Datasets:

liangzid

/

robench2024b_all_setmathSCP-p

like 0

\operatorname{Struct}(\upsigma)}\rangle⟨ italic_τ start_POSTSUBSCRIPT ⊆ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∩ ⟦ sansserif_FO [ roman_σ ] ⟧ start_POSTSUBSCRIPT roman_Struct ( roman_σ ) end_POSTSUBSCRIPT ⟩.

consider a sentence φ∈𝖥𝖮⁢[σ]𝜑𝖥𝖮delimited-[]σ\varphi\in\mathsf{FO}[\upsigma]italic_φ ∈ sansserif_FO [ roman_σ ] defining an open

Consider for this a monotone sentence φ∈𝖥𝖮⁢[σ]𝜑𝖥𝖮delimited-[]σ\varphi\in\mathsf{FO}[\upsigma]italic_φ ∈ sansserif_FO [ roman_σ ]

φ∈𝖥𝖮⁢[σ]𝜑𝖥𝖮delimited-[]σ\varphi\in\mathsf{FO}[\upsigma]italic_φ ∈ sansserif_FO [ roman_σ ]. However,

sentence φ∈𝖥𝖮⁢[σ]𝜑𝖥𝖮delimited-[]σ\varphi\in\mathsf{FO}[\upsigma]italic_φ ∈ sansserif_FO [ roman_σ ] such that A⊧φmodels𝐴𝜑A\models\varphiitalic_A ⊧ italic_φ.

B

Although 𝖫𝖢𝖴𝖫𝖢𝖴\mathsf{LCU}sansserif_LCU is a particular case of 𝖢𝖮𝖡𝖢𝖮𝖡\mathsf{COB}sansserif_COB, they are used with different roles in our applications, so it is more practical to use different notations.

Fix a directed partial order S=⟨S,≤⟩𝑆𝑆S=\langle S,\leq\rangleitalic_S = ⟨ italic_S , ≤ ⟩ and a relational system 𝐑=⟨X,Y,R⟩𝐑𝑋𝑌𝑅\mathbf{R}=\langle X,Y,R\ranglebold_R = ⟨ italic_X , italic_Y , italic_R ⟩. Define the property:

Let 𝐑𝐑\mathbf{R}bold_R be a relational system, S𝑆Sitalic_S a directed partial order and let L𝐿Litalic_L be a linear order without greatest element. Then

Let 𝐑=⟨X,Y,R⟩𝐑𝑋𝑌𝑅\mathbf{R}=\langle X,Y,R\ranglebold_R = ⟨ italic_X , italic_Y , italic_R ⟩ be a relational system of the reals, S=⟨S,≤S⟩𝑆𝑆subscript𝑆S=\langle S,\leq_{S}\rangleitalic_S = ⟨ italic_S , ≤ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⟩ a directed partial order, L=⟨L,≤L⟩𝐿𝐿subscript𝐿L=\langle L,\leq_{L}\rangleitalic_L = ⟨ italic_L , ≤ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ⟩ a linear order, and let ℙℙ\mathbb{P}blackboard_P be a forcing notion. Define the following properties.

Let 𝐑𝐑\mathbf{R}bold_R be a relational system of the reals, S𝑆Sitalic_S a directed partial order without greatest element, and let ℙℙ\mathbb{P}blackboard_P be a forcing notion. If μ=cp(S)V\mu=\operatorname{cp}(S)^{V}italic_μ = roman_cp ( italic_S ) start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT and λ=cof(S)V\lambda=\operatorname{cof}(S)^{V}italic_λ = roman_cof ( italic_S ) start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT, then

B

Note that our argument in establishing the Gromov hyperbolicity of (X,j~)𝑋~𝑗(X,{\widetilde{j}})( italic_X , over~ start_ARG italic_j end_ARG ) is different with [17]. Recently, the authors in [32] proved a result similar to Theorem 1.3 in Euclidean spaces setting. Our approach is also different with that of [32]. According to [21, Theorem 3.1], the identity map id:(X,j~)→(X,h):id→𝑋~𝑗𝑋ℎ{\operatorname{id}}\colon(X,{\widetilde{j}})\to(X,h)roman_id : ( italic_X , over~ start_ARG italic_j end_ARG ) → ( italic_X , italic_h ) is roughly similar. We demonstrate that roughly similar maps preserve the Gromov hyperbolicity of metric spaces (not necessarily geodesic), which induces a quasisymmetric correspondence between the punctured Gromov boundaries equipped with Hamenstädt metrics, see Lemma 3.4. Therefore, Theorem 1.3 follows immediately from Theorem 1.1 and the above facts.

It is known that every simply connected planar proper sub-domain admits a Gromov hyperbolic metric by pulling back the classical hyperbolic metric of the unit disk via the Riemann mapping. Recently, Ibragimov introduced a new metric that hyperbolizes (in the sense of Gromov) all locally compact incomplete metric spaces in [21]. Subsequently, it is shown that the Gromov boundary ∂hXsubscriptℎ𝑋\partial_{h}X∂ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_X of (X,h)𝑋ℎ(X,h)( italic_X , italic_h ) can be identified with the metric boundary ∂X𝑋\partial X∂ italic_X of a bounded locally compact incomplete metric space (X,d)𝑋𝑑(X,d)( italic_X , italic_d ) via a quasisymmetric map; for the unbounded case, the extended boundary ∂X∪{∞}𝑋\partial X\cup\{\infty\}∂ italic_X ∪ { ∞ } is equipped with a chordal metric, see [22, Theorem 3.1].

In [1], Bonk et al. further studied the relationship between Gromov hyperbolic domains and uniform domains of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Recall that a proper domain in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is called Gromov hyperbolic if it is δ𝛿\deltaitalic_δ-hyperbolic with respect to its quasihyperbolic metric for some δ≥0𝛿0\delta\geq 0italic_δ ≥ 0. They showed that a bounded domain in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is uniform if and only if it is both Gromov hyperbolic and its Euclidean boundary is naturally quasisymmetrically equivalent to the Gromov boundary, see [1, Theorem 1.11]. Subsequently, Väisälä [30] generalized this result to Banach spaces and obtained a dimensional free result. Instead of bounded domains he considered also unbounded domains, where the quasisymmetric equivalence is replaced by quasimöbius equivalence.

It was shown in [1, Theorem 3.6] by Bonk et al. that a locally compact bounded uniform metric space is Gromov hyperbolic in the quasihyperbolic metric and the metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary.

In this paper, with the aid of Theorem 1.5, we establish an unbounded analogue of [1, Theorem 1.11] and prove that an unbounded domain in a Banach space is uniform if and only if it is both Gromov hyperbolic and its norm boundary is naturally quasisymmetrically equivalent to the punctured Gromov boundary equipped with a Hamenstädt metric.

C

Then [BS64, § 6.2. Théorème] yields the finiteness of H1⁢(Γ,Autk¯G⁡(X))superscriptH1ΓsuperscriptsubscriptAut¯𝑘𝐺𝑋\mathrm{H}^{1}(\Gamma,\operatorname{Aut}_{\overline{k}}^{G}(X))roman_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Γ , roman_Aut start_POSTSUBSCRIPT over¯ start_ARG italic_k end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT ( italic_X ) ), which finishes the proof.

The proofs of these two results are detailed in § 3.3. The first result is proven by a straightforward calculation. As for the Theorem 1.3, the first part is a special case of a result of Borovoi and Gagliardi (see [BG21, Theorem 1.6]). We include a direct proof in the case of homogeneous spaces, relying only on Proposition 1.1. First of all, this gives an explicit instance of Borovoi and Gagliardi’s result, and it also allows us to give a self-contained presentation.

In this subsection we make explicit the ΓΓ\Gammaroman_Γ-actions introduced in § 4.2, on the colored equipment (𝒱G⁢(G/H),𝒟B⁢(G/H))superscript𝒱𝐺𝐺𝐻superscript𝒟𝐵𝐺𝐻(\mathcal{V}^{G}(G/H),\mathcal{D}^{B}(G/H))( caligraphic_V start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT ( italic_G / italic_H ) , caligraphic_D start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_G / italic_H ) ) of G/H𝐺𝐻G/Hitalic_G / italic_H, and then specialize the Luna-Vust theory in this setting (Theorem 5.7).

Let us mention that forms of spherical homogeneous spaces (see Definition 4.15) over an arbitrary base field of characteristic zero were studied by Borovoi and Gagliardi in [Bor20, BG21]. The reader is referred to [MJT21a, § 3] and [BG21, § 11] for examples of spherical homogeneous spaces for which versions of Proposition 1.1 and Theorem 1.3 are applied to determine their (k,F)𝑘𝐹(k,F)( italic_k , italic_F )-forms. Other results, based in part on a weaker version of Proposition 1.1, concerning the real forms of complex symmetric spaces can be found in [MJT21b].

In this subsection, we present a cohomological invariant introduced by Borovoi and Gagliardi in [BG21], and also used in the real setting in [MJT21a].

D

Moreover, we note that based on its definition, it is also possible to use the recursive relation of 𝕎~⁢(⋅)~𝕎⋅\widetilde{\mathbb{W}}(\cdot)over~ start_ARG blackboard_W end_ARG ( ⋅ ), given in Corollary 1, to obtain explicit values of ℤ~⁢(z,⋅)~ℤ𝑧⋅\widetilde{\mathbb{Z}}(z,\cdot)over~ start_ARG blackboard_Z end_ARG ( italic_z , ⋅ ).

The paper is organised as follows: In Section 2 we define the MAP in discrete time and space and derive the so-called occupation mass matrix formula, from which we obtain some useful identities to be used in the following sections. In Section 3, we introduce some fundamental matrices associated to the discrete MAP, identify the first of two discrete scale matrices and derive matrix expressions for the one and two-sided upward exit problem. In Section 4, we derive results for the corresponding one and two-sided reflected processes, including the over-shoot and under-shoot upon exit which are then used in Section 5 to derive expressions for the one and two-sided downward exit problems of the original (non-reflected) discrete MAP.

As discussed in the previous section, the downward exit problems can be solved using an auxiliary result for the one-sided (lower) reflected process. As such, let us define

In this section, we deviate from the basic MAC described above and consider the associated two-sided reflection of the process {Xn}n⩾0subscriptsubscript𝑋𝑛𝑛0\{X_{n}\}_{n\geqslant 0}{ italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n ⩾ 0 end_POSTSUBSCRIPT with respect to a strip [−d,0]𝑑0[-d,0][ - italic_d , 0 ] with d>0𝑑0d>0italic_d > 0. The choice of strip is purely for notational convenience and can easily be converted to the general strip [−b,a]𝑏𝑎[-b,a][ - italic_b , italic_a ] by shifting the process appropriately. The main result of this section is given in Theorem 4 which is interesting in its own right, but is also used to derive the aforementioned downward exit problems of the original (un-reflected) MAC.

Although the result of Theorem 4 is interesting in its own right, its main importance in this paper is as a stepping stone for proving a similar result for the associated one-sided reflected process (see Section 4.1 below) and consequently, the two-sided and one-sided (as a limiting case) downward exit problems for the original (non-reflected) MAC.

D

Let ∂dμ1subscript𝑑subscript𝜇1\partial_{d}\mu_{1}∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT be the derivative of μ1subscript𝜇1\mu_{1}italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with respect to d𝑑ditalic_d, then ∂dμ1≥0subscript𝑑subscript𝜇10\partial_{d}\mu_{1}\geq 0∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ 0. Furthermore, if h^^ℎ\hat{h}over^ start_ARG italic_h end_ARG is non-constant, then ∂dμ1>0subscript𝑑subscript𝜇10\partial_{d}\mu_{1}>0∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0 for all d>0𝑑0d>0italic_d > 0.

as }t\to\infty.≤ italic_C roman_exp ( - ( italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_t ) → 0 as italic_t → ∞ .

then the partial derivative ∂dH1⁢(t)subscript𝑑subscript𝐻1𝑡\partial_{d}H_{1}(t)∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) of H1⁢(t)subscript𝐻1𝑡H_{1}(t)italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ), with respect to the diffusion coefficient d𝑑ditalic_d, satisfies

Let ∂dμ1subscript𝑑subscript𝜇1\partial_{d}\mu_{1}∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT be the derivative of μ1subscript𝜇1\mu_{1}italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with respect to d𝑑ditalic_d, then ∂dμ1≥0subscript𝑑subscript𝜇10\partial_{d}\mu_{1}\geq 0∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ 0. Furthermore, if h^^ℎ\hat{h}over^ start_ARG italic_h end_ARG is non-constant, then ∂dμ1>0subscript𝑑subscript𝜇10\partial_{d}\mu_{1}>0∂ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0 for all d>0𝑑0d>0italic_d > 0.

Indeed, if we differentiate (4.1) with respect to d𝑑ditalic_d (and denote the derivative as ′), we get

D

(2) Hk⁢(Xn)TLn=0subscript𝐻𝑘superscriptsubscript𝑋𝑛subscriptTL𝑛0H_{k}(X_{n})^{\operatorname{TL}_{n}}=0italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_TL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = 0 for all k,n≥1𝑘𝑛1k,n\geq 1italic_k , italic_n ≥ 1 (invariants are trival)

ℱ𝓃=ℛ1(𝓃)⊇ℛ2(𝓃)⊇…⊇ℛ⌊𝓃2⌋(𝓃)subscriptℱ𝓃subscriptsuperscriptℛ𝓃1superset-of-or-equalssubscriptsuperscriptℛ𝓃2superset-of-or-equals…superset-of-or-equalssubscriptsuperscriptℛ𝓃𝓃2\euscr{F}_{n}=R^{(n)}_{1}\supseteq R^{(n)}_{2}\supseteq...\supseteq R^{(n)}_{%

Tcn⊇S1⊇{*}⊇…⊇{*}superset-of-or-equalssuperscript𝑇subscript𝑐𝑛superscript𝑆1superset-of-or-equalssuperset-of-or-equals…superset-of-or-equalsT^{c_{n}}\supseteq S^{1}\supseteq\{*\}\supseteq...\supseteq\{*\}italic_T start_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⊇ italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⊇ { * } ⊇ … ⊇ { * }

ℱ=∨𝓃−1𝒮1∨𝒸𝓃−(𝓃−1)𝒮1⊇𝒮1⊇{*}⊇…⊇{*}ℱsubscriptsubscript𝒸𝓃𝓃1subscript𝓃1superscript𝒮1superscript𝒮1superset-of-or-equalssuperscript𝒮1superset-of-or-equalssuperset-of-or-equals…superset-of-or-equals\euscr{F}=\vee_{n-1}S^{1}\vee_{c_{n}-(n-1)}S^{1}\supseteq S^{1}\supseteq\{*\}%

ℱ⁢(𝒳)=𝒜1⊇𝒜1∩𝒜3⊇…⊇𝒜1∩𝒜3∩…⁢𝒜2⋅⌊𝓃2⌋−1ℱ𝒳subscript𝒜1superset-of-or-equalssubscript𝒜1subscript𝒜3superset-of-or-equals…superset-of-or-equalssubscript𝒜1subscript𝒜3…subscript𝒜⋅2𝓃21\euscr{F}(X)=A_{1}\supseteq A_{1}\cap A_{3}\supseteq...\supseteq A_{1}\cap A_{%

A

L⁢Uj=2,4,6,⋯𝐿superscript𝑈𝑗246⋯\displaystyle LU^{j=2,4,6,\cdots}italic_L italic_U start_POSTSUPERSCRIPT italic_j = 2 , 4 , 6 , ⋯ end_POSTSUPERSCRIPT

:[S≥0]288→[Sj]288⋅23⁢j+1:absent→subscriptdelimited-[]subscript𝑆absent0288subscriptdelimited-[]subscript𝑆𝑗⋅288superscript23𝑗1\displaystyle:[S_{\geq 0}]_{288}{\color[rgb]{0,0,1}\definecolor[named]{%

:[S≥0]288→[S≥j]288⋅23⁢j:absent→subscriptdelimited-[]subscript𝑆absent0288subscriptdelimited-[]subscript𝑆absent𝑗⋅288superscript23𝑗\displaystyle:[S_{\geq 0}]_{288}{\color[rgb]{0,0,1}\definecolor[named]{%

:[S≥0]288→[Sj]288⋅23⁢j:absent→subscriptdelimited-[]subscript𝑆absent0288subscriptdelimited-[]subscript𝑆𝑗⋅288superscript23𝑗\displaystyle:[S_{\geq 0}]_{288}{\color[rgb]{0,0,1}\definecolor[named]{%

:[S≥0]288→[S≥j]288⋅23⁢j+1:absent→subscriptdelimited-[]subscript𝑆absent0288subscriptdelimited-[]subscript𝑆absent𝑗⋅288superscript23𝑗1\displaystyle:[S_{\geq 0}]_{288}{\color[rgb]{0,0,1}\definecolor[named]{%

C

+Y_{1}&\delta_{2}&\alpha_{1}\delta_{1}\end{pmatrix}\cdot 1=1-1=0.blackboard_b start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = bold_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT { 1 } , { 2 , 3 } end_POSTSUBSCRIPT + bold_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT { 3 } , { 2 , 3 } end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + italic_Y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) ⋅ 1 = 1 - 1 = 0 .

For the A3subscript𝐴3A_{3}italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT case, one can also compute

One can check that the above coefficients 𝕒Iu,IvIwsuperscriptsubscript𝕒subscript𝐼𝑢subscript𝐼𝑣subscript𝐼𝑤\mathbb{a}_{I_{u},I_{v}}^{I_{w}}blackboard_a start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUPERSCRIPT agree with the formula (10).

Consider the A3subscript𝐴3A_{3}italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT-case. Consider Iu=(2,3,1,2,1),Iv=(1,2,3,2,1)formulae-sequencesubscript𝐼𝑢23121subscript𝐼𝑣12321I_{u}=(2,3,1,2,1),I_{v}=(1,2,3,2,1)italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT = ( 2 , 3 , 1 , 2 , 1 ) , italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = ( 1 , 2 , 3 , 2 , 1 ), then 𝕔Iu,IvIw=0subscriptsuperscript𝕔subscript𝐼𝑤subscript𝐼𝑢subscript𝐼𝑣0\mathbb{c}^{I_{w}}_{I_{u},I_{v}}=0blackboard_c start_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 unless w=w0𝑤subscript𝑤0w=w_{0}italic_w = italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the longest element. Fix Iw0=(1,2,3,1,2,1)subscript𝐼subscript𝑤0123121I_{w_{0}}=(1,2,3,1,2,1)italic_I start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ( 1 , 2 , 3 , 1 , 2 , 1 ), in which case we have

Note that when computing 𝕒1,1Iw0subscriptsuperscript𝕒subscript𝐼subscript𝑤011\mathbb{a}^{I_{w_{0}}}_{1,1}blackboard_a start_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT, one needs to compute the following coefficients:

A

\text{(see Prop.\ \ref{prop:octuple})}\end{subarray}start_ARG start_ROW start_CELL Octuple ident. end_CELL end_ROW start_ROW start_CELL (see Prop. ) end_CELL end_ROW end_ARG

\text{ident.\ (Prop.\ \ref{prop:nonuple})}\end{subarray}start_ARG start_ROW start_CELL First nonuple end_CELL end_ROW start_ROW start_CELL ident. (Prop. ) end_CELL end_ROW end_ARG

\text{ident.\ (Prop.\ \ref{prop:nonuple2})}\end{subarray}start_ARG start_ROW start_CELL Second nonuple end_CELL end_ROW start_ROW start_CELL ident. (Prop. ) end_CELL end_ROW end_ARG

First nonupleident. (Prop. 1.1)First nonupleident. (Prop. 1.1)\begin{subarray}{c}\text{First nonuple}\\

Second nonupleident. (Prop. 4.8)Second nonupleident. (Prop. 4.8)\begin{subarray}{c}\text{Second nonuple}\\

C

We say that a variety X𝑋Xitalic_X is ℚℚ\mathbb{Q}blackboard_Q-factorial if for every Weil divisor D⊂X𝐷𝑋D\subset Xitalic_D ⊂ italic_X, there is a positive integer k>0𝑘0k>0italic_k > 0 such that k⁢D𝑘𝐷kDitalic_k italic_D is a Cartier divisor, and for the canonical sheaf ωXsubscript𝜔𝑋\omega_{X}italic_ω start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT, there is a positive integer m>0𝑚0m>0italic_m > 0 such that (ωX⊗m)**superscriptsuperscriptsubscript𝜔𝑋tensor-productabsent𝑚absent(\omega_{X}^{\otimes m})^{**}( italic_ω start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊗ italic_m end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT is a line bundle. It is well known that if X𝑋Xitalic_X is a ℚℚ\mathbb{Q}blackboard_Q-factorial 3-fold and X⇢X′⇢𝑋superscript𝑋′X\dasharrow X^{\prime}italic_X ⇢ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is a flip or a divisorial contraction, then X′superscript𝑋′X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is also ℚℚ\mathbb{Q}blackboard_Q-factorial.

We say that a divisorial sheaf ℒℒ\mathcal{L}caligraphic_L is ℚℚ{\mathbb{Q}}blackboard_Q-Cartier (or a ℚℚ{\mathbb{Q}}blackboard_Q-line bundle) if (ℒ⊗m)**superscriptsuperscriptℒtensor-productabsent𝑚absent(\mathcal{L}^{\otimes m})^{**}( caligraphic_L start_POSTSUPERSCRIPT ⊗ italic_m end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT is a line bundle for some m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N.

If X𝑋Xitalic_X is a complex manifold, then every reflexive rank 1111 sheaf on X𝑋Xitalic_X is a line bundle. In particular, X𝑋Xitalic_X is strongly ℚℚ\mathbb{Q}blackboard_Q-factorial.

We will say that a variety X𝑋Xitalic_X is strongly ℚℚ\mathbb{Q}blackboard_Q-factorial if every divisorial sheaf ℒℒ\mathcal{L}caligraphic_L is a ℚℚ\mathbb{Q}blackboard_Q-line bundle. Note that a complex manifold is an example of a strongly ℚℚ\mathbb{Q}blackboard_Q-factorial variety (see Lemma 2.3).

If X𝑋Xitalic_X is a normal variety then we say that a coherent sheaf ℒℒ\mathcal{L}caligraphic_L is divisorial if it is reflexive of rank 1.

D

Given α𝛼\alphaitalic_α on C𝐶Citalic_C, consider its pull-back p′*⁢αsuperscriptsuperscript𝑝′𝛼{p^{\prime}}^{*}\alphaitalic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_α to Tn−1×[0,∞)superscript𝑇𝑛10T^{n-1}\times[0,\infty)italic_T start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT × [ 0 , ∞ ). We conclude using the multiplicativity under covers of both sides in the inequality in (66).

7 L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-Harmonic Forms on Complex-Hyperbolic Cusps

We now study the growth of L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology on towers of real- and complex-hyperbolic manifolds with cusps.

6 L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-Harmonic Forms on Real-Hyperbolic Cusps

We can now estimate from above the dimension of the space of L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-harmonic forms on a complete finite volume complex-hyperbolic manifold.

A

H⁢(|f^⁢(A)|2)≤K⁢I⁢(f).𝐻superscript^𝑓𝐴2𝐾𝐼𝑓H(|\hat{f}(A)|^{2})\leq KI(f).italic_H ( | over^ start_ARG italic_f end_ARG ( italic_A ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ italic_K italic_I ( italic_f ) .

In version 1 of this note, which can still be found on the ArXiv, we showed that the analogous version of the conjecture for complex functions on {−1,1}nsuperscript11𝑛\{-1,1\}^{n}{ - 1 , 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT which have modulus 1111 fails. This solves a question raised by Gady Kozma some time ago (see [K], comment from April 2, 2011). More specifically, we proved

where for A⊆[n]𝐴delimited-[]𝑛A\subseteq[n]italic_A ⊆ [ italic_n ], |A|𝐴|A|| italic_A | denotes the cardinality of A𝐴Aitalic_A. This object, especially for boolean functions, is a deeply studied one and quite influential (but this is not the reason for the name…) in several directions. We refer to [O] for some information. A recent paper dealing with

Here we give an embarrassingly simple presentation of an example of such a function (although it can be shown to be a version of the example in the previous version of this note). As was written in the previous version, an anonymous referee of version 1 wrote that the theorem was known to experts but not published. Maybe the presentation below is what was known.

For the significance of this conjecture we refer to the original paper [FK], and to Kalai’s blog [K] (embedded in Tao’s blog) which reports on all significant results concerning the conjecture. [KKLMS] establishes a weaker version of the conjecture. Its introduction is also a good source of information on the problem.

