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Acknowledgements. BG was, for some of the period during which this work was carried out, a fellow of the Radcliffe Institute at Harvard. <|MaskedSetence|> TT is supported by NSF Research Award DMS-0649473, the NSF Waterman award and a grant from the MacArthur Foundation. <|MaskedSetence|> <|MaskedSetence|> This work was largely completed during that week. 3. Basic notation.

**A**: TZ is supported by ISF grant 557/08, an Alon fellowship and a Landau fellowship of the Taub foundation. **B**: He is very grateful to the Radcliffe Institute for providing excellent working conditions. **C**: All three authors are very grateful to the University of Verona for allowing them to use classrooms at Canazei during a week in July 2009.

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been already be verified independently in [H] and [GGV2]. We will verify the case (m,n)=(50,75)𝑚𝑛5075(m,n)=(50,75)( italic_m , italic_n ) = ( 50 , 75 ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We do not provide proofs for.

**A**: Due to this we take an alternative strategy. **B**: Doing this directly using (1.10) amounts to solving a system of 123123123123 equations and 123 variables. **C**: The first part of this procedure is similar to the one used in [GGV1]*Section 8, and is inspired by [M].

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Using the calculations in [Law95], one sees by inspection that the number of Jordan blocks of unipotent elements on the adjoint module is independent of good characteristic; this is reflecting the fact that the centralisers of unipotent elements Gusubscript𝐺𝑢G_{u}italic_G start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT are smooth. <|MaskedSetence|> The phenomenon that centralisers are usually smooth holds much more generally; see [BMRT10] and more recently, [Her13]. It was also noted in [Law95] when the number of Jordan blocks of a unipotent element on Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT was the same as in characteristic zero. (It turned out that this held in good characteristic.) Thus in good characteristic the scheme of fixed points (Vmin)usuperscriptsubscript𝑉min𝑢(V_{\mathrm{min}})^{u}( italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_u end_POSTSUPERSCRIPT is smooth. We discuss the complementary question in our context, which is possibly more natural. Beforehand we must first be a little more precise about Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT. <|MaskedSetence|> The two exceptions are when (G,p)=(F4,3)𝐺𝑝subscript𝐹43(G,p)=(F_{4},3)( italic_G , italic_p ) = ( italic_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , 3 ) or (G2,2)subscript𝐺22(G_{2},2)( italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , 2 ). In this case there are essentially two ways in which one may construct a lattice in (Vmin)ℤsubscriptsubscript𝑉minℤ(V_{\mathrm{min}})_{\mathbb{Z}}( italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT blackboard_Z end_POSTSUBSCRIPT. For one of these, the resulting module after reduction modulo p𝑝pitalic_p, henceforth Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT, has a 1111-dimensional trivial module in its head with an irreducible (dimVmin−1)dimensionsubscript𝑉min1(\dim V_{\mathrm{min}}-1)( roman_dim italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT - 1 )-dimensional socle; this is an induced, co-standard or dual-Weyl module for G𝐺Gitalic_G, and Vmin∗superscriptsubscript𝑉minV_{\mathrm{min}}^{*}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is the corresponding standard or Weyl module for G𝐺Gitalic_G. <|MaskedSetence|>

**A**: Most of the time Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT is irreducible and the theory of high weights identifies Vminsubscript𝑉minV_{\mathrm{min}}italic_V start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT uniquely up to isomorphism (possibly after twisting with a graph automorphism in the case of E6subscript𝐸6E_{6}italic_E start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT). **B**: (Note that for the purposes of computing ranks of powers of matrices, hence Jordan blocks, it matters not whether one works with a module V𝑉Vitalic_V or its dual.). **C**: Another way of stating this is that the orbit G⋅u⋅𝐺𝑢G\cdot uitalic_G ⋅ italic_u of u𝑢uitalic_u is separable, or that Lie⁡(CG⁢(u)⁢(k))=𝔠𝔤⁢(u)Liesubscript𝐶𝐺𝑢𝑘subscript𝔠𝔤𝑢\operatorname{Lie}(C_{G}(u)(k))=\mathfrak{c}_{\mathfrak{g}}(u)roman_Lie ( italic_C start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_u ) ( italic_k ) ) = fraktur_c start_POSTSUBSCRIPT fraktur_g end_POSTSUBSCRIPT ( italic_u ).

