Papers

Reduced and mixed precision have moved from a niche optimization to a central design axis in scientific computing and engineering, driven by energy constraints, heterogeneous accelerators, and the convergence of simulation and machine learning. This paper organizes the landscape around seven coupled themes—number formats, floating-point emulation, emerging architectures, hardware/software co-design, relation to other approximations, software design, and precision as a multilevel resource —and, for each theme, synthesizes the state of the art, future directions, and open questions. We emphasize energy per trusted solution as the core objective, and we frame \say{recklessly responsible} computing as a pragmatic doctrine: exploit low precision aggressively, but with systematic detection, escalation, and certification pathways.

We prove dimension-independent Hyers–Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here $\varepsilon\geq0$ is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most $9\varepsilon/2$ in the trace metric and $2\sqrt2\varepsilon$ in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to $3\varepsilon$ even under the hypothesis of $\delta$-surjectivity, independently of $\delta$. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as $\varepsilon\to0$, but no such modulus can be chosen independently of dimension.

The Wagner-Preston theorem states that every inverse semigroup can be embedded into an inverse semigroup of partial bijections on some set. However, this embedding does not respect lattice operations in the natural ordering. In this paper, we show that an abstract pseudogroup can be faithfully represented by partial bijections on a set if and only if its frame of idempotents is spatial. More generally, we show that any abstract pseudogroup can be faithfully represented by partial homeomorphisms on a locale.

Generalized reaction-diffusion systems encompass diverse transport mechanisms and coupled reaction kinetics. A central question for neural PDE solvers is what should be learned so that a common interface can accommodate phase-field and degenerate transport, local reactions, and multispecies coupling. We propose the Neural Constitutive Laws–Mass-Compression-Transport (NCL-MCT) Solver, which learns PDE-specific constitutive responses while retaining temporal evolution in a shared MCT integrator. Transport is represented through mobility and thermodynamic driving force, and reaction through relative reaction rates. These constitutive responses depend on the current density rather than explicitly on the initial condition or elapsed time, motivating their reuse across different initial conditions and time horizons. The same interface supports velocity-data supervision and known-law supervision, neither of which requires time integration during training. When constitutive laws are known, supervision can be evaluated on independently sampled density fields, enabling trajectory-free constitutive learning without generating solution trajectories. Across seven systems, separately trained constitutive modules share the same interface and MCT integrator and achieve relative rollout $L^2$ errors of $10^{-4}$ to $10^{-2}$. Tests with unseen initial-condition families and an extended time horizon assess reuse beyond training conditions, while separate experiments demonstrate trajectory-free constitutive learning. These results support constitutive responses as an effective learning target for a shared neural PDE framework.

We study the $\mathrm{v}$-numbers of ordinary and symbolic powers of the facet ideal and the Stanley–Reisner ideal of a chessboard complex. Exact formulas are established for the $\mathrm{v}$-numbers of ordinary and symbolic powers of the Stanley–Reisner ideal, as well as symbolic powers of the facet ideal. Furthermore, we show that the $\mathrm{v}$-number function of the ordinary powers is asymptotically linear.

Momentum-dependent symmetry actions can turn the even-integer constraint of a $2\mathbb{Z}$ classification of topological insulators and superconductors into an odd-integer constraint without changing the symmetry algebra. For internal and order-two crystalline symmetries in up to four spatial dimensions, we derive parity relations between the Chern or winding number of a gapped Hamiltonian and topological invariants of its momentum-dependent antiunitary symmetry matrices. Whenever the symmetry action enforces odd parity, any symmetry-preserving gapped phase must be topologically nontrivial. Using cellwise actions, which are represented by momentum-independent matrices in a fixed unit-cell basis, as a reference, we determine how locality constrains the realization of odd topological numbers. Explicit model constructions and no-go theorems distinguish three cases: odd values can be realized by finite-range symmetry actions, can be realized by exponentially decaying quasilocal actions but not by finite-range actions, or are impossible in finite-dimensional Bloch systems.

