Papers

Manipulation under time constraints requires both accurate actions and an execution rhythm that matches the evolving scene. This becomes critical when a robot must intercept moving objects or complete a sequence of adjustments before a deadline. Although one-step policies reduce generation cost, their directly predicted action sequences leave temporal allocation implicit. We propose Shared Execution-Clock Drifting (SECD), which makes execution rhythm an explicit part of one-step action generation. Conditioned on an observation and a latent sample, the policy jointly predicts a progress-indexed action curve and a shared monotone clock that maps fixed control times to locations on the curve. Demonstration-derived alignment anchors this decomposition, which is trained jointly through drifting on the decoded actions. The resulting policy retains a fixed-rate control interface and requires one network evaluation. We evaluate SECD across four real-robot tasks with inference on NVIDIA Thor. Across 300 trials, it achieves 77.00% task-averaged success and outperforms the evaluated one-step baselines on every task, including 91% success in cup retrieval from a 16 m/min conveyor and 54% in restoring and folding a crumpled shirt within 90 s. A fixed-clock variant reaches 79% on the same conveyor protocol. Complementary state-based RoboMimic experiments, including cross-seed ablations on Transport and Square, further support the joint design of the temporal representation and demonstration alignment. Project page: https://secd-anonymous-ewn.pages.dev/

We prove congruence theorems for capillary hypersurfaces of Euclidean space supported on a hyperplane or on a sphere, extending a result of Ros for closed hypersurfaces. We also establish a Heintze–Karcher type inequality for capillary hypersurfaces in substatic manifolds.

Many structural and dynamic economic models imply that key parameters are identified by conditional quantile restrictions. Building on the exponential-weighting approach of Bierens (1990) and penalized maximum statistics for conditional moment restrictions developed by Chen et al. (2025), we develop a framework for uniform inference on such parameters. We transform the conditional restriction into a continuum of unconditional moment conditions and construct an adaptive $\ell_1$-penalized supremum statistic that aggregates evidence across quantile indices. The penalty regularizes maximization over the weighting direction. Under the stated conditions, adaptive penalty selection asymptotically preserves or improves worst-case local power relative to the unpenalized test over a prespecified set of alternatives. The improvement is strict whenever a positive candidate penalty raises the population maximin criterion above its value at zero. We extend the theory to settings with pre-estimated nuisance parameters, characterizing the additional terms induced by the plug-in step in the limiting process. We derive an analytically corrected variance estimator that accounts for this estimation uncertainty and establish the validity of a Gaussian multiplier bootstrap for null inference and its shifted counterpart for analysis of local power. Monte Carlo comparisons favor CvM-KS for size control under pre-estimation and show that adaptive penalization can improve empirical power relative to the unpenalized test.

We present a statistical study of the hierarchical structure of molecular clouds in the second quadrant of the Galactic midplane (139.75 deg <= l <= 159.75 deg, -5.25 deg <= b <= 8.25 deg), based on high-sensitivity 12CO and 13CO J = 1-0 data from the Milky Way Imaging Scroll Painting survey. Using DBSCAN and a non-binary Dendrogram algorithm, we identify 2,912 13CO clouds. Type 2 (multicloud, single-tracer) scaling relations show arm-dependent trends: the power-law index of the linewidth-size relation is about 0.32 in the Local Arm and about 0.42 in the Perseus and Outer Arms. The classical Keto-Heyer relation is weak in the Local Arm but present in the Perseus and Outer Arms, with a slope of about 0.3. The virial parameter is anti-correlated with cloud size, mass, and surface density, consistent with previous surveys. Type 4 (single-cloud, single-tracer) scaling relations exhibit stronger internal correlations. The linewidth-size index spans about 0.3-0.8 and correlates positively with sonic Mach number, indicating that the relation is not universal but depends on cloud compressibility. While substructures do not follow the conventional surface-density-based Keto-Heyer relation, a volume-density formulation restores a clear virial scaling, with evolutionary tracks aligning along lines of constant virial parameter. Moreover, the structural complexity of molecular clouds, characterized by the relationship between the number of substructures and the min_delta parameter, correlates with surface density rather than sonic Mach number, contrary to results from turbulence-only magnetohydrodynamic simulations. Together, these results are more consistent with a gravity-regulated hierarchical dynamical picture than with a purely turbulence-supported scenario.

