Papers

In this paper, a natural duality on the family of biperiodic Fibonacci words $\mathfrak{F}^{(a,b)}$ generated by the directive sequence $(a,b,a,b,\dots)$. Both of $\mathfrak{F}^{(a,b)}$ and $\mathfrak{F}^{(b,a)}$ are related by the explicit morphism $\sigma_a:0\mapsto 0^a1$, $1\mapsto 0$, establishing a precise substitutional correspondence between them. We compute the exact letter frequencies, give a complete description of the return words for each letter, prove the existence of arbitrarily long palindromic prefixes, and determine the continued fraction expansion of the slope $\theta^{(a,b)}$. These findings reveal that the apparent asymmetry in several invariants arises uniformly from the length-redistribution mechanism induced by the morphism $\sigma_a$.

Given a closed set $E \subset \partial {\mathbb D}$ of measure zero and a continuous function $\varphi : E \to {\mathbb C}$, the classical Rudin-Carleson interpolation theorem states that there exists a continuous function $F : \overline {\mathbb D} \to {\mathbb C}$ that is holomorphic on ${\mathbb D}$ and satisfies $F\rvert_E = \varphi$. In this paper we obtain a generalisation of the Rudin-Carleson theorem for maps $\varphi : E \to X$ into arbitrary connected complex manifolds $X$ that combines interpolation of $\varphi$ on $E$ with uniform approximation on compact subsets of $\overline {\mathbb D} \setminus E$ of another given continuous map $f:\overline {\mathbb D} \to X$ that is holomorphic on ${\mathbb D}$. Under the further assumption that $X$ is an Oka manifold we obtain a corollary that combines Rudin-Carleson interpolation of a continuous map $\varphi : \overline {\mathbb D} \to X$ on $E$ with Runge approximation of $\varphi$ on a compact set $K\subset {\mathbb D}$ without any holes on which $\varphi$ is holomorphic.

Cartan's method classifies the class of linearizable system of two second-order ODEs into many branches. This paper investigates Branch I of the classification, characterized by a rank-one generalized Wilczynski invariant matrix and the vanishing of two relative invariants $K_1$ and $L_1$. It is demonstrated that any linearizable system belonging to this branch admits an eight-dimensional Lie point symmetry algebra. The canonical form for this class is provided and the invariant characterizations based on the obtained rank-zero invariant coframe and the corresponding constant structure equations are established. Also, a systematic procedure for constructing the linearizing point transformation is derived. The theoretical results are illustrated by several examples.

We show that Khovanov homology cannot distinguish the roll-spun slice disk from the trivial slice disk bounding the connected sum of a knot and its mirror when composed with a Morse 1-handle.

We study Poisson Galton–Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon–Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous–Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n/n \to {\mathbb E}_{\pi}[1/T_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1/\infty=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $$ \int_S q_u(x)\,\pi(dx) - {u\over 2}\int_{S\times S}\kappa(x,y)q_u(x)q_u(y)\,\pi(dx)\pi(dy). $$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.

We introduce ideal Gårding polynomials, a convexity-enhanced subclass of Gårding polynomials whose Gårding components are recursively convex under partial differentiation. This class strictly contains real stable polynomials and, after translation and homogenization, lies in the Lorentzian class. Our main result is that ideal Gårding polynomials still admit a robust structure theory despite this additional convexity: they are preserved under polarization, satisfy natural closure properties, and support a linear preserver theory. A key contribution of this paper is a universal model for univariate Gårding polynomials, described by monotone root sequences and equivalently by volume polynomials of Pitman–Stanley polytopes. We establish quotient concavity, and Newton–Maclaurin type inequalities, which leads to the polarization theorem, and suggests further connections with convex geometry and Lorentzian polynomials.

A linear involution is an injective piecewise isometry defined on a pair of disjoint intervals. They are defined by a combinatorial data given by a generalized permutation and a length vector. As it was done for interval exchange transformations, it is conjectured that, except for some combinatorial data, a typical linear involution is measure-theoretically weakly mixing. In this direction, one may first explore the question of topological weak mixing for topological models of linear involutions. In this article we prove topological weak mixing for the natural symbolic codings of typical linear involutions defined by some generalized permutations of hyperelliptic type. We consider the generalized permutation \[ \sigma_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix}. \] We prove that the natural symbolic coding of a typical linear involution defined by a generalized permutation in the Rauzy class of $\sigma_{s,r}$ is topologically weakly mixing, provided that it has at least a simple letter and its associated genus is sufficiently large.

