Papers

We prove several new results in the theory of $\mu$-AECs, focusing mainly on (almost) stability, with the primary objective of undertaking a systematic study of $\mu$-AECs of $R$-modules. Our main results are the following. 1. We show that, under suitable syntactic assumptions, all tame $\mu$-AECs of $R$-modules (where $R$ is a ring) are almost stable, and are stable if they additionally satisfy a strong amalgamation property. This extends the work of the second author and Shelah [49] to the setting of $\mu$-AECs. 2. We then turn to applications to concrete $\mu$-AECs of $R$-modules. Our main result in this direction is that $(R$-Mod$, \leq_{pp}^\mu)$ has a stable independence relation and is a stable and tame $\mu$-AEC, where $\leq_{pp}^\mu$ denotes the $\mu$-pure submodule relation. We also prove similar stability results for various classes of abelian groups, including the $\aleph_1$-AEC of torsion-free abelian groups with the balanced subgroup relation. Moreover, we prove the almost stability of all $\mu$-AECs of modules of the form $(R$-Mod$, \preccurlyeq)$, where $\preccurlyeq$ refines the direct summand relation and satisfies a strong form of coherence. 3. Finally, we study $\mu$-AECs of the form $(K, \leq_\oplus)$, where $K$ is a class of pure-injective $R$-modules (note that this is, in general, not an AEC), and use our results to show that, for many natural choices of $K$, the class $(K, \leq_\oplus)$ has a stable independence relation and is therefore stable and tame. We use these results to give a sufficient condition for abstract classes of modules of the form $(K, \leq_{pp})$ to be stable when $K$ is closed under pure-injective envelopes. This generalizes, by a substantially different proof, results of Mazari-Armida [45].

We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions $\overline{U}$ for which both $\overline{U}$ and $\beta \overline{U}$ are self-similar profiles for some nontrivial $\beta$. Homothetic solutions are, in addition, the only solutions for which a singular limit argument can be used to prove non-uniqueness of Leray-Hopf solutions along the lines of the Jia-Šverák program. In three dimensions, and for sufficiently regular initial data, we prove a Liouville theorem that rules out the existence of non-trivial homothetic solutions. In two dimensions, our Liouville theorem reproves a result of Luo and Shvydkoy that the only decaying homothetic solution with continuous angular data is the Oseen vortex. In addition, we prove that the Euler equations linearized around the Oseen vortex are stable in some subspaces of $L^2(\mathbb{R}^2)$, for example in the space of $2$-fold rotationally symmetric functions. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler equations exists.

Every human brain folds differently, and such natural variation confounds the search for imaging biomarkers of neurodevelopmental disorders. Physics-based simulation can help determine the causal mechanisms that underpin this variability. Yet every simulation must be initiated from a volumetric mesh of the brain's interior, tetrahedral or hexahedral, and it is the worst element in that mesh, not the average, that decides whether the simulation runs at all. Building that mesh from fetal MRI currently requires labour-intensive manual intervention. We therefore present CORTET (CORtical TETrahedral meshing): a fully automated pipeline that converts a triangulated cortical surface into a solver-ready tetrahedral mesh whose worst-element quality meets a strict quality target with no manual repair. By benchmarking against a general-purpose tetrahedral mesher on the same input surfaces, we isolate the pipeline's contribution from that of the input geometry, and we validate quality across a cohort of nearly 200 fetal subjects spanning the folding period. A mesh taken straight from the pipeline sustains a numerically stable morphoelastic folding simulation of a real fetal subject.

We introduce fkcompute, an open-source package for computing the Gukov–Manolescu invariant of links from a braid presentation. fkcompute implements Park's inverted state sum through a three-phase pipeline. First, a search is performed for a suitable braid presentation and for an additional inversion data on the braid. Then, the state space of the inverted sum is encoded as a polytope, bounded by the associated linear constraint system. Finally, the invariant is constructed by multiplication of R-matrices associated to the states. Benchmarks show that prime knots up to 12 crossings, and prime links up to 10 crossings and of at most 3 components, are comfortably within reach. As a result, fkcompute is used to compile the first public database of the Gukov–Manolescu invariant. The package is available as a Python library, a command-line tool, and a Mathematica paclet.

