Let H(A) be the Hattori space associated with A ⊆ R, and put B = R \ A. Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández asked ([9], Problem 7.8) for characterizations of when H(A)³, all finite powers H(A)ⁿ, or H(A)^ω are Lindelöf. Using the square theorem and the perfect set property dichotomy, we show that if B has the perfect set property, then the following conditions are equivalent: B is countable, H(A) is second countable, H(A)² is Lindelöf, H(A)³ is Lindelöf, H(A)ⁿ is Lindelöf for every integer n ≥ 2, and H(A)^ω is Lindelöf. This applies in particular when B is analytic or Borel. We also show that the perfect set property assumption cannot be removed for higher powers. In ZFC there is a set A ⊆ R such that B has cardinality c and contains no subspace homeomorphic to 2^ω in the Euclidean topology, while H(A)² is Lindelöf and H(A)ⁿ is not Lindelöf for every n ≥ 4. In particular, H(A)^ω is not Lindelöf. The corresponding implication for the cube remains open.

In this paper, we study the singular limit $q\to \infty$ for non-negative weak solutions of a doubly nonlinear parabolic Cauchy problem with absorption. Under the assumptions $m>1$, $m(p-1)>1$ and $q>p(m+2)$, we first prove that the family $\left\{u^{\left(q\right)}\right\}$ is precompact, uniformly on compact subsets of ${\mathbb{R}}^N\times (0,\infty )$, and that the whole family converges to the unique bounded weak solution of the homogeneous problem with initial datum $v_0\left(x\right)=min\{u_0\left(x\right),1\}$. We then show that the limit operator ${\mathcal{T}}_t$ is given by the composition of the homogeneous semigroup with the truncation $u_0\mapsto min\{u_0,1\}$; it forms an order-preserving semigroup, satisfies the $L^1$-contraction property, and exhibits an initial layer precisely characterized by the excess mass ${(u_0-1)}_{+}$. Finally, we compare the singular limit with the homogeneous solution and give a sharp description of the corresponding convergence defect.

We give a partial answer to a problem of Hernández-Hernández, Ramírez-Chávez and Rojas-Hernández concerning the Lindelöf property of function spaces over one-point extensions of countable free sums. For a free filter \(\mathcal{F}\) on \(\omega\), let \(L_{\mathcal{F}}\) denote the space of real sequences that converge to zero along \(\mathcal{F}\). A coding lemma identifies the descriptive complexity of this auxiliary space by showing that \(L_{\mathcal{F}}\) is \(K\)-analytic if and only if \(\mathcal{F}\) is analytic as a subspace of \(2^\omega\). If \(\mathcal{F}\) is analytic and \((X_n)_{n<\omega}\) is a sequence of non-empty compact spaces, then\[ C_p\left(\bigoplus_{n<\omega} X_n\right)\text{ is Lindelöf} \quad\Longrightarrow\quad C_p(X_{\mathcal{F}})\text{ is Lindelöf}. \]Moreover, if \(C_p(X_{\mathcal{F}})\) is Lindelöf, then \(C_p\left(\bigoplus_{n\in\omega\setminus A} X_n\right)\) is Lindelöf for every \(A\in\mathcal{F}\). Consequently, for constant compact fibres and analytic free filters, \(C_p(X_{\mathcal{F}}(K))\) is Lindelöf and only if \(C_p(K)^\omega\) is Lindelöf. If the free sum has a countable network, no descriptive-set-theoretic restriction on the filter is needed.

Ori's vacuum--dust spacetime has a regular initial hypersurface whose vacuum-core portion is umbilical. The core develops a chronology horizon.Let \(\theta_0\) be the angular cutoff of the core and choose \(0<a<b\) with \(b+2\log(2+\sqrt{3})<\theta_0\). Fix a nonzero real \(p\in C_c^\infty((a,b))\) that is odd about \((a+b)/2\). We construct a smooth, nontrivial one-parameter family of finite-amplitude vacuum constraint deformations, compactly supported in a fixed annulus of the core and defined for every real \(\varepsilon\) with \(\varepsilon^2\|p^2+(p')^2\|_\infty<1\).The family satisfies both nonlinear vacuum Einstein constraints exactly within the core and extends, with Ori's unchanged dust envelope and exterior, to global Einstein--dust initial data satisfying the full constraints. We propagate its axial first variation using the established Kodama--Ishibashi formalism. Under the specified core-size and support conditions, the resulting Weyl Pontryagin first variation satisfies a logarithmic \(L^2\) lower bound as the background horizon is approached. A local hyperbolic equation and the curvature reconstruction establish finite propagation for the invariant curvature observable, even though the auxiliary master potential need not be compactly supported. A radial Wronskian gives a nonzero horizon coefficient on a set of positive spectral measure. We also compute the exact initial twist and the leading term in the initial-slice Weyl Pontryagin expansion of the finite-amplitude data. The growth estimate concerns the linearized Einstein equations, not the curvature of any fixed nonzero-amplitude development. No nonlinear chronology-protection statement is implied.

Graphical discreteness is not preserved under quasi-isometry in general [17] [Corollary 1.3]. Nevertheless, for every finite thick generalized m-gon Γ with m ≥ 3, graphical discreteness is constant among finitely generated groups quasi-isometric to the right-angled Coxeter group WΓ. Such a group Λ is graphically discrete if and only if Γ is nonflexible, meaning that no nontrivial graph automorphism of Γ fixes a closed star pointwise [22] [p. 1]. The proof combines Fuchsian-building quasi-isometric rigidity with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, the standard Cayley-graph automorphism group is compact-by-discrete if and only if it is discrete, and both conditions are equivalent to nonflexibility of the defining diagram. Further results use one-vertex doublings to obstruct graphical discreteness, characterize when the full automorphism groups of standard chamber graphs of graph products of finite groups are compact-by-discrete, and give applications to finite projective planes and rightangled Coxeter groups with Menger curve boundary. The arguments do not determine whether any nonflexible finite thick generalized polygon exists.