Distributed tracing in large-scale AI infrastructure fails silently when clock accuracy is insufficient: causal events are misordered, fault attribution is corrupted, and performance diagnoses are unreliable. We present TempoTrace, a system that co-designs IEEE 1588v2 PTP time synchronization with distributed tracing to preserve causal ordering across heterogeneous, multi-tenant GPU clusters. We formally prove that NTP-grade clocks produce causal inversions at 25-30% of directed operation pairs under WAN/cloud conditions (11.3% under LAN NTP). A Laplace heavy-tail noise model extends the analysis beyond Gaussian assumptions, giving a weighted expected misorder rate of 27.76% vs. 29.37% Gaussian, confirming robustness to tail shape. TempoTrace reduces GPU-to-host timestamp uncertainty from 2.1 us to a design target of 0.056 us residual standard deviation via GPUDirect RDMA hardware timestamping, with sub-100 ns PTP synchronization. A bounded Sketch Vector Clock tracks causal relationships with false-positive rate below 10^-5 for up to four participants. A hybrid rule-based and XGBoost diagnosis engine achieves macro-F1 0.974 in controlled validation (1,800-incident corpus; McNemar p=1.25e-23). A formally proven multi-tenant model provides isolated logical clock domains. For RoCEv2/ECMP fabrics, P4-based in-band network telemetry corrects path-delay asymmetry, yielding 99.1% attribution accuracy. Application Confidence Policies let workloads specify precision requirements and degraded-mode fallback. Evaluation combines physical five-node measurements (NTP inversion rate 44.978%, Laplace fit preferred over Gaussian by AIC under congestion) and controlled synthetic validation with illustrative configurations from 512 to 16,384 H100 GPUs across InfiniBand and RoCEv2 fabrics.

We present a phenomenological study of the near-threshold $\pi^- p \to J/\psi n$ reaction, focusing on the interference effects between the $s-$channel $P_{c \bar c}$ resonances and non-resonance background. This reaction provides a complementary probe to photoproduction for investigating the nature of the hidden-charm pentaquark states. Using an effective Lagrangian approach, we construct scattering amplitudes for $J^P=1/2^-$ and $3/2^-$ $P_{c\bar c}$ states under two spin-parity scenarios, with the background dominated by the $t-$channel $\rho$ meson exchange. We find that the interference can produce peaks, dips, or complex line shapes depending on the relative phase, in contrast to the simple resonance peaks commonly reported in previous theoretical studies. The two spin-parity scenarios yield qualitatively similar patterns, indicating that our main conclusion is robust. Given the current scarcity of experimental data for the pion-induced channel, we emphasize the urgent need for future measurements at J-PARC to understand the dynamics of the $J/\psi N$ interaction near threshold and to search for the heavy pentaquark states.

We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes. Unlike conventional black-box neural surrogates, the proposed network architecture is fully dictated by finite element discretization rules: local computations stem from simplex geometry and barycentric coordinate transformations, while global consistency is enforced through shared degrees of freedom. For linear Lagrange elements, local basis functions are directly implemented via affine barycentric layers, and high-order polynomial bases are explicitly decomposed into barycentric product compositions realized by specially designed $\mathrm{ReLU}^p$ modules. Equipped with element indicator branches and multiplication units, these modular local components are globally assembled into a sparsely connected neural network whose function space coincides exactly with the target finite element space, thereby inheriting the complete classical finite element approximation theory. By prescribing customized backward differentiation rules for piecewise activations, function values and their spatial gradients can be simultaneously extracted via automatic differentiation within a unified computational graph, eliminating the separate gradient calculation subroutines required in standard finite element implementations. Numerical experiments verify the accuracy of the neural network representation of Lagrange finite elements. Furthermore, by virtue of the intrinsic mesh-free nature of this neural network representation, finite element functions can be interpolated between non-matching meshes, and the proposed scheme can be applied to adaptive finite element methods for solving parabolic partial differential equations. An open-source code implementation of the proposed architecture is made publicly available.