We study the interaction between integral structures, automorphisms, and tautological relations for the cohomology of character varieties. Based on this, we propose a method to detect differences in the homotopy types of Galois conjugate character varieties. As an application, we find the first example of a pair of Galois conjugate character varieties that are not homotopy equivalent, answering negatively a 2005 question of Hausel.

Splitting methods are among the most classical and fundamental tools for the simulation of quantum dynamics, and their importance has grown further with the rise of quantum computing. In this work, we analyze the Schrödinger equation with Yukawa potential, a physically relevant and widely used model potential. It may be viewed as a Coulomb interaction with exponential decay at spatial infinity, preserving the Coulomb singularity at the origin while removing the long-range Coulomb tail. We prove that the operator splitting for this unbounded Hamiltonian achieves a global $1/4$-order convergence rate in the time step for many-body Yukawa interactions, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in $H^2(\mathbb R^{3N})$, the natural domain of the Hamiltonian, and our numerical experiments are consistent with the theoretical estimates. To identify the sharp obstruction behind this rate, we prove a short-time lower bound in the one-body setting of order $t^{5/4}$ for the one-step error, which rules out any uniform global estimate of order better than $1/4$ in general. This agreement with the optimal $1/4$ rate in the Coulomb case is particularly interesting, as Yukawa potential is short-ranged compared to Coulomb potential. For the many-body upper bound, one of the new technical ingredients is the explicit polynomial-in-system-size Sobolev estimates of many-body Yukawa systems. These estimates are crucial for obtaining fully a priori bounds that depend only on the norms of the initial states, rather than on the solution at time $t$. For the one-body lower bound, we leverage a new analysis argument based on Fourier analysis and Kato smoothing.

Guessing random additive noise decoding (GRAND) can efficiently decode any moderately redundant code with near maximum likelihood (ML) performance via noise effect guessing. For binary linear codes, Rowshan and Yuan's Segmented GRAND was the first to show that constrained guessing can reduce guesswork. Although powerful, their approach requires a specific parity-check matrix structure that limits the number of constraints that can be exploited as well as the class of applicable codes. Here we introduce GSegGRAND, a generalization of Segmented GRAND that circumvents its limitations. Built on a novel parity check structure and a transformation that maps codes into this structure, GSegGRAND efficiently incorporates up to log2(n) constraints for a wide range of codes, reducing guesswork by an additional 75% over Segmented GRAND. To leverage that advantage for soft-output decoding, we derive an accurate soft-output (SO) equation for GSegGRAND by extending soft-output GRAND (SOGRAND) to incorporate constrained guessing. Applying this SO to turbo product decoding, GSegGRAND achieves up to 88% guesswork reduction, making it a promising candidate for low-latency decoding in future communication systems.

We develop explicit computational tools for the recent homological approach to the construction of $2$-dimensional integrable field theories on $\Sigma$ from $4$-dimensional semi-holomorphic Chern-Simons theory on $\Sigma \times C$. In this framework, the operation of integrating out the spectral curve $C$ is realized by homotopy transfer of a cyclic $L_\infty$-algebra associated with the $4$-dimensional theory with prescribed singularities and boundary conditions. We construct explicit strong deformation retracts for divisor-twisted Dolbeault complexes on $C=\mathbb{C}P^1$ and use them to make the transferred $L_\infty$-structure computationally accessible. As an application, we study the choice of meromorphic $1$-form corresponding to the principal chiral model with a Wess-Zumino term. We compute the transferred Maurer-Cartan action and the associated Lax connection, showing that the former resums to the standard principal chiral model action with a Wess-Zumino term and that the latter reproduces the usual Lax connection.

