Papers

The Ekedahl sieve is a powerful tool for enumerating arithmetic objects, but traditional formulations relying on inductive steps often yield suboptimal bounds when applied to highly skew boxes. This limitation is particularly restrictive when introducing large modular conditions that compete with the tail-end variables of a binary form. In this paper, we develop a specialized variant of the Ekedahl sieve tailored to the singular locus of the discriminant polynomial. By exploiting the specific non-degeneracy properties of the discriminant outside its two most extreme coefficients, we bypass the standard inductive framework, reducing the sieve to a highly efficient two-step process in some cases, and a one step process in others. We establish robust generic tail-end estimates as well as squarefree, power-saving bounds that seamlessly incorporate external modular conditions. This optimization maximizes the permissible range of the sieving modulus, yielding improved error terms for the enumeration of bounded squarefree values of certain polynomials and providing the foundational geometric sieve estimates required for the weighted enumeration of number fields by discriminant.

We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of irreducible orders. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals. We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields.

We prove two sliding-window inequalities for maximal convolutions. The first concerns the multiplicative maximal convolution. If $f$ and $g$ are nonnegative continuous functions on $[0,A]$ and $[0,B]$, respectively, define \[ h(x)=\max_{\substack{0\le u\le A\\0\le x-u\le B}} f(u)g(x-u),\qquad 0\le x\le A+B. \] Then there exists a window $[a,a+B]$ of length $B$ such that \[ \frac1B\int_a^{a+B}h(x)\,dx\ge \left(\frac1A\int_0^A f(x)\,dx\right) \left(\frac1B\int_0^B g(x)\,dx\right). \] The second concerns the additive maximal convolution. Let $f$ and $g$ be nonnegative continuous functions on $[0,C]$, and define \[ H(x)=\max_{\substack{0\le u\le C\\0\le x-u\le C}}\{f(u)+g(x-u)\},\qquad 0\le x\le 2C. \] Then, for every $p\ge1$, there exists a window $[a,a+C]$ of length $C$ such that \[ \left(\int_a^{a+C}H(x)^p\,dx\right)^{1/p} \ge \left(\int_0^C f(x)^p\,dx\right)^{1/p} + \left(\int_0^C g(x)^p\,dx\right)^{1/p}. \] We also record discrete analogues. The main point is that, in one-dimensional maximal-convolution settings, certain global Brunn–Minkowski or Prékopa–Leindler type phenomena admit natural sliding-window localizations.

We develop a contact Tulczyjew formalism for dissipative dynamics on skew algebroids. Starting from the Tulczyjew morphism of an skew algebroid, we identify its contact extension in a local line-bundle trivialization. The local representative is obtained by adding to the ordinary Tulczyjew morphism the Euler vector field contribution on $E^*$. This gives an intrinsic explanation of the contact term appearing in the local contact Tulczyjew morphism. For a contact generating object, the construction produces an implicit dissipative dynamics on the contact phase side. In local coordinates, the matching condition gives the Euler-Lagrange-Herglotz equations on the skew algebroid. In the hyperregular case, the corresponding contact Hamiltonian equations are recovered by Legendre transformation. We also develop the discrete counterpart of the construction. After fixing a discrete admissibility relation, a discrete contact generating object defines a discrete contact Tulczyjew relation on the contact phase space. Discrete Herglotz extremals are obtained by matching consecutive contact momenta, with the usual conormal interpretation in the constrained case. In the regular tangent-bundle case, this recovers standard contact variational integrators, while in the singular or skew algebroid setting the same construction remains meaningful as an implicit discrete relation rather than an a priori update map.

We derive a generating function identity for the joint distribution of the numbers of peaks of a permutation and its inverse, via enriched $P$-partitions. The coefficients of the corresponding peak polynomial $W_n(s,t)$ satisfy a second-order recurrence. A martingale formulation of this recurrence yields a bivariate central limit theorem, showing that the two statistics are asymptotically independent. We also give an exact closed form for their covariance, which is of order $n^{-1}$.