We study continuous data assimilation for the two-dimensional incompressible Navier-Stokes equations on the periodic torus using a single signed scalar velocity projection in a prescribed spatially varying direction. We prove exponential synchronization in both L2 and H1 through two complementary mechanisms. In the regular resonant regime, nontrivial invisible currents are controlled through a moving-frame expansion, adapted coordinates, and Farhat-Lunasin-Titi logarithmic estimates, provided the resulting geometric shear defect is viscously absorbable. This recovers the periodic one-component mechanism for constant rational directions and applies to genuinely nonconstant projection fields. In the kernel-free regime, qualitative injectivity and compactness yield observability with arbitrarily small viscous leakage, giving synchronization for every sufficiently large gain without an upper gain restriction. For sufficiently regular projection fields, both mechanisms extend to L2-stable Type-I coarse scalar observations satisfying a first-order approximation property, under the usual gain-resolution condition. A common parabolic smoothing argument then upgrades all four L2-synchronization results to H1-synchronization without additional observation hypotheses.

We construct the Sasaki metric on the unit tangent bundle $U_g M$ of a Riemannian manifold $(M , g)$ and describe the unit tangent bundle $U\mathbb{H}^n$ of real hyperbolic space as a homogeneous space, both under $SO_0 (1, n)$ and under the larger group $SO_0 (1, n) \times SO_0 (1, 1)$, yielding explicit of $G$-invariant metrics. Using Hopf coordinates and Busemann functions, we then construct a Riemannian metric $g_{Hopf}$ on $U\mathbb{H}^n$ that is invariant under the geodesic flow, and we identify the horospherical cylinders as totally geodesic leaves of a natural foliation associated to a Busemann function, with respect to an explicit metric connection with torsion.

In this paper, we prove that forward ideal sub-Finslerian manifolds support interpolation inequalities for optimal transport, extending the results of Barilari and Rizzi, arXiv:1705.05380, from the sub-Riemannian to the sub-Finslerian setting. A key role is played by the introduction of sub-Finslerian Jacobi fields and the establishment of optimal transport theory on sub-Finslerian manifolds. By combining this transport framework with sub-Finslerian Jacobian estimates, we characterize the generalized distortion coefficients. As an application, we deduce several fundamental geometric inequalities, including the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities. Finally, for the case of the Randers sub-Finslerian Heisenberg group, whose metric is defined by a sub-Riemannian metric perturbed by a drift term, we explicitly show that it satisfies the measure contraction property.

Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identify the active physical group even though the training data cannot distinguish the physical rays within it. We develop inference for active physical rays, unit atoms modulo sign, after latent dictionary learning. In a fixed-dimensional Gaussian train-test experiment, we retain all dictionaries compatible with a robust training-moment region, profile the test representation over them, and project surviving configurations onto a permutation-invariant support space. The resulting confidence correspondence can report cross-sheet inconclusiveness, group resolution with child ambiguity, or fine-support resolution. We characterize both its statistical cost and decision-theoretic benefit. Residual block orientation first affects the latent training density at cubic order, yielding information of order $s^6$, where $s$ is the within-block collision scale. The correspondence provides high-probability-over-training conditional test coverage, with resolution governed separately by parent detectability, test-time support separation, and learned-dictionary orientation. In the resolved fixed-shell regime, its projective Hausdorff diameter contracts at the minimax-optimal rate $s \wedge (\sqrt{N}s^2)^{-1}$, up to constants. A restricted-task theorem further determines when coefficient asymmetry allows test replication to supplement training information and when calibration uncertainty remains irreducible. The framework thus yields honest, resolution-adaptive support statements and guides the allocation of training versus test measurements.

