2 Note that in **the** approximation in which we are work-
ing, **the** shape of **the** Wilson line is irrelevant, and **the** fram- ing is entirely specified by its **boundary** endpoint(s).
∂χ(z) = j(z). **The** **OPE** of χ with various op- erators can be derived **from** **the** known j OPEs. As explained at length in [12], while generically χ does not exist as an operator in **the** holographic dual CFT (since its zero mode is unphysical), it is nevertheless a useful computational tool. In terms of χ, **the** expression for **the** gauge-invariant **bulk** scalar reads

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3.2. AC field applied perpendicularly to pre-existing magnetization
Now we investigate **the** results obtained in **the** “crossed field” configuration. A typical result of **the** decay of **the** c-axis average magnetic flux density against **the** transverse field AC field is plotted in figure 3. This graph shows how **the** average flux density (probed by **the** sensing coil) is affected drastically when cycles of transverse field are applied to **the** single domain. This behaviour is in excellent agreement with other results of crossed field experiments carried out on other superconducting materials [27-31]. It should be emphasized that **the** DC signal plotted in figure 3 is **the** average flux density, which may differ **from** **the** true magnetization due to finite-size effects, as shown clearly in ref. [39]. On figure 4, **the** results of similar experiments are displayed, but what is plotted is **the** central flux density against **the** surface of **the** sample, as recorded by a miniature Hall probe stuck against **the** sample surface. Two experiments were carried out in order to investigate **the** influence of parameters of **the** experimental system on **the** measured data. On figure 4(a), **the** Hall probe was intentionally misoriented by a small angle (4 degrees) while in figure 4(b) **the** Hall probe was purposefully placed near **the** edge of **the** sample. **The** results are always compared to those obtained with a correctly placed Hall probe. **The** two set of curves plotted in figure 4(b) are very close to each other, showing that **the** exact positioning of **the** Hall probe against **the** surface is not a critical parameter of **the** experiment. Figure 4(a), however, shows that a correct Hall probe angular orientation is essential. Following similar sets of experiments carried out at different angles, it was determined in practice that **the** (unavoidable) experimental misorientation that can be tolerated should be 0.5° or less.

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displacement used as **boundary** conditions is thus picked up **from** nodes in **the** **bulk** of **the** DIC mesh. **The** displacement **fields** measured by DIC and MIC using **the** mesh in Figure 9(a) for **the** last loading step are depicted in Figure 11 . **The** scatter between **the** two displacements is difficult to observe by **the** naked eyes. For assessing **the** accuracy of **the** displacement field obtained by MIC, **the** correlation error map are plotted for **the** two analyses in Figure 12 . These two error maps are again difficult to distinguish what confirms that **the** MIC displacement leads to a registration of **the** images of **the** same accuracy as **the** DIC analysis. To further assess this point, **the** average correlation error for DIC and MIC is plotted as a function of **the** steps in Figure 13 . **The** error level is slightly higher for MIC, what is expected because only 500 modes are used to measure **the** displacement instead of 2 × N . **The** search of **the** optimal solution in terms of grey level conservation is thus more constraint and **the** minimum error level reached with **the** reduced parametrization is higher. These results ensure that **the** displacement parametrization adopted for MIC leads to measured **fields** globally as accurate as thoses obtained for a DIC analysis.

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As expected, **the** transverse field reduces **the** -axis induction but perceptible differences appear between **the** and direc- tions. First we examine **the** field distribution along **the** axis (Fig. 4). **The** plots are slightly tilted as indicated by **the** arrows in Fig. 4. This behavior can be understood qualita- tively by considering demagnetization effects and by taking into account that **the** Hall probe is located at a finite distance **from** **the** sample top surface. After **the** first sweep, **the** sample is per- manently magnetized along **the** -axis. Since **the** induction lines are necessarily closed, they find a return path in **the** free space, in particular above **the** top-surface of **the** disk. This produces an additional -component of **the** induction measured by **the** Hall probe. This component is positive for and negative for , resulting in a distorted flux profile. Measurements with **fields** directed toward (not shown here) yield **the** oppo- site phenomenon. Therefore **the** behavior depicted in Fig. 4 is merely a geometric effect caused by **the** trapped induction in **the** transverse field direction.