This work is concerned with the numerical computation of Gross-Pitaevskii ground states using the tensor train (TT) format. We regard the problem as the minimization of the Gross-Pitaevskii energy functional under a unit mass constraint, and propose a first-order Riemannian optimization scheme that preserves both the mass constraint and the low-rank representation throughout the optimization, by operating on the manifold of unit-norm functions of fixed TT-rank. On this manifold, we derive the energy-adaptive Riemannian gradient and show how to compute it efficiently in the TT format. Spatial discretization is performed using a spectral method with numerical integration, which allows the required computations to be carried out efficiently while preserving the low-rank format. The method is shown to substantially reduce computational time compared to full-rank computations, while maintaining accuracy for both single- and multicomponent Gross-Pitaevskii equations.

This paper extends the formalism of stochastic path-space thermodynamics by systematically expanding the space of thermodynamic variables with a spectrum of boundary and relative functionals of random processes. Generalizing the approaches introduced in preprints arXiv:2607.24078 and arXiv:2609.08477, we investigate the fluctuation statistics of a one-dimensional Ornstein-Uhlenbeck process with an asymmetric linear drift and a step penalty potential at the origin, which models the waiting phase of a molecular Brownian motor. Utilizing the Feynman-Kac formalism and the parabolic cylinder theory, a numerical scheme based on matching logarithmic derivatives via the secant method is constructed. A high-precision eigenvalues of the effective Hamiltonian is obtained, which define the cumulant generating function. We formulate a modified path-space fluctuation theorem of the Gallavotti-Cohen type for the conjugate drift current at a fixed integral occupation time of the system in the dissipative half-plane. The physical and thermodynamic significance of the new variables is elucidated. A combination of "residence time" and "integral current" functionals is applied to describe an ion channel sensitive to mechanical or electrical stimuli.

Ding \cite{Ding2016} proposed ten conjectures on binary linear codes arising from Boolean functions. Four of them, namely Conjectures 38–41, were subsequently proved by Göloğlu and Krasnayová \cite{GologluKrasnayova2019}. In this paper, we investigate the remaining six conjectures, namely Conjectures 19, 27, 30, 33, 34, and 37. For Conjectures~19 and~27, we obtain common weight restrictions and several infinite five-weight families. For Conjecture~30, we prove that every admissible code has three, four, or five nonzero weights, and an explicit four-weight example disproves the original “three or five weights” assertion. For Conjecture~33, an infinite five-weight family is obtained. For Conjecture~34, we obtain a general $(2h+1)$-weight upper bound and give an explicit six-weight counterexample, showing that the original “three or five weights” assertion is false in general, where $h$ is a positive integer. The case $h=3$ with $3\nmid m$ is also completely determined. Finally, Conjecture~37 is completely resolved by combining the known results of Ahmadi and Shafaeiabr \cite{AhmadiShafaeiabr2023} with the treatment of the two remaining classes $(a)$ and $(b)$.

We study how actuator rank and control horizon jointly determine worst-case control performance for finite-dimensional linear systems under a fixed total actuator-gain budget. The design variable is the input covariance $X=BB^\top$. Scalar actuators give rank-one matrices $X=bb^\top$; dropping the rank constraint yields their convex hull and a semidefinite benchmark. We identify regimes in which low-rank designs necessarily fall short and others in which they attain the benchmark. At small time, the relaxed maximizer is unique and full rank. If $A$ is cyclic, then, for every $k

For a controllable single-input linear system in dimension $n$, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders $T, T^3, \ldots, T^{2n-1}$. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near $T=0$ for the eigenvalues divided by their leading powers of $T$, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein–Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker–Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct $A$ and determine $b$ up to its global sign.

Neural network conformal theories were first introduced in \cite{Halverson:2024axc} and exact two-, three- and four-point correlators were computed from an ensemble of infinite-width neural networks. In particular, the four-point correlator of any scalar operator satisfied crossing-symmetry and conformal block decomposition across different channels were in agreement without any constraint on moments. An infinite tower of degenerate operators with the same scaling dimension were found. In this note, we take a closer look at the structure of correlators and relation between operators. We also layout a recipe to construct the algebra of the theory.

We analyze exponential backoff, an algorithm used to share a single communication channel in a distributed manner between several users, in a similar way as many networking standards such as 802.11. We demonstrate that this algorithm has a meta-stable behavior, in the sense that the system oscillates over long time-scales between meta-stable configurations where only a subset of the sources effectively access the channel. In other words, meta-stability creates a form of implicit admission control. We provide both theoretical tools and numerical experiments to understand this phenomenon further. Our experiments are fully reproducible and the code is publicly available as a Python package available at https://pypi.org/project/slotted-aloha-simulator/.

