Perfect resource placement in dense Gaussian networks partitions the network into Lee balls centered at resource nodes. The fault-free placement problem is already classified; this paper studies the complementary post-deployment problem of repairing such placements after resource faults. The paper gives exact local repair theorems for the dense Gaussian placement generated by $t+(t+1)i$; by conjugation and rotation symmetry, the same results hold for the companion generator $(t+1)+ti$. For one failed resource, we prove failure-cell locality, derive the exact replacement number $ρ_G(1)=3$ and $ρ_G(t)=2$ for all $t\ge2$, and prove the sharp minimum-overlap formula $Ω_G(t)=t+1$ among minimum-size repairs. The overlap lower bound is proved from the corner structure of equal-size Lee balls in the rotated coordinates $u=x+y$ and $v=x-y$, where Gaussian Lee balls become parity-constrained squares. For two failed resources, we prove exact additivity: every pair of failed resource cells requires exactly four local replacements for $t\ge2$, and four always suffice. The two-fault lower bound reduces all relevant resource displacements to two canonical neighboring cases and exhibits four mutually incompatible failed-cell corners in each case. For multi-failure repairs, we prove a general inclusion--exclusion identity for overlap inside the failed region; hence the formula remains exact for arbitrary higher-order dense cores. When a canonical repair instance is certified to have maximum multiplicity three, the identity reduces to the compact correction $Ω_{\rm extra}=P_2-A-C_3$. A ground-truth audit over 7,494 Gaussian cases recomputes coverage from lattice geometry, verifies all exact formulas, and records reproducible multiplicity witnesses.

This paper presents a unified framework for exploiting the boundaries of reconfigurable intelligent surfaces (RIS) joint optimization in multi-user wireless systems, where a single RIS accommodates diverse objectives.We first propose an adaptive gradient-scaling mechanism that accelerates the convergence of the underlying optimization algorithm while maintaining stable performance across varying channel and system parameters. The proposed mechanism enables the solver to reach a reasonably good solution rapidly without requiring manual tuning of step sizes or algorithmic hyperparameters when system inputs change. We then propose a low-complexity beamformer recovery method tailored for single-user scenarios, which circumvents the full matrix decomposition required by traditional approaches, thereby significantly reducing computational overhead. Building on these foundations, we develop an element allocation strategy that enables user-specific prioritization through assignment of RIS subsets. This is further extended by a modular add-drop mechanism that supports partial-panel optimization in general multi-user settings. The framework is evaluated across three representative scenarios: (i) signal amplification for all users, (ii) signal suppression for all users, and (iii) selective amplification and suppression. To characterize performance limits, we derive power trade-off boundaries using scalarized joint optimization, which closely align with Monte Carlo simulations. Our unified joint optimization method consistently yield solutions near these boundaries, confirming its near-optimality. Extensive simulations under realistic channel models demonstrate that the proposed approach outperforms conventional semidefinite relaxation techniques, offering a scalable and effective RIS control strategy for cooperative and competitive multi-user environments.

In this paper we study the class of graphs without cycles of size 4 and independent sets of size 4 as induced subgraphs: $\mathop{Free}\{C_4, 4K_1\}$. This is one of the three minimal minimal open cases for the complexity of the colouring problem when restricted to classes defined by excluding induced subgraphs of order 4. We investigate the clique width of some subclasses of $\mathop{Free}\{C_4, 4K_1\}$. We introduce a new framework: the $(k,l,m)$-decomposition and prove that if all the graphs of a class $\cal G$ are $(k,l,m)$-decomposable, then graphs in $\cal G$ have bounded clique width. We give a few examples of such class, found with the help of a program we designed. We also show, for any graph $G \in \mathop{Free}\{C_4, 4K_1\}$ that is 3 cliques coverable, an infinite family in $\mathop{Free}\{C_4, 4K_1\}$ of supergraphs of $G$ which have unbounded clique width.

In survival analysis, dependent censoring poses significant challenges in accurately estimating model parameters and survival functions. This study introduces a novel framework leveraging Extended Generalized Marshall-Olkin (EGMO) models to address dependent censoring mechanisms. Geometric optimization techniques are employed to develop efficient estimation procedures that capture dependencies between failure and censoring times. We establish their asymptotic properties. Simulation studies and real data applications illustrate the method's robustness and effectiveness.

