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1}{4}x^{-m-2}R^{\prime\prime};divide start_ARG italic_m ( italic_m + 2 ) end_ARG start_ARG 4 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 4 end_POSTSUPERSCRIPT italic_R - divide start_ARG 1 + 2 italic_m end_ARG start_ARG 4 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 3 end_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 2 end_POSTSUPERSCRIPT italic_R start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ;

(i) fast calculation of f′′/f′superscript𝑓′′superscript𝑓′f^{\prime\prime}/f^{\prime}italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT from f/f′𝑓superscript𝑓′f/f^{\prime}italic_f / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT,

F′′superscript𝐹′′\displaystyle F^{\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT

Rnm′′′⁢(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′′′𝑥\displaystyle{R_{n}^{m}}^{\prime\prime\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ( italic_x )

F′′′superscript𝐹′′′\displaystyle F^{\prime\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT

D

\eta_{2},\eta_{3}).italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( over~ start_ARG italic_η end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , over~ start_ARG italic_η end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_η start_POSTSUBSCRIPT 123 end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

case when n=2𝑛2n=2italic_n = 2: Let X𝑋Xitalic_X be a 2222-groupoid object in 𝒞𝒞\mathcal{C}caligraphic_C and let Z1⇒Z0⇒subscript𝑍1subscript𝑍0Z_{1}\Rightarrow Z_{0}italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⇒ italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be in 𝒞𝒞\mathcal{C}caligraphic_C with structure maps as in (1) up to

object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ), as we have demonstrated in the

In all our cases, for a set-theoretical map to be a morphism in 𝒞𝒞\mathcal{C}caligraphic_C, we only have to

Z→X→𝑍𝑋Z\to Xitalic_Z → italic_X is an epimorphism in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) and Y→X→𝑌𝑋Y\to Xitalic_Y → italic_X is a morphism in 𝒞𝒞\mathcal{C}caligraphic_C, then the

C

𝒫⁢(A)≡A→𝒫⁢(1)𝒫𝐴𝐴→𝒫1{\cal P}(A)\,\equiv\,A\rightarrow{\cal P}(1)caligraphic_P ( italic_A ) ≡ italic_A → caligraphic_P ( 1 )

we can prove that for any subset W∈𝒫⁢(A)𝑊𝒫𝐴W\in{\cal P}(A)italic_W ∈ caligraphic_P ( italic_A ) we can derive

Therefore, a subset of A𝐴Aitalic_A, being an element of 𝒫⁢(A)𝒫𝐴{\cal P}(A)caligraphic_P ( italic_A ),

the power collection 𝒫⁢(A)𝒫𝐴{\cal P}(A)caligraphic_P ( italic_A ) of a set A𝐴Aitalic_A is interpreted as the corresponding

𝒫⁢(A)≡A→𝒫⁢(1)𝒫𝐴𝐴→𝒫1{\cal P}(A)\,\equiv\,A\rightarrow{\cal P}(1)caligraphic_P ( italic_A ) ≡ italic_A → caligraphic_P ( 1 )

B

By construction, G/Γ𝐺ΓG/\Gammaitalic_G / roman_Γ is isomorphic to G𝐧0/Γ𝐧0subscript𝐺subscript𝐧0subscriptΓsubscript𝐧0G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for some 𝐧0subscript𝐧0{\bf n}_{0}bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, so we may assume without loss of generality that G/Γ=G𝐧0/Γ𝐧0𝐺Γsubscript𝐺subscript𝐧0subscriptΓsubscript𝐧0G/\Gamma=G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}italic_G / roman_Γ = italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT; since all smooth Riemannian metrics on a compact manifold are equivalent, we can also assume that dG/Γ=dG𝐧0/Γ𝐧0subscript𝑑𝐺Γsubscript𝑑subscript𝐺subscript𝐧0subscriptΓsubscript𝐧0d_{G/\Gamma}=d_{G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}}italic_d start_POSTSUBSCRIPT italic_G / roman_Γ end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT. We may also normalise F𝐹Fitalic_F to be bounded in magnitude by 1111. But this contradicts (5.1) for 𝐧𝐧{\bf n}bold_n sufficiently large, and the claim follows.

The purpose of this paper is to establish the general case of a conjecture named the Inverse Conjecture for the Gowers norms by the first two authors in [23, Conjecture 8.3]. If N𝑁Nitalic_N is a (typically large) positive integer then we write [N]:={1,…,N}assigndelimited-[]𝑁1…𝑁[N]:=\{1,\dots,N\}[ italic_N ] := { 1 , … , italic_N }. For each integer s⩾1𝑠1s\geqslant 1italic_s ⩾ 1 the inverse conjecture GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ), whose statement we recall shortly, describes the structure of 1111-bounded functions f:[N]→ℂ:𝑓→delimited-[]𝑁ℂf:[N]\rightarrow\mathbb{C}italic_f : [ italic_N ] → blackboard_C whose (s+1)stsuperscript𝑠1st(s+1)^{\operatorname{st}}( italic_s + 1 ) start_POSTSUPERSCRIPT roman_st end_POSTSUPERSCRIPT Gowers norm ‖f‖Us+1⁢[N]subscriptnorm𝑓superscript𝑈𝑠1delimited-[]𝑁\|f\|_{U^{s+1}[N]}∥ italic_f ∥ start_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_s + 1 end_POSTSUPERSCRIPT [ italic_N ] end_POSTSUBSCRIPT is large. These conjectures together with a good deal of motivation and background to them are discussed in [19, 21, 23]. The conjectures GI⁡(1)GI1{\operatorname{GI}}(1)roman_GI ( 1 ) and GI⁡(2)GI2{\operatorname{GI}}(2)roman_GI ( 2 ) have been known for some time, the former being a straightforward application of Fourier analysis, and the latter being the main result of [21] (see also [51] for the characteristic 2222 analogue). The case GI⁡(3)GI3{\operatorname{GI}}(3)roman_GI ( 3 ) was also recently established by the authors in [28]. The aim of the present paper is to establish the remaining cases GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ) for s⩾3𝑠3s\geqslant 3italic_s ⩾ 3, in particular reestablishing the results in [28].

Thus, to establish Theorem 1.3, it will suffice to establish Conjecture 5.3 for s⩾3𝑠3s\geqslant 3italic_s ⩾ 3. This is the objective of the remainder of the paper.

In our previous paper [28] it was already rather painful to keep proper track of such notions as “many” and “correlates with”. Here matters are even worse, and so to organise the above tasks it turns out to be quite convenient to first take an ultralimit of all objects being studied, effectively placing one in the setting of nonstandard analysis. This allows one to easily import results from infinitary mathematics, notably the theory of Lie groups and basic linear algebra, into the finitary setting of functions on [N]delimited-[]𝑁[N][ italic_N ]. In §5 and Appendix A we review the basic machinery of ultralimits that we will need here; we will not be exploiting any particularly advanced aspects of this framework. The reader does not really need to understand the ultrafilter language in order to comprehend the basic structure of the paper, provided that he/she is happy to deal with concepts like “dense” and “correlates with” in a somewhat informal way, resembling the way in which analysts actually talk about ideas with one another (and, in fact, analogous to the way we wrote this paper). It is possible to go through the paper and properly quantify all of these notions using appropriate parameters δ𝛿\deltaitalic_δ and (many) growth functions ℱℱ\mathcal{F}caligraphic_F. This would have the advantage of making the paper on some level comprehensible to the reader with an absolute distrust of ultrafilters, and it would also remove the dependence on the axiom of choice and in principle provide explicit but very poor bounds. However it would cause the argument to be significantly longer, and the notation would be much bulkier.

The inverse conjecture GI⁡(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ), Conjecture 1.2, has been formulated using linear nilsequences F⁢(gn⁢x⁢Γ)𝐹superscript𝑔𝑛𝑥ΓF(g^{n}x\Gamma)italic_F ( italic_g start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x roman_Γ ). This is largely for compatibility with the earlier paper [23] of the first two authors on linear equations in primes, where this form of the conjecture was stated in precisely this form as Conjecture 8.3. Subsequently, however, it was discovered that it is more natural to deal with a somewhat more general class of object called a polynomial nilsequence F⁢(g⁢(n)⁢Γ)𝐹𝑔𝑛ΓF(g(n)\Gamma)italic_F ( italic_g ( italic_n ) roman_Γ ). This is particularly so when it comes to discussing the distributional properties of nilsequences, as was done in [24]. Thus, we shall now recast the inverse conjecture in terms of polynomial nilsequences, which is the formulation we will work with throughout the rest of the paper.

B

HΦ,𝒪⁢(N)subscript𝐻Φ𝒪𝑁H_{\Phi,\mathcal{O}}(N)italic_H start_POSTSUBSCRIPT roman_Φ , caligraphic_O end_POSTSUBSCRIPT ( italic_N ) acts trivially on ℋ⁢(N)ℋ𝑁\mathcal{H}(N)caligraphic_H ( italic_N ).

So let 𝔞∈I⁢(N⁢F)𝔞𝐼𝑁𝐹\mathfrak{a}\in I(NF)fraktur_a ∈ italic_I ( italic_N italic_F ) be an integral ideal

𝔞∈I⁢(N⁢F)𝔞𝐼𝑁𝐹\mathfrak{a}\in I(NF)fraktur_a ∈ italic_I ( italic_N italic_F ), if [𝔞]delimited-[]𝔞[\mathfrak{a}][ fraktur_a ]

It remains to prove that if 𝔞∈I⁢(N⁢F)𝔞𝐼𝑁𝐹\mathfrak{a}\in I(NF)fraktur_a ∈ italic_I ( italic_N italic_F ) acts trivially

acts trivially on all f∈ℱN𝑓subscriptℱ𝑁f\in\mathcal{F}_{N}italic_f ∈ caligraphic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT.