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Abstract
**Bulk** melt-processed Y-Ba-Cu-O (YBCO) has significant potential for a variety of high-field permanent-magnet- like applications, such as **the** rotor of a brushless motor. When used in rotating devices of this kind, however, **the** YBCO can be subjected to both transient and alternating magnetic **fields** that are not parallel to **the** direction of magnetization and which have a detrimental effect on **the** trapped field. These effects may lead to long-term decay of **the** magnetization of **the** **bulk** sample. In **the** present work, we analyze both experimentally and numerically **the** remagnetization process of a melt-processed YBCO single domain that has been partially demagnetized by a magnetic field applied orthogonal to **the** initial direction of trapped flux. Magnetic torque measurements are used as a tool to probe changes in **the** remanent magnetization during various sequences of applied field. **The** application of a small magnetic field between **the** transverse cycles parallel to **the** direction of original magnetization results in partial remagnetization of **the** sample. Rotating **the** applied field, however, is found to be much more efficient at remagnetizing **the** **bulk** material than applying a magnetizing field pulse of **the** same amplitude. **The** principal features of **the** experimental data can be reproduced qualitatively using a two- dimensional finite-element numerical model based on an E-J power law. Finally, **the** remagnetization process is shown to result **from** **the** complex modification of current distribution within **the** cross-section of **the** **bulk** sample.

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As expected, these results are completely different **from** **the** transport properties of **the** same model in **the** absence of **bulk** dephasing [16], where **the** quantum coherence of **the** **bulk** dynamics maintains **the** localized character via a step-magnetization profile and an exponentially decaying current with **the** system size [16].
[1] “50 years of Anderson Localization”, E. Abrahams Ed, World Scientific (2010). [2] R. Nandkishore and D. A. Huse, Ann. Review of Cond. Mat. Phys. 6, 15 (2015). [3] E. Altman and R. Vosk, Ann. Review of Cond. Mat. Phys. 6, 383 (2015). [4] S. A. Parameswaran, A. C. Potter and R. Vasseur, arXiv:1610.03078. [5] J. Z. Imbrie, V. Ros and A. Scardicchio, arXiv:1609.08076.

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1.6 Conclusion
In this chapter, we have given a brief survey of active contours in segmentation as well as techniques for road and hydrographic network extraction **from** images. Active contour methods are categorized into two different classes: edge-based and region-based. **The** for- mer has many limitations because it considers local information around **the** **boundary** of **the** object in question. Between **the** two types of edge-based active contours, geometric active contours have many advantages over parametric active contours, such as computational simplicity and **the** ability to change curve topology during evolution. **The** introduction of region-based methods tends to give better segmentation results because they use global in- formation about **the** object. **The** corresponding energy functionals tends to have fewer local minima which makes reasonable **the** use of local optimization algorithms. Moreover, we have mentioned a family of active contour methods which incorporates specific knowledge about **the** shape of **the** object: **the** so-called shape priors. Our interest fits this family of methods because we aim to model network-like regions which have complex shapes and need specific and sophisticated shape priors for automatic (or semi-automatic) solution of **the** problem of extraction.