The direct static capacity region of a bipartite quantum state describes its asymptotic conversion into classical communication, quantum communication, and entanglement, with all communication directed from Alice to Bob. This paper gives a unified entropic converse for this region. The proof treats generated and consumed resources simultaneously and derives all three information inequalities for one local instrument. Its main ingredients are no-signalling, the chain rule, strong subadditivity, and continuity of conditional entropy. A supporting-hyperplane characterization leads to the static capacity formula, which is helpful in determining single-letterization of the capacity region. For a pure state of entanglement entropy $h$ subjected to erasure with probability $p$, the complete region is the convex hull of $0$ and $(0,-ph,(1-p)h)$, plus the unit-resource cone. This statement holds for arbitrary collective instruments by using subset-entropy inequalities related to earlier methods used for the dynamic capacity region of the erasure channel. Complete characterizations also hold for symmetrically extendible states and locally flagged pure-state mixtures. For maximally correlated states, the optimization over local instruments admits an exact matrix formulation and an additive outer bound. An explicit qubit example shows that a nonorthogonal discarded quantum memory can outperform every efficient instrument and every instrument with conditionally commuting discarded states. For states obtained by qubit dephasing of Schmidt-aligned pure states, an exact threshold characterizes when the static capacity formula vanishes, at every blocklength and after regularization. Full single-letter characterizations for general Hadamard states and for the remaining trade-offs for dephased states remain open, with sufficient additivity conditions identified.

We prove the sign-isotypic components of the coinvariant rings $R_n^{(2,1)}$ and $R_n^{(2,0)} \otimes R_n^{(0,1)}$ are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial $S_n(q,t,a)$, divided by $1+a$. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of $R_n^{(2,1)}$. Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky–Mellit (2026) which relates the Khovanov–Rozansky homology of the $(n,n+1)$-torus knot to $R_n^{(2,0)} \otimes R_n^{(0,1)}$, by showing that the associated Poincaré series for this knot can be computed from the sign component of $R_n^{(2,1)}$.

Tokamak equilibria are commonly exchanged as gridded G-EQDSK files whose conventions, resolutions, and machine-specific formats impede cross-device comparisons and data-driven modeling. Here, we present VEQDB, an open, compact, and reconstructible fixed-boundary equilibrium database built on continuous MXH–Chebyshev geometry and independent physical-profile roots. By decoupling authoritative equilibrium physics from rectangular meshes, VEQDB enables continuous evaluation and metric differentiation at arbitrary application-demanded resolutions. Backed by an automated numerical validation pipeline, VEQDB is structured as an extensible repository for ongoing community expansion. Its inaugural release provides 13,291 accepted equilibria across 267 conventional and spherical tokamaks, encompassing parameter-sampled Grad–Shafranov solutions, G-EQDSK projections spanning EAST, MAST-U, and ITER scales, and controlled variation families with explicit provenance. Benchmark projections reproduce normalized flux maps with RMS errors between $1.09 \times 10^{-3}$ and $1.45 \times 10^{-3}$, while compact JSON representations achieve an 89–96-fold size reduction relative to standard $129 \times 129$ G-EQDSK files. The complete initial release occupies 41~MB in raw JSON and 18~MB in compressed archives, and all records successfully passed independent reload and evaluation tests. VEQDB establishes an extensible, provenance-preserving foundation for equilibrium studies, reduced-order surrogate modeling, and cross-machine workflows.

Exciton-polaritons, hybrid light-matter quasiparticles formed when a material interacts with a confined electric field, have experimentally been shown to exhibit mesoscale coherent quantum propagation that remains robust at room temperature. However, an accurate and direct quantum dynamical simulation of this phenomenon that does not resort to semi-classical approximations is prohibitively expensive computationally, limiting the microscopic understanding of the rich dynamical interplay among phonons, photons, and electrons under collective light-matter coupling. To address this fundamental challenge, we develop a non-Markovian master equation approach which enables the fully quantum mechanical simulation of non-equilibrium exciton-polariton dynamics and captures phonon-induced decoherence and dissipation beyond the conventional Markovian limit. To carry out this task, a procedure is developed in which the wave vector space is coarse-grained and each diagonal element of the density matrix is evolved in parallel. To demonstrate the utility of this approach, we simulate exciton-polariton transport in TIPS-pentacene. We find that our approach reasonably captures the experimentally observed renormalization of the polariton group velocity, which originates from the phonon-induced non-Markovian Lamb shift. We further show that this renormalization cannot be reproduced within conventional Markovian theories.

