Papers

Decision-focused learning (DFL) trains predictors for downstream decision quality, but often relies on optimal reference decisions that are expensive to obtain. We present PolyStepOR, which trains directly from realized decision costs without pre-computed optima and extends to in-constraint predictions through repair or infeasibility penalties. To handle piecewise-constant losses, PolyStepOR perturbs predictor parameters, evaluates the resulting decisions, and uses optimal transport to favor lower-cost directions, requiring no derivatives. Without task-specific tuning, PolyStepOR performs strongly on classical optimization benchmarks and competitively on predicted-constraint and real-world problems. Theoretically, we characterize decision-preserving perturbations and boundary detection, bound sensitivity to cost errors, and establish stationarity guarantees for a smoothed objective. PolyStepOR thus replaces optimal reference decisions and derivatives with forward evaluations.

Multi-objective Bayesian optimization (MOBO) commonly relies on independent Gaussian processes (GPs) with stationary kernels, limiting its ability to represent nonstationary structure and share information between objectives. However, expressive nonstationary GPs do not necessarily make reliable BO decisions. We study this mismatch for the multi-output low-rank nonstationary (MO-LRN) GP: strong training fit can coexist with large off-design errors and optimistic acquisition predictions. We introduce MOLRN-BO, which combines a regularized shared-spectral surrogate with objective-specific residuals, prequential mean correction and tempered covariance scaling, and Pareto-local qLogEHVI optimization with periodic global search. Experiments on 12 deterministic bi-objective benchmarks show that MOLRN-BO substantially improves upon the original MO-LRN and achieves the best average problem ranks for final normalized hypervolume and normalized inverted generational distance among nine evaluated algorithms. It also achieves the strongest adverse-tail performance while remaining competitive with the leading baselines in anytime optimization. Ablation studies further show that the shared spectral construction improves off-design prediction, the local–global decision policy improves optimization performance, and hierarchical calibration reduces systematic candidate bias. These results demonstrate that nonstationary multi-output surrogates can deliver strong and robust MOBO performance when their structure and use are explicitly adapted to the demands of sequential optimization.

We show that the constraint satisfaction problem of any finite conservative structure that can not pp-construct st-Con (whose CSP is NL-complete) or 3Lin2 (whose CSP is Mod2L-complete) is solved by a symmetric linear Datalog program and therefore in L.

Motion-language models are typically scored by motion-language evaluators, but how well these evaluators ground language structure remains unclear. Here, we introduce the Structure grounding via Temporal-order, Reflection, and Identity Diagnostic Evaluation (STRIDE) benchmark to systematically evaluate the ability of evaluators to track temporal order, mirror reflection, and action identity. STRIDE comprises $5{,}869$ triples, each consisting of a motion, its original caption, and a perturbed caption, spanning both short and long descriptions. We likelihood-balance caption pairs to reduce text-only bias and estimate each evaluator's caption preference under unrelated motions to measure the discrimination gain from matched motions relative to this baseline. Our experiments reveal weak structural grounding and severe deficits in mirror sensitivity among the audited evaluators, which commonly used evaluation protocols fail to expose. To understand why these limitations go undetected in standard tests, we examine the evaluators more closely. We find that text-only priors alone can solve naive perturbation tests on existing datasets, while common retrieval and distributional metrics barely respond to structural corruption introduced by mirroring ground-truth motions. These findings suggest a natural intervention: structural hard negatives. Our experiments show that a simple modification to contrastive learning substantially improves performance on temporal order and mirror reflection. The benchmark and code will be released.

We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d = \Theta(n^{4/5})$ exist by constructing a graph in the split Cayley hexagon, matching the known upper bound. This resolves the first open case for $K_k$-free graphs after $k=3$ for which Alon gave a tight construction in 1994. This has a variety of implications for $K_4$-free pseudorandom graphs. Furthermore, it implies an explicit lower bound on the Ramsey number of $r(4, t) \geq t^{1.\overline{6} - o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4, t) \geq t^{1.6-o(1)}$.

