Volume 32, Issue 3

We obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space Bn{B_{n}}, and we prove that the size of the space dim⁡(Bn){\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions d=2,3{d=2,3}, d=4,5{d=4,5} and d≥6{d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case λ=μ{\lambda=\mu} of bilinear quasimode estimates improves L4{L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on LpL^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when d≥8{d\geq 8}. And on this basis, we give approximation bounds in H-1{H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in L2{L^{2}}-norm using the results of Lp{L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

We give necessary and sufficient conditions for the group of a rational maximal bifix code Z to be isomorphic with the F -group of Z∩F{Z\cap F}, when F is recurrent and Z∩F{Z\cap F} is rational. The case where F is uniformly recurrent, which is known to imply the finiteness of Z∩F{Z\cap F}, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of F .

The main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic product Hardy spaces Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} (0<p≤1{0<p\leq 1}): Tm⁢f⁢(x,y)=∫ℝn1×ℝn2m⁢(ξ,η)⁢f^⁢(ξ,η)⁢e2⁢π⁢i⁢(x⋅ξ+y⋅η)⁢d⁢ξ⁢d⁢η.T_{m}f(x,y)=\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}}m(\xi,\eta)\hat{f% }(\xi,\eta)e^{2\pi i(x\cdot\xi+y\cdot\eta)}\mathop{}\!d\xi\mathop{}\!d\eta. Our main theorem is the following: Assume that m⁢(ξ,η){m(\xi,\eta)} is a function on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} satisfying supj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→)<∞\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})}<\infty with s1>ζ1,--1⁢(1p-12){s_{1}>\zeta_{1,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, s2>ζ2,--1⁢(1p-12){s_{2}>\zeta_{2,-}^{-1}(\frac{1}{p}-\frac{1}{2})}, where ζ1,-{\zeta_{1,-}} and ζ2,-{\zeta_{2,-}} depend only on the eigenvalues and are defined in the first section. Then Tm{T_{m}} is bounded from Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} to Hp⁢(ℝn1×ℝn2;A→){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}};\vec{A})} for all 0<p≤1{0<p\leq 1} and ∥Tm∥Hp⁢(A→)→Hp⁢(A→)≤CA→,s1,s2,p⁢supj,k∈ℤ⁡∥mj,k∥W(s1,s2)⁢(A→),\lVert T_{m}\rVert_{H^{p}(\vec{A})\to H^{p}(\vec{A})}\leq C_{{\vec{A},s_{1},s_% {2},p}}\sup_{j,k\in\mathbb{Z}}\lVert m_{j,k}\rVert_{W^{(s_{1},s_{2})}(\vec{A})}, where W(s1,s2)⁢(A→){W^{(s_{1},s_{2})}(\vec{A})} is a bi-parameter anisotropic Sobolev space on ℝn1×ℝn2{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}} with CA→,s1,s2,p{C_{{\vec{A},s_{1},s_{2},p}}} is a positive constant that depends on A→,s1,s2,p{\vec{A},s_{1},s_{2},p}. Here we use the notations mj,k⁢(ξ,η)=m⁢(A1∗j⁢ξ,A2∗k⁢η)⁢φ(1)⁢(ξ)⁢φ(2)⁢(η){m_{j,k}(\xi,\eta)=m(A_{1}^{\ast j}\xi,A_{2}^{\ast k}\eta)\varphi^{(1)}(\xi)% \varphi^{(2)}(\eta)}, where φ(1)⁢(ξ){\varphi^{(1)}(\xi)} is a suitable cut-off function on ℝn1{\mathbb{R}^{n_{1}}} and φ(2)⁢(η){\varphi^{(2)}(\eta)} is a suitable cut-off function on ℝn2{\mathbb{R}^{n_{2}}}, respectively.

We consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ:M→End⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}. If an abstract kernel factors through SEnd⁡(A)Inn⁡(A){\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where SEnd⁡(A){\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A , then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group U⁢(Z⁢(A)){U(Z(A))} of invertible elements of the center Z⁢(A){Z(A)} of A , on which M acts via Φ. An abstract kernel Φ:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U⁢(Z⁢(A)){U(Z(A))}.

We introduce an intrinsic filtration to the magnitude chain complex of a metric space, and study basic properties of the associated spectral sequence of the magnitude homology. As an application, the third magnitude homology of the circle is computed.

In this paper, we study the varieties of nilpotent Lie superalgebras of dimension ≤5{\leq 5}. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a byproduct, we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

Baer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R -projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.

Let the self-affine measure μM,D{\mu_{M,D}} be generated by an expanding real matrix M=diag⁡(ρ1-1,ρ2-1){M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}{D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with α1⁢β2-α2⁢β1≠0{\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for L2⁢(μM,D){L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.

Let the 2×2{2\times 2} expanding matrix Rk{R_{k}} be an integer Jordan matrix, i.e., Rk=diag⁡(rk,sk){R_{k}=\operatorname{diag}(r_{k},s_{k})} or Rk=J⁢(pk){R_{k}=J(p_{k})}, and let Dk={0,1,…,qk-1}⁢v{D_{k}=\{0,1,\ldots,q_{k}-1\}v} with v=(1,1)T{v=(1,1)^{T}} and 2≤qk≤pk,rk,sk{2\leq q_{k}\leq p_{k},r_{k},s_{k}} for each natural number k . We show that the sequence of Hadamard triples {(Rk,Dk,Ck)}{\{(R_{k},D_{k},C_{k})\}} admits a spectrum of the associated Moran measure provided that lim infk→∞⁡2⁢qk⁢∥Rk-1∥<1{\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert<1}.

The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space 𝐄13{\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane ℒ2{\mathcal{L}^{2}}). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in 𝐄13{\mathbf{E}_{1}^{3}}. We shall discuss the causal character of the second derivative of unit speed spacelike curves in 𝐄13{\mathbf{E}_{1}^{3}} in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.

We show that there exists a constant a such that, for every subgroup H of a finite group G , the number of maximal subgroups of G containing H is bounded above by a|G:H|3/2{a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most a⁢n3/2{an^{3/2}} maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most |G:H|-1{\lvert G:H\rvert-1}.

In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDEs, where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

Let D be a division ring of fractions of a crossed product F⁢[G,η,α]{F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order ≤{\leq}. For D we introduce the notion of freeness with respect to ≤{\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F -vector space F⁢((G)){F((G))} of all formal power series in G over F with respect to ≤{\leq}. From this we obtain that all division rings of fractions of F⁢[G,η,α]{F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G . Moreover, F⁢[G,η,α]{F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of F⁢[G,η,α]{F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F -vector space F⁢((G)){F((G))} is a skew field.

We show that unitary groups of II1{\mathrm{II}_{1}} factors and of properly infinite von Neumann algebras have uncountable cofinality and the Bergman property. In particular, we obtain a short alternative proof for the strong uncountable cofinality of U⁢(ℓ2⁢(ℕ)){\mathrm{U}(\ell^{2}(\mathbb{N}))}, which was first proven by Ricard and Rosendal.

We study groups of germs of complex diffeomorphisms having a property called irreducibility . The notion is motivated by the similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [D. Cerveau and F. Loray, Un théorème de Frobenius singulier via l’arithmétique élémentaire, J. Number Theory 68 1998, 2, 217–228]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.

Given a monoid S acting (on the left) on a set X , all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X . Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family 𝒜σ,X{\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any y1,y2∈Y{y_{1},y_{2}\in Y}, there exists a third element y∈Y{y\in Y} whose closure contains both y1{y_{1}} and y2{y_{2}}. More in detail, relying on some specific properties of the maximal members of the family 𝒜σ,X{\mathcal{A}_{\sigma,X}}, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.