Papers

Pair arithmetic omits the renormalization stage performed at the end of each operation of multi-component multiple-precision arithmetic, reducing the operation count and, by eliminating conditional branches, easing SIMD vectorization. Renormalization cannot be omitted entirely, however, because iterative solvers may then fail to converge. We propose RenormBF-pair, a branch-free fixed-trip-count renormalizer parameterized by the word count \(K\) and the input length \(n\): a VecSum sweep, a tail fold, and \(r\) rounds of a FastTwoSum chain, costing \(6(n-1)+(n-K)+3r(K-1)\) flops. Over \(200{,}000\) trials per condition, the smallest round count producing no non-overlap violation was \(r=1\) for \(K \leq 3\) and \(r=2\) for \(K=4\), whereas the same cost spent on a VecSum\(K\)-style TwoSum chain fails for \(K=4\). We delimit with explicit counterexamples both the input condition under which the sum relation holds and the fact that the scheme does not, in general, guarantee a correctly rounded lowest component. Integrating it into CG and BiCGStab over pair arithmetic in binary64 and binary32, we sweep \(240\) paired runs including real matrices screened from SuiteSparse. The proposed scheme and VecSum\(K\) returned bitwise identical words wherever both succeeded, so they are indistinguishable for CG in binary64; differences appear only where VecSum\(K\) breaks, namely binary32 and BiCGStab. There the \(22\)–\(16\) win–loss record is not significant (sign test, \(p=0.42\)); the clear bias is structural, non-overlap violations dropping from \(66\) rows to \(37\). The cost is \(+3.4\%\) for the whole CG solver.

We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.

Every solution for the classical four body problem in space which ends in total collision has a limiting shape. We prove that when that shape is planar the entire solution lies in a fixed plane and when that shape is collinear the entire solution lies in a fixed line.

Let $g=w|dz|^2$ be an affine special Kähler metric on a punctured disc and let $\Xi=q(z)\,dz^3$ be its associated holomorphic cubic differential. We prove that if the $g$-area of concentric punctured neighbourhoods grows at most polynomially, then $q$ is meromorphic at the puncture. More precisely, an area bound of order $\eps^{-N}$ forces every pole of $q$ to have order strictly less than $N+3$. Hence an essential singularity of the cubic differential rules out every polynomial area upper bound. The proof associates to $(g,\Xi)$ a curvature $-1$ conformal pseudometric and applies the Ahlfors–Schwarz lemma, followed by the submean inequality for holomorphic functions. We also discuss the identically zero cubic differential and the borderline logarithmic growth responsible for the strict pole-order bound.

A square is a word $xx$, where $x$ is nonempty. We show that a circular word of length $n$ contains at most $\lfloor 3n/2 \rfloor$ distinct squares of length at most $n$. The proof combines known results relating squares to circuits in Rauzy graphs. The coefficient $3/2$ agrees with the known lower bound.

Deploying AI systems requires deciding when to delegate tasks and when humans should intervene to monitor and mitigate risk induced by AI operations. These decisions are challenging when failures cluster: a hallucination or harmful output can trigger further errors, creating periods of elevated risk. We introduce a continuous-time framework for learning adaptive human oversight under such turbulent AI risk. Existing oversight and delegation formulations condition on history but do not model incident clustering, or its suppression by supervision effort, jointly with the delegation decision and this study fixes this gap. The self-exciting dynamics capture how risk events increase the likelihood of subsequent events, making their timing and history central to decision-making. We formulate a stochastic control problem combining human actions, monitoring effort, and switching between human-AI-assisted operation and full AI delegation, balancing operational rewards against oversight costs, and cascading AI-failures and induced uncertainty. Human participation is an endogenous component of risk management: the policy determines both when oversight is needed and how much effort to allocate. We study a relaxed switching formulation and propose Hawkes-PPO, a policy-gradient method that uses a bank of exponential filters of observed incident times. In a synthetic environment it attains a higher risk-adjusted objective than either fixed regime and approaches an approximate full-information oracle. We illustrate our results with numerical simulations by examining how cascade risks influence intervention and delegation, connecting reinforcement learning with adaptive human oversight of AI systems. In particular, we illustrate the benefit of our switching strategy and Hawkes-PPO algorithm to monitor the project efficiently along time, reducing turbulent risks occurrences and costs.