The rapid emergence of sixth-generation (6G) networks and the low-altitude economy has accelerated the evolution of wireless infrastructures toward air-ground integrated coverage networks (AGICNs), which seamlessly fuse terrestrial and aerial communication resources. However, existing AGICN studies primarily focus on coverage enhancement, while ignoring sustainability. Pursuing sustainable AGICNs introduces new challenges due to the multidimensional resource coupling across heterogeneous air-ground segments. In view of this, this paper presents a comprehensive survey and tutorial on sustainable AGICNs, aiming to balance coverage capacity with carbon efficiency in low-altitude economies. An integrated sensing, communication, and computation (ISCC)-driven architecture, which enables dynamic resource orchestration through closed-loop control, is proposed. We thus introduce a multi-dimensional sustainability metric system, which covers operational efficiency, task-oriented performance, and full lifecycle carbon emissions, to quantify energy and carbon footprints. We review enabling technologies, including artificial intelligence, hybrid precoding, integrated sensing and communication, and simultaneous wireless information and power transfer, and discuss their integration into the ISCC framework to minimize energy consumption while maintaining robust coverage. Experimental results on a real-world testbed demonstrate a 20% reduction in power consumption while achieving over 90% coverage probability, highlighting the feasibility of sustainable AGICNs for future green networks.

Given a graph $G$, its chromatic polynomial $P (G, k)$ counts proper $k$-colorings, while the corresponding list color function $P_{\ell} (G, k)$ counts the minimum number of proper colorings across all assignments of $k$ colors to each vertex. While it is clear that $P_{\ell} (G, k) \leq P (G, k)$, Donner showed in 1992 that $P_{\ell} (G, k) = P (G, k)$ whenever $k$ is sufficiently large. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph. A list version of Hanlon's notion was introduced in 2024 by Kaul and Mudrock, who further raised the question of whether the analog of Donner's result holds in the unlabeled case. While they proved this for all connected point-determining graphs, even the case of the edgeless graph on $n$ vertices remained open and was posed as a conjecture. We prove this conjecture and show that it implies that, more generally, a disconnected graph satisfies the unlabeled analog of Donner's result if all of its connected components do.

Variational Quantum Algorithms (VQAs) are a leading paradigm for near-term quantum computing, yet their training suffers from sensitivity to circuit depth, initialization, and landscape pathologies such as barren plateaus. We study progressive depth training (PDT) – a layerwise curriculum that trains a shallow circuit before appending new layers – and identify a fundamental obstacle: fixed entangling gates (CNOTs) in hardware-efficient ansätze cause initialization shock, an energy spike when new layers are added. We propose identity-paired progressive depth training (IP-PDT), which appends forward/inverse block pairs – each consisting of a standard rotation$+$CNOT block followed by its reverse – that compose to the identity at initialization. Because the adjacent CNOT rings cancel, the effective circuit retains only a single entangling layer surrounded by overparameterized local rotations. We prove a simple Reachable Set Saturation Theorem: under this construction the variational manifold expands exactly once (when post-entangler rotations are first introduced) and then saturates; all subsequent depth increases provide pure overparameterization of single-qubit unitaries. Despite this saturation, progressive addition of rotation parameters can continue to improve optimization outcomes – a phenomenon we term trainability beyond expressibility. We formalize IP-PDT as a continuation method on nested manifolds, prove monotone energy guarantees under an acceptance rule, and connect energy error to ground-state fidelity through spectral-gap inequalities. A detailed resource analysis shows that IP-PDT achieves lower total gate cost than both baselines by eliminating most CNOT gates.

We propose a couple of novel tools for benchmarking optimization algorithms which possibly converge to different solutions on a test set: the quality profiles and the test set profiles. Their aim is to assess and compare algorithms in terms of quality (i.e. value of the objective function) of the obtained solutions, as well as to assess the consistency of the test set. A key distinguishing feature of the quality profiles we propose is its comparative deterministic procedure that emphasizes the accuracy of the solution, rather than the computational burden of solvers. In this regard, several test set–dependent approaches for both comparing and ranking algorithms have already been proposed in the literature, representing widely used benchmarking procedures. We believe that the joint use of such procedures, along with the novel quality profiles detailed here, should enhance the benchmarking process, in all those cases where the comparison encompasses exact methods as well as heuristics. Moreover, the literature on numerical optimization seems to have paid less attention, in the last decade, to determining how appropriate a test set used for benchmarking the selected solvers may be. This motivates the introduction of test set profiles, which assess the appropriateness of a test set and represent the flip side of evaluating the robustness of the solvers on that test set. This paper also includes extensive numerical experiments, showing the usefulness of quality profiles in both smooth and nonsmooth (derivative–free) optimization, along with the reference to a MATLAB code for plotting quality profiles and test set profiles.

We give a Graham-positive expansion for the product of two double Schubert polynomials indexed by two inverse Grassmannian permutations. Surprisingly, the nonzero structure constants are double Schubert polynomials in two disjoint sets of equivariant variables. We also give a positive expansion for the product of two single Schubert polynomials indexed by a $321$-avoiding permutation (e.g., a Grassmannian permutation) and an inverse Grassmannian permutation. Unexpectedly, the nonzero structure constants are Edelman–Greene coefficients.

