An Index Formula in Connection with Meromorphic Approximation

Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .     

Open Gromov–Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X , \omega )$ . We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$ . Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$ , every relative homology class $d \in H_2 (X , L ; \mathbb{Z })$ and adequate number of incidence conditions in $L$ or $X$ , the weighted number of $J$ -holomorphic discs with boundary on $L$ , homologous to $d$ , and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$ , provided that at least one incidence condition lies in $L$ . These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring $A$ .     

The distribution of the zeros of the Goss zeta-function for A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)

Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Optimal relationships between L^p-norms for the Hardy operator and its dual

We obtain sharp two-sided inequalities between $L^p$ -norms $(1<p<\infty )$ of functions $\textit{Hf}$ and $H^*f$ , where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relationships between $L^p$ -norms of functions $H\varphi -\varphi $ and $\varphi $ , where $\varphi $ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\varphi (+\infty )=0.$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $p=2k \,\,(k\in \mathbb N )$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $p\ge 2$ , and gives a sharp version of this result for $1<p<2$ .     

Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .     

The Saito–Kurokawa lifting and Darmon points

Let $E_{/_\mathbb{Q }}$ be an elliptic curve of conductor $Np$ with $p\not \mid N$ and let $f$ be its associated newform of weight $2$ . Denote by $f_\infty $ the $p$ -adic Hida family passing though $f$ , and by $F_\infty $ its $\varLambda $ -adic Saito–Kurokawa lift. The $p$ -adic family $F_\infty $ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde{A}_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$ . We relate explicitly certain global points on $E$ (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their $p$ -adic derivatives, evaluated at weight $k=2$ .     

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,?) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2?I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$ , by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$ , which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$ , including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$ -function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$ . Moreover, for the Musielak–Orlicz–Hardy space H _φ,L (? ⁿ ) associated with the Schrödinger operator L:=?Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ , the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ?L ^?1/2 is bounded from H _φ,L (? ⁿ ) to the Musielak–Orlicz space L ^φ (? ⁿ ) when i(φ)∈(0,1], and from H _φ,L (? ⁿ ) to the Musielak–Orlicz–Hardy space H _φ (? ⁿ ) when $i(\varphi)\in(\frac{n}{n+1},1]$ , where i(φ) denotes the uniformly critical lower type index of φ.     

L^{\infty } estimation of tensor truncations

Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $\ell ^{2}$ or $L^{2}$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require $\ell ^{\infty }$ or $L^{\infty }$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\| \cdot \right\| _{\infty }$ by $\left\| \cdot \right\| _{2}$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to $\left\| \cdot \right\| _{\infty }$ can be derived from the Gagliardo–Nirenberg inequality because of the special nature of the singular value decomposition truncation.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.     

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke–Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda $ . It is shown that $\left||f \right||_\infty \ll _{\lambda ,\epsilon } N^{-\frac{1}{6}+\epsilon } \left||f \right||_2$ for any $\epsilon >0$ .     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\mathcal{O }$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$ for each differential form $\beta $ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\mathcal{C }$ is defined on the universal cover of $\mathcal{O }.$ A particular instance of $\mathcal{C }$ recovers the Calabi homomorphism. If $\beta $ is the imaginary part of a holomorphic volume form, the critical points of $\mathcal{C }$ are special Lagrangian submanifolds. We present evidence that $\mathcal{C }$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $\mathcal{C }$ is convex on an open subspace $\mathcal{O }^+ \subset \mathcal{O }.$ As a prerequisite, we define a Riemannian metric on $\mathcal{O }^+$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.     

Geodesics,distance, and the CAT(0) property for the manifold of Riemannian metrics

Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$      

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Étale cohomology of a DM curve-stack with coefficients in \mathbb G _m

We compute étale cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb G _m)$ in several cases, where $X$ is a connected smooth tame Deligne–Mumford stack of dimension $1$ over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give partial results and examples in the general case. In particular, we show that if the stabilizers are abelian then $H_{\acute{\mathrm{e}}\mathrm{t}}^2(X, \mathbb{G }_m)$ does not depend on $X$ but only on the underlying orbicurve $Y$ and on the generic stabilizer. We show with two examples that, in general, the higher cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb{G }_m)$ cannot be computed knowing only the base of the gerbe $X \rightarrow Y$ and the banding group.