Perfect resource placements in dense Eisenstein--Jacobi (EJ) networks partition the network into hexagonal radius-$t$ service cells. This paper studies local repair of such placements after resource failures. For one failed resource, we prove that one replacement cannot cover the failed hexagon and two always suffice, giving $ρ_{\mathrm{EJ}}(t)=2$ for all $t\ge1$. Among minimum-size repairs, the sharp minimum-overlap formula $Ω_{\mathrm{EJ}}(t)=t^2$ follows from the three-strip geometry of EJ balls. For two failed resources, independent repair gives a four-replacement upper bound, but unlike the Gaussian case EJ repair is not always additive: two infinite neighboring displacement families admit three-replacement repairs, proved optimal by a two-ball impossibility argument. Additive behavior is established algebraically via endpoint-rigidity and diagonal-corridor theorems. For $q$ failed resources, independent canonical repair gives a universal $2q$ upper bound, exact when failed cells are pairwise more than $4t$ apart. Dense cluster subadditivity is proved for infinite four-fault and six-fault families with exact repair numbers four and five, giving savings of four and seven over independent repair. An exact inclusion--exclusion identity governs repeated coverage for arbitrary multi-fault repairs. An audit over 19,400 instances confirms widespread subadditivity. EJ local repair is structurally distinct from the Gaussian case: the one-fault overlap is quadratic, two-fault repair can be non-additive, and clustered repairs reuse replacement balls across multiple failed cells.

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter--importer--product--year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational--Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

Farrow structures based on linear--phase FIR subfilters provide an efficient realization of variable fractional--delay (VFD) filters with reduced implementation complexity. While the all--linear--phase configuration admits a decoupled least--squares (LS) formulation with an analytical solution, this decoupling fails when branches of mixed types, linear--phase and general FIR, are required, as occurs when a group--delay constraint is imposed. This letter presents a unified LS design for Farrow structures via a nullspace parameterization of the per--branch symmetry constraints, yielding an analytical solution that accommodates arbitrary per--branch types. Numerical results demonstrate that the proposed framework satisfies group--delay constraints that the all linear--phase approach cannot meet, while substantially reducing the number of free parameters relative to the unconstrained general FIR baseline.

We formulate and analyze a sex- and stage-structured model for Anopheles dynamics under the sterile insect technique (SIT), motivated by the need for tools robust to insecticide resistance and outdoor transmission. The model tracks aquatic stages, adult males, unmated females, and females mated with wild or sterile males; includes egg-laying capacity and larval competition; and uses a refractory period followed by density-dependent mate search. The resulting Holling type-II mating term generates a mate-finding Allee effect. After establishing well-posedness, we prove that this Allee effect makes the mosquito-free equilibrium locally stable for all admissible parameters and globally asymptotically stable when a quick-mate-search reproduction number $R_0^q$ is below one. When $R_0^q>1$, habitat capacity is large, and larval competition is weak, two positive equilibria arise through a saddle-node bifurcation: a stable natural equilibrium and an unstable Allee equilibrium separating persistence from extinction. For a reduced model, a Goh-Volterra Lyapunov functional estimates the persistence basin. We then show how constant and population-responsive sterile-male releases reshape this bistability. Sufficiently large releases annihilate the positive equilibria in a second saddle-node bifurcation, while a sufficiently large constant release drives local elimination from every admissible initial state. Thus SIT need only push the population across the Allee separatrix, after which mate-finding failure can complete extinction. In a free-horizon optimization framework with an Allee-threshold stopping rule, a hybrid release strategy reduces the sterile-male requirement by about $5\%$ relative to the best constant-only strategy and $39\%$ relative to the best population-responsive-only strategy. These results recast the Allee effect as a control lever for vector suppression.

