Volume 36, Issue 2

We consider nonlinear equations having generalized Orlicz growth (also known as Musielak–Orlicz growth). We prove that if differential operators 𝒜 i {\mathcal{A}_{i}} converge locally uniformly to an operator 𝒜 {\mathcal{A}} , then the sequence of solutions ( u i ) {(u_{i})} has a subsequence converging to the solution u of the limit operator in Sobolev and Hölder norms.

Let Γ be a subgroup of finite index of SL 2 ⁡ ( 𝐙 ) {\operatorname{SL}_{2}(\mathbf{Z})} . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ {J_{\Gamma}} of the corresponding modular curve X Γ {X_{\Gamma}} . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles . The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ ( 2 ) {\Gamma\subset\Gamma(2)} , where we can consider convenient generalized Jacobians instead of J Γ {J_{\Gamma}} . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves.

We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 ⁡ ( ℝ ) {\operatorname{SL}_{2}(\mathbb{R})} . The conjugation action of SL n ⁡ ( ℝ ) {\operatorname{SL}_{n}(\mathbb{R})} is not Asplund representable for every n ≥ 4 {n\geq 4} . The linear action of GL n ⁡ ( ℝ ) {\operatorname{GL}_{n}(\mathbb{R})} on ℝ n {{\mathbb{R}}^{n}} , for every n ≥ 2 {n\geq 2} , is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 {l_{1}} ). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct, we obtain some counterexamples about Banach representations of homogeneous G -actions G / H {G/H} .

We initiate a systematic study of abelian-by-cyclic Moufang loops, that is, Moufang loops Q with an abelian normal subgroup X such that Q / X {Q/X} is a cyclic group. Among other results, we construct all split abelian-by-cyclic Moufang loops in which both X and Q / X {Q/X} are 3-divisible, using so-called Moufang permutations on X , which are permutations that deviate from an automorphism of X by an alternating biadditive mapping. Additional abelian-by-cyclic Moufang loops are obtained from so-called construction pairs. As an aside, we show that, in a Moufang loop Q obtained from a construction pair, the abelian normal subgroup X induces an abelian congruence of Q if and only if Q is a group.

We establish the equivalence of Gromov ellipticity and subellipticity in the algebraic category.

Let { A n } n = 1 ∞ {\{A_{n}\}_{n=1}^{\infty}} be a sequence of expanding matrices with A n ∈ M 2 ⁢ ( ℤ ) {A_{n}\in M_{2}(\mathbb{Z})} , and let { D n } n = 1 ∞ {\{D_{n}\}_{n=1}^{\infty}} be a sequence of three-element digit sets with { x ∈ ( 0 , 1 ) 2 : ∑ d ∈ D n e 2 ⁢ π ⁢ i ⁢ 〈 d , x 〉 = 0 } = { ± 1 3 ⁢ ( 1 , i ) t } {\{x\in(0,1)^{2}:\sum_{d\in D_{n}}{e^{2\pi i\langle d,x\rangle}}=0\}=\{\pm% \frac{1}{3}(1,i)^{t}\}} , i ∈ { 1 , 2 } {i\in\{1,2\}} . The associated Moran measure generated by the infinite convolution μ { A n } , { D n } = δ A 1 - 1 ⁢ D 1 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ D 2 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ A 3 - 1 ⁢ D 3 * ⋯ . \mu_{\{A_{n}\},\{D_{n}\}}=\delta_{A_{1}^{-1}D_{1}}*\delta_{A_{1}^{-1}A_{2}^{-1% }D_{2}}*\delta_{A_{1}^{-1}A_{2}^{-1}A_{3}^{-1}D_{3}}*\cdots. In this paper, we give some necessary and sufficient conditions for μ { A n } , { D n } {\mu_{\{A_{n}\},\{D_{n}\}}} to be a spectral measure under some suitable conditions on A n {A_{n}} and D n {D_{n}} .