Insurance pricing combines pooling with differentiation: a tariff may leave benchmark differences in expected loss partly mutualized or translate them into policy-level premium differences. We propose a benchmark-relative pooling–profiling scale with two complementary coordinates. The coupled $L^p$ coordinate measures policy-level alignment between an evaluated tariff and a stated benchmark pure premium, whereas the marginal Wasserstein coordinate compares their exposure-weighted premium distributions. The difference between their residual $p$-costs defines an allocation mismatch. Under portfolio balance, the coupled $L^1$ coordinate has an exact actuarial interpretation: it is the fraction of the transfer volume induced by full pooling that the tariff removes. Synthetic and motor-insurance applications show that broad classes, proxies, shrinkage and tail caps can affect marginal differentiation, policy-level allocation and transfers differently. A barycentric group-parity intervention further shows that conditional premium disparities can fall mainly through reallocation and restored benchmark-relative transfers, with little change in marginal differentiation.

In the A* search algorithm, the tie-breaking strategies for nodes with the same $f$-value determines which states A* expands on the final $f$-layer. For nine standard tie-breaking strategies, we show that under a consistent heuristic, every pair has positive-cost instances favoring each strategy over the other by an arbitrarily large additive expansion gap. A parameterized unit-cost grid example also gives unbounded expansion-count ratios between low-$h$ with FIFO and LIFO. In unit-cost search with $h > 0$ at non-goals, exact heuristic values near the goal lead to complementary extremal results: low-$h$ minimizes the number of remaining expansions from a common configuration within the perfect region, while high-$h$ maximizes the total number of expansions when every final-plateau state with $h=1$ is a goal predecessor. Finally, with the evaluation function $f_{\alpha} = g + \alpha h$, when $h>0$ at non-goals, every heuristic weight $0 \leq \alpha<1$ eliminates tie-breaking sensitivity, and all tie-breaking strategies expand the same set of states.

We investigate QCD thermodynamics in the intermediate-temperature regime using a gas of confining strings supplemented by excluded volume (EV) corrections. In this description, the discrete hadron resonance spectrum is replaced by an exponentially growing open-string spectrum characterized by a Hagedorn temperature $T_H$, related to the confining string tension. With a common EV parameter shared by mesons and baryons, the model gives a good description of bulk lattice-QCD thermodynamics above the pseudocritical temperature. Among the values considered, $T_H \simeq 300\,\text{MeV}$ provides the best agreement, lower than the value $T_H \simeq 320\!-\!340\,\text{MeV}$ inferred from ideal-string-gas fits to the vacuum hadron spectrum. A simultaneous calibration of the mesonic and baryonic EV parameters gives $b_M \simeq 0.19\,\text{fm}^3$ and $b_B \simeq 0.73\,\text{fm}^3$. Within the present spectral ansatz, the larger baryonic EV parameter indicates that stronger effective suppression is required in the baryon sector, although noticeable tensions remain in several conserved-charge observables.

We exhibit a graph $G$ with $14$ vertices and list chromatic number equal to $3$ such that there is a $2$-list assignment $L$ of $G$ such that at most $9$ vertices of $G$ can be properly coloured from $L$. This disproves the Partial List Colouring Conjecture of Albertson, Grossman and Haas. This counterexample was discovered and fully verified by ChatGPT 6 Astra Ultra after some persistent prompting, but almost no mathematical input, from the author.

High-dimensional count data are common in scientific applications, but most diffusion and flow models are designed for continuous or categorical data, and generation often requires many sequential model evaluations. Here, we propose Count Flow Map, a generative model that learns finite-time transitions directly in count space for one- or few-step generation and transport between count distributions. Our model directly learns stochastic transitions over finite time intervals, using Poisson births and Binomial deaths to preserve nonnegative integer counts without a predefined maximum. These transition models are trained to match the underlying local birth–death dynamics and to maintain consistency across step sizes. We characterize the connection between local dynamics and finite-time transition consistency and derive a bound on the generation error. After validating Count Flow Map in several simulations, we apply it to single-cell drug perturbation prediction and neural population forecasting, where it captures perturbation responses and supports forecasts of high-activity events with only one or a few model evaluations. Together, these experiments demonstrate that Count Flow Map enables high-quality generation directly in count space across inference budgets, from one-step to few-step generation, using a single trained model.

