Papers

This expository note is about simulating a target distribution $Q$ from observations of a proposal distribution $P$. In the model suggested by Harsha, Jain, McAllester and Radhakrishnan, we observe an infinite sequence of i.i.d. samples $X_1,X_2,\ldots$ from $P$. The goal is to find some index $I \in \{1,2,\ldots\}$ such that $X_I$ is distributed like $Q$ while minimizing $\mathbb{E} \log I$. Following the Poisson-race approach developed by Maddison, Li and El Gamal, and others, this note shows that if $D(Q||P) < \infty$ then there is a $P$ to $Q$ simulator $I$ such that $\mathbb{E} [\log I] \leq D(Q||P) + 1.45 \|Q-P\|_1$. In the other direction, for every $P$ to $Q$ simulator $I$, the cost is at least $\mathbb{E}[\log I] \geq \frac{1}{2} \max \{ D(Q||P), \|Q-P\|_1 \}$.

We prove that the reduced crossed product $C^*$-algebras of almost periodic actions of countable discrete groups having a topologically free extreme boundary on unital, separable, simple, exact, $\mathcal{Z}$-stable and monotracial $C^*$-algebras are selfless. We combine Ozawa's "tree-graded” space method with the theory of strong convergence of indexed families for the proof.

We study long-horizon Wasserstein-penalized minimax control for discrete-time stochastic linear systems with empirical disturbance data, in which adversarial disturbance distributions induce time-varying mean and covariance dynamics, making standard turnpike arguments not directly applicable. For possibly uncentered data, we characterize the generally nonzero mean reference through a reduced convex-concave Hamiltonian saddle problem. We prove horizon-uniform, two-sided exponential turnpike estimates for the mean state, adjoint, control, worst-case disturbance mean, and closed-loop covariance, showing that they spend the majority of time near static references when the horizon is long. We further construct a hybrid policy combining time-independent affine feedback with finite-horizon steering over a terminal layer, proving that its worst-case cost gap decays exponentially with the terminal-layer length uniformly in the horizon, which helps reducing the computation cost for long-horizon robust controls. Numerical examples illustrate the estimates and their dependence on the Wasserstein penalty.

We establish quantitative rigidity results for pseudo-rotations of the two-torus under a $(C,\delta)$-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a $(C,\delta)$-deviation condition, we show that Hölder continuous super-Liouvillean irrational pseudo-rotations are $C^0$-rigid with an exponential decay rate and that $C^k$ semi-irrational pseudo-rotations of strong non-Brjuno type exhibit $C^{k-1}$-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that Hölder continuous skew products on $\mathbb{T}^2$ over circle rotations are $C^0$-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.

Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density.

We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<\beta<-\frac12$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the probability law, there is a stationary density $h_\varepsilon\in W^{2,1}(I)$, unique among stationary densities belonging to $W^{1,1}(I)$. Moreover, the map $\varepsilon\mapsto h_\varepsilon$ is differentiable at $\varepsilon=0$ in $W^{1,1}(I)$, and we obtain an explicit linear response formula.

Graphs are invariant under node permutations, motivating permutation-equivariant architectures in generative models. In flow matching, however, symmetry also affects the source-target coupling: one must choose both which graphs to pair and which node representatives to align. We study these two forms of alignment, termed outer and inner alignment, and their effect on equivariant graph flow matching. We connect inner alignment to transport on the graph quotient space, whose Euclidean quotient metric coincides with Gromov-Monge distance, and show that quotient couplings admit aligned representative lifts while symmetrization yields equivariant flow-matching minimizers, including for categorical endpoints. In practice, we compare random augmentation, permutation-blind minibatch optimal transport, approximate Gromov-Wasserstein alignment, and combinations of inner and outer alignment. Across continuous graph and molecular generation, alignment can substantially simplify trajectories and improve few-step generation, while its benefits depend on the alignment and computational budget. Our results highlight that effective graph flow matching benefits from alignment both inside and across graph pairs.

