robench-2024b
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48 items • Updated
context stringlengths 100 6.94k | A stringlengths 100 4.68k | B stringlengths 100 5.83k | C stringlengths 100 5.59k | D stringlengths 100 4.72k | label stringclasses 4
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−m2x−m−2R+12x−m−1R′;𝑚2superscript𝑥𝑚2𝑅12superscript𝑥𝑚1superscript𝑅′\displaystyle-\frac{m}{2}x^{-m-2}R+\frac{1}{2}x^{-m-1}R^{\prime};- divide start_ARG italic_m end_ARG start_ARG 2 end_ARG italic_x start_POSTSUPERSCRIPT - italic_m - 2 end_POSTSUPERSCRIPT italic_R + divide start_ARG 1 end_ARG start_ARG 2 end_AR... | Rnm′′′(x)superscriptsuperscriptsubscript𝑅𝑛𝑚′′′𝑥\displaystyle{R_{n}^{m}}^{\prime\prime\prime}(x)italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT ( italic_x ) | F′′superscript𝐹′′\displaystyle F^{\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT | F′′′superscript𝐹′′′\displaystyle F^{\prime\prime\prime}italic_F start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT | (i) fast calculation of f′′/f′superscript𝑓′′superscript𝑓′f^{\prime\prime}/f^{\prime}italic_f start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT from f/f′𝑓superscript𝑓′f/f^{\prime}italic_f / italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, | B |
The nerve X𝑋Xitalic_X of a 2222-groupoid object X2⇛X1⇒X0⇛subscript𝑋2subscript𝑋1⇒subscript𝑋0X_{2}\Rrightarrow X_{1}\Rightarrow X_{0}italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⇛ italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⇒ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathca... | 2222-groupoid objects in (𝒞,𝒯′′)𝒞superscript𝒯′′(\mathcal{C},\mathcal{T}^{\prime\prime})( caligraphic_C , caligraphic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) in the sense of Prop-Def. 2.17; | object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ), as we have demonstrated in the | Prop-Def. 2.17 is a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Def. 1.3. | The first three layers of a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Def. 1.3 is a 2222-groupoid object in (𝒞,𝒯)𝒞𝒯(\mathcal{C},\mathcal{T})( caligraphic_C , caligraphic_T ) as in Prop-Def. 2.17. | C |
Then we define the quotient category of Q(mTT) over canonical isomorphisms in which we will interpret emTT. The equivalence relation generated by canonical isomorphisms coincides with extending the previous collection of canonical isomorphisms by including all identity morphisms between objects of Q(mTT) 888We still ge... | We call Q(mTT)/≃similar-to-or-equals\simeq≃ the category obtained by quotienting Q(mTT) over canonical isomorphisms: | we can interpret emTT (and emTTdp) judgements in the category Q(mTT)/≃similar-to-or-equals\simeq≃ (Q(mTTdp)/≃similar-to-or-equals\simeq≃) by following the idea behind the naive interpretation of dependent types | emTT in a category Q(mTT)/≃similar-to-or-equals\simeq≃ obtained by quotienting Q(mTT) only over suitable | Indeed, if we take the category Q(mTT)/≃similar-to-or-equals\simeq≃ obtained from Q(mTT) by quotienting it over isomorphisms, then this category enjoys a unique choice | A |
By construction, G/Γ𝐺ΓG/\Gammaitalic_G / roman_Γ is isomorphic to G𝐧0/Γ𝐧0subscript𝐺subscript𝐧0subscriptΓsubscript𝐧0G_{{\bf n}_{0}}/\Gamma_{{\bf n}_{0}}italic_G start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT / roman_Γ start_POSTSUBSCRIPT bold_n start_POSTSUBSCRIPT 0 end_POSTSU... | Thus, to establish Theorem 1.3, it will suffice to establish Conjecture 5.3 for s⩾3𝑠3s\geqslant 3italic_s ⩾ 3. This is the objective of the remainder of the paper. | The purpose of this paper is to establish the general case of a conjecture named the Inverse Conjecture for the Gowers norms by the first two authors in [23, Conjecture 8.3]. If N𝑁Nitalic_N is a (typically large) positive integer then we write [N]:={1,…,N}assigndelimited-[]𝑁1…𝑁[N]:=\{1,\dots,N\}[ italic_N ] := { 1 ,... | The inverse conjecture GI(s)GI𝑠{\operatorname{GI}}(s)roman_GI ( italic_s ), Conjecture 1.2, has been formulated using linear nilsequences F(gnxΓ)𝐹superscript𝑔𝑛𝑥ΓF(g^{n}x\Gamma)italic_F ( italic_g start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x roman_Γ ). This is largely for compatibility with the e... | In our previous paper [28] it was already rather painful to keep proper track of such notions as “many” and “correlates with”. Here matters are even worse, and so to organise the above tasks it turns out to be quite convenient to first take an ultralimit of all objects being studied, effectively placing one in the sett... | A |
π0π0¯=pmsubscript𝜋0¯subscript𝜋0superscript𝑝𝑚\pi_{0}\overline{\pi_{0}}=p^{m}italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT over¯ start_ARG italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = italic_p start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT. | Let A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG be an abelian variety over a number field k𝑘kitalic_k | A=((a,bt~;c,dt~)modN)𝐴modulo𝑎𝑏~𝑡𝑐𝑑~𝑡𝑁A=((a,b\widetilde{t};c,d\widetilde{t})\bmod{N})italic_A = ( ( italic_a , italic_b over~ start_ARG italic_t end_ARG ; italic_c , italic_d over~ start_ARG italic_t end_ARG ) roman_mod italic_N ). | an abelian variety A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG in characteristic zero with a nice | that the endomorphisms of A~~𝐴\widetilde{A}over~ start_ARG italic_A end_ARG over k¯¯𝑘\overline{k}over¯ start_ARG italic_k end_ARG are defined over k𝑘kitalic_k. | A |
\text{ }0 < italic_α ( italic_t ) ≤ italic_y start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_t , italic_ω ) ≤ italic_β ( italic_t ) . | solution set 𝒮b(0).superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right).caligraphic_S start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ) . Observe that in Proposition 3, the convergence in law is rather weak. However, thanks to the | We have found the limiting solutions, i.e., 𝒮fb(0)superscriptsubscript𝒮𝑓𝑏0\mathcal{S}_{f}^{b}\left(0\right)caligraphic_S start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ). However, the following example shows that the probability | fixed ω∈Ω/𝒩,𝜔Ω𝒩\omega\in\Omega/\mathcal{N},italic_ω ∈ roman_Ω / caligraphic_N , we will explore what the limit of Xε(⋅,ω)superscript𝑋𝜀⋅𝜔X^{\varepsilon}(\cdot,\omega)italic_X start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( ⋅ , italic_ω ) in 𝒮b(0)superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right)caligraphic_S st... | We now prove (21) holds. All the trajectories in 𝒮b(0)superscript𝒮𝑏0\mathcal{S}^{b}\left(0\right)caligraphic_S start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ( 0 ) are analyzed and verified seriously in the following | D |
Determining whether two given admissible trees are (topologically) equivalent is the problem we want to solve. The problem of determining whether two given admissible trees are isomorphic is combinatorial in nature and easy to solve algorithmically simply by enumerating all maps from the set of vertices of T1subscript�... | There is an algorithm which to any admissible tree T𝑇Titalic_T associates a reduced admissible tree equivalent to T𝑇Titalic_T. | In this section, we simply call 2-automaton a planar topological 2-automaton. To any such 2-automaton 𝒳𝒳\mathcal{X}caligraphic_X, we shall associate a decorated graph G(𝒳)𝐺𝒳G(\mathcal{X})italic_G ( caligraphic_X ) which contains enough information to recover the space of ends of the surface associated to 𝒳𝒳\mat... | We now introduce three moves that can be used to simplify an admissible tree without changing its equivalence class. | We will show how to modify an admissible tree without changing its topological equivalence class, until it belongs to a special class of admissible trees, called reduced, for which topological equivalence will turn out to be equivalent to isomorphism. In order to motivate the construction, we first give some simple exa... | D |
The collapsing of manifolds studied in this paper are closely related to multi-scale models in manifold learning where the intrinsic dimension of the data set is modelled by a function that depends on the scale, see [69]. | For example, this occurs when the cryogenic electron-microscopy [57] is applied to image the structure of a large molecule which is connected to a small part of the molecule with a connection that allows a rotation (roughly | recall the counting function, NX(E)subscript𝑁𝑋𝐸{{N}}_{X}(E)italic_N start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) of X𝑋Xitalic_X, which is the number of the | The structure of collapsing in the moduli space 𝔐(n,Λ,D)𝔐𝑛Λ𝐷{\mathfrak{M}}(n,\Lambda,D)fraktur_M ( italic_n , roman_Λ , italic_D ), with respect to the Gromov-Hausdorff distance, | speaking, the molecule has a moving ‘tail’). In this case, the problem of imaging the molecule is to find a manifold diffeomorphic to, e.g. SO(3)×S1(ϵ)𝑆𝑂3superscript𝑆1italic-ϵSO(3)\times S^{1}(\epsilon)italic_S italic_O ( 3 ) × italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_ϵ ) which almost collaps... | A |
In 2011, Hong-Huang-Wang [7] studied a class of degenerate elliptic Monge-Ampère equation in a smooth, bounded and strictly convex domain | respect to the normal direction −Du𝐷𝑢-Du- italic_D italic_u, we have the following formula on the m𝑚mitalic_m-th curvature of the level sets of the solution u𝑢uitalic_u, | Let uΩj,j=0,1,formulae-sequencesubscript𝑢subscriptΩ𝑗𝑗01u_{\Omega_{j}},j=0,1,italic_u start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_j = 0 , 1 , be the solution to the problem | When they proved the existence of global smooth solutions to the homogeneous Dirichlet problem, they introduced the key auxiliary function ℋℋ\mathcal{H}caligraphic_H, | which is the product of curvature κ𝜅\kappaitalic_κ of the level line of u𝑢uitalic_u and the cubic of |Du|𝐷𝑢|Du|| italic_D italic_u |, and got the uniformly lower bound of ℋℋ\mathcal{H}caligraphic_H | C |
Now let d𝑑ditalic_d be even. The same results for the transvections t21(ωℓ)subscript𝑡21superscript𝜔ℓt_{21}(\omega^{\ell})italic_t start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) and t12(ωℓ)subscript𝑡12superscript𝜔ℓt_{12}(\omega^{\ell})italic_t start_POSTSUB... | as a word in the LGO standard generators of SL(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ). | Our aim is to determine the length and memory quota for an MSLP for the Bruhat decomposition of an arbitrary matrix g∈SL(d,q)𝑔SL𝑑𝑞g\in\textnormal{SL}(d,q)italic_g ∈ SL ( italic_d , italic_q ) via the above method, with the matrices u1subscript𝑢1u_{1}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, u2subscript𝑢2u... | We now compute upper bounds for the length and memory quota of an MSLP for expressing an arbitrary diagonal matrix h∈SL(d,q)ℎSL𝑑𝑞h\in\textnormal{SL}(d,q)italic_h ∈ SL ( italic_d , italic_q ) as a word in the LGO generators, i.e. the computation phase of the algorithm. | Finally, we construct a second MSLP, described in Section 3.5, that writes a diagonal matrix h∈SL(d,q)ℎSL𝑑𝑞h\in\textnormal{SL}(d,q)italic_h ∈ SL ( italic_d , italic_q ) as a word in the standard generators of SL(d,q)SL𝑑𝑞\textnormal{SL}(d,q)SL ( italic_d , italic_q ) (when evaluated with these generators as input)... | C |
C=x+C−1x−1+C−2x−2+⋯with each C−i∈K[y].𝐶𝑥subscript𝐶1superscript𝑥1subscript𝐶2superscript𝑥2⋯with each C−i∈K[y].C=x+C_{-1}x^{-1}+C_{-2}x^{-2}+\cdots\qquad\text{with each $C_{-i}\in K[y]$.}italic_C = italic_x + italic_C start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIP... | and, again by Proposition 1.7, there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λj∈K×subscript𝜆𝑗superscript𝐾\lambda_{j}\in K^{\times}italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT | then there exists j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and λ∈K×𝜆superscript𝐾\lambda\in K^{\times}italic_λ ∈ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT such that | ℤ/eℤℤ𝑒ℤ\mathds{Z}/e\mathds{Z}blackboard_Z / italic_e blackboard_Z acts on 𝒮0subscript𝒮0\mathcal{S}_{0}caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. In fact, if | Since m+n>2𝑚𝑛2m+n>2italic_m + italic_n > 2, by Proposition 1.7 there exist j∈ℤ𝑗ℤj\in\mathds{Z}italic_j ∈ blackboard_Z and | D |