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The referee’s contribution is significant in this paper. Some explicitly appear in the text, but others got hidden in the revision process. The contributions of the latter nature include the following. (1) The author originally worked exclusively with motivic cohomology with compact supports. It was the referee who explained how to develop the axiomatic framework as in Section 2, and then apply it to the case of motives. Most importantly, Definition 2.3 and Axiom 2.6 are due to the referee. These simplified the author’s previous arguments, particularly the proofs of Propositions 2.4 and 2.10, and unified cases that had been treated separately. (2) The author originally worked only over an algebraically closed base field. It was the referee who suggested working over a perfect (sometimes even non-perfect) field whenever possible and explained how to do it. This turned out fruitful in the study of algebraic part, and we obtained the current version of Propositions 3.2 and 3.4. The use of the symmetric power construction and [Mil86, Theorem 3.13] in the proof of Proposition 3.4 is also due to the referee. <|MaskedSetence|> <|MaskedSetence|> For this subsection, we would also like to thank Charles Vial for pointing out independently that the method in ibid. is applicable. <|MaskedSetence|> Subsection 3.2 should be attributed to him or her. The idea of unpointed regular homomorphisms enables us to state the relation to Ayoub and Barbieri-Viale’s work (Remark 3.11). It also made the comparison with Serre-Ramachandran’s Albanese varieties simpler and conceptual (Subsection 3.3). Convention. From now on, schemes are assumed separated and of finite type over a base field k,𝑘k,italic_k , and morphisms of schemes are those over the base field except in Subsection 3.5. Group schemes are also assumed to be defined over k.𝑘k.italic_k ..

**A**: The referee also directed the author to the paper [ACMV17a]. **B**: Finally, Subsection 3.5 would have been nonexistent, had it not been for the referee’s explicit question on the topic. **C**: (3) The definition of unpointed regular homomorphisms is due to the referee (Definition 3.7).

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<|MaskedSetence|> <|MaskedSetence|> The first remarkable application is that Buff and Chéritat used it as one of the main tools to prove the existence of Julia sets of quadratic polynomials with positive area [BC12]. Recently, Cheraghi and his collaborators have found several other important applications. In [Che13] and [Che19], Cheraghi developed several elaborate analytic techniques based on Inou-Shishikura’s results. <|MaskedSetence|> For examples, the Feigenbaum Julia sets with positive area (which is different from the examples in [BC12]) have been found in [AL22], the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type has been proved in [CC15], the local connectivity of the Mandelbrot set at some infinitely satellite renormalizable points was proved in [CS15], some statistical properties of the quadratic polynomials was depicted in [AC18], the topological structure and the Hausdorff dimension of high type irrationally indifferent attractors were characterized in [Che22a] and [CDY20] respectively. .

**A**: 1.4. **B**: The tools in [Che13] and [Che19] have led to part of the recent major progresses on the dynamics of quadratic polynomials. **C**: Some observations There are several applications of Inou-Shishikura’s invariant class.

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The paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In Section 4.1 we find a simple formula for a local Kähler potential of the elliptic metric, and in Section 4.2 we show that in the limit when the order of the elliptic element tends to ∞\infty∞ the elliptic metric coincides with the corresponding cusp metric. Finally, in Section 4.3 we give a simple example of a relation between the elliptic metric and special values of Selberg zeta function for Fuchsian groups of signature (0;1;2,2,2)..

**A**: Section 2 contains the necessary background material. **B**: Specifically, we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces. Since the cases of smooth (both compact and punctured) Riemann surfaces have been well understood by us quite a while ago [15, 10], in Section 3.2 we mainly emphasize the computation of the contribution from conical points corresponding to elliptic elements. **C**: In Section 3 we prove the local index theorem for families of ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups.

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<|MaskedSetence|> <|MaskedSetence|> This is not the only manner in which the two are connected. <|MaskedSetence|> they satisfy the same Lie algebra and by virtue of Stone-von Neumann, may be realized in some way as harmonic oscillators. If one takes γ=−δ𝛾𝛿\gamma=-\deltaitalic_γ = - italic_δ, then the coupled SUSY equations become .

