and Applied Analysis 3 2. Preliminaries LetM : R → R be anN-function, that is,M is continuous, convex, withM t > 0 for t > 0, M t /t → 0 as t → 0 and M t /t → ∞ as t → ∞. The N-function M conjugate to M is defined byM t sup{st −M s : s > 0}. Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q; that is, for each ε > 0, P t Q εt −→ 0 as t −→ ∞. 2.1 The N-functio...

Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆n−1 and then adding each k-simplex independently with probability p. Let Hk−1(Y ;R) denote the (k−1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ ...

For the usual norm on spaces C(K) and C b (Ω) of all continuous functions on a compact Hausdorff space K (all bounded continuous functions on a locally compact Hausdorff space Ω), the following equalities are proved: lim t→0+ ||f + tg|| C(K) − ||f || C(K) t = max x∈{z | |f (z)|=||f ||} Re(e −i arg f (x) g(x)). and lim t→0+ ||f + tg|| C b (Ω) − ||f || C b (Ω) t = inf δ>0 sup x∈{z | |f (z)|≥||f |...

and Applied Analysis 3 = [− 0.5k 1 x 1.5 1 + 0.5k 1 x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 )] dt + [− 0.5k 2 x 1.5 2 + 0.5k 2 x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 )] dt + [0.5 (x −0.5 1 − x −1 1 ) ε 12 x 2 + 0.5 (x −0.5 2 − x −1 2 ) ε 21 x 1 ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) . (11) There e...

Hard scattering processes involving hadrons at small x are described by a kT -factorization formula driven by a BFKL gluon. We explore the equivalence of this description to a collinear-factorization approach in which the anomalous dimensions γgg and γqg/αS are expressed as power series in αS log(1/x), or to be precise αS/ω where ω is the moment index. In particular we confront the collinear-fa...

and Applied Analysis 3 Lemma 2. For the polynomial equation (10), one has the following results. (i) If r < 0, then (10) has at least one positive root. (ii) If r ≥ 0 and 󳵻 = p − 3q ≤ 0, then (10) has no positive roots. (iii) If r ≥ 0 and󳵻 = p−3q > 0, then (10) has positive roots if and only if z 1 = (1/3)(−p +√󳵻) > 0 and h(z 1 ) ≤ 0. Suppose that (10) has positive roots, without loss of genera...

For PDE examples illuminating the feedback (Redheffer product) theory of conservative boundary control systems, we need to verify conservativity of various hyperbolic PDE’s in domains Ω that have Lipschitz boundaries. A properly rigged version of Green’s Identity is then required on such domains where technical assumptions on functions at the corners of ∂Ω cannot be tolerated; such as present i...

and Applied Analysis 3 It is worth mentioning that in the absence of the white noise, the existence of a global attractor for 1.3 1.4 was established in 40 . The main contribution of this paper is to extend the method of tail estimates to stochastic retarded LDSs and prove the existence of a random attractor for the infinite dimensional random dynamical system generated by stochastic retarded L...