Let $G$ be a finite group. Using a new kind of operad, we axiomatize and explore an infinite loop space machine that constructs (genuine) $G$-spectra from suitably structured functors on orbital presheaves, which are just contravariant functors from the orbit category of $G$ to based spaces. The theory leads unexpectedly to a new operadic description of Mackey functors and hence to a definition of “topological Mackey functors" and a construction of their associated $G$-spectra. It also leads to Picard $G$-spectra, Azumaya ring $G$-spectra, and Brauer $G$-spectra. These constructions raise many unanswered questions.

Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood. We consider systems of causal SDEs that are equipped with an explicit causal semantics. We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions. As core results we establish the $\sigma$-separation Markov property and the do-calculus in terms of the system's causal graph for probabilistic independence and interventions on the level of sample paths. For a class of additive-noise SDEs we prove a stronger $d$-separation Markov property, even if the system is cyclic. As a corollary of the do-calculus, we obtain an explicit causal interpretation of the graph: that the absence of a directed path implies the absence of a causal effect. We further introduce time-split systems, which consider the causal relations between the processes when evaluated on disjoint intervals or time-points, and use them to reason about subsampled time-series, continuous-time Granger non-causality and local independence. Finally, we discuss how constraint-based causal discovery algorithms (PC, FCI, CCD, CCI) apply directly to SDEs within our framework when conditional independence between sample paths can be consistently tested.

We study the automorphism groups of complete intersections of hypersurfaces of strictly increasing degrees in projective space. Under a combinatorial rigidity condition on the tuple of defining polynomials, we show that every automorphism of the complete intersection extends to an automorphism of each defining hypersurface, so that its automorphism group is the intersection of the automorphism groups of the defining hypersurfaces. We apply this principle to two natural families of complete intersections of two hypersurfaces of different degrees. For complete intersections of two Fermat hypersurfaces, we determine the automorphism group in every smooth case. For complete intersections of a Klein hypersurface with the reverse-order Klein hypersurface, we describe the automorphism group under an explicit arithmetic condition relating the two degrees, with Klein hypersurfaces of Wagstaff type as a natural source of examples.

Finite-element mass matrix solves and approximate inverses arise often in explicit time integration as well as Schur-complement-based block preconditioning. A diagonal approximation of the mass matrix is cheap and widely used, but can be a poor approximation, particularly for high-order elements. This paper introduces a broken-space additive Schwarz (BRAS) mass inverse approximation, formed by applying exact element-local inverse mass matrices on the broken finite-element space and averaging the result back to the conforming space. The construction uses the same element matrices and local-to-global maps as standard mass assembly, has the same element-adjacency sparsity graph as the conforming mass matrix, and is symmetric positive definite for any conforming space, mesh geometry, and polynomial basis. We prove spectral bounds for the preconditioned mass matrix, and wide-ranging numerical experiments for \(H^1\), \(H(\operatorname{curl})\), and \(H(\operatorname{div})\) finite elements on two- and three-dimensional simplicial meshes show that BRAS reduces spectral condition numbers, Krylov iterations, and solve times relative to diagonal preconditioning. For preconditioned conjugate gradient (CG), BRAS yields a 1.1–4.7$\times$ speedup over diagonal/Jacobi preconditioning across all cases tested over finite-element orders $p\in[1,4]$. Further, theory and numerical experiments show that in the block-preconditioning case, BRAS can improve Schur-complement approximations and reduce outer solve times. On mixed Poisson and biharmonic systems, BRAS yields a 1.5–3$\times$ speedup in time-to-solution over a standard diagonal-based preconditioning approach.

We prove that there exist functions $f:\mathbb N^2\to\mathbb N$ and $g:\mathbb N\to\mathbb N$ such that for all positive integers $k$, $d$, and $\ell\ge3$, every graph $G$ either contains $k$ cycles of length at least $\ell$ that are pairwise at distance greater than $d$, or admits a subset of vertices $X$ with $|X|\le f(k,\ell)$ such that $G-B_G(X,g(d))$ contains no cycle of length at least $\ell$, where $B_G(X,r)$ denotes the ball of radius $r$ around $X$. This generalizes a theorem of Dujmović, Joret, Micek, and Morin (2024), which established the $\ell=3$ case. Moreover, we prove that the theorem holds with $f(k,\ell)\in\mathcal{O}(\ell k\log k)$ and $g(d)\in\mathcal{O}(d)$. The linear bound on $g$ is best possible, while the bound on $f$ is optimal as a function of $k$ for every fixed $\ell$. In particular, for $\ell=3$ our result improves the previous bound of $\mathcal{O}(k^{18}\mathsf{polylog} k)$ by Dujmović et al.