Recent video generation models produce realistic videos and show potential as a foundation for world models. These advances create opportunities for generating medical teaching demonstrations, which requires both convincing visual quality and precise procedural actions. However, whether current generators can meet these requirements has not been measured. To address this problem, we introduce Readiness Evaluation of Medical Education Demonstration sYnthesis (REMEDY), to our knowledge, the first benchmark for AI-generated medical teaching demonstrations. REMEDY provides 900 first frames from real demonstration videos, covering 12 tasks across four scenarios: operating room, imaging, clinic and bedside, and resuscitation. Five contemporary open-source video generation models produce 4,500 videos from these frames. We combine task-specific clinical checklists with video and motion quality metrics. Evaluation covers four dimensions: clinical action following, clinical profiles, video quality, and motion quality. Our results show that realistic appearance and temporal consistency do not ensure correct clinical actions. Even the most advanced MiniMax-H3 achieves only 28.25% on strict clinical success rate, and fine-grained clinical actions remain challenging. These findings establish a foundation and roadmap for developing future medical education world models.

Large Language Model (LLM)-based agents are increasingly used for software engineering tasks, yet their performance is not determined by the base model alone. The agent harness substantially shapes how SE agents interact with repositories, execute actions, and validate solutions. However, the role of harness design remains insufficiently understood, especially across different models, tasks, and harness components. In this paper, we present a systematic empirical study of harness effects in SE agents. We first evaluate two representative harnesses, mini-SWE-agent and OpenCode, with ten models from two prominent open-weight model families, Qwen and DeepSeek, on three benchmarks: SWE-bench Pro, ProgramBench, and GitTaskBench. We then construct NanoHarness, a lightweight modular harness built on top of mini-SWE-agent, and use it to analyze five representative harness components: tool registry, context compression, explicit planning, subagents, and lazy skills. Experimental results show that harness effectiveness depends jointly on model capability and task type. Complex harnesses provide diminishing marginal gains on SWE-style issue repair as model capability improves, but can benefit stronger models on more complex and open-ended repository-level tasks. Component-level analysis on ProgramBench further shows that structured tool use and task-specific subagents provide the most stable improvements, while context compression and general subagents can hurt repository-generation performance. When combined, NanoHarness improves over mini-SWE-agent by 7.37 and 6.21 percentage points on Qwen3.7-Max and DeepSeek-V4-Pro, respectively, recovering most of the gains of product-level harnesses. These findings highlight harness design as a first-class factor in SE-agent performance and provide insights for building more effective and efficient coding agents.

Reflective prompt optimization improves large language model (LLM) systems without updating model weights, but fixed architectures constrain how self-refinement is organized. We introduce Workflow-Designing Agents (WDA), a framework for streamlined reflective evolution of task-adaptive self-refinement pipelines. Starting from a minimal prompt, WDA jointly evolves stage instructions and their sequential structure. During evolution, we find that repeated revisions can accumulate redundant instructions in a single prompt. In WDA, we propose to address this problem with SPLIT, which redistributes these instructions across specialized stages. Three-example reflection and local screening guide selective search, while calibration scores guide Pareto admission and rollback of unhelpful trailing updates. The resulting pipelines are task-adaptive: their instructions and depth are learned from task data, then fixed for all test inputs within that task. We evaluate WDA on five benchmarks spanning knowledge, mathematical reasoning, multi-hop question answering, and instruction following. On Qwen3.5-9B, WDA achieves an average score of 51.24%, improving over the initial solver by 8.63 percentage points and the variant without SPLIT by 2.60 points. On GPT-4.1-mini, it achieves 49.00%, with corresponding gains of 5.62 and 3.69 points. These results support task-adaptive self-refinement as a complementary direction to broader agentic workflow search.

Transformers process information strictly forward through depth, preventing deeper computation from revisiting and refining earlier representations. To augment the standard forward pass, existing approaches either re-execute depth, incurring additional computation, or modify the residual stream using predefined directions, limiting their instance-level adaptation. Recently, inference-time feedback offers a direct mechanism for recycling endogenously produced computation by writing deeper residual states back to earlier layers, yet what should be fed back remains unclear. We argue that the depth increment Delta, capturing newly accumulated computation between two layers, provides a more effective, composable, and scalable feedback signal than the full state. Building on this observation, we introduce ReFlux, a learnable feedback graph that dynamically selects and composes increment-carrying routes. ReFlux supports synchronous feedback to the same token and streaming feedback to subsequent tokens. Extensive experiments across various models, corpora, and benchmarks show that synchronous ReFlux consistently reduces perplexity across ten language-modeling corpora, and improves accuracy by 2.1-2.3 points, with gains reaching 4.7 points on multi-hop reasoning. Streaming ReFlux further retains most of these gains while preserving the base model's 1x theoretical backbone FLOPs. These results establish ReFlux as an efficient paradigm for unlocking the latent computational potential of LLMs, allowing them to revisit the same tokens with fresh representations. Code implementation can be found at https://github.com/gooogleshanghai/reflux.

