The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is between $4n+1$ and $12n-2$. We improve both of these bounds, showing that $5n-\frac{1+(-1)^{n}}{2}\leq R(W_{2n})\leq 8n+664$ for all integers $n\geq 2$. The main focus of the paper concerns two general results on the Ramsey numbers of stars versus even wheels and even cycles versus even wheels, from which the above bounds are obtained as a corollary. That is, we asymptotically determine $R(K_{1,m}, W_{2n})$ and $R(C_{2m}, W_{2n})$ for all sufficiently large $m$ and $n$, both of which were open problems for most regimes. As for odd wheels, we note that the analogous values for stars versus odd wheels and odd cycles versus odd wheels were already known exactly, from which it follows that $6n+4=R(K_{1,2n+1}, W_{2n+1})\leq R(W_{2n+1})\leq 2\cdot R(C_{2n+1}, W_{2n+1})=12n+2$. Very recently, Zhang and Chen improved the upper bound to $R(W_{2n+1})\leq \frac{32n}{3}+O(1)$. We are able to refine their proof, further improving the upper bound to $R(W_{2n+1})\leq 10n+O(1)$.

We present GreyRing, a new model for the post-merger signal in black-hole binary coalescences based on the greybody factor of the remnant. The model accurately reproduces the full frequency-domain ringdown signal of a large set of comparable-mass, aligned-spin numerical relativity waveforms, achieving mismatches of order ${\cal O}(10^{-6})$ for the dominant $(\ell,m)=(2,2)$ mode, and typically outperforming state-of-the-art time-domain models. Building on this model, we introduce a novel consistency test of strong gravity based on the greybody factor: the remnant mass and spin inferred from GreyRing can be compared with those obtained through standard black hole spectroscopy. This agnostic test relies exclusively on the post-merger signal and does not require the inclusion of overtones or the choice of very early ringdown starting times, combining the advantages of inspiral-merger-ringdown consistency tests and traditional black hole spectroscopy. We apply the test to GW250114 and find that the remnant mass and spin inferred from GreyRing are consistent with those measured from the full signal. Remarkably, the inferred parameters can be measured with a precision comparable to, or slightly better than, that achieved with standard black-hole spectroscopy. Our greybody-factor waveform model allows for new precision tests of strong gravity using the ringdown signal.

Numerous software analysis tools exist today, yet applying them to diverse open-source projects remains challenging due to environment setup, dependency resolution, and tool configuration. LLM-based agents offer a potential solution, yet no prior work has systematically studied their effectiveness on the specific task of automated software analysis, which, unlike issue solving or general environment setup, requires installing and configuring a separate analysis tool alongside the target project, generating tool-specific prerequisites, and validating that the tool produces meaningful analysis outputs. We introduce AnalysisBench, a benchmark of 35 tool-project pairs spanning seven analysis tools and ten diverse C/C++ and Java projects, each with a manually constructed reference setup. Using AnalysisBench, we evaluate four agent architectures across four LLM backends. Our custom agent, AnalysisAgent, achieves manually verified success rates of 94% (Gemini-3-Flash, 33/35 tasks), compared to 77% for the best baseline (ExecutionAgent). Beyond quantitative results, we identify key limitations in existing agents, including stage mixing, poor error localization, and premature termination, and show that agentic architecture matters more than LLM capability alone. We further find that whole-program analyses and symbolic execution are the most difficult tasks, that Java toolchains pose greater challenges than C/C++, and that LLM-self-validated success consistently overstates manually verified success.

We study when a programming language can emulate programs written in that same language without delegating the guest program back to the host evaluator or compiler. We call this property emulation-completeness. The central observation is that Turing-completeness by itself is not enough: a self-emulator must not only compute the guest program's result, but must also account for the guest-visible state on which realistic programs depend, including control flow, exceptions, callbacks, timing, memory usage, and runtime metadata such as stack traces or line numbers. This paper is a systematization paper. Its contribution is not a new emulator implementation, but a precise vocabulary and a structured taxonomy for reasoning about self-emulation. We distinguish source-level evaluation from compiled-code emulation, define syntactic and compiled-code emulation-completeness, and separate weak from strong emulation-completeness according to how much observable runtime behavior must be preserved. We then organize the requirements into two classes: language-side requirements, which determine whether the guest semantics can be represented explicitly inside the language, and emulator-side requirements, which determine whether the resulting emulator can faithfully mask or reproduce relevant observations. The discussion is grounded by concrete examples, including publicly documented details from Erlang, where argument limits, bitstring pattern matching, and message reception expose subtle mismatches between direct execution and self-emulation. The resulting framework is intended as guidance for language designers, implementers of evaluators and emulators, and researchers interested in secure sandboxing, decompilation, and reflective execution.