C

\text{ }X^{x,\varepsilon}\left(0\right)&=&x\in\mathbb{R}^{d},\end{array}\right.{ start_ARRAY start_ROW start_CELL d italic_X start_POSTSUPERSCRIPT italic_x , italic_ε end_POSTSUPERSCRIPT ( italic_t ) end_CELL start_CELL = end_CELL start_CELL italic_b ( italic_X start_POSTSUPERSCRIPT italic_x , italic_ε end_POSTSUPERSCRIPT ( italic_t ) ) d italic_t + italic_ε d italic_W ( italic_t ) , italic_t ≥ 0 , end_CELL end_ROW start_ROW start_CELL italic_X start_POSTSUPERSCRIPT italic_x , italic_ε end_POSTSUPERSCRIPT ( 0 ) end_CELL start_CELL = end_CELL start_CELL italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , end_CELL end_ROW end_ARRAY

The whole point is then to identify this shortest exit time with the limit of the exit times of the solutions of the SDEs as the viscosity tends to zero. However, this is rather difficult as known examples show that the points of the boundary of K𝐾Kitalic_K that can be reached in the fastest way are not the only ones to be reached by the limiting solutions of the vanishing SDEs. One effective trick is then to normalize K𝐾Kitalic_K in such a way that all the points of the boundary are reached in the same time. In this framework, it is indeed possible to give a relevant notion of fastest solutions. Nevertheless, how to normalize K𝐾Kitalic_K is also a intractable problem unless some concrete examples..

solution to the SDE (2). Then, as ε→0+,→𝜀superscript0\varepsilon\rightarrow 0^{+},italic_ε → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ,

A natural question concerns the behavior of the limit of perturbed SDEs (2) with respect to the ODE (1), as ε→0+→𝜀superscript0\varepsilon\rightarrow 0^{+}italic_ε → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT. In the classical Lipschitz case we refer to Friedlin and Wentzell

zero is excluded immediately. Indeed, our objective is to prove that the limiting solutions of the SDEs are supported by the set of solutions to the corresponding ODE that are the fastest ones to escape from the singularity. Intuitively, this is well understood: The system lives for some time in the neighborhood of the singularity. Under the action of the noise, it visits for some time the entire local vicinity. Consequently, the fastest solutions are then the solutions that are the

C

When n=1𝑛1n=1italic_n = 1, given the fact that there are only two 1111-manifolds up to homeomorphism, and that they are both automata 1111-manifolds, it is tempting to dismiss this phenomenon as an artefact caused by the use of Turing machines. In higher dimensions, however, this should probably be taken seriously. Notions of computable manifolds have been proposed; we refer to work of Calvert-Miller [8] and Aguilar-Conde [1] and the references therein for further discussion.

Determining whether two given admissible trees are (topologically) equivalent is the problem we want to solve. The problem of determining whether two given admissible trees are isomorphic is combinatorial in nature and easy to solve algorithmically simply by enumerating all maps from the set of vertices of T1subscript𝑇1T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to that of T2subscript𝑇2T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT until a suitable bijection is found (or not.)

In this paper we are concerned with n=2𝑛2n=2italic_n = 2, i.e. surfaces. For us, the interesting case is that of surfaces of infinite type. These have been studied at least since the 1970s since they appear as leaves of foliations on compact 3-manifolds (see e.g. [25, 9, 11].) They are also studied from the point of view of Teichmüller theory (see e.g. [14, 19] and the references therein) and translation surfaces [26].

It would be interesting to define intermediate classes between automata surfaces and computable surfaces and determine whether the homeomorphism problem—or other algorthmic problems—are decidable for them.

We now give some examples and non-examples of automata surfaces. All of them are orientable, so E′′=∅superscript𝐸′′E^{\prime\prime}=\emptysetitalic_E start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = ∅. According to [2], five infinite-type surfaces are sufficiently important to have received names in the literature. They are all automata surfaces:

C

Let G𝐺Gitalic_G be a Lie group, and M𝑀Mitalic_M be a Riemannian manifold of class C∗ksubscriptsuperscript𝐶𝑘C^{k}_{*}italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with k≥1𝑘1k\geq 1italic_k ≥ 1.

the pseudo-group action of G𝐺Gitalic_G on B𝐵Bitalic_B is of class C∗3subscriptsuperscript𝐶3C^{3}_{*}italic_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT.

where it is stated that the G𝐺Gitalic_G-action on M𝑀Mitalic_M is of class Cksuperscript𝐶𝑘C^{k}italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT.

Since the action of G𝐺Gitalic_G on (B′,h~0)superscript𝐵′subscript~ℎ0(B^{\prime},\widetilde{h}_{0})( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is isometric, the action of G^^𝐺\widehat{G}over^ start_ARG italic_G end_ARG on F⁢(B′,h~0)𝐹superscript𝐵′subscript~ℎ0F(B^{\prime},\widetilde{h}_{0})italic_F ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )

Suppose that the action of G𝐺Gitalic_G on M𝑀Mitalic_M is isometric. Then the G𝐺Gitalic_G-action on M𝑀Mitalic_M,

D

In 2011, Hong-Huang-Wang [7] studied a class of degenerate elliptic Monge-Ampère equation in a smooth, bounded and strictly convex domain

When they proved the existence of global smooth solutions to the homogeneous Dirichlet problem, they introduced the key auxiliary function ℋℋ\mathcal{H}caligraphic_H,

respect to the normal direction −D⁢u𝐷𝑢-Du- italic_D italic_u, we have the following formula on the m𝑚mitalic_m-th curvature of the level sets of the solution u𝑢uitalic_u,

which is the product of curvature κ𝜅\kappaitalic_κ of the level line of u𝑢uitalic_u and the cubic of |D⁢u|𝐷𝑢|Du|| italic_D italic_u |, and got the uniformly lower bound of ℋℋ\mathcal{H}caligraphic_H

Let uΩj,j=0,1,formulae-sequencesubscript𝑢subscriptΩ𝑗𝑗01u_{\Omega_{j}},j=0,1,italic_u start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_j = 0 , 1 , be the solution to the problem

A

Instruction type (i) above simply copies an element already in memory to a different memory slot. These instructions can arguably be disregarded for the purpose of determining the length of an MSLP, because in a practical implementation they could be handled via relabelling.

of our approach by constructing MSLPs for the Bruhat decomposition of d×d𝑑𝑑d\times ditalic_d × italic_d

To demonstrate how MSLPs function in practice, we discuss MSLPs for some fundamental group operations that arise in our more involved examples in the subsequent sections.

For the purposes of determining the cost of Taylor’s algorithm in terms of matrix operations, namely determining the length of an MSLP for the algorithm, we assume that the field elements −gi⁢c⁢gr⁢c−1subscript𝑔𝑖𝑐superscriptsubscript𝑔𝑟𝑐1-g_{ic}g_{rc}^{-1}- italic_g start_POSTSUBSCRIPT italic_i italic_c end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_r italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT in (11) (and similarly in (12)) are given to us as polynomials of degree at most f−1𝑓1f-1italic_f - 1 in the primitive element ω𝜔\omegaitalic_ω, where q=pf𝑞superscript𝑝𝑓q=p^{f}italic_q = italic_p start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT for some prime p𝑝pitalic_p.

Therefore, for simplicity we ignore instructions of the form (i) when determining the lengths of our MSLPs in Sections 3 and 4.

D

C=x+C−1⁢x−1+C−2⁢x−2+⋯with each C−i∈K⁢[y].𝐶𝑥subscript𝐶1superscript𝑥1subscript𝐶2superscript𝑥2⋯with each C−i∈K⁢[y].C=x+C_{-1}x^{-1}+C_{-2}x^{-2}+\cdots\qquad\text{with each $C_{-i}\in K[y]$.}italic_C = italic_x + italic_C start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT + italic_C start_POSTSUBSCRIPT - 2 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT + ⋯ with each italic_C start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT ∈ italic_K [ italic_y ] .

ℤ/e⁢ℤℤ𝑒ℤ\mathds{Z}/e\mathds{Z}blackboard_Z / italic_e blackboard_Z acts on 𝒮0subscript𝒮0\mathcal{S}_{0}caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. In fact, if

and, again by Proposition 1.7, there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λj∈K×subscript𝜆𝑗superscript𝐾\lambda_{j}\in K^{\times}italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT

then there exists j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λ∈K×𝜆superscript𝐾\lambda\in K^{\times}italic_λ ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT such that

Since m+n>2𝑚𝑛2m+n>2italic_m + italic_n > 2, by Proposition 1.7 there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and

D

Evidently, every PD action is WD at each point x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X and the restriction of p𝑝pitalic_p to every G𝐺Gitalic_G-wandering neighborhood is a homeomorphism onto.

Suppose X𝑋Xitalic_X is path connected and the action of G𝐺Gitalic_G on X𝑋Xitalic_X is WD at some x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X.

Suppose that the action of G𝐺Gitalic_G on X𝑋Xitalic_X is WD at x𝑥xitalic_x, and the action of H𝐻Hitalic_H on Y𝑌Yitalic_Y is WD at y𝑦yitalic_y, so G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H is WD at w𝑤witalic_w, see (4.4) and Lemma 4.1.1.

If the actions of G𝐺Gitalic_G and H𝐻Hitalic_H are PD, then the action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H on W𝑊Witalic_W is PD as well.

Moreover, it is well known and is easy to see that if X𝑋Xitalic_X is Hausdorff and G𝐺Gitalic_G is a finite group freely acting on X𝑋Xitalic_X, then this action is also PD, e.g. [4, 11.1.3].

D

164,134,42,32,2,1superscript164superscript134superscript42superscript322116^{4},13^{4},4^{2},3^{2},2,116 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , 13 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , 4 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 2 , 1

23,19,17,152,112,9,7,32231917superscript152superscript11297superscript3223,19,17,15^{2},11^{2},9,7,3^{2}23 , 19 , 17 , 15 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 11 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 9 , 7 , 3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

19,173,9719superscript173superscript9719,17^{3},9^{7}19 , 17 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , 9 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT

27,195,173,98,1327superscript195superscript173superscript98superscript1327,19^{5},17^{3},9^{8},1^{3}27 , 19 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT , 17 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , 9 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT

27,19,173,942719superscript173superscript9427,19,17^{3},9^{4}27 , 19 , 17 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT , 9 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT

B

We continue with the assumption that H0⁢(ΩL¯1)=0=H1⁢(𝒪L¯)superscript𝐻0superscriptsubscriptΩ¯𝐿10superscript𝐻1subscript𝒪¯𝐿H^{0}(\Omega_{\overline{L}}^{1})=0=H^{1}(\mathcal{O}_{\overline{L}})italic_H start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUBSCRIPT over¯ start_ARG italic_L end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) = 0 = italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( caligraphic_O start_POSTSUBSCRIPT over¯ start_ARG italic_L end_ARG end_POSTSUBSCRIPT ).

