Papers

Soft Actor-Critic (SAC) is widely used for entropy-regularised reinforcement learning with continuous action spaces, and practical implementations perform only a few actor steps towards an evolving target. In this work, we prove convergence guarantees when the target policy arises from policy mirror descent and compare it with the classical Gibbs target. We derive sufficient conditions for the strong convexity and smoothness of the actor objective, characterised by the curvature of the $Q$-function estimate through the Legendre differential operator, and establish an $\mathcal{O}\!\left(N^{-\frac{1}{5}}\right)$ best-iterate finite-time convergence rate up to actor and critic approximation errors. Moreover, the mirror-descent step size $\lambda$ directly controls the target drift and hence actor tracking error, whereas the analogous Gibbs bound contains a non-vanishing tracking term.

We solve Conjecture 1.6 of Furio and Lombardo [On $7$-adic Galois representations for elliptic curves over $\Q$, Proc.\ London Math.\ Soc.\ (3) 133 (2026), e70193.]. As a consequence, we complete the classification of $7$-adic Galois images for non-CM elliptic curves over $\Q$. This is the first solved case in Problem 6.2 from Balakrishnan [Chabauty and beyond: Explicit p-adic methods for rational points, Proceedings of the International Congress of Mathematicians 2026 - Volume 3: Invited Lectures (Sections 1-4), 321-341, 2026]. The proof is the result of a long AI-training period by the author from June 2026 to September 2026. The author has been working on this project alone without the help of any other.

Zhang and Yeung (IEEE Trans. Inf. Theory, 1998) established the first non-Shannon-type inequality that holds for all entropic vectors. Chan and Yeung (IEEE Trans. Inf. Theory, 2002) showed that there is a one-to-one relation between linear entropy inequalities and multiplicative inequalities involving cardinalities of subgroups of a finite group. The Shannon inequality admits a simple proof in the group theoretic setting, but a direct proof of the translation of Zhang–Yeung's inequality to the group-theoretic setting remained elusive. We resolve this open problem, giving a direct proof of the group-theoretic Zhang–Yeung inequality for cardinalities of subgroups of finite groups, using elementary group-theoretic and counting arguments.

Let C be a non-singular real algebraic curve of genus g. The real structure on C induces a real structure on the configuration space of n distinct unlabeled points on C. We prove that this configuration space is maximal (in the sense of the Smith-Thom inequality) if and only if C is maximal or the genus of C is zero and n is even.

Pólya and Szegő conjectured in 1945 that the disk achieves the largest electrostatic capacity among all planar sets with given logarithmic capacity. This question generalizes naturally to balls and Newtonian capacities in higher dimensions. The conjecture remains open. This paper establishes a conditional result: the conjecture would follow if among all lower dimensional sets in a horizontal strip with Neumann boundary conditions, the transverse component of the Dirichlet integral of the equilibrium potential could be shown minimal for the lower dimensional ball. As supporting evidence toward this latter property, the paper shows extremality of the ball for $L^p$-norms of the potentials, via Baernstein star-function symmetrization techniques.

The existence of orbifold structures on mapping spaces between orbifolds was established in previous work, notably by W. Chen and B. Chen-Du-Liao. In this paper, we construct an infinite-dimensional orbifold structure on the mapping space in a more geometric way, using bibundles (or Hilsum-Skandalis morphisms), based on Lerman's description of orbifold morphisms. We construct a proper étale (or Fréchet) Lie groupoid representing the mapping space in the $C^k$, $C^\infty$, and $W^{k,p}$ settings, and show that its Morita equivalence class is independent of the choices of groupoid presentations. Locally, the mapping groupoid is isomorphic to the action groupoid of a finite group, where the finite group is identified with the automorphism group of the corresponding bibundle. This provides a concrete geometric framework for further studies of mapping spaces and moduli spaces on orbifolds, with applications to Floer theory on symplectic orbifolds.

