This paper introduces an inequality on vectors in $mathbb{R}^n$ which compares vectors in $mathbb{R}^n$ based on the $p$-norm of their projections on $mathbb{R}^k$ ($kleq n$). For $p>0$, we say $x$ is $d$-projectionally less than or equal to $y$ with respect to $p$-norm if $sum_{i=1}^kvert x_ivert^p$ is less than or equal to $ sum_{i=1}^kvert y_ivert^p$, for every $dleq kleq n$. For...

and Applied Analysis 3 To complete the following algorithm, we need the following elementary property. Lemma 7. Let F(x, y) ∈ R[x, y] and p(x) be an irreducible polynomial in R[x]. Consider the module equation F (x, y) = 0 (mod p (x)) . (20) The number of all solutions y = y(x) (modp(x)) is thus at most deg y F. Assumption. Throughout this algorithm, for any F(x, y) ∈ R[x, y], one can solve all...

In this paper, we introduce the notion of an action $Y_X$as a generalization of the notion of a module,and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$  the set of all real numbers. A great many standard mappings on algebr...

Letm(y)=∑j=1yj/n and s(y)= √ m(y2)−m2(y) be themean and the standard deviation of the components of the vector y = (y1, y2, . . . , yn−1, yn), where yq = (y 1 , y 2 , . . . , y n−1, y n) with q a positive integer. Here, we prove that if y ≥ 0, then m(y2p) + (1/√n− 1)s(y2p) ≤ √ m(y2 ) + (1/ √ n− 1)s(y2 ) for p = 0,1,2, . . . . The equality holds if and only if the (n − 1) largest components of y...

i−1 1) Fig. 1. Relay channel. The user, aided by the relay, sends the message M to the receiver. The relay channel is a channel in which there is one sender and one receiver with an intermediate node that acts as relay to help the communication. The channel model consists of four finite sets X , X 1 , Y, Y 1 , and a collection of conditional pmfs p(y, y 1 |x, x 1) on X × X 1 × Y × Y 1. The send...

In this paper, we give an another proof of Theorem 3 in [4]. Theorem 3 in [4] is the following theorem. Theorem 1. Let ζ(s) be the Riemann zeta function and let ζ * (s) = π −s/2 Γ(s/2)ζ(s). For each y ≥ 1 the constant term of the Eisenstein series a 0 (y, s) := ζ * (2s)y s + ζ * (2 − 2s)y 1−s (1) is a meromorphic function that satisfies the modified Riemann hypothesis. More precisely , there is...

Proof of Lemma 1: This retailer would see (y/L) of the total demand, hence the cumulative distribution function of its demand is H(s) = F(sL/y) where F() gives the cumulative distribution function for total market demand, x. Differentiating H(s) gives the required result. () parts) by on (integrati) () () () () () () () () () () () 0 , max(0 0 0 0 0 0 ds s F L y ds s f s y Lq L y ds s f s y Lq ...