λ𝜆\lambdaitalic_λ-connection over R⁢[λ]/λn𝑅delimited-[]𝜆superscript𝜆𝑛R[\lambda]/\lambda^{n}italic_R [ italic_λ ] / italic_λ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Then there is

and then a family of λ𝜆\lambdaitalic_λ-connections over over R⁢[[λ]]𝑅delimited-[]delimited-[]𝜆R[[\lambda]]italic_R [ [ italic_λ ] ];

λ𝜆\lambdaitalic_λ-connection over R⁢[[λ]]𝑅delimited-[]delimited-[]𝜆R[[\lambda]]italic_R [ [ italic_λ ] ]

over R⁢[[λ]]𝑅delimited-[]delimited-[]𝜆R[[\lambda]]italic_R [ [ italic_λ ] ]. The algebra 𝒟λ^:=lim𝒟λ/λnassignsubscript𝒟^𝜆subscript𝒟𝜆superscript𝜆𝑛\mathcal{D}_{\widehat{\lambda}}:=\lim\mathcal{D}_{\lambda}/\lambda^{n}caligraphic_D start_POSTSUBSCRIPT over^ start_ARG italic_λ end_ARG end_POSTSUBSCRIPT := roman_lim caligraphic_D start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT / italic_λ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT

C

HomDMN⁢i⁢se⁢f⁢f⁢(k)⁡(M⁢(T),A)subscriptHomsuperscriptsubscriptDM𝑁𝑖𝑠𝑒𝑓𝑓𝑘𝑀𝑇𝐴\displaystyle\operatorname{Hom}_{{\mathrm{DM}}_{Nis}^{eff}(k)}(M(T),A)roman_Hom start_POSTSUBSCRIPT roman_DM start_POSTSUBSCRIPT italic_N italic_i italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e italic_f italic_f end_POSTSUPERSCRIPT ( italic_k ) end_POSTSUBSCRIPT ( italic_M ( italic_T ) , italic_A )

HomShN⁢i⁢s⁢(Sm/k;Set)⁡(T~,A),subscriptHomsubscriptSh𝑁𝑖𝑠Sm𝑘Set~𝑇𝐴\displaystyle\operatorname{Hom}_{{\mathrm{Sh}}_{Nis}(\operatorname{Sm}/k;%

L:D⁢(ShN⁢i⁢st⁢r⁢(k))⟶DMN⁢i⁢se⁢f⁢f⁢(k).:𝐿⟶𝐷superscriptsubscriptSh𝑁𝑖𝑠𝑡𝑟𝑘superscriptsubscriptDM𝑁𝑖𝑠𝑒𝑓𝑓𝑘L\colon D({\mathrm{Sh}}_{Nis}^{tr}(k))\longrightarrow{\mathrm{DM}}_{Nis}^{eff}%

HomD⁢(ShN⁢i⁢st⁢r⁢(Sm/k))⁡(ℤt⁢r⁢(T),A)subscriptHom𝐷superscriptsubscriptSh𝑁𝑖𝑠𝑡𝑟Sm𝑘subscriptℤ𝑡𝑟𝑇𝐴\displaystyle\operatorname{Hom}_{D({\mathrm{Sh}}_{Nis}^{tr}(\operatorname{Sm}/%

Γ⁢(M⁢(X))=HomDMN⁢i⁢se⁢f⁢f⁢(k)⁡(ℤ,M⁢(X))≅HomD⁢(ShN⁢i⁢st⁢r⁢(k))⁡(ℤ,C*⁢ℤt⁢r⁢(X))≅h0N⁢i⁢s⁢(X)⁢(k).Γ𝑀𝑋subscriptHomsuperscriptsubscriptDM𝑁𝑖𝑠𝑒𝑓𝑓𝑘ℤ𝑀𝑋subscriptHom𝐷superscriptsubscriptSh𝑁𝑖𝑠𝑡𝑟𝑘ℤsubscript𝐶subscriptℤ𝑡𝑟𝑋superscriptsubscriptℎ0𝑁𝑖𝑠𝑋𝑘\Gamma(M(X))=\operatorname{Hom}_{{\mathrm{DM}}_{Nis}^{eff}(k)}({\mathbb{Z}},M(%

C

\ast})roman_δ start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT : caligraphic_A → roman_P ( caligraphic_A start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) is a

This is the reason instead to denote simply by 𝟣1\mathsf{1}sansserif_1 the unit of 𝒜𝒜\mathcal{A}caligraphic_A.

𝟣𝔇subscript1𝔇\mathsf{1}_{\mathfrak{D}}sansserif_1 start_POSTSUBSCRIPT fraktur_D end_POSTSUBSCRIPT defined in Def. 0.4.1 is the unit morphism

of the object 𝒜𝒜\mathcal{A}caligraphic_A, i.e. the identity map from 𝒜𝒜\mathcal{A}caligraphic_A to itself.

In the following definition 𝟣1\mathsf{1}sansserif_1 is the unit of the unital algebra 𝒜𝒜\mathcal{A}caligraphic_A,

D

By (6.16) and (6.18), we have xm+k>2⁢π⁢ym+k+C0subscript𝑥𝑚𝑘2𝜋subscript𝑦𝑚𝑘subscript𝐶0x_{m+k}>2\pi y_{m+k}+C_{0}italic_x start_POSTSUBSCRIPT italic_m + italic_k end_POSTSUBSCRIPT > 2 italic_π italic_y start_POSTSUBSCRIPT italic_m + italic_k end_POSTSUBSCRIPT + italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Let kn=kfn⩾1subscript𝑘𝑛subscript𝑘subscript𝑓𝑛1k_{n}=k_{f_{n}}\geqslant 1italic_k start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⩾ 1 be the integer introduced in Proposition 2.3, D3>0subscript𝐷30D_{3}>0italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT > 0 be a constant introduced in Lemma 2.11 and 𝒟n=𝒟fnsubscript𝒟𝑛subscript𝒟subscript𝑓𝑛\mathcal{D}_{n}=\mathcal{D}_{f_{n}}caligraphic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = caligraphic_D start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT be the set defined in (2.23).

Let D3>0subscript𝐷30D_{3}>0italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT > 0 be the constant introduced in Lemma 2.11.

Let D3>0subscript𝐷30D_{3}>0italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT > 0 be introduced in Lemma 2.11.

Let D3>0subscript𝐷30D_{3}>0italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT > 0 be the constant introduced in Lemma 2.11.

B

\right)^{2}\;.divide start_ARG ( italic_z - over¯ start_ARG italic_z end_ARG ) start_POSTSUPERSCRIPT 2 italic_k end_POSTSUPERSCRIPT ( italic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_z ) start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT end_ARG start_ARG ( italic_z - italic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_z ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over¯ start_ARG italic_z end_ARG - italic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_z ) start_POSTSUPERSCRIPT 2 italic_k end_POSTSUPERSCRIPT end_ARG = divide start_ARG italic_ω start_POSTSUPERSCRIPT italic_i ( italic_k + 1 ) end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_ω start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ⋅ divide start_ARG ( 1 - | italic_u | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 italic_k end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_ω start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT | italic_u | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 italic_k end_POSTSUPERSCRIPT end_ARG ( divide start_ARG italic_d italic_u end_ARG start_ARG italic_d italic_z end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

The integrand in the right hand side is smooth and the integral is absolutely convergent, cf. (2.7). We need to show that

In particular, μ⁢(0)=0𝜇00\mu(0)=0italic_μ ( 0 ) = 0 for m>2𝑚2m>2italic_m > 2 and ∂μ∂u⁢(0)=0𝜇𝑢00\dfrac{\partial\mu}{\partial u}(0)=0divide start_ARG ∂ italic_μ end_ARG start_ARG ∂ italic_u end_ARG ( 0 ) = 0 for m=2𝑚2m=2italic_m = 2.

Since μ⁢(0)=0𝜇00\mu(0)=0italic_μ ( 0 ) = 0 for m>2𝑚2m>2italic_m > 2, see (2.1), the integral in the left hand side of (3.3) is absolutely convergent, and we have

Since for m>2𝑚2m>2italic_m > 2 we have μ⁢(0)=0𝜇00\mu(0)=0italic_μ ( 0 ) = 0, cf. (2.1), this yields I3=0subscript𝐼30I_{3}=0italic_I start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0. When m=2𝑚2m=2italic_m = 2, we use Lemma 2, the fact that ΦΦ\Phiroman_Φ is holomorphic [1], and the formulas

C

=−2⁢(δ−γ)⁢(a∗⁢a−γ2).absent2𝛿𝛾superscript𝑎𝑎𝛾2\displaystyle=-2(\delta-\gamma)\left(a^{*}a-\frac{\gamma}{2}\right).= - 2 ( italic_δ - italic_γ ) ( italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a - divide start_ARG italic_γ end_ARG start_ARG 2 end_ARG ) .

the commutation relations of 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) [26], hence the Lie algebra associated to coupled SUSY is found to be isomorphic to the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) Lie algebra.

Thus the triple generates a Lie algebra as it is closed under commutation. After adding a multiple of the identity to a∗⁢asuperscript𝑎𝑎a^{*}aitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a and rescaling,

With ladder operators established for coupled SUSYs, it is natural to inquire about the eigenvalues of the coupled SUSY Hamiltonians a∗⁢asuperscript𝑎𝑎a^{*}aitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a, a⁢a∗𝑎superscript𝑎aa^{*}italic_a italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, b∗⁢bsuperscript𝑏𝑏b^{*}bitalic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_b, and b⁢b∗𝑏superscript𝑏bb^{*}italic_b italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. As in standard SUSY, a∗⁢asuperscript𝑎𝑎a^{*}aitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a and a⁢a∗𝑎superscript𝑎aa^{*}italic_a italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT share the same eigenvalues—up to a possible eigenvalue of 00. Likewise, b∗⁢bsuperscript𝑏𝑏b^{*}bitalic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_b and b⁢b∗𝑏superscript𝑏bb^{*}italic_b italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT share the same eigenvalues—up to a possible eigenvalue of 00. Moreover, the spectra of a∗⁢asuperscript𝑎𝑎a^{*}aitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a and b⁢b∗𝑏superscript𝑏bb^{*}italic_b italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT are related by a shift of γ𝛾\gammaitalic_γ since a∗⁢a=b⁢b∗+γsuperscript𝑎𝑎𝑏superscript𝑏𝛾a^{*}a=bb^{*}+\gammaitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a = italic_b italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + italic_γ. Thus it is sufficient to study one of a∗⁢asuperscript𝑎𝑎a^{*}aitalic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_a and a⁢a∗𝑎superscript𝑎aa^{*}italic_a italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT to fully understand the eigenvalues of any of the Hamiltonians in a coupled SUSY.

Traditionally the QMHO is associated to the 1D Heisenberg-Weyl Lie algebra as this is the Lie algebra which corresponds to the canonical commutation relations which is reflected in the algebra generated by the ladder operators. This is not the only Lie algebra which may be associated to the QMHO. There are two other treatments of the QMHO: Schwinger’s “spinification” of the two-particle QMHO and the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) treatment of the QMHO. Coupled SUSY is to some degree a unification of the two treatments.

