Non-limiting Case Research Articles

Optimal default options

Previous studies of optimal default options demonstrate that either opt-out minimization or maximization is optimal under restrictive conditions. We obtain a general characterization of the solution by studying optimal defaults when one of the problem’s parameters approaches a limiting value. We interpret these “asymptotic optima” as approximate optima for non-limiting cases and justify this interpretation through numerical simulations. When the designer and choosers agree about the activity’s value, simple forms of weighted opt-out minimization are asymptotically optimal. Additional results encompass Pigouvian fees, normative ambiguity, and cases in which the designer and choosers disagree about the activity’s value.

Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator

In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, text {id}_tau : {B}_{p_1,q_1}^{s_1,tau _1}(Omega ) hookrightarrow {B}_{p_2,q_2}^{s_2,tau _2}(Omega ) and text {id}_tau : {F}_{p_1,q_1}^{s_1,tau _1}(Omega ) hookrightarrow {F}_{p_2,q_2}^{s_2,tau _2}(Omega ), where Omega subset {{{mathbb {R}}}^d} is a bounded domain, obtaining necessary and sufficient conditions for the continuity of text {id}_tau . This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces.

Signal recovery of a Fabry–Pérot interferometric x-ray pulse detector based on the RadOptic effect

The signal recovery of a Fabry–Pérot interferometric x-ray pulse detector based on the RadOptic effect in the non-limiting case was investigated in this research. A Fe-doped InP with an invariant excess carrier recombination mechanism was used as the interference cavity material to achieve a constant temporal instrumental response function (tIRF). A linear and time-invariant detection system described by the convolution of the time-varying x-ray pulse and the constant tIRF was established based on the transient refractive index variation model determined by the three effects of band filling, band shrinkage, and free-carrier absorption. For the non-limiting case, the accumulation of excess carriers enhanced the sensitivity but altered the fluctuations of the real x-ray pulse. To realistically reconstruct the x-ray pulse, two-photon absorption of the infrared ultrashort pulse was used to simulate the ultrashort x-ray excitation to obtain the tIRF. Finally, using the conjugate gradient method, the original signal recorded by the detection system was deconvoluted to recover the signal. The success of signal recovery in the non-limiting case provided the basis for the development of detectors with adjustable sensitivity controlled by carrier lifetime.

Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED

Compared with the quantum trajectory equation (QTE), the quantum Bayesian approach has the advantage of being more efficient to infer a quantum state under monitoring, based on the integrated output of measurements. For weak measurement of qubits in circuit quantum electrodynamics (cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest. Elegant work towards this task was carried out by Korotkov in ‘bad-cavity’ and weak-response limits (Korotkov 2011 Quantum Bayesian approach to circuit QED measurement (arXiv:1111.4016)). In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of Korotkov to more general system parameters. The obtained Bayesian rule is in full agreement with Korotkovʼs result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.

Some [formula omitted]-numbers of embeddings in function spaces with polynomial weights

In this paper, we investigate the asymptotic behavior of the Gelfand, Kolmogorov and Weyl numbers of Sobolev embeddings in weighted function spaces of Besov and Triebel–Lizorkin type with polynomial weights. The exact estimates for the Gelfand and Kolmogorov numbers are obtained in the so-called non-limiting case and complement those previous results in the quasi-Banach setting. Also, the asymptotic order of the Weyl numbers is established by means of the basic Weyl estimates in the finite-dimensional spaces improved recently as well as flexible uses of the operator ideal technique.

Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces

In this paper we study the Gelfand and Kolmogorov numbers of Sobolev embeddings between weighted function spaces of Besov and Triebel–Lizorkin type with polynomial weights. The sharp asymptotic estimates are determined in the so-called non-limiting case.

