We consider Itô’s stochastic differential equation (SDE). First, we would review a standard theory under standard assumptions. Then we would see how such a standard theory should be modified under different and more general assumptions and how and what new notions need be introduced to discuss such modifications. Consider the following SDE on R dX (t) = σ k(X(t))dB (t) + b(X(t))dt, X(0) = x (1)...

where A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t))t≥0, on a Banach space X; the histories xt : (−∞, 0] → X, xt(θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically; Ω ⊂ B is open; 0 ≤ σ < T ; σ < t0 < t1 < .... < tn ≤ T ; and q : Bn → B; F, G : [σ, T ] × Ω → X are appropriate continuous functions. There exist a extensive l...

The fundamental laws of physics are described from the modern point of view by gauge theories with continuous gauge groups. With some modification the gauge principle can (and should) be applied to discrete dynamical systems with nontrivial symmetries also. We consider specifics of introduction of gauge invariance into W -equivariant discrete dynamical systems with finite sets of states Σ, wher...

Wireless and wireline communication applications demand A/D converters with MHz signal bandwidth and 14b or better resolution. Compared with switched-capacitor (SC) implementations, continuous-time (CT) Σ∆ modulators have the advantages of lower power consumption and intrinsic anti-alias filtering. Moreover, as the sampler is inside of the CT Σ∆ loop (see the upper half of Fig. 3.5.1), any samp...

and Applied Analysis 3 Combining Lemma 2 in [45] and Lemma 1 in [28], one can obtain the remaining proof easily. Remark 4. WhenT → 0, system (2) degenerates to a general continuous-time positive switched system as follows: ?̇? (t) = A σ(t) x (t) + A dσ(t) x (t − d (t)) + D σ(t) w (t) , x (t) = φ (t) , t ∈ [−d, 0] z (t) = C σ(t) x (t) + E σ(t) w (t) , (5) where d(t) denotes the time-varying delay...

Sigma-delta (Σ∆) modulators are widely used for analogue-to-digital and digital-to-analogue conversion in both discrete-time and continuous-time implementations [1]. A Σ∆ modulator is an example of a nonlinear system, the nonlinearity being introduced by the presence of the quantiser. In [2] we applied results of our work on driven interval shifts to explain the steady-state dynamics of the fir...

The reduction method provides an algorithm to compute large deviation estimates of (possibly non reversible) Markov processes with exponential transition rates. It replaces the original graph minimisation equations of Freidlin and Wentzell by more tractable path minimisation problems. When applied to study the metastability of the dynamics, it gives a large deviation principle for the loop eras...

For any abelian Polish σ-compact group H there exist an Fσ ideal Z ⊆ P (N) and a Borel Z -approximate homomorphism f : H → HN which is not Z -approximable by a continuous true homomorphism g : H → HN .