D

},s_{i})\to(u_{i+1},s_{i+1})\big{]}\Big{|}\,.| sansserif_Wgt start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_n , roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t → 0 end_POSTSUPERSCRIPT ) - sansserif_Wgt start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) | ≤ ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT | sansserif_Wgt start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT [ ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) → ( italic_u start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ] - sansserif_Wgt start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) → ( italic_u start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ] | .

Now note that, if the segments of ρntsuperscriptsubscript𝜌𝑛𝑡\rho_{n}^{t}italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT and ρn0superscriptsubscript𝜌𝑛0\rho_{n}^{0}italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT between points (ui,si)subscript𝑢𝑖subscript𝑠𝑖(u_{i},s_{i})( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) and (ui+1,si+1)subscript𝑢𝑖1subscript𝑠𝑖1(u_{i+1},s_{i+1})( italic_u start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) pass through a good box, then the corresponding last-displayed summand is small. Thus, we bound the number of bad boxes that ρntsuperscriptsubscript𝜌𝑛𝑡\rho_{n}^{t}italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT passes through. Indeed, finding such an upper bound is a coarse last passage percolation problem with Bernoulli weights—each box becomes a vertex, with weight one if it is bad and weight zero if it is good. The boxes overlap and these assignations are not independent, but this problem is minor and may be addressed by decomposing the system of boxes into suitable disjoint sets. Further, Bernoulli LPP is not integrable, but its Bernoulli variables may be dominated with a Poisson cloud of points, so that a suitably sharp tail bound is offered by the integrable Poissonian LPP model: see Lemma 7.8.

Theorem 6.1: most such boxes are stable in the sense that, uniformly over a class of endpoint pairs that vary rather freely near the base and top of the box, the weight of the interpolating polymer changes little between times zero and t𝑡titalic_t. The boxes’ aspect ratio will be such that, with high probability, polymers starting and ending in the bottom middle and the top middle of a box do not exit the box. If a box satisfies both of these typical properties, we will call it ‘good’ in this overview; otherwise, we will call it ‘bad’.

In the mentioned coupling, we will replace vertices in ℤ2superscriptℤ2\mathbb{Z}^{2}blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT by unit boxes and dominate the Bernoulli variable associated to a vertex by a Poisson cloud of points in the corresponding box. In analysing Poissonian LPP, we will consider oriented paths whose angle with the vertical axis at any point is at most π/4𝜋4\pi/4italic_π / 4. Since the Poisson points are arbitrarily located inside the boxes, we need to separate the boxes a little to ensure that this constraint is satisfied.

We conclude then that no zigzag between (0,0)00(0,0)( 0 , 0 ) and (0,1)01(0,1)( 0 , 1 )—in particular ρntsuperscriptsubscript𝜌𝑛𝑡\rho_{n}^{t}italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT—passes through too many bad boxes. For such a bad box, we bound the corresponding summand by a crude uniform upper bound on the point-to-point weight between two elements in the box. Since the number of such summands is low, the cumulative resulting error is manageable, and we obtain the sought upper bound on |𝖶𝗀𝗍0⁢(ρn,ℓt→0)−𝖶𝗀𝗍t⁢(ρnt)|superscript𝖶𝗀𝗍0superscriptsubscript𝜌𝑛ℓ→𝑡0superscript𝖶𝗀𝗍𝑡superscriptsubscript𝜌𝑛𝑡\Big{|}\mathsf{Wgt}^{0}\big{(}\rho_{n,\ell}^{t\to 0}\big{)}-\mathsf{Wgt}^{t}%

B

Iso⁡(M,g)Iso𝑀𝑔\operatorname{Iso}(M,g)roman_Iso ( italic_M , italic_g ) is virtually a Lie group extension of ΛΛ\Lambdaroman_Λ by a compact Lie group.

Theorem 2.3 is a straightforward consequence of Theorems 2.2 and 1.1, and can be proved following the lines of the deduction of [4, Theorem B] from [4, Theorem A]. For the sake of completness, we provide a proof below.

In [4], this result is essentially obtained as a corollary of Theorem 1.1 and [4, Theorem A], which we recall for convenience:

At the time when [4] was published, Theorem 1.1 had only been stated (and proved) for finitely generated groups, which is enough for Theorem 2.1. However, as outilned in [4], the full version of Theorem 1.1 can be used to prove the following result.

But it is only in 2021 that this point of view was successfully exploited to yield important information on the structure of Iso⁡(Mn+1)Isosubscript𝑀𝑛1\operatorname{Iso}(M_{n+1})roman_Iso ( italic_M start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ). Indeed, in a recent breakthrough, Charles Frances managed to prove the following result (see [4, Theorem B]).

C

Several improvements have been presented, including Floatman–Hormann interpolation [16, 38], that reach better approximation quality than splines.

scale sub-exponential with space dimension. As the numerical experiments in Section 8 suggest, we believe that only non-tensorial grids are able to lift the curse of dimensionality, which requires

whenever the optimal rate for p=2𝑝2p=2italic_p = 2 applies. This can be observed in the numerical experiments presented in the next section.

We demonstrate the numerical stability of such an approach in Section 8.2. But before that, we discuss the approximation power of polynomial interpolation in m𝑚mitalic_mD.

However, all of them share the above weaknesses (A,B,C), as we demonstrate in the numerical experiments of Section 8.

D

In §1 we shall prove our main Theorem 0.1 on smoothability of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-connections with prescribed curvature. In §2 we prove Corollary 0.3 and Proposition 0.4, which are direct consequences of Theorem 0.1. Next, in §3 we discuss the application of our smoothability Theorem 0.1 to W2,psuperscript𝑊2𝑝W^{2,p}italic_W start_POSTSUPERSCRIPT 2 , italic_p end_POSTSUPERSCRIPT-isometric immersions. Finally, we conclude the note with remarks in §4.

1. Smoothability of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-connections

0.1. Smoothability of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-connections

\mathscr{F}})roman_Ω start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT → roman_Ω ∈ fraktur_A start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_M , italic_κ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , script_F ) strongly in Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. Under the assumption of uniqueness,

In §1 we shall prove our main Theorem 0.1 on smoothability of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-connections with prescribed curvature. In §2 we prove Corollary 0.3 and Proposition 0.4, which are direct consequences of Theorem 0.1. Next, in §3 we discuss the application of our smoothability Theorem 0.1 to W2,psuperscript𝑊2𝑝W^{2,p}italic_W start_POSTSUPERSCRIPT 2 , italic_p end_POSTSUPERSCRIPT-isometric immersions. Finally, we conclude the note with remarks in §4.

A

Any permutation polynomial f⁢(x)𝑓𝑥f(x)italic_f ( italic_x ) decomposes the finite field 𝔽qsubscript𝔽𝑞\mathbb{F}_{q}blackboard_F start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT into sets containing mutually exclusive orbits, with the cardinality of each set being equal to the cycle length of the elements in that set. The cycle structure111The cycle structure of a permutation ΣfsubscriptΣ𝑓\Sigma_{f}roman_Σ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT in general is the set containing information about cycle (or orbit) lengths along with their multiplicities. But in this work, we use this term to only denote the orbit lengths of distinct cycles without considering their multiplicity. ΣfsubscriptΣ𝑓\Sigma_{f}roman_Σ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT of a permutation polynomial f⁢(x)𝑓𝑥f(x)italic_f ( italic_x ) is the set of all cycle lengths of the permutation. Computing the cycle structure of permutations represented by PPs is an important problem encountered in cryptography, coding theory, and communication systems [15, 16] with no known efficient algorithm for a general class of PPs. Computing the cycle structure of specific forms of PP is a well-studied problem in the theory of finite fields [17, 18, 19].

Univariate polynomials f⁢(x):𝔽→𝔽:𝑓𝑥→𝔽𝔽f(x):\mathbb{F}\to\mathbb{F}italic_f ( italic_x ) : blackboard_F → blackboard_F that induces a bijection over the field 𝔽𝔽\mathbb{F}blackboard_F are called permutation polynomials (in short, PP) and have been studied extensively in the literature. For instance, given a general polynomial f⁢(x)𝑓𝑥f(x)italic_f ( italic_x ) over 𝔽𝔽\mathbb{F}blackboard_F deciding whether it is a PP is a well-researched problem in literature [10]. Though computational verification of a given polynomial f⁢(x)𝑓𝑥f(x)italic_f ( italic_x ) to be PP is a polynomial time problem in its degree d𝑑ditalic_d, conditions for any polynomial to be PP is well understood only for certain polynomials with specific structures such as monomials, linearized polynomials, and Dickson polynomials, to name a few.

There has been extensive study about a family of polynomial maps defined through a parameter a∈𝔽𝑎𝔽a\in\mathbb{F}italic_a ∈ blackboard_F over finite fields. Some well-studied families of polynomials include the Dickson polynomials and reverse Dickson polynomials, to name a few. Conditions for such families of maps to define a permutation of the field 𝔽𝔽\mathbb{F}blackboard_F are well studied and established for special classes like Dickson polynomials [20], linearized polynomials [21] and few other specific forms [13, 14] to name a few.

Given an n𝑛nitalic_n-dimensional vector space 𝔽nsuperscript𝔽𝑛\mathbb{F}^{n}blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT over finite field 𝔽𝔽\mathbb{F}blackboard_F, maps F:𝔽n→𝔽n:𝐹→superscript𝔽𝑛superscript𝔽𝑛F:\mathbb{F}^{n}\to\mathbb{F}^{n}italic_F : blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT are ubiquitous in the representation of Finite Automata [1, 2], recurrence sequences through Feedback Shift Registers (FSR) [3, 4], mathematical models of Stream Ciphers [5], state updates of Genetic Networks [6, 7, 8, 9] to name a few. In these applications, computations of compositions and inverses of such maps, as well as their representations in polynomials over the finite field 𝔽𝔽\mathbb{F}blackboard_F, play an important role.

The paper primarily addresses the problem of linear representation, invertibility, and construction of the compositional inverse for non-linear maps over finite fields. Though there is vast literature available for invertibility of polynomials and construction of inverses of permutation polynomials over 𝔽𝔽\mathbb{F}blackboard_F, this paper explores a completely new approach using the Koopman operator defined by the iterates of the map. This helps define the linear representation of non-linear maps, which translates non-linear compositions of the map to matrix multiplications. This linear representation naturally defines a notion of linear complexity for non-linear maps, which can be viewed as a measure of computational complexity associated with computations involving such maps. The framework of linear representation is then extended to parameter dependent maps over 𝔽𝔽\mathbb{F}blackboard_F, and the conditions on parametric invertibility of such maps are established, leading to a construction of the parametric inverse map (under composition). It is shown that the framework can be extended to multivariate maps over 𝔽nsuperscript𝔽𝑛\mathbb{F}^{n}blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, and the conditions are established for invertibility of such maps, and the inverse is constructed using the linear representation. Further, the problem of linear representation of the group generated by a finite set of permutation maps over 𝔽nsuperscript𝔽𝑛\mathbb{F}^{n}blackboard_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT under composition is also solved by extending the theory of linear representation of a single map. This leads to the notion of complexity of a group of permutation maps under composition.

B

At the beginning of Sec. 5 we defined turns back and forward and the twist number T=T⁢(σ)𝑇𝑇𝜎T=T(\sigma)italic_T = italic_T ( italic_σ ) of an IRE scheme σ.𝜎\sigma.italic_σ .

The total number of cycles in two mutually dual schemes has the same parity as d.𝑑d.italic_d . The twist total is always an even number.

At the beginning of Sec. 5 we defined turns back and forward and the twist number T=T⁢(σ)𝑇𝑇𝜎T=T(\sigma)italic_T = italic_T ( italic_σ ) of an IRE scheme σ.𝜎\sigma.italic_σ .

For an IRE scheme σ𝜎\sigmaitalic_σ, an instance of σ⁢(α⁢b)=β⁢e,𝜎𝛼b𝛽e\sigma(\alpha{\mathrm{b}})=\beta{\mathrm{e}},italic_σ ( italic_α roman_b ) = italic_β roman_e , α,β∈𝒜,𝛼𝛽𝒜\alpha,\beta\in{\mathcal{A}},italic_α , italic_β ∈ caligraphic_A , is called a turn back at β⁢e,𝛽e\beta{\mathrm{e}},italic_β roman_e , and an instance of σ⁢(β⁢e)=α⁢b,𝜎𝛽e𝛼b\sigma(\beta{\mathrm{e}})=\alpha{\mathrm{b}},italic_σ ( italic_β roman_e ) = italic_α roman_b , α,β∈𝒜,𝛼𝛽𝒜\alpha,\beta\in{\mathcal{A}},italic_α , italic_β ∈ caligraphic_A , is called a turn forward at α⁢b.𝛼b\alpha{\mathrm{b}}.italic_α roman_b . It is obvious that every cycle in the permutation σ𝜎\sigmaitalic_σ contains an equal number ob turns back and turns forward. We call a cycle twisted, if it contains more than one turn back. Define the twist nummber of a cycle as the number of turns back in it minus one. The twist number T=T⁢(σ)𝑇𝑇𝜎T=T(\sigma)italic_T = italic_T ( italic_σ ) of an IRE scheme σ𝜎\sigmaitalic_σ is the sum of the twist numbers of all its cycles; in other words, it is the total number of turns back in this scheme minus the number N=N⁢(σ)𝑁𝑁𝜎N=N(\sigma)italic_N = italic_N ( italic_σ ) of its cycles.

The total number of turns back (as well as turns forward) in two mutually dual IRE schemes is d.𝑑d.italic_d .

D

202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)

202020203030303040404040505050501111333355557777m𝑚mitalic_m (size of the matrix)Normalized objective value

solution found by the algorithms in terms of the normalized objective value (the objective value of the found solution divided by the otimal objective value) for SNR 6 and SNR 1, respectively. As Gurobi’s spatial branch-and-bound algorithm finds certifiably globally optimal solution to (SR), its normalized objective value is always 1, though the runtime is orders of magnitude slower than glmnet and NExOS (see the next paragraph). The closer the normalized objective value is to 1, better is the quality of the solution in terms of minimizing the objective value. We see

202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)

202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)202020203030303040404040505050506060606080808080100100100100m𝑚mitalic_m (size of the matrix)Support recovery (%)

A

\Lambda_{4}^{(l)}|\xi|^{-2}=\frac{1}{2}\operatorname{diag}(1,1,-2)|\xi|^{-2}.italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 , italic_l ) end_POSTSUPERSCRIPT | italic_ξ | start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT - [ italic_N start_POSTSUBSCRIPT 5 , italic_l end_POSTSUBSCRIPT , roman_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT ] | italic_ξ | = : roman_Λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT | italic_ξ | start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_diag ( 1 , 1 , - 2 ) | italic_ξ | start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT .

To sum up, by carrying out five steps of diagonalization procedure, we have derived the diagonal matrix in the final coupled system

In short, by carrying out four steps of diagonalization procedure, we have derived the diagonal matrix in the last coupled system

In the above treatment, we applied the two steps diagonalization procedure (see, for example, [20, Section 4]). Consequently, we derive

In this paper, we are going to derive some large time qualitative properties for the hyperbolic-parabolic coupled system (1.1). We first are going to adapt the diagonalization procedure developed by [36, 33, 37, 17] to the model (1.1), which allows us to derive an asymptotic representation of the energy term W𝑊Witalic_W defined in (2.1). That is the key to generalize the results in [35] for higher dimensions. For another, by reducing the coupled system (1.1) to the higher order evolution equation, we may deduce the sharp representation of the solution u𝑢uitalic_u in the Fourier space, in which the non-summable singularity for small frequencies can be compensated. Finally, by using the WKB analysis and the Fourier analysis, we derive some regularity-loss type decay estimates (especially, the optimal estimates under higher regularities for the Cauchy data) and large time asymptotic profiles for the energy term W𝑊Witalic_W and the solution u𝑢uitalic_u. These profiles consist of the solutions to the biharmonic parabolic equation and the diffusion wave equation with the Riesz potential operator. As a byproduct, motivated by the singular limit process, under some conditions for the initial data, we discover the coupled system (1.1) can be approximated by the wave equation with the Riesz potential dissipation (see Appendix A for its large time behavior). It is a new explanation for the regularity-loss structure of our model (1.1) as well as the role of thermal equation.

C

Setting f⁢(n)𝑓𝑛f(n)italic_f ( italic_n ) equal to the indicator function for S⊆ℕ𝑆ℕS\subseteq\mathbb{N}italic_S ⊆ blackboard_N, then that the first limit in Corollary 1.6 is equal to favg=dSsubscript𝑓avgsubscript𝑑𝑆f_{\operatorname{avg}}=d_{S}italic_f start_POSTSUBSCRIPT roman_avg end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT, re-proves Theorem 3.6 of [8].

where the sum is taken over all partitions, and sm⁡(λ)sm𝜆\operatorname{sm}(\lambda)roman_sm ( italic_λ ) denotes the smallest part of λ∈𝒫𝜆𝒫\lambda\in\mathcal{P}italic_λ ∈ caligraphic_P.

Somewhat surprisingly, if one replaces “smsm\operatorname{sm}roman_sm” with “lglg\operatorname{lg}roman_lg” in Theorem 1.4, the limit still holds.

For each k≥1𝑘1k\geq 1italic_k ≥ 1, the factor (q;q)k−1−1superscriptsubscript𝑞𝑞𝑘11(q;q)_{k-1}^{-1}( italic_q ; italic_q ) start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT generates partitions with largest part strictly <kabsent𝑘<k< italic_k. The factor qn⁢ksuperscript𝑞𝑛𝑘q^{nk}italic_q start_POSTSUPERSCRIPT italic_n italic_k end_POSTSUPERSCRIPT adjoins a largest part k𝑘kitalic_k with multiplicity n𝑛nitalic_n to each partition, for every n≥1𝑛1n\geq 1italic_n ≥ 1. The qk⁢(k−1)/2=q1+2+3+…+(k−1)superscript𝑞𝑘𝑘12superscript𝑞123…𝑘1q^{k(k-1)/2}=q^{1+2+3+...+(k-1)}italic_q start_POSTSUPERSCRIPT italic_k ( italic_k - 1 ) / 2 end_POSTSUPERSCRIPT = italic_q start_POSTSUPERSCRIPT 1 + 2 + 3 + … + ( italic_k - 1 ) end_POSTSUPERSCRIPT factor guarantees at least one part of each size <kabsent𝑘<k< italic_k. Thus (10) is the generating function for partitions γ𝛾\gammaitalic_γ with every natural number <lg⁡(γ)absentlg𝛾<\operatorname{lg}(\gamma)< roman_lg ( italic_γ ) appearing as a part, weighted by (−1)lg⁡(γ)⁢f⁢(mlg⁢(γ))=(−1)k⁢f⁢(n)superscript1lg𝛾𝑓subscript𝑚lg𝛾superscript1𝑘𝑓𝑛(-1)^{\operatorname{lg}(\gamma)}f\left(m_{\operatorname{lg}}(\gamma)\right)=(-%

Let 𝒫𝒫\mathcal{P}caligraphic_P denote the integer partitions [3]. For λ∈𝒫𝜆𝒫\lambda\in\mathcal{P}italic_λ ∈ caligraphic_P, let |λ|𝜆|\lambda|| italic_λ | denote the size of λ𝜆\lambdaitalic_λ (sum of parts), ℓ⁢(λ)ℓ𝜆\ell(\lambda)roman_ℓ ( italic_λ ) denote the length (number of parts), and let sm⁡(λ)sm𝜆\operatorname{sm}(\lambda)roman_sm ( italic_λ ) and lg⁡(λ)lg𝜆\operatorname{lg}(\lambda)roman_lg ( italic_λ ) denote the smallest part and largest part of λ𝜆\lambdaitalic_λ, respectively, noting |∅|=ℓ⁢(∅)=sm⁡(∅)=lg⁡(∅):=0ℓsmlgassign0|\emptyset|=\ell(\emptyset)=\operatorname{sm}(\emptyset)=\operatorname{lg}(%

B

\in\mathcal{D}\}caligraphic_C × caligraphic_D ≔ { italic_P × italic_Q : italic_P ∈ caligraphic_C , italic_Q ∈ caligraphic_D } is a local covering of the product topology.

Let P𝑃Pitalic_P and Q𝑄Qitalic_Q be two countably piecewise hyperdefinable sets with global logic topologies. Then, P×Q𝑃𝑄P\times Qitalic_P × italic_Q is a countably piecewise hyperdefinable set and its product topology is its global logic topology.

By Proposition 1.34(1), Cartesian projection maps are continuous between the logic topologies. Therefore, the A𝐴Aitalic_A\hyplogic topology of a finite product of piecewise A𝐴Aitalic_A\hyphyperdefinable sets is at least as fine as the product topology of the A𝐴Aitalic_A\hyplogic topologies. In the case of local hyperdefinable sets with global logic topologies, we have that they coincide.

Similarly, in the case of countably piecewise hyperdefinable sets with global logic topologies, we also conclude that they coincide.

Let P𝑃Pitalic_P be a countably piecewise A𝐴Aitalic_A\hyphyperdefinable set. Then, P𝑃Pitalic_P is normal with the A𝐴Aitalic_A\hyplogic topology. In particular, global logic topologies of countably piecewise hyperdefinable sets are Hausdorff.

C

}/X))\|∥ script_F ( script_L / { italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } ) ∥ = ∥ script_F ( script_L / italic_X ) ) ∥ and rk⁡(X)=rk⁡({e1,…,em})rk𝑋rksubscript𝑒1…subscript𝑒𝑚\operatorname{rk}(X)=\operatorname{rk}(\{e_{1},\ldots,e_{m}\})roman_rk ( italic_X ) = roman_rk ( { italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } ).

The poset of connected components of intersections of 𝒜αsubscript𝒜𝛼\mathscr{A}_{\alpha}script_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is isomorphic to the quotient poset F⁢(ℒ)/G𝐹ℒ𝐺{F}(\mathscr{L})/Gitalic_F ( script_L ) / italic_G. Moreover, every intersection is topologically a torus (S1)d−rsuperscriptsuperscript𝑆1𝑑𝑟(S^{1})^{d-r}( italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_d - italic_r end_POSTSUPERSCRIPT where r𝑟ritalic_r is the rank of the corresponding element in F⁢(ℒ)/G𝐹ℒ𝐺{F}(\mathscr{L})/Gitalic_F ( script_L ) / italic_G.

Comparing with the definition of F⁢(ℒ)/G𝐹ℒ𝐺{F}(\mathscr{L})/Gitalic_F ( script_L ) / italic_G gives the claim.

Now, α𝛼\alphaitalic_α is translative by Lemma 5.5, and by Theorem A.16 the characteristic polynomial of the poset F⁢(ℒ)/G𝐹ℒ𝐺F(\mathscr{L})/Gitalic_F ( script_L ) / italic_G equals (−1)d⁢Tα⁢(1−t,0)superscript1𝑑subscript𝑇𝛼1𝑡0(-1)^{d}T_{\alpha}(1-t,0)( - 1 ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( 1 - italic_t , 0 ), thus the claim follows.

The goal is to study the homeomorphism type of covector posets of FAOMs by comparing them with those of restrictions to unions of parallelism classes of elements of a basis.

B

For example, the Chow and K0superscriptK0\mathrm{K}^{0}roman_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-motives with integral and finite coefficients are nowadays a common tool

One of the most striking applications of motives was Voevodsky’s proof [Vo03, OVV] of the Milnor conjecture which relies on Rost’s computation [Ros] of

The main result of the present article is a computation of the ring of the Morava K-theory for the prime 2

The Rost nilpotence principle is proven for the Chow motives of projective quadrics by Rost in [Ros]. Alternative proofs of this result are given in [Vi98] and [Br03]. The Rost nilpotence principle for Chow motives of projective homogeneous varieties is proven in [CGM] and [Br05], for generically split Chow motives of smooth projective varieties in [ViZa], and for Chow motives of surfaces in [Gi10, Gi14]. By [ViYa, Corollary 2.8] the Rost nilpotence principle holds in these cases for algebraic cobordism or connective K-theory motives as well.

the motive of a Pfister quadric. More generally, the structure of Chow motives of norm varieties plays a crucial role in the proof of the Bloch–Kato conjecture by Rost and Voevodsky (see [Vo11]).

A

Additionally, the performance of these algorithms is highly sensitive to hyperparameters and requires careful selection for optimum results in each experiment.

Graph temporal logic (GTL) is introduced in [16] to impose high-level task specifications as a constraint to the Markov chain synthesis. Markov chain synthesis is formulated as mixed-integer nonlinear programming (MINLP) feasibility problem and the problem is solved using a coordinate descent algorithm. In addition, an equivalence is proven between the feasibility of the MINLP and the feasibility of a mixed-integer linear program (MILP) for a particular case where the agents move along the nodes of a complete graph. While this study assumes homogeneous swarms for Markov chain synthesis subject to finite-horizon GTL formulas, an improved version of the formulation is presented in [17] to enable probabilistic control of heterogeneous swarms subject to infinite-horizon GTL formulas. Instead of solving the resulting MINLP using a coordinate descent algorithm, a sequential scheme, which is faster, more accurate, and robust to the choice of the starting point, is developed in the aforementioned paper.