We investigate numerically fluid flow through a thin porous layer, which is modelled as a lower-dimensional interface separating two bulk domains. At this interface, we impose effective transmission conditions involving homogenized coefficients rigorously derived via two-scale methods in previous works. These coefficients are determined by cell problems encoding the microscale properties of the porous layer. In a first step, we analyze qualitatively how the effective coefficients depend on the pore geometry. Based on these computations, we then present numerical simulations of the macroscopic models incorporating the effective interface conditions. Although replacing the thin porous layer by an effective interface substantially reduces the computational cost, the resulting nonstandard transmission conditions pose additional challenges for the numerical implementation. We employ a Taylor–Hood-type mixed finite element discretization with discontinuous pressure and tangential velocity across the interface and prove stability as well as error estimates for the resulting scheme. Finally, for microscale geometries that are still computationally resolvable, we compare the results of the macroscopic models with those obtained from direct simulations of the microscopic models.

We study when logarithmic regret is attainable in structured adaptive linear-quadratic regulation. The state and input matrices depend affinely on a common unknown parameter. We consider neighborhoods of a known nominal model that is stabilizable and observable through the output associated with the state cost, with radii proportional to the inverse fourth root of the horizon. The key condition is controller identifiability, which requires the optimal gain derivative to vanish in every parameter direction that leaves the closed-loop dynamics unchanged under nominal optimal feedback. Under suitable regularity conditions on the noise density, this condition at the nominal parameter yields logarithmic local minimax regret when the gain derivative there is nonzero. Failure of the condition yields square-root local minimax regret. If the nominal gain derivative vanishes, local minimax regret remains bounded. We prove that controller identifiability is equivalent to the optimal gain remaining unchanged under sufficiently small invisible perturbations of the nominal model. Motivated by this invariance, we construct a certainty-equivalent policy that estimates only the visible parameter component and attains the logarithmic upper bound under controller identifiability. The policy requires neither the horizon nor the noise law, and its guarantee holds for independent, identically distributed noise with zero mean and finite positive definite covariance.

This paper revisits Willis' elastodynamic homogenization (Mechanics of Materials, 41 (2009) 385-393) of periodic laminate composites and uses the equivalent inclusion-based method (EIM) to evaluate the unique effective constitutive properties with eigen-fields. The microstructure-specific Green's function is essential in Willis's exact homogenization method, which was derived from the corresponding periodic laminates. Our re-examination identifies several algebraic differences in the published formulae for the Floquet number and the related effective properties. In parallel, this paper simulates inhomogeneities with a comparison medium containing two eigen-fields, namely eigenstrain and eigen-momentum, to simulate stiffness and mass density mismatch, respectively. Therefore, the general Green's function can be directly used. The exact homogenization method with the updated specific Green's function is compared to the proposed EIM with eigen-fields homogenization scheme, and excellent agreement is obtained. The present paper provides a base to extend the homogenization framework to multi-dimensional Willis-type homogenization.

Evolutionary program search driven by large language models (LLMs) has produced record-breaking constructions for open problems in combinatorics and beyond. We apply this approach to the longstanding problem of improving the best-known bounds for binary linear codes. Building on the EvoTune evolutionary framework and the ShinkaEvolve codebase, we introduce LinCodeEvolve, which evolves code-construction programs against an exact minimum-distance evaluator. A strategy loop combines diversity-driven search and expert supervision: when progress plateaus, new strategies are used to redirect the search. LinCodeEvolve discovers seven record-breaking codes, $[172,21,66]$, $[173,20,68]$, $[176,21,68]$, $[181,21,70]$, $[184,21,72]$, $[189,22,72]$ and $[200,21,77]$, six of which have concise quasi-cyclic descriptions. With standard code modification techniques, they improve $22$ entries of the tables. Every code is verified by exhaustive enumeration. These results suggest that LLM-guided search can help find improved codes and complement existing methods in coding theory.