“Block what you can and randomize what you cannot” is the core principle for treatment effect estimation in randomized controlled trials. Although a wealth of allocation strategies has been developed, an explicit trade-off between the covariate balance achieved by blocking and the robustness guaranteed by randomization is seldom quantified. Motivated by the second law of thermodynamics, this work posits a new criterion that lowers the covariate imbalance while maximizing the entropy that quantifies contrast and allocation diversity. The resulting optimal strategy, termed the minimum free energy randomized design, is then derived, thereby formally achieving such a trade-off. To facilitate practical implementation, we further develop a computationally efficient dynamic allocation algorithm with theoretical guarantees. Using a finite-sample variance decomposition, the proposed randomization strategy is shown to control covariate imbalance while preventing unobserved heterogeneity from dominating the mean squared error, thus retaining minimax efficiency under the prescribed design constraints. Extensive numerical simulations demonstrate that our method achieves superior statistical efficiency and greater robustness than existing approaches.

We prove lower bounds for the Hausdorff dimensions of various natural sets associated with the Liouville quantum gravity (LQG) metric. We prove that the set of 3-star points (i.e., starting points of three disjoint geodesics) has Hausdorff dimension at least two with respect to the LQG metric, which is conjectured to be optimal. Our proof works for a general class of planar length metrics which also includes, e.g., Kendall's Poisson roads metric. We additionally prove a dimension lower bound of one for the set of 2-star points intersected with the boundary and for the metric net intersected with the boundary, as well as a dimension lower bound of two for the intersection of two metric nets. In the particular setting of LQG, we obtain sharper lower bounds for the Hausdorff dimensions of the set of 2-star points and the LQG metric net, with respect to both the Euclidean metric and the LQG metric. Our proofs are primarily topological. The key idea is to express the sets of interest in terms of non-constancy sets of Lipschitz functions.

We consider a finite-horizon, zero-sum stochastic differential game in which one player controls a path-dependent stochastic system, while the opponent is given the opportunity to terminate the game prematurely. We introduce a control randomization formulation, which allows us to establish that the upper and lower value functions coincide. Our approach also yields a representation of the common game value in terms of a nonlinear Snell envelope, where the underlying stopped process is given by the unique maximal solution of a backward stochastic differential equation (BSDE) with constrained jumps.

The fractional Laplacian has been widely used to model non-local diffusion on graphs, allowing interactions that extend beyond immediate neighbors. However, it suffers from a structural inconsistency as it breaks compatibility with the topology of the original network. To address this issue, a combination of the standard and fractional Laplacians aimed at restoring compatibility while retaining the spectral richness of the fractional operator was recently proposed. In this work, we provide a thorough analysis of the diffusion properties of the resulting regularized operator. We prove that it yields superdiffusive behavior independently of whether the underlying graph is weighted or unweighted -a property not generally satisfied by other nonlocal Laplacian variants. An efficient construction of the regularized operator which preserves the same asymptotic computational cost of the fractional Laplacian is also given. Our numerical experiments demonstrate the effectiveness and computational practicality of the regularized operator for modeling non-local diffusion on real-life networks.

This paper develops a mean-field theory for a simplified single-layer causal multi-head self-attention model trained by cross-entropy minimization. Each attention head is treated as a particle in parameter space, and the empirical law of the heads is used as the large-head state variable. In the infinite-head limit, the averaged attention logits define a risk functional on probability measures, whose first variation generates a nonlinear Wasserstein gradient-flow equation. Unlike classical mean-field analyses of shallow networks that often focus on square-loss regression, the present model contains the softmax residual from the cross-entropy objective and the query-key-value structure of masked self-attention. We prove a static finite-head approximation bound for the optimal risk, characterize global minimizers through a variational support condition, and establish a quantitative finite-time propagation-of-chaos estimate comparing finite-head stochastic gradient descent with the limiting PDE. We then study the long-time behavior of the PDE: energy dissipation, convergence to the stationary set under compactness, convergence to a single stationary measure under topological or Kurdyka–Łojasiewicz assumptions, and explicit convergence rates under gradient-domination conditions. Finally, we prove local exponential stability under a Wasserstein strong-monotonicity condition and give verifiable stability and instability criteria for Dirac stationary measures. The results provide a rigorous baseline mean-field framework for attention-head training and clarify the additional compactness, landscape, and curvature assumptions needed to pass from stationarity to convergence and stability.