The shortest-path algorithm is used for the synthesis of the Markov chain for the transient states while the Markov chain for the recurrent states is synthesized by the DSMC algorithm.

For the fastest mixing Markov chain synthesis, the problem is formulated as a convex optimization problem in [5], assuming that the Markov chain is symmetric. This paper also presents an extension to the method that involves synthesizing the fastest mixing reversible Markov chain with a given desired distribution. Furthermore, the number of variables in the optimization problem is reduced in [6] by exploiting the symmetries in the graph.

The synthesis of the Markov matrix using the DSMC algorithm and the shortest-path algorithm for the recurrent and transient states of the desired distribution is given as {dmath*} M[i,j] =

A

Then there is a Weinstein embedding Xd→2⁢n⁢↪𝑊⁢Xd′→2⁢nsubscriptsuperscript𝑋2𝑛→𝑑𝑊↪subscriptsuperscript𝑋2𝑛→superscript𝑑′X^{2n}_{\vec{d}}\overset{W}{\hookrightarrow}X^{2n}_{\vec{d^{\prime}}}italic_X start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over→ start_ARG italic_d end_ARG end_POSTSUBSCRIPT overitalic_W start_ARG ↪ end_ARG italic_X start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over→ start_ARG italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG end_POSTSUBSCRIPT if and only if d→⪯d′→precedes-or-equals→𝑑→superscript𝑑′\vec{d}\preceq\vec{d^{\prime}}over→ start_ARG italic_d end_ARG ⪯ over→ start_ARG italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG.

Conversely, any manifold X𝑋Xitalic_X of dimension at least six which admits a nondegenerate two-form and an exhausting Morse function with critical points of index at most half the ambient dimension is diffeomorphic to a Weinstein manifold (see [CE, Thm. 13.2]).

Hence, we intuitively view a more general Weinstein domain as the cotangent disk bundle of its singular skeleton, and Liouville embeddings of Weinstein domains as singular generalizations of exact Lagrangian embeddings.

Note that Theorem 1.11 obstructs Liouville embeddings, hence a fortiori Weinstein embeddings. Since the constructions provided by Theorem 1.15 are Weinstein embeddings,

We note, however, that Weinstein embeddings have more restricted topology compared to Liouville embeddings. Namely, the complementary cobordism must admit a Morse function with all critical points having index at most half the dimension (see e.g. §2.1). Consequently, many of the obstructions involved in Corollary 1.17 follow simply from singular homology considerations (c.f. Remark 6.4).

D

[durrRealisticTheoryQuantum1992, samolsStochasticModelQuantum1995, berndlNonlocalityLorentzInvariance1996, durrHypersurfaceBohmDiracModels1999]

theory of quantum mechanics (QM) to the (special-)relativistic setting.333Recall that Maxwell’s electromagnetism

[bornZurQuantenmechanikII1926, diracQuantumTheoryEmission1927, wallerStreuungKurzwelligerStrahlung1929, jordanZurQuantenmechanikGasentartung1927, jordanMehrkoerperproblemQuantentheorie1927, jordanUberPaulischeAquivalenzverbot1928, fermiSopraElettrodinamicaQuantistica1929, fermiSopraElettrodinamicaQuantistica1930, weisskopfBerechnungNatuerlichenLinienbreite1930, heisenbergZurQuantendynamikWellenfelder1929, heisenbergZurQuantentheorieWellenfelder1930]

[nafeHyperfineStructureAtomic1947, lambFineStructureHydrogen1947, nafeHyperfineStructureHydrogen1948, kuschPrecisionMeasurementRatio1947, foleyIntrinsicMomentElectron1948], that created the need for an

[takabayasiRemarksFormulationQuantum1953, hollandSymmetriesConservationLaws2012, poirierBohmianMechanicsPilot2010]

D

(for any λ∈ℝ∖{0}𝜆ℝ0\lambda\in\mathbb{R}\setminus\{0\}italic_λ ∈ blackboard_R ∖ { 0 } and K>0𝐾0K>0italic_K > 0).

and thus are incompatible with the Wick-ordered L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff.

RNsubscript𝑅𝑁R_{N}italic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT is as in (1.8) with λ∈ℝ∖{0}𝜆ℝ0\lambda\in\mathbb{R}\setminus\{0\}italic_λ ∈ blackboard_R ∖ { 0 } and k=3𝑘3k=3italic_k = 3.

L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff) for the Benjamin-Ono equation (1.19) with k=3𝑘3k=3italic_k = 3.

focusing Gibbs measure with an L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff:

C

The result implies that for a convex PL-function ℓℓ\ellroman_ℓ supported on 𝒩𝒩\mathcal{N}caligraphic_N,

For a PL-function ℓ∈PL⁢(𝒩)ℓPL𝒩\ell\in\mathrm{PL}(\mathcal{N})roman_ℓ ∈ roman_PL ( caligraphic_N ) we will write λ⁢(ℓ)𝜆ℓ\lambda(\ell)italic_λ ( roman_ℓ ) for the

such that λ⁢(ℓ)+μ⁢λ⁢(hP)>0𝜆ℓ𝜇𝜆subscriptℎ𝑃0\lambda(\ell)+\mu\lambda(h_{P})>0italic_λ ( roman_ℓ ) + italic_μ italic_λ ( italic_h start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ) > 0. That is, the function

λ=λ⁢(ℓ)𝜆𝜆ℓ\lambda=\lambda(\ell)italic_λ = italic_λ ( roman_ℓ ). Denote the connected components of the

In particular, λ=λ⁢(q0)𝜆𝜆subscript𝑞0\lambda=\lambda(q_{0})italic_λ = italic_λ ( italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) for some q0subscript𝑞0q_{0}italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT if and only if

C

\gamma_{2}^{*}E\leq\pi^{\prime*}(\mathcal{Y}^{\prime}).italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_C ≤ italic_π start_POSTSUPERSCRIPT ′ ∗ end_POSTSUPERSCRIPT ( caligraphic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E ≤ italic_π start_POSTSUPERSCRIPT ′ ∗ end_POSTSUPERSCRIPT ( caligraphic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .

as C𝐶Citalic_C and E𝐸Eitalic_E are closed subschemes of the scheme-theoretic union C∪′Esuperscript′𝐶𝐸C\cup^{\prime}Eitalic_C ∪ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_E. As closed immersions are stable under base change we get that there are closed immersions

By [gortz2010algebraic, Corollary 11.49], the two conditions can be reformulated to that there are closed immersions for the underlying scheme structures

A morphism X→S→𝑋𝑆X\to Sitalic_X → italic_S of scheme is called projective if there is a closed immersion X↪ℙSnabsent↪𝑋subscriptsuperscriptℙ𝑛𝑆X\xhookrightarrow{}\mathbb{P}^{n}_{S}italic_X start_ARROW start_OVERACCENT end_OVERACCENT ↪ end_ARROW blackboard_P start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT over S𝑆Sitalic_S for some n≥0𝑛0n\geq 0italic_n ≥ 0. (See [gortz2010algebraic, Summary 13.71] for the comparison of different conventions.)

where the right hand side is the scheme-theoretic inverse image. Hence we obtain that there exist closed immersions

B

In the sequel, for any functional F:𝒫2⁢(𝒳)→ℝ:𝐹→subscript𝒫2𝒳ℝF\colon\mathcal{P}_{2}(\mathcal{X})\rightarrow\mathbb{R}italic_F : caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( caligraphic_X ) → blackboard_R, we let grad⁡Fgrad𝐹\operatorname{{\mathrm{grad}}}Froman_grad italic_F denote the the gradient of F𝐹Fitalic_F with respect to W2subscript𝑊2W_{2}italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

Here the statistical error is incurred in estimating the Wasserstein gradient by solving the dual maximization problem using functions in a reproducing kernel Hilbert space (RKHS) with finite data, which converges sublinearly to zero as the number of particles goes to infinity.

To study optimization problems on the space of probability measures, we first introduce the background knowledge of the Riemannian manifold and the Wasserstein space. In addition, to analyze the statistical estimation problem that arises in estimating the Wasserstein gradient, we introduce the reproducing kernel Hilbert space.

Another important tool we need in the following analysis is the reproducing kernel Hilbert space (RKHS).

In addition to the above assumptions, we impose the following regularity condition on the RKHS kernel K𝐾Kitalic_K.

C

In the rest of this section, let us fix h∉ℚℎℚh\notin\mathbb{Q}italic_h ∉ blackboard_Q and u∈𝔥reg𝑢subscript𝔥regu\in\mathfrak{h}_{\rm reg}italic_u ∈ fraktur_h start_POSTSUBSCRIPT roman_reg end_POSTSUBSCRIPT.

In Section 2.1-2.3, we introduce the unique formal solution, the canonical solutions and Stokes matrices of the equation (1) respectively.

The radius of convergence of the formal power series H^⁢(z)^𝐻𝑧\hat{H}(z)over^ start_ARG italic_H end_ARG ( italic_z ) is in general zero. However, it follows from the general principle of differential equations with irregular singularities that (see e.g., [6, 44, 48] or the proof of Proposition 6.3) the Borel resummation (Borel-Laplace transform) of H^^𝐻\hat{H}over^ start_ARG italic_H end_ARG gives a holomorphic function in each Stokes supersector around z=∞𝑧z=\inftyitalic_z = ∞. In this way, one gets actual solutions of (23) on each Stokes supersector. These sectors are determined by the irregular term h⁢uℎ𝑢huitalic_h italic_u of the differential equation as follows.

the other sectors. Then the following theorem follows from the general principle of differential equations with irregular singularities (see e.g., [6, 44, 48] or the proof of Proposition 6.3).

Similar to Section 2.2, the following definitions and theorem follow from the general principle of the Borel resummation of the formal solutions of the meromorphic differential equations.

D

Let L≤Q𝐿𝑄L\leq Qitalic_L ≤ italic_Q be a Levi subgroup of Q𝑄Qitalic_Q, such that the centre A𝐴Aitalic_A of L𝐿Litalic_L is contained in P𝑃Pitalic_P. By Lemma 6.2, Λ⊂LΛ𝐿\Lambda\subset Lroman_Λ ⊂ italic_L almost surely. Since ΛΛ\Lambdaroman_Λ is P𝑃Pitalic_P-invariant and Q=L⁢P𝑄𝐿𝑃Q=LPitalic_Q = italic_L italic_P we can upgrade this inclusion to Λ⊂⋂p∈PLp=⋂q∈QLqΛsubscript𝑝𝑃superscript𝐿𝑝subscript𝑞𝑄superscript𝐿𝑞\Lambda\subset\bigcap_{p\in P}L^{p}=\bigcap_{q\in Q}L^{q}roman_Λ ⊂ ⋂ start_POSTSUBSCRIPT italic_p ∈ italic_P end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT = ⋂ start_POSTSUBSCRIPT italic_q ∈ italic_Q end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT. Let J𝐽Jitalic_J be the intersection of all Levi subgroups of Q𝑄Qitalic_Q. The set of real points of Q𝑄Qitalic_Q is Zariski dense so J𝐽Jitalic_J coincides with the real points of the intersection of all complex Levi subgroups of Q⁢(ℂ)𝑄ℂQ(\mathbb{C})italic_Q ( blackboard_C ). By Lemma 6.3 we deduce that ΛΛ\Lambdaroman_Λ is supported on Subd⁢(H)subscriptSub𝑑𝐻\text{Sub}_{d}(H)Sub start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_H ) where H=⋂q∈QLq𝐻subscript𝑞𝑄superscript𝐿𝑞H=\bigcap_{q\in Q}L^{q}italic_H = ⋂ start_POSTSUBSCRIPT italic_q ∈ italic_Q end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT is a proper semisimple factor of G𝐺Gitalic_G almost surely.

The following results (see Theorem 4.1 and Corollary 4.4 below) generalize to invariant random subgroups the classical decomposition-to-irreducible-factors theorem for lattices in semisimple groups (see [Rag89, Theorem 5.22]):

For semisimple Lie groups with Kazhdan’s property (T) we deduce the following generalization of the Stuck–Zimmer theorem for discrete stationary random subgroups:

The following decomposition theorem generalizes Theorem 4.1 from invariant random subgroups to discrete stationary random subgroups:

It is obvious that a limit of stationary measures is stationary and it follows from Theorem 3.3 that a limit of discrete stationary random subgroups is also discrete.

C

{a}}(x)}\|f\|_{\infty}.∥ italic_f ( italic_x ) - italic_v start_POSTSUBSCRIPT a end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ⩽ divide start_ARG square-root start_ARG italic_C italic_N ( italic_δ - italic_σ ) end_ARG end_ARG start_ARG italic_w start_POSTSUBSCRIPT a end_POSTSUBSCRIPT ( italic_x ) end_ARG ∥ italic_f ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT .

Using the definition of the weights wa,σsubscript𝑤a𝜎w_{\textbf{a},\sigma}italic_w start_POSTSUBSCRIPT a , italic_σ end_POSTSUBSCRIPT in (38) together with the distortion estimates from Proposition 2.4, we obtain

The uniform hyperbolicity property in Proposition 2.4 implies that for all a∈WNj,asuperscriptsubscript𝑊𝑁𝑗\textbf{a}\in W_{N}^{j},a ∈ italic_W start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ,

We will drop the subscripts from the inner product and from the norm whenever it is clear which space is being referred to.

It remains to bound the weights from below. From their definition in (38) and from the uniform hyperbolicity property in Proposition 2.4 it follows that

D

   >> parameter = {A,S};   % parameters of the mapping g

   >> [z,res,fcond] = ...   % Call Newton on the mapping g with J in rank-9 projection from

   >> [Z,res,fcd] = Newton({g,{ones(4,1),ones(4,1)},{F,K,x,y}}, {gjac,7},{u0,v0}, 1)   % rank-7 Newton

   >> [z,res,fcond] = Newton({f,domain,parameter},{fjac,3},z0,1);   % call rank-3 Newton iteration on f

   >> [z,res,fcond] = Newton({f,domain,parameter},{J,8},{u0,v0,w0},1);   % Newton on f with J in

A

\rm ent}(\Gamma_{U},S)roman_ent ( italic_X ) ≥ divide start_ARG 1 end_ARG start_ARG 2 roman_diam start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_U ) + italic_ε end_ARG roman_ent ( roman_Γ start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT , italic_S )

We can now prove the following result complementing Corollary 2.11 under some mild combinatorial assumptions.

Then, relying on the notion of Urysohn width, we show that the minimal volume entropy of simplicial complexes satisfying the covering non-collapsing assumption and some mild combinatorial conditions is positive.

We can now prove the following result complementing Theorem 2.10 when the fundamental group is thick.

Since the subexponential growth rate of a group with polynomial growth is zero, we immediately derive the following corollary.

A

Hence by [19, Prp. 3.16 and Remark 15, part (ii)] the penalty function ΦcsubscriptΦ𝑐\Phi_{c}roman_Φ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is globally

Note that by Corollary 2 the functional ℐℐ\mathcal{I}caligraphic_I is Lipschitz continuous on bounded sets in

Under the assumptions of the theorem the functional ℐℐ\mathcal{I}caligraphic_I is Lipschitz continuous on bounded sets by

X𝑋Xitalic_X. Therefore by [20, Corollary 2] the functional ℐℐ\mathcal{I}caligraphic_I is Lipschitz continuous on any

(x,y)∈X𝑥𝑦𝑋(x,y)\in X( italic_x , italic_y ) ∈ italic_X in the case p=+∞𝑝p=+\inftyitalic_p = + ∞. Suppose also that the functional ℐℐ\mathcal{I}caligraphic_I is Lipschitz continuous on

A

In practice, not all nodes that enter the network have the same degree, and thus it would be interesting to extend our result to the case of a random initial degree distribution. Promising results on this model have been obtained in [7, 8]. Moreover, in this paper we assume that the parameters of the model are known, but in many practical situations one is given a realization of the graph and the task is estimating the unknown parameters, see [10, 20, 21]. If we consider a more general class of preferential attachment graphs, for which a model-free approach is used and therefore the exact distribution of the graph is not known (see for instance [11]), we expect that the techniques presented in this paper could be used to derive central limit theorem for all the degree counts. This is an interesting open problem.

Here Γ⁢(⋅)Γ⋅{\Gamma}(\cdot)roman_Γ ( ⋅ ) is the Gamma function. When the graph size goes to infinity, the probability that a new vertex forms a self-loop tends to zero and thus it easy to check that (2.3) and (2.4) hold still for our model without self-loops following the proof proposed in [19, Section 8.6]. Here we choose to update the degrees during the attachment of a new vertex, but it is possible to consider also the case in which we update the degrees of the vertices only when the m𝑚mitalic_m-th edge is added. In this case, after constructing a suitable martingale with respect to the filtration (ℱs)s≥1subscriptsubscriptℱ𝑠𝑠1({\cal F}_{s})_{s\geq 1}( caligraphic_F start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_s ≥ 1 end_POSTSUBSCRIPT generated by the construction of the preferential attachment graph until time s𝑠sitalic_s, it is possible to reproduce the same computations. Thus we are able to prove a similar result for this model by using the techniques presented here.

In practice, not all nodes that enter the network have the same degree, and thus it would be interesting to extend our result to the case of a random initial degree distribution. Promising results on this model have been obtained in [7, 8]. Moreover, in this paper we assume that the parameters of the model are known, but in many practical situations one is given a realization of the graph and the task is estimating the unknown parameters, see [10, 20, 21]. If we consider a more general class of preferential attachment graphs, for which a model-free approach is used and therefore the exact distribution of the graph is not known (see for instance [11]), we expect that the techniques presented in this paper could be used to derive central limit theorem for all the degree counts. This is an interesting open problem.

Here, we consider the PA model without self-loops described for example in [8]. In particular, each graph in the sequence is connected. We make this choice in order to simplify the calculations, but we believe that our result also holds, for example, for the PA model with self-loops considered in the standard reference [19, Chapter 8]. We argue this in more detail later on. We assume that each new node has a fixed number m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N of edges attached to it. This is a particular case of the model considered in [8], where m𝑚mitalic_m is a random variable and is sampled for each new node. Our main result is a central limit theorem for the proportion of nodes with a given degree. In fact, we prove this jointly for all degree counts. In particular, we give an explicit expression for the asymptotic covariance. The first results concerning the study of the asymptotic normality of degree counts in the preferiantal attachment models without self-loops is given in [13] by using martingale central limit theorems. Our results generalize those obtained in [17] for the preferential attachment tree. More precisely, in [17] the authors consider a PA model with self-loops and such that m=1𝑚1m=1italic_m = 1. Note that here we consider the PA model with m≥1𝑚1m\geq 1italic_m ≥ 1 and without self-loops. However, this does not influence the asymptotic behavior of the degree counts, since as the graph size goes to infinity, the probability that a new vertex forms a self-loop tends to zero. Because of this one would expect to recover the results in [17] when plugging m=1𝑚1m=1italic_m = 1 in our result. Indeed this is the case if one takes into account a few minor mistakes in [17] which we will discuss later. Note that a major difference between the two models is the resulting connectivity structure. Our model produces a connected graph with probability 1111 (w.p.1111), while the model in [17] is disconnected w.p.1111. However, this does not play a role in the distribution of the degree counts. In [15] the authors studied the joint degree counts in linear preferential attachment random graphs. The results are stated in terms of weight of vertices, but they can be tought of as degree, since each time a vertex receives a new edge, its weight increases by one. The main difference between our and their model is that we consider the attachment probabilities proportional to a linear function of the degree of an old vertex (see (2.1)), while in [15] they are proportional to the degree of an old vertex.

The outline of the paper is as follows: in Section 2 we define the model rigorously and we state our main result. In Section 3 we give the proof of our main result.

D

Note that the above procedure do not communicate on variables pisubscript𝑝𝑖p_{i}italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and qisubscript𝑞𝑖q_{i}italic_q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, because they are local and unique for each node, in contrast with variables x𝑥xitalic_x and y𝑦yitalic_y. It is easy to check that Algorithm 1 also satisfies the definition of the procedure above.

The global function f⁢(x,y)𝑓𝑥𝑦f(x,y)italic_f ( italic_x , italic_y ) in (66) is exactly an example of a bilinear function from the work on non-distributed lower bounds for the strongly convex-strongly concave case [60], but with strong convexity and strong concavity constants μ∼εR2similar-to𝜇𝜀superscript𝑅2\mu\sim\tfrac{\varepsilon}{R^{2}}italic_μ ∼ divide start_ARG italic_ε end_ARG start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (see (9) in Appendix 9). In fact, if we write the dual function for this f⁢(x,y)𝑓𝑥𝑦f(x,y)italic_f ( italic_x , italic_y ) from (9), we have exactly the problem from [46], but with a smoothness constant ∼L2μsimilar-toabsentsuperscript𝐿2𝜇\sim\tfrac{L^{2}}{\mu}∼ divide start_ARG italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_μ end_ARG (see Lemma 9.2 in Appendix 9). The problem (66) is decomposed into summands fisubscript𝑓𝑖f_{i}italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in such a way that if we want to get closer to the solution, we need to communicate (see also [51]). This idea is explained in more detail in Appendix 9. Here we only give a simplified version of the main theorem.

The idea of proving any lower bounds for optimization problems comes from the first results on lower bounds for convex minimization [46]. The essence is to give an example of a ”bad” function.

We provide the lower bounds on the number of communication steps and oracle calls per node for convex-concave SPPs

In this section we present results on lower bounds. In obtaining them, we focus on the Euclidean case. Before proceeding directly to the lower bounds, we need to identify the class of algorithms for which they are valid.

B

The linearization based relaxation finds the best possible linearizable matrix for the given QMSTP instance, by solving a linear programming (LP) problem.

linearizable if and only if it is a symmetric weak sum matrix. In particular, the authors show that for a symmetric weak sum matrix

In particular, it searches for the best under-estimator of the quadratic objective function that is in the form of a weak sum matrix.

is a linearization vector. Thus, solving the QMSTP in which the cost matrix is a symmetric weak sum matrix

This implies that the optimal solution of the dual provides the best skew symmetric matrix that is added to the quadratic cost to improve the GL type bound.

B

The Chow ring CH∙⁡(X,R)superscriptCH∙𝑋𝑅\operatorname{CH}^{\bullet}(X,R)roman_CH start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( italic_X , italic_R ) for any ring R𝑅Ritalic_R.

Moreover, the mixed Hodge structures of the moduli spaces are pure and if d′′superscript𝑑′′d^{\prime\prime}italic_d start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT is any integer coprime with r𝑟ritalic_r and (ξ′′,δ′′)∈ℳℒ′⁢(1,d′′)superscript𝜉′′superscript𝛿′′subscriptℳsuperscriptℒ′1superscript𝑑′′(\xi^{\prime\prime},\delta^{\prime\prime})\in{\mathcal{M}}_{{\mathcal{L}}^{%

,d)][ caligraphic_M start_POSTSUBSCRIPT caligraphic_L end_POSTSUBSCRIPT ( italic_r , italic_d ) ] = [ caligraphic_M start_POSTSUBSCRIPT caligraphic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_r , italic_d ) ] in K^⁢(𝒱⁢a⁢rℂ)^𝐾𝒱𝑎subscript𝑟ℂ\hat{K}(\mathcal{V}ar_{\mathbb{C}})over^ start_ARG italic_K end_ARG ( caligraphic_V italic_a italic_r start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ). Moreover, if d′superscript𝑑′d^{\prime}italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is any integer coprime with r𝑟ritalic_r, then E⁢(ℳℒ⁢(r,d))=E⁢(ℳℒ′⁢(r,d′))𝐸subscriptℳℒ𝑟𝑑𝐸subscriptℳsuperscriptℒ′𝑟superscript𝑑′E({\mathcal{M}}_{{\mathcal{L}}}(r,d))=E({\mathcal{M}}_{{\mathcal{L}}^{\prime}}%

By Theorem 5.7, the motive of the moduli space is independent of the Lie algebroid structure. Thus, if L𝐿Litalic_L and L′superscript𝐿′L^{\prime}italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are the underlying line bundles to ℒℒ{\mathcal{L}}caligraphic_L and ℒ′superscriptℒ′{\mathcal{L}}^{\prime}caligraphic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT respectively and d𝑑ditalic_d and d′superscript𝑑′d^{\prime}italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are coprime with r𝑟ritalic_r, then

Moreover, the mixed Hodge structures of the moduli spaces are pure and, if d′superscript𝑑′d^{\prime}italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is any integer coprime with r𝑟ritalic_r, then

D

{c↔d}∪{ce↔de:e∈E−Ts−T}.\{c\leftrightarrow d\}\cup\{c_{e}\leftrightarrow d_{e}:e\in E-T_{s}-T\}.{ italic_c ↔ italic_d } ∪ { italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ↔ italic_d start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT : italic_e ∈ italic_E - italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - italic_T } .