We develop a differential graded model for finite non-singular semisimplicial sets and their cellular sheaves. To a finite non-singular semisimplicial set \(S\), we associate an exterior DGA \(\Omega^\bullet_S\), extending the classical anti-equivalence between finite simple directed graphs and first-order differential calculi to higher degrees. We characterize the essential image of this construction, obtaining an equivalence of categories that recovers the graph-FODC correspondence in dimension one. For a fixed \(S\), we then characterize a category of differential graded \(\Omega^\bullet_S\)-modules equivalent to the category of cellular sheaves on $S$, thereby giving a differential refinement of the usual incidence-algebra description. Finally, we study connections and curvature in this framework. Connections are described by edgewise linear maps and their curvature on a 2-dimensional simplex is the difference between direct and composite edge transport. When the edge transports are invertible, vanishing of this curvature on every \(2\)-simplex can equivalently be seen as a gluing criterion to extend the given edge transports to a connection sheaf on \(P_S\).

Inverse inequalities are an important tool in the stability and convergence analysis of kernel approximation methods. In a multiscale setting, however, the trial space changes with the kernel scale $\delta$, and inverse estimates for a fixed kernel are not sufficient. The constants must remain controlled as $\delta\to0$. In this paper, we study scale-uniform inverse inequalities for spaces generated by scaled positive definite kernels whose native spaces are Sobolev spaces. We first establish inverse estimates in scale-dependent Sobolev spaces. On bounded domains, this leads to a scale-uniform inverse inequality in standard Sobolev norms with $L^2$ as the weaker norm. The estimate requires only that the separation distance of the centers be bounded above by a fixed multiple of the kernel scale. This allows the kernel scale to decrease more slowly than the separation distance, as is relevant in multiscale refinement. We then establish more general scale-uniform Bernstein inequalities on the whole space. Our result covers a broad range of weaker and stronger Sobolev indices under the same relation between the separation distance and the kernel scale. The proof works directly with the Fourier representation of functions in the kernel space and combines a low-high frequency decomposition with frame estimates for separated exponential polynomials. This avoids the additional scale factors that arise from separate comparisons of scaled and standard Sobolev norms.

Ambient IoT (A-IoT) devices rely on energy harvesting and duty cycling to sustain operation, thereby fundamentally changing collaborative sensing compared with traditional always-ON sensor networks. In this paper, we study the joint deployment and sensing scheduling of A-IoT devices equipped with directional sensing. We explore four solution strategies: (i) a grid deployment with static duty cycling, (ii) a centralized policy-gradient reinforcement learning (RL) approach that begins with a grid deployment and learns energy-aware device relocation and duty-cycling policies, (iii) a mixed-integer linear programming (LP) approach that couples static deployment design with duty-cycle allocation, and (iv) a hybrid LP+RL that combines optimization-based initialization with learning-based refinement. Using representative A-IoT use cases, we evaluate coverage as a function of device density, field-of-view, and maximum feasible duty cycle, determined by harvested energy and device consumption. Numerical results indicate that the proposed LP+RL and RL policies consistently outperform both the grid baseline and the LP-based method, achieving up to 2x higher mean effective coverage in low and medium energy harvesting (EH) regimes. In contrast, the standalone LP method remains limited by its conservative static duty cycle allocation under tight EH constraints. Moreover, the structured initialization of the LP+RL method substantially accelerates convergence, reducing the total offline optimization time by up to 10x compared to the standalone RL.

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require “anti-collapse mechanisms” to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-\Delta)^{\alpha}$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $\alpha>0$, we prove that the fractional Hardy inequality $$(-\Delta)^{\alpha}\geq\frac{4^{\alpha}\Gamma^2(\alpha+1/2)}{\pi}\frac{\Gamma(2\,\cdot\,-1)}{\Gamma(2\,\cdot\,-1+2\alpha)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.

The Andreadakis-Johnson filtration and its associated construction, known as the Johnson homomorphism, are useful tools in group theory. They provide a step-by-step approach to studying the automorphisms of a given group. After explaining the basics we survey both classical and recent results around the Andreadakis-Johnson filtration, with emphasis on the mapping class group of a once-bordered surface.

We obtain a complete classification of rank two semistable sheaves with maximal third Chern class on the Fano threefold $X_5$ of index 2 and degree 5. Also, we describe moduli spaces of such sheaves in the general case of big second Chern class. These moduli spaces are smooth rational varieties. The study uses the theory of tilt-stability and Bridgeland stability conditions.