We study tight conic relaxations for a quadratically constrained quadratic programming (QCQP) formulation of rank-one doubly nonnegative (DNN) matrix completion. Motivated by sparse QCQPs whose lifted matrix variables include elements not directly specified by the objective or constraints, we interpret tightness as a rank-one completion property for the unspecified elements. For sparsity patterns whose blocks consist of cycles and edges, we prove that the dual formulations associated with the DNN and completely positive (CP) relaxations are equivalent. For cycle-type sparsity patterns, we derive explicit sufficient conditions under which the semidefinite programming (SDP) and DNN relaxations are tight. These sufficient conditions are stated explicitly in terms of local ratio bounds and cumulative-difference conditions on a rank-one optimal solution. We also show that adding suitable edges to the sparsity pattern relaxes the ratio conditions required for tightness. The results provide tractable certificates for when conic relaxations recover a rank-one optimal solution of the underlying QCQP.

Let (a_k) be the positive coefficient sequence of the normalized Riemann xi-function, and let D_{r,k} denote its consecutive Toeplitz minors. The Riemann Hypothesis is equivalent to (a_k) being a Polya frequency sequence of infinite order, and hence to nonnegativity of all Toeplitz minors. We prove that D_{r,k} > 0 for every r >= 2 and k >= 10^18 r^3. This gives an explicit cubic tail scale uniform in r, in contrast with Katkova's fixed-order asymptotic positivity. The proof does not use numerically verified zeros of the Riemann zeta-function. It combines a certified complex saddle-point analysis of the moment transform, an exact q-Pascal dilation semigroup controlling every degree simultaneously, and a weighted Banach-algebra majorant for the nonlinear remainder, closed by an inertia-preservation argument. All analytic constants are certified with directed rounding in Arb ball arithmetic, and all algebraic identities are verified in exact rational arithmetic. The ancillary files reproduce every certificate. The result concerns only the tail regime k much larger than r^3 and makes no progress on the Riemann Hypothesis, which concerns the complementary region.

We study the uniform-in-time stability and mean-field limit of an infinite Cucker–Smale-type (ICS-type) model in a fully noncompact setting. First, we clarify the main difficulties arising from the original ICS model and its corresponding kinetic Cucker–Smale (KCS) model, when the spatial support is noncompact. In particular, we show that the velocity diameter may remain constant in time, which invalidates the classical approach based on position and velocity diameters. Moreover, we construct a counterexample showing that the original KCS model does not satisfy uniform-in-time stability in the noncompact spatial setting. To overcome these obstacles, we introduce a new ICS-type model whose communication weight depends on a pairwise spatial moment. Under suitable assumptions, we also establish the uniform-in-time stability of the ICS-type model with respect to initial data in the noncompact setting. Finally, we derive the uniform-in-time mean-field limit for the ICS-type model, together with uniform-in-time stability for the corresponding KCS-type model.

In this paper, we study the low-regularity Cauchy problem for the Chern–Simons gauged $O(3)$ sigma model in $\mathbb{R}^{1+d}$ ($d=1,2$) under the Lorenz gauge. For $d=1$, we establish local well-posedness for initial data $(\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_1}(\mathbb{R})\times H^{s_1-1}(\mathbb{R})$ with $s_1>\frac12$. This improves the previous result of Jin and Huh \cite{HJ} by one quarter of a derivative and is almost optimal in view of the scaling-invariant regularities $\dot H^{1/2}(\mathbb{R})$ for the matter field and $\dot H^{-1/2}(\mathbb{R})$ for the gauge field. For $d=2$, we establish local well-posedness for initial data $(\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_2}(\mathbb{R}^2)\times H^{s_2-\frac34}(\mathbb{R}^2)$ with $s_2>1$. This improves the previous result of Jin and Zhang \cite{JZ} by one quarter of a derivative and brings the regularity threshold close to the scaling-invariant exponents $\dot H^{1}(\mathbb{R}^2)$ and $\dot H^{0}(\mathbb{R}^2)$ for the matter and gauge fields, respectively. The analysis relies on two main ingredients. In two space dimensions, we identify the complete null structure of the derivative nonlinearities, allowing the entire system to be treated within a unified null-form framework. In one space dimension, we establish a direct energy estimate in the function space introduced by Keel and Tao, avoiding the finite-propagation reduction to a small-data problem and enabling the low-regularity iteration for general initial data.