This paper studies low-altitude question answering (LAQA), in which distributed unmanned aerial vehicle (UAV) memories are aggregated at a ground server to answer questions about observations over a long horizon. Unlike conventional resource allocation based on sensing, communication, control, or computation metrics, LAQA requires an explicit measure of memory value. We propose a generative adversarial exam (GAE) that uses forward simulation to evaluate memory retrieval and exam scores to quantify memory quality. This enables the downstream QA value of candidate memories to be measured and optimized without accessing the internal mechanisms of the black-box captioning, retrieval, and reasoning pipeline. Building on this metric, we develop a memory-centric (MemCen) framework that jointly selects UAVs and allocates transmit power to maximize memory quality under communication constraints. In the noise-limited regime, we derive a QoM-aware capped water-filling law that explicitly connects task utility with physical-layer power allocation. We further develop penalty successive optimization (PSO) and learning to memorize (L2M) solvers. MemCen achieves QA accuracies of 92.4% and 84.0% in CARLA Town04 and Town05 under static and dynamic communication conditions, respectively. In real-world experiments, MemCen achieves 88.5% QA accuracy on the panoramic multi-agent system (PMAS) benchmark. Finally, UAV-to-robot-dog demonstrations further validate the practical utility of the acquired memories for environmental understanding and navigation.

We study classical and quantum indistinguishability of sums of independent random permutations and related transformations from permutations to functions. Let $G$ be a finite abelian group of order $N$, and let $\pi^k_+(x)=\pi_1(x)+\cdots+\pi_k(x)$ for $k\geq2$ independent uniform random permutations of $G$. We give a unified Fourier analytic treatment in which the construction is represented by its probability density and a distinguisher by its acceptance function, with the classical and quantum query models imposing different restrictions on the Fourier support of the latter. Classically, we obtain the bound $O_k(q/N^{k-1/2})$ for every $q<N$, and refine it below the birthday threshold to $O_k(q^2/N^k)$. In the quantum model, a simulation argument gives $O_k(N^{-(k-3/2)})$ for $q\leq(N-1)/2$, while Fourier interpolation gives concrete finite bounds up to $q\leq4N/15$ and the query-dependent bounds $O\left(\min\left\{N^{-1/2},q^3/N^2 + 1/N\right\}\right)$ and $O_k\left(\min\left\{q^3/N^k,N^{-(k-3/2)}\right\}\right)$, for $k=2$ and $k \geq 3$, respectively, throughout $1\leq q\leq(N-1)/2$. For $q = 1$, the first bound sharpens to $O(N^{-2})$. Over $G=\mathbb F_2^n$, a one-query Fourier attack matches the order of our one-query bound, while an $N/2$-query parity attack with advantage $1/2$ shows that our bounds reach the constant-advantage query threshold. We further study two variants of sum of permutations over binary vector spaces. First, we allow arbitrary surjective linear postprocessing, which includes truncation, and obtain classical and quantum bounds that retain the output-size dependence. Second, we analyse Dinur's variable-output single-permutation construction, $\mathsf{LXoP}$, for every fixed output width, and derive its classical and quantum security bounds; for one- and two-block outputs, we give concrete quantum security bounds.

We construct the first convex planar counterexample to the Schiffer and Pompeiu conjectures: a bounded strictly convex non-disc domain with real-analytic boundary carrying a nonconstant solution of $\Delta u+u=0$ with $u=1$ and $\partial_\nu u=0$ on the boundary. Together with the higher-dimensional constructions, this gives convex non-ball counterexamples in every dimension from two to eighteen, and in dimensions twenty and twenty-one, with real-analytic boundaries diffeomorphic to spheres and indicator Fourier transforms vanishing on the unit sphere. In dimensions $4$, $6$, $8$, $10$ and $14$, planar reductions through compact Lie algebras of rank two use Harish-Chandra's radial part formula and Kostant's convexity theorem. In dimensions $3$, $5$ and $7$, an axisymmetric formulation on the unit ball gives a quartic equation in weighted coefficient spaces. The exact two-sided inverse of the residual-corrected linearisation combines a Dirichlet Helmholtz inverse, a holomorphic boundary real-part problem and the material-derivative identity; it gives the planar Schiffer domain and, by one dimension-parametrised argument, the domains in dimensions $5$, $9$, $11$ to $13$, $15$ to $18$, $20$ and $21$. We also construct the first convex planar counterexample to Berenstein's conjecture: a strictly convex non-disc domain with real-analytic boundary carrying a sign-changing solution of $\Delta u+u=0$ with $u=0$ and $\partial_\nu u=1$ on the boundary. Planar existence for this problem is due to Colbrook, Sadeghi and Stepaniants. All existence proofs in this paper are computer-assisted.