World modeling under partial observation requires reasoning about the complete worlds that remain compatible with limited visual evidence. Occluded objects and unseen regions can leave several world states possible; additional views can exclude alternatives and strengthen the conclusions supported by the observations. We introduce WorldScope to study this process through possible-world semantics, evidence-grounded data, and learned visual representations. WorldScope-1.2M provides 1.2 million English question-answer pairs spanning eight world properties, ten task interfaces, and three observation protocols. Its answers encode confirmed facts, supported bounds, and unresolved possibilities. Complementary supervision comprises 4,800 certified counterworld groups with equivalent base observations and different hidden object configurations and query answers. These groups provide physical witnesses of ambiguity and training-only labels for world compatibility and view-induced exclusions. We propose WorldFlow, which composes cross-view entity evidence and support-surface coverage into an image-subset evidence lattice. Counterworld compatibility and transition objectives train subset representations to reflect how new observations constrain possible worlds. A shared answer generator uses these representations to predict the strongest supported conclusion. WorldScope-Bench evaluates claim judgments and evidence-dependent conclusions as views are selected, combined, removed, or ordered. On its 5,000-question test set, WorldFlow reaches 64.34% exact accuracy, improving over the same backbone trained on QA alone by 24.88 percentage points. It retains 50.43% accuracy on the 3,000 questions from structure-disjoint scenes.

Camera pose estimation from multi-view images remains a challenge in computer vision. Traditional methods often address this problem using Structure-from-Motion (SfM) with bundle adjustment. However, camera poses estimated from sparse views are inherently ambiguous due to insufficient geometric constraints. Recent work leverages probabilistic models, such as diffusion models, to generate multiple camera pose hypotheses and therefore capture this uncertainty better. Most of these methods represent camera rotations using unit quaternions, but treat them as unconstrained 4D vectors during the generative processes, thereby ignoring the unit-norm constraint of quaternions. Unconstrained quaternions create non-smooth and suboptimal generation trajectories. To this end, we propose *QuacamFM*, a quaternion-constrained flow matching framework for camera pose estimation that preserves unit quaternion representations throughout the entire flow trajectory. We design the optimal transport of the quaternion flows using smooth spherical linear interpolation. Experiments on CO3Dv2 demonstrate our method's advantage in camera pose accuracy over diffusion-based methods and classical SfM approaches. We further show that our quaternion-constrained formulation outperforms the naive application of standard flow matching to 4D quaternion vectors on sparse-view camera pose estimation. Finally, it is observed that QuacamFM generalizes well across datasets and in-the-wild examples.

Skeleton sequences can represent both individual actions and multi-entity interactions, encompassing human bodies, hands, objects, and robots. Existing approaches to recognize skeleton-based actions and interactions usually adopt a late fusion strategy, which expects individuals are independent and identically distributed to train a robust weight-shared entity encoder. However, observed entity bias in various skeletal data violates this assumption, leading to suboptimal optimization of backbone models that might produce wrong recognition results. This bias arises from the world coordinate system's initial configuration, where the choice of origin often creates bias in representation. To this end, we propose a Convex Hull Adaptive Shift based normalization method to reduce Entity bias (CHASE), improving performance across a variety of skeleton-based action and interaction recognition tasks. To adaptively apply plausible shifts to the input skeletons, we formulate a plug-and-play parameterized network that ensures the relocated world origin lies within the skeleton convex hull, which avoids non-convergence by limiting the search space. To further minimize entity bias, we incorporate an auxiliary objective that leverages pair-wise distribution distances to guide network optimization. To support both single- and multi-entity actions, we propose a sub-entity strategy that offers a consistent formulation for both scenarios. Moreover, CHASE demonstrates compatibility with various intra-skeleton modalities, such as bones and velocities, highlighting its adaptability. Essentially, our method works as a normalization approach to reduce entity bias, enabling subsequent classifiers to achieve improved recognition performance across diverse settings. Extensive experiments on 7 datasets verify our approach by seamlessly integrating with various backbones and significantly boosting their performance.

