12. Let ax + by = 0, a, b ∈ R and a + b 6= 0 be a fixed line in R through the origin. We know u = ( −b √ a2+b2 , a √ a2+b2 ) is a point on line ax + by = 0 and v = ( a √ a2+b2 , b √ a2+b2 ) is a unit vector orthogonal to ( −b √ a2+b2 , a √ a2+b2 ). Thus the set B = {( −b √ a2+b2 , a √ a2+b2 ), ( a √ a2+b2 , b √ a2+b2 )} is an orthonormal basis for R. To find the reflection T (x, y) of (x, y) wi...

Let A be a factor von Neumann algebra and ? preserve n-tuple new product derivations on A, that is, for every A1,A2,... ,An ?(A1 A2 ?...? An) = ?(A1) An + A1 ?(A2) ?... ... ?(An) where Ai ?Aj A*iAj ?A*jAi i,j N, then is additive *-derivation, the condition ?(?I/2) self-adjoint operator {1,i}.

I’ll now sketch a proof. It basically follows by by recalling the definition of the probability of an even conditioned on X2 = a2. We first note that the set of choices for (X 1, X2, 3) given that X2 = a2 is def Sa2 = {(X1, a2, X3) : (X1, a2) ∈ C12 and (a2, X3) ∈ C23} . Conditioning on X2 = a2, we obtain a sample chosen uniformly from Sa2 . Thus, for (a 1, a2, a3) ∈ Sa2 , 1 P [(X1, X2, X3) = (a...

Let θ(a1, a2, · · · , ak) denote the graph obtained by connecting two distinct vertices with k independent paths of lengths a1, a2, · · · , ak respectively. Assume that 2 ≤ a1 ≤ a2 ≤ · · · ≤ ak. We prove that the graph θ(a1, a2, · · · , ak) is chromatically unique if ak < a1 + a2, and find examples showing that θ(a1, a2, · · · , ak) may not be chromatically unique if ak = a1 + a2.

Let m be a fixed positive integer. For any positive integer n, we define the arithmetical function am(n) as the integer part of the m-th root of n. That is, am(n) = [n 1 m ], where [x] denotes the greatest integer not exceeding to x. For example, a2(1) = 1, a2(2) = 1, a2(3) = 1, a2(4) = 2, a2(5) = 2, a2(6) = 2, a2(7) = 2, a2(8) = 2, a2(9) = 3, a2(10) = 3, · · · . In [1], Professor F. Smarandach...

This note investigates πT,c1,c2(a1,a2)=P(supt∈[0,T](σ1B(t)−c1t)>a1,supt∈[0,T](σ2B(t)−c2t)>a2), where {B(t):t≥0} is a standard Brownian motion, with T>0,σ1,σ2>0,c1,c2∈R. We derive explicit formula for the probability πT,c1,c2(a1,a2) and find its asymptotic behavior both in so called many-source high-threshold regimes.

Phospholipase A2 values increase in serum in various inflammatory states, infections, and postoperatively in surgical patients. Several organs, including the liver and spleen have been suggested as sources of circulating phospholipase A2. The purpose of the present work was to examine the possible role of the spleen as a source of elevated serum concentrations of phospholipase A2 after surgery....

Let A2(n, d) be the maximum size of a binary code of length n and minimum distance d. In this paper we present the following new lower bounds: A2(18, 4) ≥ 5632, A2(21, 4) ≥ 40960, A2(22, 4) ≥ 81920, A2(23, 4) ≥ 163840, A2(24, 4) ≥ 327680, A2(24, 10) ≥ 136, and A2(25, 6) ≥ 17920. The new lower bounds are a result of a systematic computer search over transitive permutation groups.