We develop a distributionally robust formulation of principal component analysis that minimizes worst-case reconstruction risk over distributions lying within a Wasserstein neighborhood of the empirical measure. The Wasserstein neighborhood, viewed as an ambiguity set of distributions, is adaptively calibrated through a transport matrix $G$ to capture heterogeneous uncertainty across dimensions. The homogeneous case, in which G is a scalar multiple of the identity matrix, recovers classical PCA. Under a general transport matrix G, we derive a dual characterization of the associated minimax optimization problem and introduce a tractable surrogate objective function consisting of the square-root empirical reconstruction error plus a geometry-dependent residual exposure penalty. The exact and surrogate estimators are shown to be consistent for the population PCA subspace and asymptotically equivalent at the projector level. The transport geometry is allowed to be data adaptive, while the Wasserstein radius is calibrated via robust Wasserstein profile inference, yielding a data-driven radius of order $n^{-1/2}$. Comprehensive theoretical guarantees are established, including consistency and local Grassmannian asymptotics exhibiting an explicit Wasserstein-induced drift determined by the limiting transport geometry and calibration level. Numerical experiments and a real-data application demonstrate that the proposed method can substantially improve finite-sample out-of-sample performance under structured covariance shifts, moderate contamination, and certain same-distribution regimes.

We derive the optimal quantizer of a real-valued random variable $W$ with distribution $P_W$ such that 1) the distribution of the quantization output $X$ that can take $k$ values follows any specified distribution $P_X$ over $\{1,\ldots,k\}$, and 2) the minimum mean squared error (MMSE) of estimating $W$ from $X$ is minimized. It is shown that the optimal quantizer takes the form $X=\sigma\big(F_{\sigma^{-1}(X)}^{-1}(F_W(W))\big)$, where $\sigma$ is the optimal permutation of $\{1,\ldots,k\}$ among all permutations to minimize the MMSE, and $F$ is the cumulative distribution function. When $P_W$ is uniform over an interval or $P_X$ is uniform over $\{1,\ldots,k\}$, the quantizer takes a simple form $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.

Let $\mathcal{C}$ be any stable presentably symmetric monoidal $\infty$-category. In this paper, we characterize $\mathrm{Shv}(-,\mathcal{C})$ on locally compact Hausdorff spaces as the unique six functor formalism satisfying a list of very natural properties. As a consequence, we deduce that every continuous six functor formalism $D$ in the sense of Zhu is equivalent to $\mathrm{Shv}(-, D(\mathrm{pt}))$.

We survey recent progress in the classification of hyperkähler and Hermitian gravitational instantons (i.e., complete noncompact 4 dimensional Ricci-flat manifolds with quadratic curvature decay), as well as the construction of non-Hermitian gravitational instantons via harmonic maps. We also present a list of open questions related to gravitational instantons.

We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.

In 2023, Greaves, Syatriadi, and Yatsyna found a set of $57$ equiangular lines in $\mathbb{R}^{18}$, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of $57$ equiangular lines in $\mathbb{R}^{18}$ as affine equiangular sets in an integral overlattice of $A_9^2 \oplus A_1$. In this paper, we construct further sets of $57$ equiangular lines in $\mathbb{R}^{18}$ from Latin squares of order $6$ and Pasch configurations, realized as affine equiangular sets in an integral overlattice of $A_5^3 \oplus A_1^4$. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.

We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \[V\] on \[\mathbb S^{n-1}\]. For graph domains \[r<R-V(\theta)\], the first band satisfies \[ \lambda_{j+1}=\alpha^2+\frac{\pi^2}{R^2} +\frac{2\pi^2}{R^3}\bigl(\eta_j(T_n+V)-b_0\bigr)+o(R^{-3}), \qquad j=0,1, \] where \[T_n\] is the nonlocal spherical operator with multiplier \[\psi(\ell+\alpha)-\psi(\alpha)\]. Consequently the horoconvex fundamental gap has the sharp \[D^{-3}\] polynomial scale, and the leading large-diameter constant is characterized by the compact variational formula \[ 16\pi^2\inf_{V\in\mathcal A_n} \bigl(\eta_1(T_n+V)-\eta_0(T_n+V)\bigr). \] Geodesic balls realize the polynomial scale, but an explicit admissible axial perturbation lowers the reduced leading-constant value at first order.