We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , ( x , t ) ∈ ℝ 2 × ℝ + , κ > 0 , \partial_{t}\theta+u\cdot\nabla\theta+\kappa\Lambda^{2\beta}\theta=0,\quad(x,t% )\in\mathbb{R}^{2}\times\mathbb{R}^{+},\;\kappa>0, where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ {u=-\nabla^{\bot}\Lambda^{-2+2\alpha}\theta} , α ∈ [ 0 , 1 2 ] {\alpha\in[0,\frac{1}{2}]} and β ∈ ( 0 , 1 ] {\beta\in(0,1]} . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space.

In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ ( x ) = ∫ R n e i ⁢ ϕ ⁢ ( x , ξ ) ⁢ a ⁢ ( x , ξ ) ⁢ f ^ ⁢ ( ξ ) ⁢ d ξ T_{\phi,a}f(x)=\int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\hat{f}(\xi)\,d\xi with 𝑎 in the forbidden Hörmander class S ρ , 1 m S^{m}_{\rho,\smash{1}} and ϕ ∈ Φ 2 \phi\in\Phi^{2} satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 0\leq\rho\leq 1 , set m ⁢ ( n , ρ , p ) = − ( n − ρ ) ⁢ ( 1 2 − 1 p ) + n p ⁢ ( ρ − 1 ) . m(n,\rho,p)=-(n-\rho)\biggl{(}\frac{1}{2}-\frac{1}{p}\biggr{)}+\frac{n}{p}(\rho-1). When 2 ≤ p ≤ ∞ 2\leq p\leq\infty , 1 ≤ q ≤ ∞ 1\leq q\leq\infty and s > m − m ⁢ ( n , ρ , p ) s>m-m(n,\rho,p) , we show that T ϕ , a T_{\phi,a} is bounded from the Besov space B p , q s B_{p,q}^{s} to B p , q s − m + m ⁢ ( n , ρ , p ) B_{p,q}^{s-m+\smash{m(n,\rho,p)}} . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a T_{a} with a ∈ S 1 , 1 m a\in S^{m}_{1,1} , and indices 𝑠 and m ⁢ ( n , ρ , p ) m(n,\rho,p) are sharp in some cases.

Let 0 < ρ < 1 {0<\rho<1} and let { a j , b j , n j } j = 1 ∞ {\{a_{j},b_{j},n_{j}\}_{j=1}^{\infty}} be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } {\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}} generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , \mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}}=\delta_{\rho^{n_{1}}\{0,a_{1},b_{1}\}}% \ast\delta_{\rho^{n_{1}+n_{2}}\{0,a_{2},b_{2}\}}\ast\delta_{\rho^{n_{1}+n_{2}+% n_{3}}\{0,a_{3},b_{3}\}}\ast\cdots, where gcd ⁡ ( a j , b j ) = 1 {\gcd(a_{j},b_{j})=1} for all j ∈ ℕ {j\in{\mathbb{N}}} . In this paper, we show that L 2 ⁢ ( μ ρ , { 0 , a j , b j } , { n j } ) {L^{2}(\mu_{\rho,\{0,a_{j},b_{j}\},\{n_{j}\}})} admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ ( mod ⁢ 3 ) {\{a_{j},b_{j}\}\equiv\{\pm 1\}~{}(\mathrm{mod}~{}3)} for all j ≥ 1 {j\geq 1} ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ {\rho^{-r}\in 3{\mathbb{N}}} and n j ∈ r ⁢ ℕ {n_{j}\in r{\mathbb{N}}} for all j ≥ 2 {j\geq 2} .