Hypergraph data are often projected onto a weighted graph by constructing an adjacency matrix whose $(i,j)$ entry counts the number of hyperedges containing both nodes $i$ and $j$. This reduction is computationally convenient, but it can lose information: distinct hypergraphs may induce the same matrix, and the matrix entries are generally dependent because each hyperedge contributes to multiple pairs. We study the planted clique problem under this matrix-only observation model. For detection, we show that a spectral norm test is asymptotically powerful at the $\sqrt{n}$ scale, with explicit dependence on the background hyperedge probability $p$. For recovery, we analyze a polynomial-time spectral method based on the leading eigenvector and prove exact recovery at the canonical $\sqrt{n}$ scale, again with explicit dependence on $p$. We also extend both results to sparse regimes in which the hyperedge probability may depend on \(n\). Our analysis adapts a leave--one--out eigenvector framework to this setting. These results provide rigorous detection and recovery guarantees when only the adjacency matrix is observed.

Motivated by the recent Dark Energy Spectroscopic Instrument (DESI) DR2 preference for dynamical dark energy, we study interacting dark energy models in which a canonical quintessence field couples to cold dark matter through a field-dependent mass $m(ϕ)$. In such scenarios, the effective equation of state inferred under the assumption of non-interacting dark sectors, $w_{\rm eff}(z)$, can differ from the intrinsic scalar-field equation of state $w_ϕ(z)$, making an apparent phantom crossing $w_{\rm eff}<-1$ possible without introducing a phantom scalar. We show that a viable realization of this mechanism requires the scalar field to originate from a frozen phase deep in the radiation era, in order for the effective coupling to remain sufficiently suppressed before recombination to evade cosmic microwave background constraints, and for the late-time evolution to become strong enough to reproduce the apparent behavior of $w_{\rm eff}(z)$ preferred by DESI. We identify the general conditions that allow these requirements to be satisfied simultaneously, and present an illustrative phenomenological realization in which $w_{\rm eff}(z)$ evolves from $w_{\rm eff}\approx -1.2$ at $z \approx 1.0$ to $w_{\rm eff}\approx -0.9$ at $z\approx 0.4$. These conditions and requirements serve as a guide for designing future models of this kind which can safely navigate the phantom divide at $w=-1$ in an effective way without phantom fields.

Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $λ: A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $λ$. We provide a criterion for the nonnegativity of the coefficients of these multivariate polynomials and apply it to show that marked order polytopes of skew shapes are Ehrhart positive in a multivariate sense. This extends recent results of Ferroni-Morales-Panova on order polytopes of skew shapes and proves conjectures on the Ehrhart positivity of skew Gelfand-Tsetlin polytopes and $m$-generalized Pitman-Stanley polytopes due to Alexanderson-Alhajjar and Dugan-Hegarty-Morales-Raymond, respectively.

Modern search systems rely on a fast first stage retriever to fetch relevant items from a massive catalog of items. Deployed search systems often use user engagement signals to supervise bi-encoder retriever training at scale, because these signals are continuously logged from real traffic and require no additional annotation effort. However, engagement is an imperfect proxy for semantic relevance. Items may receive interactions due to popularity, promotion, attractive visuals, titles, or price, despite weak query-item relevance. These limitations are further accentuated in Walmart's e-commerce sponsored search. User engagement on ad items is often structurally sparse because the frequency with which an ad is shown depends on factors beyond relevance such as whether the advertiser is currently running that ad, the outcome of the auction for available ad slots, bid competitiveness, and advertiser budget. Thus, even highly relevant query ad pairs can have limited engagement signals simply due to limited impressions. We propose a bi-encoder training framework for Walmart's sponsored search retrieval in e-commerce that uses semantic relevance as the primary supervision signal, with engagement used only as a preference signal among relevant items. Concretely, we construct a context-rich training target by combining 1. graded relevance labels from a cascade of cross-encoder teacher models, 2. a multichannel retrieval prior score derived from the rank positions and cross-channel agreement of retrieval systems running in production, and 3. user engagement applied only to semantically relevant items to refine preferences. Our approach outperforms the current production system in both offline evaluation and online AB tests, yielding consistent gains in average relevance and NDCG.

