Starting and unstarting of hypersonic air inlets

This thesis concerns a technical hurdle that must be overcome in relation to air-breathing propulsion technologies for future space access vehicles--it discusses the flow starting process in supersonic and hypersonic air-inlets. A study is conducted, with the aid of numerical simulations, based on an inviscid model of a thermally perfect gas. Effects of boundary-imposed temporal and spatial gradients on the inlet starting phenomenon are documented for the first time. It is shown that purely accelerative starting is generally not possible, for inlets of any positive contraction, unless thousands of g 's of acceleration are imposed. It is proposed that removal of frangible structures, such as fast rupturing diaphragms, be used to impose sufficiently high spatial gradients, as necessary to permit starting beyond Kantrowitz' limit. It is shown that, for a perforated diffuser, starting takes place if a sonic line, at the leading edge of a slit, occurs at an area ratio equal to, or higher than, that corresponding to Kantrowitz' limit.

Starting and unstarting of hypersonic air inlets

This thesis concerns a technical hurdle that must be overcome in relation to air-breathing propulsion technologies for future space access vehicles--it discusses the flow starting process in supersonic and hypersonic air-inlets. A study is conducted, with the aid of numerical simulations, based on an inviscid model of a thermally perfect gas. Effects of boundary-imposed temporal and spatial gradients on the inlet starting phenomenon are documented for the first time. It is shown that purely accelerative starting is generally not possible, for inlets of any positive contraction, unless thousands of g 's of acceleration are imposed. It is proposed that removal of frangible structures, such as fast rupturing diaphragms, be used to impose sufficiently high spatial gradients, as necessary to permit starting beyond Kantrowitz' limit. It is shown that, for a perforated diffuser, starting takes place if a sonic line, at the leading edge of a slit, occurs at an area ratio equal to, or higher than, that corresponding to Kantrowitz' limit.

Pfaffian Point Processes from Free Fermion Algebras: Perfectness and Conditional Measures

The analogy between determinantal point processes (DPPs) and free fermionic calculi is well-known. We point out that, from the perspective of free fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show that a positive contraction acting on a ''doubled'' one-particle space with an additional structure defines a unique PfPP. Recently, Olshanski inverted the direction from free fermions to DPPs, proposed a scheme to construct a fermionic state from a quasi-invariant probability measure, and introduced the notion of perfectness of a probability measure. We propose a method to check the perfectness and show that Schur measures are perfect as long as they are quasi-invariant under the action of the symmetric group. We also study conditional measures for PfPPs associated with projection operators. Consequently, we show that the conditional measures are again PfPPs associated with projection operators onto subspaces explicitly described.

Invariant coupling of determinantal measures on sofic groups

To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a countable sofic finitely generated group and $G=({\rm\Gamma},\mathsf{E})$ be a Cayley graph of ${\rm\Gamma}$. We show that if $Q_{1}$ and $Q_{2}$ are two ${\rm\Gamma}$-equivariant positive contractions on $\ell ^{2}({\rm\Gamma})$ or on $\ell ^{2}(\mathsf{E})$ with $Q_{1}\leq Q_{2}$, then there exists a ${\rm\Gamma}$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination $\mathbf{P}^{Q_{1}}\preccurlyeq \mathbf{P}^{Q_{2}}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when ${\rm\Gamma}$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures $\mathbf{P}^{Q}$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when ${\rm\Gamma}$ is amenable, $\mathbf{P}^{Q}$ is isomorphic to a Bernoulli shift, which was known before only when ${\rm\Gamma}$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.

Exponential Asymptotic Stability of a Repairable System with Two Identical Components

The repairable system solution’s exponential asymptotic stability was discussed in this paper, First we prove that the positive contraction strongly continuous semigroup which is generated by the operator corresponding to these equations describing a system with two identical components is a quasi-compact operator. Following the result that 0 is an eigenvalue of the operator with algebraic index one and the strongly continuous semi-group is contraction, we deduce that the spectral bound of the operator is zero. By the above results we obtain easily the exponential asymptotic stability of the solution of the repairable system.

On the a.s. convergence of the one-sided ergodic Hilbert transform

We show that for T a Dunford–Schwartz operator on a σ-finite measure space (X,Σ,μ) and f∈L1(X,μ), whenever the one-sided ergodic Hilbert transform ∑ n≥1(Tnf/n) converges in norm, it converges μ-a.s. A similar result is obtained for any positive contraction of some fixed Lp(X,Σ,μ), p>1. Applying our result to the case where T is the (unitary) operator induced by a measure-preserving (invertible) transformation, we obtain a positive answer to a question of Gaposhkin.

ON PERTURBED POSITIVE SEMIGROUPS ON THE BANACH SPACE OF TRACE CLASS OPERATORS

Let [Formula: see text] be the generator of a positive (i.e. leaving invariant [Formula: see text]) contraction semigroup on [Formula: see text], the space of self-adjoint trace class operators on a Hilbert space H, endowed with the trace norm, and let [Formula: see text] be a positive linear operator such that [Formula: see text], [Formula: see text]. We show that there exists a minimal positive contraction semigroup generated by some [Formula: see text] and provide a systematic study of the total mass carried by individual trajectories [Formula: see text] with non-negative initial data ρ. The analysis relies on mathematical properties of two (a priori) different extensions [Formula: see text] of the functional [Formula: see text] to [Formula: see text].

EXISTENCE OF FELLER COCYCLES ON A C*-ALGEBRA

The quantum stochastic differential equation dkt = kt ∘ θαβdΛβα(t) is considered on a unital C*-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.