Robotic manipulation is inherently history-dependent, yet most pretrained robotic policies condition on only the current observation or a short temporal window. Equipping such policies with long-term memory remains challenging: existing approaches either feed the backbone multi-frame observation windows, which substantially increase inference cost, or rely on pre-defined semantic features, which limit task generality and may also require the retraining of the backbone to adapt to the memory. We introduce DRAM (Delta-rule Recurrent Associative Memory), a plug-and-play memory module that can be attached to a wide range of pretrained robotic policies, endowing them with long-horizon memory without architectural modification or backbone retraining, requiring only task-specific post-training of the memory module and action expert. DRAM maintains a fixed-size associative memory using gated delta-rule linear attention, with a modified update that incorporates all tokens within each frame in parallel. An architecture-agnostic readout integrates historical context into action prediction across different policy architectures. Experiments show that DRAM consistently improves frozen pretrained policies over short-context baselines and alternative compact memory designs, validating its effectiveness as a fixed-size, post-hoc memory module trained with the backbone frozen.

Reflective prompt optimization revises instructions using examples of a model's behavior, but which evidence the reflector should receive remains unclear. We study evidence composition, visibility of examples, candidate selection and domain-knowledge policy within a single-parent Pareto-guided search. Using Qwen3.5-9B as both task model and reflector, we evaluate nine reflection strategies on five datasets. From these experiments, we find three distinct patterns. For performance improvement, Failures-only produces the largest mean test gain (+8.0 percentage points), while Balanced-mix and No-examples+Val share the best mean performance rank. For reflection effectiveness, No-examples achieves the best rank for improving sampled parents, yet yields only a 1.4-point mean test gain: local reflection success does not necessarily produce a stronger final prompt. For overfitting assessment, Failures-only and Balanced-mix share the lowest mean calibration-gap rank, while larger gaps on GPQA and IFBench show that calibration gains can overstate held-out improvement. This gap is a descriptive indicator, not a direct measure of overfitting. Together, these results show why reflection strategies should be assessed separately on final performance, parent improvement and calibration-to-test transfer.

Galactic activity has been shown to play a fundamental role in galaxy and cluster evolution throughout the history of the Universe. In the last decades, a number of works have tackled the determination of radio luminosity functions and their evolution, with the aim to understand and quantify this role. In this work we propose a different approach, based on the application of a semi-analytical model to produce mock populations of radio loud active galaxies throughout several orders of magnitude in power, and the comparison with observational samples. The aim of this work is to test the ability of our semi-analytical model to reproduce luminosity and size distributions of large numbers of sources from the radio-loud active galactic nuclei sample from the LoTSS survey. We apply our semi-analytical model for the evolution of radio galaxies and generate hundreds of thousands of sources applying random selection of parameters within probability distributions for each of them. Our results show that the model is able to reproduce the main features of the observed sample, and identifies jet mass-load included in our model and activity periods as critical parameters that should be addressed in more detail in future work. These initial results allow us to establish a maximum activity period ranging from 10 to 200 Myr within an active time distribution probability p(t_max) \propto 1/t_{max}. We also reproduce the radio luminosity function for the z<0.6 population, and the jet power spectrum from 10^{34.5} W to 10^{38} W. Our results show that mass-load is a fundamental parameter in the determination of radio source luminosities with distance, mainly in the low-power regime. Our model predicts that at least 3-4 x 10^{5} radio galaxies must exist in the LoTSS field in order to obtain the 17045 observed ones. This population is vastly dominated by low power sources.

Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $\phi_{a,b}$ defined by $0 \mapsto aa$ and $1 \mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \le x < y \le 1$, we investigate the order relations between the corresponding Lagrange constants $\mathcal{L}([\phi_{a,a+1}(G(x))])$ associated with the mechanical words $G(x)$. While an inequality is established for pairs of the second type (type (2)) whenever $a \ge 1$, the comparisons for type (1) and type (3) pairs hold for $a \ge 2$. Consequently, for any integer $a \ge 2$, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map $x \mapsto \mathcal{L}([\phi_{a,a+1}(G(x))])$ is injective on $\mathbb{Q} \cap [0, 1]$. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary $b > a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.