Non-thermal fermions can be produced non-perturbatively in the early universe during coherent oscillations of a scalar field. We explore fermion production in $λϕ^{4}$ inflation through this mechanism and analyze the momentum spectrum of the fermions produced, which depends on a coupling parameter $q$. For $q \gtrsim 0.01$, the main contribution to the total number density comes from an approximately half-filled Fermi sphere as a result of non-adiabaticity. For $q\lesssim 0.01$, we find that the major contributions instead come from resonance peaks at higher momentum values. We find a simple relation to predict the momentum values corresponding to resonance peaks for any $q$. We also obtain analytic power-law approximations for the total number density of fermions and find that it is proportional to $q^{1/2}$ for $q\lesssim 0.01$ and proportional to $q^{3/4}$ for $q\gtrsim 10$. If fermions produced by this mechanism make up the entirety of dark matter, we estimate lower bounds on their mass.

Dynamical decoupling is a long-established and effective way to suppress unwanted interactions in qubit systems, enabling advances in fields ranging from quantum metrology to quantum computing. For general qudit systems, however, comparable protocols remain rare, mainly because Hamiltonian engineering in higher dimensions lacks the geometric intuition available for qubits. Here we present a general framework for dynamical decoupling in qudit systems, based on Lie group representation theory. By extending the group theory approach to dynamical decoupling, we show how decoupling groups can be systematically identified among the finite subgroups of SU(d) by analyzing their access to the irreducible components of the operator space. As an application, we construct new pulse sequences for interacting qutrit systems based on finite subgroups of SU(3), and show how subgroup factorizations and group orientations can be exploited to obtain shorter and more experimentally practical protocols for spin-1 systems with large zero-field splitting. We further show that the same symmetry-based framework yields quantum error-correcting codes: whenever a finite subgroup of SU(d) acts as a decoupling group for the relevant set of operators, the associated one-dimensional symmetry sectors define codespaces satisfying the Knill-Laflamme conditions, thereby unifying dynamical decoupling and quantum error correction in multi-level quantum systems.

We consider a quenched SU(2)$\times$U(1) gauge Higgs theory on the lattice, coupled to a static vector-like fermion which, in this case, is in the same gauge group representation as the Higgs field. Physical (i.e. locally gauge invariant) electrically charged and electrically neutral states of matter particles in the electroweak theory were described decades ago, but those constructions do not exhaust all the possibilities, and new types of electrically charged/neutral states, orthogonal to former constructions, are described here. The difference has to do with how the static source, which by itself does not create a physical state, is dressed by dynamical fields. We find that, unsurprisingly, the neutral static fermion is much lighter than any of the charged fermion states. But a lattice study of the propagation of the charged fermion states indicates the existence of (at least) two particle states with different masses in charged particle spectrum.

We define the exact category of contraherent cosheaves of contramodules on a locally Noetherian formal scheme, as well as the exact categories of locally contraherent cosheaves of contramodules (with respect to a given open covering). We also construct the direct image and inverse image functors of locally contraherent cosheaves of contramodules under morphisms of locally Noetherian formal schemes. The exposition in the section of preliminaries in adic commutative algebra is worked out in the greater generality of arbitrary commutative rings with adic topologies (of finitely generated ideals).

The quantum Mpemba effect is a phenomenon characterized by an exponential relaxation from a non-equililbrium state to a steady state. This effect was predicted with an analysis of the Liouvillian superoperator and experimentally demonstrated in a three-level system. In this work, the system dynamics of the Mpemba effect is predicted within the steepest-entropy-ascent quantum thermodynamics framework considering a single constituent three-level isolated system. The system is projected from a four-dimensional Hilbert space onto a three-dimensional one using the Feshbach projection in order to compare the theoretical results with experimental data. Since the quantum Mpemba effect is characterized by a dissipative acceleration, the relaxation parameter, $τ_D$, plays a fundamental rol in the dissipative dynamics predicted by the model and is determined using machine learning methods, resulting in a model that thermodynamically describes this phenomenon at the quantum level.