Let d ≥ 1 {d\geq 1} be an integer and let 𝒲 d {\mathcal{W}_{d}} be the Witt algebra. For any admissible 𝒲 d {\mathcal{W}_{d}} -module P and any 𝔤 ⁢ 𝔩 d {\mathfrak{gl}_{d}} -module V , one can form a 𝒲 d {\mathcal{W}_{d}} -module ℱ ⁢ ( P , V ) {\mathcal{F}(P,V)} , which as a vector space is P ⊗ V {P\otimes V} . Since 𝒲 d {\mathcal{W}_{d}} has a natural subalgebra isomorphic to 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} , we can view ℱ ⁢ ( P , V ) {\mathcal{F}(P,V)} as an 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -module. Taking P = Ω ⁢ ( 𝝀 ) {P=\Omega(\boldsymbol{\lambda})} , the rank-1 U ⁢ ( 𝔥 ) {U(\mathfrak{h})} -free 𝒲 d {\mathcal{W}_{d}} -module, and V = V ⁢ ( 𝐚 , b ) {V=V({\mathbf{a}},b)} , the simple cuspidal module over 𝔤 ⁢ 𝔩 d {\mathfrak{gl}_{d}} , we get the special 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -modules ℱ ⁢ ( 𝝀 ; 𝐚 , b ) = ℱ ⁢ ( Ω ⁢ ( 𝝀 ) , V ⁢ ( 𝐚 , b ) ) \mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)=\mathcal{F}(\Omega(% \boldsymbol{\lambda}),V({\mathbf{a}},b)) which are U ⁢ ( 𝔥 ) {U(\mathfrak{h})} -free modules of infinite rank. We determine the necessary and sufficient condition for the 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -module ℱ ⁢ ( 𝝀 ; 𝐚 , b ) {\mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)} to be simple, and for the non-simple case we construct their proper submodules explicitly. At last, using the above results, we deduce an explicit simplicity criterion for the generalized Verma modules induced from V ⁢ ( 𝐚 , b ) {V({\mathbf{a}},b)} and obtain a family of simple affine modules from ℱ ⁢ ( 𝝀 ; 𝐚 , b ) {\mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)} , which can be viewed as the non-weight version of loop modules.

The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi-equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi-equivelar map embedded on a torus. If a map has k edge orbits under its automorphism group, it is referred to as a k -edge orbital or k -orbital. Specifically, it is referred to as an edge-transitive map if k = 1 {k=1} . If any two edges have the same edge-symbol, a map is said to be edge-homogeneous. Every edge-homogeneous toroidal map has an edge-transitive cover, as proved in [A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 2011, 2, 385–402]. In this article, we show the existence and give a classification of k -edge covers of semi-equivelar toroidal maps.

We construct a two-state Mealy automaton A over the three-letter alphabet generating a regular branch group G ⁢ ( A ) {G(A)} , which surjects onto the infinite cyclic group. Some algebraic and geometric properties of the group G ⁢ ( A ) {G(A)} are derived. In particular, this group has a nearly finitary subgroup of index two, is amenable, just non-solvable, has exponential growth, and its action on the corresponding regular rooted tree is self-replicating, contracting, but it does not have the congruence subgroup property. We also derive in detail an ascending finite L -presentation for the group G ⁢ ( A ) {G(A)} .

This article focuses on the Cauchy problem of the Benjamin–Bona–Mahony–Burgers (BBMB) equation with nonlocal control constraints in ℝ 3 {\mathbb{R}^{3}} . We employ the Phragmén–Lindelöf decomposition method and Green’s function to investigate global classical solutions and their long-term behaviors, including optimal estimates for finite density initial perturbation. Additionally, optimal solutions near the minimal mass ground state are analyzed.

We introduce and study a notion of module nuclear dimension for a C * \mathrm{C}^{*} -algebra A which is a C * \mathrm{C}^{*} -module over another C * \mathrm{C}^{*} -algebra 𝔄 {\mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄 {\mathfrak{A}} -NF. The converse is shown to hold when 𝔄 {\mathfrak{A}} is a C ⁢ ( X ) {C(X)} -algebra with simple fibers, with X compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when 𝔄 {\mathfrak{A}} is unital and simple, if the module decomposition rank of A is finite then A is 𝔄 {\mathfrak{A}} -QD. We study the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} of 𝔄 {\mathfrak{A}} -valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} . Along the way, we also prove a module Choi–Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.