Language-model self-reports are evidence about behavior in a prompt environment, not by themselves evidence of a self-model. We investigate counterfactual reports about affect-like states under activation interventions and ask whether the report remains bound to the named intervention when the demonstration environment changes. Across three open instruction models, wrong-source demonstrations move reports toward the source answer family, while explicit mechanism binding reduces this pull. Self-report benchmarks should include environment-shift invariance tests under fixed intervention before treating accuracy as evidence for an autonomous report mechanism.

Autoregressive Next-Token Prediction (NTP) has enabled strong reasoning capabilities in language models, while Diffusion Language Models (DLMs) offer flexible token orders and parallel generation. We ask whether DLMs can acquire NTP-style reasoning through distillation without giving up their native generation process. Direct distillation, however, faces a fundamental mismatch: an autoregressive teacher predicts from a left prefix, whereas a DLM can condition on tokens on both sides. We introduce ForkLeft, a distillation framework that resolves this mismatch by separating the student's rollout from teacher supervision. During training, the student first performs entropy-first rollouts that commit uncertain positions and expose potential forks. We then fix the resulting student prefix and distill an NTP teacher under the same context, with answer correctness determining the supervision source. At inference, the student returns to its native confidence-first parallel decoding. With Qwen3-30B-A3B-Base, ForkLeft improves Efficient-DLM-4B on all ten benchmarks, raising MATH500 from 72.60% to 79.60% and consistently outperforming three alternative designs. The gains scale with teacher strength and generalize to SDAR-4B with only $500$ updates. At matched scale, the distilled 4B and 8B students exceed the published SDAR-Chat and OPDLM models on seven benchmarks, showing that DLMs can learn NTP-style reasoning without sacrificing native parallel generation. Code and datasets will be released upon acceptance.

Learning robust and scalable finite-state tracking is fundamental to sequence processing. While linear recurrent neural networks (RNNs), linear attention, and state space models enable scalable parallel training through affine recurrences, their theoretical expressivity guarantees assume idealized arithmetic and do not extend to finite precision, where the parallel scan that makes them fast is itself a source of perturbation. We formalize finite-state tracking at finite precision and characterize length independence: tracking that stays correct at every sequence length, at a precision cost that does not grow with the length. We show that length-independent state tracking requires two competing dynamics within a single map: contraction to suppress numerical perturbations and separation to keep distinct states apart. We prove that affine recurrences, which offer a single rate at each step to serve both roles, realize at most definite automata at finite precision. Instead of treating scan compatibility as a restriction on the update map, we reinterpret it as a computational budget and introduce the Neural Finite-State Machine (NFSM): a nonaffine, scan-compatible RNN built for length-independent finite-state tracking. On synthetic benchmarks spanning abelian and nonabelian groups, noninvertible monoids, and textual state-tracking tasks, affine baselines fail on every nondefinite task, most of them within a few hundred steps. A single NFSM layer instead learns the exact transition tables of every algebraic task, which certifies correctness beyond the tested lengths, and a stack of NFSMs keeps perfect accuracy on the textual tasks at every tested length.

In sampling, different flavours of annealing, tempering, or other successive approximation approaches are widely used and studied. In this work we investigate annealed underdamped Langevin sampling for logconcave distributions of the form $\pi(x) \propto e^{-\Psi(x)}$ for a potential $\Psi:\mathbb{R}^d \rightarrow \mathbb{R}$. That is, we assume access to the score of a general approximating family of distributions $(\pi_{\tau})_{\tau\in [0,1]}$ with $\pi_0=\pi$. This family is used in a time-inhomogeneous underdamped Langevin process with moving target $\pi_{\tau(t)}$ and an appropriate annealing schedule $t\mapsto \tau(t)$ such that $\tau(t)\downarrow 0$ as $t\to T$ for some $T>0$. While the degeneracy of the Brownian motion prohibits conventional Girsanov arguments, we are able to circumvent this issue by relying on an appropriately designed auxiliary process. To the best of our knowledge these are the first (quantitative) results for sampling using time-inhomogeneous underdamped Langevin dynamics. Moreover, we obtain an iteration complexity of $\mathcal{O}(\epsilon^{-2})$ in total variation distance in comparison to $\mathcal{O}(\epsilon^{-6})$ for annealed overdamped Langevin sampling. While we currently assume convexity, our proof strategies are amenable to generalization for non-logconcave targets and enable for several future research directions.