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices $(A, B, C)$. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: $A$ and $C$ are described by a classical weighted graph called the strange graph, while $B$ provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.

We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set, and that a large subclass of directed complexes with frame-acyclic molecules is compatible with the Gray tensor product. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension $3$. As a corollary of our main results, the pasting theorem also applies to the class of regular $3$-polygraphs.

We prove that all geometric helices in the derived category of coherent sheaves on a del Pezzo surface are related by a sequence of elementary operations: rotation, shifting, orthogonal reordering, tensoring by a line bundle, and tilting. As a consequence, any two non-commutative crepant resolutions of the affine cone over a del Pezzo surface are related by mutations. The proof relies on a geometric interpretation of tilting operations as cluster transformations acting on toric models of a log Calabi--Yau surface mirror to the del Pezzo surface.

We analyse the action of the basic Dehn twists on the essential curves, $γ$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology $H_1({\rm T})=\mathbb Z^2$ of the branched covering torus with a hole, ${\rm T}\to \mathbb D_3$. Our explicit description of orbits of the action of the pure mapping class group ${\rm PMod}(\mathbb D_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_γ$. We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps.

In this paper, we consider the unfolding of the real-analytic and generic zero-Hopf bifurcation of co-dimension two. It is well-known that in an open set of parameter space the splitting of one-dimensional stable and unstable manifolds is beyond all orders. This paper provides a new geometric dynamical-systems-oriented proof for the exponentially small splitting. As a novel aspect, we relate the exponentially small splitting to the lack of analyticity of center-like manifolds of generalized saddle-nodes. Moreover, the blowup method plays an important technical role as a systematic way to relate dynamics on different orders of magnitude. Finally, as our approach takes place in the (complexified) phase space, we do not rely on an explicit time-parametrization of the unperturbed heteroclinic connection. We therefore believe that our approach has general interest.

Valence-bond-based wavefunctions, such as the spin-coupled generalized valence bond (SCGVB) ansatz, provide compact and chemically interpretable descriptions of strong correlation. However, their non-orthogonal determinant structure poses a major challenge for quantum computing implementations. Although recent fermion-qubit mappings allow non-orthogonal orbitals to be encoded on qubit registers, the evaluation of overlap and Hamiltonian matrix elements remains a bottleneck on NISQ devices due to the need for ancilla qubits, controlled operations, and deep circuits. We present a measurement-driven quantum framework for evaluating these quantities in SCGVB wavefunctions. Instead of preparing the full wavefunction, we reformulate the problem in terms of vacuum expectation values of Pauli-string operators, enabling evaluation with shallow, ancilla-free circuits based on local Clifford rotations and computational-basis measurements. Unlike Hadamard-test-based approaches, this method avoids controlled operations by reducing the task to local Pauli measurements, yielding a low-depth strategy suitable for near-term devices. We demonstrate the framework on the H4 cluster along a dissociation pathway from square geometry to the separated-fragment limit, considering five nuclear configurations via quantum-circuit emulation. The overlap and Hamiltonian matrices agree well with classical Lowdin-based references, and Chirgwin-Coulson weights remain chemically consistent. These results highlight the robustness of the approach and its suitability as a NISQ-compatible building block for SCGVB-based quantum algorithms.

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.

Centered weighted composition operators on $L^2$-spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally half-centered operators is introduced, and it is shown that unbounded weighted composition operators are spectrally half-centered provided their powers are closed and densely defined. A criteria for centered weighted shifts on directed trees of types I--IV are provided. Various examples are presented.