This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like $\delta^{-\gamma}$ in the accuracy $\delta$. When $\gamma>2$, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on $\gamma$, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy $\delta$ by an unbiased estimator of it, whose variance $\sigma^2$ becomes a second, independently priced dial: the cost of one call drops from $\delta^{-\gamma}$ to $\delta^{2-\gamma}\sigma^{-2}$. Plain inexact gradient descent driven by that oracle reaches loss $\varepsilon$ at expected compute $\Theta(\varepsilon^{-\gamma})$ in the convex case, against $\Theta(\varepsilon^{-(\gamma+1)})$ for the same method run at a fixed accuracy: randomization buys a full power of $\varepsilon$. Under $\mu$-strong convexity the exponent halves, to $\varepsilon^{-\gamma/2}$, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.

This paper gives a readable and accessible version of the profile-construction part of OpenAI's manuscript "Finite Time Blowup for Navier-Stokes". We regard OpenAI's work as a major advance on the Navier-Stokes Millennium Prize Problem. The profiles are smooth and axi-symmetric. Inserting them into the Navier-Stokes equations gives a residual consisting of a divergence-form term and a remainder vanishing to infinite order in $1-t$ on fixed similarity sectors. The associated stress and radial shear satisfy the admissible cone condition. The stress need not be small. We introduce a new linear model that provides a clearer explanation of the inner core construction. The cancellation by oscillatory pulses will be treated in the companion paper "Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part II: Residual Correction via Oscillatory Pulses".

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible $\mathbb{F}_{\ell}$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$ is zero for all $n\geq0$ and all $i>\dim\operatorname{Supp}F$. In particular $\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$ for $i>\dim\operatorname{Supp}F$, which answers a question of Bhatt–Schnell–Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier–Mellin transform of a perverse sheaf. With $\ell$-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault–Kerz. The corresponding estimates for the completed transform with $\mathbb{F}_{\ell}$-coefficients are new and hold even when Hard Lefschetz fails.

Connecting mathematical formalism in open quantum systems to its underlying physics necessitates the notion of a dilation, a way to bridge stochastic dynamics with deterministic Schrödinger evolution on a larger space. Although dilations are well known in the literature for states, channels, and quantum combs, there is a substantial gap when it comes to the fully general setting of non-Markovian dynamics on a continuous interval. In an accompanying work, we introduce a continuous process tensor (cPT) framework wherein these processes are represented by vectors in multi-species bosonic Fock spaces of $L^2$ functions. This accommodates all multi-time statistics of a quantum system on an interval. Here, we present a corresponding dilation theorem and show that all cPTs can be approximated arbitrarily well by an explicit closed description with system and environment coupled by a bounded Hamiltonian. As well as its physical appeal and demonstrating completeness of the framework, we show that such regular representations actually underpin the formalism, defining a dense subspace of all cPTs that is easy to handle. This leads to a continuum-suitable version of Choi duality, and permits the rigorous treatment of continuum controls.

We consider the problem of minimizing the Ginzburg-Landau functional among $\mathbb{R}^n$-valued maps from the unit ball $B \subset \mathbb{R}^n$ with vortex boundary condition $u(x) = x$ on $\partial B$. We show that, for every Ginzburg-Landau parameter $\varepsilon > 0$, there exists a unique minimizer given exactly by the rotationally symmetric vortex solution in dimension $n\geq 2$.

We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.