We consider the semilinear fully fractional heat equation \[ (\partial_t-\Delta)^\sigma u = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < \sigma < 1. \] For $n\leq 2\sigma$ or $1<p\leq \frac{n+2\sigma}{n-2\sigma}$, we generalize the monotonicity formula and Liouville-type theorem when $\sigma=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $\sigma=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

We study thermal transport across copper-graphene-copper interfaces using Non-Equilibrium Molecular Dynamics (NEMD), focusing on the influence of finite domain length and domain configuration, including lattice initialisation and associated graphene wrinkling, on the predicted thermal conductivity and Kapitza resistance. In the literature, NEMD simulations identified trends in the Kapitza resistance of graphene-copper interfaces. However, the simulation outcomes and reliability may depend heavily on configuration choices that are underexplored in the literature. We identify a strong sensitivity of the Kapitza resistance to domain configuration choices that affect the lattice constants and atomic density. We show that two conventional lattice initialisation strategies yield a factor of two difference in the Kapitza resistance, despite differences of only a few per cent in the lattice parameters. This behaviour is accompanied by strain-dependent shifts in the graphene and copper phonon spectra, and by increased phonon overlap at lower strain. Counter to conventional expectations, greater phonon-mode overlap coincides with higher Kapitza resistance, showing that spectral overlap alone cannot capture the interfacial heat-transfer dynamics. We suggest that in lattices initialised with lower residual strain, a damping boundary layer develops near the interface, and increases thermal resistance, as indicated by increased local structural disorder and local spectral broadening over a wider interfacial region. Beyond strain- and density-related effects associated with the lattice constants, Kapitza resistance shows no significant dependence on domain length or boundary temperature enforcement in this study. By contrast, the copper lattice conductivity exhibits clear domain-size and temperature dependence, consistent with phonon mean-free-path limitations and supported by phonon spectral analysis.

We introduce a modified Benamou–Brenier (MBB) formulation of optimal transport that incorporates Euclidean invariance at the dynamical level. We establish existence of minimizers for the resulting variational problem and prove its equivalence to a static formulation defining the Procrustes–Wasserstein distance. In the Gaussian setting, we show that this distance admits a closed-form expression, reducing to the Euclidean distance between the vectors of square roots of the ordered eigenvalues of the covariance matrices. On the computational side, we formulate a primal–dual scheme for the discretized problem. We prove a local conditional subsequential convergence result through an abstract analysis of a class of parameter-dependent saddle-point problems and illustrate the method's performance numerically.

We consider a fourth order singularly perturbed boundary value problem with two small parameters, in one dimension, under the assumption of analytic input data. We show that the solution may be decomposed into a smooth part, two different width boundary layers, and a negligible remainder. We provide estimates for arbitrary order derivatives of each term of the decomposition, which are explicit in the differentiation order and the singular perturbation parameters, and are needed for proving the convergence of high order numerical methods, such as the $p/hp$ versions of the Finite Element Method. We also provide classical differentiability results, which show that the solution will be analytic, if the data are analytic, but negative powers of the singular perturbation parameter(s) show up once we start differentiating.

Treewidth is the standard measure for how “tree-like” a graph is. This paper studies how the treewidth of a product graph depends on the treewidth of its factors. Kozawa, Otachi, and Yamazaki [2014] and Hickingbotham and Wood [2025] independently showed that $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)\text{had}(H)-1$ for all graphs $G$ and $H$, where $\text{had}(H)$ is the Hadwiger number of $H$. We improve this bound to $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)(\text{tw}(H)+1)-1$, thereby solving an open problem of Hickingbotham and Wood. We also prove analogous product inequalities for pathwidth, Cartesian products, and strict bramble number, which is a parameter that is tied to treewidth. As an application of our results, we show that products of expanders have large subgraphs that are expanders.

The classical Ingleton inequality is known to hold for entropic points under specific exact conditional independence constraints. Recently, Matveev and Romashchenko (2026) investigated the stability of these implications, quantifying the extent to which the Ingleton inequality can be violated when a group of conditional mutual information terms is small but non-zero. While their proofs relied fundamentally on the complex framework of tropical probability spaces, we revisit these stability results using a completely tropicalization-free approach. By developing an alternative framework, we significantly streamline the underlying concepts and proofs, derive explicit error terms, and improve some estimates. Furthermore, we resolve an open problem posed in prior work by exhibiting a new infinite family of entropy inequalities that establishes the stability of the sum of two Ingleton expressions.