B

16161616;  P1⁢(3,1)superscript𝑃131P^{1}(3,1)\ italic_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( 3 , 1 )  32323232; (16+32)/48=𝟏1632481(16+32)/48={\bf 1}( 16 + 32 ) / 48 = bold_1.

4[1]⁢1[2]superscript4delimited-[]1superscript1delimited-[]24^{[1]}1^{[2]}4 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

(4[1],1[2])superscript4delimited-[]1superscript1delimited-[]2(4^{[1]},1^{[2]})( 4 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT )

4[1]⁢1[2]superscript4delimited-[]1superscript1delimited-[]24^{[1]}1^{[2]}4 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

4[1]⁢1[2]superscript4delimited-[]1superscript1delimited-[]24^{[1]}1^{[2]}4 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT:

A

What are the explicit minimal generators of the first syzygy of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ], in terms of the the lattice ℒℒ\mathcal{L}caligraphic_L?

The first Betti number β1⁢(ℒ)subscript𝛽1ℒ\beta_{1}(\mathcal{L})italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( caligraphic_L ) of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ] is

The following theorem gives us the linear generators for the first syzygy of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ].

Find an exact formula for the first Betti number of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ].

For a planar distributive lattice ℒℒ\mathcal{L}caligraphic_L, the number of strip type generators of first syzygy of R⁢[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ], we denote it by

C

It is therefore desirable to have a robust theory at hand which is insensitive to the details of the

Another motivation to introduce the framework developed here stems from the fact that a common way to derive properties of a stochastic PDE is to approximate it by discrete systems for which the desired property holds and then show

In the specific case where the solution to the stochastic PDE at hand is a function of time, [MatetskiDiscrete] developed a framework adapting the theory of regularity structures to allow for certain spatial discretisations.

The theory of regularity structures is a framework developed by the second author in [Regularity] which

the results in the article show that, given a stochastic PDE such that the theory of regularity structures

A

In [17, Theorem 3.3], Jahnke-Simon show that any theory of separably algebraically maximal Kaplansky fields of a fixed finite degree of imperfection is dependent if and only if the residue field and value group are.

Let (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) be either an algebraically maximal Kaplansky valued field or a henselian valued field of residue characteristic 00. Then (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is strongly dependent if and only if the residue field and the value group are. The result remains valid under strongly dependent expansions of the residue field.

For the last part, recall that the residue field of a strongly dependent pure valued field (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is a stably embedded pure field provided that (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is either algebraically maximal Kaplansky or henselian of residue characteristic 00, see for example [12, Corollary 4.4].

Let (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) be either an algebraically maximal Kaplansky valued field or a henselian valued field of residue characteristic 00. Since strong dependence is preserved under interpretations, it is clear that if (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is strongly dependent so are the value group and the residue field with all their induced structure. So we prove the reverse implication.

A valued field (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) of residue characteristic p>0𝑝0p>0italic_p > 0 is a Kaplansky field if the value group is p𝑝pitalic_p-divisible, the residue field is perfect and the residue field does not admit any finite separable extensions of degree divisible by p𝑝pitalic_p.

A

Since Toda-Uehara’s construction consists of complicated inductive step, it is difficult to find an explicit description of the resulting tilting bundle in general.

3.2. Toda-Uehara’s assumptions for Y𝑌Yitalic_Y and Y′superscript𝑌′Y^{\prime}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT

In the present subsection, we check that Toda-Uehara’s assumptions (Assumption 2.10 and Assumption 2.11) hold for Y𝑌Yitalic_Y and Y′superscript𝑌′Y^{\prime}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

Y𝑌Yitalic_Y and Y′superscript𝑌′Y^{\prime}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT satisfy Assumption 2.11.

Y𝑌Yitalic_Y and Y′superscript𝑌′Y^{\prime}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT satisfy Toda-Uehara’s assumptions.

D

We denote by m:G(2)→G:𝑚→superscript𝐺2𝐺m\colon G^{(2)}\rightarrow Gitalic_m : italic_G start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT → italic_G the partial multiplication and let ι:G→G:𝜄→𝐺𝐺\iota\colon G\rightarrow Gitalic_ι : italic_G → italic_G be the inversion.

In the following we will always identify M𝑀Mitalic_M with an embedded submanifold of G𝐺Gitalic_G via the unit map 1:M→G:1→𝑀𝐺1\colon M\rightarrow G1 : italic_M → italic_G (in the following we suppress the unit map in the notation without further notice).

In the following we establish a new characterization of the elements of S𝒢⁢(α)subscript𝑆𝒢𝛼S_{\mathcal{G}}(\alpha)italic_S start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT ( italic_α ).

The source map α𝛼\alphaitalic_α is the unit element of S𝒢subscript𝑆𝒢S_{\mathcal{G}}italic_S start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT.555Here we identify M⊆G𝑀𝐺M\subseteq Gitalic_M ⊆ italic_G via the unit map, i.e. without this identification the unit is 1∘α1𝛼1\circ\alpha1 ∘ italic_α!

f,g)\mapsto f\circ g∘ : italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_Y , italic_Z ) × italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_X , italic_Y ) → italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_X , italic_Z ) , ( italic_f , italic_g ) ↦ italic_f ∘ italic_g. However, this mapping will in general not be smooth (not even continuous!) if X𝑋Xitalic_X is non-compact. Following B.13, we can obtain a smooth map if we restrict to the subset of smooth proper maps Prop⁡(X,Y)Prop𝑋𝑌\operatorname{Prop}(X,Y)roman_Prop ( italic_X , italic_Y ). It turns out that the S𝒢subscript𝑆𝒢S_{\mathcal{G}}italic_S start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT will be contained in this set if the Lie groupoid satisfies the following condition.

A

We consider M𝑀Mitalic_M as a G𝐺Gitalic_G-bornological coarse space with the bornological and coarse structures induced by the Riemannian distance. In the following we show that certain invariant open subsets Z𝑍Zitalic_Z with the induced bornological coarse structures of M𝑀Mitalic_M are also very proper. The argument in 5 above does not apply directly since bounded subsets of Z𝑍Zitalic_Z need not be relatively compact. A simple idea is to intersect the partition constructed for M𝑀Mitalic_M with Z𝑍Zitalic_Z. But then in general the condition

In this section we introduce the C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-categories of locally finite equivariant controlled Hilbert spaces and define the equivariant coarse K𝐾Kitalic_K-homology functor.

3.7.4 is violated. In the following we introduce an assumption on Z𝑍Zitalic_Z which ensures 3.7.4 in this procedure.

We now observe that with this procedure we increase every projection at most a finite number of times (by Assumption 3.7.3).

in Assumption 3.7.5. This still ensures that H⁢(B)𝐻𝐵H(B)italic_H ( italic_B ) is separable for every bounded subset B𝐵Bitalic_B of X𝑋Xitalic_X.

B

It then follows from Lemma 1 that 1≤αiF≤α1superscriptsubscript𝛼𝑖𝐹𝛼1\leq\alpha_{i}^{F}\leq\alpha1 ≤ italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT ≤ italic_α for all the local eigenvalues. Thus, Λ~h△=Λ~hfsuperscriptsubscript~Λℎ△superscriptsubscript~Λℎ𝑓\tilde{\Lambda}_{h}^{\triangle}={\widetilde{\Lambda}}_{h}^{f}over~ start_ARG roman_Λ end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT △ end_POSTSUPERSCRIPT = over~ start_ARG roman_Λ end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT and Λ~hΠ=∅superscriptsubscript~ΛℎΠ\tilde{\Lambda}_{h}^{\Pi}=\emptysetover~ start_ARG roman_Λ end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Π end_POSTSUPERSCRIPT = ∅ from (33) and (36).

The key to approximate (25) is the exponential decay of P⁢w𝑃𝑤Pwitalic_P italic_w, as long as w∈H1⁢(𝒯H)𝑤superscript𝐻1subscript𝒯𝐻w\in{H^{1}({\mathcal{T}_{H}})}italic_w ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( caligraphic_T start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ) has local support. That allows replacing P𝑃Pitalic_P by a semi-local operator Pjsuperscript𝑃𝑗P^{j}italic_P start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT. That works fine for low-contrast coefficients and is the subject of Section 3.2. For high-contrast coefficients however, the exponential decay rate is smaller, and to circumvent that we consider in Section 3.1 a spectral decomposition of Λ~hfsuperscriptsubscript~Λℎ𝑓\tilde{\Lambda}_{h}^{f}over~ start_ARG roman_Λ end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT.

Of course, the numerical scheme and the estimates developed in Section 3.1 hold. However, several simplifications are possible when the coefficients have low-contrast, leading to sharper estimates. We remark that in this case, our method is similar to that of [MR3591945], with some differences. First we consider that T~~𝑇\tilde{T}over~ start_ARG italic_T end_ARG can be nonzero. Also, our scheme is defined by a sequence of elliptic problems, avoiding the annoyance of saddle point systems. We had to reconsider the proofs, in our view simplifying some of them.

The remainder of the this paper is organized as follows. Section 2 describes a suitable primal hybrid formulation for the problem (1), which is followed in Section 3 by its a discrete formulation. A discrete space decomposition is introduced to transform the discrete saddle-point problem into a sequence of elliptic discrete problems. The analysis of the exponential decay of the multiscale basis function is considered in Section 3.2. To overcome the possible deterioration of the exponential decay for high-contrast coefficients, in Section 3.1 the Localized Spectral Decomposition (LSD) method is designed and fully analyzed. To allow an efficient pre-processing numerical scheme, Section LABEL:ss:findim discusses how to reduce the right-hand side space dimension without losing a target accuracy, and also develops L2⁢(Ω)superscript𝐿2ΩL^{2}(\Omega)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) a priori error estimates. Section LABEL:s:Algorithms gives a global overview of the LSD algorithm proposed. Appendix LABEL:s:Auxiliaryresults provides some mathematical tools and Appendix LABEL:s:Notations refers to a notation library for the paper.

As in many multiscale methods previously considered, our starting point is the decomposition of the solution space into fine and coarse spaces that are adapted to the problem of interest. The exact definition of some basis functions requires solving global problems, but, based on decaying properties, only local computations are required, although these are not restricted to a single element. It is interesting to notice that, although the formulation is based on hybridization, the final numerical solution is defined by a sequence of elliptic problems.