We study filter-Laver forcing on omegaxomega with successor sets containing final quadrants. The observation retaining the minimum coordinate and orientation is a complete projection. Its finite nontrivial hidden-label quotients are Cohen, whereas the full offset quotient has Boolean density equal to the ground-model dominating number dV . For the associated grid-meager ideal I_box, we prove cov(I_box) = add(M) for the usual meager ideal M, and obtain further cardinal bounds. We also analyze threshold games, oracle degrees and the distinction between positive and filter-large certificate sets: recurrent filter-large acceptance admits refinements whose every branch succeeds. As an arithmetic application, we prove Wilfs inequality c \leq 13e for numerical semigroups with thirteen elements below the conductor c, where eis the embedding dimension. A complete arithmetic proof, including a bounded exact computation, is given in the appendix. Its terminating certificate procedures supply a computable quadrant tree and uniform refinements for every forcing condition. Finite support kernels describe persistence of the registered proofs; arithmetic absoluteness explains precisely why forcing organizes these certificates without replacing their arithmetic justification.

We study the behavior of the characteristic and chromatic polynomials of certain families of graphs and their relation to homomorphism counts, focusing on polynomial dependence of the coefficients at each fixed codegree on the graph family parameter. We prove that the characteristic polynomials of strongly regular graphs with polynomial parameters form a two-level polynomial of infinite depth in the sense of Bogart and Woods. In contrast, the chromatic polynomials of Paley graphs have maximal constant depth two: their coefficient of codegree three is not eventually polynomial in the number of vertices. We also prove that both the chromatic and characteristic polynomials of every strongly polynomial graph sequence, in the sense of de La Harpe and Jaeger, have infinite depth. Finally, we investigate homomorphism counts between two strongly polynomial graph sequences. We show that infinite depth need not hold in general, but we construct target sequences for which it holds for every strongly polynomial source sequence.

This paper presents a constrained iterative Linear Quadratic Regulator (Box-iLQR) method for nonlinear aerospace optimal-control problems with bounded states and controls. Logarithmic barrier functions are used with a two-fold objective: the barrier parameter is reduced toward zero to recover the constrained optimal trajectory, while a finite barrier is retained to construct a smooth, constraint-aware feedback policy for closed-loop operation. The method is demonstrated on spacecraft attitude control, a minimum-fuel planar orbit transfer, and an L2 to L2 halo-orbit transfer. Box-iLQR respects the imposed constraints and produces nominal trajectories together with time-varying feedback policies. Closed-loop simulations under disturbance, measurement noise, and control uncertainty show substantial reductions in terminal error. These results demonstrate that Box-iLQR provides both high-accuracy constrained trajectory optimization and practical constraint-aware feedback within a single framework.

Experiments have shown that modifications in communication hardware can enable unauthorized exfiltration of data from transmitters. We study the resulting Transmitter Trojan channel, in which a malicious encoder embedded within a legitimate transmitter perturbs the transmitted codeword on a symbol-by-symbol basis. The model reflects the limited footprint, delay, and computational complexity available to such hardware modifications. We derive achievable rogue message rate and achievable rates for the legitimate message under the mismatch induced by the Trojan. We also analyze decoding error exponents for the legitimate and rogue receivers, including operational exponent lower bounds obtained through channel-resolvability transfer arguments. Finally, we study Trojan detection at the legitimate receiver under different levels of statistical knowledge, including likelihood-ratio detection when the ensemble-averaged induced channel is known and goodness-of-fit detection when only the nominal channel is known. The resulting detection error exponents, together with the achievable rate and decoding error exponents, quantify tradeoffs among rogue message rate, legitimate message rate, decoding reliability, and Trojan detectability. We specialize the analysis to the binary symmetric channel and the additive white Gaussian noise channel, and illustrate the resulting tradeoffs numerically.

We survey Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type, with particular emphasis on sharp constants and extremal functions. We discuss both classical results and recent developments, including limit relations between Nikolskii constants for polynomials and entire functions of exponential type, weighted and multidimensional versions, and inequalities on constrained classes of functions. Special attention is given to the difficult case $p=1$ and to methods based on reproducing kernels, duality, and extremal approximation. We also describe connections of sharp Bernstein–Nikolskii inequalities with approximation theory, the Riemann zeta function, spherical designs, Turán–Delsarte type problems, and sphere packing. Several open problems concerning sharp constants, extremal functions, and multidimensional extensions are formulated.