In that case W𝑊Witalic_W can be identified with Xm×Ysuperscript𝑋𝑚𝑌X^{m}\times Yitalic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT × italic_Y, where m𝑚mitalic_m is the order of H𝐻Hitalic_H, and is contractible as well with respect to the usual product topology. | The wreath product G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H acts from the left on W𝑊Witalic_W by the following rule: if (ϕ:H→G,h)∈G≀H=Map(H,G)⋊H({\phi}\colon H\to G,h)\in G\wr H=\mathrm{Map}(H,G)\rtimes H( italic_ϕ : italic_H → italic_G , italic_h ) ∈ italic_G ≀ italic_H = roman_Map ( italic_H , italic_G ) ⋊ italic_H and (α:... | Moreover, the above action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H on W𝑊Witalic_W also turns out to be free. | If the actions of G𝐺Gitalic_G and H𝐻Hitalic_H are PD, then the action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H on W𝑊Witalic_W is PD as well. | If the actions of G𝐺Gitalic_G and H𝐻Hitalic_H are free, then the action of G≀H≀𝐺𝐻G\wr Hitalic_G ≀ italic_H is also free. | B |
A~1subscript~𝐴1\tilde{A}_{1}over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 64,510,416,320,220,110superscript64superscript510superscript416superscript320superscript220superscript1106^{4},5^{10},4^{16},3^{20},2^{20},1^{10}6 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , 5 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT , 4 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 3 start_POSTSUPERSCRIPT 20 en... | 220,116superscript220superscript1162^{20},1^{16}2 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT | 216,120superscript216superscript1202^{16},1^{20}2 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT | 216,120superscript216superscript1202^{16},1^{20}2 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT | C |
Yksubscript𝑌𝑘Y_{k}italic_Y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Suppose that the p𝑝pitalic_p-curvature of ψ𝜓\psiitalic_ψ is 00 for all | Concretely, this means that if Y¯¯𝑌\overline{Y}over¯ start_ARG italic_Y end_ARG is a smooth compactification | support equal to Lℂsubscript𝐿ℂL_{\mathbb{C}}italic_L start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT means that ℳksubscriptℳ𝑘\mathcal{M}_{k}caligraphic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is | compactification which we will denote T∗(X~k′)¯¯superscript𝑇subscript~𝑋superscript𝑘′\overline{T^{*}(\tilde{X}_{k^{\prime}})}over¯ start_ARG italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( over~ start_ARG italic_X end_ARG start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRI... | where L¯¯𝐿\overline{L}over¯ start_ARG italic_L end_ARG is a suitable compactification of L𝐿Litalic_L. The | A |
(This remark was originally stated for an equidimensional scheme X.𝑋X.italic_X . We thank the referee for pointing out that the equidimentionality is unnecessary if we work with a Chow group indexed by dimension, not by codimension.) | For an arbitrary X∈Sch/k,𝑋Sch𝑘X\in\operatorname{Sch}/k,italic_X ∈ roman_Sch / italic_k , there is a pullback diagram by [ILO, Exposé 0, Theorem 3 (1)] | Suppose k𝑘kitalic_k is perfect and let X∈Sch/k.𝑋normal-Sch𝑘X\in\operatorname{Sch}/k.italic_X ∈ roman_Sch / italic_k . Then, there is an inclusion Γalg(M(X))⊂H0(X,ℤ)0.subscriptnormal-Γ𝑎𝑙𝑔𝑀𝑋subscript𝐻0superscript𝑋ℤ0\Gamma_{alg}(M(X))\subset H_{0}(X,{\mathbb{Z}})^{0}.roman_Γ start_POSTSUBSCRIPT italic_a ita... | If k𝑘kitalic_k is algebraically closed, then there exists a universal regular homomorphism for Γalg(M(X))subscriptnormal-Γ𝑎𝑙𝑔𝑀𝑋\Gamma_{alg}(M(X))roman_Γ start_POSTSUBSCRIPT italic_a italic_l italic_g end_POSTSUBSCRIPT ( italic_M ( italic_X ) ) for any X∈Sch/k.𝑋normal-Sch𝑘X\in\operatorname{Sch}/k.italic_X ∈ ... | Let us assume the resolution of singularities. In this case, for an arbitrary X∈Sch/k,𝑋normal-Sch𝑘X\in\operatorname{Sch}/k,italic_X ∈ roman_Sch / italic_k , we have a comparison isomorphism between the | D |
Let b∈sQG4b𝑠𝑄subscript𝐺4\mathrm{b}\in sQG_{4}roman_b ∈ italic_s italic_Q italic_G start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, T∈ℭ𝔫𝔱(a4,b)Tℭ𝔫𝔱superscripta4b\mathrm{T}\in\mathfrak{Cnt}(\mathrm{a}^{4},\mathrm{b})roman_T ∈ fraktur_C fraktur_n fraktur_t ( roman_a start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , roman_... | for any T,J∈MorB𝑇𝐽subscriptMor𝐵T,J\in\mathrm{Mor}_{B}italic_T , italic_J ∈ roman_Mor start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, such that (T,J)∈Dom(∘)𝑇𝐽𝐷𝑜𝑚(T,J)\in Dom(\circ)( italic_T , italic_J ) ∈ italic_D italic_o italic_m ( ∘ ), | Let α𝛼\alphaitalic_α be a geodesic on M(x,f)𝑀𝑥𝑓M(x,f)italic_M ( italic_x , italic_f ) such that 0∈dom(α)0𝑑𝑜𝑚𝛼0\in dom(\alpha)0 ∈ italic_d italic_o italic_m ( italic_α ) eventually by a reparametrization, | and an open neighbourhood K0subscript𝐾0K_{0}italic_K start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of 00 in dom(α)𝑑𝑜𝑚𝛼dom(\alpha)italic_d italic_o italic_m ( italic_α ) such that | Assume that 0∈dom(α)0𝑑𝑜𝑚𝛼0\in dom(\alpha)0 ∈ italic_d italic_o italic_m ( italic_α ) eventually by a reparametrization, | D |
On the other hand, note that any Siegel disk cannot contain a critical point. Hence the second question on the Siegel disk boundary is: | The answer is no. Ghys and Herman gave the first examples of polynomials having a Siegel disk whose boundary does not contain a critical point (see [Ghy84], [Her86] and [Dou87]). | On the other hand, note that any Siegel disk cannot contain a critical point. Hence the second question on the Siegel disk boundary is: | Suppose f𝑓fitalic_f is an analytic function which has a Siegel disk properly contained in the domain of holomorphy. Ghys proved that if the rotation number belongs to ℋℋ\mathcal{H}caligraphic_H and the boundary of the Siegel disk is a Jordan curve, then f𝑓fitalic_f has a critical point in the boundary of the Siegel d... | Suppose the closure of the Siegel disk of f𝑓fitalic_f is compactly contained in the domain of definition of f𝑓fitalic_f. One may wonder what phenomena near the boundary of a Siegel disk prevents f𝑓fitalic_f from having a larger linearization domain. Obviously, the presence of periodic cycles near the boundary is one... | A |
We will be interested in orbifolds X=Γ\ℍ𝑋\ΓℍX=\Gamma\backslash\mathbb{H}italic_X = roman_Γ \ blackboard_H (or X=Γ\𝔻𝑋\Γ𝔻X=\Gamma\backslash\mathbb{D}italic_X = roman_Γ \ blackboard_D, if we treat ΓΓ\Gammaroman_Γ as acting on 𝔻𝔻\mathbb{D}blackboard_D) for Fuchsian groups ΓΓ\Gammaroman_Γ of the first kind. Such an or... | Here c1(λk,||⋅||kQ)c_{1}(\lambda_{k},||\cdot||_{k}^{Q})italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , | | ⋅ | | start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT ) is the first Chern form of the determinant lin... | we show that the contribution to the local index formula from elliptic elements of Fuchsian groups is given by the symplectic form of a Kähler metric on the moduli space of orbisurfaces. | We proceed with the basics of the deformation theory of Fuchsian groups. Let ΓΓ\Gammaroman_Γ be a Fuchsian group of the first kind of signature (g;n;m1,…,ml)𝑔𝑛subscript𝑚1…subscript𝑚𝑙(g;n;m_{1},\ldots,m_{l})( italic_g ; italic_n ; italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT i... | The dimension of the space of square integrable meromorphic (with poles at punctures and conical points) k𝑘kitalic_k-differentials on X𝑋Xitalic_X, or cusp forms of weight 2k2𝑘2k2 italic_k for ΓΓ\Gammaroman_Γ, is given by Riemann-Roch formula for orbifolds: | D |
The above uncertainty principles agree exactly with the traditional Heisenberg uncertainty principle in the case of the QMHO since γ=−1𝛾1\gamma=-1italic_γ = - 1 and δ=1𝛿1\delta=1italic_δ = 1, giving an uncertainty bound of 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG for each with the minimizers bein... | By considering the QMHO in the context of SUSY, the coupled SUSY structure unifying the QMHO and SUSY was developed. Coupled SUSY has many of the desirable properties of both: true ladder operators exist, there are two sectors, and charge operators exist between the sectors. The existence of true ladder operators led t... | The remainder of this paper is organized as follows. In Section 2, we develop the coupled SUSY structure which expands the relationship among the QMHO, SUSY, and the corresponding coupled SUSYs. In Section 3, we establish eigenvalues and eigenfunctions of the corresponding coupled SUSY. In Section 4, we establish the c... | Traditionally the QMHO is associated to the 1D Heisenberg-Weyl Lie algebra as this is the Lie algebra which corresponds to the canonical commutation relations which is reflected in the algebra generated by the ladder operators. This is not the only Lie algebra which may be associated to the QMHO. There are two other tr... | As previously noted, the quantum mechanical harmonic oscillator is a specific instance of a coupled supersymmetry, albeit a somewhat trivial case in which the two coupled SUSY equations are identical. This is not the only manner in which the two are connected. Indeed, a special class of coupled SUSYs may be realized as... | D |
24242424; H21(2,1)subscriptsuperscript𝐻1221H^{1}_{2}(2,1)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 2 , 1 ) | 3[1]1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT: | 3[1]1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT: | (3[1],1[3])superscript3delimited-[]1superscript1delimited-[]3(3^{[1]},1^{[3]})( 3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT [ 3 ] end_POSTSUPERSCRIPT ) | 3[1]1[2]superscript3delimited-[]1superscript1delimited-[]23^{[1]}1^{[2]}3 start_POSTSUPERSCRIPT [ 1 ] end_POSTSUPERSCRIPT 1 start_POSTSUPERSCRIPT [ 2 ] end_POSTSUPERSCRIPT: | A |
With the same notation as above, kerϕitalic-ϕ\phiitalic_ϕ is generated by the σijsubscript𝜎𝑖𝑗\sigma_{ij}italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT. | The paper is structured as follows. In Section 2, we collect basic notations, terminology, and results that will be used in the paper. The first syzygy of Hibi rings is discussed in Section 3. Explicit expression for the first Betti number for planar distributive lattices has been discussed in Section 4. | In this section, we find explicitly the generators of the first syzygy of Hibi ring for a planar distributive lattice and also find a condition for which the first syzygy linear. | For a planar distributive lattice ℒℒ\mathcal{L}caligraphic_L, the number of strip type generators of first syzygy of R[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ], we denote it by | The following theorem gives us the linear generators for the first syzygy of R[ℒ]𝑅delimited-[]ℒR[\mathcal{L}]italic_R [ caligraphic_L ]. | B |
To cast this into the framework of regularity structures we first work with an abstract version of uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT, i.e., we write uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT as a generalised Taylor expansion... | The “Taylor coefficients” can be thought of as the “derivatives” of an abstract version of a 𝒞γsuperscript𝒞𝛾{\mathcal{C}}^{\gamma}caligraphic_C start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT-function and they are given by | To cast this into the framework of regularity structures we first work with an abstract version of uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT, i.e., we write uεsubscript𝑢𝜀u_{\varepsilon}italic_u start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT as a generalised Taylor expansion... | is a sequence of linear spaces 𝒳εsubscript𝒳𝜀{\mathcal{X}}_{\varepsilon}caligraphic_X start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT that can be viewed as subspaces of 𝒟′(ℝd)superscript𝒟′superscriptℝ𝑑{\mathcal{D}}^{\prime}(\mathbb{R}^{d})caligraphic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R sta... | In particular we define an abstract notion of convolution 𝒦γεsuperscriptsubscript𝒦𝛾𝜀\mathcal{K}_{\gamma}^{\varepsilon}caligraphic_K start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT against the Green function Kεsuperscript𝐾𝜀K^{\varepsilon}italic_K start_POSTSUPERSCR... | A |
Since char(K)=pchar𝐾𝑝\mathrm{char}(K)=proman_char ( italic_K ) = italic_p it is perfect and so vK𝑣𝐾vKitalic_v italic_K is p𝑝pitalic_p-divisible. | Moreover, as K𝐾Kitalic_K is dependent it follows from the proof of [22, Proposition 5.3] that Kv𝐾𝑣Kvitalic_K italic_v is Artin-Schreier closed, and therefore infinite. | If Kv𝐾𝑣Kvitalic_K italic_v is separably closed and perfect it is algebraically closed and hence strongly dependent. If it is not perfect then by an argument of Scanlon’s [16, Proposition 3.7] v𝑣vitalic_v is definable in K𝐾Kitalic_K and hence (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) is strongly dependent, so that Kv�... | if [0,v(p)]0𝑣𝑝[0,v(p)][ 0 , italic_v ( italic_p ) ] is infinite then Kv𝐾𝑣Kvitalic_K italic_v is infinite. | Keeping the same notation, assume that [0,v(p)]0𝑣𝑝[0,v(p)][ 0 , italic_v ( italic_p ) ] is infinite. As before, Δ0subscriptΔ0\Delta_{0}roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is non trivial. The proof of [22, Proposition 5.3] shows that if K𝐾Kitalic_K is dependent and (K,v)𝐾𝑣(K,v)( italic_K , italic_v ) i... | A |