Spatially distributed model for the analysis of laser beam induced current (LBIC) measurements of thin film silicon solar modules

A 3D distributed model is developed and implemented based on circuit analysis software for the investigation of spatial variation in performance due to the distributed nature and non-uniformity of solar cell properties. This is applied to LBIC measurements where it is used for sensitivity analysis of the measurements with respect to certain parameters in series connected thin film PV modules. The model is used to explain the differences in dark and illuminated measurements, which clearly shows the illuminated LBIC signal is largely dependent on the homogeneity of the background illumination. The dark LBIC is largely affected by the shunt resistance of the neighbouring cells rather than by the signal strength of the cell under test. It is required to bring the cell into limiting conditions, which then gives a signal one order of magnitude stronger than that in the non-limiting case. The simulations are validated against measurements taken in these regimes.

Multiple hydrodynamic states in trickle flow: Quantifying the extent of pressure drop, liquid holdup and gas–liquid mass transfer variation

It is well established that pressure drop and liquid holdup under trickle flow conditions are functions of the flow history. However, the extent of possible variation of these and other critical hydrodynamic parameters has not been fully quantified. In this study, specifically defined prewetting procedures are used as limiting cases for hydrodynamic hysteresis. These are: • Non-prewetted. • Levec prewetted: the bed is flooded and drained and after residual holdup stabilisation the gas and liquid flows are introduced. • Kan L prewetted: the bed is operated in the pulse flow regime (by increasing liquid velocity) after which liquid flow rate is reduced to the desired set point (all at the desired gas flow rate). • Kan G prewetted: the bed is operated in the pulse flow regime (by increasing gas velocity) after which gas flow rate is reduced to the desired set point (all at the desired liquid flow rate). • Super prewetted: the bed is flooded and gas and liquid flows are introduced once draining commences. It is shown that the upper limiting case for pressure drop is the Kan L mode of operation. The lower limiting cases are the non-prewetted and Levec prewetted modes (these coincide). Pressure drop may vary by as much as 700% even for prewetted beds. Liquid holdup is different in all five prewetting modes. The upper limiting case is the Kan G mode of operation, while the lower limiting case is the non-prewetted mode ( Kan G holdup is approximately 160% that of non-prewetted mode holdup at U G = 90 mm / s ). At low gas velocities the Kan L holdup can be 400% of that of the non-prewetted beds. Importantly, the lower limiting case for prewetted beds is the Levec mode. Holdup in the Kan G mode may be as much as 130% of the holdup in the Levec mode (at U G = 90 mm / s ). The effect of hydrodynamic multiplicity of the volumetric mass transfer coefficient is measured by the desorption of oxygen from water into nitrogen. In this case the different prewetting procedures result in three distinct regions, the upper region being the Kan and Super prewetted beds, the intermediate region being the Levec prewetted bed and the lower region being the dry bed. Mass transfer coefficients in the upper region can be as much as 600% of that of the lower region and 250% of that of the intermediate region. Evidently, prewetting (and even pulsing flow prewetting) does not guarantee that the bed is operating at the maximum values of pressure drop, holdup and mass transfer coefficient. Evidence of operation in between the limiting cases is presented. These non-limiting cases can be reached in multiple ways.

Simplified Model for the Influence of Inclusion Aspect Ratio on the Stiffness of Aligned Reinforced Composites

A simplified model for the Young’s Modulus of aligned composites possessing spheroidal inclusions, of any aspect ratio α, is developed herein based on a triply periodic array of parallelpiped inclusions. The Young’s Modulus of continuous unidirectional fiber composites (α→∞), particulate composites (α = 1) and periodically bilaminated composites (α = 0) are obtained as limiting cases based on the Parallel-Series and Series-Parallel schemes of analyses. A reinforcing parameter, R, of the inclusion is then identified from the limiting cases such that R ∊ [0,1]. By curve-fitting for a relationship between the reinforcement parameter and the inclusion aspect ratio for α∊ [0,∞], the Young’s Modulus for aligned short fiber composites (α > 1) and aligned flake composites (α < 1) are obtained. For verification, the present model at limiting cases (α =0, 1, ∞) were reduced to other simple models and/or compared with experimental results. For nonlimiting cases (α <1, α > 1), comparison with other theoretical models show reasonable agreement.