A separating curve is a real curve whose real part disconnects its complex part. It also admits totally real morphisms to the projective line, and the separating gonality is the minimal possible degree for such morphisms. The separating gonality is bounded from below by the number of connected components of the real part of the curve. In this paper, we build on a strategy of Manzaroli to improve this lower bound for some curves embedded in the projective plane or in a Hirzebruch surface.

Let $S$ be a smooth projective surface with $q(S)=0$. We give numerical criteria for the nef cones of $S^{[n,n+1]}$ and $S^{[1,n]}$ to decompose as sums of pullbacks of nef cones under their natural morphisms. These criteria recover the known decompositions for the projective plane, Hirzebruch surfaces, and Picard rank one K3 surfaces, and apply to del Pezzo surfaces with $2\le K_S^2\le7$. For a del Pezzo surface of degree one, we show that the corresponding pullback decompositions fail for both $S^{[n,n+1]}$ and $S^{[1,n]}$.

We construct a three-parameter family of rational approximations to values of $q$-hypergeometric series. Using these approximations, we prove that, for every integer $x$ with $|x|\geq2$, the values at $r=x^{-1}$ of Ramanujan's theta function $\psi(r)=\sum_{n\geq0}r^{n(n+1)/2}$, the generating function $\Delta(r)=\sum_{m\geq0}d(2m+1)r^m$ of the divisor function restricted to odd integers, and the generating function $B_4(r)=\sum_{n\geq0}b_4(n)r^n$ for $4$-regular partitions are irrational. We further obtain the upper bounds $18/7$, $18\pi^2/(7\pi^2-24)$, and $3$, respectively, for their irrationality measures. We also show that one of the constructed approximations coincides with the Padé approximation to a Lambert series due to Coussement–Smet.

This paper studies the minimum rate required for a partially observable linear system with side information. The Linear Quadratic Gaussian(LQG) plant and the Gaussian-Markov source are considered. We show that a class of linear policies is sufficient for optimizing the conditional directed information lower bound. We also show that the resulting optimization problem is convex for the scalar case in both time-varying and time-invariant systems. Our results generalize the past works that consider the case with full or partial observation only, and the case with full observation and side information. Numerical simulations are presented to illustrate the effect of side information for partially observable systems.

We study properties of the transcendental numerical dimension on compact Kähler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact Kähler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao–Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(\Omega_X^1)^{\otimes m},~m \geq 1\). We prove that \[ \nu(L,X)\leq\nu(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.

Peer two-step methods are attractive for the numerical solution of stiff initial value problems because they combine favorable features of Runge-Kutta and multi-step methods, including high-order accuracy, strong stability properties, and the avoidance of order reduction. This paper develops rigorous norm estimates for the propagation operators of variable-stepsize diagonally-implicit $L(\alpha)$-stable Peer methods applied to stiff semi-linear systems. The analysis relies on a class of Peer methods whose propagation matrices possess a grid-independent left eigenvector, enabling the construction of suitable matrix norms for entire coefficient families. We show that this class contains $L(\alpha)$-stable methods of stage order $q=s$ and super-convergence of order $s+1$ with $s$ stages, exceeding the classical stage-order barrier $q\le 2$ of irreducible $s$-stage diagonally-implicit Runge-Kutta methods, and construct such methods up to order 5. Achieving high stage order is one of the principal advantages of Peer methods, especially for stiff differential equations where it helps mitigate order reduction. The resulting theory provides uniform stability bounds for stiff semi-linear problems on variable grids and establishes a foundation for further applications to optimal control and related boundary value problems. Comparative numerical results are presented for three classical stiff benchmark problems, including three novel Peer two-step methods of order 3, 4, 5, and four singly diagonally implicit Runge-Kutta methods with a first explicit step of stage order 2 and order 3, 4, 5, and 6.