This paper investigates the control of discrete-time linear time-invariant (LTI) systems subject to incomplete and corrupted measurements. Specifically, we focus on designing a Linear Quadratic Gaussian (LQG) controller without relying on explicit state estimation. By leveraging minimum variance duality, our approach allows the current control input to be represented as a linear function of available measurements and previously applied inputs, successfully reducing the task to a tractable deterministic optimization problem. We provide theoretical justification for this framework and demonstrate its practical effectiveness through numerical experiments.

A separable bilinear layer relaxation assigns a population cost to each layer and a nonpositive bilinear coupling to each pair of layers, then excludes a profile span whenever the relaxed minimum exceeds a reference value. We ask what a Kuznetsov-Sahinidis-type relaxation can certify at best. Let gamma = g(1) be the singleton-layer cost. For every valid relaxation with gamma > -1/2, every reference bounded per particle, and every underlying system satisfying two elementary low-energy witness conditions, we prove that the certified deficit N - rho(N) is at most of order sqrt(N), with an explicit bound depending only on the reference bound and gamma. Thus no relaxation in this class can certify a larger asymptotic deficit, independently of the detailed layer cost, interaction kernel, or reference. The threshold is sharp in the abstract witness class: at gamma = -1/2 we construct a valid relaxation with a bounded sound reference and deficit N - 1. The geometric Lennard-Jones system behaves differently. A three-atom chain restores the square-root ceiling at the endpoint, while sufficiently negative singleton costs allow valid relaxations with linear deficit. For still lower costs, beyond the stability constant, no sound relaxation excludes any span. These regimes define a certification spectrum for layer relaxations and reveal a reentrant transition between no exclusion, linear exclusion, and the universal square-root ceiling.

The $p$-value and the Bayes factor are measures of evidence that are often considered to be philosophically and mathematically incompatible: The $p$-value quantifies conflict between data and $H_0$ ("surprise"), whereas the Bayes factor quantifies the relative predictive accuracy of $H_0$ versus $H_1$ ("evidence"). We revisit Jeffreys's Approximate Bayes factor (JAB) – a simple, largely overlooked approximation dating back to the 1930s – which connects these two paradigms for objective hypothesis testing of the existence of an effect. Under a unit-information prior the approximation requires only the $p$-value and the effective sample size $n_\text{eff}$. We clarify the core assumptions and boundary conditions for the application of JAB and show across 704 published $t$-tests and 39 comparisons of proportions that JAB approximates objective Bayes factors remarkably well. The connection between $p$-values and JAB has a practical implication: The evidence implied by a $p$-value depends strongly on $n_\text{eff}$. Conventional verbal labels for $p$-values (e.g., "strong surprise" for .001 < $p$ < .01) correspond to similarly graded Bayes factors only around $n_\text{eff} \approx 8$; for larger samples the same $p$-value implies weaker evidence. In moderately sized to large samples, $p > .10$ can amount to moderate or even strong evidence for $H_0$. JAB offers a cheap, sample-size-sensitive supplement to $p$-values, computable from routinely reported statistics, that remains valid even under optional stopping.

Strongly regular signed graphs are an extension of strongly regular graphs to the realm of signed graphs, that is, graphs where each edge is positive or negative. Unlike with ordinary strongly regular graphs, most kinds of signed counterparts with girth 4 or higher are describable in terms of known structures. We prove that those with girth 4 that are bipartite are classified by designs of two kinds: weighing matrix designs and symmetric block designs. Those of girth 5 are few and readily described. There are none of higher girth. Those with girth 4 that are not bipartite are unsolved.

