For simplicity, we adopt the following rules: i, j, m, n, k denote natural numbers, x denotes a set, K denotes a field, a, a1, a2 denote elements of K, D denotes a non empty set, d, d1, d2 denote elements of D, M , M1, M2 denote matrices over D, A, A1, A2, B1, B2 denote matrices over K, and f , g denote finite sequences of elements of N. One can prove the following propositions: (1) Let K be a ...

One can prove the following propositions: (1) Let A, B be finite sequences and f be a function. Suppose rngA∪rngB ⊆ dom f. Then there exist finite sequences f1, f2 such that f1 = f · A and f2 = f · B and f · (A a B) = f1 a f2. (2) For every bag b of 0 holds decomp b = 〈〈∅, ∅〉〉. (3) For all natural numbers i, j and for every bag b of j such that i ¬ j holds b↾i is an element of Bags i. (4) Let i...

The Lemaître-Tolman-Bondi (LTB) model represents an inhomogeneous spherically symmetric universefilledwithfreelyfallingdustlikematterwithoutpressure. First,wehaveconsideredaFinslerian anstaz of (LTB) and have found a Finslerian exact solution of vacuum field equation. We have obtained the R(t,r) and S(t,r) with considering establish a new solution of Rµν = 0. Moreover, we attempttouseFinslergeo...

rH..il:ribllltlton.-tr'ee two-sample test is proposed tha t is an extension of the Wilcoxon test with arbitrary censoring on the right. The test is conditional on the pattern ~ha.t~TV:&l>ll~l..lll. The null hypothesis is H0 : F1(t) = F2(t) (t ~ T) agai nst either H1 : F1(t) < F 2(t) (t ~ T) or H2 : F1(t) < F2(t) or F1(t) > F2 (t) (t ~ T), '~here .l\,F2 are cumulative distributions (discrete or ...

We follow the rules: D is a non empty set, f1, f2 are finite sequences of elements of D, and i, n, n1, n2, n3, n4, n5, n6 are natural numbers. We now state two propositions: (1) If n ¬ len f1, then (f1 a f2)⇂n = ((f1)⇂n) a f2. (2) (f1 a f2)↾(len f1 + i) = f1 a (f2↾i). A Gene-Set is a non-empty non empty finite sequence. Let S be a Gene-Set. We introduce GA− SpaceS as a synonym of UnionS. Let f ...

One can prove the following propositions: (1) For all functions f , g, h such that dom f ∩ dom g ⊆ domh holds f+·g+·h = g+·f+·h. (2) For all functions f , g, h such that f ⊆ g and rng h ∩ dom g ⊆ dom f holds g · h = f · h. (3) For all functions f , g, h such that dom f ⊆ rng g and dom f misses rng h and g domh misses dom f holds f · (g+·h) = f · g. (4) For all functions f1, f2, g1, g2 such that...

The scheme FuncIdx correctness deals with a set A , a non empty set B, and a unary functor F yielding an element of B, and states that: (i) There exists a function F from A into B such that for every x such that x ∈ A holds Fx = F (x), and (ii) for all functions F1, F2 from A into B such that for every x such that x ∈ A holds (F1)x = F (x) and for every x such that x∈A holds (F2)x = F (x) holds...