B

Let K𝐾Kitalic_K be a metric space with the isometry class in 𝒦ℓsubscript𝒦ℓ\mathcal{K}_{\ell}caligraphic_K start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT.

areaBi⩾π⋅ε24.areasubscript𝐵𝑖⋅𝜋superscript𝜀24\mathop{\rm area}\nolimits B_{i}\geqslant\tfrac{\pi{\hskip 0.5pt\cdot\hskip 0.%

areaK⩽π⋅ℓ2.area𝐾⋅𝜋superscriptℓ2\mathop{\rm area}\nolimits K\leqslant\pi{\hskip 0.5pt\cdot\hskip 0.5pt}\ell^{2}.roman_area italic_K ⩽ italic_π ⋅ roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Denote by areaAarea𝐴\mathop{\rm area}\nolimits Aroman_area italic_A the two-dimensional Hausdorff measure of A⊂K𝐴𝐾A\subset Kitalic_A ⊂ italic_K.

lengths∘γ⩾lengths′∘γlength𝑠𝛾lengthsuperscript𝑠′𝛾\mathop{\rm length}\nolimits s\circ\gamma\geqslant\mathop{\rm length}\nolimits

C

Definition 3.1, in fact QMsubscript𝑄𝑀Q_{M}italic_Q start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT is a vacuum state). This is

Λ3⁢ℓPlanck2≈0.96×10−122Λ3superscriptsubscriptℓPlanck20.96superscript10122\frac{\Lambda}{3}\ell_{\rm Planck}^{2}\approx 0.96\times 10^{-122}divide start_ARG roman_Λ end_ARG start_ARG 3 end_ARG roman_ℓ start_POSTSUBSCRIPT roman_Planck end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≈ 0.96 × 10 start_POSTSUPERSCRIPT - 122 end_POSTSUPERSCRIPT

measurements Λ≈2.89×10−122⁢ℓPlanck−2Λ2.89superscript10122superscriptsubscriptℓPlanck2\Lambda\approx 2.89\times 10^{-122}\>\ell_{\rm Planck}^{-2}roman_Λ ≈ 2.89 × 10 start_POSTSUPERSCRIPT - 122 end_POSTSUPERSCRIPT roman_ℓ start_POSTSUBSCRIPT roman_Planck end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT

1−γ⁢(K⁢3)≈1.70×10−271𝛾𝐾31.70superscript10271-\gamma(K3)\approx 1.70\times 10^{-27}1 - italic_γ ( italic_K 3 ) ≈ 1.70 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT and the general asymptotics of

≈0.96×10−122absent0.96superscript10122\approx 0.96\times 10^{-122}≈ 0.96 × 10 start_POSTSUPERSCRIPT - 122 end_POSTSUPERSCRIPT of the cosmological constant.

D

For later use we need to be able to adjust how wide the mountain is (using a parameter c𝑐citalic_c) and how narrow the mountain pass is (using a parameter δ𝛿\deltaitalic_δ). To make this precise we will need a few explicit constructions.

For any smooth map φ:Sk−1→ℝk−{0}:𝜑→superscript𝑆𝑘1superscriptℝ𝑘0\varphi:S^{k-1}\to\mathbb{R}^{k}-\{0\}italic_φ : italic_S start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - { 0 } we call a map

Let 𝒞klsuperscriptsubscript𝒞𝑘𝑙\mathcal{C}_{k}^{l}caligraphic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT be the set of smooth maps f:ℝ2⁢k→ℝ:𝑓→superscriptℝ2𝑘ℝf:\mathbb{R}^{2k}\to\mathbb{R}italic_f : blackboard_R start_POSTSUPERSCRIPT 2 italic_k end_POSTSUPERSCRIPT → blackboard_R such that

Pick a smooth even function φ:ℝ→ℝ:𝜑→ℝℝ\varphi:\mathbb{R}\to\mathbb{R}italic_φ : blackboard_R → blackboard_R such that

We now set out to define the missing maps in the diagram in Equation (9.5) at level (k,l)𝑘𝑙(k,l)( italic_k , italic_l ). Consider any smooth even function q¯:ℝ→ℝ:¯𝑞→ℝℝ\overline{q}:\mathbb{R}\to\mathbb{R}over¯ start_ARG italic_q end_ARG : blackboard_R → blackboard_R such that

C

\end{array};\frac{(1-c)^{1/r}}{r^{r}}\right).start_FLOATSUBSCRIPT italic_r + 2 end_FLOATSUBSCRIPT italic_F start_POSTSUBSCRIPT 2 italic_r end_POSTSUBSCRIPT ( start_ARRAY start_ROW start_CELL 1 , roman_Δ ( italic_r , - italic_k + italic_s ) , divide start_ARG italic_β + italic_s + italic_i end_ARG start_ARG italic_r end_ARG end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL roman_Δ ( italic_r , italic_β + italic_s ) , roman_Δ ( italic_r , italic_s + 1 ) end_CELL start_CELL end_CELL end_ROW end_ARRAY ; divide start_ARG ( 1 - italic_c ) start_POSTSUPERSCRIPT 1 / italic_r end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_ARG ) .

The authors would like to thank Professor Ali Zaghouani for his important remarks and suggestions that significantly improved this work.

The proof of this proposition proceeds by induction on n𝑛nitalic_n. Note that, it is sufficient to apply (4.3) for n=d𝑛𝑑n=ditalic_n = italic_d to obtain

The operator S𝑆Sitalic_S and its matrix elements will be the focal points of our study for the remainder of this paper. It is important to note that S𝑆Sitalic_S is invertible, with its inverse given by S−1=e−Q⁢(J−)⁢e−J+superscript𝑆1superscript𝑒𝑄subscript𝐽superscript𝑒subscript𝐽S^{-1}=e^{-Q(J_{-})}e^{-J_{+}}italic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_Q ( italic_J start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUPERSCRIPT. The matrix elements of S𝑆Sitalic_S and S−1superscript𝑆1S^{-1}italic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT are denoted and defined by

To facilitate the analysis of our operator S𝑆Sitalic_S and its impact on the d𝑑ditalic_d-orthogonal polynomial sequence, we will need to work with the associated monic polynomials P^n⁢(k)subscript^𝑃𝑛𝑘\widehat{P}_{n}(k)over^ start_ARG italic_P end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_k ). These polynomials are explicitly defined by the formula:

A

_{G}(w)}(\lambda))\in({\mathbb{C}}[\![q^{-1}]\!])P= roman_gch roman_Γ ( bold_Q start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_w ) , caligraphic_O start_POSTSUBSCRIPT bold_Q start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_w ) end_POSTSUBSCRIPT ( italic_λ ) ) ∈ ( blackboard_C [ [ italic_q start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ] ] ) italic_P

H>0⁢(𝐐G⁢(w),𝒪𝐐G⁢(w)⁢(λ))={0}.superscript𝐻absent0subscript𝐐𝐺𝑤subscript𝒪subscript𝐐𝐺𝑤𝜆0\displaystyle H^{>0}(\mathbf{Q}_{G}(w),{\mathcal{O}}_{\mathbf{Q}_{G}(w)}(%

Γ⁢(𝐐G⁢(si),𝒪𝐐G⁢(si)⁢(λ))∨↪Γ⁢(𝐐G,𝒪𝐐G⁢(λ))∨⁢λ∈P+↪Γsuperscriptsubscript𝐐𝐺subscript𝑠𝑖subscript𝒪subscript𝐐𝐺subscript𝑠𝑖𝜆Γsuperscriptsubscript𝐐𝐺subscript𝒪subscript𝐐𝐺𝜆𝜆subscript𝑃\Gamma(\mathbf{Q}_{G}(s_{i}),{\mathcal{O}}_{\mathbf{Q}_{G}(s_{i})}(\lambda))^{%

ℂ[𝔑]⊃Γ(𝐐G,𝒪𝐐G(ϖi))⟶→Γ(𝐐G(tβ′),𝒪𝐐G⁢(tβ′)(ϖi))⊂ℂ[𝕆(tβ′)],{\mathbb{C}}[\mathfrak{N}]\supset\Gamma(\mathbf{Q}_{G},{\mathcal{O}}_{\mathbf{%

Γ⁢(𝐐G,𝒪𝐐G⁢(λ))∨=𝕎⁢(−w0⁢λ).Γsuperscriptsubscript𝐐𝐺subscript𝒪subscript𝐐𝐺𝜆𝕎subscript𝑤0𝜆\Gamma(\mathbf{Q}_{G},{\mathcal{O}}_{\mathbf{Q}_{G}}(\lambda))^{\vee}=\mathbb{%

A

The analogous topological result for the left and right geometric dimension of a one-relator special monoid is also obtained.

In particular this results says that for every special one-relator monoid whose defining relator is not a proper power admits an equivariant classifying space of dimension at most 2222.

In fact, in this case it turns out that the Cayley complex of the monoid gives an equivariant classifying space of dimension at most 2222, as the following result demonstrates.

In fact, it will follow from our results that when w𝑤witalic_w is not a proper power then the Cayley complex of the one-relator monoid M𝑀Mitalic_M is an equivariant classifying space for M𝑀Mitalic_M of dimension at most 2222.

Let X𝑋Xitalic_X be the 2222-complex obtained by filling in each loop labeled by w𝑤witalic_w in the Cayley graph Γ⁢(M,A)Γ𝑀𝐴\Gamma(M,A)roman_Γ ( italic_M , italic_A ) of M𝑀Mitalic_M. Then X𝑋Xitalic_X is left equivariant classifying space for M𝑀Mitalic_M with dimension at most 2222.

C

For the opposite direction, it is relatively simple to check that ϱKsuperscriptitalic-ϱ𝐾\varrho^{K}italic_ϱ start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT satisfies the properties of a p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measure.

The p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measures are extensions of convex risk measures in Föllmer and Schied (2002).

The conditional p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measures (with partial order ≤Ksubscript𝐾\leq_{K}≤ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT) are extensions of conditional convex risk measures introduced in Detlefsen and Scandolo (2005).

A special example of p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measures which is so called OCE, is discussed in the next section. Finally, in Sect. 5, the p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measures are used to study the dual representation of the p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-dynamic risk measures.

Now, the dual representation of p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-convex risk measures is provided and which will be used in the proof of p⁢(⋅)𝑝⋅p(\cdot)italic_p ( ⋅ )-dynamic risk measures in Sect. 5.