This paper investigates foundational quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category FHilb. While conventional Hilbert-space formalisms often conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework directly encodes algorithmic functionalities into their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein–Vazirani and Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanisms of oracles. Going beyond the standard qubit model, we construct explicit representations for the qutrit-adapted topological Deutsch–Jozsa and single-shot Grover algorithms. In particular, we establish a necessary and sufficient condition for the single-shot Grover search. We further implement CNOT gates via complementary Frobenius structures and investigate a diagrammatic decomposition scheme for the W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes a composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem.

We give an axiomatic homotopical analog of the classical categorical Beck monadicity theorem. It often holds when classical monadicity fails. This grew out of an understanding of a general context for recognition principles in iterated loop space theory, as treated in the logical sequel ArXiv 2402.03649, but the present result applies differently and more generally. An example gives a new perspective on the old equivalence between simplicial sets and topological spaces: both are equivalent to simplicial topological spaces, and the equivalence implies a curiously close relationship between realizations of simplicial spaces and realizations of their underlying simplicial sets, viewed as discrete simplicial spaces.

We define the notion of an exchange theorem and show that any two functors satisfying an exchange theorem are canonically related via twisted norm maps. This is done by identifying the universal category receiving a pair of functors satisfying an exchange theorem. Additionally, we show that the twists occurring are $K$-theoretic in nature, parametrized by a categorified analogue of virtual vector bundles. As an application, we show that every 3-functor formalism has a canonical extension which encodes Poincaré duality and Thom twists internal to the formalism. This gives a 1-categorical realization of the "coherent six operations" outlined by Hoyois. In the process of proving universality, techniques for computing categories associated to bi- and $n$-simplicial spaces are developed. Many of the results in this direction may be viewed as model-independent rederivations of work of Liu–Zheng.

We prove Kleber's rectangular-complement conjecture for Schur functions over an arbitrary commutative ring $R$, showing that, for a fixed rectangle, the products $s_\lambda s_{\lambda^\vee}$, indexed by unordered complementary pairs, are linearly independent in $\Lambda_R$. The proof rests on a general independence theorem for componentwise splittings, which asserts that for every partition $\theta$, the products $s_\alpha s_\beta$ are linearly independent as $\{\alpha,\beta\}$ ranges over unordered pairs of partitions satisfying $\alpha+\beta=\theta$. The independence of the products $s_\lambda s_{\lambda^\vee}$ also yields linear independence of the Koike–Terada universal-character products over any field, answering a question of Gao–Orelowitz–Yong. We also prove the analogous result for monomial symmetric functions over fields of characteristic zero, as well as integral linear independence over $\mathbb{Z}$.

This article describes a novel numerical algorithm for tracking the motion of a collection of open curves in two space dimensions. The proposed strategy combines two complementary representations of these curves at each iteration of the evolution process: on the one hand, they are meshed explicitly, as a sub-collection of the entities of a mesh of the total computational domain. Concurrently, using a variant of the Level Set Method, they are captured implicitly as algebraic combinations of the negative, zero and positive subsets of two auxiliary scalar functions, defined on the whole ambient space. This coupling of representations allows to perform accurate geometric or mechanical computations on these curves, while leaving room for large evolution of their shape. After the description of its main numerical ingredients, several applications examples of this methodology are proposed, where it is used to simulate the motion of open physical discontinuities, such as a vortex sheet roll-up, and to optimize the shape of open-ended curves, e.g. with respect to their anisotropic length, with the aim to improve the trajectory of a laser acting on a powder bed in the context of additive manufacturing, or to reconstruct fracture sets.

This article establishes a new numerically viable technique for solving a class of constrained, nonconvex, continuous-time optimal control problems (OCPs) for linear systems that commonly arise in aerial and aerospace applications. The lossless convexification technique is employed to translate the original nonconvex OCP with annular control magnitude constraints into a convex problem, and then by finitely parametrizing the control space with piecewise constant functions, an efficient numerical approach is established that guarantees exact solutions while ensuring the satisfaction of an uncountable family of constraints over a compact time interval. The effectiveness of the approach is demonstrated on a spacecraft landing problem involving three degrees of freedom (DoF), underscoring its potential for real-world aerospace guidance and control tasks.