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Crossed-magnetic-field effects on **bulk** high-temperature superconductors have been studied both experi- mentally and numerically. **The** sample geometry investigated involves finite-size effects along both 共crossed-兲 magnetic-field directions. **The** experiments were carried out on **bulk** melt-processed Y-Ba-Cu-O single domains that had been premagnetized with **the** applied field parallel to their shortest direction 共i.e., **the** c axis兲 and then subjected to several cycles of **the** application of a transverse magnetic field parallel to **the** sample ab plane. **The** magnetic properties were measured using orthogonal pickup coils, a Hall probe placed against **the** sample surface, and magneto-optical imaging. We show that all principal features of **the** experimental data can be reproduced qualitatively using a two-dimensional finite-element numerical model based on an E-J power law and in which **the** current density flows perpendicularly to **the** plane within which **the** two components of magnetic field are varied. **The** results of this study suggest that **the** suppression of **the** magnetic moment under **the** action of a transverse field can be predicted successfully by ignoring **the** existence of flux-free configura- tions or flux-cutting effects. These investigations show that **the** observed decay in magnetization results **from** **the** intricate modification of current distribution within **the** sample cross section. **The** current amplitude is altered significantly only if a field-dependent critical current density J c 共B兲 is assumed. Our model is shown to be quite appropriate to describe **the** cross-flow effects in **bulk** superconductors. It is also shown that this model does not predict any saturation of **the** magnetic induction, even after a large number 共⬃100兲 of transverse field cycles. These features are shown to be consistent with **the** experimental data.

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where G = (H − iω) −1 and ~ k = (k x , k y ) is **the** momen-
tum. This can be implemented numerically, and some examples are given in Fig. 1 . **The** Chern number can be as large as -5, which is unusually high for such a model. **The** phase diagrams in Fig. 1 show changes in Chern numbers away **from** **the** analytically calculated **bulk** gap closing lines for **the** Γ and Dirac points. To be certain that this is not a numerical error we track **the** **bulk** gap across some of these transitions, see Fig. 2 . For all **the** changes in **the** Chern number we can see that **the** gap does close at some point in **the** BZ as required. One can also explicitly check that **the** **bulk**-**boundary** correspon- dence holds, demonstrating that these regions are not caused by numerical errors. However, as we shall see in what follows, these high Chern numbers do not necessar- ily lead to large numbers of protected MBS.

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1.3. Outline of **the** paper: In Section 2 we give **the** precise definition of our micro- scopic dynamics and its invariant measures, we also introduce all **the** spaces of test functions where **the** microscopic fluctuation **fields**, namely **the** density and **the** height, will be defined and we give **the** proper definition of these **fields**. Section 3 contains all **the** rigorous definitions of solutions to **the** SPDEs that we obtain, namely **the** OU /SBE with Dirichlet **boundary** conditions, **the** KPZ equation with Neumann **boundary** condi- tions and **the** SBE with linear Robin **boundary** conditions, and we explain how these equations are linked. In Section 4 we prove **the** convergence of **the** microscopic **fields** to **the** solutions of **the** respective SPDEs, namely Theorems 3.17 and 3.20 . In Section 5 we prove **the** second order Boltzmann-Gibbs principle, which is **the** main technical result that we need at **the** microscopic level in order to recognize **the** macroscopic limit of **the** density fluctuation field as an energy solution to **the** SBE. Finally, in Section 6 we give **the** proof of **the** uniqueness of solutions to **the** aforementioned SPDEs. **The** appendices contain some important aside results that are needed along **the** paper, but to facilitate **the** reading flow we removed them **from** **the** main body of **the** text. In particular, Ap- pendix E sketches how one could prove that **the** microscopic Cole-Hopf transformation of **the** microscopic density fluctuation field converges to **the** SHE, and in particular we show that already at **the** microscopic level **the** Cole-Hopf transformation changes **the** **boundary** conditions **from** Neumann to Robin’s type.