Then, an R𝑅Ritalic_R-module ϕ∗ℱsubscriptitalic-ϕℱ\phi_{*}\mathcal{F}italic_ϕ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT caligraphic_F is Cohen-Macaulay. | A reflexive R𝑅Ritalic_R-module M𝑀Mitalic_M is say to be a modifying module if EndR(M)subscriptEnd𝑅𝑀\operatorname{End}_{R}(M)roman_End start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( italic_M ) is a (maximal) Cohen-Macaulay R𝑅Ritalic_R-module. | Let R𝑅Ritalic_R be a normal Cohen-Macaulay domain and M,N𝑀𝑁M,Nitalic_M , italic_N (maximal) Cohen-Macaulay R𝑅Ritalic_R-modules. | Then, an R𝑅Ritalic_R-module ϕ∗ℱsubscriptitalic-ϕℱ\phi_{*}\mathcal{F}italic_ϕ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT caligraphic_F is Cohen-Macaulay. | Let R𝑅Ritalic_R be a normal Cohen-Macaulay domain and M𝑀Mitalic_M a (maximal) Cohen-Macaulay R𝑅Ritalic_R-module. | D |
Here we study a related monoid, replacing continuous mappings with smooth ones and topological groupoids by Lie groupoids. | Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid. | Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid, then the map | Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid, then we define the set | Let 𝒢=(G⇉M)𝒢⇉𝐺𝑀\mathcal{G}=(G\nobreak\rightrightarrows\nobreak M)caligraphic_G = ( italic_G ⇉ italic_M ) be a Lie groupoid then we define | C |
,H_{0},\phi_{0},n)}{C_{\mathit{lc}}(\{Z^{c}\}\cap\{A\},H_{0},\phi_{0},n)}\ .divide start_ARG italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( italic_Z ∩ { italic_A } , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_n ) end_ARG start_ARG italic_C st... | Here we define the Roe algebra of a big family as the C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-subalgebra of the Roe algebra of the ambient space generated by operators which are supported on members of the family. | The family big family {Zc}∩Zsuperscript𝑍𝑐𝑍\{Z^{c}\}\cap Z{ italic_Z start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT } ∩ italic_Z is the coarse geometric version of the boundary of Z𝑍Zitalic_Z. | The second part is the transition from the K𝐾Kitalic_K-theories of Roe algebras to the K𝐾Kitalic_K-theory of Roe categories which are in the background of the construction of the equivariant coarse K𝐾Kitalic_K-homology functor K𝒳G𝐾superscript𝒳𝐺K\!\mathcal{X}^{G}italic_K caligraphic_X start_POSTSUPERSCRIPT itali... | We define the Roe algebra C𝑙𝑐(𝒴,H0,ϕ0,n)subscript𝐶𝑙𝑐𝒴subscript𝐻0subscriptitalic-ϕ0𝑛C_{\mathit{lc}}({\mathcal{Y}},H_{0},\phi_{0},n)italic_C start_POSTSUBSCRIPT italic_lc end_POSTSUBSCRIPT ( caligraphic_Y , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ita... | A |
\boldsymbol{n}}^{\tau}italic_λ = caligraphic_A bold_∇ italic_u ⋅ bold_italic_n start_POSTSUPERSCRIPT italic_τ end_POSTSUPERSCRIPT for all elements τ𝜏\tauitalic_τ. | One difficulty that hinders the development of efficient methods is the presence of high-contrast coefficients [MR3800035, MR2684351, MR2753343, MR3704855, MR3225627, MR2861254]. When LOD or VMS methods are considered, high-contrast coefficients might slow down the exponential decay of the solutions, making the method ... | In the spirit of the Multiscale Hybrid Methods [AHPV, HMV, HPV, MR3584539] and FETI methods [FETI, totalfeti, MR1285024, MR2282408, MR2104179], we consider the decomposition | It is hard to approximate such problem in its full generality using numerical methods, in particular because of the low regularity of the solution and its multiscale behavior. Most convergent proofs either assume extra regularity or special properties of the coefficients [AHPV, MR3050916, MR2306414, MR1286212, babuos85... | mixed finite elements. We note the proposal in [CHUNG2018298] of generalized multiscale finite element methods based on eigenvalue problems inside the macro elements, with basis functions with support weakly dependent of the log of the contrast. Here, we propose eigenvalue problems based on edges of macro element remov... | B |
Moreover, τfsubscript𝜏𝑓\tau_{f}italic_τ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT does not have to be injective, an example is given in [23, 4.2]. | Metric-minimizing discs include many well-studied maps from 𝔻𝔻\mathbb{D}blackboard_D to metric spaces. | Assume that a map f:𝔻→Y:𝑓→𝔻𝑌f\colon\mathbb{D}\to Yitalic_f : blackboard_D → italic_Y has no bubbles. | Let Y𝑌Yitalic_Y be a metric space and s:𝔻→Y:𝑠→𝔻𝑌s\colon\mathbb{D}\to Yitalic_s : blackboard_D → italic_Y be a metric-minimizing map. | However, for metric-minimizing discs f:𝔻→Y:𝑓→𝔻𝑌f\colon\mathbb{D}\to Yitalic_f : blackboard_D → italic_Y both statements hold true; see | D |
(⋅,⋅)⋅⋅(\>\cdot\>,\>\cdot\>)( ⋅ , ⋅ ) renders C(M)𝐶𝑀C(M)italic_C ( italic_M ) a complex Hilbert space what we shall | C(M)𝐶𝑀C(M)italic_C ( italic_M ) we shall write A∈ℜ𝐴ℜA\in{\mathfrak{R}}italic_A ∈ fraktur_R but B^∈ℋ^𝐵ℋ\hat{B}\in{\mathscr{H}}over^ start_ARG italic_B end_ARG ∈ script_H from now on as usual. | 𝔅(ℋ)𝔅ℋ{\mathfrak{B}}({\mathscr{H}})fraktur_B ( script_H ). Multiplication in C(M)𝐶𝑀C(M)italic_C ( italic_M ) from the left on | closure of the image of C(M)𝐶𝑀C(M)italic_C ( italic_M ) under π𝜋\piitalic_π within 𝔅(ℋ)𝔅ℋ{\mathfrak{B}}({\mathscr{H}})fraktur_B ( script_H ) or | \>\>.fraktur_R := ( italic_π ( italic_C ( italic_M ) ) ) start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⊂ fraktur_B ( script_H ) . | B |
Let H(M)⊂P(M)𝐻𝑀𝑃𝑀H(M)\subset P(M)italic_H ( italic_M ) ⊂ italic_P ( italic_M ) denote Waldhausen’s partitions that are also hℎhitalic_h-cobordisms. Using the diffeomorphism we may identify the pair ((𝒯2)kl,(ℋ2)kl)superscriptsubscriptsuperscript𝒯2𝑘𝑙superscriptsubscriptsuperscriptℋ2𝑘𝑙((\mathcal{T}^{2})_{k}^{l... | As before, the inclusion of these partition spaces are highly connected when the inclusion inducing them are highly connected and the relative homotopy type is highly connected (in this case contractible). | The first inclusion is highly connected as this corresponds to an induced inclusion of Waldhausen’s partition spaces. Indeed, (𝒲′)klsuperscriptsubscriptsuperscript𝒲′𝑘𝑙(\mathcal{W}^{\prime})_{k}^{l}( caligraphic_W start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POST... | is a homotopy fibration sequence with this particular null homotopy. As the inclusion of these spaces into the original sequence are homotopy equivalences (Lemma 6.2) and the map to the homotopy fiber is again compatible with this inclusion, the result follows. | The second inclusion is highly connected as it is a highly connected choice to pick a point in M+subscript𝑀M_{+}italic_M start_POSTSUBSCRIPT + end_POSTSUBSCRIPT for M∈𝒯kl𝑀superscriptsubscript𝒯𝑘𝑙M\in\mathcal{T}_{k}^{l}italic_M ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ita... | A |
S=eJ+eQ(J−).𝑆superscript𝑒subscript𝐽superscript𝑒𝑄subscript𝐽S=e^{J_{+}}e^{Q(J_{-})}.italic_S = italic_e start_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_Q ( italic_J start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT . | The synergy between Lie theory and orthogonal polynomials is well-recognized, highlighted by pivotal references such as [19, 12, 6, 7, 8]. These polynomials notably emerge as matrix elements associated with the generators, paving the way for an algebraic framework that facilitates the derivation of generating functions... | The study of this operator not only extends the classical Meixner polynomials but also uncovers essential features such as generating function, recurrence relations, and differential equations through the lens of Barut-Girardello coherent states, as referenced in [4, 9, 22]. | The paper is structured as follows: Section 2 revisits the foundational concepts and key findings regarding d𝑑ditalic_d-orthogonal polynomials. Section 3 introduces the 𝔰𝔲(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) algebra and its representations, detailing the Barut-Girardello coherent states and e... | Turning our attention to the operator S𝑆Sitalic_S, we examine its effect on the vectors |z,β⟩𝑧𝛽\lvert z,\beta\rangle| italic_z , italic_β ⟩. This analysis leads us to the generating function F(z,k)𝐹𝑧𝑘F(z,k)italic_F ( italic_z , italic_k ), which encapsulates the properties of the normalized polynomials {P^n(k)}... | B |
In particular, every two (closed) points of 𝒬̊(β)̊𝒬𝛽\mathring{\mathscr{Q}}(\beta)over̊ start_ARG script_Q end_ARG ( italic_β ) are transferred to each other by the G[[z]]𝐺delimited-[]delimited-[]𝑧G[\![z]\!]italic_G [ [ italic_z ] ]-action. In the same vein, every two points of 𝒬̊(β,w)⊂𝕆(w)̊𝒬𝛽𝑤𝕆𝑤\mathrin... | By the dimension comparison using (4.3), we deduce that this embedding must be open dense in 𝒬(β)=𝒬(β,e,w0)𝒬𝛽𝒬𝛽𝑒subscript𝑤0\mathscr{Q}(\beta)=\mathscr{Q}(\beta,e,w_{0})script_Q ( italic_β ) = script_Q ( italic_β , italic_e , italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). The locus Y𝑌Yitalic_Y on which t... | We have ℬ=𝒬(0)ℬ𝒬0\mathscr{B}=\mathscr{Q}(0)script_B = script_Q ( 0 ) by the Plücker embedding. By expanding the map | of 𝒵(β)𝒵𝛽\mathscr{Z}(\beta)script_Z ( italic_β ). The map (4.5) can be also obtained as the formal completion of the map | For the first assertion, combine Theorem 2.12 and Corollary 3.13 to obtain the map Ψ−1∘ΦsuperscriptΨ1Φ\Psi^{-1}\circ\Phiroman_Ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ roman_Φ, that have dense image. Note that the both sides are rings and the identity [𝒪Gr0]delimited-[]subscript𝒪subscriptGr0[{\mathcal{O}}_{\m... | B |
A number of other interesting homological and homotopical finiteness properties have been studied in relation to monoids defined by complete rewriting systems; see [3, 55, 28]. | on the behaviour of FPnsubscriptFP𝑛{\rm FP}_{n}roman_FP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT under free products and certain rather restricted free products of monoids with amalgamation. The proofs in Cremanns and Otto are quite long and technical, as is often the case for results in this area. The results i... | The Hochschild cohomological dimension of M𝑀Mitalic_M, written dimMdimension𝑀\dim Mroman_dim italic_M, is the length of a shortest projective resolution of ℤMℤ𝑀\mathbb{Z}Mblackboard_Z italic_M as a ℤ[M×Mop]ℤdelimited-[]𝑀superscript𝑀𝑜𝑝\mathbb{Z}[M\times M^{op}]blackboard_Z [ italic_M × italic_M start_POSTSUPER... | In general, the left- and right-cohomological dimensions of a monoid are not equal. In fact they are completely independent of each other; see [27]. One immediate corollary of the above result is that if M𝑀Mitalic_M is a finitely presented special monoid with left- and right-cohomological dimensions both at least equa... | The cohomological dimension of monoids has also received attention in the literature; see for example [13, 27, 47]. | D |
Cheridito et al. (2006) considered dynamic coherent, convex monetary and monetary risk measures for discrete-time processes modelling the evolution of financial values. Acciaio et al. (2012) extended dynamic convex risk measures in Cheridito et al. to take the timing of cash flow into consideration. Sun and Hu (2018) i... | Nowadays, as the digital economy and cryptocurrencies develop rapidly, they have a great impact on the financial market. The volatility of cryptocurrencies is a distinctive characteristic defined by rapid and substantial price fluctuations within relatively short periods. Compared to traditional financial assets, crypt... | Although there are many different studies of risk measure, a common aspect of these studies is the space of financial positions was considered as the space or subspace of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. It is crucial to note that our focus diverges from this convention... | Recently, many external factors including changes in international situations, increase of war risk and significant environmental changes, all cause the financial market becomes much more volatile than before, and the financial market also has different orders of risk data mixed over a short period of time. Therefore, ... | As economic landscapes evolve, traditional risk metrics may prove insufficient in capturing the complexities and nuances of contemporary risks. In particular, from the global financial crisis of 2008 to the more recent disruptions caused by geopolitical tensions and public health crises, which have underscored the need... | A |
