We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: Theorem. Let M be a connected oriented closed smooth 3-manifold. Let L1(M) be the set of framed links in M up to a framed cobordism. Let deg : L1(M) → H1(M ;Z) be the map taking a framed link to its homology class. Then for each α ∈ H1(M ;Z) there is a 1-1 ...

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finitedimensional vector space H1(M,R) are stable norms of a Riemannian metric on M . If the dimension of M is at least three, I. Babenko and F. Bala...

E3 c(x, y, z)dH(x)dH(y)dH(z) where H1 is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E ⊂ Rn is said to be “purely unrectifiable” if for any Lipschitz function γ : R → Rn, H1(E ∩ γ(R)) = 0 whereas it is said to ...

B7-H1, a member of the B7 family of proteins, is hypothesised to play an important role in the immune escape of tumours through its binding to the PD-1 receptor on activated T and B cells. By inducing T lymphocyte apoptosis, tumour cells can suppress an effective antitumour immune response. Although the immunosuppressive effect of B7-H1 has been studied in many tumours, its other biological fun...

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...

and Applied Analysis 3 Definition 2.1. A function u x, t is called a weak solution of 2.1 if and only if i u ∈ L2 0, T ;H1 0 Ω , with u′ ∈ L2 0, T ;H−1 Ω , ii u| −τ,0 u0 ∈ L2 Ω , iii ∫T 0 〈ut, φ〉 Du,Dφ dt ∫T 0 f u t , u t − τ , φ dt, for each φ ∈ L2 0, T ;H1 0 Ω . Here 〈, 〉 and , denote the pair of H−1 Ω and H1 0 Ω , the inner product in L2 Ω , respectively. Next we will give two very important...