We extend quasi-projective dimension and quasi-projective length from finitely generated modules to homologically finite complexes by using finite filtrations in the derived category. Our definitions recover the original invariants of Gheibi–Jorgensen–Takahashi for modules and behave well under exact functors, which simplifies the proofs of several results. We establish the Auslander–Buchsbaum formula, the derived depth and width formulas, and the dependency formula for complexes of finite quasi-projective dimension. Extending a result of Gheibi–Jorgensen–Takahashi, we show that every homologically finite complex has finite quasi-projective dimension over a suitable complete intersection ring. We also prove a new intersection theorem and a descent theorem for Serre's conditions. Finally, we obtain vanishing results for Tor, Ext, and Tate (co)homology, and also symmetry of eventual Ext vanishing.

For over two decades, Na I D ($\lambda \lambda5891.58, 5897.56$ Å) absorption has been the primary optical tracer of cool, neutral gas in galaxies. In contrast, Ca II H&K ($\lambda \lambda$3934.78, 3969.59 Å), which traces both neutral and ionized gas, has been comparatively unexplored. Here, we present a comparative study of the Na I D and Ca II H&K interstellar absorption lines using spatially resolved MaNGA IFU spectroscopy of 140 red geyser galaxies (local quiescent systems hosting low-luminosity AGN with bisymmetric ionized outflows) and a control sample of 140 quiescent galaxies. Using Gaussian fits to the Na I D and Ca II K lines, we measure gas velocity and dispersion. We find that the Ca II reservoirs are up to ~ 6 times larger in area, twice as radially extended, and have ~ 1.7 - 1.8x broader line widths than Na I D. Furthermore, Na I D shows a direct dependence on dust reservoir size, but the correlation vanishes for Ca II. In only ~ 20% of galaxies do the Na I D detections extend beyond dust regions, but this fraction jumps to ~70% for Ca II. Using these findings, we argue that dust plays a significant role in shielding Na I D from ionizing radiation, whereas Ca II, with its higher ionization potential, is not as dependent on shielding and can more easily survive beyond regions of dust. Na I D and Ca II trace distinct gas components, and studying both together provides a more complete picture of gas flows in galaxies.

Let $v$ be a finite place of a rational function field with finite constant field. We establish an Ogus-type $v$-adic Chowla-Selberg formula for $t$-motives with "complex multiplication" by subfields of Carlitz cyclotomic function fields. For this purpose, we first introduce the $v$-adic crystalline action of the Weil group on the de Rham spaces of $t$-motives via the de Rham-crystalline comparison isomorphism of Hartl-Kim, and use it to give a "$v$-isocrystal" period interpretation of special values of Thakur's $v$-adic geometric gamma function. Our Ogus-type formula is then based on the isogeny theorem for CM $t$-motives, together with a recipe for constructing "abelian CM types" given in the $\infty$-adic setting over function fields.

3D reconstruction from a lengthy video stream input poses a dilemma for feed-forward reconstruction models (FFRMs), that a whole-stream inference context cannot be retained under limited GPU memory.Recent studies seek to resolve this problem via a trade-off between the integrity of inference context and GPU memory usage, which either suffer from a rapid memory inflation or degraded context integrity due to artificially capping memory usage.Driven by our key observation that the initial saliency of a token reliably dictates its long-term importance across the stream, we propose RegVGGT, a training-free token regulation method which aggressively regulates the tokens of incoming frames.By admitting at most 1% of tokens per frame to update the context memory, our method dramatically suppresses memory inflation as the stream progresses.Equipped with a FlashAttention-compatible token saliency estimation scheme, RegVGGT is capable of processing thousands of frames on a consumer-grade GPU with negligible compromise to reconstruction quality.Extensive experiments demonstrate that RegVGGT achieves state-of-the-art performance on long-horizon benchmarks across diverse FFRM prediction tasks, surpassing prior FFRM-based stream reconstruction baselines by a large margin.

An expert may misreport her forecast to influence a decision she cares about. We study contracts that elicit both her belief about a verifiable state and her stake in a binary action. If every belief is possible at two distinct stakes, exact incentive compatibility prevents a belief-only mechanism from letting the action depend on the report. Asking about both belief and stake allows the action to respond, but may require implementation rent. For threshold rules, we show that minimum rent equals the cutoff's positive curvature weighted by the belief distribution. The principal therefore chooses a cutoff by balancing virtual surplus against the implementation rent of truthful reporting. We characterize the best cutoff rule and give sufficient conditions under which it is optimal even among randomized mechanisms.