Let $\mathcal{O}$ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of $\mathrm{GL}_n(\mathcal{O})$ modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of Gérardin's construction for $\mathrm{GL}_n(\mathcal{O})$. As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.

We study the problem of extracting reliable and interpretable linear models from nonlinear dynamical systems with periodic behavior. Dynamic Mode Decomposition (DMD) is a widely used data-driven method for approximating dynamics by a linear system and is commonly interpreted through Koopman operator theory. However, because standard DMD assumes linear evolution in the chosen observables, it can produce misleading results on nonlinear systems. Recent extensions address this limitation by modeling the dynamics as a linear system driven by a nonlinear forcing term, but their relationship to Koopman theory and their reliability remain unclear. We observe that the Koopman theoretical framweork allows for explicit, finite-dimensional, forced linear representations of dynamical systems, independently of the chosen observable space. The main question is therefore how to choose a useful, computable, and interpretable model for the forcing term, since forced linear representations are generally nonunique. In the framework considered here, this modeling question is related to the choice of a defect space: a complementary space that closes the action of the Koopman operator on the chosen observables. Existing forced-DMD approaches can be related to different modeling choices for this defect space, where the structure of the learned linear operator depends on the choice of the defect space. As an example of defect-space modeling for algorithm design, we introduce Correlation-Basis enhanced DMD (CB-DMD), a data-driven method whose goal is to learn an effective autonomous linear operator for periodic nonlinear dynamics. The forcing term is used to absorb nonlinear components that cannot be represented by the learned linear operator. Experiments on three nonlinear periodic systems demonstrate that CB-DMD yields more reliable linear models than standard DMD and related variants in the tested examples.

We show that an almost-everywhere upper bound $b\geq 1$ for the Hausdorff dimension of visible parts of all compact planar sets implies the upper bound $d-2+b$ for visible parts of all compact sets $K$ in $\mathbb{R}^d$, $d\geq3$. Inserting the sharp planar bound $b=3/2$ of Orponen and Rutar \cite{OrponenRutar2026} gives $\dim_H \mathrm{Vis}_e(K)\leq d-1/2$ for almost every $e\in S^{d-1}$. This would answer their Question 1.9 sharply in all dimensions.

We study the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $\mathrm{GL}_N(\mathbb{C})$ with general initial conditions. Building on earlier work [arXiv:2605.06429], we prove that, for every initial condition in the enhanced state space, the limiting dynamics are determinantal and derive an explicit double-contour formula for their extended multitime correlation kernel. The dependence on the initial state, including information not determined by the particle coordinates, is encoded by the associated Laguerre–Pólya entire function. We further extend the corresponding infinite-dimensional stochastic differential equation description to arbitrary initial configurations, including those with coinciding coordinates.

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $\omega=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(\tau)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(\tau)}=\sum_{m\ge0}\Phi_m(\tau)q_z^m \] defines a family $\{\Phi_m\}_{m\ge0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{\Phi_0\}\cup\{\Phi_m\}_{m\ge2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{\Phi_m\}_{m\ge0}$ with the generating function of the Niebur–Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.

We study the supervised classification problem for interacting particle systems (IPS) from trajectory observations. The IPS belongs to one of K classes, each characterized by a distinct interaction drift. Given a learning dataset, the goal is then to predict the class label of a newly observed system based on the trajectory of a single particle. This setting raises several statistical challenges. First, the particles within each system interact and are therefore dependent. Second, although the full particle system is Markovian, the trajectory of a single particle is not. To address these difficulties, we exploit the McKean-Vlasov limit of the IPS, which describes the dynamics of a typical particle as the number of particles tends to infinity. We propose a plug-in classification procedure based on estimating the interaction drift associated with each class from discretely observed trajectory data. Hence, for each class, we provide a new nonparametric estimator of the drifts by minimizing a ridge-regularized least-squares contrast over a B-spline basis. In particular, our theoretical findings reveal that the convergence rate of the resulting classifier is of order N -1/6+$\epsilon$ for any $\epsilon$ > 0, where N denotes the number of particles in each system. Numerical experiments illustrate the performance of the proposed method and show that it outperforms an end-to-end neural network baseline that does not exploit the underlying particle-system structure.