Direct Preference Optimization (DPO) aligns language models by optimizing over sequence-level sums of token-wise implicit reward differences. However, we identify a pervasive pathology in this formulation: a disproportionately small subset of high-frequency token types dominates cumulative sequence scores while appearing symmetrically across both preferred and dispreferred responses. Specifically, under canonical Qwen tokenization on Anthropic HH-RLHF, merely 69 token types account for $55.1\%$ of all response tokens and $85.9\%$ of within-pair shared token mass, exhibiting substantially lower preference-side specificity than the remaining vocabulary. This symmetric ubiquity induces gradient entanglement and dilutes the discriminative preference signal propagated through the objective. To resolve this issue, we introduce Anisotropic DPO (ADPO) and its canonical realization, Frequency-Hard DPO. Using a fixed, label-agnostic vocabulary mask, our method zeroes the implicit reward contribution of high-frequency response tokens while assigning unit weight to informative positions, thereby suppressing gradient interference without modifying preference pairs, discarding context, or introducing learned parameters. Here, anisotropy designates non-uniform token-level objective weighting rather than representational geometry. Extensive empirical evaluations on AlpacaEval, MT-Bench, and Arena-Hard demonstrate that Frequency-Hard DPO consistently outperforms standard DPO across Qwen-2.5-7B-Instruct and Llama-3-8B-Instruct, establishing that selectively masking shared high-frequency tokens offers an effective, zero-overhead mechanism for robust preference alignment.

We study training-inference mismatch in reinforcement learning with verifiable rewards (RLVR) for large language models, where rollouts are sampled by an inference engine while gradients are computed by a training engine, and the two engines assign different probabilities to the same tokens. To account for this discrepancy in policy updates, we introduce calibrated importance sampling (CIS). CIS is motivated by an empirically supported logit-displacement characterization that expresses the mismatch as an additive displacement $\varepsilon_t$ in log-odds, determined by the per-logit perturbation before the softmax, whose distribution is approximately invariant to token confidence. This characterization motivates a confidence-aware truncation: large positive displacements are truncated at a single constant threshold, which maps back to an importance-ratio cap that tightens as token confidence increases. Theoretically, we show that CIS replaces the unbounded second moment that governs the error of exact importance sampling with a term bounded by a constant, at the cost of a bias controlled by the truncated excess. In evaluation across three mixture-of-experts models and five mathematical reasoning benchmarks, CIS achieves the highest five-benchmark average on all three models among the evaluated baselines. Diagnostic analyses show that CIS places less truncation bias on low-confidence tokens than truncated importance sampling, while upward clipping of small importance weights reduces held-out accuracy.

For area-$4\pi$ metrics on the two-sphere, we obtain quantitative refinements of classical eigenvalue inequalities and study the joint behavior of the first two Laplace eigenvalues. We prove a sharp quadratic improvement of Hersch's reciprocal-sum inequality in terms of the first eigenvalue, together with a simultaneous quadratic estimate for the first three eigenvalues and a geometric stability result for admissible measures. We also establish a quantitative refinement of Nadirashvili's second-eigenvalue inequality with sharp exponential scale. Finally, using explicit spectral families due to Petrides, a degree argument, and a refined finite-dimensional matrix method, we obtain nontrivial inner and outer bounds for the joint range of $(\lambda_1,\lambda_2)$.

[An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on $d$-dimensional tori, $d \le 2$, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If $d=1$, fully periodic symmetric bases exist across all AZC classes. If $d=2$, topological obstructions appear as non-trivial phases characterized by $\mathbb{Z}$ (Chern number) or $\mathbb{Z}_2$ (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.