**A**: 7. **B**: Indeed, a special class of coupled SUSYs may be realized as harmonic oscillator-like systems, i.e. **C**: Realizing Harmonic Oscillators With Coupled Supersymmetry As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical.

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<|MaskedSetence|> By definable we will mean definable with parameters. We will need a few results from [27]. <|MaskedSetence|> <|MaskedSetence|> The next sub-section is dedicated to a quick overview of (parts) of the language we are using, and to the basic properties of definable sets. 2.1. Ordered abelian groups.

**A**: Since this text is not readily available, we try to keep the present work as self contained as possible, referring to more accessible sources whenever we are aware of such. **B**: Throughout the text G𝐺Gitalic_G will denote a group, usually abelian and often ordered, ℭℭ\mathfrak{C}fraktur_C will denote a sufficiently saturated model of Th⁢(G)Th𝐺\mathrm{Th}(G)roman_Th ( italic_G ). **C**: In particular, for the study of ordered abelian groups we chose the language of [4], rather than the language used by Schmitt.

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One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254]. <|MaskedSetence|> <|MaskedSetence|> Additionally, the spectral techniques remove macro-elements corner singularities that occur in LOD methods based on mixed finite elements. <|MaskedSetence|> Here, we propose eigenvalue problems based on edges of macro element removing the dependence.

**A**: Here in this paper, in the presence of rough coefficients, spectral techniques are employed to overcome such hurdle, and by solving local eigenvalue problems we define a space where the exponential decay of solutions is insensitive to high-contrast coefficients. **B**: We note the proposal in [CHUNG2018298] of generalized multiscale finite element methods based on eigenvalue problems inside the macro elements, with basis functions with support weakly dependent of the log of the contrast. **C**: When LOD or VMS methods are considered, high-contrast coefficients might slow down the exponential decay of the solutions, making the method not so practical.

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The seminal work of Artzner et al. (1999) has bestowed upon the field of risk assessment a set of four pivotal axioms that stand as the cornerstones of coherence for any reputable risk measure. Building upon this foundational framework, Föllmer and Schied (2002), in tandem with the pioneering efforts of Frittelli and Rosazza-Gianin (2002) expanded the purview of risk measures. They introduced the broader class which are convex risk measures by dropping one of the coherency axioms. <|MaskedSetence|> Shushi and Yao (2020) proposed two multivariate risk measures based on conditional expectation and derived explicit formulae for exponential dispersion models. Zuo and Yin (2022) considered the multivariate tail covariance for generalized skew-elliptical distributions. Cai et al. (2022) defined a new multivariate conditional Value-at-Risk risk measure based on the minimization of the expectation of a multivariate loss function. While these advancements have introduced sophisticated risk measures, it’s important to highlight that their theoretical foundation frequently exists within a static framework. <|MaskedSetence|> Dynamic risk measures represent a sophisticated and evolving field within risk management, extending the analysis beyond static frameworks to account for temporal changes in risk. Unlike traditional static risk measures that provide a snapshot assessment, dynamic risk measures recognize the fluid nature of financial markets and aim to capture how risk evolves over time. Introduced by Riedel (2004), dynamic coherent risk measures offer a framework that allows for a more nuanced understanding of risk dynamics. <|MaskedSetence|> Additionally, the introduction of dynamic convex risk measures by Detlefsen and Scandolo (2005) further enriched the field, providing insights into the time consistency properties of risk measures over different time horizons..

**A**: The conventional depiction of these theories operates within a fixed temporal frame, offering a foundational understanding of risk. Over the past two decades, not only has the study of static risk measures flourished, but also dynamic theories of risk measurement have developed into a thriving and mathematically refined area of research. **B**: This advancement enables a comprehensive assessment of risk in the context of changing market conditions and evolving investment portfolios. **C**: Song and Yan (2009) gave an overview of representation theorems for various static risk measures.