A

It is interesting to remark that Zhu [Zh] proved that for any line bundle on the moduli stack ℬ⁢u⁢n𝒢ℬ𝑢subscript𝑛𝒢\mathscr{B}un_{\mathscr{G}}script_B italic_u italic_n start_POSTSUBSCRIPT script_G end_POSTSUBSCRIPT for a ‘reasonably good’ parahoric Bruhat-Tits group scheme 𝒢𝒢\mathscr{G}script_G over a curve Σ¯¯Σ\bar{\Sigma}over¯ start_ARG roman_Σ end_ARG, the pull-back of the line bundle to the twisted affine Grassmannian at every point of Σ¯¯Σ\bar{\Sigma}over¯ start_ARG roman_Σ end_ARG is of the same central charge. It matches the way we define the space of covacua, i.e., we attach integrable highest weight representations of twisted affine Lie algebras of the same central charge at every point.

Our work was initially motivated by a conjectural connection predicted by Fuchs-Schweigert [FSc] between the trace of diagram automorphism on the space of conformal blocks and certain conformal field theory related to twisted affine Lie algebras. A Verlinde type formula for the trace of diagram automorphism on the space of conformal blocks has been proved recently by the first author [Ho1, Ho2], where the formula involves the twisted affine Kac-Moody algebras mysteriously.

The Wess-Zumino-Witten model is a type of two dimensional conformal field theory, which associates to an algebraic curve with marked points and integrable highest weight modules of an affine Kac-Moody Lie algebra associated to the points, a finite dimensional vector space consisting of conformal blocks. The space of conformal blocks has many important properties including Propagation of Vacua and Factorization. Deforming the pointed algebraic curves in a family, we get a sheaf of conformal blocks. This sheaf admits a flat projective connection when the family of pointed curves is a smooth family. The mathematical theory of conformal blocks was first established in a pioneering work by Tsuchiya-Ueno-Yamada [TUY] where all these properties were obtained. All the above properties are important ingredients in the proof of the celebrated Verlinde formula for the dimension of the space of conformal blocks (cf. [Be, Fa1, Ku2, So1, V]).This theory has a geometric counterpart in the theory of moduli spaces of principal bundles over algebraic curves and also the moduli of curves and its stable compactification.

12. Identification of twisted conformal blocks with the space of global sections of line bundles on moduli stack

In particular, when ΓΓ\Gammaroman_Γ has nontrivial stabilizers at the marked points, general twisted affine Kac-Moody Lie algebras and their representations occur naturally in this twisted theory of conformal blocks. Damiolini’s work dealt with the untwisted affine Lie algebras since only the unramified points are marked in her setting. In our work, Kac’s theory of twisted affine Lie algebras associated to finite order automorphisms and related Sugawara operators in the twisted setting are extensively employed. These new features bring considerably more Lie theoretic complexity for the results stated above, which enriches the twisted theory in a most natural way. Notably, the proof of Theorem A (or Theorem 4.3) is highly technical, where we have to introduce the technical condition that the finite group ΓΓ\Gammaroman_Γ stabilizes a Borel subalgebra in 𝔤𝔤\mathfrak{g}fraktur_g. Furthermore, the Hurwitz stack of ΓΓ\Gammaroman_Γ-curves with only unramified points marked is in general not proper, i.e., such a pointed smooth ΓΓ\Gammaroman_Γ-curve may degenerate to a ΓΓ\Gammaroman_Γ-curve with non-free nodal ΓΓ\Gammaroman_Γ-orbits. Accordingly, it is desirable to have factorization theorem (Theorem B) for the ΓΓ\Gammaroman_Γ-curves with general nodal ΓΓ\Gammaroman_Γ-orbits, which naturally involves the twisted conformal blocks with ramified points marked. Our more general theory of twisted conformal blocks fits perfectly with the compactification of Hurwitz stacks, and marking ramified points is very crucial towards a Verlinde type formula for twisted conformal blocks of any kind.

A

M𝑀Mitalic_M is quasi-graphic with framework G𝐺Gitalic_G and ℬℬ\mathcal{B}caligraphic_B is the set of cycles of G𝐺Gitalic_G that are circuits of M𝑀Mitalic_M.

Thus the class of biased-graphic matroids differs from that of quasi-graphic matroids only in that a biased-graphic matroid is permitted to consist of 2-sums of frame matroids and lifted-graphic matroids, while for such a 2-sum to remain in the class of quasi-graphic matroids, either all summands must be frame or all summands must be lifted-graphic.

In [10] Zaslavsky showed that the class of frame matroids is precisely that of matroids arising from biased graphs, as follows.

In the class of frame matroids, as is the case with cycle matroids of graphs, a connected matroid cannot be represented by a disconnected graph.

Just as with frame matroids, Zaslavsky showed that given any graph G𝐺Gitalic_G and partition (ℬ,𝒰)ℬ𝒰(\mathcal{B},\mathcal{U})( caligraphic_B , caligraphic_U ) of its cycles with ℬℬ\mathcal{B}caligraphic_B obeying the theta property, there is a lifted-graphic matroid M=L⁢(G,ℬ)𝑀𝐿𝐺ℬM=L(G,\mathcal{B})italic_M = italic_L ( italic_G , caligraphic_B ) whose circuits are precisely those biased subgraphs described above, and that all lifted-graphic matroids arise from biased graphs in this way.

B

N_{i^{\prime}}\end{pmatrix},italic_w start_POSTSUBSCRIPT { italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } end_POSTSUBSCRIPT = ∏ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( start_ARG start_ROW start_CELL over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_N start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - 1 - ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + italic_m - 1 end_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_N start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL end_ROW start_ROW start_CELL italic_N start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) ,

In Sections 3 and 4, we have shown how we may construct the basis of multi-τ⁢τ¯𝜏¯𝜏\tau\bar{\tau}italic_τ over¯ start_ARG italic_τ end_ARG Hilbert space on a disk and obtain the generalized exclusion principle on a disk by understanding the matrix of mutual exclusion statistics on a disk. In Section 6, we have marched in the opposite direction. Namely, we directly construct the basis of multi-τ⁢τ¯𝜏¯𝜏\tau\bar{\tau}italic_τ over¯ start_ARG italic_τ end_ARG space on a multi-boundary surface by using the bases on a disk as building blocks. Along with this construction, we derive explicitly the statistical weight and prove that the generalized Pauli exclusion principle holds. We derive explicitly the exclusion statistics parameters on the multi-boundary surface. Therefore, both the extended anyonic exclusion statistics and the basis of a multi-τ⁢τ¯𝜏¯𝜏\tau\bar{\tau}italic_τ over¯ start_ARG italic_τ end_ARG space characterize the Hilbert space structure completely and equivalently. Since the sphere case can be obtained from the disk case by the topological operation introduced in Section 5, we have a more general equivalence relation as follows.

In this section, we study the extended anyonic exclusion statistics on a disk with a gapped boundary. We shall show that the extended statistical weight indeed satisfies (in fact a special case of) Eq. (8). We shall reveal the true meaning of pseudo-species too.

In this paper, we extend Haldane’s generalized exclusion principle to systems with gapped boundaries, by proposing that the number of available single particle states for additional particles linearly and mutually depends on the number of (every species of) existing anyons and the number of (every boundary type of) existing gapped boundaries. The dependency coefficients are called the statistics parameters. In terms of these extended statistics parameters, we extend Wu’s formula for statistical weight, which lays the foundation of the statistical mechanics of anyons on systems with gapped boundaries. Clearly, such exclusion statistics put the boundary components and bulk anyons on an equal footing. We shall dub this kind of exclusion statistics by extended anyonic exclusion statistics.

The paper is structured as follows. Section 2 introduces our extended anyonic exclusion statistics, accompanied by briefing of the concept of anyon exclusions statistics. Section 3 computes the state counting and derive the statistical weight of doubled Fibonacci anyons on a disk with a gapped boundary, using the extended Levin-Wen model. Motivated by the results of Section 3, Section 4 systematically constructs the bases of the Hilbert spaces of Fibonacci system on a disk, from which the statistics parameters can be immediately read off. Section 5 addresses the physical meaning of pseudo-species and the boundary effect on state counting and pseudo-species. Section 6 generalizes the story to surfaces with multiple gapped boundaries by two topological operations. Section 7 brings up an important equivalence relation between statistical weight and fusion basis. Appendices collect certain reviews, details, and more examples, including an Abelian example—the ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT toric code order, which further corroborate our results.

B

ΠFD⁢(U,u0)⟶ΠFD⁢(X,u0)⟶superscriptΠFD𝑈subscript𝑢0superscriptΠFD𝑋subscript𝑢0\Pi^{\rm FD}(U,u_{0})\,\longrightarrow\,\Pi^{\rm FD}(X,u_{0})roman_Π start_POSTSUPERSCRIPT roman_FD end_POSTSUPERSCRIPT ( italic_U , italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⟶ roman_Π start_POSTSUPERSCRIPT roman_FD end_POSTSUPERSCRIPT ( italic_X , italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )

Let U⟶X⟶𝑈𝑋U\,\longrightarrow\,Xitalic_U ⟶ italic_X be an open embedding of k𝑘kitalic_k-schemes. Assume that X𝑋Xitalic_X is projective and that U𝑈Uitalic_U is big in X𝑋Xitalic_X. Then, if U𝑈Uitalic_U is connected, it is CPC.

Ureg⊂Usuperscript𝑈reg𝑈U^{\rm reg}\,\subset\,Uitalic_U start_POSTSUPERSCRIPT roman_reg end_POSTSUPERSCRIPT ⊂ italic_U are big open subsets, the assumption implies that

We first assume that the proposition is true for big open subsets and deduce the general case from it.

It follows from Proposition 4.4 and Lemma 4.6 that the functor {ℰn}⟼x0∗⁢ℰ0⟼subscriptℰ𝑛superscriptsubscript𝑥0subscriptℰ0\{\mathcal{E}_{n}\}\,\longmapsto x_{0}^{*}\,\mathcal{E}_{0}{ caligraphic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } ⟼ italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT caligraphic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is exact, as it is the combination of the forgetful functor of Proposition 4.4

C

GCH holds, κ=ℵ2𝜅subscriptℵ2\kappa=\aleph_{2}italic_κ = roman_ℵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and the special Aronszajn tree property at ℵ2subscriptℵ2\aleph_{2}roman_ℵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (and hence Souslin’s Hypothesis at ℵ2subscriptℵ2\aleph_{2}roman_ℵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) holds.

