We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

We study a mean field game (MFG) of state and control with state dynamics described by stochastic differential equations driven by both idiosyncratic and common noise, and subject to the constraint that the terminal state variable belongs to a nonempty, convex, closed set. The mean-field interaction enters both the dynamics and the costs through state and control. We formulate an equivalent auxiliary MFG problem and derive the stochastic maximum principle for an auxiliary optimization problem under fixed flows. By means of a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type, we establish a characterization of MFG solutions via such a system. Moreover, we prove the existence and uniqueness of solutions for the FBSDE system in the specific case where the running cost is zero and the state coefficients are linear. Additionally, we obtain uniqueness results in the sense of Lasry-Lions and we apply our findings to MFGs of optimal investment with a quadratic relative performance criterion.

We propose a new approach to unbiased estimation of the gradients of the stationary means associated with parametrized families of Markov chains. Our estimators are particularly efficient when the Markov chains have slow mixing rate. Our approach does not require a specific parametrization except for an oracle to evaluate the transition density and its gradient at a given data point without any additional knowledge about the density function itself. It makes our estimator suitable for parametrizations associated with neural networks. The estimator can potentially achieve large improvement in terms of efficiency. Numerical experiments confirm the good performance predicted by the theory.

Efficiently decomposing finite element stiffness matrices into the Pauli basis is challenging due to the exponential growth of Pauli strings with problem size. A naive Pauli expansion requires $\Theta(8^{\lceil \log_2 N \rceil})$ operations, where $N$ denotes the number of degrees of freedom, rendering direct decomposition infeasible for large systems. Existing approaches exploit algebraic sparsity or operator structure but do not incorporate the geometric organization intrinsic to finite element discretizations, and consequently exhibit poor scaling for stiffness matrices. To address this problem, we introduce PHASE, a hierarchical, geometry-aware Pauli decomposition algorithm that leverages recursive mesh partitioning to organize element contributions across multiple spatial scales. PHASE employs a hybrid strategy that combines full- and reduced-space Tensorized Pauli Decomposition with Fast Walsh-Hadamard Transform-based aggregation to assemble global Pauli coefficients efficiently. We show that this approach yields a dimension-dependent reduction in the exponential scaling exponent of Pauli assembly asymptotic complexity relative to existing methods, reducing the cost from $2^{2{\lceil \log_2 N \rceil}}$ to $2^{\gamma_d{\lceil \log_2 N \rceil}}$ with $\gamma_d < 2$ under standard mesh regularity and balanced partition assumptions. These results substantially improve the feasibility of quantum-compatible operator synthesis for large-scale finite element models.

The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh--sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection--diffusion, and open quantum systems.

We study the task of density estimation, where we hope to accurately estimate a probability density from $n$ samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.

We prove a conjecture on the minimal number of generators of licci zero-dimensional ideals that are either monomial or have small Loewy colength.

Let $G$ be the group of power series $x+a_2x^2+a_3x^3+\cdots\in R[[x]]$ under substitution, where $R$ is a commutative ring with $1\neq 0$ of prime characteristic $p$. Given any $n\geq 1$, the subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in R\}$ is normal in $G$, and the quotient $G_n=G/K_n$ is the group of truncated polynomials over $R$ of degree $\leq n$ under substitution. In this paper, we compute the exponent of the image of $K_r$ in $G_n$, for all $r,n\geq 1$, indicating in every case a family of elements realizing this exponent.

The aim of this note is to present a simple strategy to answer negatively a decomposition question on tori posed by Colliot-Thélène and Sansuc in the article \textit{Principal Homogeneous Spaces under Flasque Tori: Applications} of 1987. We then deduce a torus $T$ over $\mathbb{Q}$ and a prime number $p$ such that $T(\mathbb{Z}_p)\,T(\mathbb{Q})\not=T(\mathbb{Q}_p)$, where $T(\mathbb{Z}_p)$ denotes the maximal compact subgroup of $T(\mathbb{Q}_p)$.

We establish explicit subconvex bounds for central values of Rankin--Selberg $L$-functions $L(1/2,\pi\times\pi')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.

A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme.