We establish absolute and relative almost-linear Zarankiewicz bounds for semilinear relations in valued vector spaces. For every fixed arity and description complexity, a $K_{t,\ldots,t}$-free semilinear $r$-partite hypergraph has at most \[ O\!\left(n^{r-1}(\log n)^c\right) \] edges, where $c$ depends only on the arity and the number of valuative literals. In the bipartite case a separate arbitrary-trace argument gives the explicit bound $O(n(\log n)^{2s})$ for description complexity $(\rho,s)$. We also prove a relative extension theorem: intersecting any relation with a hereditary almost-linear profile by $s$ affine moving-radius comparisons increases the logarithmic exponent by at most $2s$. For the additive affine-valuative structures on $\mathbb Q_p$ and $\mathbb C_p$, quantifier elimination converts these semilinear results into bounds for all definable relations. Finally, over every valued field with infinite value group, we construct $K_{2,2}$-free semilinear point–box graphs of description complexity $(1,4)$ with $\Omega(n\log n/\log\log n)$ edges.

We consider the nearest-neighbor Edwards–Anderson Ising model on cubic boxes with random perturbations supported at the boundary. We prove that any perturbation admitting an energy envelope that is negligible compared with the volume, both in expectation and almost surely, leaves the limiting quenched specific free energy unchanged. For independent finite-variance bulk disorder in dimension at least two, an Efron–Stein estimate also yields almost-sure self-averaging along the full sequence of boxes. The hypotheses are verified for i.i.d. scalar surface fields, fixed random exterior spins, and periodic wrap-around bonds, with an $ O(L^{-1}) $ comparison of quenched means. The result concerns the specific free energy and does not assert convergence of finite-volume Gibbs measures.

We study the existence of solutions for systems of both elliptic and parabolic partial differential equations with potentially singular right-hand sides and free boundaries, posed in general smooth domains. Our results are established within a framework of "meta-theorems." This approach hinges on specific strong properties of the operators and their solutions (in approximate smooth settings) to guarantee the existence of a limit as the approximation parameter tends to zero. The primary challenge lies in applying these meta-theorems to prototype cases, which requires verifying that the necessary strong properties hold. For our analysis, we focus on fully nonlinear and $p$-Laplacian operators and a mixing of these operators, in both elliptic and parabolic contexts. While we focus on these specific cases, the meta-theorems remain valid for any other operators that satisfy the required properties. Beyond the complex proofs of our meta-theorems, and their applications to specific operators, a major challenge is the technical handling of the $p$-parabolic case, which requires proving the regularity of solutions of the $p$-parabolic equation with singular or degenerate right-hand side–addressed in the Appendix–along with several (new) properties, which is missing in the literature.

We study continuous planar facility location in which a demand point's captured value decays with distance, with the per-point decay scale varying across points. This heterogeneity is ubiquitous yet underexploited, and one nearest-facility objective unifies decay, clustering, and median goals, containing k-means, the Weber/p-median problem, and maximum covering as special cases. We make four contributions. (i) A tractability classification: the discrete objective is always monotone submodular, so the (1-1/e) greedy guarantee holds regardless of decay shape or heterogeneity, and the continuous cooperative objective is concave if and only if the decay is concave in distance; the clip max(0,d) in common coverage specifications is what destroys concavity, and the classification is tight. (ii) An exact discrete method: the candidate-discretized maximum-cover MIP has an empirically tight LP relaxation (~0% gap) and is solved by branch-and-bound in seconds for n <= 500. (iii) A force-as-gradient / large-neighborhood-search heuristic, within 0.5% of the discrete optimum, that outperforms the (1-1/e) greedy, Cooper-style alternating location-allocation, particle swarm optimization, and weighted k-means (30/30 per-instance wins at K=30, p<10^-9) and is competitive with bespoke solvers on k-means, Weber/p-median, and shape-demand instances. (iv) A real-world study: on 592,667 urban-delivery orders, ignoring the calibrated decay variation loses up to 9.7% of captured demand and relocates facilities by up to 37% of the map; a retail dataset calibrates the decay as exponential with scale R ~ 1.4 km.