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1. Introduction
**The** spontaneous or magnetic field-induced reversal of **the** magnetisation direction in materials and nanostruc- tures has been **the** subject of intensive studies since many decades. **The** use of magnetic materials in applications ranging **from** compass needles to electrical motors, truck brakes, cellular phones and personal computers has triggered **the** search for materials with particular properties concerning both their static and dynamic behaviour. Materials used e.g. in transformers need to switch their magnetisation direction under very small values of applied magnetic **fields** to minimise losses; these are so-called soft magnetic materials. On **the** other hand, permanent magnets used in electrical motors and generators need their magnetisation to be as stable as possible against both magnetic **fields** and thermal effects; these are so-called hard magnetic materials. Consider however information written on magnetic storage media in **the** form of small magnetic grains. In this case, a compromise must be found for **the** grains need to be stable for years, but on **the** other hand their magnetisation direction should respond quickly to moderate magnetic field values to allow information to be written fast. Thus for each particular case an understanding of how magnetisation reversal proceeds is needed. Magnetisation reversal can take place in different ways, depending on **the** size of **the** object and physical parameters like **the** exchange interaction and magnetic anisotropy. In **the** atomic case, a magnetic field along a direction other than **the** initial magnetisation will cause a torque on **the** magnetisation given by M × H ef f , where H ef f is **the** local effective field. This torque will induce a

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tical transitions which can be brought into resonance with electromagnetic **fields** in plasmonic nanostructures and microcavities 15–18 ; v) in monolayer TMDs, electron-
hole (e-h) interactions are much stronger than in con- ventional semiconductors, due to **the** reduced effective screening 19,20 and carrier confinement in a single atomic layer - common features of all **the** 2D systems- in combi- nation with large carrier effective masses (see Ref. 21 and references therein); that leads to **the** impossibility to use **the** standard hydrogenic model in 2D limit and makes meaningless **the** usual classification of strong and weak exciton based on **the** binding energy. On **the** contrary, **the** investigation of **the** exciton dispersion as a function of momentum q, which can be experimentally accessed by electron energy loss spectroscopy 22 , is capable to reveal

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top of this, we cannot exclude that **the** object obtained by braid translation of **the** open **boundary** conditions for **the** gl(1|1) is not some sort of version of symplectic fermions with topological defects: this requires further study [2].
To muddy **the** waters some more, a crucial point in our discussion is that we have insisted on braid translating models with open, local **boundary** conditions. In **the** RSOS case, this was all that was needed to understand **the** periodic models. It turns out, however, that **the** relationship between representations of **the** periodic Temperley–Lieb algebra and **the** blob algebra is much more general [24]. One can, for instance, imagine starting **from** **the** representation of **the** affine TL algebra in **the** periodic gl(1|1) spin chain, and try to “invert” **the** braid translation (we provide details for this in App. E) in order to obtain a representation of **the** blob algebra on **the** open spin chain, with much more complicated modules than **the** ones encountered so far. This representation would then be one that, like in **the** RSOS case, becomes relevant to **the** understanding of **the** periodic model. But in doing so one would get a very non-local expression for **the** blob generator, see **the** derivation in App. E.2:

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We first introduce **the** concept of regularized curvature lines that model **the** lines designers draw over curved surfaces, encompassing curvature lines and their extension as geodesics over flat or umbilical regions. We build on this concept to define **the** orthogonal cross field that assigns two regularized curvature lines to each point of a 3D surface. Our algorithm first estimates **the** projection of this cross field in **the** drawing, which is non-orthogonal due to foreshortening. We formulate this estimation as a scattered interpo- lation of **the** strokes drawn in **the** sketch, which makes our method robust to sketchy lines that are typical for design sketches. Our interpolation relies on a novel smoothness energy that we derive **from** our definition of regular- ized curvature lines. Optimizing this energy subject to **the** stroke constraints produces a dense non-orthogonal 2D cross field, which we then lift to 3D by imposing orthogonality. Thus, one central concept of our approach is **the** generalization of existing cross field algorithms to **the** non-orthogonal case. We demonstrate our algorithm on a variety of concept sketches with vari- ous levels of sketchiness. We also compare our approach with existing work that takes clean vector drawings as input.

