Suppose M is a complete n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). For any 0 < p < 1, (p0, t0) ∈ M × (0, T ), q ∈ M , we define the Lp-length between p0 and q, Lp-geodesic, the generalized reduced distance lp and the generalized reduced volume Ṽp(τ), τ = t0 − t, corresponding to the Lp-geodesic at the point p0 at time t0. ...

5. Uk = rUk_1 + sUk„2\ U0; U-L arbitrary (J/jff + £/0s#)(l ra s^)" = £7-̂ + (^ + sUQ)x + ••• or (£/0 + (U1 i/0)̂ )(l 2W sx)' = [/0 + U±x + (rU1 + s£/Q)x + ••• 6. Tn = p^.3. + sTn_2 rsTn_3; TQ9 T±s T2 arbitrary (T2x + (sT1 rsTQ)x rsT^) (1 rx s# + rsx)' = T2x + (vT2 + s ^ PS^Q)^ + ••• or (T0 + (Si -rT0)x + (T2 r ^ sT0)x)(l rx sx + P 2 X 3 ) 1 = T0 + T±x + T2x + (rT2 + sTx r TQ)x + •••• From the sol...

Vertebral samples of little skate (Leucoraja erinacea) and winter skate (Leucoraja ocellata) were collected from Cape Hatteras, USA, to Canadian waters to estimate age, growth, and length at weight relationships for both species throughout this range. Maximum observed age was 12.5 and 20.5 years for little skate and winter skate, respectively. Significant length at weight relationships were fou...

By a classical result of P. L evy, the Brownian motion (B t) t0 on R may be characterized as a continuous process on R such that (B t) t0 and (B 2 t ? t) t0 are martingales. Generalizations of this result are usually obtained in the setting of the so-called martingale problem. This paper contains a variant of the martingale problem for stochastic processes on locally compact groups with indepen...

The role played by the additives salicylic acid, L-ascorbic acid and oxalic acid in promoting the catalytic activity of [MnIV2(O)3(tmtacn)2](PF6)2 (1(PF6)2, where tmtacn = N,N',N''-trimethyl-1,4,7-triazacyclononane) in the epoxidation and cis-dihydroxylation of alkenes with H2O2 and in suppressing the catalysed decomposition of H2O2 is examined. Whereas aliphatic and aromatic carboxylic acids e...

and Applied Analysis 3 aij , bij , cik, γik : R → 0, ∞ are all continuous functions bounded above and below by positive constants, and τik t ≥ 0 are bounded continuous functions, ri max1≤j≤l{supt∈Rτij t } > 0, and i, j 1, 2, . . . , n, k 1, 2, . . . , l. Furthermore, in the case Dij t,N aij t − bij t e−N , to guarantee the meaning of mortality terms we assume that aij t > bij t for t ∈ R and i,...

SA.1. Proof of Proposition 1 We prove the result more generally for contracts with lockouts. Fix a contract C = ( ,W0, b, l). The result is trivial if = ;, so assume 6= ;. Let T = max . For any period t 2 with t < T , define the smallest successor period in as (t) = min{t0 : t0 > t, t0 2 }; moreover, let (0) = min . Given any action profile for the agent, the agent’s time-zero expected discount...

To a strongly continuous semigroup ? T(t) t0 on a Banach space X we will associate semigroups ? T n (t) t0 on new Banach spaces X n for each n 2 Z. This construction is inspired by the classical Sobolev spaces and, due to its simplicity, of great help in understanding abstract and concrete semigroups. We start with a strongly continuous semigroup ? T(t) t0 on a Banach space X for which we assum...

We study the Hilbert manifold structure on T0(1) — the connected component of the identity of the Hilbert manifold T (1). We characterize points on T0(1) in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators B1 and B4, associated with the points in T0(1) via conformal welding, are Hilbert-Schmidt. We define a “universal Liouville action” — a real-valued function S1 on T...