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\"Written at the advanced undergraduate level, the book will serve equally well as a text for students and as a reference for instructors and users of separation of variables. It requires a background in engineering mathematics, but no prior exposure to separation of variables. The abundant worked examples provide guidance for deciding whether and how to apply the method to any given problem, help in interpreting computed solutions, and give insight into cases in which formal answers may be useless\"--Jacket.

The concepts G-g-closedness and G-g-openness were used to popularize presented modern classes of separation axioms in grill topological spaces. Many relationships between multiple kinds of these classes are summarized, too.

Objectives To prospectively evaluate a 3D-multiecho-Dixon sequence with inline calculation of proton density fat fraction (PDFF) and R2* (qDixon), and an improved version of it (qDixon-WIP), for the MR-quantification of hepatic iron in a clinical setting. Methods Patients with increased serum ferritin underwent 1.5-T MRI of the liver for the evaluation of hepatic iron overload. The imaging protocol for R2* quantification included as follows: (1) a validated, 2D multigradient-echo sequence (initial TE 0.99 ms, R2*-ME-GRE), (2) a 3D-multiecho-Dixon sequence with inline calculation of PDFF and R2* (initial TE 2.38 ms, R2*-qDixon), and optionally (3) a prototype (works-in-progress, WIP) version of the latter (initial TE 1.04 ms, R2*-qDixon-WIP) with improved water/fat separation and noise-corrected parameter fitting. For all sequences, three manually co-registered regions of interest (ROIs) were placed in the liver. R2* values were compared and linear regression analysis and Bland-Altman plots calculated. Results Forty-six out of 415 patients showed fat-water (F/W) swap with qDixon and were excluded. A total of 369 patients (mean age 52 years) were included; in 203/369, the optional qDixon-WIP was acquired, which showed no F/W swaps. A strong correlation was found between R2*-ME-GRE and R2*-qDixon ( r 2 = 0.92, p < 0.001) with Bland-Altman revealing a mean difference of − 3.82 1/s (SD = 21.26 1/s). Correlation between R2*-GRE-ME and R2*-qDixon-WIP was r 2 = 0.95 ( p < 0.001) with Bland-Altman showing a mean difference of − 0.125 1/s (SD = 30.667 1/s). Conclusions The 3D-multiecho-Dixon sequence is a reliable tool to quantify hepatic iron. Results are comparable with established relaxometry methods. Improvements to the original implementation eliminate occasional F/W swaps and limitations regarding maximum R2* values. Key Points • The 3D-multiecho-Dixon sequence for 1.5 T is a reliable tool to quantify hepatic iron. • Results of the 3D-multiecho-Dixon sequence are comparable with established relaxometry methods. • An improved version of the 3D-multiecho-Dixon sequence eliminates minor drawbacks.

Purpose Multifunctionality in life-cycle assessment (LCA) is solved with allocation, for which many different procedures are available. Lack of sufficient guidance and difficulties to identify the correct allocation approach cause a large number of combinations of methods to exist in scientific literature. This paper reviews allocation procedures for recycling situations, with the aim to identify a systematic approach to apply allocation. Methods Assumptions and definitions for the most important terms related to multifunctionality and recycling in LCA are given. The most relevant allocation procedures are identified from literature. These procedures are expressed in mathematical formulas and schemes and arranged in a systematic framework based on the underlying objectives and assumptions of the procedures. Results and discussion If the LCA goal asks for an attributional approach, multifunctionality can be solved by applying system expansion—i.e. including the co-functions in the functional unit—or partitioning. The cut-off approach is a form of partitioning, attributing all the impacts to the functional unit. If the LCA goal asks for a consequential approach, substitution is applied, for which three methods are identified: the end-of-life recycling method and the waste mining method, which are combined in the 50/50 method. We propose to merge these methods in a new formula: the market price-based substitution method. The inclusion of economic values and maintaining a strict separation between attributional and consequential LCA are considered to increase realism and consistency of the LCA method. Conclusions and perspectives We identified the most pertinent allocation procedures—for recycling as well as co-production and energy recovery—and expressed them in mathematical formulas and schemes. Based on the underlying objectives of the allocation procedures, we positioned them in a systematic and consistent framework, relating the procedures to the LCA goal definition and an attributional or consequential approach. We identified a new substitution method that replaces the three existing methods in consequential LCA. Further research should test the validity of the systematic framework and the market price-based substitution method by means of case studies.