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Proof. Decompose σ=τ⁢ϵad⁢h𝜎𝜏superscriptitalic-ϵadℎ\sigma=\tau\epsilon^{{\rm ad}h}italic_σ = italic_τ italic_ϵ start_POSTSUPERSCRIPT roman_ad italic_h end_POSTSUPERSCRIPT as in (3). <|MaskedSetence|> By Lemma 2.1, since m𝑚mitalic_m divides s¯⁢c¯𝑠𝑐\bar{s}cover¯ start_ARG italic_s end_ARG italic_c (by assumption), λ⁢(αi∨)∈ℤ𝜆subscriptsuperscript𝛼𝑖ℤ\lambda(\alpha^{\vee}_{i})\in\mathbb{Z}italic_λ ( italic_α start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∈ blackboard_Z for all the simple coroots αi∨subscriptsuperscript𝛼𝑖\alpha^{\vee}_{i}italic_α start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of 𝔤τsuperscript𝔤𝜏\mathfrak{g}^{\tau}fraktur_g start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT, i.e., λ𝜆\lambdaitalic_λ belongs to the weight lattice of 𝔤τsuperscript𝔤𝜏\mathfrak{g}^{\tau}fraktur_g start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT. So, if Gτsuperscript𝐺𝜏G^{\tau}italic_G start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT is simply-connected, λ𝜆\lambdaitalic_λ gives rise to a character of Hτsuperscript𝐻𝜏H^{\tau}italic_H start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT. <|MaskedSetence|> Recall that for a diagram automorphism τ𝜏\tauitalic_τ, Gτsuperscript𝐺𝜏G^{\tau}italic_G start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT is simply-connected unless (𝔤,r)=(A2⁢n,2)𝔤𝑟subscript𝐴2𝑛2(\mathfrak{g},r)=(A_{2n},2)( fraktur_g , italic_r ) = ( italic_A start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT , 2 ), where r𝑟ritalic_r is the order of τ𝜏\tauitalic_τ. <|MaskedSetence|>

**A**: In this case Gτ=superscript𝐺𝜏absentG^{\tau}=italic_G start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT = SO(2⁢n+1)2𝑛1(2n+1)( 2 italic_n + 1 ) and following the notation of the identity (6), . **B**: To prove that V⁢(λ)𝑉𝜆V(\lambda)italic_V ( italic_λ ) integrates to a Gσsuperscript𝐺𝜎G^{\sigma}italic_G start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT-module, it suffices to show that the torus Hσ=Hτsuperscript𝐻𝜎superscript𝐻𝜏H^{\sigma}=H^{\tau}italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_H start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT acts on V⁢(λ)𝑉𝜆V(\lambda)italic_V ( italic_λ ), where H𝐻Hitalic_H is the maximal torus of G𝐺Gitalic_G with Lie algebra 𝔥𝔥\mathfrak{h}fraktur_h (𝔥𝔥\mathfrak{h}fraktur_h being a σ𝜎\sigmaitalic_σ-stable Cartan subalgebra). **C**: Thus, in this case Hτsuperscript𝐻𝜏H^{\tau}italic_H start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT acts on V⁢(λ)𝑉𝜆V(\lambda)italic_V ( italic_λ ).

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Both HaldaneHaldane (1991) and WuWu (1994) considered ideal gases with a fixed boundary condition, which is in fact the periodic boundary condition, which renders the anyon systems being on spaces without actual boundaries. Neither did Ref. Hu et al. (2014) consider boundaries. Nevertheless, materials with boundaries are much easier to fabricate than closed ones. <|MaskedSetence|> For such a system to have a well-defined, topologically protected, degenerate ground-state Hilbert space, which may support a robust quantum memory and quantum computingKitaev (2003, 2006), the systems with gapped boundaries are of most interest. <|MaskedSetence|> The fusion space structure of multiple anyons is closely related to the boundary conditions, which select only certain anyons that can move to the gapped boundary without any energy cost333In other words, these anyons condense at the boundaryKitaev and Kong (2012); Levin (2013); Hung and Wan (2013, 2015).. <|MaskedSetence|>

**A**: A recent workHung and Wan (2015) has shown how gapped boundary conditions of a topological order dictate the ground state degeneracy of the topological order and how certain anyons in the bulk may connect to the gapped boundary. **B**: Hence we expect that the boundary conditions of a topological order would affect the state counting of the anyons. . **C**: Understanding the anyonic exclusion statistics in topologically ordered states with boundaries is thus important.