In this paper, we obtain order estimates for the Kolmogorov widths of an intersection of a finite family of Besov classes $SB_{p_j,\theta_j}^{\overline{r}_j}(\mathbb{T}^d)$ with dominating mixed smoothness in a Besov space $B_{q,\sigma}^{\overline{l}}(\mathbb{T}^d)$ in the case $2<q, \, \sigma <\infty$ when the parameters satisfy certain conditions of general position.

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

The cumulative distribution transform (CDT) is a quantile-based transport representation that exactly linearizes one-dimensional translations of positive densities. We study how this structure behaves under additive perturbations and how it can be exploited for shift recovery. Under a local nondegeneracy condition, we derive a first-order expansion showing that additive noise in physical space induces a nonlocal perturbation in CDT space through the primitive of the noise, weighted by the reciprocal density. This yields an explicit description of transform-domain sensitivity and shows, in particular, that perturbations are amplified in low-density regions. When the physical-space perturbation is modeled as a centered Gaussian random field, the induced first-order CDT perturbation is again Gaussian, with an explicit covariance kernel. We then use this structure to study recovery in CDT coordinates. In the known-template setting, the transport shift is obtained by projection onto the constant mode, giving an explicit estimator together with exactness in the noiseless case and a stability bound under perturbations. In the unknown-template setting, multiple observations permit joint recovery of the shifts and a common template up to the natural constant-mode gauge, leading to a simple de-shift--and--average procedure. We also consider a signed-signal analogue based on the signed cumulative distribution transform (SCDT), where shifts are estimated numerically by feature matching and unknown templates are recovered by alternating alignment and averaging. Numerical experiments validate the perturbation analysis and illustrate effective recovery for both density-valued and signed signals.

While long treated as empirical fits, Hill functions have been postulated to be the universal Hopfield barrier for sharpness of input-output responses by Martinez-Corral, Nam, DePace, and Gunawardena. A Hopfield barrier is a fundamental limit on how well biological systems can process information without expending energy. Their case rested on numerical findings for Hill coefficients $4$ and $6$. We give a precise formulation and proof of this: measuring sharpness by the supremum of the derivative in semi-log scale, any rational function $r(x)=(\alpha_0+\alpha_1 x+ \cdots +\alpha_n x^n)/(\beta_0 + \beta_1 x+ \cdots + \beta_n x^n)$ with real coefficients $0\leq \alpha_i\leq \beta_i$ has sharpness at most $n/4$, with equality if and only if $r$ is a Hill function with Hill coefficient $n$.

For primes $p\geq 5$, we determine those $q$ for which the projective special linear group $\psl{2}{q}$ over the finite field of order $q$ is $(2,p)$-generated -- that is, there exist two elements of $\psl{2}{q}$ of orders $2$ and $p$, respectively, that generate $\psl{2}{q}$.

We prove that optimally weighted second-order least squares (SLS) and the degree-two generalized polynomial maximization method (PMM) are the same population estimating equation for linear regression with conditionally homoskedastic non-Gaussian errors: they choose the same optimal linear combination of the first two centered residual moments, solve one population normal system, share one influence function, and attain the common asymptotic variance $c_2g_2/N$ -- the ordinary-least-squares slope-variance factor $c_2$ scaled by the PMM variance-reduction coefficient $g_2=1-\gamma_3^2/(2+\gamma_4)$ (with $\gamma_3,\gamma_4$ the error skewness and excess kurtosis). Feasible plug-in implementations are therefore first-order equivalent, with only higher-order finite-sample differences. The identity is sharp: under heteroskedasticity the unconditional PMM body and the conditional SLS weighting separate, costing efficiency for symmetric errors and consistency for asymmetric errors. Beyond degree two, PMM holds an efficiency reserve that SLS cannot reach within its second-moment span. For symmetric platykurtic errors SLS collapses to ordinary least squares for the slope, while degree-three PMM exploits kurtosis information outside the SLS moment span through a closed-form coefficient $g_3$; for canonical asymmetric laws this reserve is $30$--$50\%$ within the degree-three polynomial moment class. The Lean 4 development machine-checks the degree-specific algebraic core -- the closed forms for $g_2$ and $g_3$, the $g_2\le1$ result, the design cancellations, and the symmetric collapse -- while the general monotonicity $g_{S+1}\le g_S\le1$ is proved analytically by nesting. A Monte Carlo study illustrates the equivalence, the reserve, and the heteroskedastic boundary at finite samples.