Over-the-air federated learning (FL) leverages the superposition property of multiple-access channels to enable communication-efficient distributed model training. Existing integrated sensing, communication, and computation (ISCC)-enabled over-the-air FL systems typically require dedicated resources for the sensing module, inevitably compromising FL performance due to resource competition. In this paper, we propose a sensing-native over-the-air FL framework that explores built-in distributed wireless sensing capability with zero overhead per model aggregation. Specifically, the high-dimensional local gradient signals possessing favorable autocorrelation property are concurrently leveraged for target distance estimation, while the gradient statistics already required for over-the-air FL serve as a ready-made gateway to deliver locally-sensed results to the edge server for cooperative localization. To combat inter-device interference, channel fading, and communication noise, we put forth a robust trilateration-based target positioning method building upon an efficient matched-filtering-based distance estimation. Then, by explicitly characterizing the impact of imperfect model aggregation and noisy gradient-statistics transmission on the sensing-native over-the-air FL convergence, we develop a statistics-aware communication-learning co-design approach. We first derive the closed-form optimal power budgets allocated to local gradients and their statistics, based on which an efficient successive convex approximation method is proposed for receiver beamforming optimization. Simulation results show that the proposed framework simultaneously achieves superior learning and sensing performance compared to representative baselines.

Let $(A,\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\ge 1$, $N$ a perfect module of dimension $t\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\ge 1$, it is well known that the functions $n \mapsto \ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \mapsto \ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\le t-1$ and $s^i_{I,N}(M)\le t-1$ and when $I$ is the maximal ideal $\mathfrak{m}$, both the inequalities become equalities. We also show that $r_i^{I,N}(M)\le r_1^{I,N}(Ω^dk)$, $s^i_{I,N}(M)\le s^1_{I,N}(Ω^dk)$ and $r_i^{I,N}(Ω^dk)=r_1^{I,N}(Ω^dk)=s^1_{I,N}(Ω^dk)=s^i_{I,N}(Ω^dk)$. \end

We study derived categories associated with log del Pezzo surfaces whose singularities are of type $\frac{1}{3}(1,1)$. For such a surface $X$, we consider the canonical smooth Deligne--Mumford stack $π:\mathcal X\to X$ and compare it with the singular coarse surface $X$. Our main result proves that, if $X$ is a complex log del Pezzo surface whose singularities are all of type $\frac{1}{3}(1,1)$, then $D^b(\operatorname{coh}\mathcal X)$ admits a full exceptional collection. The proof combines rationality of log del Pezzo surfaces, Orlov's blow-up formula, and the special McKay correspondence of Ishii--Ueda. We then specialize to a general degree $10$ hypersurface $X_{10}\subset \mathbb P(1,2,3,5)$. The Corti--Heuberger cascade identifies its minimal resolution as $\widetilde{X}_{10}\cong \operatorname{Bl}8\mathbb F_3$, and therefore the canonical stack $\mathcal X_{10}$ has a full exceptional collection of length $13$. We also discuss the singular coarse category through the approach of Karmazyn--Kuznetsov--Shinder.

Solutions of hyperbolic conservation laws exhibit both smooth structures across large scales and sharp localized features such as shocks and contact discontinuities, making them difficult to approximate accurately with existing neural operators. The Fourier Neural Operator (FNO) captures long-range interactions well but tends to smear localized structures through excessive numerical dissipation. To address this, we propose a Local-Global Neural Operator (LGNO) that learns a one-step discrete flow map by combining a global FNO branch for representing smooth dynamics at large scales with a local multiresolution branch for enhancing localized discontinuities and nonsmooth features. The model is trained with a one-step loss that combines a physical space prediction term and a spectral penalty on high frequencies to suppress spurious oscillations near steep fronts. On a large collection of benchmarks in one and two dimensions, LGNO consistently outperforms FNO baselines with matched parameter counts, reducing one-step errors by factors of 2-5 and remaining significantly more accurate over long autoregressive rollouts. Most strikingly, although it is trained only on short-time data from a high-order WENO-Z scheme, the long-time rollout of LGNO on a coarse $256^2$ grid exhibits lower numerical dissipation than the same WENO-Z scheme run on a finer $512^2$ grid, while being orders of magnitude cheaper to evaluate. These results suggest that, with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps. They further suggest that such learned operators have the potential to control long-time numerical dissipation better than the conventional shock-capturing schemes that generate the training data.

We construct non-rigid tensor-triangulated categories with non-distributive lattice of thick tensor-ideals.

We study infinite-horizon optimal stopping for normally reflected two-dimensional diffusions in the positive quadrant with max payoff \(G(x_1,x_2)=x_1\veeαx_2\). The non-smooth payoff produces a singular stopping-gain measure on the kink set \(Δ=\{x_1=αx_2\}\). We prove $\displaystyle Γ^Δ(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+α^2}}\,σ_Δ(dx)$, with $n=(1,-α)$, so the diagonal component is non-positive and strictly negative under local ellipticity. This implies that every interior kink point lies in the continuation region. We further show that the correct value representation uses the resolvent killed at first entry into the stopping set, $\displaystyle V=G-R_r^{\mathcal C}Γ$, and give a closed-form reflected Brownian counter-example showing that the unrestricted reflected resolvent is generally wrong. A reflected Brownian benchmark and numerical experiments illustrate the local-time, resolvent-gap, and diagonal-avoidance mechanisms.