Observe that Im(β)Im𝛽\operatorname{\mathrm{Im}}(\beta)roman_Im ( italic_β ) is 𝔤Γq⊕𝔤Γqdirect-sumsuperscript𝔤subscriptΓ𝑞superscript𝔤subscriptΓ𝑞\mathfrak{g}^{\Gamma_{q}}\oplus\mathfrak{g}^{\Gamma_{q}}fraktur_g start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⊕ frakt... | By the Residue Theorem, β𝛽\betaitalic_β is indeed a Lie algebra embedding. (Observe that Lemma 7.3 has been used to show that β𝛽\betaitalic_β is an embedding.) | From the surjectivity of β𝛽\betaitalic_β, we get that the map F𝐹Fitalic_F is surjective by combining the equation | From the definition of β𝛽\betaitalic_β (cf. equation (48)), it is easy to see that, under the identification of V(μ∗)⊗V(μ)tensor-product𝑉superscript𝜇𝑉𝜇V(\mu^{*})\otimes V(\mu)italic_V ( italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ⊗ italic_V ( italic_μ ) with Endℂ(V(μ))subscriptEndℂ𝑉𝜇\operatorname{... | Combining (132) and (133), we get λ≡0𝜆0\lambda\equiv 0italic_λ ≡ 0. This proves that σ˙˙𝜎\dot{\sigma}over˙ start_ARG italic_σ end_ARG is the identity map. | B |
By Lemma 1.2, G[C]𝐺delimited-[]𝐶G[C]italic_G [ italic_C ] is either a cycle in ℬℬ\mathcal{B}caligraphic_B, or one of the following biased subgraphs: a theta with no cycle in ℬℬ\mathcal{B}caligraphic_B, tight handcuffs, loose handcuffs, or a bracelet. | Then T𝑇Titalic_T is dependent in F(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ) and so dependent in M𝑀Mitalic_M, while every proper subset of T𝑇Titalic_T is independent in L(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ) and so independent in M𝑀Mitalic_M. | That is, M𝑀Mitalic_M is biased-graphic if there is a biased graph (G,ℬ)𝐺ℬ(G,\mathcal{B})( italic_G , caligraphic_B ) such that M𝑀Mitalic_M is intermediate between L(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ) and F(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ), | This implies that every dependent set of M𝑀Mitalic_M is dependent in L(G,ℬ)𝐿𝐺ℬL(G,\mathcal{B})italic_L ( italic_G , caligraphic_B ). | By Lemma 2.7 every dependent set of F(G,ℬ)𝐹𝐺ℬF(G,\mathcal{B})italic_F ( italic_G , caligraphic_B ) is dependent in M𝑀Mitalic_M. | C |
Recall for a bosons, α=0𝛼0\alpha=0italic_α = 0; hence, a vacancy box for bosons can be occupied arbitrary number of times. For a fermion, however, α=1𝛼1\alpha=1italic_α = 1; hence, a vacancy box for fermions can be occupied at most once. Now we have (α~ττ¯D1)11=2subscriptsubscriptsuperscript~𝛼superscript𝐷1𝜏¯𝜏112... | Recall for a bosons, α=0𝛼0\alpha=0italic_α = 0; hence, a vacancy box for bosons can be occupied arbitrary number of times. For a fermion, however, α=1𝛼1\alpha=1italic_α = 1; hence, a vacancy box for fermions can be occupied at most once. Now we have (α~ττ¯D1)11=2subscriptsubscriptsuperscript~𝛼superscript𝐷1𝜏¯𝜏112... | In the exclusion rule above, we define a box as a vacancy and call it a vacancy box. A ■■\boxed{\blacksquare}■ is a vacancy box occupied, a ×\boxed{\times}× is a vacancy box that cannot be occupied (or unhabitable), and ✓✓\boxed{\checkmark}✓ is a vacancy box that may be occupied (or habitable). In other words, for the ... | Here, □□\boxed{\square}□ is a wildcard, meaning the box is either occupied or unoccupied; the RHS of (c)𝑐(c)( italic_c ) forces the two boxes in the column to be both occupied (unoccupied) if the box on the LHS is occupied (unoccupied). | Now the question is: How we may decide which vacancy boxes in the disk basis Fig. 4(e) can be occupied and which cannot be? To find the answer, let us further simplify the basis into the form in Fig. 5 by removing the tree and the boundary but keeping the boxes only. This procedure is correct because the degrees of fre... | C |
Following Deligne [Del90, 1.2], a tensor category 𝒯𝒯\mathcal{T}caligraphic_T over k𝑘kitalic_k is a k𝑘kitalic_k-linear | ⊗tensor-product\otimes⊗-category which is ACU, rigid, and abelian with k=End(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ). A category is | condition k=End(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ) is superfluous.) | We start by noting that, in our particular case, the ring End(𝟙)End1\mathrm{End}(\mathds{1})roman_End ( blackboard_1 ) is isomorphic to | For simplicity, in this work a tensor category 𝒯𝒯\mathcal{T}caligraphic_T over k𝑘kitalic_k is a k𝑘kitalic_k-linear ⊗tensor-product\otimes⊗-category which is ACU, rigid, and abelian with k=End(𝟙)𝑘End1k=\mathrm{End}(\mathds{1})italic_k = roman_End ( blackboard_1 ). This amounts to say that ⊗tensor-product\otimes⊗ ... | A |
For each β≤κ+𝛽superscript𝜅\beta\leq\kappa^{+}italic_β ≤ italic_κ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, ℚβsubscriptℚ𝛽{\mathbb{Q}}_{\beta}blackboard_Q start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT has the κ𝜅\kappaitalic_κ-chain condition. | As we will see, the following lemma will enable us to ease our path through the proof of Claim 4.7, in Section 4, in a significant way. | As we will see, the weak compactness of κ𝜅\kappaitalic_κ is used crucially in order to prove Lemma 4.1. | The preservation of all higher cardinals proceeds by showing that the construction has the κ𝜅\kappaitalic_κ-chain condition. For this, we use the weak compactness of κ𝜅\kappaitalic_κ in an essential way. The proof of the κ𝜅\kappaitalic_κ-c.c. of ℚβsubscriptℚ𝛽{\mathbb{Q}}_{\beta}blackboard_Q start_POSTSUBSCRIPT ital... | Let ℱℱ\mathcal{F}caligraphic_F be the weak compactness filter on κ𝜅\kappaitalic_κ, i.e., the filter on κ𝜅\kappaitalic_κ generated by the sets | B |
Now we claim that ℰc=0subscriptℰ𝑐0\mathcal{E}_{c}=0caligraphic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 0 in our associated graded algebra. | According to [19], to define the RTT Yangian YV(𝔤)subscript𝑌𝑉𝔤Y_{V}({\mathfrak{g}})italic_Y start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( fraktur_g ), one can use any finite-dimensional representation V𝑉Vitalic_V of Y(𝔤)𝑌𝔤Y({\mathfrak{g}})italic_Y ( fraktur_g ) which is not a sum of trivial representations... | 2.19. Filtration on YV(𝔤)subscript𝑌𝑉𝔤Y_{V}({\mathfrak{g}})italic_Y start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( fraktur_g ) | It follows from relation [19, (5.10)] (which is still true for our filtration) and from [19, Lemma 4.13]. | Note that from Theorem 2.4 it follows that this is indeed a filtration and grYnew(𝔤)grsubscript𝑌𝑛𝑒𝑤𝔤{\rm gr}\,Y_{new}({\mathfrak{g}})roman_gr italic_Y start_POSTSUBSCRIPT italic_n italic_e italic_w end_POSTSUBSCRIPT ( fraktur_g ) is commutative. | C |
We note that Dai, Mallick and Stoffregen have independently found examples of slice disks with large stabilization distance [DMSEquivariant]. Additionally, they use some of the techniques of this paper in their work to study equivariant knots. | and properly embedded surfaces in B4superscript𝐵4B^{4}italic_B start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT with boundary a knot K𝐾Kitalic_K with a type of metric | We proceed to define the stabilization distance of a pair of surfaces with boundary a given knot, which | of a pair of surfaces S𝑆Sitalic_S, S′∈Surf(K)superscript𝑆′Surf𝐾S^{\prime}\in\operatorname{{Surf}}(K)italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Surf ( italic_K ), which is similar to the stabilization distance, | We define the stabilization distance of the pair (S,S′)𝑆superscript𝑆′(S,S^{\prime})( italic_S , italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), for which we write | B |
Both stacks are Pic(Σ~b(1))Picsuperscriptsubscript~Σ𝑏1\operatorname{Pic}(\widetilde{\Sigma}_{b}^{(1)})roman_Pic ( over~ start_ARG roman_Σ end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT )-torsors. Since τ~*superscript~𝜏\widetilde{\tau}^{*}over~ start_ARG italic_... | Let k𝑘kitalic_k be an algebraically closed field. Let ℬℬ\mathcal{B}caligraphic_B be a k𝑘kitalic_k-scheme locally of finite type. Let Y𝑌Yitalic_Y be a stack locally of finite type over ℬℬ\mathcal{B}caligraphic_B. Let Y~⟶Y⟶~𝑌𝑌\widetilde{Y}\longrightarrow Yover~ start_ARG italic_Y end_ARG ⟶ italic_Y be a 𝔾msubscript... | In this subsection we review the Fourier-Mukai transforms on commutative group stacks, following [8]. Let k𝑘kitalic_k be an algebraically closed field. Let ℬℬ\mathcal{B}caligraphic_B be an irreducible k𝑘kitalic_k-scheme that is locally of finite type. | Let 𝒢𝒢\mathcal{G}caligraphic_G be a commutative group stack locally of finite type over ℬℬ\mathcal{B}caligraphic_B. The dual commutative group stack 𝒢∨superscript𝒢\mathcal{G}^{\vee}caligraphic_G start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT classifies 1-morphisms of group stacks from 𝒢𝒢\mathcal{G}caligraphic_G to B... | Let S𝑆Sitalic_S be a k𝑘kitalic_k-scheme. Let b𝑏bitalic_b be an S𝑆Sitalic_S-point of (BP0)(1)superscriptsuperscriptsubscript𝐵𝑃01(B_{P}^{0})^{(1)}( italic_B start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT. Consider the foll... | B |
We can say more about (ii), and in D we give a sample result for slightly more abstract semigroups, for which we would not a priori know the associated equations. | This section is devoted to the proofs of the results of Section 3. We will prove them in a certain order to arrive at Corollaries 25 and 37, thus concluding by the optimality of the weight and the interpretation in terms of dual nonlinear semigroup. The proofs of Propositions | The proofs of Theorem 35 and Corollary 36 are given in Section 4.2, while Theorem 33 is proved in Section 4.5. | The rest of this paper is organized as follows. We recall basic facts in Section 2, we state our main results in Section 3, and prove them in Section 4. For completeness, some results for minimal discontinuous viscosity solutions are proved in A, a complete proof of well-posedness for L∞superscript𝐿L^{\infty}italic_L ... | In this section we precisely state our results: the weighted L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT contraction estimate for (1) in Section 3.1, the optimality of the weight in Section 3.2, and the interpretation in terms of dual nonlinear semigroup in Section 3.3. Section 3.3 contain... | A |
Throughout this paper function 𝒜()𝒜\mbox{${{{\cal A}}}$}()caligraphic_A ( ) will denote the arithmetic mean of any number of variables. We will also use the notation 𝒜(ai:1≤i≤n)\mbox{${{{\cal A}}}$}(a_{i}:1\leq i\leq n)caligraphic_A ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_n... | If H⊂ℝ𝐻ℝH\subset\mathbb{R}italic_H ⊂ blackboard_R is a finite set then 𝒜(H)𝒜𝐻\mbox{${{{\cal A}}}$}(H)caligraphic_A ( italic_H ) denotes the arithmetic mean of its distinct points. | Let (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) be a sequence and a≤b(a,b∈ℝ)𝑎𝑏𝑎𝑏ℝa\leq b\ (a,b\in\mathbb{R})italic_a ≤ italic_b ( italic_a , italic_b ∈ blackboard_R ) are two of its accumulation points. Then [a,b]⊂AAR(an)𝑎𝑏𝐴𝐴subscript𝑅subscript𝑎𝑛[a,b]\subset AAR_{(a... | Throughout this paper function 𝒜()𝒜\mbox{${{{\cal A}}}$}()caligraphic_A ( ) will denote the arithmetic mean of any number of variables. We will also use the notation 𝒜(ai:1≤i≤n)\mbox{${{{\cal A}}}$}(a_{i}:1\leq i\leq n)caligraphic_A ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : 1 ≤ italic_i ≤ italic_n... | Let (an)subscript𝑎𝑛(a_{n})( italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) be a sequence such that it has at least 4 accumulation points: a,b,−∞,+∞(a,b∈ℝ,a<b)a,b,-\infty,+\infty\ (a,b\in\mathbb{R},a<b)italic_a , italic_b , - ∞ , + ∞ ( italic_a , italic_b ∈ blackboard_R , italic_a < italic_b ). Let Z⊂[a,b]�... | A |
\frac{3}{8}-\frac{5}{4p},&\quad 4<p<6.\end{aligned}\right.italic_σ ( italic_p ) = { start_ROW start_CELL 0 , end_CELL start_CELL 2 < italic_p ⩽ 3 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG - divide start_ARG 3 end_ARG start_ARG 4 italic_p end_ARG , end_CELL start_CELL 3 < ita... | It is worth noting that the spacial support of the amplitude appearing in the right-hand side of (2.12) and (2.13) is slightly larger than that appearing in the left-hand side. | The results obtained above generalize its constant coefficient counterpart in [14, 12]. It is worth noting that the results in the case 2<p≤32𝑝32<p\leq 32 < italic_p ≤ 3 are sharp, except for possibly arbitrarily small regularity loss. | Very recently, Guth-Wang-Zhang [10] established the sharp square function estimate in the Euclidean case in 2+1212+12 + 1 dimensions. As a result, the corresponding local smoothing conjecture is resolved. For the variable coefficient setting, the Kakeya compression phenomena will happen which leads to the difference in... | The following stability lemma makes the variable coefficient case and its constant counterpart comparable at sufficiently small scales. | B |