For a commutative ring $R$ with unity, its algebraic $K$-theory space $K(R)$ may be obtained by group-completing the symmetric monoidal category of finitely generated free $R$-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of $K(R)$, up to $\pi_0$. We also discuss how a similar group-completion would give the $p$-perfection and, more generally, the localization of $K(R)$ at any non-trivial multiplicatively closed subset $S \subseteq \mathbb{Z}_{> 0}$. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.

For many trees $T$, the Ramsey number of $T$, denoted by ${\mathcal R}(T)$, is determined by the sizes of the partition classes in its unique bipartition. In 1976, Burr proved that when $T$ has partition classes of size $t_1$ and $t_2$ with $t_1 \le t_2$, the Ramsey number is at least $\max(2t_2-1,2t_1+t_2-1)$, and conjectured that this is tight. While counterexamples have been found for some pairs $(t_1, t_2)$, a main focus of research on this problem has been determining ratios $t_2/t_1$ or bounds on the maximum degree of $T$ for which Burr's bound is either exactly or asymptotically tight. We essentially resolve these questions for lopsided trees. Specifically, we show that (a) there are counterexamples whenever $t_2 \ge 2t_1$, with the order of magnitude of the difference between the largest Ramsey numbers and Burr's bound being $\max \left( t_1^2/t_2, \sqrt{t_1} \right)$, and (b) for $t_2 \ge 500 t_1$, Burr's bound is tight when $\Delta(T) \le t_2 - t_1$, but is off by at least $C \log t_2$ (even when $t_2 \ge 2 t_1$) when $\Delta(T) \gtrsim t_2 - t_1$. In particular, this shows that Burr's bound need not hold for $t$-vertex trees $T$ with $\Delta(T) \approx t/3$.

We prove the Harish-Chandra isomorphism theorem for supersymmetric spaces, describing the polynomial algebra of eigenvalues of invariant differential operators. The polynomials obtained satisfy novel invariance conditions, which remain somewhat mysterious. We also prove the Harish-Chandra isomorphism for ghost distributions, which satisfy a `square root' of the invariance conditions coming from invariant differential operators. All proofs are algebraic, and rely on a rank-one reduction argument and the Chevalley restriction theorem.

Several recent works have addressed the problem of deconvolution under a linear model, where the goal is to estimate a completely unknown $G_0$ from a vector of noisy observations $\boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\epsilon}$, assuming the coefficients $\beta_j$ are i.i.d. unobserved realizations from $G_0$. Assuming $G_0$ has a density $g_0$, we study theoretically a Bayesian nonparametric method proposed in Weinstein et al. (2025) that postulates a Pólya tree prior $\Pi$ on $g_0$ and bases a deconvolution estimate on the posterior distribution $\Pi(\cdot|\boldsymbol{Y})$. Our main result asserts that under the true model (fixed and unknown $g_0$), and under a suitable condition on the minimum eigenvalue of $X^\top X$, the posterior $\Pi(\cdot|\boldsymbol{Y})$ concentrates around $g_0$ in sup-norm. The analysis presented builds on and extends results from Castillo (2017), where posterior consistency of Pólya trees was proved for density estimation, the simpler problem of estimating $g_0$ when observing the coefficients $\beta_j$ directly.

In this note, we derive a large-deviation-type scaling limit for a discretization of the Martingale Representation Theorem. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.

Let $\lambda = \lambda(\beta,\sigma,a,b)$ denote the top Lyapunov exponent for the linearization along trajectories of the noisy damped non-linear oscillator $\ddot{x}+\beta \dot{x} + ax+bx^3 = \sigma \dot{W}_t$, where $a$, $b$ and $\beta$ are all positive and $\sigma \neq 0$. In 2004 Arnold, Imkeller and Sri Namachchivaya stated without proof that $\lambda(\varepsilon^2 \beta,\varepsilon \sigma,a,b) \sim \overline{\lambda} \varepsilon^{2/3}$ as $\varepsilon \to 0$ with $\overline{\lambda} > 0$. This paper contains a proof of this assertion.