{c↔d}∪{ce↔de:e∈E−Ts−T}.\{c\leftrightarrow d\}\cup\{c_{e}\leftrightarrow d_{e}:e\in E-T_{s}-T\}.{ italic_c ↔ italic_d } ∪ { italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ↔ italic_d start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT : italic_e ∈ italic_E - italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - italic_T } .

Case 1: the pair induced by the edge replacement, {c↔d}↔𝑐𝑑\{c\leftrightarrow d\}{ italic_c ↔ italic_d }.

c′∈∩T(c)superscript𝑐′subscript𝑇𝑐c^{\prime}\in\cap_{T}(c)italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ ∩ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_c ) to be a tree-cycle induced by a cycle-edge

or to w𝑤witalic_w and ce↔de↔subscript𝑐𝑒subscript𝑑𝑒c_{e}\leftrightarrow d_{e}italic_c start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ↔ italic_d start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT its corresponding pair of

B

Can Proposition 2.4 be strengthened by adding additional constraints on X𝑋Xitalic_X, for example if X𝑋Xitalic_X has stationary increments?

The measure μ∗superscript𝜇normal-∗\mu^{\ast}italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT in Theorem 2.1 should be thought of as the distribution of the location of the minima under the limit u→∞normal-→𝑢u\to\inftyitalic_u → ∞. For an example of a precise statement, see [19] Theorem 4.1.

The assumption that γ⁢(t)𝛾𝑡\gamma(t)italic_γ ( italic_t ) has no minima on (0,h)0ℎ(0,h)( 0 , italic_h ) in Theorem 2.12 can be verified numerically easily. It is however difficult to verify in generality. See the figures at the end of the article for numerical verification.

Can the assumption that γ⁢(t)𝛾𝑡\gamma(t)italic_γ ( italic_t ) has no minima on (0,h)0ℎ(0,h)( 0 , italic_h ) in Theorem 2.12 be verified in generality under Assumption 2.8?

By assumption, we can verify that ϕ⁢(t)italic-ϕ𝑡\phi(t)italic_ϕ ( italic_t ) has no minima on (0,h)0ℎ(0,h)( 0 , italic_h ). Thus by symmetry ϕitalic-ϕ\phiitalic_ϕ can have at most one extrema on the interval (h,2⁢h)ℎ2ℎ(h,2h)( italic_h , 2 italic_h ) which would also be a maxima, as well. Therefore there are no minima in the interval (h,2⁢h)ℎ2ℎ(h,2h)( italic_h , 2 italic_h ) either. Then we just need to check that

C

This is [24, Theorem 2.1] with δ=0𝛿0\delta=0italic_δ = 0 and x𝑥xitalic_x replaced with x/2𝑥2x/2italic_x / 2.

In Section 3, we recall basic properties of standard L𝐿Litalic_L-functions and Rankin–Selberg L𝐿Litalic_L-functions that we will use in our proofs. In Section 4, we prove a large sieve inequality for the Dirichlet coefficients of L⁢(s,π×π′)−1𝐿superscript𝑠𝜋superscript𝜋′1L(s,\pi\times\pi^{\prime})^{-1}italic_L ( italic_s , italic_π × italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and a corollary on mean values of Dirichlet polynomials, which we use in our proof of Theorem 1.1 in Section 5. We then prove Theorem 2.1 in Section 6, Theorem 2.3 in Section 7, and Theorem 2.5 in Section 8.

In this section, we prove Theorem 2.1. Let π′∈𝔉n′superscript𝜋′subscript𝔉superscript𝑛′\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ fraktur_F start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and Q≥1𝑄1Q\geq 1italic_Q ≥ 1. In (1.6), we rescale ε𝜀\varepsilonitalic_ε to ε/72𝜀72\varepsilon/72italic_ε / 72 and define α=ε/(7.2⁢max⁡{n,n′})𝛼𝜀7.2𝑛superscript𝑛′\alpha=\varepsilon/(7.2\max\{n,n^{\prime}\})italic_α = italic_ε / ( 7.2 roman_max { italic_n , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } ) so that

Lemma 7.2 informs the following choices of parameters that we will use throughout the rest of this section. Let 0<ε<10𝜀10<\varepsilon<10 < italic_ε < 1, let Q𝑄Qitalic_Q be sufficiently large with respect to ε𝜀\varepsilonitalic_ε, and let

Let 0<ε<10𝜀10<\varepsilon<10 < italic_ε < 1, and let β∈(1−ε,1)𝛽1𝜀1\beta\in(1-\varepsilon,1)italic_β ∈ ( 1 - italic_ε , 1 ). By (3.13) and the above discussion, we have

C

In [FV13] it is proven that the fundamental group of a surface bundle of the type described is large. In particular, this implies that v⁢b1⁢(X)=∞𝑣subscript𝑏1𝑋vb_{1}(X)=\inftyitalic_v italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_X ) = ∞. Then let X~→X→~𝑋𝑋\widetilde{X}\to Xover~ start_ARG italic_X end_ARG → italic_X be a finite regular cover, as described in the diagram of Equation 5, with b1⁢(X~)>2subscript𝑏1~𝑋2b_{1}(\widetilde{X})>2italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG ) > 2. As B~~𝐵{\widetilde{B}}over~ start_ARG italic_B end_ARG is also a torus, the fibration F~↪X~→B~↪~𝐹~𝑋→~𝐵{\widetilde{F}}\hookrightarrow\widetilde{X}\to{\widetilde{B}}over~ start_ARG italic_F end_ARG ↪ over~ start_ARG italic_X end_ARG → over~ start_ARG italic_B end_ARG has excessive homology.

In [Mark22] the author proves that NFO(2,0,02002,0,02 , 0 , 0) fails. (As often happens, the case of genus 2222 is special because the mapping class group is entirely composed of hyperelliptic classes.) This entails the following:

Marković [Mark22] has shown that the conjecture fails when the genus of the surface is 2222. Combining this with Theorem 1, we have the following, that gives a partial answer to a question of Hillman ([Hi15, Section 11, Question 4]):

The “only if” part of Theorem 1 uses an epimorphism from the base of the extension to the mapping class group ModgsubscriptMod𝑔\operatorname{Mod}_{g}roman_Mod start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT, so that we can make the construction of the groups deciding the conjecture NFO(g,0,0𝑔00g,0,0italic_g , 0 , 0) very explicit, especially in the case where g≥3𝑔3g\geq 3italic_g ≥ 3. We illustrate this in the surface-by-free case: Let F𝐹Fitalic_F be a surface of genus g≥3𝑔3g\geq 3italic_g ≥ 3; this admits a cyclic automorphism of order 4⁢g+24𝑔24g+24 italic_g + 2, and denote by ΠΠ\Piroman_Π the fundamental group of the mapping torus of this automorphism. (This is a Seifert–fibered manifold which is finitely covered by a product.) Any two such automorphisms are conjugate in ModgsubscriptMod𝑔\operatorname{Mod}_{g}roman_Mod start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT, so that ΠΠ\Piroman_Π is uniquely determined as a group. By [Kor05] ModgsubscriptMod𝑔\operatorname{Mod}_{g}roman_Mod start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT can be generated by two such automorphisms, related by conjugation by the automorphism δ:K→K:𝛿→𝐾𝐾\delta\colon K\to Kitalic_δ : italic_K → italic_K induced by a Dehn twist along a nonseparating curve. (See Section 3 for references and more details),

Given any group G𝐺Gitalic_G as in Equation 1, we can ask whether it admits a finite-index subgroup with excessive homology. By the work of [FV23, KW22] this is equivalent to the property that G𝐺Gitalic_G virtually algebraically fibers, namely a finite index subgroup of G𝐺Gitalic_G admits an epimorphism to ℤℤ\mathbb{Z}blackboard_Z with finitely generated kernel. Our main aim is to show that in the case where K=π1⁢(F)𝐾subscript𝜋1𝐹K=\pi_{1}(F)italic_K = italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_F ) is a surface group there is a natural relation between this question and a conjecture by Putman and Wieland [PW13, Conjecture 1.2]. We refer the reader to Section 2 and the original source for more detail. The case labeled as NFO(g,0,0𝑔00g,0,0italic_g , 0 , 0) of that conjecture posits that the action of the mapping class group Modg1subscriptsuperscriptMod1𝑔\operatorname{Mod}^{1}_{g}roman_Mod start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT of the surface F𝐹Fitalic_F (where g𝑔gitalic_g is the genus of F𝐹Fitalic_F) on the rational homology of finite-index characteristic covers of F𝐹Fitalic_F (sometimes referred to as higher Prym representation) has no finite orbits. We will show the following.

A

In other words, boxm⁡(u,v)subscriptbox𝑚𝑢𝑣\operatorname{box}_{m}(u,v)roman_box start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u , italic_v ) can be written as the sum of the boxes spanned by

By (5), it suffices to show that h⁢(box⁡(u,v))=0ℎbox𝑢𝑣0h(\operatorname{box}(u,v))=0italic_h ( roman_box ( italic_u , italic_v ) ) = 0 for all u=γ⁢(z)𝑢𝛾𝑧u=\gamma(z)italic_u = italic_γ ( italic_z ) and v=γ⁢(z+𝟏)𝑣𝛾𝑧1v=\gamma(z+\mathbf{1})italic_v = italic_γ ( italic_z + bold_1 ) with z∈V⁢(G⁢[ℓ−1]m)𝑧𝑉𝐺superscriptdelimited-[]ℓ1𝑚z\in V(G[\ell-1]^{m})italic_z ∈ italic_V ( italic_G [ roman_ℓ - 1 ] start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ). We can assume that u𝑢uitalic_u and v𝑣vitalic_v do not lie in a common axis parallel hyperplane, as otherwise boxm⁡(u,v)=0subscriptbox𝑚𝑢𝑣0\operatorname{box}_{m}(u,v)=0roman_box start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u , italic_v ) = 0 and the claim immediately follows from the fact that hℎhitalic_h is a homomorphism. Put

h⁢(box⁡(u,v))=∑w∈Wh⁢(box⁡(𝟏,w))=∑w∈Wχh⁢(w)=χh⁢(u)⁢|W|=0,ℎbox𝑢𝑣subscript𝑤𝑊ℎbox1𝑤subscript𝑤𝑊subscript𝜒ℎ𝑤subscript𝜒ℎ𝑢𝑊0h(\operatorname{box}(u,v))=\sum_{w\in W}h(\operatorname{box}(\boldsymbol{1},w)%

(When m𝑚mitalic_m is clear from the context, we omit it and write boxbox\operatorname{box}roman_box instead of boxmsubscriptbox𝑚\operatorname{box}_{m}roman_box start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT.) Note that boxm⁡(x,y)subscriptbox𝑚𝑥𝑦\operatorname{box}_{m}(x,y)roman_box start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_x , italic_y ) is an m𝑚mitalic_m-chain and it is nontrivial if and only if x𝑥xitalic_x and y𝑦yitalic_y do not lie in a common axis parallel hyperplane.

𝟏1\mathbf{1}bold_1 and each of the corners of the m𝑚mitalic_m-dimensional box with opposite vertices u𝑢uitalic_u and v𝑣vitalic_v.

D

\bigoplus\limits_{v\in V_{K}}X^{*}(\widehat{Z})\right]_{\mathrm{Gal}(K/F)},italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_alg end_POSTSUBSCRIPT ( caligraphic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_K / italic_F ) , italic_Z ( blackboard_A start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ) ≅ [ ⨁ start_POSTSUBSCRIPT italic_v ∈ italic_V start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( over^ start_ARG italic_Z end_ARG ) ] start_POSTSUBSCRIPT roman_Gal ( italic_K / italic_F ) end_POSTSUBSCRIPT ,

Hbas1⁢(ℰ1⁢(K/F),T⁢(𝔸K)/T⁢(K))≅X*⁢(T)Gal⁢(K/F)subscriptsuperscript𝐻1bassubscriptℰ1𝐾𝐹𝑇subscript𝔸𝐾𝑇𝐾subscript𝑋subscript𝑇Gal𝐾𝐹H^{1}_{\mathrm{bas}}(\mathcal{E}_{1}(K/F),T(\mathbb{A}_{K})/T(K))\cong X_{*}(T%

Hbas1(ℰ1(K/F),G′(𝔸K)/ZG′(K))→Hbas1(ℰ1(K/F),G′′(𝔸K)/ZG′′(K))→Hbas1(ℰ1(K/F),C(𝔸K)/C(K)H^{1}_{\mathrm{bas}}(\mathcal{E}_{1}(K/F),G^{\prime}(\mathbb{A}_{K})/Z_{G^{%

Hbas1⁢(ℰ1⁢(K/F),Z⁢(𝔸K)/Z⁢(K))≅X*⁢(Z^)Gal⁢(K/F),subscriptsuperscript𝐻1bassubscriptℰ1𝐾𝐹𝑍subscript𝔸𝐾𝑍𝐾superscript𝑋subscript^𝑍Gal𝐾𝐹H^{1}_{\mathrm{bas}}(\mathcal{E}_{1}(K/F),Z(\mathbb{A}_{K})/Z(K))\cong X^{*}(%

Hbas1⁢(ℰ1⁢(K/F),G⁢(𝔸K)/ZG⁢(K))≅Hbas1⁢(ℰ1⁢(K/F),G′⁢(𝔸K)/ZG′⁢(K))/Hbas1⁢(ℰ1⁢(K/F),Z⁢(𝔸K)/Z⁢(K)).subscriptsuperscript𝐻1bassubscriptℰ1𝐾𝐹𝐺subscript𝔸𝐾subscript𝑍𝐺𝐾subscriptsuperscript𝐻1bassubscriptℰ1𝐾𝐹superscript𝐺′subscript𝔸𝐾subscript𝑍superscript𝐺′𝐾subscriptsuperscript𝐻1bassubscriptℰ1𝐾𝐹𝑍subscript𝔸𝐾𝑍𝐾H^{1}_{\mathrm{bas}}(\mathcal{E}_{1}(K/F),G(\mathbb{A}_{K})/Z_{G}(K))\cong H^{%

C

By Krull’s intersection theorem, ⋂N≥1𝔫N=0subscript𝑁1superscript𝔫𝑁0\bigcap_{N\geq 1}\mathfrak{n}^{N}=0⋂ start_POSTSUBSCRIPT italic_N ≥ 1 end_POSTSUBSCRIPT fraktur_n start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT = 0, which implies all the fitting ideals of 𝔐𝔐\mathfrak{M}fraktur_M are trivial for k<m𝑘𝑚k<mitalic_k < italic_m i.e., Fitk⁢(𝔐)=0subscriptFit𝑘𝔐0\text{Fit}_{k}(\mathfrak{M})=0Fit start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( fraktur_M ) = 0 for k<m𝑘𝑚k<mitalic_k < italic_m, and since 𝔐𝔐\mathfrak{M}fraktur_M has m𝑚mitalic_m generators {a1,a2,⋯,am}subscript𝑎1subscript𝑎2⋯subscript𝑎𝑚\{a_{1},a_{2},\cdots,a_{m}\}{ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋯ , italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } by definition Fitm⁢(𝔐)=𝔖ℛ.subscriptFit𝑚𝔐subscript𝔖ℛ\text{Fit}_{m}(\mathfrak{M})=\mathfrak{S}_{\mathscr{R}}.Fit start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( fraktur_M ) = fraktur_S start_POSTSUBSCRIPT script_R end_POSTSUBSCRIPT . Hence by 2.13, 𝔐𝔐\mathfrak{M}fraktur_M is projective 𝔖ℛsubscript𝔖ℛ\mathfrak{S}_{\mathscr{R}}fraktur_S start_POSTSUBSCRIPT script_R end_POSTSUBSCRIPT-module.

In literature, there are also slightly different versions of Breuil-Kisin modules used by different authors (e.g., [3], [7]) in different perspectives. Some authors preferred only either the freeness property or the finiteness property. Some authors assumed 𝔐𝔐\mathfrak{M}fraktur_M to be u𝑢uitalic_u-torsion free while some authors inverted E⁢(u)𝐸𝑢E(u)italic_E ( italic_u ) in defining 𝔖⁢[1/E⁢(u)]𝔖delimited-[]1𝐸𝑢\mathfrak{S}[1/E(u)]fraktur_S [ 1 / italic_E ( italic_u ) ]-linearization map 𝔖⊗φ,𝔖𝔐⁢[1/E⁢(u)]→𝔐⁢[1/E⁢(u)]→subscripttensor-product𝜑𝔖𝔖𝔐delimited-[]1𝐸𝑢𝔐delimited-[]1𝐸𝑢\mathfrak{S}\otimes_{\varphi,\mathfrak{S}}\mathfrak{M}[1/E(u)]\to\mathfrak{M}[%

The authors are grateful to Olivier Brinon for valuable comments in an old version of this paper. The first author acknowledges CSIR, Government of India, for the award of Senior Research Fellowship with File no.-09/025(0249)/2018-EMR-I.

We further continue our investigation toward the projectiveness property of 𝔐∈Mod/𝔖ℛφ𝔐superscriptsubscriptModabsentsubscript𝔖ℛ𝜑\mathfrak{M}\in\text{Mod}_{/\mathfrak{S}_{\mathscr{R}}}^{\varphi}fraktur_M ∈ Mod start_POSTSUBSCRIPT / fraktur_S start_POSTSUBSCRIPT script_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_φ end_POSTSUPERSCRIPT. For this, we need the help of the following three results:

A core in the Breuil-Kisin classification theory of p𝑝pitalic_p-divisible groups by BK-module is the Frobenious structure on BK-modules, and if a p𝑝pitalic_p-divisible group has a faithful action of ℛℛ\mathscr{R}script_R, then the resulting BK-module attached to p𝑝pitalic_p-divisible not only has a ℛℛ\mathscr{R}script_R-action but also the action is compatible with the Frobenious map φ𝔖⁢(u)=upsubscript𝜑𝔖𝑢superscript𝑢𝑝\varphi_{\mathfrak{S}}(u)=u^{p}italic_φ start_POSTSUBSCRIPT fraktur_S end_POSTSUBSCRIPT ( italic_u ) = italic_u start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. Exploiting the compatibility of the Frobenious and ℛℛ\mathscr{R}script_R-action, we produce the following Theorem 2.14. Before proving our result, we recall the definition of fitting ideal of a finitely generated module and some properties:

B

Suppose that the maximal existence interval is I=(−T*,T*)𝐼subscript𝑇superscript𝑇I=(-T_{*},T^{*})italic_I = ( - italic_T start_POSTSUBSCRIPT * end_POSTSUBSCRIPT , italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ). We proceed by contradiction and assume that T*=+∞superscript𝑇T^{*}=+\inftyitalic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT = + ∞.

We now may choose T0>0subscript𝑇00T_{0}>0italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 sufficiently large such that I′⁢(0)<2⁢δ0⁢C0⁢T0superscript𝐼′02subscript𝛿0subscript𝐶0subscript𝑇0I^{\prime}(0)<2\delta_{0}C_{0}T_{0}italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) < 2 italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. From this and (3.5),

Let 4−2⁢bN<p<2b∗42𝑏𝑁𝑝subscriptsuperscript2normal-∗𝑏\frac{4-2b}{N}<p<2^{\ast}_{b}divide start_ARG 4 - 2 italic_b end_ARG start_ARG italic_N end_ARG < italic_p < 2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, then for all t∈I𝑡𝐼t\in Iitalic_t ∈ italic_I, there exits δ0>0subscript𝛿00\delta_{0}>0italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0, such that

Let 4N<p<2b∗4𝑁𝑝subscriptsuperscript2normal-∗𝑏\frac{4}{N}<p<2^{\ast}_{b}divide start_ARG 4 end_ARG start_ARG italic_N end_ARG < italic_p < 2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, then there exists α>1𝛼1\alpha>1italic_α > 1, such that

First, by the Lemma 3.1, there exists C0>0subscript𝐶00C_{0}>0italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that

D

\operatorname{WH}^{*}_{\operatorname{QFC}}(M,\phi,0).italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT caligraphic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_M ∖ ∂ italic_M , italic_g start_POSTSUBSCRIPT roman_QFC end_POSTSUBSCRIPT ) ≅ roman_WH start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_QFC end_POSTSUBSCRIPT ( italic_M , italic_ϕ , 0 ) .

The main result of [23] roughly relies on two intermediate results. The first one, topological in nature, consists in identifying weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology groups of fibered boundary and fibered cusp metrics with suitable intersection cohomology groups. Since fibered cusp and fibered boundary metrics are conformally related, one in fact only needs to establish such a result for the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of a fibered cusp metric. These weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology groups can be understood as the cohomology groups of a sheaf on an associated stratified space, so that using Mayer-Vietoris long exact sequences, it suffices to identify these weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology groups with intersection cohomology for local models. These local identifications in turn can be achieved thanks to the Künneth formula of Zucker [45, Corollary 2.34] for the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of warped products. Except for certain types of fibered cusp metrics, notice that L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology itself is infinite dimensional and cannot be identified with some intersection cohomology, hence the importance to introduce a weight to obtain such an identification in general.

Finally, to obtain the identification with upper middle intersection cohomology, notice that (3.59) gives in this case

which, since misubscript𝑚𝑖m_{i}italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is odd, is the same as the local behavior of upper middle and lower middle intersection cohomology [23, (8)]. The result therefore follows from [23, Proposition 1].

In middle degree, the intersection cohomology with lower or upper middle perversity can in fact be described in terms of the usual cohomology as the next proposition shows.

B

\bm{x};\bm{\alpha},\bm{\beta}).= ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ , italic_μ end_POSTSUBSCRIPT ( bold_italic_α , bold_italic_β ) italic_G start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( bold_italic_x ; bold_italic_α , bold_italic_β ) .

Using the Schur expansions of Gλ⁢(𝒙;𝜶,𝜷)subscript𝐺𝜆𝒙𝜶𝜷G_{\lambda}(\bm{x};\bm{\alpha},\bm{\beta})italic_G start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( bold_italic_x ; bold_italic_α , bold_italic_β ) and gλ⁢(𝒙;𝜶,𝜷)subscript𝑔𝜆𝒙𝜶𝜷g_{\lambda}(\bm{x};\bm{\alpha},\bm{\beta})italic_g start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( bold_italic_x ; bold_italic_α , bold_italic_β ), we show the following duality, which justifies the name “refined

where the second equality is obtained by first considering the sum over all k∈ℤ𝑘ℤk\in\mathbb{Z}italic_k ∈ blackboard_Z and then using the fact hm⁢(𝒙)=0subscriptℎ𝑚𝒙0h_{m}(\bm{x})=0italic_h start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_x ) = 0 for m<0𝑚0m<0italic_m < 0.

Using the duality in Theorem 3.5, the skew operator f⟂superscript𝑓perpendicular-tof^{\perp}italic_f start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT, and the fact

in Theorem 1.6 and Corollary 4.5, and for gλ⁢(𝒙;𝜶,𝜷)subscript𝑔𝜆𝒙𝜶𝜷g_{\lambda}(\bm{x};\bm{\alpha},\bm{\beta})italic_g start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( bold_italic_x ; bold_italic_α , bold_italic_β ) in Theorem 1.6 and

C

(β,1,0,1−β,0,1)𝛽101𝛽01(\beta,1,0,1-\beta,0,1)( italic_β , 1 , 0 , 1 - italic_β , 0 , 1 ) together with (1+β,0,1,β,1,0)1𝛽01𝛽10(1+\beta,0,1,\beta,1,0)( 1 + italic_β , 0 , 1 , italic_β , 1 , 0 ), as

(−β,1,0,1+β,0,1)𝛽101𝛽01(-\beta,1,0,1+\beta,0,1)( - italic_β , 1 , 0 , 1 + italic_β , 0 , 1 ) and (1−β,0,1,−β,1,0)1𝛽01𝛽10(1-\beta,0,1,-\beta,1,0)( 1 - italic_β , 0 , 1 , - italic_β , 1 , 0 ), with

β=p⁢q𝛽𝑝𝑞\beta=\sqrt{p\hskip 0.5ptq\hskip 0.5pt}italic_β = square-root start_ARG italic_p italic_q end_ARG, so β=α⁢q𝛽𝛼𝑞\beta=\alpha\hskip 0.5ptqitalic_β = italic_α italic_q and

.5pt\mathbb{Z}}+\frac{m}{q}\beta\bigr{)}italic_α blackboard_Z = over˙ start_ARG ⋃ end_ARG start_POSTSUBSCRIPT italic_m = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q - 1 end_POSTSUPERSCRIPT ( italic_β blackboard_Z + divide start_ARG italic_m end_ARG start_ARG italic_q end_ARG italic_β ). This gives

(β,1,0,1−β,0,1)𝛽101𝛽01(\beta,1,0,1-\beta,0,1)( italic_β , 1 , 0 , 1 - italic_β , 0 , 1 ) together with (1+β,0,1,β,1,0)1𝛽01𝛽10(1+\beta,0,1,\beta,1,0)( 1 + italic_β , 0 , 1 , italic_β , 1 , 0 ), as

A

\|_{H^{s_{0}}}caligraphic_E start_POSTSUPERSCRIPT italic_i italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := ∥ ( italic_ρ start_POSTSUPERSCRIPT italic_i italic_n end_POSTSUPERSCRIPT - italic_ρ start_POSTSUBSCRIPT # end_POSTSUBSCRIPT , fraktur_u start_POSTSUPERSCRIPT italic_i italic_n end_POSTSUPERSCRIPT , italic_T start_POSTSUPERSCRIPT italic_i italic_n end_POSTSUPERSCRIPT - italic_T start_POSTSUBSCRIPT # end_POSTSUBSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT.