We study two inverse problems for a nonlinear dynamical Schrödinger operator with magnetic and electric potentials. Under suitable analyticity assumptions, we show that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials. We establish the uniqueness of these potentials from both full data and partial data. In particular, for the partial data problem, the desired uniqueness is established by assuming that the potentials are known near the boundary, and the Neumann data is measured on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem, where we obtain the optimal Sobolev regularity for solutions.

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar z}-μw_z+\mathcal{A}w+\mathcal{B}\bar w=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $Θ=|\mathcal{B}|^2/(1-|μ|^2)\,dx\,dy$ and its total mass are invariants under multiplicative gauges $w\mapstoϕw$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=φv'+ψ\bar v'$, the complex gauges being the case $ψ\equiv0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tildeφ=-iλ/(φ-ψ)$, where $λ$ is the spectral root of the structure polynomial.

We review the current state of the Cherlin-Zilber Algebraicity Conjecture on simple groups of finite Morley rank, which states that every such group is the group of $K$-rational points of an algebraic group for some algebraically closed field $K$. We will explain the relevance of sharply 2-transitive groups as a potential source of counterexamples and how the Burnside problem necessarily comes into the picture.

For a self-similar graph $(G, E)$, we find a distinguished subgroupoid of the associated path groupoid $\mathcal{G}_{G,E}$ -- the symmetric cycline subgroupoid $\mathcal{S}_{\text{sym}}$. If the acting group $G$ is abelian, we show that $\mathcal{S}_{\text{sym}}$ is open, abelian, and normal. For $G=\mathbb{Z}$, we describe the dual bundle $\hat{\mathcal{S}}_{\text{sym}}$ of $\mathcal{S}_{\text{sym}}$ which can be used to provide a different groupoid model for the self-similar graph $C^*$-algebra $\mathcal{O}_{\mathbb{Z}, E}\cong C^*_r(\mathcal{G}_{\mathbb{Z},E})$. For a large class of self-similar graphs $(\mathbb{Z}, E)$, we further prove that $\mathcal{S}_{\text{sym}}$ is maximal among open abelian subgroupoids of $\mathrm{Iso}(\mathcal{G}_{\mathbb{Z},E})^{\circ}$ and closed in $\mathcal{G}_{\mathbb{Z},E}$, so that it gives rise to a Cartan subalgebra of $\mathcal{O}_{\mathbb{Z}, E}$. This result seems new even for genuine actions. Our proofs heavily rely on careful studies of dynamical behaviours of cycline triples of $(\mathbb{Z}, E)$ and on a dynamical-flavour classification for the vertices of $E$. Some results hold in more general settings and may be of independent interest.

This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-α$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

We propose a machine-learning framework for detecting whether a given matrix is a perturbation of a matrix with a large Jordan block. The proposed model achieves high classification accuracy on synthetically generated, robustly perturbed data and outperforms a classical numerical baseline. Moreover, we demonstrate that the learned model generalizes to several classes of matrices not seen during training. These results suggest that the architecture captures structural properties associated with matrix defectiveness.

We prove a Trudinger-Moser type inequality in fractional Sobolev spaces with singularities on smooth compact sets of codimension $k$, where $1 < k < d$ and $sp = d$. The singular term is given by the inverse $d$-th power of the distance to the submanifold. The proof is based on a fractional Hardy inequality adapted to smooth submanifolds, and we show the sharpness of the constant. We also establish the equivalence of two natural fractional Sobolev spaces vanishing on the singular set.

