For independent events A and B, the probability P (A & B) is equal to the product of the corresponding probabilities: P (A&B) = P (A) · P (B). It is well known that the product f(a, b) = a ·b has the following property: once n ∑ i=1 P (Ai) = 1 and m ∑ j=1 P (Bj) = 1, the probabilities P (Ai & Bj) = f(P (Ai), P (Bj)) also add to 1: n ∑

For independent events A and B, the probability P (A & B) is equal to the product of the corresponding probabilities: P (A&B) = P (A) · P (B). It is well known that the product f(a, b) = a ·b has the following property: once n ∑ i=1 P (Ai) = 1 and m ∑ j=1 P (Bj) = 1, the probabilities P (Ai & Bj) = f(P (Ai), P (Bj)) also add to 1: n ∑

Let G be a graph with p vertices and q edges and A = {0, 1, 2, . . . , [q/2]}. A vertex labeling f : V (G) → A induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the indu...

Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions finite degree (IFFD) in some variable exponent Lebesgue space real defined on $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this, we employ transference theorem which produce norm starting from $\mathcal{C}(\boldsymbol{R})$, the class bounded uniformly continuous $\...

1.1 The Riemann integral on the real line Let [a, b] be a compact interval of the real line and f : [a, b] → R be a (bounded) function. For the Riemann-integral of f over [a, b] one defines lower and upper (Darboux) sums with respect to a subdivision p of the form a = k0 < k1 < . . . < kn = b: S(f, p) := ∑n i=1(ki − ki−1) inf {f(x) | ki−1 ≤ x ≤ ki} S(f, p) := ∑n i=1(ki − ki−1) sup {f(x) | ki−1 ...

In this paper, we investigate cyclic code over the ring Fpk + vFpk + v 2 Fpk + ...+ v r Fpk , where v = v, p a prime number, r > 1 and gcd(r, p) = 1, we prove as generalisation of [9] that these codes are principally generated, give generator polynomial and idempotent depending on idempotents over this ring as response to an open problem related in [11]. we also give a gray map and proprieties ...

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A Latin square of order n $n$ is an × $n\times n$ matrix symbols, such that each symbol occurs exactly once in row and column. For odd prime power q $q$ let F ${{\mathbb{F}}}_{q}$ denote the finite field . quadratic a L [ , b ] ${\rm{ {\mathcal L} }}[a,b]$ defined by ( ) i j = + − if residue otherwise ${({\mathscr{L}}[a,b])}_{i,j}=\left\{\begin{array}{cc}i+a(j-i) & \,\text{if}\,\,j-i\,\,\text{i...

• for all non-empty algebras A, B over S such that A and B are isomorphic and P [A] holds P [B] • for every non-empty algebra A over S and for strict non-empty subalgebra B of A such that P [A] holds P [B] • for every non-empty algebra A over S and for every congruence R of A such that P [A] holds P [A/R] • Let I be a set and F be an algebra family of I over A . Suppose that for every set i suc...