We now outline the construction of the truncated expansion (1.20), whose details will be given in Section 2 later. Denote by

of the fluid variables of 𝔐[ρ,𝔲,T],Fk,Fkb⁢(1≤k≤4)subscript𝔐𝜌𝔲𝑇subscript𝐹𝑘subscriptsuperscript𝐹𝑏𝑘1𝑘4\mathfrak{M}_{[\rho,\mathfrak{u},T]},F_{k},F^{b}_{k}(1\leq k\leq 4)fraktur_M start_POSTSUBSCRIPT [ italic_ρ , fraktur_u , italic_T ] end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_F start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( 1 ≤ italic_k ≤ 4 ) are all given. The precise expressions will be derived later.

whose parameters (ρ,𝔲,T)𝜌𝔲𝑇(\rho,\mathfrak{u},T)( italic_ρ , fraktur_u , italic_T ) satisfy the compressible Euler system. In particular, for the domain with boundary, the key (also the difficulty) is to determine the boundary condition for the compressible Euler system. It was formally derived in Sone’s book [39] that for the Boltzmann equation with the complete diffusive boundary condition (1.3), the limiting Euler system would be imposed on the impermeable boundary condition (see also [37], for instance). The goal of the this paper is to rigorously justify this limit by using the method of Hilbert expansion, in the context of short time smooth solutions. We first state the main result of this paper in an informal way. The precise statement will be given in the later part of Introduction, after we introduce some required notations.

In this subsection, the truncation of the Hilbert expansion (2.1) will be done. Our goal is to prove the compressible Euler limit from the scaled Boltzmann equation by above Hilbert type expansion. The key point is to prove that the remainders of expansion will go to zero as the Knudsen number ε→0→𝜀0\varepsilon\to 0italic_ε → 0. Mathematically, this means we search for a special class of solutions of the original scaled Boltzmann equation for sufficiently small Knudsen number ε𝜀\varepsilonitalic_ε. Since the more terms are expanded, the more special the solutions are. We hope that the terms in the expansion are as less as possible.

A

Taking a step in this direction, we study the existence of a random dynamical system induced by solutions of a system of Navier-Stokes equations on the three-dimensional torus T3superscriptT3\textbf{T}^{3}T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT subject to a rough transport noise. The noise arises from perturbing the transport advecting velocity field by a space-time dependent noise and is, at least formally, energy conservative.

To prove this result we adapt to the rough path setting the selection procedure introduced in [9] and further generalized in [5, 6, 7]. The key property which then permits to deduce the existence of a measurable random dynamical system is the rough path stability in the spirit of a Wong-Zakai approximation result, see Theorem 3.3. Indeed, this permits to go back to probability and consider random driving rough paths which satisfy a suitable cocycle property as introduced in [3]. An example is a fractional Brownian motion with Hurst parameter H>13𝐻13H>\frac{1}{3}italic_H > divide start_ARG 1 end_ARG start_ARG 3 end_ARG.

Here we prove the remaining two main ingredients for the construction of a semiflow, the shift invariance property and the continuation property of the set of solutions.

In this subsection we address the issue of sequential stability which will allow us to show the compactness of the set 𝒰⁢[u0,E0,Z]𝒰subscript𝑢0subscript𝐸0Z\mathcal{U}[{u_{0},E_{0},\textbf{Z}}]caligraphic_U [ italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , Z ] as well as the required measurability of the semiflow selection. It is also essential for proving the measurability of the random dynamical system constructed in Section 4.

The rough path philosophy of splitting analysis from probability, as well as a Wong-Zakai stability result are the key ingredients of our construction. For the first time,

D

This paper is organized as follows. Theorem 1.1 is to be proved in Section 2. Exact solutions will be discussed in Section 3.

after the use of the Sobolev imbedding theorem and (2.6). Here and in what follows, C𝐶Citalic_C is a generic constant independent of ν𝜈\nuitalic_ν.

The proof is developed from Lin and Xu Lin together with Constantin et al.Const . Let C𝐶Citalic_C be a generic constant and the condition α>1𝛼1\alpha>1italic_α > 1 is assumed throughout this section.

The present study is motivated by the work of Lin and Xu Lin , where they noticed that linearized the Navier-Stokes flow (1.5) is close to the linearized Euler flow

When a=0𝑎0a=0italic_a = 0, this is comparable with an enhanced damping result given by Lin and Xu Lin in the following.

B

Since hnsubscriptℎ𝑛h_{n}italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is chosen to satisfy n⁢hnd→∞→𝑛superscriptsubscriptℎ𝑛𝑑nh_{n}^{d}\rightarrow\inftyitalic_n italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → ∞, then n−1/2⁢hn−d/4=o⁢(n−1/4)superscript𝑛12superscriptsubscriptℎ𝑛𝑑4𝑜superscript𝑛14n^{-1/2}h_{n}^{-d/4}=o(n^{-1/4})italic_n start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_d / 4 end_POSTSUPERSCRIPT = italic_o ( italic_n start_POSTSUPERSCRIPT - 1 / 4 end_POSTSUPERSCRIPT ), and we conclude that the Chatterjee’s test, studied in Auddy, Deb and Nandy (2024), is powerless along the local alternative (4.1) compared with the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-test. Shi, Drton and Han (2022b) showed that Hoeffding’s test, Blum-Kiefer-Rosenblatt’s test and Bergsma-Dassios-Yanagimoto’s test are more powerful, with the optimal rate of order O⁢(n−1/2)𝑂superscript𝑛12O(n^{-1/2})italic_O ( italic_n start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ), than L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-test, L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-test and Chatterjee’s test for the local alternative (4.1), but presenting some difficulties in estimating quantiles of the null distribution in high dimensional setting. The main question is now to find other local alternatives for which the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-test is more powerful than the others.

To our best knowledge, this is the first time that the general context L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm for testing the independence appeared in the literature and gives the main motivation of the present work by responding to the open problems mentioned in Gretton and Györfi (2010). The main contribution of this paper is to establish the asymptotic distribution of the proposed test statistic under the null hypothesis and under the local alternatives that converge to the null at the rate of n−1/2⁢hn−d/4superscript𝑛12superscriptsubscriptℎ𝑛𝑑4n^{-1/2}h_{n}^{-d/4}italic_n start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - italic_d / 4 end_POSTSUPERSCRIPT. As an important feature, the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-based

Samworth (2021). Although many existing tests for independence focus on two random vectors, they frequently do not have clear methods to evaluate mutual independence among more than two random vectors with different dimensions. This paper presents a new method to assess the independence of several random vectors, regardless of their dimensionality. Our method uses the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-distance between the joint density and the product of the marginal densities. Under the null hypothesis, we use the Poissonization techniques to find, for the first time, the asymptotic normal approximation of the corresponding statistic. We do this without making assumptions about the regularity of the underlying Lebesgue density f⁢(⋅)𝑓⋅f(\cdot)italic_f ( ⋅ ).

This work presents a different approach to the problem of testing the mutual independence of random vectors in arbitrary dimensions based on the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm between the joint density and the product of p𝑝pitalic_p the marginal densities, for p≥2𝑝2p\geq 2italic_p ≥ 2. The motivation for L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm is connected with the fact that ∫ℝd|f⁢(𝐱~)−g⁢(𝐱~)|⁢𝑑𝐱~subscriptsuperscriptℝ𝑑𝑓~𝐱𝑔~𝐱differential-d~𝐱\int_{\mathbb{R}^{d}}|f(\tilde{\mathbf{x}})-g(\tilde{\mathbf{x}})|d\tilde{%

This paper develops a general framework for testing the independence among continuous random vectors based on L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm. Our test achieves the following desirable properties:

D

Let μ1<0subscript𝜇10\mu_{1}<0italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 and 2≤n<42𝑛42\leq n<42 ≤ italic_n < 4. Suppose h∈C2⁢((0,∞))∩C1⁢([0,∞))ℎsuperscript𝐶20superscript𝐶10h\in C^{2}((0,\infty))\cap C^{1}([0,\infty))italic_h ∈ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ( 0 , ∞ ) ) ∩ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( [ 0 , ∞ ) ) is a solution of (1.11). Let u𝑢uitalic_u be given by (4.16). Suppose there exists a constant s0>0subscript𝑠00s_{0}>0italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that (4.22) holds. Then (4.23) holds.

Let μ1<0subscript𝜇10\mu_{1}<0italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 and 2≤n<42𝑛42\leq n<42 ≤ italic_n < 4. Suppose h∈C2⁢((0,∞))∩C1⁢([0,∞))ℎsuperscript𝐶20superscript𝐶10h\in C^{2}((0,\infty))\cap C^{1}([0,\infty))italic_h ∈ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ( 0 , ∞ ) ) ∩ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( [ 0 , ∞ ) ) is a solution of (1.11). Let u𝑢uitalic_u be given by (4.16). Suppose there exists a constant s0>0subscript𝑠00s_{0}>0italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that (4.22) holds. Then (4.23) holds.

Let 2≤n<42𝑛42\leq n<42 ≤ italic_n < 4. Suppose (4.23) does not hold. Then there exists a constant 0<δ<10𝛿10<\delta<10 < italic_δ < 1 and a sequence

Let μ1<0subscript𝜇10\mu_{1}<0italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 0 and 2≤n<42𝑛42\leq n<42 ≤ italic_n < 4. Suppose h∈C2⁢((0,∞))∩C1⁢([0,∞))ℎsuperscript𝐶20superscript𝐶10h\in C^{2}((0,\infty))\cap C^{1}([0,\infty))italic_h ∈ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ( 0 , ∞ ) ) ∩ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( [ 0 , ∞ ) ) is a solution of (1.11). Let q𝑞qitalic_q be given by (4.3). Suppose there exists a constant s0>0subscript𝑠00s_{0}>0italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that (4.22) holds. Then there exists a constant C>0𝐶0C>0italic_C > 0 such that

Let λ,μ1∈ℝ𝜆subscript𝜇1ℝ\lambda,\mu_{1}\in\mathbb{R}italic_λ , italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R and 2≤n≤42𝑛42\leq n\leq 42 ≤ italic_n ≤ 4. Then there exists a constant ε>0𝜀0\varepsilon>0italic_ε > 0 such that

B

Then (Δ⁢U,U)Δ𝑈𝑈(\Delta U,U)( roman_Δ italic_U , italic_U ) is a horizontal ΔΔ\Deltaroman_Δ-semimartingale on 𝒪⁢(M)𝒪𝑀\mathcal{O}(M)caligraphic_O ( italic_M ). In fact, by Lemma 2.17 and (3.25), we have

∫θ⁢(U−)∘𝑑U𝜃subscript𝑈differential-d𝑈\displaystyle\int\theta(U_{-})\circ dU∫ italic_θ ( italic_U start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ∘ italic_d italic_U

∫θ⁢(U)∘𝑑U𝜃𝑈differential-d𝑈\displaystyle\int\theta(U)\circ dU∫ italic_θ ( italic_U ) ∘ italic_d italic_U

∫(σ,t]θ∘𝑑Usubscript𝜎𝑡𝜃differential-d𝑈\displaystyle\int_{(\sigma,t]}\theta\circ dU∫ start_POSTSUBSCRIPT ( italic_σ , italic_t ] end_POSTSUBSCRIPT italic_θ ∘ italic_d italic_U

∫θ⁢(U−)⁢γ~⁢𝑑U𝜃subscript𝑈~𝛾differential-d𝑈\displaystyle\int\theta(U_{-})\ \tilde{\gamma}dU∫ italic_θ ( italic_U start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) over~ start_ARG italic_γ end_ARG italic_d italic_U

B

\mathcal{A}})italic_p ⊩ italic_φ start_POSTSUPERSCRIPT italic_V ( over˙ start_ARG caligraphic_A end_ARG ) end_POSTSUPERSCRIPT ( overroman_ˇ start_ARG italic_s end_ARG , overroman_ˇ start_ARG italic_v end_ARG , over˙ start_ARG over¯ start_ARG italic_a end_ARG end_ARG , over˙ start_ARG caligraphic_A end_ARG ). Since p∈G𝑝𝐺p\in Gitalic_p ∈ italic_G we have that φV⁢(𝒜)⁢(s,v,a¯,𝒜)superscript𝜑𝑉𝒜𝑠𝑣¯𝑎𝒜\varphi^{V(\mathcal{A})}(s,v,\bar{a},\mathcal{A})italic_φ start_POSTSUPERSCRIPT italic_V ( caligraphic_A ) end_POSTSUPERSCRIPT ( italic_s , italic_v , over¯ start_ARG italic_a end_ARG , caligraphic_A ) holds in V⁢[G]𝑉delimited-[]𝐺V[G]italic_V [ italic_G ], and therefore, φ⁢(s,v,a¯,𝒜)𝜑𝑠𝑣¯𝑎𝒜\varphi(s,v,\bar{a},\mathcal{A})italic_φ ( italic_s , italic_v , over¯ start_ARG italic_a end_ARG , caligraphic_A ) holds in V⁢(𝒜)𝑉𝒜V(\mathcal{A})italic_V ( caligraphic_A ), and so s∈S𝑠𝑆s\in Sitalic_s ∈ italic_S.

The converse direction S⊆S′𝑆superscript𝑆′S\subseteq S^{\prime}italic_S ⊆ italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT follows from the next claim.

We will conclude the proof of the lemma by showing that S=S′𝑆superscript𝑆′S=S^{\prime}italic_S = italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.

{\}}italic_S = { italic_s ∈ italic_V ∣ italic_V ( caligraphic_A ) ⊧ italic_φ ( italic_s , italic_v , over¯ start_ARG italic_a end_ARG , caligraphic_A ) }.

To see that S′⊆Ssuperscript𝑆′𝑆S^{\prime}\subseteq Sitalic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_S, let s∈S′𝑠superscript𝑆′s\in S^{\prime}italic_s ∈ italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and take any condition p∈G𝑝𝐺p\in Gitalic_p ∈ italic_G such that n0,…,nk−1∈dom⁢(p)subscript𝑛0…subscript𝑛𝑘1dom𝑝n_{0},...,n_{k-1}\in\mathrm{dom}(p)italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_n start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∈ roman_dom ( italic_p ). By definition of S′superscript𝑆′S^{\prime}italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and by the choice of ξ𝜉\xiitalic_ξ, p⊩φV⁢(𝒜˙)⁢(sˇ,vˇ,a¯˙,𝒜˙)forces𝑝superscript𝜑𝑉˙𝒜ˇ𝑠ˇ𝑣˙¯𝑎˙𝒜p\Vdash\varphi^{V(\mathcal{\dot{A}})}(\check{s},\check{v},\dot{\bar{a}},\dot{%

A

A motion picture of F𝐹Fitalic_F is a 1111-parameter family {F[t]}t∈ℝsubscriptsubscript𝐹delimited-[]𝑡𝑡ℝ\{F_{[t]}\}_{t\in\mathbb{R}}{ italic_F start_POSTSUBSCRIPT [ italic_t ] end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t ∈ blackboard_R end_POSTSUBSCRIPT.

Plat⁢(F)=1Plat𝐹1\mathrm{Plat}(F)=1roman_Plat ( italic_F ) = 1 if and only if F𝐹Fitalic_F is either a trivial 2222-knot or a trivial non-orientable surface-knot.

A surface-knot is trivial if it is equivalent to a connected sum of standardly embedded 2222-spheres, tori, and projective planes ([8]).

Here standardly embedded projective planes P+subscript𝑃P_{+}italic_P start_POSTSUBSCRIPT + end_POSTSUBSCRIPT and P−subscript𝑃P_{-}italic_P start_POSTSUBSCRIPT - end_POSTSUBSCRIPT are illustrated in Figure 6.

A surface-link is a closed surface embedded in ℝ4superscriptℝ4\mathbb{R}^{4}blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, and a surface-knot is a connected surface-link.

B

Note that the energy E⁢(t)𝐸𝑡E(t)italic_E ( italic_t ) defines a norm in ℋℋ\mathcal{H}caligraphic_H which is equivalent to ∥.∥ℋ\|.\|_{\mathcal{H}}∥ . ∥ start_POSTSUBSCRIPT caligraphic_H end_POSTSUBSCRIPT.

It is worth mentioning that the above mentioned results precise some of the necessary conditions that lead to the exponential stabilization of the model (depending on the choice of the boundary conditions, the equality or non-equality of the wave speeds and the nature of damping terms). Here, the challenge consists in considering nonlinear damping which are distributed everywhere in the domain and acting only on the rotational angles. Taking into account the work of Alabau-Boussouira [2, 4], we aim here to establish a general and explicit decay result for the energy associated with the system (1.5)–(1.7) below. More precisely, we prove that the energy decay rate, as introduced in [2] for a nonlinearly damped hyperbolic system coupled by velocities, can be extended to our plate model. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method used in [1, 2]. Indeed, the latter method is originally developed in [1] where the author completed the study carried out in [23] and improved the results in [25]. In the present work we assume a convexity assumption on the feedback as we will see later on, and we prove the asymptotic behavior of the energy in higher dimensions. For more details, we refer the reader to [5] for the wave equation and to [2] for the one-dimensional Timoshenko system.

The first main result of this section is given by the following theorem. The method used for the proof is adapted from the general approach in [1] and applied here to the plate coupled system.

In the following theorem, we formulate our result on the existence and uniqueness of the solution of (1.5)–(1.7) associated with (2.3) and (2.4).

the system (1.5)–(1.7) is a decreasing function of t𝑡titalic_t. Finally, in Section 4, we show that the optimal-weight convexity method allows us to obtain an explicit decay rate formula of the total energy associated with the solution of the problem (1.5)–(1.7).

C

The assumption that B𝐵Bitalic_B is at the same time a module for the monoid C𝐶Citalic_C and a comodule for the comonoid A𝐴Aitalic_A entails the existence of an action map a:C⊗B→B:𝑎→tensor-product𝐶𝐵𝐵a:C\otimes B\to Bitalic_a : italic_C ⊗ italic_B → italic_B and a coaction map c:B→A⊗B:𝑐→𝐵tensor-product𝐴𝐵c:B\to A\otimes Bitalic_c : italic_B → italic_A ⊗ italic_B, which plugged into the profunctor P𝑃Pitalic_P yield

\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_I ⊗ italic_Ai2subscript𝑖2\scriptstyle{i_{2}}italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTC𝐶\textstyle{C}italic_CI𝐼\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_II⊗Itensor-product𝐼𝐼\textstyle{I\otimes I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_I ⊗ italic_II⊗Itensor-product𝐼𝐼\textstyle{I\otimes I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_I ⊗ italic_II𝐼\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_I

⟨C,A⟩𝐶𝐴\textstyle{\langle C,A\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}⟨ italic_C , italic_A ⟩∫QSet⁢(Q⁢(A,C),Q⁢(C,A))subscript𝑄Set𝑄𝐴𝐶𝑄𝐶𝐴\textstyle{\int_{Q}\textsf{Set}(Q(A,C),Q(C,A))\ignorespaces\ignorespaces%

β[B]A⁢C=P⁢(a,c):P⁢(B,B):superscriptsubscript𝛽delimited-[]𝐵𝐴𝐶𝑃𝑎𝑐𝑃𝐵𝐵\textstyle{\beta_{[B]}^{AC}=P(a,c):P(B,B)\ignorespaces\ignorespaces%

S𝑆\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_Su𝑢\scriptstyle{u}italic_uM⊗Atensor-product𝑀𝐴\textstyle{M\otimes A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_M ⊗ italic_AM⊗ftensor-product𝑀𝑓\scriptstyle{M\otimes f}italic_M ⊗ italic_fM⊗Btensor-product𝑀𝐵\textstyle{M\otimes B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}italic_M ⊗ italic_Bv𝑣\scriptstyle{v}italic_vT𝑇\textstyle{T}italic_T

C

Note that any disc associated to a contractible component of \ul⁢Lm\ulsuperscript𝐿𝑚\ul L^{m}italic_L start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT as in (iii) must be a connected component of Σ∖\ul⁢LmΣ\ulsuperscript𝐿𝑚\Sigma\setminus\ul{L}^{m}roman_Σ ∖ italic_L start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT: indeed, if it contained a component of \ul⁢Lm\ulsuperscript𝐿𝑚\ul L^{m}italic_L start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT then this component would have to be contractible and then the disc associated with it would violate the uniqueness property in (iii). It also follows from (iv) that all these discs have equal area. We denote this common area by αmsubscript𝛼𝑚\alpha_{m}italic_α start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. Note that the other components of Σ∖\ul⁢LmΣ\ulsuperscript𝐿𝑚\Sigma\setminus\ul L^{m}roman_Σ ∖ italic_L start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT all have area smaller than or equal to αmsubscript𝛼𝑚\alpha_{m}italic_α start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT.

\ul⁢L⊂Σ\ul𝐿Σ\ul{L}\subset\Sigmaitalic_L ⊂ roman_Σ to be a smooth embedding of finitely many pairwise disjoint circles (see Figure 1 below). We emphasize, because it contrasts the setup for many other works about Floer theory on surfaces, that the individual components of the link are not required to be Floer theoretically non-trivial; for example, they can be small contractible curves. Whenever \ul⁢L\ul𝐿\ul{L}italic_L satisfies a certain monotonicity assumption, see Definition 1.7, we define a link spectral invariant c\ul⁢L:C∞⁢([0,1]×Σ,ω)→\R:subscript𝑐\ul𝐿→superscript𝐶01Σ𝜔\Rc_{\ul{L}}:C^{\infty}([0,1]\times\Sigma,\omega)\to\Ritalic_c start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT : italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( [ 0 , 1 ] × roman_Σ , italic_ω ) →. The properties of the invariants c\ul⁢Lsubscript𝑐\ul𝐿c_{\ul{L}}italic_c start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT are summarized in Theorem 1.4 below. We have the following result for suitable sequences of Lagrangian links which always exist and which we refer to as equidistributed links, see Section 3.1 for the precise definition.

it is straightforward to check that x=(1,…,1)𝑥1…1x=(1,\dots,1)italic_x = ( 1 , … , 1 ) is a critical point of W\Sym⁢(\ul⁢L)⁢(x,JX)subscript𝑊\Sym\ul𝐿𝑥subscript𝐽𝑋W_{\Sym(\ul{L})}(x,J_{X})italic_W start_POSTSUBSCRIPT ( italic_L ) end_POSTSUBSCRIPT ( italic_x , italic_J start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ).

It is straightforward to check that equidistributed sequences of Lagrangian links exist; see Figure 2.

We call a sequence of Lagrangian links \ul⁢Lm\ulsuperscript𝐿𝑚\ul{L}^{m}italic_L start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT equidistributed if

C

(i) follows by letting s=3𝑠3s=3italic_s = 3 and replacing n𝑛nitalic_n with 2⁢n+32𝑛32n+32 italic_n + 3 and (ii) by letting s=4𝑠4s=4italic_s = 4 and then replacing n𝑛nitalic_n with 2⁢n+42𝑛42n+42 italic_n + 4. Use Theorem 5.2 in both cases.

This short section is organized similarly to the previous section. We will begin by stating a few definitions of the p𝑝pitalic_p-k𝑘kitalic_k gamma function which can be found in [4]. Next, we will establish a p𝑝pitalic_p-k𝑘kitalic_k gamma function analog of Ramanujan’s Master Theorem in the form of Theorem 3.1 which will be derived classically from the original Ramanujan’s Master Theorem rather than from the integral representation of p𝑝pitalic_p-k𝑘kitalic_k gamma function which we did in the case of k𝑘kitalic_k-gamma function for Theorem 2.1 in the previous section. Furthermore, we will state some examples and applications of our derived analog in the form of solutions of certain definite integrals. Since theorems and applications in this section are partially motivated by the previous section, for the sake of simplicity and saving our precious time, most of the proofs in this section will be short.