Let $G$ be a finite simple graph with oriented incidence matrix $D$. The signed graph on edge set $E(G)$ with adjacency matrix \[ A_{\mathcal A(G)}=D^{\mathsf T}D-2I \] is classical in the signed-line-graph literature. In this paper we study its canonical switching class as a source of invariants of the underlying unsigned graph. We prove that the switching class of $\mathcal A(G)$ determines $G$ up to isomorphism modulo isolated vertices, and we relate the frustration index $\ell(\mathcal A(G))$ to classical bipartization parameters. In particular, we show \[ \operatorname{def}(G)\le \ell(\mathcal A(G))\le (Δ(G)-1)\operatorname{def}(G), \] and, for cubic graphs, \[ \ell(\mathcal A(G))=2\,\operatorname{oct}(G). \] We then prove the exact optimization identity \[ \ell(\mathcal A(G)) = \frac14\sum_{v\in V(G)} d(v)^2 -\frac14\max_{x\in\{\pm1\}^{E(G)}}\|Dx\|^2, \] so $\ell(\mathcal A(G))$ is exactly a Boolean edge-space Laplacian optimization problem. This yields a spectral lower bound in terms of the largest Laplacian eigenvalue, a cubic spectral lower bound on odd cycle transversal, and explicit family-level comparisons showing that the spectral and defect bounds govern different regimes: on odd cycles the spectral bound is asymptotically vacuous, while on complete multipartite graphs it already captures exactly $3/4$ of the true value of $\ell(\mathcal A(G))$. Thus the paper uses a classical signed line graph in a new way: as a source of combinatorial invariants of ordinary graphs, especially through frustration and odd-cycle-transversal phenomena.

Exciton transfer dynamics between chromophores depends on excitonic coupling, which is governed by relative orientation between the chromophores. While the excitonic coupling is treated as a static parameter in many cases, structural dynamics can introduce time-dependence on the excitonic coupling. However, influence of the dynamics of excitonic coupling on the exciton transfer has been scarcely understood. In the present study, exciton transfer under dynamical fluctuation in excitonic coupling was investigated via combined use of non-adiabatic molecular dynamics simulations, exciton density analysis, and a simple two-state model, for inter-ligand exciton transfer in bis(dipyrrinato)Zn(II) as the example case. The reaction coordinate for the exciton transfer was obtained a posteriori via regression analysis where the target and explanatory variables are diabatic energy gaps and atomic displacements, respectively. The results suggest that dynamical angular fluctuation between the two dipyrrinato ligands breaks the symmetry to incidentally increase the excitonic coupling, accelerating the exciton transfer between the ligands.

Fast Ewald summation efficiently evaluates Coulomb interactions and is widely used in molecular dynamics simulations. It is based on a split into a short-range and a long-range part, where evaluation of the latter is accelerated using the fast Fourier transform (FFT). The accuracy and computational cost depend critically on the mollifier in the kernel split and the window function used in the spreading and interpolation steps that enable the use of the FFT. The first prolate spheroidal wavefunction (PSWF) has optimal concentration in real and Fourier space simultaneously, and is used when defining both a mollifier and a window function. We provide a complete description of the method and derive rigorous error estimates. In addition, we obtain closed-form approximations of the Fourier truncation and aliasing errors, yielding explicit parameter choices for the achieved error to closely match the prescribed tolerance. Numerical experiments confirm the analysis: PSWF-based Ewald summation achieves a given accuracy with significantly fewer Fourier modes and smaller window supports than Gaussian- and B-spline-based approaches, providing a superior alternative to existing Ewald methods for particle simulations.

We derive two model-independent results for spacetimes with globally bounded tidal fields. These are operational resolution scales of the local-inertial approximation and tidal dynamics; no spacetime discreteness is implied. Given an invariant bound $λ_{\max}\leλ_{\rm bound}$ on the electric Riemann eigenvalues $E_{ij}\equiv R_{\hat{0}i\hat{0}j}$ along freely falling worldlines, we prove (i)~a rigorous upper bound on accumulated geodesic deviation through any bounded curvature interior, controlled by $τ_*\equivλ_{\max}^{-1/2}$, and (ii)~the existence of a critical wavenumber $k_*\simτ_*^{-1}$ separating adiabatic from non-adiabatic perturbation transfer through high-curvature epochs, with Bogoliubov coefficients exponentially suppressed for $k\,τ_*\gg 1$. Both results depend only on the tidal bound (and, for mode transfer, on a mild timescale assumption for the curvature-driven effective potential) and are otherwise insensitive to metric details. For preparation, we collect the standard operational consequences of bounded curvature, including the accuracy-dependent local-inertial domain $L_{\rm LI}(\varepsilon)\sim\sqrt{\varepsilon}\, λ_{\max}^{-1/2}$ and, for conformally flat cores in four dimensions, the benchmark ratio $τ_*/L_*=24^{1/4}$ with $L_*\equiv K_{\max}^{-1/4}$. We quantify the robustness of this coefficient under departures from maximal symmetry via the Weyl-to-Kretschmann ratio $ε_C$. The general framework is validated numerically in the extremal Hayward geometry.