We introduce \verb|LipschitzSaturation|, a package for the computer algebra system Macaulay2 that implements algorithms for computing Lipschitz saturations of modules and toric varieties. In the module setting, the package handles three distinct saturation notions for an $\mathcal{O}_{X}$-submodule $\mathcal{M}\subseteq\mathcal{O}_{X}^{p}$: the 1-, 2-, and 3-Lipschitz saturations $\mathcal{M}_{S_{1}}$, $\mathcal{M}_{S_{2}}$ and $\mathcal{M}_{S_{3}}$, together with the auxiliary construction of the double module $\mathcal{M}_{D}$. To bypass the computationally intractable multivariate calculations for $\mathcal{M}_{S_{1}}$, we implemented a curve-based membership test, achieving near-constant runtime on parametric families that cause the purely algebraic method to time out or exhaust memory. For Toric Singularities, we implemented a construction algorithm. The package is freely available and requires Macaulay2 version 1.22 or later.

In this work, the Maxwell–Cattaneo–Vernotte (MCV) equation is used to model the one-dimensional hyperbolic Stefan problem in the limit of a small Stefan number (Ste $\ll$ 1). The solutions are approximated with perturbation series expansions using a reformulation in which time is expressed as a function of the solid-liquid interface position. The first proposed solution is derived in a framework that considers diffusive heat transfer at the phase change interface, for analytic tractability. Two rectification strategies are proposed to address the asymptotic divergence present in this formulation: a rescaled inner solution which is then combined with the outer solution to yield a composite solution, and size-dependent thermo-physical system parameters for better capture of hyperbolic effects at the phase change interface. The resulting interface profiles exhibit a characteristic parabolic-like shape, consistent with diffusive Stefan problem findings, with pronounced early-time hyperbolic effects at larger thermal relaxation times. Parametric studies are done over three pertinent variables in the dimensionless system: the Stefan number ($\mathrm{Ste}$), the dimensionless thermal relaxation time ($\widetilde \tau$), and the thermal diffusivity ($\alpha$). The studies suggest that model error scales with the Stefan number in accordance with the theoretical truncation error of the perturbation expansion. Additionally, larger values of $\widetilde \tau$ amplify early-time hyperbolic effects, thereby increasing model error, while larger $\alpha$ extends the relative temporal domain over which these hyperbolic effects remain significant, also corresponding to an increase in model error.

An ideal $\mathcal{I}$ on $\omega$ has the Ramsey property if $\mathcal{I}^{+}\to(\mathcal{I}^{+})^2_2$: every $2$-colouring of the pairs of an $\mathcal{I}$-positive set has an $\mathcal{I}$-positive homogeneous subset. Whether a tall Borel ideal can have the Ramsey property is an open question of Hrušák, Meza-Alcántara, Thümmel and Uzcátegui; a coanalytic example exists in ZFC, so a negative answer must use definability essentially. Our main theorem, a synthesis of the results of the paper, states that a tall Borel Ramsey ideal admits no countable local reading. Below every positive set, such an ideal is not a countable intersection of topologically represented (or tall analytic $P$-) ideals, and its quotient has no countable dense subset. Moreover, every quotient name for a new real has uncountable width, the colouring witnessing non-selectivity of the generic ultrafilter is never read continuously on a positive condition, and hereditary tall subfamilies saturate every finite window of barrier dimensions coherently but never all dimensions at once. We prove separately that a weakly selective $q^+$ ideal admits no positive $\mathcal{ED}_{\mathrm{fin}}$-carrier. The converse question – must Borelness force a properness-like countable reading on some positive condition? – is stated in three precise forms with proved consequences: two of them would refute tall Borel Ramsey ideals outright, the third the strictly weaker Nash–Williams class. The main question remains open.

