ERGODIC THEOREMS* BY ALEXANDRA IONESCU TULCEA AND CASSIUS IONESCU TULCEA UNIVERSITY OF PENNSYLVANIA Communicated by Einar Hille, December 21, 1961 1. Let (Z. £, , be a complete totally a-finite measure space and E a Banach space. For each 1 < p < o denote by 4C. the vector space of all (Bochner) measurable mappings f of Z into E for which z oflf(z)|P is 4-integrable; here JE is endowed with the...

M. Ablikim, J. Z. Bai, Y. Ban, J. G. Bian, X. Cai, H. F. Chen, H. S. Chen, H. X. Chen, J. C. Chen, Jin Chen, Y. B. Chen, S. P. Chi, Y. P. Chu, X. Z. Cui, Y. S. Dai, Z. Y. Deng, L. Y. Dong, Q. F. Dong, S. X. Du, Z. Z. Du, J. Fang, S. S. Fang, C. D. Fu, C. S. Gao, Y. N. Gao, S. D. Gu, Y. T. Gu, Y. N. Guo, Y. Q. Guo, Z. J. Guo, F. A. Harris, K. L. He, M. He, Y. K. Heng, H. M. Hu, T. Hu, G. S. Huan...

where N ≥ 1 is an integer, ∆ is the forward difference operator defined by ∆u(t) = u(t + 1) − u(t), ∆u(t) = u(t), ∆u(t) = ∆(∆u(t)) for i ≥ 1, p : [0, N ]Z → R with p(0) = p(N), q : [1, N ]Z → R, f : [1, N ]Z × R → R is continuous in its second argument, and λ is a positive parameter. By a solution of BVP (1.1), we mean a function u : [−1, N + 2]Z → R such that u satisfies (1.1). Difference equa...

M. Ablikim, M.N. Achasov, O. Albayrak, D. J. Ambrose, F. F. An, Q. An, J. Z. Bai, R. Baldini Ferroli, Y. Ban, J. Becker, J. V. Bennett, M. Bertani, J.M. Bian, E. Boger,* O. Bondarenko, I. Boyko, R. A. Briere, V. Bytev, H. Cai, X. Cai, O. Cakir, A. Calcaterra, G. F. Cao, S. A. Cetin, J. F. Chang, G. Chelkov,* G. Chen, H. S. Chen, J. C. Chen, M. L. Chen, S. J. Chen, X. Chen, Y. B. Chen, H. P. Che...

M. Ablikim, M.N. Achasov, X. C. Ai, O. Albayrak, D. J. Ambrose, F. F. An, Q. An, J. Z. Bai, R. Baldini Ferroli, Y. Ban, J. Becker, J. V. Bennett, M. Bertani, J.M. Bian, E. Boger,* O. Bondarenko, I. Boyko, R. A. Briere, V. Bytev, H. Cai, X. Cai, O. Cakir, A. Calcaterra, G. F. Cao, S. A. Cetin, J. F. Chang, G. Chelkov,* G. Chen, H. S. Chen, J. C. Chen, M. L. Chen, S. J. Chen, X. Chen, Y. B. Chen,...

Preparation First, based on the definitions of A t , Y t , ˜ Z t and Z t , we can write g t = G(s t , r t , X t) = p −1 st p −1 rt (I − X t X ⊤ t)(E st ⊙ A)(E ·rt ⊙ X) = (I − X t X ⊤ t)A t Y t. Then from (6), we have X t+1 = X t + α t g t W t − α 2 t 2 X t g ⊤ t g t W t. Since W t = (I + α 2 t 4 g ⊤ t g t) −1 = I − α 2 t 4 g ⊤ t g t + O(α 4 t), we get X t+1 = X t + α t A t Y t − α t X t X ⊤ t A...

Oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form r(t)ψ(x(t)) ∣z′(t) ∣∣α−1 z′(t) + b ∫ a q(t, ξ)f(x(g(t, ψ)))dσ(ξ) = 0, t ≥ t0, where α > 0 and z(t) = x(t) + p(t)x(t − τ). Our results improve and extend some known results in the literature. Some illustrating examples are also provided to show the importance of our...

Let $\mathcal{J}_{\field}$ be the Jordan triple system of all $p \times q$ ($p\neq q$; $p,q >1)$ rectangular matrices over a field $\field$ characteristic 0 with product $\{x,y,z\}= x y^t z+ z $, where $y^t$ is transpose $y$. We study universal associative envelope $\mathcal{U}(\mathcal{J}_{\field})$ and show that $\mathcal{U}(\mathcal{J}_{\field}) \cong M_{p+q p+q}(\field)$, $M_{p+q\times p+q}...

By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations z t n q t f x σ t 0, t ≥ t0, where n is even, z t x t p t x τ t , 0 ≤ p t ≤ p0 < ∞, and q t ≥ 0. These oscillation criteria, at least in some sense, complement and improve those of Zafer 1998 and Zhang et al. 2010 . An example is considered to illust...

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on coordinates; but together with a charge conjugation C, total C-P-R-T have enriched active transformations fields representations spacetime-internal symmetry groups quantum field theories (QFTs). In this work, we derive that these can be furth...