+1}\|_{Z_{s}})\tilde{G}_{n}(u^{k+1}(s)))\,dW(s),~{}\mathbb{P}\textrm{-a.s..}+ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cos ( ( italic_k italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_s ) s... | To prove the existence of a local mild solution to problem (5.4)-(5.5) in the sense of Definition 5.13. | Thus, in light of (6.43)-(6.44), in order to prove (6.42) it is sufficient to prove equality (6.46) below, i.e., | In this section we will prove the deterministic Strichartz type estimate, see Theorem 3.2 below, which is a generalization of [42, Theorem 1.2] and is essential to tackle, both, the Dirichlet and the Neumann boundary case. | This section is devoted to prove a stochastic Strichartz inequality, which is sufficient to apply the Banach Fixed Point Theorem in the proof of a local well-posedness result for problem (1.2), see Theorem 7.1 in Section 7. | B |
In constrast, in the S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-equivariant version we replace H(X)𝐻𝑋H(X)italic_H ( italic_X ) by HS1(X)≅H(X)⊗H(ℂℙ∞)subscript𝐻superscript𝑆1𝑋tensor-product𝐻𝑋𝐻ℂsuperscriptℙH_{S^{1}}(X)\cong H(X)\otimes H(\mathbb{CP}^{\infty})italic_H start_POSTSU... | This allows us to define capacities for larger class of domains which are not necessarily exact, and it also allows us to discuss Maurer–Cartan theory (see §4). | Before discussing our main results, we first motivate and illustrate the concept of “higher symplectic capacities” from the perspective of Floer theory, showing how to construct new capacities based on algebraic structures on symplectic cohomology. | Let us now observe that symplectic cohomology has various additional algebraic structures which we can try to exploit to define further capacities. | It is tempting to also use these non-equivariant structures to define symplectic capacities, but unfortunately | C |
}}\Big{|}^{s+t}\Big{)}^{\frac{1}{s+t}}.start_ROW start_CELL ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ( italic_q start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_s + italic_t - 1 end_ARG start_ARG italic_... | We multiply both inequalities by b(1−q)𝑏1𝑞b(1-q)italic_b ( 1 - italic_q ), then raise them to the power s+t𝑠𝑡s+titalic_s + italic_t. Thus, we obtain | Multiplying this inequality by bp+1(1−q)p+1superscript𝑏𝑝1superscript1𝑞𝑝1b^{p+1}(1-q)^{p+1}italic_b start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT ( 1 - italic_q ) start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT, and relying on the formula for the sum of the first n𝑛nitalic_n terms of the geometric ... | in q𝑞qitalic_q-calculus was given. Here we eliminate some inaccuracies by simplifying and modifying the proofs of the theorems. | In view of f(bqn)=∑j=0n−1f(bqj+1)−f(bqj)𝑓𝑏superscript𝑞𝑛superscriptsubscript𝑗0𝑛1𝑓𝑏superscript𝑞𝑗1𝑓𝑏superscript𝑞𝑗f(bq^{n})=\sum\limits_{j=0}^{n-1}{f(bq^{j+1})-f(bq^{j})}italic_f ( italic_b italic_q start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRI... | A |
ψ∗E:=N×ψEassignsuperscript𝜓∗𝐸subscript𝜓𝑁𝐸\psi^{\ast}E:=N\times_{\psi}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E := italic_N × start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT italic_E over N𝑁Nitalic_N. In explicit terms | ψ∗Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E over the open set ψ−1(U)superscript𝜓1𝑈\psi^{-1}\left(U\right)italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_U ). Then | ∇ψsuperscript∇𝜓\nabla^{\psi}∇ start_POSTSUPERSCRIPT italic_ψ end_POSTSUPERSCRIPT over ψ∗Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E satisfies the formula | ψ∗Esuperscript𝜓∗𝐸\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E over ψ−1(U)superscript𝜓1𝑈\psi^{-1}\left(U\right)italic_ψ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_U ). We compute the local | ψ∗Esuperscript𝜓∗𝐸\displaystyle\psi^{\ast}Eitalic_ψ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_E | D |
Remark: henceforth we will deal only with oriented elementary allowed paths, for this reason we will call them simply by allowed elementary paths, with certain abuse of language. | A disorientation of the allowed elementary (p+1)𝑝1(p+1)( italic_p + 1 )-paths of G𝐺Gitalic_G is a choice of orientation G+p+1subscriptsuperscript𝐺𝑝1G^{p+1}_{+}italic_G start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT for these paths such that no pair of (p+1)𝑝1(p+1)( i... | We highlight that the notions of p𝑝pitalic_p-paths and of p𝑝pitalic_p-forms were introduced in grig without the notion of orientation. But such orientation will be essential to relate our Laplace operators to interesting Markov chains, as in its definition we have an alternating sum. | We close this paper with the consideration that we used the notion of up-adjacency to define our Markov chain. One may define a similar process using the notion of down-adjacency. But it is not trivial to relate this new walk to the spectrum of the Laplacian. If it can be done, one is one step closer to understand the ... | But, in higher orders, one can define to (p+1)𝑝1(p+1)( italic_p + 1 )-paths to be neighbors if they “intersect” in a p𝑝pitalic_p-path, as follow. | D |
\mathord{\mbox{}\,\overline{\!y\!\!\;}\!\>}\mbox{}\rangle⟨ italic_x , italic_y ⟩ = ⟨ start_ID over¯ start_ARG italic_x end_ARG end_ID , start_ID over¯ start_ARG italic_y end_ARG end_ID ⟩. | For all x,y∈𝕆𝑥𝑦𝕆x,y\in\mathord{\mathbb{O}}italic_x , italic_y ∈ blackboard_O the following identities hold: | for all x,y∈S𝑥𝑦𝑆x,y\in Sitalic_x , italic_y ∈ italic_S and for all z∈𝕆𝑧𝕆z\in\mathord{\mathbb{O}}italic_z ∈ blackboard_O we have (xy)z=x(yz)𝑥𝑦𝑧𝑥𝑦𝑧(xy)z=x(yz)( italic_x italic_y ) italic_z = italic_x ( italic_y italic_z ). | For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O, the following identities hold: | For all x,y,z∈𝕆𝑥𝑦𝑧𝕆x,y,z\in\mathord{\mathbb{O}}italic_x , italic_y , italic_z ∈ blackboard_O the following identities hold: | D |
In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structur... | In [7], coarse proximity spaces were introduced to axiomatize the “at infinity” perspective of coarse geometry, providing general definitions of coarse neighborhoods (whose metric space specific definition was given by Dranishnikov in [3]), asymptotic disjointness, and closeness “at infinity.” Coarse proximity structur... | Every coarse proximity map between coarse proximity spaces induces a map between the boundaries of these coarse proximity spaces, as the following definition shows. | In this section, we provide the basic definitions and theorems surrounding proximity spaces and coarse proximity spaces. The definitions and theorems about proximity spaces come from [11], and the definitions and theorems about coarse proximity spaces come from [9]. | The collection of coarse proximity spaces and closeness classes of coarse proximity maps makes up the category of coarse proximity spaces. For details, see [7]. | C |
(a), 9μ9μ9\upmu9 roman_μs (b), 18μ18μ18\upmu18 roman_μs (c), 45μ45μ45\upmu45 roman_μs (d), 65μ65μ65\upmu65 roman_μs | at 0μ0μ0\upmu0 roman_μs (a), 9μ9μ9\upmu9 roman_μs (b), 45μ45μ45\upmu45 roman_μs (c), 65μ65μ65\upmu65 roman_μs (d) | 9μ9μ9\upmu9 roman_μs (b), 18μ18μ18\upmu18 roman_μs (c), 45μ45μ45\upmu45 roman_μs (d), 65μ65μ65\upmu65 roman_μs (e), | at 0μ0μ0\upmu0 roman_μs (a), 9μ9μ9\upmu9 roman_μs (b), 45μ45μ45\upmu45 roman_μs (c), 65μ65μ65\upmu65 roman_μs (d). | (a), 9μ9μ9\upmu9 roman_μs (b), 18μ18μ18\upmu18 roman_μs (c), 45μ45μ45\upmu45 roman_μs (d), 65μ65μ65\upmu65 roman_μs | B |
_{\text{int},c}({\mathbb{C}}).over¯ start_ARG italic_π end_ARG start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT int , italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT : fraktur_G start_POSTSUBSCRIPT italic_R , over˙ start_ARG italic_R end_ARG , italic_σ end_POSTSUBSCRIPT ( blackboard_C ) → italic_G start_POSTSUBSCRIPT int ... | Proof. It immediately follows from [16, Lemma 3.2.32] and the explanation after its proof on Page 130. □□\Box□ | By the above adjustment, the next proposition is a generalization of [16, Proposition 3.2.41]. Its proof is also similar to the proof | Proof. Let (A)𝐴(A)( italic_A ) denote the field of fractions of A𝐴Aitalic_A. The above relations hold | Proof. When R˙=An˙𝑅subscript𝐴𝑛\dot{R}=A_{n}over˙ start_ARG italic_R end_ARG = italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT (for n≥2𝑛2n\geq 2italic_n ≥ 2) this is [16, Proposition 3.3.1]. We now consider all other possible cases. Similar to the proof of [16, Proposition 3.3.1], it is clear | B |
In particular, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is non-logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02]. | By Theorem 6.4 and by 2.9.2, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is logarithmically bounded by I𝐼Iitalic_I if and only if | Gal(L/K)∗I={1}\operatorname{Gal}(L/K)_{*}^{I}=\{1\}roman_Gal ( italic_L / italic_K ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT = { 1 } if and only if the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is ∗*∗-bounded by I𝐼Iitalic_I. | In particular, the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is non-logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02]. | By [Sa09, 1.2.6], this shows that the ramification of L/K𝐿𝐾L/Kitalic_L / italic_K is logarithmically bounded by I𝐼Iitalic_I in our sense if and only if it is logarithmically bounded by r𝑟ritalic_r in the sense of Abbes-Saito [AS02]. | A |
Explicitly, a set in g(ℒr)𝑔subscriptℒ𝑟g(\mathcal{L}_{r})italic_g ( caligraphic_L start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) is a counter-example to a FD X→A𝑋→𝐴X\operatorname{\rightarrow}Aitalic_X → italic_A if it contains | counter-example in all realities where A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B does not hold. | In particular, it satisfies C→A𝐶→𝐴C\operatorname{\rightarrow}Aitalic_C → italic_A and A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B. | along with the abstract tuples which will be interpreted as counter-examples to A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B or B→A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A. | First, remark that both A→B𝐴→𝐵A\operatorname{\rightarrow}Bitalic_A → italic_B and B→A𝐵→𝐴B\operatorname{\rightarrow}Aitalic_B → italic_A are possible. | D |
For edges e∈E∞𝑒subscript𝐸e\in E_{\infty}italic_e ∈ italic_E start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT, we define | ℐe={V0⊂V:the subgraph of (V,E0∪Eopen) induced by V0 makes e occupied}.subscriptℐ𝑒conditional-setsubscript𝑉0𝑉the subgraph of (V,E0∪Eopen) induced by V0 makes e occupied\mathcal{I}_{e}=\{V_{0}\subset V:\text{the subgraph of | G0subscript𝐺0G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Gopensubscript𝐺openG_{\text{\rm open}}italic_G start_POSTSUBSCRIPT open end_POSTSUBSCRIPT with their subgraphs induced by Iesubscript𝐼𝑒I_{e}italic_I start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT, so that | induced by $V_{0}$ makes $e$ occupied}\}.caligraphic_I start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊂ italic_V : the subgraph of ( italic_V , italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_E start_POSTSUBSCRIPT open end_POSTSUBSCRIPT ) induced by italic_... | an oriented subgraph G0=(V,E0)subscript𝐺0𝑉subscript𝐸0G_{0}=(V,E_{0})italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( italic_V , italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). We then | A |
