) od prostej, wspó lrze↪dne punktów której spe lniaja↪ równanie ax+by+c = 0 . Oczywíscie po to, by to równanie przedstawia lo prosta↪ trzeba za lożyć, że wektor [a, b] nie jest wektorem zerowym, czyli że [a, b] 6= [0, 0] . Odleg lość punktów (x1, y1) i (x2, y2) to √ (x1 − x2) + (y1 − y2) . Wynika to natychmiast z twierdzenia Pitagorasa zastosowa...

Notes on Variance, Covariance, and Summation Operator By Hiro Kasahara Properties of Summation Operator For a sequence of the values {x1, x2, ..., xn}, we write the sum of x1, x2, ..., xn−1, and xn using the summation operator as x1 + x2 + ... + xn = n ∑ i=1 xi. (1) Given a constant c, n ∑ i=1 cxi = cx1 + cx2 + ... + cxn = c× (x1 + x2 + ... + xn) = c n ∑ i=1 xi. (2) • For example, consider the ...

The bivariate confluent hypergeometric function kind 1 distribution is defined by the probability density function proportional to x1 1 x2 2 1 F1(α; β; −x1 − x2). In this article, we study several properties of this distribution and derive density functions of X1/X2, X1/(X1 + X2), X1 + X2 and 2 √ X1X2. The density function of 2 √ X1X2 is represented in terms of modified Bessel function of the s...

A complex number is nothing more than a point in the xy–plane. The first component, x, of the complex number (x, y) is called its real part and the second component, y, is called its imaginary part, even though there is nothing imaginary about it. The sum and product of two complex numbers (x1, y1) and (x2, y2) are defined by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) (x1, y1) (x2, y2) = (x1x2 − ...

Figure 1: Factor Graph In particular, consider a setup where (U1, U2) are two equiprobable bits that are encoded into (X1, X2) = (U1 ⊕ U2, U2). Then, (X1, X2) are mapped to (Y1, Y2) by two independent BMS channels with transition probabilities P(Y1 = y |X1 = x) = P(Y2 = y |X2 = x) = W (y|x). The factor graph for this setup is shown in Figure 1. Since the mapping from (U1, U2) to (X1, X2) is inv...