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, \chi)$ where $T$ is a tree and $\chi : V(T) \rightarrow 2^{V(G)}$ is a function satisfying two axioms: for every edge $uv \in E(G)$ there is an $x \in V(T)$ such that $\{u,v\} \subseteq \chi(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) | u \in \chi(x)\}$ induces a non-empty and connected subtree of $T$. The sets $\chi(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the decomposition. A graph $H$ is an induced minor of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by vertex deletions and edge contractions. We prove that for every $t\in\mathbb{N}$ there exists an $\epsilon > 0$ such that every graph $G$ either contains the complete bipartite graph $K_{t,t}$ or the wall $W_{t\times t}$ as an induced minor, or has tree-independence at most $O(2^{O((\log n)^{1-\epsilon})})$. This leads to algorithms with running time $2^{n^{o(1)}}$, for a wide range of problems on $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs. Our result is a substantial generalization of existing bounds for the tree-independence and tree-width on various graph classes, and a partial resolution of the conjecture of Chudnovsky, E S, and Lokshtanov [Arxiv, 2025] that $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs have poly-logarithmic tree independence number. The generality comes at the cost of a sub-polynomial, rather than poly-logarithmic upper bound. Our result leads to a complete classification of induced-minor closed classes into ones that have sub-polynomial tree-independence, tree-independence equal to $\tilde{O}(\sqrt{n})$, and linear tree-independence.

We study exponentially damped Möbius approximants in $\mathscr H=L^2([1,\infty),dt/t^{-2})$. With \[ \gamma_n(t)=\left\lfloor\frac tn\right\rfloor -\frac{\lfloor t\rfloor}{n},\qquad f(u)(t)=\sum_{n\ge1}\mu(n)e^{-nu}\gamma_n(t),\] we compute the relevant scalar products, characterize the Möbius coefficients as the unique coefficients giving pointwise convergence to the constant function, and prove $\langle1 \mid f(u)\rangle\to1$. Vasyunin's formula expresses $F(e^{-u})=\|f(u)\|_2^2$ as an arithmetic cotangent sum. To analyze $F(x)$ as $x\uparrow 1$, we define the canonical third-order truncation $\mathcal F_{[3]}$ of $F$ by deleting the sole remainder $\rho_3$. We prove exact edge and residue-character cancellations, initial-edge asymptotics, finite-scale formulas, and \[ \mathcal F_{[3]}(x)\ll \frac{\log^2\!\bigl(e/(1-x)\bigr)}{1-x}. \] For the terms containing $\rho_3$, we prove initial-edge asymptotics, and a finite-scale criterion. The unresolved boundedness problem is thereby reduced to explicit global bilinear cancellation.

We introduce a tangent-space multiscale manifold method for nonlinear heterogeneous elliptic problems. The method represents the fine-scale solution by a nonlinear reconstruction of a coarse state. The ideal reconstruction eliminates fine scales through a constrained variational problem, and the computable reconstruction approximates this map by localized nonlinear patch solves blended with a partition of unity. Because the approximation set is a nonlinear manifold, the coarse equation is posed with tangent multiscale test functions. We also formulate a network-interpolated variant in which only the restricted patch outputs used by the partition-of-unity blend, together with their tangent actions, are approximated by local learned maps. For heterogeneous monotone nonlinear diffusion, we record the structural monotonicity, differentiability, patch-map regularity, and conditional perturbation estimates that separate the geometric stability mechanism from localization, residual, and optional learning defects. A rigorous a priori theory for the decay of the localization defect, and the resulting convergence rates in the coarse mesh size, is deferred to a separate analysis; here these defects are controlled conditionally and their decay is demonstrated numerically.

We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\dots,v_n \in \mathbb{R}^k, $ we prove \[ \operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}. \] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every radial simplex with the Gaussian solid angle of its positive cone, and then use that the radial cones form a complete fan. As a consequence, the volume of the projection of $\crosp^n$ onto any $k$-dimensional subspace is at most $2^k/k!$, with equality only for coordinate subspaces.