Let ∥s∥2subscriptdelimited-∥∥𝑠2\lVert s\rVert_{2}∥ italic_s ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be the input norm in [K23, Thm. 3.9] with respect to the Ohsawa measure dV[Ψ]Y𝑑𝑉subscriptdelimited-[]Ψ𝑌dV[\Psi]_{Y}italic_d italic_V [ roman_Ψ ] start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT. From Theorem 6.2, w... | Let Y𝑌Yitalic_Y be a maximal lc center of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) with a unique lc place. | Now let Y𝑌Yitalic_Y be a maximal lc center of an lc pair (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) as above. Let ΨΨ\Psiroman_Ψ be a quasi-psh function on X𝑋Xitalic_X with analytic singularities determined by the relation e−Ψh=e−ψsuperscript𝑒Ψℎsuperscript𝑒𝜓e^{-\Psi}h=e^{-\psi}italic_e start_POSTSUPERSCRIPT - roman_... | Let Y𝑌Yitalic_Y be a maximal lc center of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) with a unique lc place. If | A log canonical center (or an lc center) of (X,ψ)𝑋𝜓(X,\psi)( italic_X , italic_ψ ) is an irreducible subvariety Y⊂X𝑌𝑋Y\subset Xitalic_Y ⊂ italic_X that is the image of a prime divisor E𝐸Eitalic_E in a log resolution X′→X→superscript𝑋′𝑋X^{\prime}\to Xitalic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_X... | A |
Define the operators999In fact we will prove stronger bounds where the low frequency inputs in (3.19) are replaced by vL1†,⋯vL2r†superscriptsubscript𝑣subscript𝐿1†⋯superscriptsubscript𝑣subscript𝐿2𝑟†v_{L_{1}}^{\dagger},\cdots v_{L_{2r}}^{\dagger}italic_v start_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 1 end_POSTS... | 𝒫−(w)superscript𝒫𝑤\displaystyle\mathcal{P}^{-}(w)caligraphic_P start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_w ) | Here the random averaging operator 𝒬=𝒫(1−𝒫)−1𝒬𝒫superscript1𝒫1\mathcal{Q}=\mathcal{P}(1-\mathcal{P})^{-1}caligraphic_Q = caligraphic_P ( 1 - caligraphic_P ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, and 𝒫𝒫\mathcal{P}caligraphic_P has the form | \mathrm{HS}}).∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT ≤ ∥ caligraphic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_OP end_POSTSUBSCRIPT ∥ caligraphic_P start_POSTSUBSCRIPT 2 en... | 𝒫+(w)superscript𝒫𝑤\displaystyle\mathcal{P}^{+}(w)caligraphic_P start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_w ) | D |
We provide some applications of this in Section 4 before proving our main theorem in Section 5. The paper proper concludes with an examination of other circumstances under which domination implies isolation in Section 6. | This paper has an extensive Appendix that records a number of definitions and facts from basic geometric stability theory. | The results of this paper are concerned with non-locally modular regular types over models. It is natural to ask for other circumstances under which leaves are constructible. This is always possible if T𝑇Titalic_T is ω𝜔\omegaitalic_ω-stable, as such a theory has constructible models over any set. | We are grateful to the anonymous referee, as his/her thorough reading of this paper has led to many expositional improvements. | a witness to non-modularity over M′superscript𝑀′M^{\prime}italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. To see this, note that abcd | C |
After dropping nodes in 𝒱−superscript𝒱\mathcal{V}^{-}caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and all their incident edges, the resulting graph is likely to be disconnected. | Therefore, we use a link construction procedure to obtain a connected graph supported by the nodes in 𝒱+superscript𝒱\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT. | By means of the rounding procedure in (5), the nodes in 𝒱𝒱\mathcal{V}caligraphic_V are partitioned in two sets, 𝒱+superscript𝒱\mathcal{V}^{+}caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and 𝒱−=𝒱∖𝒱+superscript𝒱𝒱superscript𝒱\mathcal{V}^{-}=\mathcal{V}\setminus\mathcal{V}^{+}caligraphic_V start_POST... | The cost of inverting 𝐋𝒱−,𝒱−subscript𝐋superscript𝒱superscript𝒱{\mathbf{L}}_{\mathcal{V}^{-},\mathcal{V}^{-}}bold_L start_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is 𝒪(|𝒱−|3)𝒪superscriptsuperscript𝒱3\m... | where 𝐋𝒱+,𝒱−subscript𝐋superscript𝒱superscript𝒱{\mathbf{L}}_{\mathcal{V}^{+},\mathcal{V}^{-}}bold_L start_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , caligraphic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_POSTSUBSCRIPT identifies a sub-matrix of 𝐋𝐋{\mathbf{L}}bold_L with rows... | A |
By Lemma 6.3, s−a†≻1succeeds𝑠superscript𝑎†1s-a^{\dagger}\succ 1italic_s - italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ≻ 1 for all a∈L×𝑎superscript𝐿a\in L^{\times}italic_a ∈ italic_L start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT. | Giving L(f)𝐿𝑓L(f)italic_L ( italic_f ) the valuation and ordering from Lemma 6.4 makes it a pre-H𝐻Hitalic_H-field extension of L𝐿Litalic_L with gap 00 of type (v)(). | Then giving L(f)𝐿𝑓L(f)italic_L ( italic_f ) the ordering and valuation from Lemma 6.5 makes it a pre-H𝐻Hitalic_H-field extension of L𝐿Litalic_L with gap 00 of type (vi)(). | Let K𝐾Kitalic_K be a pre-H𝐻Hitalic_H-field and L𝐿Litalic_L be a pre-H𝐻Hitalic_H-field extension of K𝐾Kitalic_K with gap 00. | If K𝐾Kitalic_K is a pre-H𝐻Hitalic_H-field with gap 00 and L𝐿Litalic_L is moreover a differential field extension of K𝐾Kitalic_K with small derivation, then L𝐿Litalic_L is also a pre-H𝐻Hitalic_H-field with gap 00. | B |
The associators in 𝒞𝒞\mathcal{C}caligraphic_C will descend to morphisms in 𝒞←←𝒞\underleftarrow{\mathcal{C}}under← start_ARG caligraphic_C end_ARG and still satisfy the pentagon equations. We have to convince ourselves that these morphisms define a natural isomorphism, with respect to the extra morphisms in the enri... | Since we made a choice to use β𝛽\betaitalic_β rather than β−1superscript𝛽1\beta^{-1}italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT in Definition 17, we also have: | In defining the 2-category 𝐁𝐓𝐂(𝒜)𝐁𝐓𝐂𝒜\mathbf{BTC}(\mathcal{A})bold_BTC ( caligraphic_A ) of braided tensor categories containing 𝒜𝒜\mathcal{A}caligraphic_A, there are several choices to be made, we use the following definition: | If β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT is indeed a functor, then it is clearly invertible with inverse given by using the opposite crossings in Equation (23). So we need to check that β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT preserve... | the composite of the functor β−2superscript𝛽2\beta^{-2}italic_β start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT (Definition 21) with the symmetry in 𝒜𝒜\mathcal{A}caligraphic_A. | A |
Let α=1+κ′𝛼1superscript𝜅′\alpha=1+\kappa^{\prime}italic_α = 1 + italic_κ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for 0<κ′≪1,0superscript𝜅′much-less-than10<\kappa^{\prime}\ll 1,0 < italic_κ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≪ 1 ,then | converges in the 𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT topology, see e.g. | 𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT norm of ΞΞ\Xiroman_Ξ (see Definition 3.5) it was | 𝒳αsuperscript𝒳𝛼\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT in probability. | 𝒳αsuperscript𝒳𝛼\displaystyle\mathcal{X}^{\alpha}caligraphic_X start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT | D |
{gr}}}}_{2^{s}}))/2= ( italic_E start_POSTSUBSCRIPT 1 , 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( start_OPFUNCTION over¯ start_ARG italic_gr end_ARG end_OPFUNCTION start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERS... | ≤v<1(𝑔𝑟s) by (18)absentsubscript𝑣absent1subscript𝑔𝑟𝑠 by (18)\displaystyle\leq v_{<1}(\operatorname{\mathit{gr}}_{s})\text{ by \eqref{shade% | ≤v<1(𝑔𝑟3⋅2s) by (18)absentsubscript𝑣absent1subscript𝑔𝑟⋅3superscript2𝑠 by (18)\displaystyle\leq v_{<1}(\operatorname{\mathit{gr}}_{3\cdot 2^{s}})\text{ by % | ≤(v<1(𝑔𝑟2s)+eh(𝑔𝑟¯2s))/2 using (18).absentsubscript𝑣absent1subscript𝑔𝑟superscript2𝑠subscript𝑒ℎsubscript¯𝑔𝑟superscript2𝑠2 using (18)\displaystyle\leq(v_{<1}(\operatorname{\mathit{gr}}_{2^{s}})+e_{h}(% | =(E1,2s+eh(𝑔𝑟¯2s))/2absentsubscript𝐸1superscript2𝑠subscript𝑒ℎsubscript¯𝑔𝑟superscript2𝑠2\displaystyle=(E_{1,2^{s}}+e_{h}(\operatorname{\overline{\operatorname{\mathit% | C |
A linear parametrization using the Bott-Samelson coordinates for the finite dimensional Schubert cells (Proposition 4.2). | We obtain the defining equations for Kazhdan-Lusztig varieties by pulling back equations from (A) to coordinates determined by (B). Following the strategies developed in [WY12], we prove that these equations form a Gröbner basis of the Kazhdan-Lusztig ideals by an inductive argument on subword complexes. Our proof reli... | There is extensive literature on the geometry of Kazhdan-Lusztig varieties in the type A𝐴Aitalic_A setting. In [LY12b], Li and Yong studied Hilbert–Samuel multiplicity for points of Schubert varieties in the complete flag variety by Gröbner degeneration of Kazhdan–Lusztig ideals, and gave an explicit combinatorial int... | 4.2. Equations for Kazhdan-Lusztig varieties in Fl0(V)𝐹subscript𝑙0𝑉Fl_{0}(V)italic_F italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_V ) | In [WY12], Woo and Yong gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A𝐴Aitalic_A flag variety, generalizing the Gröbner basis theorem [KM05, Theorem B] for Schubert determinantal ideals. In this paper, we present a Gröbner basis for Kazhdan-Lusztig ideals in the affine type A𝐴Aitalic_A flag variety, ge... | A |
Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. In the usual story of moduli spaces of wild harmonic bundles ove... | Since our identification will use a certain set of (framed) wild harmonic bundles, we remark that this set does not match any of the usual wild moduli spaces ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT. In the usual story of moduli spaces of wild harmonic bundles ove... | We hope that our interpretation of the Ooguri-Vafa space in terms of wild harmonic bundles serves as a first step to establish part of the conjectural picture of [GMN10, GMN13] and [Nei13, Section 7] mentioned above. For now, we leave the question of the specific relation between the Ooguri-Vafa metric and the hyperkäh... | We remark that this proposed picture of the hyperkähler metric is very similar to the one given by Gross-Wilson for the hyperkähler metric of K3333 surfaces (see [GW00]). In the picture of [GW00], we have a generic elliptic fibration of a K3333 surface f:X→ℂP1:𝑓→𝑋ℂsuperscript𝑃1f:X\to\mathbb{C}P^{1}italic_f : italic... | Motivated by these facts, our goal in this paper is to relate the Ooguri-Vafa space with the objects present in ℳHitsubscriptℳHit\mathcal{M}_{\text{Hit}}caligraphic_M start_POSTSUBSCRIPT Hit end_POSTSUBSCRIPT, namely harmonic bundles (E,∂¯E,θ,h)𝐸subscript¯𝐸𝜃ℎ(E,\overline{\partial}_{E},\theta,h)( italic_E , over¯ sta... | B |
Applying Proposition 1 for the SBM X𝑋Xitalic_X and the function f:=fσassign𝑓subscript𝑓𝜎f:=f_{\sigma}italic_f := italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT and taking into account the latter equalities complete the proof. | We conclude the section with the lemma proving that the assumptions of Theorem 1 are stronger than the ones of Proposition 1, as stated in Remark 9. | Since the estimator proposed relies mostly on the behavior of the process around the threshold, it is slower than the ones based on quadratic variations and occupation times of the positive and negative part of the process proposed in [36] for OBM which exploit the entire trajectory. | The decomposition is introduced in Proposition 4, where it is stated that the discrete martingale satisfies the assumptions of Proposition 5 (a reformulation of a special case of Theorem 3.2 in [25]) which entails the limits in Theorem 1. | study of the number of crossings for a more general class of processes in Theorem 2, estimation of the skewness parameter of SBM in Theorem 3, and estimation of the parameters of OSBM in Proposition 3. | A |
We study the sample efficiency of policy-based reinforcement learning in the episodic setting of linear MDPs with full-information feedback. We proposed an optimistic variant of the proximal policy optimization algorithm, dubbed as OPPO, which incorporates the principle of “optimism in the face of uncertainty” into pol... | Theoretically, we establish the sample efficiency of OPPO in an episodic setting of Markov decision processes (MDPs) with full-information feedback, where the transition dynamics are linear in features (Yang and Wang, 2019b, a; Jin et al., 2019; Ayoub et al., 2020; Zhou et al., 2020). In particular, we allow the transi... | The authors would like to thank Lingxiao Wang, Wen Sun, and Sham Kakade for pointing out a technical issue in the first version regarding the covering number of value functions in the linear setting. This version has fixed the technical issue with a definition of the linear MDP different from the one in the first versi... | We establish an upper bound of the regret of OPPO (Algorithm 1) in the following theorem. Recall that the regret is defined in (2.1) and T=HK𝑇𝐻𝐾T=HKitalic_T = italic_H italic_K is the total number of steps taken by the agent, where H𝐻Hitalic_H is the length of each episode and K𝐾Kitalic_K is the total number of e... | Although the variant of linear MDPs defined in Assumption 2.1 and the one studied by Yang and Wang (2019b); Jin et al. (2019) both cover the tabular setting and the one proposed by Yang and Wang (2019a) as special cases, they are two different definitions of linear MDPs as their feature maps ψ(⋅,⋅,⋅)𝜓⋅⋅⋅\psi(\cdot,\c... | B |