In this paper, we have derived some analogs of Ramanujan’s Master Theorem. In particular, we have explored the analogs of Ramanujan’s Master Theorem that arise from the k−limit-from𝑘k-italic_k -gamma function and p,k−𝑝limit-from𝑘p,k-italic_p , italic_k -gamma function. Furthermore, we have explored some other analogs that involve the zeta function. And finally, in the last section, we have derived a double integral analog of Ramanujan’s Master Theorem. Further work in this direction is in progress.

The paper is organized as follows. In sections 2 and 3, we stated the analogs of Ramanujan’s Master Theorem for the k𝑘kitalic_k-gamma function and p𝑝pitalic_p-k𝑘kitalic_k gamma function along with their examples. In section 4, we established certain extensions of Ramanujan’s Master Theorem along with their corollaries and applications to calculate some definite integral. Next, in section 5, we have established a double integral identity of Ramanujan’s Master Theorem along with its examples and corollaries.

Further applications and examples of the theorems that we have established in this paper will be presented elsewhere in a more detailed investigation.

D

We provide numerical experiments that showcase the performance of the algorithms on generalized self-concordant objectives to complement the theoretical results.

In particular, they highlight that the simple step size strategy we propose is competitive with and sometimes outperforms other variants on many instances.

We propose an alternative step size that still ensures the monotonicity and domain-preserving properties,

Finally, we point out that the stateless step-size strategy can be viewed as a particular instance of a backtracking line search

the per iteration cost of the stateless step-size is about three times that of the simple step-size.

A

2}\theta_{k}\cdot\frac{C_{\mathrm{BS},\epsilon}^{(j)}}{k^{n+1}}.| italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_r italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - | italic_u | italic_T ( italic_r italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) | ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG italic_C start_POSTSUBSCRIPT roman_BS , italic_ϵ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_ARG start_ARG | italic_u | start_POSTSUPERSCRIPT italic_j + 1 + italic_ϵ end_POSTSUPERSCRIPT end_ARG + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_θ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋅ divide start_ARG italic_C start_POSTSUBSCRIPT roman_BS , italic_ϵ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_k start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT end_ARG .

We shall only need to assume (BS”(n)) for n=j𝑛𝑗n=jitalic_n = italic_j and, in addition, the results of Proposition 5.1. We shall follow the structure of the proof of Proposition 5.4.

Having obtained a bound on the boundary term, we can proceed as in the previous proof. We insert the bootstrap estimate (BS(n)) for n=j𝑛𝑗n=jitalic_n = italic_j and the estimate

We shall improve these estimates in the following. Note that we only assume estimates on the ∂vsubscript𝑣\partial_{v}∂ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT-derivatives, so we can just use the method of the proof of Proposition 5.4 with some adaptations. The crucial observation is that, while the differences Tn⁢(ϕ1−|u|⁢T⁢ϕ1)superscript𝑇𝑛subscriptitalic-ϕ1𝑢𝑇subscriptitalic-ϕ1T^{n}(\phi_{1}-|u|T\phi_{1})italic_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - | italic_u | italic_T italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) do not solve the wave equation, the error term is of the form ∂vTn+1⁢(r⁢ϕ1)subscript𝑣superscript𝑇𝑛1𝑟subscriptitalic-ϕ1\partial_{v}T^{n+1}(r\phi_{1})∂ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ( italic_r italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), over which we already have sharp control by Proposition 5.1.

We follow the proof of Proposition 5.4. We first need to acquire an estimate for ∂v(r⁢ϕ1)subscript𝑣𝑟subscriptitalic-ϕ1\partial_{v}(r\phi_{1})∂ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_r italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) on ΓfsubscriptΓ𝑓\Gamma_{f}roman_Γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT. We assume as a bootstrap assumption that

A

A fundamental theme in the representation theory of finite-dimensional algebras is classification of representation types. Let A𝐴Aitalic_A be a finite-dimensional algebra over an algebraically closed field 𝗄𝗄\mathsf{k}sansserif_k. Then A𝐴Aitalic_A is said to be representation-finite if there are only finitely many indecomposable A𝐴Aitalic_A-modules. Otherwise, A𝐴Aitalic_A is said to be representation-infinite. We say that a representation-infinite algebra A𝐴Aitalic_A is tame if all but finitely many d𝑑ditalic_d-dimensional indecomposable A𝐴Aitalic_A-modules can be organized in a one-parameter family, for each dimension d𝑑ditalic_d. A representation-infinite algebra A𝐴Aitalic_A is called wild if there exists a faithful exact 𝗄𝗄\mathsf{k}sansserif_k-linear functor from the module category of the free associative algebra 𝗄⁢⟨x,y⟩𝗄𝑥𝑦\mathsf{k}\langle x,y\ranglesansserif_k ⟨ italic_x , italic_y ⟩ to the module category of A𝐴Aitalic_A. Thanks to Drozd’s Finite-Tame-Wild Trichotomy (see [D]), we know that the representation type of any finite-dimensional algebra over 𝗄𝗄\mathsf{k}sansserif_k is exactly one of representation-finite, tame and wild.

The representation type of tensor product algebras has been studied in various contexts. In the 1970s, Bondarenko and Drozd [BD] considered the representation type of finite groups, while Auslander and Reiten [AR] dedicated their effort to the representation type of triangular matrix rings. Moving into the 1980s, mathematicians delved into the representation type of triangular matrix algebras over different classes of algebras, for example, see [S] for Nakayama algebras, [HM] for self-injective algebras, [L2] for radical square zero algebras, etc. The most recent progress in this field can be attributed to Leszczyn´´n\acute{\text{n}}over´ start_ARG n end_ARGski and Skowron´´n\acute{\text{n}}over´ start_ARG n end_ARGski, as clear in their series of papers [L1, LS1, LS2]. These papers provide a complete description of non-wild tensor product algebras. However, it is still open to distinguish representation-finite cases and tame cases from non-wild cases.

This paper is organized as follows. In Section 2, we review some basic materials and fix our notations, including simply connected algebras, tame concealed algebras, Tits form of algebras, etc. In Section 3, we consider the case that one of A𝐴Aitalic_A and B𝐵Bitalic_B is a representation-finite hereditary algebra, and the results here will reduce the problem to the case that one of A𝐴Aitalic_A and B𝐵Bitalic_B is a Nakayama algebra with radical square zero, which will be handled in Section 4. Finally, we give a complete list of minimal representation-infinite simply connected tensor product algebras in Section 5.

This paper is dedicated to providing a complete classification of representation-finite tensor products between simply connected algebras. We refer to Subsection 2.2 for the definitions of simply connected algebras.

It is worth mentioning that Leszczyn´´n\acute{\text{n}}over´ start_ARG n end_ARGski ([L1]) has classified weakly sincere non-wild tensor product algebras, in terms of quiver and relations. In our classification, we also rely on quiver and relations; however, we do not need the restriction on weakly sincere cases.

A

This is expected since better connectivity implies less consensus errors in each iteration, and the algorithms perform closer to the centralized gradient descent algorithm which is faster.

In this paper, we consider decentralized optimization over general directed networks and propose a novel Compressed Push-Pull method (CPP) that combines Push-Pull/𝒜⁢ℬ𝒜ℬ\mathcal{A}\mathcal{B}caligraphic_A caligraphic_B with a general class of unbiased compression operators. CPP enjoys large flexibility in both the compression method and the network topology. We show CPP achieves linear convergence rate under strongly convex and smooth objective functions.

Second, we consider a broadcast-like version of CPP (B-CPP) which also achieves linear convergence rate for strongly convex and smooth objective functions. B-CPP can be applied in an asynchronous broadcast setting and further reduce communication costs compared to CPP.

We propose CPP – a novel decentralized optimization method with communication compression. The method works under a general class of compression operators and is shown to achieve linear convergence for strongly convex and smooth objective functions over general directed graphs. To the best of our knowledge, CPP is the first method that enjoys linear convergence under such a general setting.

In this paper, we proposed two communication-efficient algorithms for decentralized optimization over a multi-agent network with general directed topology. First, we consider a novel communication-efficient gradient tracking based method, termed CPP, that combines the Push-Pull method with communication compression. CPP can be applied to a general class of unbiased compression operators and achieves linear convergence for strongly convex and smooth objective functions.

D

Certainly, we want to reduce the number of communications (or calls the regularizer gradient) as much as possible.

This is especially important when the problem (1) is a fairly personalized (λ≪Lmuch-less-than𝜆𝐿\lambda\ll Litalic_λ ≪ italic_L) and information from other nodes is not significant. To solve this problem and separate the oracle complexities for the saddle function and the composites, we base our method on sliding technique [27].

or open-source codes). This definition comes from the standard results, that for smooth functions the stepsize ∼1Lsimilar-toabsent1𝐿\sim\frac{1}{L}∼ divide start_ARG 1 end_ARG start_ARG italic_L end_ARG. We do not say that this is a good definition of L𝐿Litalic_L, but we only need it for the intuition. We also mention that we are interested in the case when λ≪Lmuch-less-than𝜆𝐿\lambda\ll Litalic_λ ≪ italic_L.

In this paper, we present a novel formulation for the Personalized Federated Learning Saddle Point Problem (1). This formulation incorporates a penalty term that accounts for the specific structure of the network and is applicable to both centralized and decentralized network settings. Additionally, we provide the lower bounds both on the communication and the number of local oracle calls required to solve problem (1). Furthermore, we have developed the novel methods (Algorithm 1, Algorithm 2, Algorithm 3) for this problem that are optimal up to logarithmic factor in certain scenarios (see Table 1). These algorithms are based on sliding or variance reduction techniques. The theoretical analysis and experimental evidence corroborate our methods. Moreover, we have customized our approach for neural network training.

To the best of our knowledge, this paper is the first to consider decentralized personalized federated saddle point problems, propose optimal algorithms and derives the computational and communication lower bounds for this setting. In the literature, there are works on general (non-personalized) SPPs. We make a detailed comparison with them in Appendix C. Due to the fact that we consider a personalized setting, we can have a significant gain in communications. For example, when λ=0𝜆0\lambda=0italic_λ = 0 or small enough in (1) the importance of local models increases and we may communicate less frequently.

A

|_{K}.∥ roman_d italic_φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_z ) italic_w ∥ → ∞ , divide start_ARG roman_d italic_φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_z ) italic_w end_ARG start_ARG ∥ roman_d italic_φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_z ) italic_w ∥ end_ARG → italic_E start_POSTSUBSCRIPT ∓ end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT .

Moreover, we have the following description of E±.subscript𝐸plus-or-minusE_{\pm}.italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT .

As the contact form α𝛼\alphaitalic_α is preserved by the flow φtsubscript𝜑𝑡\varphi_{t}italic_φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, we get by property (2) above E±⁢(z)⊂ker⁡α⁢(z)=ℝ⁢(H⁢(z)⊕V⁢(z))subscript𝐸plus-or-minus𝑧kernel𝛼𝑧ℝdirect-sum𝐻𝑧𝑉𝑧E_{\pm}(z)\subset\ker\alpha(z)=\mathbb{R}(H(z)\oplus V(z))italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_z ) ⊂ roman_ker italic_α ( italic_z ) = blackboard_R ( italic_H ( italic_z ) ⊕ italic_V ( italic_z ) ). In a first step, we will assume that E±⁢(z)∩ℝ⁢V⁢(z)={0}subscript𝐸plus-or-minus𝑧ℝ𝑉𝑧0E_{\pm}(z)\cap\mathbb{R}V(z)=\{0\}italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_z ) ∩ blackboard_R italic_V ( italic_z ) = { 0 } for every z∈Γ±𝑧subscriptΓplus-or-minusz\in\Gamma_{\pm}italic_z ∈ roman_Γ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT (we shall prove it later). Since the bundles E±subscript𝐸plus-or-minusE_{\pm}italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT are continuous, we deduce that there are two continuous functions r±:Γ±→ℝ:subscript𝑟plus-or-minus→subscriptΓplus-or-minusℝr_{\pm}:\Gamma_{\pm}\to\mathbb{R}italic_r start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT : roman_Γ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT → blackboard_R such that (2.6) holds.

As E±subscript𝐸plus-or-minusE_{\pm}italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT is preserved by the flow, we must have [X,H+r±⁢V]∈E±𝑋𝐻subscript𝑟plus-or-minus𝑉subscript𝐸plus-or-minus[X,H+r_{\pm}V]\in E_{\pm}[ italic_X , italic_H + italic_r start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT italic_V ] ∈ italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT; thus combining (2.7) and (2.6) we obtain the following Riccati equation:

E+|K=Euevaluated-atsubscript𝐸𝐾subscript𝐸𝑢E_{+}|_{K}=E_{u}italic_E start_POSTSUBSCRIPT + end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT and E−|K=Esconditionalsubscript𝐸𝐾subscript𝐸𝑠E_{-}|K=E_{s}italic_E start_POSTSUBSCRIPT - end_POSTSUBSCRIPT | italic_K = italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT and E±⁢(z)subscript𝐸plus-or-minus𝑧E_{\pm}(z)italic_E start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( italic_z ) depends continuously on z∈Γ±𝑧subscriptΓplus-or-minusz\in\Gamma_{\pm}italic_z ∈ roman_Γ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT;

A

\Uplambda^{q}(Q))fraktur_M start_POSTSUBSCRIPT bold_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ζ start_OPFUNCTION - roman_sst end_OPFUNCTION end_POSTSUPERSCRIPT ( roman_Λ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_Q ) ) can be shown to be impure in basic examples.

Let p:𝔐→ℳ:𝑝→𝔐ℳp\colon{\mathfrak{M}}\to{\mathcal{M}}italic_p : fraktur_M → caligraphic_M be the morphism to the coarse moduli space in any of the examples of categories (1)-(5) above.

The direct image of a pure complex of mixed Hodge modules along a projective morphism is pure, by Saito’s version of the decomposition theorem. So if the moduli space of objects ℳℳ{\mathcal{M}}caligraphic_M is projective, and we can define p!⁢ℚ¯𝔐subscript𝑝subscript¯ℚ𝔐p_{!}\underline{{\mathbb{Q}}}_{{\mathfrak{M}}}italic_p start_POSTSUBSCRIPT ! end_POSTSUBSCRIPT under¯ start_ARG blackboard_Q end_ARG start_POSTSUBSCRIPT fraktur_M end_POSTSUBSCRIPT in the derived category of mixed Hodge modules, then Theorem B implies purity of 𝐇c⁡(𝔐,ℚ)subscript𝐇𝑐𝔐ℚ\operatorname{\mathbf{H}}_{c}({\mathfrak{M}},{\mathbb{Q}})bold_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( fraktur_M , blackboard_Q ). This concludes case (2) of Theorem A in the case of projective S𝑆Sitalic_S.

Assume that 𝔐𝔐{\mathfrak{M}}fraktur_M is a moduli stack of objects in a category 𝒞𝒞\mathscr{C}script_C satisfying the conditions of Theorem 6.1, and assume moreover that the good moduli space ℳℳ{\mathcal{M}}caligraphic_M is a projective variety, and 𝔐𝔐{\mathfrak{M}}fraktur_M is exhausted by global quotient stacks. Then the mixed Hodge structure on 𝐇BM⁡(𝔐,ℚ)superscript𝐇BM𝔐ℚ\operatorname{\mathbf{H}}^{\operatorname{BM}}({\mathfrak{M}},{\mathbb{Q}})bold_H start_POSTSUPERSCRIPT roman_BM end_POSTSUPERSCRIPT ( fraktur_M , blackboard_Q ) is pure.

Let 𝔐𝔐{\mathfrak{M}}fraktur_M be one of the moduli stacks of objects in the 2CY categories of type (1), (2) or (3) described above, where in case (2) we assume that S𝑆Sitalic_S is projective.

D

(4)⇒(1)⇒41(4)\Rightarrow(1)( 4 ) ⇒ ( 1 ) Let x𝑥xitalic_x be a nonzero nonunit element in R⁢(+)M𝑅𝑀R\mathop{(\!+\!)}Mitalic_R start_BIGOP ( + ) end_BIGOP italic_M. If x=y⁢z𝑥𝑦𝑧x=yzitalic_x = italic_y italic_z, then without loss of generality, x∼ysimilar-to𝑥𝑦x\sim yitalic_x ∼ italic_y, and so y=x⁢t𝑦𝑥𝑡y=xtitalic_y = italic_x italic_t for some t𝑡titalic_t. Now x=x⁢t⁢z𝑥𝑥𝑡𝑧x=xtzitalic_x = italic_x italic_t italic_z, and so z𝑧zitalic_z is a unit since R⁢(+)M𝑅𝑀R\mathop{(\!+\!)}Mitalic_R start_BIGOP ( + ) end_BIGOP italic_M is présimplifiable. Therefore, any factorization of x𝑥xitalic_x is of length 1111, and so R⁢(+)M𝑅𝑀R\mathop{(\!+\!)}Mitalic_R start_BIGOP ( + ) end_BIGOP italic_M is a UFR.

The authors would like to thank the referee for careful reading of the paper and very helpful comments.

is bounded. For the element 0∈R0𝑅0\in R0 ∈ italic_R, this property coincides with the notion of being U-bounded which is defined in [agargun2001factorization, Section 4], using the language of U-factorization. Although we avoided discussing U-factorizations in this paper, nevertheless we use the term U-bounded in this case.

where Fi∈Ssubscript𝐹𝑖𝑆F_{i}\in Sitalic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S, F′∈I1superscript𝐹′subscript𝐼1F^{\prime}\in I_{1}italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, F′′∈I2superscript𝐹′′subscript𝐼2F^{\prime\prime}\in I_{2}italic_F start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (and, of course, Fi=0subscript𝐹𝑖0F_{i}=0italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 for every i⪈kgreater-than-and-not-equals𝑖𝑘i\gneq kitalic_i ⪈ italic_k). For A∈S𝐴𝑆A\in Sitalic_A ∈ italic_S let ℓ⁢(A)ℓ𝐴\ell(A)roman_ℓ ( italic_A ) denote the sum of the monomials of the least total degree and let T⁢(A)𝑇𝐴T(A)italic_T ( italic_A ) denote the total degree of A𝐴Aitalic_A. We have T⁢(ℓ⁢(G⁢H))≥2𝑇ℓ𝐺𝐻2T(\ell(GH))\geq 2italic_T ( roman_ℓ ( italic_G italic_H ) ) ≥ 2 since none of the elements G𝐺Gitalic_G and H𝐻Hitalic_H has a nonzero constant term for otherwise they would be units in R𝑅Ritalic_R. So β1subscript𝛽1\beta_{1}italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT must appear on the right hand side and the only way for this is for F1subscript𝐹1F_{1}italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to have 1111 as the constant term. After removing β1subscript𝛽1\beta_{1}italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT from each side we get

Current results on the factorization properties of R⁢(+)M𝑅𝑀R\mathop{(\!+\!)}Mitalic_R start_BIGOP ( + ) end_BIGOP italic_M are mostly for the case where R𝑅Ritalic_R is a domain (see [agargun2001factorization, anderson1996factorization, anderson2009idealization, axtell2002u, axtell2017factorizations, chang2013factorization]). The goal of this paper is to generalize some of these results to arbitrary commutative rings.

A

We mention an interesting special case of Theorem 2.2 when H𝐻Hitalic_H is a Wigner matrix and A𝐴Aitalic_A itself is chosen from a Wigner ensemble that is independent of H𝐻Hitalic_H and x𝑥xitalic_x. In this case Theorem 2.2 implies555The condition on ⟨Å2⟩delimited-⟨⟩superscript̊𝐴2\langle\mathring{A}^{2}\rangle⟨ over̊ start_ARG italic_A end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ is satisfied since ⟨A2⟩=1+o⁢(1)delimited-⟨⟩superscript𝐴21𝑜1\langle A^{2}\rangle=1+o(1)⟨ italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ = 1 + italic_o ( 1 ) and ⟨A⟩=o⁢(1)delimited-⟨⟩𝐴𝑜1\langle A\rangle=o(1)⟨ italic_A ⟩ = italic_o ( 1 ) with very high probability. Moreover, the scDos ρxsuperscript𝜌𝑥\rho^{x}italic_ρ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT is very close to a rescaled semicircle law with radius 2⁢1+x221superscript𝑥22\sqrt{1+x^{2}}2 square-root start_ARG 1 + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG with very high probability in the joint probability space of H𝐻Hitalic_H and A𝐴Aitalic_A, hence the condition (2.8) holds for all i∈[N⁢ϵ′,N⁢(1−ϵ′)]𝑖𝑁superscriptitalic-ϵ′𝑁1superscriptitalic-ϵ′i\in[N\epsilon^{\prime},N(1-\epsilon^{\prime})]italic_i ∈ [ italic_N italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_N ( 1 - italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] for some ϵ′>0superscriptitalic-ϵ′0\epsilon^{\prime}>0italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > 0. that for any fixed pair of Wigner matrices A,H𝐴𝐻A,Hitalic_A , italic_H from a high probability set, the universality of the i𝑖iitalic_i-th gap statistics of H+x⁢A𝐻𝑥𝐴H+xAitalic_H + italic_x italic_A for i∈[N⁢ϵ′,N⁢(1−ϵ′)]𝑖𝑁superscriptitalic-ϵ′𝑁1superscriptitalic-ϵ′i\in[N\epsilon^{\prime},N(1-\epsilon^{\prime})]italic_i ∈ [ italic_N italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_N ( 1 - italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ] is solely

𝐏H,A⁡(i-th gap statistics of H+x⁢A is universal)=1−𝒪⁢(N−ϵ).subscript𝐏𝐻𝐴i-th gap statistics of H+x⁢A is universal1𝒪superscript𝑁italic-ϵ\operatorname{\mathbf{P}}_{H,A}\Bigl{(}\text{$i$-th gap statistics of $H+xA$ %

so that for all H∈Ωi𝐻subscriptΩ𝑖H\in\Omega_{i}italic_H ∈ roman_Ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the statistics of the i𝑖iitalic_i-th rescaled gap of the eigenvalues λixsuperscriptsubscript𝜆𝑖𝑥\lambda_{i}^{x}italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT of Hxsuperscript𝐻𝑥H^{x}italic_H start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT is universal, i.e.

is universal}\Bigr{)}=1-\mathcal{O}(N^{-\epsilon}).bold_P start_POSTSUBSCRIPT italic_H , italic_A end_POSTSUBSCRIPT ( italic_i -th gap statistics of italic_H + italic_x italic_A is universal ) = 1 - caligraphic_O ( italic_N start_POSTSUPERSCRIPT - italic_ϵ end_POSTSUPERSCRIPT ) .

Empirically we find that the convergence towards the universal gap statistics in (2.9) is much slower for the monoparametric ensemble compared to GUE, cf. Figure 2. While even for 2×2222\times 22 × 2 GUE matrices the empirical gap distribution is already very close to the Gaudin-Mehta distribution (see Section 2.3), we observe the same effect only for large monoparametric matrices.

A

Ψ⁢(Sh,p)=ΦSh,pp=hΨsubscript𝑆ℎ𝑝subscriptsuperscriptΦ𝑝subscript𝑆ℎ𝑝ℎ\Psi(S_{h,p})=\Phi^{p}_{S_{h,p}}=hroman_Ψ ( italic_S start_POSTSUBSCRIPT italic_h , italic_p end_POSTSUBSCRIPT ) = roman_Φ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_h , italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_h.

Note that, using Remark 3 about flat Hadamard spaces, one can prove that previous corollaries (5.16 &\&& 5.17) are equivalent to the classical results about Fitzpatrick transforms in Hilbert spaces.

As a prelude to this paper, we present some preliminary and basic results about geodesic and Hadamard spaces. More details can be found in [12, 8, 36, 2, 14] and the references cited therein.