The quantum baker map is a standard model of quantum chaos, but rigorous analysis of how it spreads random perturbations remains difficult. Following the repeated “baker plus refreshed diagonal perturbation” architecture of Schack and Caves, we study a tractable strong-noise model based on the Balazs–Voros quantization: at each time step, its unitary propagator is followed by a diagonal unitary whose entries are independent and uniform on the unit circle. The resulting random propagator is a structured random walk on $\U(N)$, $N=2^k$. We quantify how quickly the baker dynamics spreads this basis-local randomness at three levels: ensemble means, two-copy statistics, and the full law of the accumulated propagator. The walk becomes an exact unitary $1$-design at time $k+1$, while its fixed-accuracy $2$-design mixing time is $\Theta(\log N)$. Thus the induced state ensembles reproduce Haar means and two-copy statistics on the Ehrenfest scale. The full law mixes much more slowly. A phase-adapted path coupling gives normalized Wasserstein mixing in $O_\varepsilon(N)$ steps, whereas metric-entropy and Haar small-ball estimates give an $\Omega_\varepsilon(N/\log N)$ lower bound and asymptotically maximal distance at every $t=o(N/\log N)$. The law remains singular with respect to Haar measure for every $t<N$. The moment results and the Wasserstein upper bound are driven by the unistochastic Markov kernel associated with $B_N$, an affine dyadic chain whose nonconstant Fourier modes vanish exactly after $k$ steps. The resulting hierarchy separates Ehrenfest-scale randomization of state ensembles from near-Heisenberg-scale exploration of the full unitary group.

In this paper, an arbitrary Lagrangian-Eulerian (ALE)-based finite element method (FEM) is developed and studied in a monolithic framework for a class of two-phase Stokes flow problems with moving interfaces and jump coefficients, where the mixed finite element approximation to Stokes moving interface problems is established and analyzed in both semi- and fully discrete schemes based on the ALE formulation. The key analytical technique involves a specific H^1-projection associated with the ALE-induced mesh motion due to the evolving interface. Optimal convergence properties of the proposed H^1-projection and its ALE temporal derivative are proved in both H^1 and L^2 norms, with which optimal error estimates are obtained for both semi- and fully discrete mixed finite element approximations to the studied Stokes moving interface problem in both H1 and L2 norms as well. Numerical experiments are carried out to validate all derived theoretical results. The developed analytical approach can be extended to Taylor-Hood and MINI mixed elements, as well as to more general two-phase flow problems.

We propose a conjecture on how to recover an algebraic curve from the isogeny classes of its Prym varieties. Motivated by this conjecture, we analyze constructions of pairs of curves with isogenous Jacobians and several isogenous Pryms.

It is shown that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb{F}_2)$ is singly generated, where $\mathbb F_2$ is the free group on two generators. With respect to the generator constructed in the paper, the topological free entropy dimension of $\mathrm{C}_r^*(\mathbb{F}_2)$ is shown to be $2$.

Recent developments in theories of partitions, quasimodular forms, multiple zeta values and $q$-multiple zeta values have led to connections between partitions and primes. Central to this progress are prime-detecting quasimodular forms, which are $q$-series whose non-negative coefficients vanish if and only if the input is prime. In this paper, we use properties of these quasimodular forms to motivate a proof of the infinitude of primes, to construct a novel partition theoretic factorization algorithm, and to show that the distribution of these coefficients relate to existence of primes in intervals. We also prove analogs of these results for primes in arithmetic progressions and discuss potential extensions of our results, including to Linnik's problem.

We study the VC-dimension of distance graphs of large subsets of vector spaces over finite fields. For $3\leq k\leq d$, we prove the distance graph of $E\subset \F_q^d$ has VC-dimension at least $k$ provided that $|E|>Cq^{d-\lfloor \frac{d-k}{3}\rfloor-1}$. We also show that this exponent is sharp in the $k=d$ case.

This work studies the expansion of the level sets of the solutions of a Fisher-KPP type reaction- diffusion equation, the diffusion being given by a nonnegative, compactly supported, even kernel. Because the latter is not assumed to be spherically symmetric, the level sets advance at an asymptotic speed given by a Freidlin-Gärtner type formula. The main contribution of this paper is to push the expansion up to terms that vanish as time goes to infinity, consisting in an asymptotically logarithmic delay corrected by an function that is constant in time, but depends on the direction of propagation.