We develop a minimizing-movement framework for parametric finite element approximations of geometric gradient flows with admissible tangential motion. At each time step, the discrete variational problem combines a metric dissipation term for the normal displacement with a surface Dirichlet energy. The metric determines the normal geometric evolution: the $L^2(Γ)$ metric gives mean curvature flow, while the $H^{-1}(Γ)$ metric gives surface diffusion flow. Tangential velocity is selected independently through weak constraints on the deformation map. The central structural condition is admissibility, namely, that the identity map satisfies the constraint. This condition keeps the identity map available as a comparison function and yields the natural stability estimate. The framework recovers the classical Barrett--Garcke--Nürnberg (BGN) scheme from the unconstrained formulation and the dual minimal-deformation-rate (MDR) scheme from the MDR constraint. We further introduce two new admissible variants: an admissible BGN scheme and a relaxed MDR scheme. For the resulting fully discrete schemes, we prove existence and uniqueness under natural nondegeneracy assumptions and establish unconditional energy stability. Numerical experiments compare the admissible and classical schemes and illustrate their stability properties and mesh-quality behavior.

This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $α,β>0$ are such that $2α+β<1$, then there are positive constants $δ= δ(α, β)$ and $Δ= Δ(α, β)$ such that if $P$ is a partial Latin square of order $n$ with $k = k(n)$ non-empty cells occupying at most $αn$ rows and $βn$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(δ/n)^k$ and $(Δ/n)^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n \to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares.

The Loewner order on Hermitian matrices is a partial order that compares matrices in terms of positive semidefiniteness. The Loewner order plays a key role in many fields such as optimization, numerical linear algebra, control theory, operator theory, and quantum information. A fundamental difficulty is that two or more Hermitian matrices do not necessarily have a unique minimal upper bound (or maximal lower bound). In this paper, we propose an iterative method to exactly compute a minimal upper bound for any finite collection of $n\times n$ Hermitian matrices. It is shown that the algorithm terminates in at most $n$ iterations. The exactitude of the algorithm is proved using standard results from finite-dimensional linear algebra. A self-contained proof of an algebraic characterization of minimality originally explored by Stott is provided. We illustrate the algorithm in examples and also provide an implementation of the algorithm in Python.

We consider the relationships between the domination number of graph, denoted $γ$, and the distance-$2$ domination number, denoted $γ_2$, and three parameters that lie between them: the packing number, denoted $ρ$, the lower packing number, denoted $ρ_L$, and the isolation number, denoted $ι$. There has been recent attention on the question of whether $γ/ρ$ is bounded or unbounded for various families of graphs. We consider similar questions for the ratios of the five parameters. In particular we show that, while $γ/ρ_L$ is unbounded in trees, it holds that $ι/γ_2$ is less than $2$ for all trees. Further, $γ/ρ_L$ is at most $3$ in interval graphs, at most~$4$ in permutation graphs, and at most $5$ in general asteroidal-triple-free graphs. We also show that every tree has a set of vertices that is both isolating and a packing, and characterize trees where $ρ=ρ_L$.

We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed integer $r\geq 2$ and every $η>0$, there exists $d_*=d_*(r,η)$ such that for every $d\geq d_*$, any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d$ satisfies \[ α(G)\geq (1-η)r^{-1/r}\left(\frac{\log d}{d}\right)^{1/r}n. \] The proof combines a cleaning procedure, which reduces the maximum $r$-degree to the average scale, with a random nibble that repeatedly extracts independent vertices while controlling all lower-order degrees created by the process. After an initial top-layer cleaning, we run a trace nibble. Since the residual hypergraph contains traces of all sizes $2,\ldots,r+1$, we track the maximum degrees in every layer. A binomial-type recurrence for this degree profile yields the stated leading constant.

We show that the von Neumann algebra generated by position operators in a 2-periodic interacting Fock space is a type $II_1$ factor. On the probabilistic side, we prove that the squared position operators have a Marchenko-Pastur distribution with respect to the vacuum state, yielding a natural realization of free Poisson laws within this framework.

We prove a weak version of the classical Hodge theorem on piecewise-algebraic spaces, a class of spaces introduced by Kontsevich and Soibelman in [KS00]. Precisely, we first prove the Poincare lemma that computes singular cohomology as a variant of de Rham cohomology. Then, as a weak Hodge theorem, we naturally embed the singular cohomology into the space of harmonic forms, instead of establishing an isomorphism (which does not hold for those spaces). Our approach in the latter is classical: Sobolev space theory. In addition, we give more detailed proofs for the claims in the appendix to [KS00]. This work is part of a program of extending arithmetic intersection theory to singular spaces. In particular, a type of currents in this singular setup is introduced.

We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $γ>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine--Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.