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Fusing Sequence Motifs and Pan-Genomic Features: Antimicrobial Resistance Prediction using an Explainable Lightweight 1D CNN-XGBoost Ensemble

Antimicrobial Resistance (AMR) is a rapidly escalating global health crisis. While genomic sequencing enables rapid prediction of resistance phenotypes, current computational methods have limitations. Standard machine learning models treat the genome as an unordered collection of features, ignoring the sequential context of Single Nucleotide Polymorphisms (SNPs). State-of-the-art sequence models like Transformers are often too data-hungry and computationally expensive for the moderately sized datasets that are typical in this domain. To address these challenges, we propose AMR-EnsembleNet, an ensemble framework that synergistically combines sequence-based and feature-based learning. We developed a lightweight, custom 1D Convolutional Neural Network (CNN) to efficiently learn predictive sequence motifs from high-dimensional SNP data. This sequence-aware model was ensembled with an XGBoost model, a powerful gradient boosting system adept at capturing complex, non-local feature interactions. We trained and evaluated our framework on a benchmark dataset of 809 E. coli strains, predicting resistance across four antibiotics with varying class imbalance. Our 1D CNN-XGBoost ensemble consistently achieved top-tier performance across all the antibiotics, reaching a Matthews Correlation Coefficient (MCC) of 0.926 for Ciprofloxacin (CIP) and the highest Macro F1-score of 0.691 for the challenging Gentamicin (GEN) AMR prediction. We also show that our model consistently focuses on SNPs within well-known AMR genes like fusA and parC, confirming that it learns the correct genetic signals for resistance. Our work demonstrates that our ensemble model overcomes the limitations of using either an order-agnostic or a standalone sequence model. Our codes are publicly available on GitHub at: https://github.com/Saiful185/AMR-EnsembleNet.

cs.LG

REIGN: Refurbished Embeddings with Integrated Guidance Networks for Efficient Context-Length Scaling

Dense retrieval over long documents is expensive. Token-level encoders scale quadratically in sequence length, and most long-context embedding models reach 32K tokens only through architectural workarounds or by stretching billion-parameter LLMs. We propose REIGN (Refurbished Embeddings with Integrated Guidance Networks), a contrastively trained bi-encoder that operates on sequences of contextualised chunk embeddings from a frozen Guidance Network (GN) rather than on raw tokens. REIGN targets multi-chunk inputs, primarily for document-to-document retrieval; single-chunk inputs stay with the GN. Decoupling token-level processing from document-level reasoning, and caching the GN embeddings to disk, cuts per-document training cost by roughly four orders of magnitude relative to chunked Transformer fine-tuning. We also release a synthetic long-document retrieval benchmark for contrastive training and evaluation at long context lengths. Across an in-distribution Wikipedia benchmark, the LoCo out-of-distribution suite, and a real-world patent retrieval case study, REIGN matches dense long-context retrievers at smaller parameter budgets in each regime. A paired significance test puts it on par with models 1.6-4.3x larger on the patent task, and it stays within 0.65 nDCG@10 of a 20x-larger model on LoCo.

cs.CL

A.X K2 Technical Report

We introduce A.X K2, a 688B-parameter Mixture-of-Experts (MoE) language model trained from scratch as a high-performance foundation for \emph{agentic} applications. Trained on approximately 8.5T tokens---fewer than its predecessor, A.X K1---on a smaller but higher-quality mixture with substantially expanded agentic and software-engineering data, it nonetheless improves over A.X K1 across the board, by over 30 percentage points on some benchmarks, reflecting large gains in token efficiency. To support long contexts efficiently, we introduce Sparse Gated Attention (SGA), which combines sparse attention with gated attention, and adopt Gated Norm (GN) to stabilize large-scale training. SGA is trained natively at 128K through a \emph{sparse} indexer warmup that optimizes the indexer against its own sparse top-$k$ selection rather than the dense attention distribution, making adaptation markedly cheaper: each query reads only 2,048 positions, yet long-context quality is unchanged and A.X K2 scores 94.6 on RULER out to 256K. The outlier suppression of GN in turn keeps 4-bit NVFP4 serving within one point of FP8 accuracy. A simple yet effective Think-Fusion recipe further lets users switch between thinking and non-thinking modes within a single unified model. Extensive evaluations show that A.X K2 performs competitively against strong open-weight baselines, matching or exceeding them on math and Korean-language benchmarks.

cs.AI

TRACE: Spatiotemporal Contact Memory Graph Network Simulator for Granular Dynamics

Learned graph simulators provide an efficient alternative to high-fidelity solvers for granular dynamics. However, granular motion depends strongly on inter-granular contact history, which is difficult to preserve when particle contacts form, break, and rearrange. Existing simulators mainly store temporal information in node features or node-level memory. Here we introduce TRACE, a graph-network simulator that stores interaction history directly on contact edges. Each edge maintains a persistent memory updated by attention-based message passing and a gated recurrent unit, while an edge-identity dictionary preserves this memory as the contact graph changes. A physics-structured decoder predicts inter-granular normal and tangential contact forces, enforces the Coulomb friction limit, and applies equal-and-opposite internal forces. The model is trained with single-step pretraining followed by autoregressive rollout fine-tuning. We evaluate TRACE on 2D and 3D granular column-collapse benchmarks. In both cases, TRACE produces stable, physically consistent long-horizon rollouts, closely reproducing the final deposit geometry and the kinetic energy released during collapse. Compared with graph network simulator (GNS) and node-memory graph neural simulator (NMGNS), TRACE reduces long-rollout position error by 31-62% and final-deposit error by 58-89% across the two benchmarks, while using fewer parameters and maintaining near-zero particle interpenetration. TRACE also achieves 12.2$\times$ and 8.9$\times$ speedups over the material point method (MPM) reference solver in 2D and 3D, respectively. Our code is available at https://github.com/Data-Driven-Computational-Geotechnics/TRACE.

cs.LG

The maximum entropy state

We give an algorithm for calculating the maximum entropy state as the least fixed point of a Scott continuous mapping on the domain of classical states in their Bayesian order.

math.PR

Eigenvalues and eigenfunctions of the fractional Laplacian on the interval

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki--Kwaśnicki--Małecki--Stós and Kwaśnicki, and confirms the conjectural $O_α(n^{-2})$ remainder suggested by the numerical simulations of Kaleta--Kwaśnicki--Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $α$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.

math.CA

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA

An Inverse Problem for Determining the Piston Speed from a Given Lipschitz Leading Shock

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

math.AP

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR

Set Theory in the Foundation of Math; Internal Classes and External Sets

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal to math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.

cs.LO

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO

Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

math.NA

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph

A counterexample to Kusner's conjecture on equilateral sets

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

math.MG