We investigate the Morse index and spectral asymptotics of least-energy solutions to the planar Choquard equation \begin{equation*} \begin{cases} -\Delta u = \displaystyle\left(\int_{\Omega} \frac{u^{p+1}(y)}{|x - y|^\alpha}dy\right) u^p & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $0<\alpha<1$. Under some geometric assumptions, we obtain sharp asymptotics for the first four eigenpairs of the linearized problem as $p\to+\infty$. The first eigenvalue is exactly $(2p+1)^{-1}$. The second and third eigenfunctions give the translation modes of the limiting bubble, and their eigenvalue expansions are determined by the Hessian of the Robin function at the concentration point. The fourth eigenfunction gives the dilation mode with eigenvalue $1+3(4-\alpha)/(2p)+o(p^{-1})$. As a consequence, we derive a comparison between the spectral indices of the solution and the critical-point indices of the Robin function.The proofs combine variational estimates, blow-up analysis and nonlocal Pohožaev identities.

We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.

Motivated by the pioneer work of Viswanath \cite{VIS2000} on random Fibonacci sequences and the development of random Euclidean addition chains in cryptography, we introduce a new family of random Binomial-Fibonacci sequences. We study the asymptotic behavior of their finite-time Lyapunov exponents. In the special case of random Bernoulli-Fibonacci sequences, we show that they convergence almost surely to an explicit Lyapunov exponent and prove a central limit theorem with an explicit formula for the variance. We also show that they satisfy a large deviation principle.

We study the backward-Euler scheme for a class of stochastic functional differential equations (SFDEs) with memory driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first establish the local consistency of the method, with a local truncation error of order H -$\rho$- $\beta$ + 1, and then, as the main result, prove the uniform global pathwise convergence of the numerical approximation, on a set of probability one, with order H -$\rho$ -$\beta$, for $\beta$ $\in$ (1-H, 1/2) and 0 < $\rho$ < H -$\beta$. A uniform high-probability formulation with deterministic constants follows as a corollary. In particular, the convergence order can be taken arbitrarily close to 2H -1, matching the order obtained for the explicit Euler scheme. The convergence proof relies on an exact integral representation of the scheme at mesh points, an a priori bound for the numerical solution, fractional calculus estimates, and a Gronwall inequality for weakly singular kernels.

Let $\operatorname{UT}_n$ denote the group of $n\times n$ real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on $\operatorname{UT}_2$ and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie–Trotter type formula for the Karcher mean on $\operatorname{UT}_2$. For $\operatorname{UT}_n$ we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from $\operatorname{UT}_2$.

Invariant manifold theory for partial differential equations (PDEs) is technically challenging as we often lack spectral gaps. Motivated by recent progress using renormalization in various areas of mathematics, we introduce a spectral renormalization method for semilinear parabolic equations that creates an artificial spectral gap at high frequencies while leaving the low modes unchanged. For the resulting family of renormalized equations, classical inertial-manifold theory yields finite-dimensional invariant manifolds whose dimension diverges as the renormalization parameter tends to zero. We show that the reduced dynamics on these manifolds approximate the original semiflow through an approximate semi-conjugacy, with an $O(\epsilon^2)$ error on bounded finite-time intervals. Under global incremental dissipativity, the semi-conjugacy is uniform in time and yields an $O(\epsilon^2)$ Hausdorff estimate for the global attractors, if they exist. We also establish further dynamical properties of the spectral renormalization such as persistence of hyperbolic equilibria and their Morse indices. We illustrate the construction in settings with and without a natural spectral gap and embed it into a two-parameter fast-slow PDE framework, where the artificial gap enables us to construct slow manifolds for systems on bounded domains with arbitrary dimension. Thus our new approach provides a finite-dimensional dynamical approximation even in regimes where an exact inertial manifold for the original equation is not available. In particular, this provides a practical balance between fully invariant manifolds and direct estimates.

Saddle-point problems play an important role in modern machine learning, including robust optimization and algorithmic fairness. We investigate structured convex–concave minimax problems where the primal variable lies in a high-dimensional, unconstrained space $\mathbb{R}^d$, while the dual variable is confined to a constrained, low-dimensional space $\mathbb{R}^J$, with $J\ll d$. Although existing theoretical frameworks establish that an accelerated rate of $\mathcal{O}(L/T^2)$ is possible, solving the associated subproblems efficiently can become a computational bottleneck in high dimensions. To address this, we introduce a decoupling technique. By restricting second-order updates to the low-dimensional dual space and employing first-order methods in the high-dimensional primal space, our framework combines Newton methods with Nesterov acceleration. For simplex-constrained dual variables and regularizers compatible with a self-concordant barrier, the resulting algorithm achieves a convergence rate of $\mathcal{O}(L/T^2)$ in primal objective suboptimality after $T$ outer iterations, without strong dual concavity. The additional linear-algebra cost per outer iteration is $\widetilde{\mathcal{O}}(dJ^2+J^{3.5})$, alongside one evaluation of the component losses and their Jacobian.