The eigenvalues are (0,8,8,8,8,8,8)0888888(0,8,8,8,8,8,8)( 0 , 8 , 8 , 8 , 8 , 8 , 8 ) and K(K8)=(ℤ/8ℤ)6𝐾subscript𝐾8superscriptℤ8ℤ6K(K_{8})=(\mathbb{Z}/8\mathbb{Z})^{6}italic_K ( italic_K start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ) = ( blackboard_Z / 8 blackboard_Z ) start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT. | Finally, we conclude with some remaining conjectures about the structure of K(G)𝐾𝐺K(G)italic_K ( italic_G ) in Section 5. | In order to deal with the case p=2𝑝2p=2italic_p = 2, we adopt the approach of Benkart, Klivans, and Reiner [reinerquiver] that gives a natural ring structure on K(G)𝐾𝐺K(G)italic_K ( italic_G ): | where K(G)𝐾𝐺K(G)italic_K ( italic_G ) is a finite abelian group, known as the sandpile group (also critical group or Jacobian in the literature) of G𝐺Gitalic_G. Kirchhoff’s Matrix–Tree Theorem shows that the size of K(G)𝐾𝐺K(G)italic_K ( italic_G ) is the number of spanning trees of G𝐺Gitalic_G. K(G)𝐾𝐺K(G)ita... | It is worth emphasizing that many of our arguments are based on the natural ring structure of K(G)𝐾𝐺K(G)italic_K ( italic_G ) coming from representation theory [reinerquiver] and the corresponding polynomial algebra. | B |
\frac{c_{2}}{L^{n}}.divide start_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_L start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_ARG ≤ ∏ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG itali... | Next we estimate from below and above the measure and the cost of compression in the following sequel. We leverage the estimates established thus far to provide these estimates. | Next we present a second moment variant inequality of the unit sum of positive integers in the following statement. | In this section we lay down one striking and a stunning consequence of the entropy of compression. One could think of these applications as analogues of the Erdós type result for the unit sums of triples of the form (x1,x2,x3)subscript𝑥1subscript𝑥2subscript𝑥3(x_{1},x_{2},x_{3})( italic_x start_POSTSUBSCRIPT 1 end_PO... | Next we present a second application of the estimates of the entropy of compression in the following sequel. | D |
In this section, we give the definitions required to more precisely state the conjectural equivalence between Fano varieties up to deformation and rigid maximally mutable Laurent polynomials up to mutation (see, for example [1] and [15]). | For the definition of rigid maximally mutable Laurent polynomials, see Definition 2.3. A Fano variety X𝑋Xitalic_X is said to be mirror to f𝑓fitalic_f if the regularised quantum period of X𝑋Xitalic_X equals the classical period of f𝑓fitalic_f. The classical period of a Laurent polynomial (Definition 2.2) is mutation... | On the A side (the Fano side) the main invariant is the quantum period. For a more detailed introduction to Gromov–Witten invariants, quantum cohomology, and the quantum period, see for example [7] (in particular, they record the formula for the quantum period of a Fano toric complete intersection). | As descendent invariants are deformation invariant, so is the quantum period. A closed form is known for smooth Fano toric complete intersections. The Abelian/non-Abelian correspondence [5, 26, 18] allows one to compute any number of terms of the quantum period of quiver flag zero loci. | The first 20 terms of the period sequence of the Laurent polynomial and the quantum period of the four dimensional Fano variety cut out of Gr(6,3)Gr63\operatorname{\mathrm{Gr}}(6,3)roman_Gr ( 6 , 3 ) by a generic section of 𝒪(1)⊕5𝒪superscript1direct-sum5\mathcal{O}(1)^{\oplus 5}caligraphic_O ( 1 ) start_POSTSUPERSC... | B |
For the curvature equations in classical geometry, the existence of hypersurfaces with prescribed Weingarten curvature was studied by Pogorelov [40], Caffarelli-Nirenberg-Spruck [4, 5], Guan-Guan [18], Guan-Ma [19] and the later work by Sheng-Trudinger-Wang [42]. The Hessian equation on Riemannian manifolds was also st... | The Yamabe problem with boundary is an important motivation for the study of the Neumann problems. The Yamabe problem on manifolds with boundary was first studied by Escobar, who shows in [13] that (almost) every compact Riemannian manifold (M,g)𝑀𝑔(M,g)( italic_M , italic_g ) is conformally equivalent to one of const... | In 1986, Lions-Trudinger-Urbas solved the Neumann problem of Monge-Ampère equations in the celebrated paper [35]. For related results on the Neumann or oblique derivative problem for some class of fully nonlinear elliptic equations can be found in Urbas [49]. Recently, the second author and G.H. Qiu [36] solved the the... | Meanwhile, the Neumann and oblique derivative problem of partial differential equations were widely studied. For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [15]. Also, we recommend the recent book written by Lieberman [33] for the Neumann and t... | These lemmas play an important role in the establishment of a priori estimates. Precisely, Lemma 2.5 is the key of the gradient estimates in Section 5, including the interior gradient estimate and the near boundary gradient estimate. Lemmas 2.4 and Lemma 2.6 are the keys of the lower and upper estimates of double norma... | C |
})(b*\psi_{\delta t})](x)m(t)\frac{dt}{t}dx.∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_ψ start_POSTSUBSCRIPT 3 , italic_t end_POSTSUBSCRIPT * italic_h ) ( italic_x ) [ ( i... | We next estimate the term B𝐵Bitalic_B: using the Cauchy-Schwartz inequality in x′superscript𝑥′x^{\prime}italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT we get that | convolutions together with the fact that for f∈L∞(ℝ2)𝑓superscript𝐿superscriptℝ2f\in L^{\infty}(\mathbb{R}^{2})italic_f ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | Using the fact that m∈L∞𝑚superscript𝐿m\in L^{\infty}italic_m ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT and the Cauchy-Schwarz inequality, this quantity can be | We note next that for x′∈[−H/2,H/2]superscript𝑥′𝐻2𝐻2x^{\prime}\in[-H/2,H/2]italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ - italic_H / 2 , italic_H / 2 ], by (16) and the fact that ‖θ‖L∞≤1subscriptnorm𝜃superscript𝐿1\|\theta\|_{L^{\infty}}\leq 1∥ italic_θ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRI... | C |
By [4, Theorem 7.4], we know that VRr(𝕊1)subscriptVR𝑟superscript𝕊1\mathrm{VR}_{r}(\mathbb{S}^{1})roman_VR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) is homotopy equivalent to 𝕊2m+1superscript𝕊2𝑚1\mathbb{S}^{2m+1}blackboard_S start_POSTSUPERSCRIPT ... | Actually, we first expected the following conjecture to be true. Observe that, if true, the conjecture would imply Theorem 7. Also, it is obvious that this conjecture is true when X𝑋Xitalic_X is a finite metric space. | Then, one might now wonder whether the conjecture holds when we restrict the range of r𝑟ritalic_r to (0,diam(X))0diam𝑋(0,\mathrm{diam}(X))( 0 , roman_diam ( italic_X ) ). But, again this new conjecture is false as the following example shows. | Since VRr(X)subscriptVR𝑟𝑋\mathrm{VR}_{r}(X)roman_VR start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_X ) is contractible for any r>diam(X)𝑟diam𝑋r>\mathrm{diam}(X)italic_r > roman_diam ( italic_X ), it is clear that length(I)≤diam(X)length𝐼diam𝑋\mathrm{length}(I)\leq\mathrm{diam}(X)roman_length ( italic... | After seeing the proof of Proposition 9.7, some readers might wonder whether one can prove a version of Hausmann’s theorem [50, Theorem 3.5] for compact ANR metric spaces. This leads to formulating the conjecture below. | B |
Let u∈c1,α(Ω,ℝ3)𝑢superscript𝑐1𝛼Ωsuperscriptℝ3u\in c^{1,\alpha}(\Omega,{\mathbb{R}}^{3})italic_u ∈ italic_c start_POSTSUPERSCRIPT 1 , italic_α end_POSTSUPERSCRIPT ( roman_Ω , blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) be an isometric immersion for 2/3≤α<123𝛼12/3\leq\alpha<12 / 3 ≤ italic_α < 1. In o... | We then show the required regularity for v𝑣vitalic_v, and proceed to prove using the Gauss equation that v𝑣vitalic_v satisfies the very weak degenerate Monge-Ampère equation as required by the assumptions of Theorem 2. Finally, we need to prove that developability of v𝑣vitalic_v, as derived from Theorem 2, implies t... | Once again applying Lemma 2.11 and (5.4) implies that the Jacobian derivative ∇u∇𝑢\nabla u∇ italic_u of the isometric immersion u𝑢uitalic_u is constant along the segments generated by the vector field η→→𝜂\vec{\eta}over→ start_ARG italic_η end_ARG in B𝐵Bitalic_B. Therefore ∇u∇𝑢\nabla u∇ italic_u satisfies conditio... | The article is organized as follows: In Section 2, we will present and prove a few statements regarding the developability properties of C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT mappings. In Section 3, we will gather a few analytical tools which deal with properties of Hölder continuous... | We apply the key developability result Theorem 2 to v𝑣vitalic_v to obtain that ∇v∇𝑣\nabla v∇ italic_v satisfies any of the equivalent conditions of Proposition 2.1 in V𝑉Vitalic_V. The developability of u𝑢uitalic_u is a consequence of Corollory 2.2 and Proposition 5.1 below. The second conclusion follows from Coroll... | D |
(b) and (c). Finally, to prove (f), from f𝑓fitalic_f we define f^=𝔦∘f∘π^𝑓𝔦𝑓𝜋\widehat{f}=\mathfrak{i}\circ f\circ\piover^ start_ARG italic_f end_ARG = fraktur_i ∘ italic_f ∘ italic_π and use | denote the principal piece of the cartesian product Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT. | of the cartesian product Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT. | Take X𝑋Xitalic_X a Hausdorff topological space, take m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N and Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT endowed with the product topology, and | Using the partition of Xmsuperscript𝑋𝑚X^{m}italic_X start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT in big puzzles one may take care of the difference between the product topology | D |
The size or length of a ladder is given by its number of rungs, i.e. the size of L𝐿Litalic_L is n𝑛nitalic_n. | With that end in view, from now on, a ladder is always supposed to be a subdivision of an elementary ladder with three rungs. | Let LAsubscript𝐿𝐴L_{A}italic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT be an elementary ladder of length lAsubscript𝑙𝐴l_{A}italic_l start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and let LCsubscript𝐿𝐶L_{C}italic_L start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT be an elementary ladder of length lCsubscript�... | In this section, we will prove that the elementary ladder with three rungs has the edge-Erdős-Pósa property. | A ladder is a subdivision of an elementary ladder. We adapt the notion of rungs, stringers and size for ladders from their counterparts in elementary ladders. | D |
=s2+S2+𝐯2+𝐕2,absentsuperscript𝑠2superscript𝑆2superscript𝐯2superscript𝐕2\displaystyle=s^{2}+S^{2}+\mathbf{v}^{2}+\mathbf{V}^{2},= italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + bold_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + bold_V start_POSTSUPER... | =2(sS+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) , | 𝖠=s+𝐯+(S+𝐕)I,𝖠𝑠𝐯𝑆𝐕𝐼\mathsf{A}=s+\mathbf{v}+(S+\mathbf{V})I,sansserif_A = italic_s + bold_v + ( italic_S + bold_V ) italic_I , | =2(sS+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) , | =2(sS+𝐯⋅𝐕),absent2𝑠𝑆⋅𝐯𝐕\displaystyle=2(sS+\mathbf{v}\cdot\mathbf{V}),= 2 ( italic_s italic_S + bold_v ⋅ bold_V ) , | A |