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Figure 6: Histograms for **the** ε xx , ε yy and ε xy distributions, calculated pixelwise as a function
of time
procedure described in [13]. They were slightly inclined to avoid aliasing in **the** corresponding images, as justified in [14] and illustrated in Figure 7. As in **the** preceding case, **the** distance between camera and specimens was adjusted in such a way that 6 pixels were used to sample one grid and checkerboard period. **The** two specimens were fixed in turn in a grip and a translation was applied to each of them according to **the** procedure described in Section 3.4 above. **The** light intensity was adjusted in such a way that **the** highest dynamic range of **the** camera sensor was used, but without saturating any pixel. Interestingly, **the** optimal lighting conditions are not **the** same for **the** two types of specimens. Indeed **the** bright spots in a 2D grid image are darker than their counterparts in a checkerboard image if exactly **the** same lighting conditions and camera settings (aperture, shutter time) are used in both cases, although **the** size of these spots is exactly **the** same. This is probably a consequence of **the** point spread function (PSF) of **the** lens of **the** camera. Indeed, white boxes are completely surrounded by black lines in 2D grid images, which is not **the** case for **the** checkerboard. **The** aperture of **the** lens was therefore changed and **the** distance between specimen and lighting sources was adjusted **from** one case to each other, in order to have a gray level distribution, which covers **the** widest range without reaching saturation at any pixel.

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Let us also mention **the** following. An acylindrically hyperbolic group Γ may admit various non-elementary acylindrical actions on hyperbolic spaces. A fruitful line of research initiated by Abbott [1] is to find **the** best possible one. Such an action Γ y X is called universal if for any element g of Γ such that there exists an acylindrical action of Γ for which g is loxodromic, **the** action of g on X also is loxodromic. One can also introduce a partial order on cobounded acylindrical actions, see [2]. When existing, a maximal action for this partial order is called a largest acylindrical action. Any largest action is necessarily a universal action and is unique. Abbott, Behrstock and Durham [3] proved that any non-elementary hierarchically hyperbolic group admits a largest acylindrical action. More precisely, they proposed a way to modify **the** hierarchical structure of **the** group so that **the** action of Γ on CS is a largest acylindrical action, where S ∈ S is **the** maximal domain. We will use this modified hierarchical structure in **the** following, see in particular **the** discussion after Proposition 3.9.

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(Figure S2a). This test showed that **the** cold/warm difference was significant for some parameters (for example, EBC, EBC bb ), however, for EBC ff no differences between cold and warm properties
were detected.
Figure 3 shows **the** variations of EBC concentration and Table S1 reports **the** seasonal medians of this parameter. EBC median concentrations are significantly higher in autumn and winter (0.34 µg m −3 (25th percentile: 0.21 µg m −3 ; 75th percentile: 0.54 µg m −3 ) and 0.32 µg m −3 (0.18 µg m −3 ; 0.56 µg m −3 )) compared to spring and summer (0.24 µg m −3 (0.16 µg m −3 ; 0.41 µg m −3 ) and 0.25 µg m −3 (0.17 µg m −3 ; 0.37 µgm −3 )), respectively. This might be a result of higher contributions of sources **from** combustion linked to conventional heating devices often observed during this time of **the** year. This may also be related to different combustion sources emitted all year long (i.e., traffic) but influenced by **the** seasonal variation of **the** **boundary** layer dynamics (represented for each season in Figure S1) trapping surface-emission during **the** cold season in a thinner layer than during summer. This is in agreement with Cesari et al., (2018) [ 33 ] at **the** Environmental-Climate Observatory of Lecce in Italy, who reported **the** highest concentrations of EBC during winter due to a possible influence of combustion sources like biomass burning and **boundary** layer dynamics. **The** hypotheses in this study will be further investigated in Section 2.2 . **The** atmospheric BL heights presented in this study were extracted **from** **the** reanalysis ERA-Interim of **the** European Center for Medium-Range Weather Forecasts (ECMWF) model with a resolution of 0.25 ◦ . During **the** day, **the** development of **the** BL is associated with enhanced vertical mixing, and therefore, a dilution of surface emitted pollutants. At night, on **the** contrary, **the** emitted particles accumulate in a thinner **boundary** layer with reduced vertical mixing leading to higher surface concentrations.