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<|MaskedSetence|> <|MaskedSetence|> This would have the intended effect. However, such a condition cannot be expressed without circularity. Fortunately, the intended content can indeed be expressed (cf. the previous footnote); our device for doing so is to phrase this content by reference to a well-defined suborder ℚα+1δsuperscriptsubscriptℚ𝛼1𝛿{\mathbb{Q}}_{\alpha+1}^{\delta}blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT of ℚα+1subscriptℚ𝛼1{\mathbb{Q}}_{\alpha+1}blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT—namely the set of conditions q∈ℚα+1𝑞subscriptℚ𝛼1q\in{\mathbb{Q}}_{\alpha+1}italic_q ∈ blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT all of whose edges of form ⟨(N0,α+1),(N1,γ1)⟩subscript𝑁0𝛼1subscript𝑁1subscript𝛾1\langle(N_{0},\alpha+1),(N_{1},\gamma_{1})\rangle⟨ ( italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_α + 1 ) , ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⟩ are such that δN0<δsubscript𝛿subscript𝑁0𝛿\delta_{N_{0}}<\deltaitalic_δ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT < italic_δ (but allowing all nodes in fq⁢(α)subscript𝑓𝑞𝛼f_{q}(\alpha)italic_f start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_α ) to be of any height below κ𝜅\kappaitalic_κ). Hence, due to the presence of this clause (7), the definition of q𝑞qitalic_q being a ℚα+1subscriptℚ𝛼1{\mathbb{Q}}_{\alpha+1}blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT-condition is ultimately to be seen as being given by recursion on the supremum of the collection of heights of models occurring in edges of the form ⟨(N0,α+1),(N1,γ1)⟩subscript𝑁0𝛼1subscript𝑁1subscript𝛾1\langle(N_{0},\alpha+1),(N_{1},\gamma_{1})\rangle⟨ ( italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_α + 1 ) , ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⟩ (i.e., those edges not coming from the restriction of q𝑞qitalic_q to α𝛼\alphaitalic_α).191919Let us reconsider for a second the situation described a few lines earlier. Suppose q∈ℚα+1𝑞subscriptℚ𝛼1q\in{\mathbb{Q}}_{\alpha+1}italic_q ∈ blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT, ⟨(N0,γ0),(N1,γ1)⟩∈τqsubscript𝑁0subscript𝛾0subscript𝑁1subscript𝛾1subscript𝜏𝑞\langle(N_{0},\gamma_{0}),(N_{1},\gamma_{1})\rangle\in\tau_{q}⟨ ( italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⟩ ∈ italic_τ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, and α𝛼\alphaitalic_α and α¯¯𝛼\bar{\alpha}over¯ start_ARG italic_α end_ARG are as in that description. Suppose x∈fq⁢(α¯)𝑥subscript𝑓𝑞¯𝛼x\in f_{q}(\bar{\alpha})italic_x ∈ italic_f start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( over¯ start_ARG italic_α end_ARG ) is of height less than δN1subscript𝛿subscript𝑁1\delta_{N_{1}}italic_δ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and y𝑦yitalic_y is a node of height at least δN0subscript𝛿subscript𝑁0\delta_{N_{0}}italic_δ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that, say, q↾μ⁢(α)↾𝑞𝜇𝛼q\restriction\mu(\alpha)italic_q ↾ italic_μ ( italic_α ) happens to force y𝑦yitalic_y to be above x𝑥xitalic_x in T∼μ⁢(α)subscriptsimilar-to𝑇𝜇𝛼\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{T}}}_{\mu(\alpha)}under∼ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_μ ( italic_α ) end_POSTSUBSCRIPT. It is then of course impossible to extend q𝑞qitalic_q to a condition q′superscript𝑞′q^{\prime}italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that x∈fq′⁢(α)𝑥subscript𝑓superscript𝑞′𝛼x\in f_{q^{\prime}}(\alpha)italic_x ∈ italic_f start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_α ). However, we can certainly pick α′superscript𝛼′\alpha^{\prime}italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that μ⁢(α′)=μ⁢(α)𝜇superscript𝛼′𝜇𝛼\mu(\alpha^{\prime})=\mu(\alpha)italic_μ ( italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_μ ( italic_α ) and such that q𝑞qitalic_q can be extended (trivially) by making fq′⁢(α′)={x}subscript𝑓superscript𝑞′superscript𝛼′𝑥f_{q^{\prime}}(\alpha^{\prime})=\{x\}italic_f start_POSTSUBSCRIPT italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_x }. <|MaskedSetence|>