Comprehensive experimental and computational investigations have revealed possible mechanisms underlying low-frequency unsteadiness observed in spanwise homogeneous shock-wave/turbulent-boundary-layer interactions (STBLI). In the present work, we extend this understanding by examining the dynamic linear response of a moderately separated Mach 2.3 STBLI to small perturbations. The statistically stationary linear response is analysed to identify potential time-local and time-mean linear tendencies present in the unsteady base flow: these provide insight into the selective amplification properties of the flow at various points in the limit cycle, as well as asymmetry and restoring mechanisms in the dynamics of the separation bubble. The numerical technique uses the synchronized large-eddy simulation method, previously developed for free shear flows, significantly extended to include a linear constraint necessary for wall-bounded flows. The results demonstrate that the STBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency for upstream shock motion. The absolute instability is maintained through constructive feedback of perturbations through the recirculation: it is self-sustaining and insensitive to external forcing. The dynamics are characterized for key frequency bands corresponding to high–mid-frequency Kelvin–Helmholtz shedding along the separated shear layer $(St_{L}\\sim 0.5)$ , low–mid-frequency oscillations of the separation bubble $(St_{L}\\sim 0.1)$ and low-frequency large-scale bubble breathing and shock motion $(St_{L}\\sim 0.03)$ , where the Strouhal number is based on the nominal length of the separation bubble, $L$ : $St_{L}=fL/U_{\\infty }$ . A band-pass filtering decomposition isolates the dynamic flow features and linear responses associated with these mechanisms. For example, in the low-frequency band, extreme shock displacements are shown to correlate with time-local linear tendencies toward more moderate displacements, indicating a restoring mechanism in the linear dynamics. However, a disparity between the linearly stable shock position and the mean shock position leads to an observed asymmetry in the low-frequency shock motion cycle, in which upstream motion occurs more rapidly than downstream motion. This is explained through competing linear and nonlinear (mass depletion through shedding) mechanisms and discussed in the context of an oscillator model. The analysis successfully illustrates how time-local linear dynamics sustain several key unsteady broadband flow features in a causal manner.

This paper presents distributed adaptive fault-tolerant control for the attitude tracking of multiple flexible spacecraft on SO ( 3 ) without modal variable measurement. Assume that the communication graph among the followers is undirected and connected and there exists at least one follower linked to the leader. To deal with the distributed tracking on SO ( 3 ) , a finite-time observer is designed to estimate the leader’s information for the followers. A distributed adaptive fault-tolerant controller is proposed to achieve the attitude tracking based on the estimation of the unmeasurable modal variables. The separation principle between the finite-time observer and the proposed controller is adopted to prove the controller convergence. Since the controllers are developed on SO ( 3 ) directly, the singularity and ambiguity associated with other attitude representations can be avoided. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed control protocols.

The generation of discrete acoustic tones in the compressible flow around an aerofoil is addressed in this work by means of nonlinear numerical simulations and global stability analyses. The nonlinear simulations confirm the appearance of discrete tones in the acoustic spectrum and, for the chosen flow case, the global stability analyses of the mean-flow dynamics reveal that the linearized operator is stable. However, the flow response to incoming disturbances exhibits important transient growth effects that culminate in the onset of aeroacoustic feedback loops, involving instability processes on the suction- and pressure-surface boundary layers together with their cross-interaction by acoustic radiation at the trailing edge. The features of the aeroacoustic feedback loops and the appearance of discrete tones are then related to the features of the least-stable modes in the global spectrum: the spatial structure of the direct modes displays the coupled dynamics of hydrodynamic instabilities on the suction surface and in the near wake. Finally, different families of global modes will be identified and the dynamics that they represent will be discussed.

The use of static or dynamic roughness elements has been shown in the past to delay the separation of a laminar boundary layer from a solid surface. Here, we examine analytically the effect of such elements on the local and breakaway separation points, corresponding respectively to the position of zero skin friction and presence of a singularity in the roughness region, for flow over a hump embedded within the boundary layer. Two types of roughness elements are studied: the first is small and placed near the point of vanishing skin friction; the second is larger and extends downstream. The forced flow solution is found as a sum of Fourier modes, reflecting the fixed frequency forcing of the dynamic roughness. Solutions for both the static and dynamic roughness show that the presence of the roughness element is able to move the separation points downstream, given an appropriate choice of roughness frequency, height, position and width. This choice is found to be qualitatively similar to that observed for leading-edge separation. Furthermore, for a negative static roughness a small region of separated flow forms at high roughness depth, although there is a critical depth above which boundary-layer breakaway moves suddenly upstream.

The ballistic impact behavior of hybrid 3-dimentional woven composites (3DWC) is predicted through a novel numerical modeling technique and validated the outcome through experimental data. Continuum shell elements with Hashin failure criterion and cohesive surface contact algorithm with traction separation law are used to predict the damage behavior and failure mechanisms during the impact process at interply and intraply levels. Z-yarns are represented by the connector elements with uniaxial behavior having stress-based failure criterion. The proposed methodology predicted the different damage mechanisms during the impact process in comparison with the experimental data and estimated the residual velocities with acceptable accuracy. Different failure mechanisms incurred due to the hybrid nature of the material are also captured by the numerical simulation and the effect of z-yarns are truly depicted by the use of simple 1D elements over the different phases of perforation process. Overall, the finite element (FE) methodology by using simplest form of the elements, their constitutive and damage behavior gives promising results by sufficiently reducing the computational cost.

A new concept called topological graph and via this concept introduces a new class of separation axioms via the concepts of graph (G-T0, G-T1, G-T2, G-T3, and G-T4), many relations among them were studied and investigated some characterization of these concepts. At last give an application for them.