Let T be a continuous linear map V → V for a (separable) Hilbert space V . Its spectrum σ(T ) is a compact subset of R, so is certainly contained in some finite interval [a, b]. As usual, for a self-adjoint continuous operator S on V , write S ≥ 0 when 〈Sv, v〉 ≥ 0 for all v ∈ V . For self-adjoint S, T , write S ≤ T when T − S ≥ 0. At the outset, with a ≤ −|T |op and b ≥ |T |op, we have, 〈a · v,...

and Applied Analysis 3 exists, when σ t t here by s → t it is understood that s approaches t in the time scale and when x is continuous at t and σ t > t, xΔ t : x σ t − x t μ t . 1.7 Note that if T R, then the delta derivative is just the standard derivative, and when T Z the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases a...

Abstract We present a 5-order continuous-time Σ∆ modulator with a hybrid active-passive loop filter consisting of only three amplifiers. The passive integrators save power, and introduce no distortion. The active integrators provide gain and minimize internal noise contributions. A single-bit switched-capacitor DAC is employed as the main feedback DAC for high clock jitter immunity. An addition...

This paper discusses higher order nonlinear neutral mixed type difference equations of the form
 Δ^{m}[x(n)+p(n)h(x(σ(n)))]+q(n)f(x(τ(n)))=0, n=0,1,2,…,
 where (p(n)), (q(n)) are sequences nonnegative real numbers, h, f:R→R continuous and nondecreasing with uh(u)>0, uf(u)>0 for all u≠0, (σ(n)) (τ(n)) integers such that
 lim_{n→∞}τ(n)=lim_{n→∞}σ(n)=∞.
 In general, we wi...

Given a k-linear operator T from a product of C(K) spaces into a Banach space X, our main result proves the equivalence between T being completely continuous, T having an X-valued separately ω∗ − ω∗ continuous extension to the product of the biduals and T having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to T being weakly compact, and ...

in this study, the role of interaction of pi electrons on the strength of simultaneous σ-hole interactions (pnicogen, chalcogen and halogen bonds) is investigated using the quantum chemical calculations. x-ben||taz∙∙∙y1,y2,y3 complexes (x = cn, f, cl, br, ch3 , oh and nh2, taz= s-triazine and y1,y2 and y3 denotes ph2f, hsf, and clf molecules) is introduced as a model. the results show that inte...

and Applied Analysis 3 Definition 2.1 see 13 . A time scale T is an arbitrary nonempty closed subset of the real setR with the topology and ordering inherited from R. The forward and backward jump operators σ, ρ : T → T, and the graininess μ, ν : T → R are defined, respectively, by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, μ t : σ t − t, ν t : t − ρ t . 2.1 The point t ∈ T is called l...

During the past decades, many integral inequalities have been established since then, for example [1-10], which have played an important role in the research of qualitative properties of solutions of dynamic equations. Our aim in this paper is to establish a new Volterra-Fredholm type delay integral inequality on time scales, which provides new bound for unknown functions. In the rest of the pa...

and Applied Analysis 3 such that a ∈⇓ d. Thus d ∈⇑ a ⊆ U. Therefore D ∩ U ̸ = 0, hence U ∈ σ(P). This proves τ(P) ⊆ σ(P). (2) Now assume that P is strongly continuous. Then P is continuous. In a strongly continuous lattice, the relation ≪ and ⇐ are the same by the Theorem 2.5 of [5]. Also by [2] in a continuous lattice, every Scott open set A satisfies the condition A = ↑≪A, it follows that ever...

Special thanks to my advisor Steffen Lempp for his guidance and conversation. Abstract Always Sometimes 2 In this talk I will present the necessary background and then describe and characterize the non-branching degrees. Significantly, I will show three distinct classes of non-branching degrees. The Medvedev lattice of Π 0 1 classes is a lattice of certain subsets of 2 ω under a natural reducti...