Let $\lambda_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $\varphi_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $\varphi_k(G)\le C\sqrt{\lambda_k(G)\log(k+1)}$ for an absolute constant $C$. The number of sets and the spectral index are both $k$, and conductance is measured in the original graph. The logarithmic dependence is optimal up to an absolute constant. The proof combines geometric partitioning of the spectral embedding with minimum-cut improvement and adaptive projections in coefficient space. A single conductance threshold is used throughout the construction. The resulting maps have disjoint supports, and the sum of their Gram matrices is bounded below by an absolute positive multiple of the identity. A dyadic maximal estimate bounds the sum of their internal energies uniformly over unit coefficient vectors. A dimension argument using local eigenvalues then yields exactly $k$ disjoint sparse cuts.

We investigate the local stability of direct Mach configurations for the two-dimensional steady compressible Euler equations. Given a piecewise constant Mach configuration that consists of an incident shock, a reflected shock, a Mach stem and a slip line, with subsonic downstream states, we prove the following: under a transversality condition on the shock polar loops and a suitable condition on the truncation segment, every sufficiently small perturbation of the incoming flow gives rise to a Mach configuration that is a small perturbation of the background one, in the sense that the reflected shock, the Mach stem and the slip line, together with the downstream subsonic flow, stay close to their background counterparts; moreover, the solution is unique in the class of solutions satisfying the a priori estimate (for incoming flows that are small in a slightly stronger norm). Lagrangian coordinates are employed to transform the unknown contact discontinuity curve into a fixed boundary and to reduce the full Euler system to a first-order elliptic system for the flow direction and the pressure. The key idea in dealing with the contact discontinuity is to solve a mixed boundary value problem, in divergence form with discontinuous coefficients, for a single elliptic equation, so that the contact discontinuity conditions are naturally preserved as compatibility conditions for the solutions of the elliptic problem.

We prove Keller's conjecture for every split finite-dimensional algebra over an arbitrary field.

Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing "some" in this definition by "every" gives rise to the significantly stronger notion of a dominating $K_t$-model introduced by Illingworth and Wood (2024). They raised the question whether every graph of chromatic number at least $t$ contains a dominating $K_t$-model. This statement is a significant strengthening of Hadwiger's conjecture and has come to be known as the dominating Hadwiger conjecture. We provide our own exposition of a disproof of this conjecture found by ChatGPT 6 Astra Ultra. The construction is the complement of a pseudorandom triangle-free graph that is obtained by randomly subsampling a block geometric graph based on the Suzuki-Tits ovoid.

Let $\Omega\subset\mathbb C$ be a bounded simply connected Lipschitz domain with conformal metric $g=\omega|dz|^2$, where $\omega\in C^2(\Omega)\cap L^\infty(\Omega)$ is positive in the interior and may tend to zero at the boundary. If the Gaussian curvature satisfies $K_g\leq K$, $K>0$, and $KM<4\pi$, where $M=\int_\Omega\omega\,dA$, then the Neumann eigenvalues, numbered by $0=\mu_1(\Omega;\omega)<\mu_2(\Omega;\omega)\leq\mu_3(\Omega;\omega)\leq\cdots$, satisfy $\frac{1}{\mu_2(\Omega;\omega)}+\frac{1}{\mu_3(\Omega;\omega)}\geq\frac{2}{\mu_2(D_K(M))}$. Here $D_K(M)$ is the geodesic disk of area $M$ in the surface of constant curvature $K$. We prove that equality holds exactly when the interiors are isometric. This completes Conjecture 1.2 of Langford–Laugesen; in particular, the model disk maximizes the first positive Neumann eigenvalue in this class.