Without interference, cross-fitting enables flexible covariate adjustment while preserving finite-sample unbiasedness under independent unit-level randomization. Under network interference, out-of-sample prediction alone no longer guarantees unbiasedness: assignments entering evaluation-fold Horvitz–Thompson weights may also affect outcomes in the training sample, inducing dependence between fitted predictions and those weights. We develop neighborhood-excluded cross-fitting, which constructs estimand- and design-specific training samples to restore the conditional independence needed for finite-sample unbiasedness without a correctly specified outcome model. We establish asymptotically valid design-based Wald inference for direct and indirect effects under Bernoulli randomization and for the global average treatment effect under Bernoulli cluster randomization. Neighborhood exclusion creates a trade-off in choosing the number of folds: unit-level splitting may require the number of folds to grow with average exclusion-neighborhood size, while cluster-level splitting can substantially relaxes this requirement, permitting a fixed number of folds under partial interference. For linear adjustment under Bernoulli randomization, we derive variance-optimal and confidence-interval-length-optimal procedures, establish explicit rate conditions allowing the covariate dimension to diverge, and show that the variance-optimal procedure is asymptotically no-harm. Simulations illustrate the bias from omitting neighborhood exclusion and the precision gains from adjustment. An application to a social network experiment yields confidence intervals substantially shorter than those from the unadjusted estimator.

In this paper, we study a class of degenerate mean field games (MFG, for short) systems with Hormander diffusion, in which the typical agent can move only along admissible direction. We establish the well-posedness of the MFG systems in intrinsic Holder spaces, which describes the Nash equilibria for a differential game with infinitely many small players. The analysis builds upon degenerate parabolic equations defined on the tours induced by Hormander vector fields without any group structure, for which we develop a global regularity theory for the first time in this general setting. A central difficulty arises from the anisotropic geometry of the state space, which induces non-commutativity and inhomogeneity of the underlying vector fields. Instead of the classical fundamental solution framework, we present an alternative method to derive a priori Schauder estimates for general Hormander degenerate parabolic equations via Campanato spaces.

We extend the study of random linear equations over finite fields to equations over finite commutative rings. We characterize precisely when the satisfiability threshold occurs at a sublinear scale; namely, when the random system become unsatisfiable with high probability with a number of constraints $m$ that is sublinear in $n$, the number of variables. In this regime, we determine the exact value of the satisfiability threshold. In the complementary regime where the satisfiability threshold is linear in $n$, we determine its precise value when $R$ is a principal ring. Interestingly, this value is independent of the choice of $R$, mirroring the same phenomenon when $R$ is a finite field. We further prove that this independence of $R$ breaks down if $R$ is nonprincipal. In particular, we investigate a classical family of nonprincipal rings and determine the satisfiability thresholds for all rings in this family. Remarkably, in this setting, the satisfiability threshold depends not only on the underlying ring, but also on other parameters defining the random linear equation model.

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over \(\I=[0,1]\). Using the Lowen fuzzy Sierpiński object \(\Sier\), which identifies \(\tau_X\) with \(C(X,\Sier)\), we explicitly determine the largest splitting topology on this mapping set. Its open weights \(\Phi:\tau_X\to\I\) are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of \(\Sier\). This yields an intrinsic characterization: \(X\) is exponentiable if and only if every \(\mu\in\tau_X\) satisfies \[ \mu=\bigvee_{\lambda\triangleleft\Phi} (\const{\Phi(\mu)}\wedge\lambda), \qquad \lambda\triangleleft\Phi \Longleftrightarrow \const{\Phi(\nu)}\wedge\lambda\leq\nu \quad(\nu\in\tau_X). \] When this condition holds, \(Y^X\) has underlying set \(C(X,Y)\), with topology generated by \([\Phi,v](f)=\Phi(v\circ f)\). We also obtain a dual closed-set formulation and three applications. Exponentiability implies that \(\tau_X\) is a continuous lattice, although the converse fails. Moreover, a classical space \(X\) is exponentiable exactly when its induced fuzzy space \(\omega X\) is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and \(N\)-compact Hausdorff spaces are exponentiable.

