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.

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.

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.

For \( \operatorname{arsinh}(x+\lambda x^3+\eta x^5) \), consider the best cubic bounds with linear term \(x\). When \(\eta=0\), a complete classification of the normalized remainder determines the optimal coefficients on \([0,1]\) for every \(\lambda\geq0\). For positive \(\lambda,\eta\), a parameter-uniform sign-variation argument gives at most two positive critical points. For fixed \(\eta\), the critical-point regimes occur in a fixed order as \(\lambda\) increases. The fold has a unique endpoint height \(\eta_c\). A positive fold threshold exists exactly for \(\eta_*<\eta<\eta_c\). The fold terminates transversely at \(\lambda=0\). Near \((\lambda_*,\eta_*)\), two analytic curves mark the appearance of an interior minimum and its equality with the boundary value. Across the second curve, the minimizing point changes discontinuously and the optimal lower coefficient is continuous but not differentiable. The local sixth-order pattern is classical. The critical-point classification is global for the full composite function. Exact sign bounds also give a rational example.