The next sections of this manuscript are organized as follows. In section 2, some basic definitions and results are collected, which are used directly or indirectly in the following sections. More precisely, the notions of geodesic spaces, CN-inequality, CAT⁢(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 )-spaces, Hadamard spaces, quasilinearization in abstract metric spaces, dual space of a Hadamard space, linear dual space of a Hadamard space and some notions related to convex analysis in Hadamard spaces, are given. Section 3 devoted to the notion of flat Hadamard spaces. A characterization result together with some new examples are considered. The most important goal of section 4 is to define the concept of p𝑝pitalic_p-Fenchel conjugate and prove the p𝑝pitalic_p-Fenchel-Young inequality. Finally, section 5 is devoted to prove a representation result for maximal monotone operators in terms of certain proper, convex, l.s.c. extended real-valued functions on X×X◆𝑋superscript𝑋◆X\times X^{\scalebox{0.7}{${}^{\lozenge}$}}italic_X × italic_X start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ◆ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT. In this section, the notion of p𝑝pitalic_p-Fitzpatrick transform for a set-valued map from a Hadamard space X𝑋Xitalic_X to its linear dual X◆superscript𝑋◆X^{\scalebox{0.7}{${}^{\lozenge}$}}italic_X start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ◆ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT is introduced and considered. Then some characterization results for maximal monotonicity of set-valued maps from a Hadamard space X𝑋Xitalic_X to its linear dual X◆superscript𝑋◆X^{\scalebox{0.7}{${}^{\lozenge}$}}italic_X start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ◆ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT are given.

Recall that [26, Definition 3.1] a Hadamard space (X,d)𝑋𝑑(X,d)( italic_X , italic_d ) is called flat if equality holds in the CN-inequality (1). Some examples of flat Hadamard spaces are Hilbert spaces and closed unit balls in any Hilbert space.

A

^{2}+z^{2}=0italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 1 - italic_α end_ARG start_ARG 1 + italic_β end_ARG italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 + italic_α end_ARG start_ARG 1 - italic_γ end_ARG italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0

is as follows. The Sklyanin algebra S⁢(α,β,γ)𝑆𝛼𝛽𝛾S(\alpha,\beta,\gamma)italic_S ( italic_α , italic_β , italic_γ ) is a free 𝐂𝐂\mathbf{C}bold_C-algebra on four generators

α+β+γ+α⁢β⁢γ=0𝛼𝛽𝛾𝛼𝛽𝛾0\alpha+\beta+\gamma+\alpha\beta\gamma=0italic_α + italic_β + italic_γ + italic_α italic_β italic_γ = 0.

The norm-closure of a self-adjoint representation of the algebra S⁢(α,β,γ)𝑆𝛼𝛽𝛾S(\alpha,\beta,\gamma)italic_S ( italic_α , italic_β , italic_γ ) by

The algebra S⁢(α,β,γ)𝑆𝛼𝛽𝛾S(\alpha,\beta,\gamma)italic_S ( italic_α , italic_β , italic_γ ) is twisted homogeneous coordinate ring of an elliptic curve in the Jacobi form ℰ⊂𝐂⁢P3ℰ𝐂superscript𝑃3\mathscr{E}\subset\mathbf{C}P^{3}script_E ⊂ bold_C italic_P start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT,

C

We first show that there exists a family of non-hamiltonian graphs 𝒢𝒢\mathcal{G}caligraphic_G for which supG∈𝒢{rG}=1subscriptsupremum𝐺𝒢subscript𝑟𝐺1\sup_{G\in\mathcal{G}}\{r_{G}\}=1roman_sup start_POSTSUBSCRIPT italic_G ∈ caligraphic_G end_POSTSUBSCRIPT { italic_r start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT } = 1. We then investigate possible values of rGsubscript𝑟𝐺r_{G}italic_r start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT; we describe an explicit construction that produces families of graphs for which rGsubscript𝑟𝐺r_{G}italic_r start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT approaches k−1k𝑘1𝑘\frac{k-1}{k}divide start_ARG italic_k - 1 end_ARG start_ARG italic_k end_ARG. This construction depends on finding a collection of edges in a non-hamiltonian graph for which every edge is connected by a hamiltonian path containing the other edges. We call such edge sets H-path connected (see Definition 2.9, Theorem 2.14).

In Section 3 we consider another extreme; rather than limiting ourselves to non-hamiltonian graphs, we construct an infinite family of hamiltonian graphs containing a minimal edge set (see Theorem 3.2).

We use a non-hamiltonian graph containing an H-path connected set of edges to construct a family of non-hamiltonian graphs which are highly pair-strung as follows.

In this section, we consider graphs that have many pairs of vertices connected by a hamiltonian path. We will quantify this in the following definition.

In the previous section, we demonstrated the existence of graphs that are as close to being hamiltonian-connected as possible without being hamiltonian. In this section, we explore an alternative extreme: we construct a family of hamiltonian-connected (and therefore hamiltonian) graphs that have a minimal number of edges and minimal number of minimum degree vertices. This partially answers a question of Modalleliyan and Omoomi ([omoomi]), which asked whether there exist minimally hamiltonian-connected graphs (i.e. hamiltonian-connected graphs such that the removal of any edge results in a non-hamiltonian-connected graph) with maximal vertex degree ΔΔ\Deltaroman_Δ so that 3≤Δ≤⌈n/2⌉.3Δ𝑛23\leq\Delta\leq\lceil n/2\rceil.3 ≤ roman_Δ ≤ ⌈ italic_n / 2 ⌉ . While preparing this paper, Zhan [Zhan] completely answered this question in the affirmative. In particular, he constructs a cubic graph which is minimally hamiltonian-connected. However, our construction is sufficiently distinct from Zhan’s that we include it out of interest to the field. Our construction is highly symmetric and inductive.

A

SIGMA enjoys a globally convergent first phase (damped phase) with linear rate followed by a local composite rate. The theorem shows that the proposed method can achieve a local super-linear or quadratic convergence rate. A complete convergence behavior, that accounts for when the method alternates between SIGMA and the Newton method is presented in Theorem 3.2.

We also study the Galerkin model that is generated by the Nyström method (sampling according to some distribution function) and we show that the coarse direction can be seen as a randomized approximation of the Newton direction. In Theorem 3.3, we prove the local super-linear convergence rate of SIGMA with probability 1−ρ1𝜌1-\rho1 - italic_ρ.

According to Lemma 8, we can infer the following about the convergence rate of the coarse model: first, note that the root of

We show that the new strategies offer convergence results with higher probabilities than when considering the uniform sampling. In addition, we verify the theoretical results through numerical experiments. We illustrate that SIGMA can achieve both super-linear and quadratic convergence rates on realistic models. We, further, report substantial improvements in the convergence rate of SIGMA with the alternative strategies against the conventional SIGMA. As a future direction, we plan to extensively examine the practical behavior of SIGMA using variants of the new adaptive techniques. In particular, we plan to introduce adaptive and iteration-dependent τksubscript𝜏𝑘\tau_{k}italic_τ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. We also consider replacing the ∥⋅∥1\|\cdot\|_{1}∥ ⋅ ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in the definition of adaptive sampling strategy with other norms. We anticipate that these modifications will offer better approximations of the Newton decrement.

The computational bottlenecks in Algorithm 1 are: (i) the construction of the coarse step in Eq. 18, and (ii) the checking process in order to avoid an ineffective search direction. To overcome both, we select the prolongation operator as in Definition 1, thus the Galerkin model is generated based on the low-rank approximation of the Hessian matrix through the Nyström method. In particular, we show that if SIGMA enters the super-linear phase, then the checking process can be omitted. Moreover, performing the Nyström method we are able to overcome the computational issues related to the construction of the coarse direction (see Section 2.5, Remark 1 and Remark 4 for details on the efficient implementation of Algorithm 1 using the Nyström method).

A

If K𝒪subscript𝐾𝒪K_{\mathcal{O}}italic_K start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT is the maximal hyperspecial compact subgroup of G𝐺Gitalic_G, so that in particular res𝒪=resK𝒪subscriptres𝒪subscriptressubscript𝐾𝒪\operatorname{res}_{\mathcal{O}}=\operatorname{res}_{K_{\mathcal{O}}}roman_res start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT = roman_res start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT end_POSTSUBSCRIPT, we expect that the only roots of unity ζ𝜁\zetaitalic_ζ that appear are the well-known Δ⁢(x¯ρ)∈{±1}Δsubscript¯𝑥𝜌plus-or-minus1\Delta(\bar{x}_{\rho})\in\{\pm 1\}roman_Δ ( over¯ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ) ∈ { ± 1 } (see [Lu84a, §6.7]) for certain families of unipotent representations of the finite groups of types E7subscript𝐸7E_{7}italic_E start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT and E8subscript𝐸8E_{8}italic_E start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT. But for other maximal compact subgroups, Proposition 13.9 shows that in 𝖲𝖫nsubscript𝖲𝖫𝑛{\mathsf{SL}}_{n}sansserif_SL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT for example, new roots of unity can appear.

Each group G′∈InnTp⁢(G)superscript𝐺′superscriptInnT𝑝𝐺G^{\prime}\in{\mathrm{InnT}}^{p}(G)italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_InnT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_G ) has a finite collection of conjugacy classes of maximal compact open subgroups max⁡(G′)superscript𝐺′\max(G^{\prime})roman_max ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). These are classified in terms of the theory of [BT72, IM65] (see Section 7). A compact group K′∈max⁡(G′)superscript𝐾′superscript𝐺′K^{\prime}\in\max(G^{\prime})italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_max ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) has a finite quotient K¯′superscript¯𝐾′\overline{K}^{\prime}over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT that is the group of k𝑘kitalic_k-points of a (possibly disconnected) reductive group over a finite field k𝑘kitalic_k. Write Run⁢(K¯′)subscript𝑅unsuperscript¯𝐾′R_{\mathrm{un}}(\overline{K}^{\prime})italic_R start_POSTSUBSCRIPT roman_un end_POSTSUBSCRIPT ( over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for the ℂℂ\mathbb{C}blackboard_C-vector space spanned by the irreducible unipotent representations of K¯′superscript¯𝐾′\overline{K}^{\prime}over¯ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. As mentioned above, for connected finite reductive groups, Lusztig [Lu84a] defined the nonabelian Fourier transform, which is the change-of-basis matrix between the basis of irreducible unipotent characters and the basis of unipotent almost characters. This is recalled in Section 5. We need to define an extension of this map to disconnected finite groups in the spirit of [Lu86] and [DM90, Section 5]. To fit with our picture, we define a nonabelian Fourier transform for the representations of the pure inner twists of the finite (possibly disconnected) reductive group K¯¯𝐾\overline{K}over¯ start_ARG italic_K end_ARG, where K∈max⁡(G)𝐾𝐺K\in\max(G)italic_K ∈ roman_max ( italic_G ). See Section 6. The point is that this transform gives an involution

In particular, while the characters of irreducible representations of a finite connected reductive group do not have good intrinsic stability properties, Lusztig’s almost characters, certain class functions defined in terms of traces of character sheaves, do. The transition matrix between characters and almost characters is Lusztig’s famous nonabelian Fourier transform ([Lu84a], [Lu18]). If we can lift this Fourier transform to the setting of p𝑝pitalic_p-adic groups, we might be able to lift stability properties of combinations of almost characters of finite reductive groups, as in [MW03, Theorem 4.3].

We remark that extra roots of unity already appear in relation with the nonabelian Fourier transform for finite reductive groups, although we do not know if this is a related issue. In that setting there are three bases of the Grothendieck group of unipotent characters:

This is similar to the calculation for the nonabelian Fourier transform in the case of finite reductive groups. We compute:

C

≥n⁢C1⁢24absent𝑛subscript𝐶142\displaystyle\geq nC_{1}\sqrt[4]{2}≥ italic_n italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT nth-root start_ARG 4 end_ARG start_ARG 2 end_ARG

The number of points that can be placed in the grid n×n𝑛𝑛n\times nitalic_n × italic_n such that no three points are collinear satisfies the lower bound

The number of points that can be placed in the grid n×n×⋯×n⁢(d⁢t⁢i⁢m⁢e⁢s)=nd𝑛𝑛normal-⋯𝑛𝑑𝑡𝑖𝑚𝑒𝑠superscript𝑛𝑑n\times n\times\cdots\times n~{}(d~{}times)=n^{d}italic_n × italic_n × ⋯ × italic_n ( italic_d italic_t italic_i italic_m italic_e italic_s ) = italic_n start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT for all d∈ℕ𝑑ℕd\in\mathbb{N}italic_d ∈ blackboard_N and with d≥2𝑑2d\geq 2italic_d ≥ 2 such that no three points are collinear satisfies the lower bound

in the grid n×n𝑛𝑛n\times nitalic_n × italic_n with no three collinear. Various upper bound to the problem had also been conjectured. For instance it is conjectured that (see [5]) the number of points that can be placed in an n×n𝑛𝑛n\times nitalic_n × italic_n grid so that no three are collinear has the optimal solution c⁢n𝑐𝑛cnitalic_c italic_n with

The number of points that can be placed in the grid n×n×n𝑛𝑛𝑛n\times n\times nitalic_n × italic_n × italic_n such that no three points are collinear satisfies the lower bound

D

This completes the proof of (6.5). To get the constant A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in (6.6), we can do the same argument but use Corollary 4.6 instead of Theorem 4.11, with A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is given by (4.15) for p=2𝑝2p=2italic_p = 2.

[BDS20] for the further analysis of its extremisers and several different approaches. For sums of squares on unimodular Lie groups, equivalent conditions for the Nash inequality have been investigated e.g. in [VSCC93, Theorem II.5.2], in particular, it implies a number of estimates for the heat operator e−t⁢Δ𝔾superscript𝑒𝑡subscriptΔ𝔾e^{-t\Delta_{\mathbb{G}}}italic_e start_POSTSUPERSCRIPT - italic_t roman_Δ start_POSTSUBSCRIPT blackboard_G end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, see e.g. [VSCC93, Theorem II.3.2]. We refer to Davies [Dav89] for a general setting and relations to the hypercontractivity of the heat semigroup.

In this section we discuss the Nash inequality on general Lie groups, and on graded groups, and its immediate application to the decay of solutions to the heat equation for the sub-Laplacian on stratified groups. For stratified Lie groups, the Nash inequality would take the form

On the Euclidean space, again it is well-known that the Nash inequality is equivalent to the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-log-Sobolev inequality. This is also the case in our setting on general graded Lie groups. In fact, our proof of Corollary 4.6 shows how Nash inequality would imply the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-log-Sobolev inequality. The converse argument is given in the proof of the next theorem, which gives the Nash inequality as its special case with a=1𝑎1a=1italic_a = 1. In particular, if 𝔾𝔾\mathbb{G}blackboard_G is a stratified group, the following theorem with a=1𝑎1a=1italic_a = 1 and ℛℛ\mathcal{R}caligraphic_R being the sub-Laplacian implies (6.1) since

As an immediate application of the Nash inequality, one can compute the decay rate for the heat equation for the sub-Laplacian, following Nash’s argument. We give a short application just to illustrate such idea, and refer to more general related analysis in the context of symmetric Markovian semigroups to e.g. [VSCC93, Section II.5].

D

Note that an alternative definition of truth-complexity is given in [5] and that our definition is stated as a theorem.

For any axiom A𝐴Aitalic_A of 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA, the truth-complexity of A𝐴Aitalic_A is less than ω+5𝜔5\omega+5italic_ω + 5.

The theory 𝖯𝖠⁢(X)𝖯𝖠𝑋\mathsf{PA}(X)sansserif_PA ( italic_X ) is the pseudo Π11subscriptsuperscriptΠ11\Pi^{1}_{1}roman_Π start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT pendant of 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA. That is, 𝖯𝖠⁢(X)𝖯𝖠𝑋\mathsf{PA}(X)sansserif_PA ( italic_X ) contains (i) the axioms of 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA and (ii) induction axioms for all formulas in the language, including those with the set variable X𝑋Xitalic_X.

Furthermore—in a large swathe of cases—a theory’s rank in this well-founded reflection ordering coincides with its proof-theoretic ordinal, where the proof-theoretic ordinal of a theory T𝑇Titalic_T is the supremum of the T𝑇Titalic_T-provably well-founded primitive recursive linear orders. Interest in proof-theoretic ordinals comes from the work of Gentzen, who, using only finitary methods and transfinite induction along a primitive recursive presentation of ε0subscript𝜀0\varepsilon_{0}italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, proved the consistency of 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA. By Gödel’s Second Incompleteness Theorem, this implies that 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA does not prove transfinite induction over any primitive recursive presentation of ε0subscript𝜀0\varepsilon_{0}italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Yet for any α<ε0𝛼subscript𝜀0\alpha<\varepsilon_{0}italic_α < italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA proves transfinite induction for some primitive recursive presentation of α𝛼\alphaitalic_α. That is ε0subscript𝜀0\varepsilon_{0}italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the supremum of ordinals for which 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA verifies transfinite induction, i.e., the proof-theoretic ordinal of 𝖯𝖠𝖯𝖠\mathsf{PA}sansserif_PA. This engendered a prominent program in proof theory—known as ordinal analysis—in which theories’ proof-theoretic ordinals are determined.

By Lemma 3.5, for each Aisubscript𝐴𝑖A_{i}italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, the truth-complexity of Aisubscript𝐴𝑖A_{i}italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is less than ω+5𝜔5\omega+5italic_ω + 5. By Proposition 3.7, the truth-complexity of 𝖳𝖨⁢(εγ,⋁i≤kψi)𝖳𝖨subscript𝜀𝛾subscript𝑖𝑘subscript𝜓𝑖\mathsf{TI}(\varepsilon_{\gamma},\bigvee_{i\leq k}\psi_{i})sansserif_TI ( italic_ε start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT , ⋁ start_POSTSUBSCRIPT italic_i ≤ italic_k end_POSTSUBSCRIPT italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is at most εγ+2subscript𝜀𝛾2\varepsilon_{\gamma}+2italic_ε start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT + 2. Thus, by Proposition 3.11

A

_{k}k[G]italic_c start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( caligraphic_F ) end_POSTSUBSCRIPT : italic_E start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( caligraphic_F ) → italic_E start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( caligraphic_F ) ⊗ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_k [ italic_G ]

On the other hand, det(V)𝑉\det(V)roman_det ( italic_V ) acquires a left k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure via the determinant representation of V𝑉Vitalic_V. By post-composing with the antipode of the Hopf algebra k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ] (i.e. pre-composing the representation with inversion on G𝐺Gitalic_G), we get a right k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure on det(V)𝑉\det(V)roman_det ( italic_V ). In a similar way as above, this induces a right k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure on the algebra Sym∙⁡(det(ℱ)∨⊗kdet(V))superscriptSym∙subscripttensor-product𝑘superscriptℱ𝑉\operatorname{Sym}^{\bullet}(\det(\mathcal{F})^{\vee}\otimes_{k}\det(V))roman_Sym start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( roman_det ( caligraphic_F ) start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ⊗ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_det ( italic_V ) ).

The vector space V𝑉Vitalic_V acquires a left k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure V→V⊗k⁢[G]→𝑉tensor-product𝑉𝑘delimited-[]𝐺V\rightarrow V\otimes k[G]italic_V → italic_V ⊗ italic_k [ italic_G ] via the representation ρ:G→GL⁡(V):𝜌→𝐺GL𝑉\rho:G\rightarrow\operatorname{GL}(V)italic_ρ : italic_G → roman_GL ( italic_V ). We equip V∨superscript𝑉V^{\vee}italic_V start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT with the dual right comodule structure c:V∨→V∨⊗k⁢[G]:𝑐→superscript𝑉tensor-productsuperscript𝑉𝑘delimited-[]𝐺c:V^{\vee}\rightarrow V^{\vee}\otimes k[G]italic_c : italic_V start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT → italic_V start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ⊗ italic_k [ italic_G ]. This induces a comodule structure ℱ⊗kV∨→(ℱ⊗kV∨)⊗k⁢[G]→subscripttensor-product𝑘ℱsuperscript𝑉tensor-productsubscripttensor-product𝑘ℱsuperscript𝑉𝑘delimited-[]𝐺\mathcal{F}\otimes_{k}V^{\vee}\rightarrow(\mathcal{F}\otimes_{k}V^{\vee})%

Suppose that the sheaf 𝒢𝒢\mathcal{G}caligraphic_G admits a k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure. This induces a comodule structure on T⁢(𝒢)𝑇𝒢T(\mathcal{G})italic_T ( caligraphic_G ), where we equip each graded component 𝒢⊗jsuperscript𝒢tensor-productabsent𝑗\mathcal{G}^{\otimes j}caligraphic_G start_POSTSUPERSCRIPT ⊗ italic_j end_POSTSUPERSCRIPT with the tensor comodule structure. The ideal used to form Sym∙⁡(𝒢)superscriptSym∙𝒢\operatorname{Sym}^{\bullet}(\mathcal{G})roman_Sym start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( caligraphic_G ) is a k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-subcomodule. It follows that the quotient Sym∙⁡(𝒢)superscriptSym∙𝒢\operatorname{Sym}^{\bullet}(\mathcal{G})roman_Sym start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( caligraphic_G ) acquires a k⁢[G]𝑘delimited-[]𝐺k[G]italic_k [ italic_G ]-comodule structure that is compatible with the multiplication and the grading.

Since this comodule structure is compatible with the algebra multiplication, it translates into a left action of the algebraic group G𝐺Gitalic_G on the scheme H⁢(ℱ,V)𝐻ℱ𝑉H(\mathcal{F},V)italic_H ( caligraphic_F , italic_V ).

D

Inspired by their works, we consider the conformal soliton version of Perelman’s conjecture, that is, rotational symmetry of gradient conformal solitons with nonnegative scaler curvature.

We consider (3)3(3)( 3 ) of Theorem 1.1. By the same argument as in the proof of Theorem 1.1, one can get formulas of the warped product manifold of the warping function (0<)|∇F|=F′(r)(0<)|\nabla F|=F^{\prime}(r)( 0 < ) | ∇ italic_F | = italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ). For a,b,c,d=2,3,⋯⁢n.formulae-sequence𝑎𝑏𝑐𝑑23⋯𝑛a,b,c,d=2,3,\cdots n.italic_a , italic_b , italic_c , italic_d = 2 , 3 , ⋯ italic_n .

As is well known, Gradient conformal solitons were studied by Cheeger-Colding ([8], see also [15]). Recently, the special case of it has been studied. It is the gradient Yamabe soliton:

Inspired by their works, we consider the conformal soliton version of Perelman’s conjecture, that is, rotational symmetry of gradient conformal solitons with nonnegative scaler curvature.

Conformal gradient solitons were studied by Cheeger and Colding [8]. They gave a characterization of warped product manifolds. Inspired by their work, we will drastically simplify the proof of the classification result of it given by Tashiro [15] (see also Catino-Mantegazza-Mazzieri’s work [7]).

D

}\rangle+L_{k+1}V(x_{k+1},x_{k})+\delta.italic_f ( italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ⩽ italic_f ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + ⟨ ∇ italic_f ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ + italic_L start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT italic_V ( italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_δ .

Let us use the reasoning in the proof of Theorem 3.1 for g⁢(x)=∇f⁢(x)𝑔𝑥∇𝑓𝑥g(x)=\nabla f(x)italic_g ( italic_x ) = ∇ italic_f ( italic_x ). Taking into account (2.3), for any x∈Q𝑥𝑄x\in Qitalic_x ∈ italic_Q, we have,

Therefore, for Lk+1⩾Lsubscript𝐿𝑘1𝐿L_{k+1}\geqslant Litalic_L start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ⩾ italic_L and taking into account (5.1) we obtain

After summing inequalities (6.17) and (6.18), we get, that the following inequalities hold for any x,y∈Q𝑥𝑦𝑄x,y\in Qitalic_x , italic_y ∈ italic_Q:

Taking into account (2.2) for α>0𝛼0\alpha>0italic_α > 0, after summing the inequalities (6.13) and (6.14), we have

D

Although our bound is more general, as it applies to all backbones H𝐻Hitalic_H as long as H⁢[C]𝐻delimited-[]𝐶H[C]italic_H [ italic_C ] is a forest, a simple comparison of both bounds shows that for trees (and forests) our bound is worse in all cases, even for small λ𝜆\lambdaitalic_λ and large χ⁢(G)𝜒𝐺\chi(G)italic_χ ( italic_G ). This is mainly because we assign a single color to I𝐼Iitalic_I naively instead of trying to optimize it.

The λ𝜆\lambdaitalic_λ-backbone coloring problem was studied for several classes of graphs, for example split graphs [5], planar graphs [3], complete graphs [6], and for several classes of backbones: matchings and disjoint stars [5], bipartite graphs [6] and forests [3].

Although our bound is more general, as it applies to all backbones H𝐻Hitalic_H as long as H⁢[C]𝐻delimited-[]𝐶H[C]italic_H [ italic_C ] is a forest, a simple comparison of both bounds shows that for trees (and forests) our bound is worse in all cases, even for small λ𝜆\lambdaitalic_λ and large χ⁢(G)𝜒𝐺\chi(G)italic_χ ( italic_G ). This is mainly because we assign a single color to I𝐼Iitalic_I naively instead of trying to optimize it.

Still, we can pose a question: is there a way to amend our approach to make it suitable also for split graphs and improve the existing bound at least for some subset of graphs?