Any genus-$g$ hyperelliptic curve $C$ defined over an algebraically closed field of characteristic $p \geq 3$ can be written in a Rosenhain form as $y^2 = x(x-1)\prod_{i=1}^{2g-1}(x-\lambda_i)$. In this paper, we first show that, if $C$ is superspecial, then each of $\lambda_i,1-\lambda_i$, and $\lambda_i-\lambda_j$ is a square in $\mathbb{F}_{p^2}$. As an application, we propose a new algorithm for enumerating superspecial hyperelliptic curves in small characteristic. By implementing our algorithm, we successfully computed the number of isomorphism classes of such curves of genera $4$ and $5$ in all characteristics $p \leq 41$, and of genus $6$ in all characteristics $p \leq 31$.

Godsil and Sun asked whether, for a strongly regular graph $X$ and any two different edges $e$ and $f$, the edge-deleted graphs $X\setminus e$ and $X\setminus f$ are degree-similar. We give an affirmative answer to the problem of Godsil and Sun. In fact, we prove the stronger statement that if $X$ is a $1$-walk-regular graph, then for any two edges $e$ and $f$ of $X$, the graphs $X\setminus e$ and $X\setminus f$ are orthogonally degree-similar. The proof is based on an edge version of the orthogonal-intertwiner method: the equality of the Gram matrices of the projected endpoint vectors in every eigenspace yields an orthogonal matrix commuting with the adjacency matrix and sending one pair of ordered endpoint vectors to the other.

We prove a conjecture of D. B. Shapiro, which asserts that no integral [10,10,16] formula can arise as a restriction of any Hurwitz-Radon formula.

We study smooth parameterized non-separable objectives of the form $\mathcal L(\theta)=F(Y(\theta))$, where a global operator couples the outputs of all training samples. We analyze two stochastic gradient constructions within this output-space framework: the ideal per-batch gradient $G$, obtained by row sampling of the full output gradient, and a cached surrogate $H$ for parameterized updates in which full-data quantities are expensive to evaluate. For uniformly sampled batches, the ideal estimator has a deterministic batch-size scaling and a second moment controlled by the full output gradient. Using this structure, we establish local high-probability confinement and convergence for fixed positive step sizes on a prescribed parameter neighborhood under a power-type gradient inequality, with linear and polynomial regimes determined by its exponent. For the cached surrogate, a Lyapunov function couples the objective gap with the cache error and yields analogous local results on the corresponding non-exit event. The theory includes explicit path-length, confinement, and rate bounds; numerical experiments illustrate finite-budget decay trends and the effects of global output-gradient sampling, batch restriction, and cache tracking.

This paper analyzes bidirectional random projections for ordinary least squares (OLS) regression under the fixed design setting. Let $(X,Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$ be a sample and $R \in \mathbb{R}^{n_1 \times n}, W \in \mathbb{R}^{p \times p_1}$ be two properly distributed random projections. We develop an expected excess loss bound for the OLS estimator built on $(WXR, WY)$. Compared to an established bound for OLS estimator built on $(XR, Y)$, the gap is approximately $O\left( p_1 + C \frac{1}{p_1} \right)$, where $C$ scales with $n_1/n$ and can be negative for small $n_1/n$. Its implications are confirmed by numerical results on real-world data.

Cancer treatment is at the core a sequential decision-making problem with partial observability, latent patient heterogeneity, and explicit constraints on the budget for medical measurements. Unlike standard Reinforcement Learning (RL) approaches that control state trajectories, cancer treatments permanently modify patients' transition dynamics, changing how states evolve over time. We model cancer treatment as a belief-space planning problem using active inference, deriving an expected free-energy objective that unifies goal-directed control and information acquisition under measurement budgets without. We implement this framework using real clinical cancer data from the AACR Project GENIE Biopharma Collaborative dataset. Results on clinical data demonstrate a simultaneous patient categorization and high treatment efficacy, under real measurement and treatment constraints.