The sharp Riesz potential estimates in metric spaces

We introduce the Riesz potential operator for functions defined on a quasi-metric space endowed with a nonnegative doubling measure, satisfying a certain lower bound for the measure of a ball. For this operator we prove sharp Lorentz-norm estimates in the spirit of the results due to O'Neil and Peetre (in the non-limiting case) and to Hansson and Brezis-Wainger (in the limiting case).

Embeddings and Entropy Numbers for General Weighted Sequence Spaces: The Non-Limiting Case

Abstract The paper deals with embeddings for sequence spaces with general weights. Our main results clarify in a rather final way the compactness of embeddings between these spaces including estimates for the entropy numbers of such compact embeddings in a generalized nonlimiting case.

A study of the instability of liquid jets and comparison with Tomotika's analysis

A novel dispersion equation describing the instability of low velocity, liquid jets is derived using an integro-differential approach. The explicit form of the derived equation enables much easier prediction of the most unstable wavenumber and disturbance growth rate than does Tomotika's implicit, complex dispersion relationship. Good agreement is observed with all limiting solutions to Tomotika's relationship and with numerical solutions to non-limiting cases. The influences of density ratio, viscosity ratio and Ohnesorge number on liquid jet instability are examined.

The embedding theorem in limiting cases and its applications in singular perturbation

In 1980, Brezis[6], using the technique of dividing the total space into two parts, proved the embedding theorem of limiting case which is very important in applications. In 1982, Ding Xiaqi improved the proof given in [6], by using of the technique of dividing the total space into three parts. In this paper, using the technique of dividing the total space into three parts, the author proves uniformly the results obtained by Ding[3,4], and gives an embedding theorem of limiting case including β (Lemma 2.2). And he also gives two kinds of examples, applying the embedding theorems (limiting case and non-limiting case) and the interpolation theorems. These examples are the singular perturbation problems in the sense of Lions[1] (for the definition of singular perturbation, see [1], Introduction). But the singular solutionU e converges uniformly to the limit solution (degenerate)U, ase→0.

Anisotropic solvent‐induced shifts of the trimethylsiloxy group

AbstractPolyorganosiloxanes of the QxMy type [Q = SiO4/2; M = (CH3)3SiO1/2] have been quantitatively analysed by 1H NMR in the presence of aromatic solvents. This has been achieved by integrating the signals due to the separation of the chemically equivalent trimethylsilyl protons. This separation is a further example of the ASIS effect (anisotropic solvent‐induced shift). The inverse relationship between observed shift and solute molar volume has been confirmed, and has been extended to include non‐limiting cases. Some evidence for solvent ordering and structure is also presented.

General mathematical model for mass transfer accompanied by chemical reaction

AbstractThe present paper concerns the mechanism of mass transfer accompanied by a first‐order irreversible chemical reaction between two phases. Based on the film‐penetration concept a general mathematical model to describe the physico‐chemical behavior at the interface has been formulated. Mass transfer mechanism may be analyzed and evaluated in terms of the dimensionless groups appearing in the derived equations.For limiting conditions the derived general equations can be reduced to those based on the simple postulations such as the film theory, the penetration theory, and the surface renewal theory. For nonlimiting cases the film‐penetration concept provides information which cannot be obtained by either the film theory or the surface renewal theory alone.Experimental results appearing in literature show that the physical mass transfer coefficient is proportional to the molecular diffusivity to the v‐th power and that v varies widely between 0.15 and 1.0. The film‐penetration concept theoretically predicts this v‐variation, whereas, in accordance with the film theory or the surface renewal theory, v has to be a certain fixed value.It is shown that if an accurate physical mass transfer coefficient is available, the film‐penetration concept, the film theory, and the surface renewal theory all predict practically the same effect of chemical reaction on the mass transfer rate. However if the chemical mass transfer coefficient is to be predicted without an accurate physical transfer coefficient, the choice of the theory or the mechanism may become important.