In [15] it was proved that the radio labeling problem can be solved in polynomial time for graphs for which k𝑘kitalic_k-coloring can be found in polynomial time for some fixed k𝑘kitalic_k, e.g. for planar graphs or graphs with bounded treewidth.

C

H𝐻Hitalic_H is contained in A4superscript𝐴4A^{4}italic_A start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT.

That is, we give a version of the Breuillard–Green–Tao theorem with polynomial bounds in bounded rank, over arbitrary fields.

this is the content of the celebrated structure theorem for approximate groups proved by Breuillard, Green, and Tao [BGT].

Theorem 1.1\wrtusdrfBGT corresponds closely to the essentially qualitative conjecture made by Lindenstrauss (Lindenstrauss, personal communication).

Breuillard, Green, and Tao described the statement of Theorem 1.1\wrtusdrfBGT as the Helfgott–Lindenstrauss conjecture.

D

Meanwhile, 𝕆¯λ∩κλ−1⁢(x,y)subscript¯𝕆𝜆superscriptsubscript𝜅𝜆1𝑥𝑦\overline{\mathbb{O}}_{\lambda}\cap\kappa_{\lambda}^{-1}(x,y)over¯ start_ARG blackboard_O end_ARG start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ∩ italic_κ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x , italic_y ) consisting of elements

Gλ⁢(x,y)={g∈Gγ∣g⋅(yu0x)=(yu′0x)⁢ for u′∈HomΩ⁡(U,W)}subscript𝐺𝜆𝑥𝑦conditional-set𝑔subscript𝐺𝛾⋅𝑔matrix𝑦𝑢0𝑥matrix𝑦superscript𝑢′0𝑥 for u′∈HomΩ⁡(U,W)G_{\lambda}(x,y)=\left\{{g\in G_{\gamma}}\mid\,{g\cdot\begin{pmatrix}y&u\\

=κZ,!⁢ι′′⁣∗⁢j′⁣∗⁢π!⁢𝕀⁢[−⟨α,β⟩]\displaystyle=\kappa_{Z,!}\iota^{\prime\prime*}j^{\prime*}\pi_{!}\mathbb{I}[-%

{E∈Fβ,γ∣E=(MνU0Mν)⁢ and ⁢E≥Mλ}conditional-set𝐸subscript𝐹𝛽𝛾𝐸matrixmissing-subexpressionsubscript𝑀𝜈𝑈missing-subexpression0subscript𝑀𝜈 and 𝐸subscript𝑀𝜆\{E\in F_{\beta,\gamma}\mid\,E=\begin{pmatrix}&M_{\nu}&U\\

(yv′0x)matrixmissing-subexpression𝑦superscript𝑣′missing-subexpression0𝑥\begin{pmatrix}&y&v^{\prime}\\

D

LNg⁢(A→B).superscriptsubscript𝐿𝑁𝑔→𝐴𝐵L_{N}^{g}(A\to B).italic_L start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ( italic_A → italic_B ) .

for the set of all lattice paths from A𝐴Aitalic_A to B𝐵Bitalic_B that have steps from 𝕊𝕊\mathbb{S}blackboard_S and obey the restrictions from 𝒞𝒞\mathcal{C}caligraphic_C. Since we only consider the set of steps 𝕊𝕊\mathbb{S}blackboard_S we will omit it from the notations. We will denote the number of paths in this set as

We will denote the set of paths from point A𝐴Aitalic_A to point B𝐵Bitalic_B that stay between axes ℋℋ\mathcal{H}caligraphic_H as the set of ”good” paths:

In this section we will count the number of paths from A𝐴Aitalic_A to B𝐵Bitalic_B on ℒℒ\mathcal{L}caligraphic_L with the set of steps 𝕊𝕊\mathbb{S}blackboard_S with one restriction 𝒞𝒞\mathcal{C}caligraphic_C. Denote such set of paths as

We will denote the paths from point A𝐴Aitalic_A to point B𝐵Bitalic_B that visit any of the axes ℋℋ\mathcal{H}caligraphic_H as the set of ”bad” paths:

D

In this section we study a version of team semantics using semirings. It can be viewed as a generalization of both ordinary and probabilistic team semantics.

The first author introduced in [1] a relational framework for developing the key notions and results on hidden variables and non-locality, which can be seen as a relational variant of the probabilistic setting of [10]. He introduced what he called “relational empirical models” and used them to show that the basic results of the foundations of quantum mechanics, usually formulated in terms of probabilistic models, can be seen already on the level of mere (two-valued) relations. Our basic observation is that we can think of the relational empirical models of [1] as teams in the sense of team semantics. The basic quantum-theoretic properties of relational empirical models can then be defined in terms of the independence atoms of independence logic [19]. We observe that the relationships between quantum theoretic properties of relational models become instances of logical consequence of independence logic in its team semantics. In fact, the existential-positive-conjunctive fragment suffices. The no-go theorems become instances of failure of logical consequence between specific formulas of independence logic. This extends also to probabilistic models with independence logic replaced by the probabilistic independence logic of [15], capturing the probabilistic notions of [10].

Semiring relations, which we can consider semiring teams, were introduced in [20] to study the provenance of relational database queries. In [18], provenance of first-order formulas was studied by defining a semiring semantics for first-order logic. A similar approach to team semantics was taken in [8]. K𝐾Kitalic_K-relations are studied in [23] as a unifying framework for conditional independence and other similar notions.

The study of a probabilistic variant of independence logic was first done in a multiteam setting in [14]. Prior to that, multiteams were studied in [29], [30] and [45]. Probabilistic teams were then introduced by Durand et al. in [15] as a way to generalize multiteams, and further investigated in [24]. They can be thought of as a special case of measure teams, another approach to probabilities in team semantics given in [30].

So far we have only been looking at possibilistic (i.e. two-valued relational) versions of the independence notions of quantum physics while these notions are usually taken to be probabilistic. To be able to discuss the probabilistic notions from the point of view of team semantics, we need a suitable framework. For this, we consider probabilistic team semantics.

B

\frac{dU}{d\epsilon}(x,u;\epsilon)\Bigr{|}_{\epsilon=0}=\eta(x,u).divide start_ARG italic_d italic_X start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_ϵ end_ARG ( italic_x ; italic_ϵ ) | start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT = italic_ξ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_x ) , divide start_ARG italic_d italic_U end_ARG start_ARG italic_d italic_ϵ end_ARG ( italic_x , italic_u ; italic_ϵ ) | start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT = italic_η ( italic_x , italic_u ) .

where τμ=Dμ⁢η−uσ⁢Dμ⁢ξσsubscript𝜏𝜇subscript𝐷𝜇𝜂subscript𝑢𝜎subscript𝐷𝜇superscript𝜉𝜎\tau_{\mu}=D_{\mu}\eta-u_{\sigma}D_{\mu}\xi^{\sigma}italic_τ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_η - italic_u start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ξ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT is the generator of the induced transformations of the derivatives uμsubscript𝑢𝜇u_{\mu}italic_u start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT. The term u~σ~⁢□⁢Xσsubscript~𝑢~𝜎□superscript𝑋𝜎\tilde{u}_{\tilde{\sigma}}\Box X^{\sigma}over~ start_ARG italic_u end_ARG start_POSTSUBSCRIPT over~ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT □ italic_X start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT gives, on differentiating with respect to ϵitalic-ϵ\epsilonitalic_ϵ at ϵ=0italic-ϵ0\epsilon=0italic_ϵ = 0, τσ⁢□⁢xσ+uσ⁢□⁢ξσsubscript𝜏𝜎□superscript𝑥𝜎subscript𝑢𝜎□superscript𝜉𝜎\tau_{\sigma}\Box x^{\sigma}+u_{\sigma}\Box\xi^{\sigma}italic_τ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT □ italic_x start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT + italic_u start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT □ italic_ξ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT, since

give, on differentiating with respect to ϵitalic-ϵ\epsilonitalic_ϵ at ϵ=0italic-ϵ0\epsilon=0italic_ϵ = 0,

Differentiating equation (3.6) with respect to ϵitalic-ϵ\epsilonitalic_ϵ at ϵ=0italic-ϵ0\epsilon=0italic_ϵ = 0 we obtain the symmetry condition

we have, on differentiating with respect to uαsubscript𝑢𝛼u_{\alpha}italic_u start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT

C

If T1subscript𝑇1T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is positive and T2subscript𝑇2T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is PPT map then T1∘T2subscript𝑇1subscript𝑇2T_{1}\circ T_{2}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∘ italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is decomposable.

The important point to note here is an observation that various classes of positive maps are used to describe quantum and useful maps in bipartite systems. In particular, quantum maps which are useless for preserving quantum entanglement are called entanglement breaking maps. T𝑇Titalic_T is an entanglement breaking map iff it can be written as

As finite dimensional models are prevailing in Quantum Information, from now on, again, we make the assumption that the considered Hilbert space ℋℋ\mathcal{H}caligraphic_H is finite dimensional. Needless to say the finite dimension of ℋℋ\mathcal{H}caligraphic_H will be assumed only for simplicity. It is possible to proceed the study of the condition 𝒯2⁢Cd⊆Cpsubscript𝒯2subscript𝐶𝑑subscript𝐶𝑝\mathcal{T}_{2}C_{d}\subseteq C_{p}caligraphic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊆ italic_C start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT in more general setting. It would require the more careful analysis of trace class operators, continuity of corresponding functionals and maps. But, we will not develop this point here. The advantage of finite dimensional approach lies in the fact that it sheds some new light on the structure of the set of positive maps, in particular, on the origin of non-decomposable maps.

In Section 2 we give definitions of the concepts just introduced. Furthermore, we will define and discuss the associated classes of positive maps. Section 3 is devoted to a study of PPT square conjecture. In particular, we will present the general approach to the problem. Keeping in mind that entanglement breaking maps are so important in Quantum Information, in Section 4 we will give a detailed analysis of the relevant tensor cones and associated classes of maps. This will be done for finite dimensional cases.

We now turn to an explanation of what is going on with the composition of a positive map with a PPT map. This will be done using the structure of positive maps, and in particular, the selection rule; for details see [9].

C

Instead, we actually deal with the geometric and non-geometric cases (where here the distinction being made is whether or not there is a closed periodic Nielsen path) at the same time. This is somewhat new, conceptually, and seems appropriate for the general free product case. (Also, Theorem Remark seems interesting and important to us, and proving it for any irreducible automorphism without adding technical conditions seems worthwhile. One of our goals was to prove Lemma 2.13 of [10] but without the ‘no twinned subgroups’ hypothesis, and in this we were successful.)

The second reason for the length of this current paper is that we do not pass to powers of the automorphism to make hypotheses and arguments simpler. For instance, we do not assume that our automorphisms are fully irreducible. (Proofs in the fully irreducible case are always easier, but the statements always seem to hold more generally.) This may seem a very minor difference, and in some situations, for instance in [10] where the goal is to prove relative hyperbolicity, passing to a power is a tame procedure. However, for more algebraic applications, like looking at centralisers, passing to powers is not benign. (Knowing something about the centraliser of a power does not yield strong information about the original one.) To take another example; in [7] there is a solution of the conjugacy problem for Dehn Twists and in [29], this is extended to roots of Dehn Twists. However, one notes that the latter paper is longer than the former because the additional complexities of looking at roots are considerable. It is probably true that our arguments can be shortened by passing to suitable powers, but this would certainly invalidate our application and would definitely be a weaker result with less scope for further applications.

It is possible that one can resolve these issues and introduce surfaces into the free product situation - using something like the improved relative train tracks of [3], especially when the underlying group is free, and the given automorphism is relatively irreducible with respect to a free factor system - which one can think of as the main case of interest for the free product theory. Taking that route would simplify our Section 6 a little, which is a generalisation of similar results in [2] and [10], but we have chosen not to do that since it seems to us that fully taking into account the subtleties of relative surfaces would add length and complexity in a different way. We would reiterate here that this process has not formally been done, and we feel that the subtleties and differences between free groups and free products require some caution in simply hand-waving through techniques which may not be entirely valid.

But this is no longer true for free products, since one can write examples of train track representative of irreducible automorphisms which are not fully irreducible (that is, some power is reducible) and whose transition matrices are not primitive (this cannot happen in the absolute case).

For the experts, one important question is - given we already have that Min⁡(ϕ)Minitalic-ϕ\operatorname{Min}(\phi)roman_Min ( italic_ϕ ) is locally finite - why is this current paper so long? There are a couple of reasons. The first, and most important, is that the proofs of this fact for Culler-Vogtmann space divide into two main arguments according to whether the irreducible automorphism is non-geometric or geometric. The second case corresponds precisely to the existence of a closed periodic Nielsen path, and then it is a well known result of [4] that the corresponding automorphism is induced from a surface homeomorphism. The arguments then proceed by appeals to surface theory in the latter case and to other arguments in the non-geometric case (which are not valid for the geometric case). A direct ancestor of our result for Culler-Vogtmann space can be found in [25], where they prove that the axis bundle is co-compact (amongst other results). As they note:

A

We now continue to the description of the category Ds⁢g⁢(𝒵n+1,W)subscript𝐷𝑠𝑔subscript𝒵𝑛1𝑊D_{sg}({\mathcal{Z}}_{n+1},W)italic_D start_POSTSUBSCRIPT italic_s italic_g end_POSTSUBSCRIPT ( caligraphic_Z start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_W ).

5.5. Category of singularities of LG∨⁡(n+1)superscriptLG𝑛1\operatorname{LG}^{\vee}\!(n+1)roman_LG start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ( italic_n + 1 )

5.3. Singular fibres of of LG∨⁡(n+1)superscriptLG𝑛1\operatorname{LG}^{\vee}\!(n+1)roman_LG start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ( italic_n + 1 )

LG∨(n+1):=(𝒴n+1,g=x2)\operatorname{LG}^{\vee}\!(n+1)\mathrel{\mathop{:}}=(\mathcal{Y}_{n+1},g=x_{2})roman_LG start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ( italic_n + 1 ) : = ( caligraphic_Y start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT , italic_g = italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )

4.3. Category of singularities of LG∨⁡(3)superscriptLG3\operatorname{LG}^{\vee}\!(3)roman_LG start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ( 3 )

A

For any ΔΔ\Deltaroman_Δ-inducing color δ𝛿\deltaitalic_δ of F𝐹Fitalic_F, we have r∈Psδ−1⁢(δ,α+1)=Psα⁢(δ,α+1)𝑟subscript𝑃subscript𝑠𝛿1𝛿𝛼1subscript𝑃subscript𝑠𝛼𝛿𝛼1r\in P_{s_{\delta-1}}(\delta,\alpha+1)=P_{s_{\alpha}}(\delta,\alpha+1)italic_r ∈ italic_P start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_δ - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ , italic_α + 1 ) = italic_P start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ , italic_α + 1 ), where sΔ−1=s1subscript𝑠Δ1subscript𝑠1s_{\Delta-1}=s_{1}italic_s start_POSTSUBSCRIPT roman_Δ - 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT if δ=Δ𝛿Δ\delta=\Deltaitalic_δ = roman_Δ.

for any β∈φ¯⁢(sℓi)𝛽¯𝜑subscript𝑠subscriptℓ𝑖\beta\in\overline{\varphi}(s_{\ell_{i}})italic_β ∈ over¯ start_ARG italic_φ end_ARG ( italic_s start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) and any i∈[1,h]𝑖1ℎi\in[1,h]italic_i ∈ [ 1 , italic_h ]. An α𝛼\alphaitalic_α-inducing color β𝛽\betaitalic_β is called a last α𝛼\alphaitalic_α-inducing color if there does not exist any α𝛼\alphaitalic_α-inducing color δ𝛿\deltaitalic_δ such that β≺δprecedes𝛽𝛿\beta\prec\deltaitalic_β ≺ italic_δ.

For any 2-inducing color δ𝛿\deltaitalic_δ of F𝐹Fitalic_F with τ≺δprecedes𝜏𝛿\tau\prec\deltaitalic_τ ≺ italic_δ,

For any 2-inducing color δ𝛿\deltaitalic_δ of F𝐹Fitalic_F with δ≺τprecedes𝛿𝜏\delta\prec\tauitalic_δ ≺ italic_τ, we have

For any ΔΔ\Deltaroman_Δ-inducing color δ𝛿\deltaitalic_δ of F𝐹Fitalic_F, we have r∈Psδ−1⁢(δ,α+1)=Psα⁢(δ,α+1)𝑟subscript𝑃subscript𝑠𝛿1𝛿𝛼1subscript𝑃subscript𝑠𝛼𝛿𝛼1r\in P_{s_{\delta-1}}(\delta,\alpha+1)=P_{s_{\alpha}}(\delta,\alpha+1)italic_r ∈ italic_P start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_δ - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ , italic_α + 1 ) = italic_P start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ , italic_α + 1 ), where sΔ−1=s1subscript𝑠Δ1subscript𝑠1s_{\Delta-1}=s_{1}italic_s start_POSTSUBSCRIPT roman_Δ - 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT if δ=Δ𝛿Δ\delta=\Deltaitalic_δ = roman_Δ.

C

}+R_{ik}w_{kj}+R_{jk}w_{ki}.italic_Q start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = divide start_ARG 2 italic_R italic_f end_ARG start_ARG 1 - italic_f end_ARG italic_w start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT + divide start_ARG ( italic_R italic_u ) start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_R italic_u ) start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_f ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + italic_R start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k italic_j end_POSTSUBSCRIPT + italic_R start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k italic_i end_POSTSUBSCRIPT .

where the constant C𝐶Citalic_C depends on n,α,ε,K0,K1𝑛𝛼𝜀subscript𝐾0subscript𝐾1n,\alpha,\varepsilon,K_{0},K_{1}italic_n , italic_α , italic_ε , italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT; compare [5], [13], and [16]. Then we proceed as in [12] and [5]. □□\Box□

Now with the help of Theorem 10 in [7] and Corollary 2.3 here, we can proceed as in Han-Zhang [9]. □□\Box□

With the help of Theorem 10 in [7], Proposition 2.2 here, and a space-time cutoff function, we can proceed as in Han-Zhang [9] with minor modifications. □□\Box□

In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow. I was inspired by Han-Zhang [9]. To state our results we first introduce some notations.

B

One of the main tools of model categories is to assume the structure to be cofibrantly generated by a pair of sets (see definition in [Hov99, 2.1.17]).

In this section, following [BHK+{}^{+}start_FLOATSUPERSCRIPT + end_FLOATSUPERSCRIPT15], we recall the dual concept of fibrantly generated model categories. We make no requirement of “cosmallness” as it is rarely satisfied in practice. Instead when fibrations and acyclic fibrations are retracts of maps built out of pullbacks and towers, we say the model category admits a Postnikov presentation. In [HKRS17, GKR20], model structures on monoidal categories are lifted to categories of coalgebras and comodules. However there is no guarantee that Postnikov presentations are lifted. We present here the notion of fibrant-compatible monoidal structure on a model category which ensures that the homotopy category is endowed with a monoidal structure via a right derived tensor product. We also introduce the notion of weak opmonoidal Quillen equivalences which provides a compatibility of the right derived tensor products.

In this case, cofibrations and acyclic cofibrations are retracts of maps built out of pushouts and transfinite compositions, and we can inductively construct a cofibrant replacement.

In particular, U𝑈Uitalic_U preserves and reflects cofibrations and weak equivalences. The model structure is combinatorial and simplicial. Every object is cofibrant.

To prove our main theorem, we dualize the methods in [SS03]. Following [Hes09] and [BHK+{}^{+}start_FLOATSUPERSCRIPT + end_FLOATSUPERSCRIPT15], we use the notion of fibrantly generated model categories. CW-complexes provide an inductive cofibrant replacement for spaces. This was generalized into the notion of cofibrantly generated model categories. We instead provide Postnikov towers as an inductive fibrant replacement. Because of the lack of “cosmallness” in practice, we cannot apply the dual of the small object argument. Instead we provide ad-hoc Postnikov presentations: (acyclic) fibrations are retracts of maps built from pullbacks and transfinite towers of generating (acyclic) fibrations.

B

1}_{i+1}-\theta\geq s-\tilde{\ell}^{n}_{i}=1.italic_s - roman_ℓ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT - italic_θ = italic_s - over~ start_ARG roman_ℓ end_ARG start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - italic_θ ≥ italic_s - over~ start_ARG roman_ℓ end_ARG start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT - italic_θ ≥ italic_s - over~ start_ARG roman_ℓ end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 .

The latter implies that none of the factors in the numerator in (2.38) vanish and so the expression in (2.38) a simple pole at z=s𝑧𝑠z=sitalic_z = italic_s, coming from the factor z−ℓn−i+1n+(N−n)⋅θ𝑧superscriptsubscriptℓ𝑛𝑖1𝑛⋅𝑁𝑛𝜃z-\ell_{n-i+1}^{n}+(N-n)\cdot\thetaitalic_z - roman_ℓ start_POSTSUBSCRIPT italic_n - italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + ( italic_N - italic_n ) ⋅ italic_θ.

The latter implies that none of the factors in the numerator in (2.28) vanish and so the expression in (2.28) has a simple pole at z=s𝑧𝑠z=sitalic_z = italic_s.

The latter implies that none of the factors in the numerator in (2.27) vanish and so the expression in (2.27) a simple pole at z=s𝑧𝑠z=sitalic_z = italic_s.

The latter implies that none of the factors in the numerator in (2.39) vanish and so the expression in (2.39) has a simple pole at z=s𝑧𝑠z=sitalic_z = italic_s, coming from the factor z−ℓk−i+1k+(N−k)⋅θ𝑧superscriptsubscriptℓ𝑘𝑖1𝑘⋅𝑁𝑘𝜃z-\ell_{k-i+1}^{k}+(N-k)\cdot\thetaitalic_z - roman_ℓ start_POSTSUBSCRIPT italic_k - italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT + ( italic_N - italic_k ) ⋅ italic_θ.

C

Therefore, 𝒫D3subscript𝒫subscript𝐷3\mathcal{P}_{D_{3}}caligraphic_P start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is a positively stable preconditioner for 𝒜𝒜\mathcal{A}caligraphic_A. We also comment here that when the A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and the Schur complements S2subscript𝑆2S_{2}italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and S3subscript𝑆3S_{3}italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are replaced by their inexact versions, some analysis on the eigenvalue distribution of the preconditioned systems are obtained in [6].

are invertible. For the twofold saddle point problem (1), following the notations in [37], we denote

The outline of the remainder of this paper is as follows. In section 2, we briefly recall the classic saddle point problem and its Schur complement, and introduce the twofold saddle point problem and the form of Schur complement, we then construct and analyze the block-triangular and block-diagonal preconditioners based on Schur complement for twofold saddle point problems. Furthermore, we extend these results to the n𝑛nitalic_n-tuple saddle point problem in Section 3. Some additive Schur complement based preconditioners are constructed and the corresponding known results in the literature are recalled in Section 4 for twofold saddle point problems. Generalizations to n𝑛nitalic_n-tuple cases are provided in Section 5. In Section 6, numerical experiments for a 3-field formulation of the Biot model are provided to justify the advantages of using positively stable preconditioners. Finally, concluding remarks are given in Section 7.

The above 3333-by-3333 block linear problems (1) and (2) can be naturally extended to the n𝑛nitalic_n-tuple cases. For example, when the system matrix in (1) is extended to the n𝑛nitalic_n-tuple case, it is the block tridiagonal systems discussed in [37]. When

Now, we will discuss preconditioners for the n𝑛nitalic_n-tuple block tridiagonal problem of the following form.

D

Changxin Mo acknowledges support from the National Natural Science Foundation of China (Grant No. 12201092), the Natural Science Foundation Project of CQ CSTC (Grant No. CSTB2022NSCQ-MSX0896), the Science and Technology Research Program of Chongqing Municipal Education Commission

Changxin Mo acknowledges support from the National Natural Science Foundation of China (Grant No. 12201092), the Natural Science Foundation Project of CQ CSTC (Grant No. CSTB2022NSCQ-MSX0896), the Science and Technology Research Program of Chongqing Municipal Education Commission

(Grant No. KJQN202200512), the Chongqing Talents Project (Grant No. cstc2022ycjh-bgzxm0040), and the Research Foundation of Chongqing Normal University (Grant No. 21XLB040), P. R. of China.

The study of T-eigenvalues has emerged as a prominent research area within the field of tensor analysis. Motivated by the aforementioned research, we pay our attention to the perturbation analysis of third-order tensors under the novel tensor-tensor multiplication (3) in this paper, encompassing both the extension of classical theoretical results and the introduction of novel pseudospectra theory.

Yimin Wei is supported by the National Natural Science Foundation of China under Grant 12271108, the Ministry of Science and Technology of China under grant G2023132005L and the Science and Technology Commission of Shanghai Municipality under grant 23JC1400501.

B