Many learning problems couple a high-dimensional block of parameters with a handful of adversarial or dual variables: worst-group risk over a few groups, or learning under a few constraints. We consider $\min_{x\in X}\max_{y\in Y} f(x,y)$ with $X\subseteq\mathbb{R}^n$, a compact convex set and strongly convex with condition number $\kappa$, and we count the oracle calls for $x$ and for $y$ separately. The textbook approach runs a cutting-plane method in $y$ and solves every inner problem to accuracy $\varepsilon$ with an accelerated method; it needs $O(m\log(1/\varepsilon))$ calls for $y$ but $O(m\sqrt{\kappa}\log^2(1/\varepsilon))$ calls for $x$. We show that the inner problems need not be solved accurately. Our method, certificate transport, keeps a strongly convex lower model of a single slice and uses it as a prior for a short accelerated run on the next slice. By concavity in $y$, every call for $y$ then either cuts the localizer or moves the lower model to a mixture of the two slices with a certified increase of the lower bound. For $m=1$ this gives an $\varepsilon$-saddle point after $O(\sqrt{\kappa}\log(1/\varepsilon))$ calls for $x$, up to an initialization term, and $O(\log(1/\varepsilon))$ calls for $y$; both counts are optimal, even though $f(x,\cdot)$ is only assumed to be concave and Lipschitz. For general $m$, with centers of gravity of the localizer treated as computable, the method needs $O((m+\sqrt{m\kappa})\log(1/\varepsilon))$ calls for $x$ and the optimal $O(m\log(1/\varepsilon))$ calls for $y$. Under strong concavity in $y$, a two-point accelerated method makes the number of calls for $x$ independent of $m$.

The facial distance of a polytope, equivalently its pyramidal width, is a geometric condition measure that appears in the convergence bounds for Frank–Wolfe algorithms. Peña and Wirth developed calculus rules for this quantity and conjectured that the ratio of the facial distance to the diameter multiplied by the square root of the dimension is bounded by a constant. We show that this quantity is uniquely maximized by a regular simplex in each dimension, which implies that the conjecture holds and that $\sqrt{2}$ is the smallest constant. We also compute the facial distance of the Birkhoff polytope.

The dominated coloring of a graph $G$ is a proper vertex coloring such that each color class is dominated by a vertex. The minimum number of colors required for a dominated coloring of a graph $G$ is called the dominated chromatic number of the graph $G$ and is denoted by $\chi_{dom}(G)$. A graph $G$ is said to be chromatic dominated if $\chi(G)=\chi_{dom}(G)$. In this paper, we characterized the perfect graphs, which are chromatic dominated. Also, we studied the dominated chromatic number of some classes of unicyclic graphs.

In this paper, we prove that for $\ell>5$, the number of $\ell$-regular conjugacy classes of a finite group is at least that of the normalizer of a Sylow $\ell$-subgroup.

This letter proposes a 20-kV 4H-SiC optical-injection-enhanced reverse-conducting IGBT (OIE-RC-IGBT) with snapback-free operation and low on-state voltage. Light-emitting diodes (LEDs) are integrated into the gate structure to provide optical injection during forward conduction. The emitted light generates a high concentration of photogenerated carriers in the carrier-storage layer (CSL) and drift region, thereby reducing the resistance of the JFET and drift regions and increasing the device current. Consequently, the collector-side P+/N junction is turned on earlier, effectively suppressing the snapback effect and facilitating the transition to bipolar conduction. After bipolar conduction is established, the photogenerated carriers further enhance conductivity modulation, resulting in a reduced on-state voltage. TCAD simulation results show that, compared with the conventional RC-IGBT (C-RC-IGBT), the proposed device achieves snapback-free operation while reducing the required number of MOS cells from 25 to 5. Moreover, the on-state voltage, turn-off fall time, and turn-off energy loss are reduced by 22.37%, 25.06%, and 19.69%, respectively. These results demonstrate the potential of optical injection for improving the conduction and turn-off performance of ultrahigh-voltage SiC RC-IGBTs.

Mechanisms for generating GNN explanations are crucial for building trust and mitigating biases in Graph Neural Networks (GNNs), especially in high-stakes scenarios. Most current methods optimize only for fidelity under the sparsity constraint. However, this discounts the need for interpretable explanations (those that consist of familiar motif patterns) and stable explanations (those that remain unchanged under structural perturbations). We propose a novel approach that optimizes GNN explanations across these metrics, exposing their relative weighing as a control. Experiments on various real-world datasets, including MUTAG, BA-2Motif, BAMultiShapes, and PROTEINS, suggest that our method produces higher-fidelity explanations than a state-of-the-art baseline on MUTAG and PROTEINS across all evaluated budgets, and on BA-2Motif at larger budgets, while being faster in the regime of small explanation budgets. They also indicate that our method produces more stable explanations for the MUTAG and BA-2Motif datasets. We also examine the relationship among different metrics via their induced trade-off surface and explore its dependence on the nature of the motif library.