Let θ𝜃\thetaitalic_θ be a large enough regular cardinal. For each i<κ+𝑖superscript𝜅i<\kappa^{+}italic_i < italic_κ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT

Rinot proved in [12] that if GCH holds, λ≥ω1𝜆subscript𝜔1\lambda\geq\omega_{1}italic_λ ≥ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is a cardinal, and □⁢(λ+)□superscript𝜆\Box(\lambda^{+})□ ( italic_λ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) holds, then there is a λ𝜆\lambdaitalic_λ-closed λ+superscript𝜆\lambda^{+}italic_λ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT-Souslin tree; on the other hand, Todorčević ([19]) proved that if κ≥ω2𝜅subscript𝜔2\kappa\geq\omega_{2}italic_κ ≥ italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is a regular cardinal and □⁢(κ)□𝜅\Box(\kappa)□ ( italic_κ ) fails, then κ𝜅\kappaitalic_κ is weakly compact in L𝐿Litalic_L.

By results of Shelah and Stanley ([15]) and of Rinot ([12]), our large cardinal assumption is optimal. Specifically:

Before moving on to the next subsection, we will briefly address the need for, and nature of, clause (7) in our definition of condition. As already mentioned in the introduction, the proof that our forcing satisfies the κ𝜅\kappaitalic_κ-chain condition is an adaptation, in our present context, of the Laver-Shelah proof that their forcing in [10] has the κ𝜅\kappaitalic_κ-chain condition. The only potential obstacles to making such an adaptation work may come from our present requirements that a condition q𝑞qitalic_q be closed under copying of all the relevant information, as dictated by the presence of edges ⟨(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 ) ⟩ in its side condition τqsubscript𝜏𝑞\tau_{q}italic_τ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT, and where this includes the information coming from the working part fqsubscript𝑓𝑞f_{q}italic_f start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT.

C

For any regular C∈G~𝐶normal-~𝐺C\in\tilde{G}italic_C ∈ over~ start_ARG italic_G end_ARG, the subalgebra B⁢(C)⊂Y⁢(𝔤)𝐵𝐶𝑌𝔤B(C)\subset Y({\mathfrak{g}})italic_B ( italic_C ) ⊂ italic_Y ( fraktur_g ) is a free polynomial algebra. Moreover, the Poincaré series of B⁢(C)𝐵𝐶B(C)italic_B ( italic_C ) coincides with the Poincaré series of H𝐻Hitalic_H.

There is a family of commutative subalgebras B⁢(C)𝐵𝐶B(C)italic_B ( italic_C ) in Y⁢(𝔤)𝑌𝔤Y({\mathfrak{g}})italic_Y ( fraktur_g ), called Bethe subalgebras, parameterized by C∈G𝐶𝐺C\in Gitalic_C ∈ italic_G. In this generality such subalgebras were first mentioned by Drinfeld [3]. For 𝔤=𝔰⁢𝔩n𝔤𝔰subscript𝔩𝑛{\mathfrak{g}}={\mathfrak{sl}}_{n}fraktur_g = fraktur_s fraktur_l start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT these subalgebras were studied by M. Nazarov and G.Olshanski in [15], for classical Lie algebras in the work of A.Molev [13]. In [12] D. Maulik and A. Okounkov define Bethe subalgebras in the R⁢T⁢T𝑅𝑇𝑇RTTitalic_R italic_T italic_T Yangian in connection with quantum cohomology of quiver varieties. We describe the associated graded of B⁢(C)𝐵𝐶B(C)italic_B ( italic_C ) in 𝒪⁢(G1⁢[[t−1]])𝒪subscript𝐺1delimited-[]delimited-[]superscript𝑡1\mathcal{O}(G_{1}[[t^{-1}]])caligraphic_O ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT [ [ italic_t start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ] ] ) as follows.

We show that Bethe subalgebras corresponding to boundary points of G¯¯𝐺\overline{G}over¯ start_ARG italic_G end_ARG are contained in some limit Bethe subalgebras. Moreover, we expect that the parameter space of limit Bethe subalgebras is some resolution of G¯¯𝐺\overline{G}over¯ start_ARG italic_G end_ARG, i.e. there is a proper birational surjective map from this parameter space to G¯¯𝐺\overline{G}over¯ start_ARG italic_G end_ARG.

Following [9] it is possible to construct new commutative subalgebras as limits of Bethe subalgebras.

Our main interest is the family of Bethe subalgebras B⁢(C)𝐵𝐶B(C)italic_B ( italic_C ) such that C𝐶Citalic_C is from the given maximal torus T⊂G𝑇𝐺T\subset Gitalic_T ⊂ italic_G. The action of such B⁢(C)𝐵𝐶B(C)italic_B ( italic_C ) preserves the weight decomposition of any Y⁢(𝔤)𝑌𝔤Y({\mathfrak{g}})italic_Y ( fraktur_g )-module. We show that all points in the closure T¯¯𝑇\overline{T}over¯ start_ARG italic_T end_ARG of T𝑇Titalic_T in the wonderful compactification G¯¯𝐺\overline{G}over¯ start_ARG italic_G end_ARG are “generic”, i.e. the subalgebras corresponding to all boundary points of T¯¯𝑇\overline{T}over¯ start_ARG italic_T end_ARG are Bethe subalgebras in the Yangians of some Levi subalgebras (Proposition 5.3). For 𝔤=𝔰⁢𝔩n𝔤𝔰subscript𝔩𝑛{\mathfrak{g}}={\mathfrak{sl}}_{n}fraktur_g = fraktur_s fraktur_l start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT this gives a new proof for [9, Theorem 5.3.1] describing limit Bethe subalgebras in the Yangian of 𝔰⁢𝔩n𝔰subscript𝔩𝑛{\mathfrak{sl}}_{n}fraktur_s fraktur_l start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as the parameter goes to the infinity.

C

The isotopy between 𝒜′superscript𝒜′\mathcal{A}^{\prime}caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and 𝒜′′superscript𝒜′′\mathcal{A}^{\prime\prime}caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT is shown in Figure 5.1.

S∖𝒜′𝑆superscript𝒜′S\setminus\mathcal{A}^{\prime}italic_S ∖ caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and S∖𝒜′′𝑆superscript𝒜′′S\setminus\mathcal{A}^{\prime\prime}italic_S ∖ caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT that intersect ∂S𝑆\partial S∂ italic_S.

(S,𝒜′)𝑆superscript𝒜′(S,\mathcal{A}^{\prime})( italic_S , caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )

(S^,𝒜′′)^𝑆superscript𝒜′′(\widehat{S},\mathcal{A}^{\prime\prime})( over^ start_ARG italic_S end_ARG , caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )

Let us write ℱ′′superscriptℱ′′\mathcal{F}^{\prime\prime}caligraphic_F start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT for (S^,𝒜′′)^𝑆superscript𝒜′′(\widehat{S},\mathcal{A}^{\prime\prime})( over^ start_ARG italic_S end_ARG , caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) and ℱ0′′=(A0,𝒜0′′)superscriptsubscriptℱ0′′subscript𝐴0superscriptsubscript𝒜0′′\mathcal{F}_{0}^{\prime\prime}=(A_{0},\mathcal{A}_{0}^{\prime\prime})caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = ( italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , caligraphic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ).

B

We denote by ℒ⁢o⁢cn,qℒ𝑜subscript𝑐𝑛𝑞\mathcal{L}oc_{n,q}caligraphic_L italic_o italic_c start_POSTSUBSCRIPT italic_n , italic_q end_POSTSUBSCRIPT the moduli stack of flat connections of rank n𝑛nitalic_n on X𝑋Xitalic_X with regular singularity at q𝑞qitalic_q. There is a canonical map from ℒ⁢o⁢cn,Pℒ𝑜subscript𝑐𝑛𝑃\mathcal{L}oc_{n,P}caligraphic_L italic_o italic_c start_POSTSUBSCRIPT italic_n , italic_P end_POSTSUBSCRIPT to ℒ⁢o⁢cn,qℒ𝑜subscript𝑐𝑛𝑞\mathcal{L}oc_{n,q}caligraphic_L italic_o italic_c start_POSTSUBSCRIPT italic_n , italic_q end_POSTSUBSCRIPT, which is defined by forgetting the P𝑃Pitalic_P-reduction.

Our definition of ℒ⁢o⁢cn,σ⁢(a¯)rℒ𝑜subscriptsuperscript𝑐𝑟𝑛𝜎¯𝑎\mathcal{L}oc^{r}_{n,\sigma(\underline{a})}caligraphic_L italic_o italic_c start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_σ ( under¯ start_ARG italic_a end_ARG ) end_POSTSUBSCRIPT and formulation of Theorem 4.3 is motivated by the work of Chen-Zhu [10] on the characteristic p𝑝pitalic_p version of the non-abelian Hodge correspondence for flat connections without singularities. The strategy of proof is similar to [10] besides the proof of the surjectivity result Proposition 4.6. The rest of this section is devoted to the proof of Theorem 4.3. We start by showing:

called parabolic weights. The parabolic weights can be used to define a stability condition on such objects, which is necessary for the construction of a moduli space. Since we focus on studying the moduli stack of such objects, we do not introduce the parabolic weights in this paper.

Note that for an arbitrary reductive group G𝐺Gitalic_G, a similar construction is used in [10] to establish the characteristic p𝑝pitalic_p version of the non-abelian Hodge correspondence for flat connections without singularities.

I would like to thank my advisor Tom Nevins for many helpful discussions on this subject, and for his comments on this paper. I would like to thank Christopher Dodd, Michael Groechenig and Tamas Hausel for helpful conversations. I would like to thank Tsao-Hsien Chen and Siqing Zhang for useful comments on an earlier version of this paper.

D

φ♭⁢(x,t)≤limν→0+φν⁢(x,t)=φ⁢(x,t),subscript𝜑♭𝑥𝑡subscript→𝜈superscript0subscript𝜑𝜈𝑥𝑡𝜑𝑥𝑡\varphi_{\flat}(x,t)\leq\lim_{\nu\to 0^{+}}\varphi_{\nu}(x,t)=\varphi(x,t),italic_φ start_POSTSUBSCRIPT ♭ end_POSTSUBSCRIPT ( italic_x , italic_t ) ≤ roman_lim start_POSTSUBSCRIPT italic_ν → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( italic_x , italic_t ) = italic_φ ( italic_x , italic_t ) ,

for any x0∈ℝdsubscript𝑥0superscriptℝ𝑑x_{0}\in\mathbb{R}^{d}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, s>0𝑠0s>0italic_s > 0, and Lebesgue point ξ𝜉\xiitalic_ξ. That is φ𝜑\varphiitalic_φ is a supersolution of (21). This completes the proof of the remaining implication in the case where φ𝜑\varphiitalic_φ is Cb∞subscriptsuperscript𝐶𝑏C^{\infty}_{b}italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT (and then φ#=φsubscript𝜑#𝜑\varphi_{\#}=\varphiitalic_φ start_POSTSUBSCRIPT # end_POSTSUBSCRIPT = italic_φ).