Safety-critical control under uncertainty requires uncertainty representations that are both statistically valid (for certifiable performance) and compatible with enforceable safety constraints. However, existing methods often assume particular distributions of uncertainty for provable safety guarantees or establish symmetric and input-agnostic prediction intervals for robust safety, which can lead to misaligned or overly conservative safety constraints in control synthesis. In this paper, we introduce a novel safe control framework with adaptive uncertainty quantification that constructs calibrated and state-dependent prediction intervals to enable high-probability safety guarantees, while improving constrained control performance. The framework leverages adaptive conformal prediction (ACP) and extends it with conformal quantile regression (CQR) to capture distribution-free, asymmetric uncertainty intervals with certifiable probabilistic coverage, and integrates the resulting uncertainty sets into a probabilistic control barrier function formulation to enforce robust safety with reduced conservativeness. This yields uncertainty-aware safe control constraints that can be incorporated within a model predictive control(MPC) framework to provide provably safe behaviors with high probability. Simulation and theoretical results are provided to demonstrate the effectiveness of our approach.

We analyze a counterexample concerning non-prox-regular functions provided by Rockafellar and Wets in their 1998 foundational monograph, Variational Analysis. By establishing a general theorem on the continuity of set-valued mappings, we show that the counterexample does not have its intended properties. We propose a slight adjustment to the construction and prove that the corrected version serves as the desired counterexample.

Trace-driven simulation is widely used to evaluate distributed training (DT) jobs in GPU clusters, but existing simulators either ignore network contention or approximate it with a fixed penalty. This misses how scheduling decisions determine which jobs share server network interfaces and inter-server links, thereby changing networking time during training. As a result, our motivating experiments demonstrate that they incur large errors, reaching up to 73.64% mean absolute percentage error (MAPE) in average job completion time (JCT). This paper introduces MoSim, a GPU-cluster simulator that models DT job execution under dynamic network contention. MoSim combines GPU-free characterization with network contention model: it obtains each job's compute time, networking time, and networking volume without GPUs, then uses the current worker assignment to estimate how shared network interfaces affect each job's iteration time. Our evaluation shows that, compared with existing simulators, MoSim reduces simulation error for average JCT by up to 3.28$\times$, tail (99th-percentile) JCT by up to 7.79$\times$, and makespan by up to 8.48$\times$, while modeling NIC contention factors with only 8.63% error on average. By avoiding real-GPU profiling, MoSim also reduces input construction overhead by 44.6$\times$.

Quantum uncertainty limits how strongly Wigner functions can concentrate in phase space. It is commonly conjectured that the vacuum Wigner function continuously majorizes that of any Wigner-positive state. However, by constructing Wigner-positive states whose localized quantum coherence lowers their Wigner entropy below the vacuum value, we show that this conjecture is incorrect. We restore vacuum majorization at finite resolution by integrating single-mode Wigner functions over concentric radial energy shells, obtaining probability vectors majorized by that of the vacuum for all Wigner-positive states and for Wigner-negative states with non-negative shell weights. Anchoring these shells to a fixed oscillator frame breaks affine symplectic invariance, rendering this relation strict for every non-vacuum Gaussian pure state. For Wigner-negative states, the persistence of negative shell weights defines a radial negativity scale with Airy-edge semiclassical asymptotics for highly excited Fock states. Operationally, we combine the majorization relation with partial transposition to obtain a two-mode entanglement criterion directly evaluable from radially binned joint homodyne outcomes without state reconstruction.

This paper investigates a six-dimensional coupled Lorenz system obtained through bidirectional asymmetric diffusive coupling of two identical Lorenz oscillators. The proposed model is analyzed from both dynamical and synchronization perspectives. Dissipativity conditions, equilibrium points, local stability properties, and Lyapunov spectra are systematically examined. The results reveal wide regions of chaotic and hyperchaotic behavior, with up to three positive Lyapunov exponents depending on the coupling parameters. These findings provide a practical framework for selecting coupling configurations that guarantee hyperchaotic dynamics while preserving the simplicity of the classical Lorenz structure. \\ A Lyapunov-based synchronization analysis is then developed. By exploiting effective attractor bounds obtained from the actual system trajectories rather than conservative worst-case estimates, a less restrictive sufficient synchronization condition is derived. Under the classical Lorenz parameters, synchronization is guaranteed for $\alpha_1+\alpha_2>15.1$. \\ The synchronization performance is further evaluated through a parametric analysis of the synchronization time. Numerical results show that increasing the coupling strength significantly accelerates convergence toward the synchronization manifold. For a broad range of admissible coupling values, synchronization times below $0.5$ second are achieved, whereas previously reported Lorenz-based synchronization schemes typically require approximately $5$ seconds.