In this paper, we study the hydrodynamics stability of the 3-D plane Couette flow $v_s=(y,0,0)$ with constant temperature state $\rho_s$ for the Boussinesq system in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$ with non-slip boundary condition in $y$ direction. We prove that if the initial velocity field $v_{in}$ and initial temperature $\rho_{in}$ satisfy \begin{align*} \|v_{in}-(y,0,0)\|_{H^2} \leq c_0 \nu, \ \ \|\rho_{in}-\rho_s\|_{H^2} \leq c_0 \nu^2, \end{align*} for some $c_0>0$ independent of the viscosity and thermal diffusion $\nu$, then the solutions to 3D Boussinesq system do not transit from the Couette flow and constant temperature state. To our best knowledge, this is the first result about the nonlinear stability of plane Couette flow for 3-D Boussinesq system in the domain with non-slip boundary condition. This result, together with [18], implies that the strong boundary layer effect does not lead to strong instability.

Bellissard's framework uses covariant observables to describe aperiodic solids. We examine how butterfly-like spectra in twisted bilayers can be related to bulk gap labels and topological response. A bulk gap permits a spectral projector and an integrated density of states. A topological response requires an additional pairing of that projector. In an illustrative finite honeycomb bilayer, an exact second-moment identity isolates the interlayer contribution to spectral variance. Angle-resolved spectra, finite state counts, an uncoupled reference, and size and broadening checks show how coupling redistributes spectral weight without assigning a bulk gap label. We then examine the requirements for atomistic electronic-structure calculations, including nonorthogonal orbitals, projected densities of states, and basis functions that move during relative sliding. The resulting framework specifies the steps needed to connect atomistic spectra with bulk state counting and topological response.

We introduce the notion of Alexandrov region. In contrast to Alexandrov spaces these are intrinsic metric spaces, which satisfy a lower curvature bound and have finite Hausdorff dimension, might however, be non-complete. We show a quantitative version of the statement that such regions are locally isometric to an Alexandrov space.

We show that the map $u_*\colon B_1\subset\mathbb R^3\to\mathbb S^2$, $x\mapsto x/|x|$, minimizes the anisotropic energy \begin{align*} \int_{B_1} \Big( k_1(\mathrm{div}\, u)^2+k_2 (u\cdot \mathrm{curl}\, u)^2 +k_3 |u\times\mathrm{curl}\, u|^2 \Big) \, dx, \end{align*} among $\mathbb S^2$-valued maps agreeing with it on the boundary, for values of $k_1,k_2,k_3>0$ beyond the known regime $k_1\leq k_2$. Our main tool is a new stability estimate for $u_*$ as a minimizing sphere-valued harmonic map.

Multistage Stochastic Programming requires scenario trees that accurately approximate the unknown underlying uncertainty process without violating the non-anticipativity constraint. Most of the existing method for scenario tree generation usually target just one of the two goals at the expense of the other, and just some recent methods try to simultaneously address both. In this paper, we propose to combine two recent distances from Optimal Transport Theory, specifically the ultrametric Gromov-Wasserstein and the Fused Gromov-Wasserstein, into a unique measure to simultaneously deal with the two goals. Our main result is a stability theorem showing that the optimal value of a Multistage Stochastic Programming problem changes by an amount controlled by a Holder-type power of the proposed distance between the true process and its scenario tree approximation, extending Wasserstein-based stability results to control marginal fidelity and non-anticipativity, simultaneously. We propose a block coordinate descent algorithm for the our scenario tree generation based on the proposed distance, and evaluate it on different inventory management test cases.

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we study necessary optimality conditions for nonsmooth bilevel programming using tangential subdifferentials. We derive a sequential enhanced tangential Fritz–John type condition for a class of nonsmooth nonlinear programs and then obtain corresponding enhanced tangential Fritz–John and enhanced tangential Karush–Kuhn–Tucker (KKT) systems by taking limits of the sequential condition. We further investigate tangential constraint qualification conditions, including T-quasinormality and a tangential cone–continuity property (T–CCP), and analyze how they guarantee the validity of tangential KKT type conditions. Finally, these results are applied to the value-function reformulation of a bilevel programming problem.