**A**: Our way around this difficulty is to incorporate, in our definition, a clause which stipulates that the above operation can be carried out. **B**: This will ensure that the generic specializing function for T∼μ⁢(α)subscriptsimilar-to𝑇𝜇𝛼\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{T}}}_{\mu(\alpha)}under∼ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_μ ( italic_α ) end_POSTSUBSCRIPT will be defined everywhere (cf. the proof of Lemma 5.3). . **C**: This is in essence what clause (7) says.181818A more naive (and simpler-looking) approach would be to require that if ⟨(N0,γ0),(N1,γ1)⟩subscript𝑁0subscript𝛾0subscript𝑁1subscript𝛾1\langle(N_{0},\gamma_{0}),(N_{1},\gamma_{1})\rangle⟨ ( italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⟩ is an edge from τqsubscript𝜏𝑞\tau_{q}italic_τ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, then ℚα+1∩N0subscriptℚ𝛼1subscript𝑁0{\mathbb{Q}}_{\alpha+1}\cap N_{0}blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT ∩ italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a complete suborder of ℚα+1subscriptℚ𝛼1{\mathbb{Q}}_{\alpha+1}blackboard_Q start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT.

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Selection 2

<|MaskedSetence|> Acknowledgements. We thank the referees for the careful reading of the first version of the text and for many helpful remarks. <|MaskedSetence|> The results of Section 4 has been obtained under support of the RSF grant 19-11-00056. The work of both authors has also been supported in part by the Simons Foundation. <|MaskedSetence|>

**A**: This research was carried out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’. **B**: The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. . **C**: 1.6.

CAB

CAB

CAB

BCA

Selection 3

<|MaskedSetence|> If cn=cn+1subscript𝑐𝑛subscript𝑐𝑛1c_{n}=c_{n+1}italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT then clearly An(ck)<An+1(ck)subscriptsuperscript𝐴subscript𝑐𝑘𝑛subscriptsuperscript𝐴subscript𝑐𝑘𝑛1A^{(c_{k})}_{n}<A^{(c_{k})}_{n+1}italic_A start_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT < italic_A start_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT. If cn≠cn+1,cn+1=k!formulae-sequencesubscript𝑐𝑛subscript𝑐𝑛1subscript𝑐𝑛1𝑘c_{n}\not=c_{n+1},\ c_{n+1}=k!italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≠ italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_k ! then An+1(ck)>k2⋅(k−1)!k!=ksubscriptsuperscript𝐴subscript𝑐𝑘𝑛1⋅superscript𝑘2𝑘1𝑘𝑘A^{(c_{k})}_{n+1}>k^{2}\cdot\frac{(k-1)!}{k!}=kitalic_A start_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT > italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ divide start_ARG ( italic_k - 1 ) ! end_ARG start_ARG italic_k ! end_ARG = italic_k. <|MaskedSetence|> Let cn≠cn+1,cn+1=k!formulae-sequencesubscript𝑐𝑛subscript𝑐𝑛1subscript𝑐𝑛1𝑘c_{n}\not=c_{n+1},\ c_{n+1}=k!italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≠ italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT = italic_k !. <|MaskedSetence|>

**A**: We show that (cn)subscript𝑐𝑛(c_{n})( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is balanced. **B**: These two statements gives the claim. We show that (cn2)subscriptsuperscript𝑐2𝑛(c^{2}_{n})( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is not balanced. **C**: Then .

ACB

ABC

ABC

ABC

Selection 2