We study families of diffeomorphisms detected by trivalent graphs via the Kontsevich classes. specify some recent results and constructions second named author to show that those non-trivial elements in homotopy groups ◂f()▸π∗⁡◂()▸(B⁢Diff∂⁢(Dd))⊗Q$\pi _*(B\mathrm{Diff}_{\partial }(D^d))\otimes {\mathbb {Q}}$ are lifted moduli space h$h$-cobordisms ◂f()▸π∗⁡◂()▸(B⁢Diff⊔⁢(Dd×I))⊗Q$\pi _*(B\mathrm{...

Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq + cq (a + b)qn+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j P∞ j=0 (b/d)(−c/bd)j q (q)j(−aq/d)j . We then use this result to deduce various corollaries, including the followi...

in this paper, we investigate the goursat problem in the class c21(d)cn0 (d  p) c00 (d q)for a third order pseudoparabolic equation. some results are given concerning the existence and uniquenessfor the solution of the suggested problem.

q 3 q 1 q 1 q 1 q 3 j j+1 i-1 i-2-1 0 D activated by P activated by P j 0 Fig. 10. Two processes entering region D. 27 q 0. .. p p q q q q q q q 1 1 1 1 1 1 1. .. .. . p 0 p p p q 3 p q j 0 q 1 q q q 1 1. .. 0 q 1 q q q 1 1. .. 0 q r 3-d r +d q 3 16 3 16 q 1 17 9-d +d q 1 9 11 9-d +d 11 q 0 9 10-b +b q 0 +ack-ack 10

We prove the nonexistence of [gq(4, d), 4, d]q codes for (a) d = q /2−q− 2q+1 for q = 2, h ≥ 3, (b) d = 2q−3q−2q+1 for q ≥ 7, and (c) d = 2q−rq−q+1 for 3 ≤ r ≤ q − q/p, q = p with p prime, where gq(k, d) = ∑k−1 i=0 ⌈ d/q ⌉