In this paper, we apply the I-method to establish global well-posedness for the fractional nonlinear Schrödinger equation with initial data u_0 \in H^s(R^d) for s <\alpha/2, i.e., below the energy threshold. Moreover, for radial initial data, we combine a modiffed Morawetz estimate-recovered via Balakrishnan's formula-with the I-method to obtain improved results. In the same spirit, we employ the "upside-down" I-method to derive polynomial-in-time growth bounds for the higher-order Sobolev norm. The main difffculty stems from the fact that Strichartz estimates for the fractional Schrödinger equation has a loss of derivatives, and the problem is always L^2-supercritical, thereby requiring more delicate analysis.

We study stochastic interacting particle systems whose interaction structure is described by dense weighted directed graphs converging to a graphon. In the thermodynamic limit, we prove a law of large numbers for the empirical measure process and derive a deterministic nonlinear master equation describing the macroscopic evolution. The limiting equation retains the heterogeneous interaction structure of the microscopic system through the limiting graphon, allowing for spatially non-homogeneous behaviors such as localized or community-type interactions.

We define and investigate two families of dual polynomials associated with the Gauss--Legendre polynomials, which have recently found interesting applications in computer graphics. Using the presented results, one can derive representations of the Gauss--Legendre polynomials, construct the dual bases for Lagrange bases and solve certain approximation problems arising, for example, in CAGD.

We study the Threshold Group Testing (TGT) problem in the noiseless and non-adaptive setting, where the objective is to exactly recover a sparse binary vector from pooled tests, using as few tests as possible. In TGT, each test applied to a subset of items returns a positive outcome if the number of 1's (defective items) in that subset meets or exceeds a specified threshold, and has a negative outcome otherwise. We investigate how the complexity of TGT compares to that of Classical Group Testing (CGT), corresponding to the special case of the threshold equal to one, and analyse the impact of increasing the threshold on the required number of tests. Our main contribution is the derivation of a sharp information-theoretic phase transition at $c_{\mathrm{inf}}^{\mathrm{TGT}}k\log(n/k)$ (non-adaptive) tests for TGT within the constant-column test design. The threshold constant $c_{\mathrm{inf}}^{\mathrm{TGT}}$ is expressed as a function of the prevalence of defectives and the threshold value. Our upper bound is derived under an analytic assumption, and we verify that this assumption is satisfied for a threshold value of 2. The value of $c_{\mathrm{inf}}^{\mathrm{TGT}}$ reveals that TGT on the constant-column design has the same information-theoretic behaviour as CGT in the low-prevalence regime. Yet, strikingly, at higher prevalences, the threshold leads to a significant reduction in the number of tests. On the other hand, we provide evidence that when the asymptotic proportion of defective items is positive, TGT actually becomes strictly harder than CGT (excluding trivial reductions).

We propose Annealed Entropic Allocation, an annealed weighted soft-min framework for sequential budget allocation in ranking and selection. The central idea is to replace the non-smooth maximin large-deviation rate objective with a weighted log-sum-exp surrogate that aggregates challenger-specific pairwise scores through soft-min weights, mitigating hard switching when several challengers are nearly active. To improve finite-budget discrimination, we incorporate the saddlepoint approximation -- a sub-exponential correction derived from refined pairwise tail asymptotics. Because these corrections are sub-exponential and the smoothing parameter is annealed to zero, the surrogate preserves the same first-order large-deviation target as the classical maximin formulation. We show that the surrogate converges uniformly to the hard minimum, that the soft-min weights concentrate on the active challengers, and that, under fixed weights, the induced target allocation map is continuous on the simplex interior. Numerical experiments on Gaussian and exponential instances demonstrate competitive performance, especially when multiple challengers are nearly tied.

We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

This paper studies a lattice structure for almost perfect matchings on certain planar, bipartite (plabic) graphs embedded in a disk. Postnikov's boundary measurement map, and subsequent related work, yielded that plabic graphs parameterize positroid cells within the totally nonnegative Grassmannian with the map itself given in terms of almost perfect matchings with fixed boundary condition. For finite planar bipartite graphs, Propp introduced a distributive lattice structure on their set of perfect matchings. Subsequently Muller--Speyer, provided this distributive lattice structure on the aforementioned almost perfect matchings with fixed boundary condition. Their work also identified the extremal matchings of this lattice for boundary conditions that coincide with face labels of the plabic graph given by the positroid structure. We extend this by giving an explicit construction of extremal matchings in terms of height functions and show that all possible boundary conditions of an almost perfect matching can be obtained within this construction.