Lossy compression is essential for managing massive scientific data, but per-element error bounds do not translate into bounds on downstream quantities of interest (QoIs) such as regional averages, neural network predictions, or multi-field derived quantities. We present TOPIQ, a statistical error-propagation framework that predicts QoI-level bias and uncertainty from compact compression metadata (less than 0.1% of original data). TOPIQ decomposes QoIs into primitive operators with closed-form propagation rules accounting for spatial error correlation and data-error coupling; new QoIs are supported by composition at runtime with no per-QoI derivation or retraining. Across 552 evaluations spanning 4 datasets, 3 compressors, 4 QoI families, and 8 error bounds, 93.1% of configurations achieve well-calibrated predictions. Pre-computed metadata enables post-hoc uncertainty quantification for arbitrary query regions at 56x-402x speedup over direct computation. A case study demonstrates integration into an AI-driven analysis pipeline with end-to-end confidence intervals for dynamically composed queries.

In this work, we study guided modes in systems of high-contrast resonators near nonzero interior Neumann frequencies, beyond the subwavelength regime. In the regular exterior regime, where the exterior Dirichlet problem is well-posed at the reference wavenumber, we introduce an infinite-dimensional frequency-dependent capacitance operator obtained by compressing the exterior Helmholtz Dirichlet-to-Neumann map to the traces of the interior resonant Neumann eigenspaces. We prove the norm-resolvent convergence of the continuous problem to this discrete effective operator as the contrast $\delta\to0$, and derive first-order asymptotic formulas for compact-defect frequencies and line-defect band functions. We then establish exponential off-diagonal decay of the capacitance coefficients by a Combes–Thomas argument, yielding an exponentially accurate truncation of the discrete operator, and show that its retained coefficients can be computed from local Helmholtz problems. At the physical frequency, this local approximation converges exponentially under a uniform stability assumption for the growing finite-cluster problems. The stability assumption can be removed by introducing a vanishing complex absorption together with a Hermitian symmetrization. In particular, an absorption parameter of order $\sqrt\delta$, together with interaction truncation and patch radii of order $|\log\delta|$, suffices to preserve the $O(\delta^2)$ accuracy of the first-order high-contrast expansion of the defect eigenfrequencies, yielding a fast computational method. Numerical experiments for dipole and quadrupole resonances illustrate the accuracy, exponential locality, and applicability of the discrete model to straight and bent waveguides generated by material or geometric detuning.

We show that there is a $\Pi^0_2$ subset $B$ of the Euclidean plane, of Hausdorff dimension 1, such that for every line $\ell$ through the origin with hyperarithmetic direction, the projection of $B$ onto $\ell$ has Hausdorff dimension 0, thus is exceptional for Marstrand's projection theorem. It follows that for any computable ordinal $\alpha$, being $\alpha$-random does not guarantee the “almost-all” statement of the projection theorem.

Let $\mathbb{S}$ denote the algebra of the sedenions, and $\Gamma_O(\mathbb{S})$ denote its orthogonality graph. We observe that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in $\Gamma_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ which has a convenient multiplication table. We describe explicitly the set of vertices of an arbitrary connected component of $\Gamma_O(\mathbb{S})$ and find its diameter. We then establish the bijection between the connected components of $\Gamma_O(\mathbb{S})$ and lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies between $3$ and $4$.

We study a free-energy minimization problem for a trapped semiclassical Bose gas with repulsive Coulomb self-interaction. Because the bosonic entropy density has zero recession slope, the natural relaxation is posed over nonnegative finite Radon measures on phase space, and a minimizer may have a singular component. We interpret this component as condensation within the relaxed semiclassical model. We prove existence and uniqueness at every positive temperature, derive the Euler-Lagrange contact condition and an obstacle-type formula for the condensed density, and establish sharp phase boundaries under only the standing continuity assumptions on the trap. At fixed mass there is a unique finite positive critical temperature, with condensation precisely below it. At fixed temperature there is a sharp critical mass, possibly infinite, separating normal and condensed minimizers. For harmonic traps we obtain a sharp confinement-strength dichotomy. We also prove an abstract conditional variational-stability statement for measure-valued curves that conserve mass and satisfy the free-energy inequality.