We study the uniform null-controllability, in the vanishing viscosity limit, of the transport–diffusion equation $u_t + M u_x = \varepsilon u_{xx}$ on $[0,L]$ with a Dirichlet boundary control at $x=0$. Our results concern the uniform null-controllability time $T_{\mathrm{unif}}$, defined as the infimum of the times for which the null-controllability cost remains bounded as $\varepsilon \to 0$. We determine this time exactly for both signs of the transport velocity: $T_{\mathrm{unif}} = 2 L / M$ in the positive-speed case $M>0$, and $T_{\mathrm{unif}} = (2 + 2 \sqrt{2}) L / |M|$ in the negative-speed case $M<0$. The positive-speed result disproves the conjecture $T_{\mathrm{unif}}=L/M$ suggested by Coron and Guerrero. For negative speed, the conjectured threshold had already been disproved by Lissy. Our result determines the exact threshold. We establish the lower bounds by constructing adjoint solutions that violate uniform observability below the respective thresholds. In the positive-speed case, the construction exploits the asymptotic structure of the Blaschke products and model spaces associated with the exponential family generated by the adjoint spectrum. The same model-space structure is used to reduce the upper-bound problems for both signs of $M$ to infinite-time observability inequalities, which are proved through a representation of the boundary-to-interior map and estimates of its Hilbert–Schmidt norm.

Virtual simulators are widely used for studying complex physical phenomena, from particle collisions to rocket propulsion. Such "computer experiments" can be highly time-intensive, and a Bayesian surrogate model can be used for efficient emulation with reliable uncertainty quantification. To train accurate surrogates with a limited sample size $n$, recent work has explored the incorporation of monotonicity (or isotonicity) information, which can often be elicited from physical systems. In practical applications with many input variables, however, existing Bayesian isotonic models can face statistical and computational limitations, which may result in worse performance compared to models that do not incorporate isotonicity. We propose a new transformed additive isotonic model (TAIM), which aims to tame this "curse-of-dimensionality". TAIM makes use of a flexible transformed additive isotonic modeling framework, which leverages a data-estimated link transformation and a monotone basis model with spike-and-slab priors on basis weights. Prediction-wise, TAIM achieves (up to log factors) a posterior contraction rate of $O(n^{-1/3})$ when the true black-box function is in a transformed additive isotonic form with mild smoothness conditions. Such a rate does not depend on the input dimension $d$ for terms involving $n$, which softens the effect of dimensionality on posterior predictions. Computation-wise, TAIM allows for efficient posterior inference via a carefully designed Gibbs sampler, where each sampling iteration requires only linear work in $d$. We further present an extension of TAIM that can model potential deviations from transformed additivity. Numerical experiments and two applications show the effectiveness of TAIM for isotonic surrogate modeling with many input variables.

We construct a Hausdorff saturated directed space whose directed path space between two distinct points has nontrivial fundamental group although the corresponding trace space is a square. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from paths to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces whose compact subspaces are metrizable, these quotients are trivial q-fibrations. We construct a globular clock containing the directed circle and admitting regular maps from all q-cofibrant globular realizations, whereas a finite cellular example admits no regular map to the circle. Normalization also gives a proof of the quotient theorem for execution paths and establishes the homotopy lifting property for nondecreasing reparametrizations.

The evaluation cost of transformers with softmax attention scales quadratically with sequence length. Kernel attention addresses this by replacing softmax with a more general kernel function. In this paper, we aim to identify kernels that retain the expressivity of attention while enabling quasi linear computation. To quantify expressivity, we introduce a capacity for each kernel, measuring the maximum sequence length for which the attention matrix can approximate the identity. A higher capacity thus indicates greater expressivity. We show that expressive kernels like softmax, Gauss, and Laplace have infinite capacity. In contrast, common quasi linear kernels, such as those derived from finite dimensional feature maps, exhibit finite capacity. As a solution, we propose additive kernels constructed from univariate spline and polynomial exponential kernels. We prove that these maintain infinite capacity while allowing quasi linear computation via sorting. Finally, we implement additive sorting kernels efficiently and benchmark them against modern softmax backends, demonstrating advantages for long sequences.