We introduce Distance-Residual Physics-Informed Neural Networks (DR-PINNs), a physics-informed learning framework for approximating solutions of ordinary and partial differential inclusions (DIs), governing laws in which a differential operator is constrained to lie in a set-valued map rather than equaling a prescribed function. The method replaces the classical pointwise PDE/ODE residual by the squared distance from the differential operator to the admissible set. This distance vanishes exactly when the inclusion is satisfied and measures the infimal correction needed for the operator to enter the admissible set. For a fixed closed convex admissible set, the squared distance is differentiable with respect to the operator value, with gradient given by the metric projection. When the admissible set also depends on the network state, that dependence is included through the chain rule. The framework encompasses ordinary and partial DIs with set-valued reaction terms. For both settings we prove consistency: under the stated closedness, measurability, convexity, and growth assumptions, any sequence of candidates satisfying the initial (and, in the parabolic case, boundary) conditions whose continuous distance-residual functional tends to zero admits a subsequence converging to an exact solution of the target inclusion. These are conditional statements for the continuous distance-residual functional; they do not cover the finite-collocation training loss, the behavior of the optimizer, or convergence rates. For admissible sets given as convex hulls of finitely many vertices, projection onto the set reduces to a small convex quadratic program, making the loss efficiently computable inside the training loop. Numerical experiments demonstrate high accuracy on the differential-inclusion benchmarks.

For $1<p<\infty$, we characterize the boundedness of the segment multiplier on $L^p(w)$ by a large-scale Muckenhoupt condition, obtained by restricting the classical $A_p$ condition to intervals of length at least one. The proof uses a discretization involving the discrete Hilbert transform on an associated weighted sequence space.

The classical method of Kaczmarz, introduced in 1937, is a textbook iterative method for solving linear systems $A x = b.$ Despite its widespread use, particularly in the context of solving inverse problems where it is commonly included in software packages within Matlab, Python, and Julia, the precise characterization of convergence has long been deemed difficult to obtain. While different bounds on convergence rates have been established, they are largely unsatisfying as they cannot explain the classical, cyclic-update method's efficient convergence in practice. In this work, we obtain a tight characterization of convergence of the classical Kaczmarz method, establishing both linear and sublinear convergence bounds. The worst-case tight (i.e., exactly attained by some instances in the considered family) bounds are expressed in terms of a fixed matrix that depends only on $A,$ but are not fully interpretable in terms of the matrix spectrum and row correlations, which had been observed to have an impact on convergence. We thus provide relaxations of these bounds that are fully expressible in terms of matrix row norms, row correlations, rank, and extremal positive singular values. The provided relaxed bounds explain one-cycle convergence in special cases where the matrix rows are all either parallel or orthogonal to each other. We further argue that the dependence on different parameters appearing in the bounds is necessary and within a small constant factor of the best attainable in the worst case. Finally, our bounds explain why the classical cyclic update is faster than the randomized one when matrix rows are weakly correlated, which is often observed in inverse problems where the cyclic method is used.

For a positive integer $\alpha$, let $p_\alpha(n)$ denote the number of $\alpha$-colored partitions of $n$. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real $\alpha>0$, but when $\alpha>24$ it involves several polar terms and a two-parameter family of exponential sums $A_k^{(\alpha)}(n,m)$. For integral $\alpha$, we prove multiplicativity and prime-power reduction formulas for these sums, expressing their local factors as classical or quadratically twisted Kloosterman sums. Together with explicit truncation and precision bounds, these formulas yield an efficient algorithm for computing $p_\alpha(n)$ exactly. Our SageMath implementation computes the 1,113,767-digit integer $p_{100}(10^{10})$ in less than one hour. As an application, we use the algorithm to certify new Ramanujan-type congruences.

This is a personal essay about how I see math, what I value about it, and what it means to me.

Physical systems evolve continuously in time, yet their states are typically observed only at discrete times. Generating trajectories consistent with their probability densities from such observations therefore requires capturing the continuous-time evolution rather than only learning transition mappings between consecutive observations. We propose PhiBE-Flow, a framework that directly estimates the probability velocity field induced by the stochastic differential equation (SDE) which governs this continuous-time distributional evolution. PhiBE-Flow learns from discrete observations using a model-free approach requiring neither known SDE coefficients nor score estimation. We establish convergence guarantees for the method, accounting for both time-discretization and finite-sample errors. We evaluate PhiBE-Flow on systems of increasing complexity, from controlled stochastic numerical systems to Navier–Stokes dynamics and real-world videos. Our results show that PhiBE-Flow accurately recovers probability flows of stochastic dynamics, preserves multiscale physical statistics, and improves video generation performance over representative baselines. The code is available at https://github.com/R1fe/PhiBE-Flow.