We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Its local penalties define only a cycle cover, so ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove connected, cost-optimal ground-state routings under explicit penalty assumptions. For $N-1$ customers and $K$ nonempty routes, the unreduced encoding uses exactly $|E|(1+\lceil\log_2(N-K+1)\rceil)$ logical problem qubits. A reversible compute-phase-uncompute implementation evaluates the flow penalties in $O(N^2\log N+N\log^2N)$ logical gates on a complete graph, using $O(\log N)$ reusable workspace and no product register. On complete loopless graphs, a position encoding that arranges all routes in one sequence separated by depot occurrences uses fewer problem qubits and written terms when the number of bits per flow register grows. Under the same reversible accounting model, the flow formulation admits a smaller logical-gate upper bound. Exact Hamiltonian and circuit audits and a benchmark of 1,200 cost matrices verify the formulation and quantify the connectivity gap. A 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits for a diagnostic $N=4$, $K=1$ counterexample. In the degree-only circuit, 78.05% of selected $p=1$ shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation, not an asymptotic advantage in solving routing problems.

Zero divisors of Cayley-Dickson algebras over an arbitrary field $\mathbb{F}$, $\mathrm{char} \, \mathbb{F} \neq 2$, are studied. It is shown that zero divisors, whose components alternate strongly pairwise and have nonzero norm, form hexagonal structures in the zero divisor graph of a Cayley-Dickson algebra. The properties of doubly alternative zero divisors, at least one of whose components has nonzero norm, are established, and an explicit form of their annihilators, orthogonalizers, and centralizers is obtained. The properties of zero divisors in Cayley-Dickson algebras with anisotropic norm are described, and it is shown that in this case directed hexagons in the zero divisor graph can be extended to undirected double hexagons in the orthogonality graph. A criterion of $C$-equivalence for elements of Cayley-Dickson algebras with anisotropic norm is obtained. Possible values of dimension for annihilators of elements of Cayley-Dickson algebras are considered.

Accurate and efficient simulation of ultrasound pressure fields and pulse-echo responses is essential for transducer design, beamforming optimization, and model-based imaging research. Conventional Spatial Impulse Response methods compute acoustic fields by explicitly sampling trapezoidal impulse responses for each rectangular aperture subdivision, which leads to high computational cost for large apertures, dense spatial grids, and high sampling frequencies. We introduce Sparse Delta Integration, a mathematical framework for the far-field SIR of rectangular apertures that expresses the trapezoidal spatial impulse response as the double integration of a sparse set of Dirac delta distributions. This formulation yields compact expressions for time-domain and frequency-domain SIRs and enables vectorized implementations whose computational cost is independent of the trapezoid duration. We further derive a spectral formulation for pulse-echo simulation that removes part of the conventional Fourier-domain convolution pipeline. Implemented in the open-source Python package eSDIva, these methods achieve speedups of up to 180x for temporal SIR computation and up to 20x for pulse-echo RF simulation relative to Field II, while maintaining high numerical accuracy with mean squared errors below 1e-5 and correlation coefficients close to unity.

The commutativity graph of the real sedenion algebra is considered. It is shown that those elements whose imaginary part is not a zero divisor correspond to isolated vertices of this graph. All other elements form a connected component whose diameter equals $3$.

Multiple scattering is a fundamental wave interaction phenomenon in acoustics and electromagnetics. The multipole expansion method (MEM) is the basis of many fast algorithms, such as the fast multipole method (FMM), for such problems. However, due to the infinitely many wave reflections involved, its convergence in three dimensions remains unexplored. In this paper, we prove spectral convergence of the MEM for time-harmonic acoustic scattering by finitely many well-separated spheres in three dimensions. Using a diagonally preconditioned single-layer formulation, we analyze the degree-$N$ truncated system in a natural spherical harmonic energy space. We split the interaction truncation into target-side and source-side high-degree parts and choose a different representation for each: a projected Green-kernel representation for the former and degree-wise estimates of a translated spherical wave family for the latter. The resulting argument, based on Parseval's identity and the spherical harmonic addition theorem, provides a general framework for the convergence analysis of MEM and reveals the geometric and physical origins of the convergence factors. We also obtain a sharper estimate through the first-transfer analysis. Numerical experiments confirm the predicted spectral decay and the geometric convergence factor. This paves the way for the convergence analysis of a large class of fast algorithms for multiple scattering.