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).
Le thème de la diffusion volontaire d’informations a été largement étudié dans la littérature. La plupart des études cherchent à expliquer les différences de niveaux de publication volontaire dans les rapports annuels via une étude des déterminants (e.g. Firth 1979 ; Cooke 1989 ; Meek et al. 1995 ; Raffournier 1995 ; Depoers 1999). Plus récemment, les recherches se sont étendues à d’autres moyens de communication tels que les conférences téléphoniques (e.g. Tasker 1998 ; Frankel et al. 1999 ; Brown et al. 2004) ou les sites Web des entreprises (e.g. Ettredge et al. 2002 ; Trabelsi et al. 2008 ; Li 2010 ; Ledoux et Cormier 2011). De manière surprenante et malgré l’intérêt des régulateurs, on constate une quasi-absence de littérature sur la diffusion volontaire d’informations lors des OPA/**OPE**. Ceci conduit Sirower et Lipin (2003, p. 26) à parler à ce sujet « d’énorme erreur ». En effet, selon ces auteurs, une stratégie de diff usion d’informations bien conçue pourrait permettre d’obtenir le soutien des actionnaires et ainsi favoriser la réussite des opérations. Les OPA/**OPE** sont des évènements particuliers dans la vie d’une entreprise, pouvant conduire les dirigeants à adopter des comportements de diffusion d’informations différents de leurs comportements habituels (Brennan 1999). Pour preuve, lors d’opérations inamicales, il n’est pas rare de voir les entreprises acquéreuse et cible se livrer à une véritable bataille de communication 3 .

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interpretability of all elements of **the** class C ′ in **the** class C. (So **the** effective interpretation
of G in F (G) is an outgrowth of **the** computable left inverse functor for F , not of F itself.) **The** results in [HKSS02] are proven largely by **the** construction of computable functors, al- though not described in that way. However, one could also ask **the** same questions about cat- egories known not to be complete. For example, there is a natural construction of a Boolean algebra F (L) **from** a linear order L, simply by taking **the** interval algebra of L, where **the** morphisms in each category are simply homomorphisms of **the** structures. On its face, this functor appears to be neither full nor faithful, based on known results, and it does not have a precise computable inverse functor on its image, although it may come close to doing so. It cannot have all of these properties, because there does exist a linear order whose spectrum is not realized by any Boolean algebra, as shown by Jockusch and Soare in [JS91, Theo- rem 1]. (Here we use a generalization of **the** result in this article, namely, that a computable equivalence of categories onto a strictly full subcategory allows one to transfer spectra **from** objects of **the** first category to objects of **the** second. This generalization appears to have a straightforward direct proof, and in any case it follows **from** effective bi-interpretability, hence **from** [HTMMM15, Theorem 12], using [Mon14, Lemma 5.3].) We suspect that similar results distinguishing **the** properties of various everyday classes of countable structures may yield further insights into effectiveness, fullness, faithfulness, and other properties of functors among these classes, especially **the** incomplete ones listed in Section 1.C.

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Roundness is relevant to characterize these asperities. A value of 100% characterizes a perfect disc and will decrease when it moves away **from** this shape.
**The** imagery was carried out by ombroscopy. To accom- plish this, fertilizer particles dispersed on glass were placed in front of a diffuse source of light. **The** selected enlargement generated a digital representation of about 3500 pixels for a particle 2.5 mm in diameter which allows a good estimate of its morphometric properties. **The** binarisation of **the** image is carried out with a fixed threshold since **the** conditions of lighting are stable. **The** minimum number of particles mea- sured by batch is 2000, which makes it possible to sufficiently minimize **the** confidence interval around **the** average of **the** granulometric and morphometric distributions.

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