This paper studies the algebraic structure underlying a broad class of information-theoretic security problems. We define the equation asymmetry degree (EAD) as $\Phi = (n - r)/n$, where $n$ is the signal embedding dimension and $r$ is the effective rank of the adversary's observation matrix. This single parameter is shown to simultaneously govern both secrecy (measured by equivocation $H(M|Y_E)$) and covertness (measured by detection error probability $P_e$). On finite fields $\mathbb{F}_q$, we establish the equivocation lower bound $H(M|Y_E) = \min(k, n - r_E) \log q$ with exact probabilistic conditions (Theorem~1), the secrecy capacity $C_s = (n - r_E) \log q$ with complete achievability and converse proofs (Theorem~2), and a strong converse (Theorem~8). In the continuous Gaussian regime, we derive a differential-entropy equivocation bound (Lemma~1), the high-SNR secrecy capacity asymptotics (Lemma~2), and a 2-Wasserstein distance covertness condition $W_2 \approx \sqrt{r_W} \cdot P / (2N\sigma) \to 0$ (Theorem~5'). The EAD-SDoF equivalence $d_s = n \cdot \Phi$ is established (Theorem~7). Both $\eta_s$ and $\eta_c$ are shown to be monotone functions of $\Phi$ (Theorem~6), with a Pearson correlation of $0.997$ in continuous-domain experiments. Seven existing security schemes – matrix embedding, MIMO wiretap, secure network coding, FRFT multi-angle transmission, traffic steganography, group-key secure summation, and MDS secure summation – are unified under the common form $C_s = (n - r) \log q$. Post-quantum security follows from the information-theoretic hardness of underdetermined linear systems (Theorem~9). All numerical experiments are reproducible with open-source code.

We study a categorical approach to the concept of varieties of chiral algebras. We prove that the class of chiral algebras in the variety defined by a binary quadratic operad Var, whose conformal structure is abelian, coincides with the class of differential algebras in the variety defined by the Manin black product of the operads Var and Com, where Com is the operad of associative commutative algebras.

In recent years, numerous polynomial invariants of knotoids have been constructed, some of which are defined with the signs of the crossings. In this paper, the coloring-allowed invariants of planar knotoids, which is a class of planar knotoid invariants defined with the coloring number are introduced. We demonstrate several basic properties of the coloring number and some examples of coloring-allowed invariants, among which the 4-phases functions are discussed in detail, including their invariance and properties.

The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler{-}Ricci flow coupled with $(1,1)$-forms, and the Laplacian flow.

Higher-order ODE solvers have shown strong empirical promise for accelerating diffusion models through the probability flow ODE, but rigorous non-asymptotic guarantees for such acceleration remain limited. In this paper, we develop a Chebyshev–Gauss–Seidel higher-order sampler and establish a non-asymptotic convergence guarantee that allows the approximation order to grow logarithmically with the number of outer iterations. In the exact-score setting, up to logarithmic factors, the proposed sampler requires at most \[ d^{1+o_T(1)}\varepsilon^{-1/K_1} \] score functions to approximate the target distribution on \(\mathbb{R}^d\) within total variation distance \(\varepsilon\), where \(o_T(1)\to 0\) as \(T\to\infty\) and \(K_1>0\) is a sufficiently large constant. The analysis assumes only a polynomial second-moment bound on the target distribution, thereby relaxing the bounded-support condition imposed in existing higher-order theory. Moreover, the guarantee is robust to score and Jacobian estimation errors and does not require higher-order smoothness assumptions on the score estimates. Numerical experiments on anisotropic Gaussian mixture benchmarks support the predicted improvement in the accuracy–cost tradeoff under finite score-evaluation budgets.

We give formulas for the $ \tau $ and $ \varepsilon $ concordance invariants of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a $ \tau $ formula for squeezed patterns as well. Towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.

We study a bilevel network interdiction problem, with an attacker interdicting (attacking) certain components of a network and a defender optimizing operations over the ensuing network. We study when the defender's optimal objective is submodular or supermodular with respect to the attacker's interdiction decisions, for optimizing the bilevel integer program more efficiently. We first consider the min-cost flow (MinCF) interdiction problem and derive necessary and sufficient conditions for the supermodularity or submodularity to hold under three types of attacks, respectively on supplies/demands, flow capacities, and cost coefficients. We extend to other variants, including capacitated facility location, maximum flow (MaxF), and shortest path (SP) interdiction. The conditions hold under general network topologies and parameter settings, and depend solely on the locations of the attacks. We further incorporate additional network information (e.g., detailed parameters and special topologies) to establish less restrictive conditions. We also derive necessary and sufficient conditions for supermodularity or submodularity in SP and MaxF interdiction in series-parallel networks. Furthermore, we explore more challenging interdiction problems where the defender may make additional binary decisions (e.g., repairing or reinforcing the network) and identify conditions that preserve submodularity or recover supermodularity. Via extensive numerical studies with diverse types of attacks, we demonstrate an order-of-magnitude computational speedup achieved by exploiting these properties and generating valid inequalities, for solving network interdiction at scale.