{\star}}_{m},P^{k,1}_{m},P^{k,2}_{m}\big{)}\Big{]}.roman_inf start_POSTSUBSCRIPT italic_ϕ ∈ italic_U start_POSTSUPERSCRIPT italic_k , italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_E [ italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_Y start_POSTSUPERSC... | Figure 1: Monte Carlo experiments for ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT with ϵk=2−ksubscriptitalic-ϵ𝑘superscript2𝑘\epsilon_{k}=2^{-k}italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT | k,2}_{m}\right)\right]=0blackboard_E [ italic_ϱ start_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_Y start_POSTSUPERSCRIPT italic_k , italic_u start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , ital... | The estimated value ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT is computed according to | Table 1: Comparison between c⋆superscript𝑐⋆c^{\star}italic_c start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT and ck,⋆superscript𝑐𝑘⋆c^{k,\star}italic_c start_POSTSUPERSCRIPT italic_k , ⋆ end_POSTSUPERSCRIPT for ϵk=2−ksubscriptitalic-ϵ𝑘superscript2𝑘\epsilon_{k}=2^{-k}italic_ϵ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRI... | C |
If a semigroup S𝑆Sitalic_S is not residually finite and embeds into a semigroup T𝑇Titalic_T, then T𝑇Titalic_T cannot be residually finite either. | While our main result significantly relaxes the hypothesis for showing that the free product of self-similar semigroups (or automaton semigroups) is self-similar (an automaton semigroup), it does not settle the underlying question whether these semigroup classes are closed under free product. It is possible that there ... | If the generating automaton is additionally complete, we speak of a completely self-similar semigroup or of a complete automaton semigroup. | Every complete automaton semigroup is residually finite [4, Proposition 3.2] and the argument can easily be extended to general self-similar semigroups. | The construction used to prove Theorem 6 can also be used to obtain results which are not immediate corollaries of the theorem (or its corollary for automaton semigroups in 8). As an example, we prove in the following theorem that it is possible to adjoin a free generator to every self-similar semigroup without losing ... | C |
φ~(a)<a~𝜑𝑎𝑎\tilde{\varphi}(a)<aover~ start_ARG italic_φ end_ARG ( italic_a ) < italic_a for all a>1𝑎1a>1italic_a > 1. | It is important to note that this inequality does not hold in general. As it is informed by the spanning method, it only holds for functions that are right continuous and of bounded variation on intervals of the form [x,x+1)𝑥𝑥1[x,x+1)[ italic_x , italic_x + 1 ) for x≥1𝑥1x\geq 1italic_x ≥ 1 with x∈ℕ𝑥ℕx\in\mathbb{N}i... | We now state an analytic property of the fractional totient invariant function. In fact, the fractional totient invariant function can be seen as a slightly continuous analogue of the Euler totient function on subsets of the reals. | By the fractional Euler totient invariant function, we mean the function φ~:[1,∞)⟶ℝ:~𝜑⟶1ℝ\tilde{\varphi}:[1,\infty)\longrightarrow\mathbb{R}over~ start_ARG italic_φ end_ARG : [ 1 , ∞ ) ⟶ blackboard_R such that | The fractional Euler totient invariant function turns out to be an interesting function that in some way extends the Euler totient function to the reals. Even though the notion of co-primality in not well-defined on the entire real line, it captures the intrinsic property of the Euler totient function defined on the po... | B |
\varepsilon}))=0.roman_lim start_POSTSUBSCRIPT italic_ε → 0 end_POSTSUBSCRIPT italic_d ( ( bold_0 , 0 ) , caligraphic_F start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( caligraphic_K start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ) = 0 . | is closed by upper semicontinuity of density and Lemma 10.5. Thus, it suffices to prove that ℬℬ\mathcal{B}caligraphic_B is countable. Define | Everything but the last claim is proven above (in the rescaled setting). The last claim follows from the fact that all limit flows are convex so [HW20] applies. | The first claim follows immediately from Lemma 7.15. To prove the second claim, it suffices (by Lemma 7.5) to show that | Claim (1) follows by construction. Claim (2) follows from the fact101010Note that the simpler statement λ(Σε)≤F(Σ)+o(1)𝜆subscriptΣ𝜀𝐹Σ𝑜1\lambda(\Sigma_{\varepsilon})\leq F(\Sigma)+o(1)italic_λ ( roman_Σ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ≤ italic_F ( roman_Σ ) + italic_o ( 1 ) as ε→0→𝜀0\varepsilon\... | C |
(α,β,γ,δ)=(2,1,2,2)𝛼𝛽𝛾𝛿2122(\alpha,\beta,\gamma,\delta)=(2,1,2,2)( italic_α , italic_β , italic_γ , italic_δ ) = ( 2 , 1 , 2 , 2 ): | Relevant indices: {i,j,n,ℓ}𝑖𝑗𝑛ℓ\{i,j,n,\ell\}{ italic_i , italic_j , italic_n , roman_ℓ }. Cases to consider: | Relevant indices: {i,j,k,n,ℓ}𝑖𝑗𝑘𝑛ℓ\{i,j,k,n,\ell\}{ italic_i , italic_j , italic_k , italic_n , roman_ℓ }. Since all the indices are relevant, we only have to consider cases where | #{i,j,k,n,ℓ}⩾4#𝑖𝑗𝑘𝑛ℓ4\#\{i,j,k,n,\ell\}\geqslant 4# { italic_i , italic_j , italic_k , italic_n , roman_ℓ } ⩾ 4. The cases left to consider are | Relevant indices: {i,k,n,ℓ}𝑖𝑘𝑛ℓ\{i,k,n,\ell\}{ italic_i , italic_k , italic_n , roman_ℓ }. Cases to consider: | A |
D1=ui(i≥3)subscript𝐷1subscript𝑢𝑖𝑖3D_{1}=u_{i}(i\geq 3)italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i ≥ 3 ), S1=v2subscript𝑆1subscript𝑣2S_{1}=v_{2}italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCR... | Hence, Staller has a strategy such that Dominator needs at least m−1𝑚1m-1italic_m - 1 steps in order to win in an | the right. By induction, Dominator needs at least k−7𝑘7k-7italic_k - 7 steps to win. In total, he needs at least | Dominator needs at least m−5𝑚5m-5italic_m - 5 more steps to win. In total, he needs at least (m−5)+3=m−2𝑚53𝑚2(m-5)+3=m-2( italic_m - 5 ) + 3 = italic_m - 2 steps to win. | At the end of each branch, we explain at least how many steps Dominator would need in order to win the game. It covers | D |
Given the initial scattering diagram 𝔇insubscript𝔇𝑖𝑛\mathfrak{D}_{in}fraktur_D start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT, there exists a unique complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D such that θ𝔇,u=idsubscript𝜃𝔇𝑢𝑖𝑑\theta_{\mathfrak{D},u}=iditalic_θ start_POSTSUBSCRIPT fraktur_D ... | Given a scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D on B𝐵Bitalic_B, u∈B0𝑢subscript𝐵0u\in B_{0}italic_u ∈ italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and λ>0𝜆0\lambda>0italic_λ > 0. For a generic counterclockwise | A scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D on B𝐵Bitalic_B is a set of 3333-tuples {(l𝔡,γ𝔡,f𝔡)}𝔡∈Isubscriptsubscript𝑙𝔡subscript𝛾𝔡subscript𝑓𝔡𝔡𝐼\{(l_{\mathfrak{d}},\gamma_{\mathfrak{d}},f_{\mathfrak{d}})\}_{\mathfrak{d}\in | The complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D is closely related to tropical discs on B𝐵Bitalic_B. | Given the initial scattering diagram 𝔇insubscript𝔇𝑖𝑛\mathfrak{D}_{in}fraktur_D start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT, there exists a unique complete scattering diagram 𝔇𝔇\mathfrak{D}fraktur_D such that θ𝔇,u=idsubscript𝜃𝔇𝑢𝑖𝑑\theta_{\mathfrak{D},u}=iditalic_θ start_POSTSUBSCRIPT fraktur_D ... | C |
V𝑉Vitalic_V homothetic to an imaginary quadratic field different from ℚ(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ) or ℚ(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ). | V𝑉Vitalic_V homothetic to ℚ(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ). | V𝑉Vitalic_V homothetic to ℚ(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ). | }2\sqrt{3}≤ | italic_w | 2 square-root start_ARG 3 end_ARG ≤ ( square-root start_ARG 2 end_ARG italic_p start_POSTSUPERSCRIPT 2 italic_m - 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT 2 square-root start_ARG 3 end_ARG | V𝑉Vitalic_V homothetic to an imaginary quadratic field different from ℚ(−1)ℚ1\mathbb{Q}(\sqrt{-1})blackboard_Q ( square-root start_ARG - 1 end_ARG ) or ℚ(−3)ℚ3\mathbb{Q}(\sqrt{-3})blackboard_Q ( square-root start_ARG - 3 end_ARG ). | B |
∂t2u+(−Δ+a|x|2)u=0superscriptsubscript𝑡2𝑢Δ𝑎superscript𝑥2𝑢0\partial_{t}^{2}u+\big{(}-\Delta+\tfrac{a}{|x|^{2}}\big{)}u=0∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u + ( - roman_Δ + divide start_ARG italic_a end_ARG start_ARG | italic_x | start_POSTSUPERSCRIP... | This paper is devoted to the study of scattering theory of solutions for (1.1) in sub-critical cases. | In this section, we shall prove a class of local Morawetz estimates for solutions in Proposition 2.5. Please note that these Morawetz estimates hold for non-radial solutions as well. | Fortunately, we can obtain scattering theory of radial finite energy solutions to the linear equation (1.6) in the appendix by the argument of [41] and a series of estimates. This fact shows us that the radial solutions of the energy critical nonlinear wave equations scatter to free waves, see [29]. | for the proof of Theorem 1.1, we need to establish the decay estimate of the energy in the interior of light cones. Since the | A |
X/\!\!/G).italic_Q italic_E italic_l italic_l start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_X ) ⊗ start_POSTSUBSCRIPT blackboard_Z [ italic_q start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ] end_POSTSUBSCRIPT blackboard_Z ( ( italic_q ) ) ≅ italic_K start_POSTSUPERSCRI... | In addition, we give an example computing quasi-elliptic cohomology, which is [Hua18, Example 3.3]. The conclusions in Example 2.6 is applied in Section 6. | In this section we review quasi-elliptic cohomology, the main reference of which is [Hua18]. It is a variant of Tate | In this section we give a loop space construction of twisted quasi-elliptic cohomology other than the twisted orbifold loop space in Example 5.3. This is a twisted version of the loop space in Definition 2.7. | In Section 2 we give a sketch of quasi-elliptic cohomology, including its definition, basic properties, and the loop space construction. In Section 3, we review Devoto’s equivariant elliptic cohomology. In Section 4, we recall the definition of twisted equivariant elliptic cohomology. In Section 5, we construct twisted... | A |
Thus, we require the number of edges is at least quartic in δ(A,B)𝛿𝐴𝐵\delta(A,B)italic_δ ( italic_A , italic_B ) to ensure | such merging only increases the Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT-degree of a vertex in U𝑈Uitalic_U by p𝑝pitalic_p. | Note that after the merging the total degree of each vertex increases by δ(A0,B0)2𝛿superscriptsubscript𝐴0subscript𝐵02\delta(A_{0},B_{0})^{2}italic_δ ( italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, | After the merging the total degree of each vertex increases by tδ(A0,B0)2𝑡𝛿superscriptsubscript𝐴0subscript𝐵02t\delta(A_{0},B_{0})^{2}italic_t italic_δ ( italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. | Thus we can perform the merging in such a way that the total degree of each vertex in the real partition increases | D |
Indeed, the latter operation indicates the existence of valuations on F𝐹Fitalic_F, which are strongly related to rigid elements. | To finish this section, let us see that from a 2222-henselian field, we can build a field with a non-trivial radical. | Finally, from valuation theory, we have pre-2222-henselian fields [9] as examples of fields with non-trivial Kaplansky radical, to be detailed in Section 4, Example 23. | Finally, in this section, we will see that rigid elements and a non-trivial Kaplansky radical usually do not exist together. | In this section, we describe the behavior of the Kaplansky radical under the basic operations (A) and (B) of Section 4, starting with free products. Since we focus on the radical, we can work only in the category of pro-2222 groups. | C |
Q†(x)=∫σ(x;θ)dν¯(θ)superscript𝑄†𝑥𝜎𝑥𝜃differential-d¯𝜈𝜃Q^{\dagger}(x)=\int\sigma(x;\theta)\,{\mathrm{d}}\bar{\nu}(\theta)italic_Q start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_x ) = ∫ italic_σ ( italic_x ; italic_θ ) roman_d over¯ start_ARG italic_ν end_ARG ( italic_θ ). We assume that Dχ2(ν¯∥ν0)<∞subs... | Under Assumptions 4.1 and 4.2, it holds for any k≤T/ϵ(k∈ℕ)𝑘𝑇italic-ϵ𝑘ℕk\leq T/\epsilon\ (k\in\mathbb{N})italic_k ≤ italic_T / italic_ϵ ( italic_k ∈ blackboard_N ) that | Under Assumptions 4.1, 4.2, and 6.3, it holds for η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT that | Under Assumptions 4.1, 4.2, and 6.1, it holds for η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT that | Upon telescoping (5.5) and setting η=α−2𝜂superscript𝛼2\eta=\alpha^{-2}italic_η = italic_α start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, we obtain that | C |
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