Let E pq be the vertical homology Hg(E 0 p∗) of this complex. In other words, if Z pq = ker(d v pq), B pq = im(d v p,q+1), then E pq = Z 1 pq/B 1 pq. We note that d h induces maps E pq → E p−1,q as follows. The condition dd = −dd shows that d(Z pq) ⊆ Z p−1,q and d(B pq) ⊆ B p−1,q; therefore, there is a homomorphism d̃pq : E 1 pq → E p−1,q. Let E pq be the horizontal homology Hp(E ∗q) of this com...

Let $G= (V,E)$ be a $(p,q)$-graph. A bijection $f: Eto{1,2,3,ldots,q }$ is called an edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ where $f^+(u) = sum_{uwin E} f(uw)$. Moreover, a bijection $f: Eto{1,2,3,ldots,q }$ is called a semi-edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ or $f^+(u)=f^+(v)$. A graph that admits an  ...

We determine the extrapolation law of rearrangement operators acting on the Haar system in vector valued H spaces: If 0 < q ≤ p < 2, then, ‖Tτ,q ⊗ IdX‖ q 2−q q ≤ A(p, q)‖Tτ,p ⊗ IdX‖ p 2−p p . For a fixed Banach space X, the extrapolation range 0 < q ≤ p < 2 is optimal. If, however, there exists 1 < p0 < ∞, so that ‖Tτ,p0 ⊗ IdE‖L0 E < ∞, for each UMD space E, then for any 1 < p < ∞, ‖Tτ,p ⊗ IdE‖...

We prove in this paper a noncommutative version of Leray Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with E 2 terms as E 2 p,q = HP p (A; HP q (B, A)) =⇒ HP p+q (B) and E 2 p,q = HP p (A; Kq(B, A)) =⇒ Kp+q(B).

We show that a necessary and sufficient condition for the existence of a Kp,q factorization of the symmetric complete bipartite multi-digraph )"K:n n is (i) m = n == 0 (mod p) for p = q and (ii) m = n == a (mod d(p' + q')p'q'/e) for p =f. q, where d = (p,q), p' = p/d, q' = q/d, e = ()..,p'q').

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We use extensions to study the semi-simple quotient of the group ring FpAut(P ) of a finite p-group P . For an extension E : N → P → Q, our results involve relations between Aut(N), Aut(P ), Aut(Q) and the extension class [E] ∈ H(Q,ZN). One novel feature is the use of the intersection orbit group Ω([E]), defined as the intersection of the orbits Aut(N) · [E] and Aut(Q) · [E] in H(Q,ZN). This gr...

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with quotient classgroup Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation Z}/p$. We establish such in most cases.