The ReD experiment measured the ionization yield Qy of argon for nuclear recoils in the 2-10 keV range using a dual-phase Time Projection Chamber irradiated with neutrons from a Cf-252 fission source. The measurement extends coverage below 7 keV, confirms consistency with previous data above 7 keV, and indicates a higher Qy at lower energies. These results are relevant for argon-based experiments searching for dark matter in the form of low-mass Weakly Interacting Massive Particles, which are very sensitive to the modeling of the detector response in this energy range.

Classical field forecast evaluation relies mainly on local scores such as RMSE or MAE. These metrics severely over-penalize small spatial or temporal displacements of coherent structures, a limitation known as the double-penalty issue and common to many forecasting domains. The present paper introduces a tolerance-based framework built on the three-dimensional gamma index, initially designed for medical dose verification, as a unified acceptance criterion for gridded forecasts. The method embeds explicit margins in space (DTA), time (TTA), and intensity (IDT), and evaluates whether predictions agree with observations within predefined physical bounds rather than through pixel-wise differences only. A synthetic illustration is first used to show why conventional metrics can misrepresent usable forecasts. The approach is then applied to satellite-derived SSI fields to demonstrate operational behaviour on a real dataset. Results confirm that the gamma criterion preserves structural consistency under minor positional noise while isolating physically significant discrepancies. The formulation is generic and can be implemented for any gridded variable provided meaningful tolerances are defined, offering a pragmatic complement to existing spatial verification tools in general forecasting workflows.

I derive the statistical relationship between a radial velocity or astrometric acceleration (a trend), a companion's mass, and the projected separation of the companion. These relationships, expressed as probability density functions, are analytic and independent of all Keplerian orbital elements so long as orbits are randomly oriented in space. I also derive a closed-form expression for the probability distribution of the ratio of the projected separation to the semimajor axis at fixed eccentricity. This expression can be numerically integrated over eccentricity for an arbitrary distribution of eccentricities. I verify my results with empirical comparisons to equivalent but more complex expressions in the literature based on the equations of Keplerian orbits. The closed-formed expressions derived here would be especially useful for any calculation that requires derivatives, e.g., Hamiltonian Monte Carlo. I also provide a Jupyter notebook including all figures and calculations.

We study the generalization error of stochastic learning algorithms from an information-theoretic perspective, with a particular emphasis on deriving sharper bounds for differentially private algorithms. It is well known that the generalization error of stochastic learning algorithms can be bounded in terms of mutual information and maximal leakage, yielding in-expectation and high-probability guarantees, respectively. In this work, we further upper bound mutual information and maximal leakage by explicit, easily computable formulas, using typicality-based arguments and exploiting the stability properties of private algorithms. In the first part of the paper, we strictly improve the mutual-information bounds by Rodríguez-Gálvez et al. (IEEE Trans. Inf. Theory, 2021). In the second part, we derive new upper bounds on the maximal leakage of learning algorithms. In both cases, the resulting bounds on information measures translate directly into generalization error guarantees.

The emergence of artificial intelligence (AI) accelerators like NVIDIA Tensor Cores offers new opportunities to speed up tensor-heavy scientific computations. However, applying them to quantum chemistry is challenging due to strict accuracy demands and irregular data patterns. We propose an adaptive precision algorithm to accelerate the density fitting (DF) method with Gaussian basis sets on AI accelerators using 8-bit integer (INT8) arithmetics. Implemented in the GPU-accelerated PySCF package, the algorithm is tested on more than twenty molecular systems with different NVIDIA GPUs. Compared to the standard FP64 code, our algorithm is up to 204\% faster on a RTX 4090 gaming GPU and up to 364\% faster on a RTX 6000 Ada workstation GPU without compromising the converged energy. This work demonstrates a practical approach to use AI hardware for reliable quantum chemistry simulations.