We consider the Lorentz positive cone of $n\times m$ matrices that map the Lorentz cone in $\Rbb^m$ into the Lorentz cone in $\Rbb^n$. The Lorentz positive cone and its dual, the cone of Lorentz separable matrices, are shown to provide polynomial-time algorithms for the problem of minimizing a bilinear objective over variables contained in ellipsoids in $\Rbb^n$ and $\Rbb^m$. We also demonstrate how these cones can be used to strengthen SDP relaxations of other nonconvex quadratic optimization problems, including the two-trust-region subproblem.

The Erdős–Sós theorem states that every graph of average degree greater than $t-2$ contains every tree on $t$ vertices. A short counting proof was discovered by GPT-6 Astra in 2026. We give a visual, reader-centered exposition of that argument whose presentation differs substantially from the original. We recast the counting objects as early neighbors in a reveal-and-stop procedure, organize the induction through explicit partitions and reversible swaps, and develop the proof through worked examples with consistent drawings. We also give direct and probabilistic conclusions, compare this formulation with other recent expositions, and record the classical consequence $R(T;q)\le q(t-2)+2$ for multicolor Ramsey numbers of trees.

We study transfer systems on partially ordered sets whose Hasse diagrams are rooted trees. We prove there is a bijection between transfer systems on a tree and order-preserving functions to the natural numbers that are bounded by the rank function. We then develop a recurrence for the set of functions and use this to enumerate transfer systems on some families of trees. In addition to our study of trees, we prove for any finite poset $\mathcal{P}$, the lattice of transfer systems on the opposite poset $\mathcal{P}^{op}$ is isomorphic to the opposite lattice of transfer systems on $\mathcal{P}$. We show this by establishing a bijection between weak factorization systems and transfer systems, which builds on the results for lattices shown in previous work of Franchere, Ormsby, Osorno, Qin, and Waugh.

The $b$-invariant of a normal graded ring was defined recently by Okuma, Watanabe, and Yoshida. We study this for invariant rings of finite groups, proving that it is nonpositive in the case of abelian groups, but not necessarily otherwise; more generally, we give a description of the anticanonical module for invariant rings of finite groups. Turning to infinite groups, we record the anticanonical module and the $b$-invariant for families of determinantal rings. We also construct standard-graded Cohen-Macaulay rings of characteristic zero, with isolated singularities and negative $b$-invariant, that are not of strongly $F$-regular type, thereby disproving a conjecture of Okuma-Watanabe-Yoshida.

We prove that, in every closed Riemannian three-manifold $(M^3,\bar g)$, there exists a constant $C>0$ such that every closed smoothly embedded minimal surface $\Sigma\subset M$ satisfies $b_1(\Sigma;\mathbb{Z}_2)\le C\bigl(\operatorname{Ind}(\Sigma)+\operatorname{Area}(\Sigma)\bigr)$. The estimate holds without orientability or two-sidedness assumptions and establishes the additive genus-index-area estimate conjectured by Song. For closed connected orientable two-sided minimal immersions, we also obtain explicit genus bounds under a lower bound on the ambient sectional curvature. Under positive ambient Ricci curvature, we prove the universal inequality $\gamma(\Sigma)\le 8\operatorname{Ind}(\Sigma)$.

Spectral risk measures, including conditional value-at-risk (CVaR), generate a family of convex criteria for portfolio estimation. We study how the criterion should be chosen when different criteria share the same population minimizer, and whether the efficient criterion can itself be learned from data. We develop a general asymptotic theory for empirical spectral-risk minimization under estimated linear constraints. Under normal scale-mixture elliptical returns, all spectral risk measures identify the same population efficient portfolio, but their empirical minimizers have different sampling distributions. Their asymptotic covariance decomposes into a common component and a positive-semidefinite component scaled by a functional of the spectral measure, reducing efficiency to an optimization over probability measures. We characterize the efficiency-optimal spectral measure and show that single-level CVaR is generally inefficient. We then construct a fully data-adaptive estimator that learns the radial distribution and optimal spectral measure from the same observations used for portfolio estimation, yet has the same first-order distribution as the infeasible oracle. Simulations and an empirical application illustrate the method.