\quad\mbox{for a fixed $t_{0}>0$,}italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x , italic_t ) := italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x , italic_t ) / square-root start_ARG ∥ italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ , italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_ARG for a fixed italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 ,

Moreover, for any fixed (x,t)𝑥𝑡(x,t)( italic_x , italic_t ), lower semicontinuity of φ𝜑\varphiitalic_φ implies that

Moreover, the above assumption on Htsubscript𝐻𝑡H_{t}italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT implies that

C

_{n}}italic_A start_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG + ⋯ + divide start_ARG italic_c start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG.

Removing or adding finitely many elements to a sequence does not change the property of being balanced.∎

By [6, Corollary 2.4] we can add the remaining elements from (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) (that are not in (bn′)||(cn)(b^{\prime}_{n})||(c_{n})( italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | | ( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )) to (dn)subscript𝑑𝑛(d_{n})( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) such that the limit in average does not change.

Then we can decompose (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) in the following way: (an)=(bn)∪(cn)∪(dn),bn→α,cn→β,(dn)formulae-sequencesubscript𝑎𝑛subscript𝑏𝑛subscript𝑐𝑛subscript𝑑𝑛formulae-sequence→subscript𝑏𝑛𝛼→subscript𝑐𝑛𝛽subscript𝑑𝑛(a_{n})=(b_{n})\cup(c_{n})\cup(d_{n}),\ b_{n}\to\alpha,\ c_{n}\to\beta,\ (d_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = ( italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∪ ( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∪ ( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) , italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_α , italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_β , ( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is the rest of the sequence (if needed). This decomposition is not unique in general. However if α=−∞,a>−∞formulae-sequence𝛼𝑎\alpha=-\infty,\ a>-\inftyitalic_α = - ∞ , italic_a > - ∞ then (bn)subscript𝑏𝑛(b_{n})( italic_b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is unique up to finitely many elements and if β=+∞,b<+∞formulae-sequence𝛽𝑏\beta=+\infty,\ b<+\inftyitalic_β = + ∞ , italic_b < + ∞ then (cn)subscript𝑐𝑛(c_{n})( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is unique up to finitely many elements.

(cn)subscript𝑐𝑛(c_{n})( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) being balanced does not imply that (cn2)subscriptsuperscript𝑐2𝑛(c^{2}_{n})( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is balanced as well.

A

\frac{3}{8}-\frac{5}{4p},&\quad 4<p<6.\end{aligned}\right.italic_σ ( italic_p ) = { start_ROW start_CELL 0 , end_CELL start_CELL 2 < italic_p ⩽ 3 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG - divide start_ARG 3 end_ARG start_ARG 4 italic_p end_ARG , end_CELL start_CELL 3 < italic_p ⩽ 4 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 3 end_ARG start_ARG 8 end_ARG - divide start_ARG 5 end_ARG start_ARG 4 italic_p end_ARG , end_CELL start_CELL 4 < italic_p < 6 . end_CELL end_ROW

It is worth noting that the spacial support of the amplitude appearing in the right-hand side of (2.12) and (2.13) is slightly larger than that appearing in the left-hand side.

Very recently, Guth-Wang-Zhang [10] established the sharp square function estimate in the Euclidean case in 2+1212+12 + 1 dimensions. As a result, the corresponding local smoothing conjecture is resolved. For the variable coefficient setting, the Kakeya compression phenomena will happen which leads to the difference in the numerology of local smoothing conjecture between the variable and constant coefficient settings in n≥3𝑛3n\geq 3italic_n ≥ 3, see [1, 2, 9] for more details. This work provides additional methods and techniques toward handling the variable coefficient case which may help advance the research in this direction.

The following stability lemma makes the variable coefficient case and its constant counterpart comparable at sufficiently small scales.

The results obtained above generalize its constant coefficient counterpart in [14, 12]. It is worth noting that the results in the case 2<p≤32𝑝32<p\leq 32 < italic_p ≤ 3 are sharp, except for possibly arbitrarily small regularity loss.

D

Secondly, these new nonlinearities are multiplied by a suitable cut-off function θn⁢(‖u‖Zt)subscript𝜃𝑛subscriptnorm𝑢subscript𝑍𝑡\theta_{n}(\|u\|_{Z_{t}})italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ∥ italic_u ∥ start_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) which depends on the whole history up to time t𝑡titalic_t of the solution. Note however that this cut-off function takes values in the interval [0,1]01[0,1][ 0 , 1 ].

The approximating problem (6.1) approximates the original problem (5.4) in a two-fold way. Firstly, the very bad non-linearities (of possibly exponential growth)

The second modification makes the coefficients of the approximating problem (6.1) not only time dependent but also random. This two-fold modification makes the coefficients in problem (6.1) enough Lipschitz so that the corresponding fixed point problem has a unique solution and problem (6.1) has a unique global solution.

The main aim of this section is to prove that problem (6.1)-(6.2) has a unique global solution, i.e. the following result.

to our auxiliary problem (6.1)-(6.2) on [0,k]0𝑘[0,k][ 0 , italic_k ] we easily can show that the process u𝑢uitalic_u is a global solution problem (6.1)-(6.2).

B

Now suppose that (X,ω)𝑋𝜔(X,\omega)( italic_X , italic_ω ) is a symplectic cobordism between (Y+,α+)superscript𝑌superscript𝛼(Y^{+},\alpha^{+})( italic_Y start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_α start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) and (Y−,α−)superscript𝑌superscript𝛼(Y^{-},\alpha^{-})( italic_Y start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , italic_α start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ).

We can also define an ℒ∞subscriptℒ\mathcal{L}_{\infty}caligraphic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT augmentation

an ℒ∞subscriptℒ\mathcal{L}_{\infty}caligraphic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT augmentation

We can define an ℒ∞subscriptℒ\mathcal{L}_{\infty}caligraphic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT homomorphism

We can also linearize ΦΦ\Phiroman_Φ to define an ℒ∞subscriptℒ\mathcal{L}_{\infty}caligraphic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT homomorphism

C

[n]q=1−qn1−q=qn−1+⋯+q+1,…,n∈ℕ.formulae-sequencesubscriptdelimited-[]𝑛𝑞1superscript𝑞𝑛1𝑞superscript𝑞𝑛1⋯𝑞1…𝑛ℕ[n]_{q}=\frac{1-q^{n}}{1-q}=q^{n-1}+\cdots+q+1,\dots,n\in\mathbb{N}.[ italic_n ] start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT = divide start_ARG 1 - italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_q end_ARG = italic_q start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT + ⋯ + italic_q + 1 , … , italic_n ∈ blackboard_N .

The real function f𝑓fitalic_f defined on [a,b]𝑎𝑏[a,b][ italic_a , italic_b ] is called q𝑞qitalic_q-increasing

Let f⁢(x)𝑓𝑥f(x)italic_f ( italic_x ) be q𝑞qitalic_q-decreasing function on [a,b]𝑎𝑏[a,b][ italic_a , italic_b ] with f⁢(b⁢q0)=0𝑓𝑏superscript𝑞00f(bq^{0})=0italic_f ( italic_b italic_q start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) = 0.

Let f𝑓fitalic_f be a function defined on an interval (a,b)⊂ℝ𝑎𝑏ℝ(a,b)\subset\mathbb{R}( italic_a , italic_b ) ⊂ blackboard_R,

On this basis, in the same paper, Jackson defined an integral on [a,b]𝑎𝑏[a,b][ italic_a , italic_b ]

C

The expression in theorem 1.5 in [Pal-Sal] (see also theorem 1.6 there for a more general statement) is important for applications to analytic micro local analysis

The expression in theorem 1.5 in [Pal-Sal] allows also to perform useful explicit global intrinsic operator

The expression in theorem 1.5 in [Pal-Sal] (see also theorem 1.6 there for a more general statement) is important for applications to analytic micro local analysis

Indeed in [Pal-Sal], theorem 1.5, we obtain a rather simple and explicit global expression for the complex

Grauert tube, an explicit formula for the complex structure such as the one in theorem 1.5 in [Pal-Sal], allows to

A

In Section 3 we use the Heat Equation and the Heat Kernel to prove the Hodge Decomposition Theorem for this cohomology. An intuitive (and naive) idea arises from statistical mechanics: if the digraph G𝐺Gitalic_G is the set of possible states to a particle and the initial condition u0:G→ℝ:subscript𝑢0→𝐺ℝu_{0}:G\to\mathbb{R}italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : italic_G → blackboard_R defines the “number of particles” or the “energy” of each state at time zero and utsubscript𝑢𝑡u_{t}italic_u start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is its evolution on time t𝑡titalic_t (as the heat dissipates), then this converges to an harmonic form u∞subscript𝑢u_{\infty}italic_u start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT that is the “thermodynamic equilibrium”. This equilibrium is the only harmonic form in the path cohomology class of the initial condition u0subscript𝑢0u_{0}italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

In Section 4, we finally introduce our stochastic process, the p𝑝pitalic_p-Lazy Random Walk on locally finite digraphs. We investigate its relationship to the spectrum of our Laplacian and the existence of high-dimensional cohomology, also proving the existence of a spectral gap.

The Laplace operator or the Laplacian is an elliptic operator that plays a fundamental role in analysis, geometry and stochastic calculus. In this section we generalize the usual Laplacian defined for graphs, whose diffusion process is the random walk. To, in the later sections of this work, relate it with the existence of homologies in the subjacent digraph to stochastic processes whose transitions depend on this operator.

In section 2 we construct high-dimensional Laplace operators, study its corresponding kernels (i.e., harmonic forms) and relate its spectrum to the existence of high homologies.

In Section 1 we generalize the aforementioned path cohomology to locally finite graphs and define notions of adjacency of paths or the graph and of orientability of such paths.

A

We call ⁢x¯⁢¯𝑥\mathord{\mbox{}\,\overline{\!x\!\!\;}\!\>}\mbox{}start_ID over¯ start_ARG italic_x end_ARG end_ID the conjugate of x𝑥xitalic_x. The following lemma summarises the properties of

For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O the following identities hold:

For all x,y∈𝕆𝑥𝑦𝕆x,y\in\mathord{\mathbb{O}}italic_x , italic_y ∈ blackboard_O the following identities hold:

For all x,y∈𝕆𝑥𝑦𝕆x,y\in\mathord{\mathbb{O}}italic_x , italic_y ∈ blackboard_O the following are true:

For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O, the following identities hold:

B