We establish an explicit upper bound on the number of stable Andrews-Curtis moves that convert thickenable balanced presentations of the trivial group to the standard one-generator presentation. We also present a proof that thickenable balanced presentations of the trivial group satisfy the (unstable) Andrews-Curtis conjecture.

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. Its continuous-time formulation can be interpreted as an interacting particle system driven by a Kalman-type preconditioned descent direction. A well-known limitation of this dynamics is the possible premature collapse of the covariance of the ensemble, which makes the method sensitive to the initial ensemble. We introduce a second-order particle system in which the particles evolve according to an inertial dynamics. The model combines a Kalman-type relaxation force with damping, attraction towards the ensemble mean, and a short-range repulsive interaction designed to counteract ensemble collapse. The resulting dynamics can be interpreted as a heavy-ball reformulation of continuous-time EKI enriched by competing attractive and repulsive mechanisms. For linear inverse problems, we analyze the induced mean and fluctuation dynamics and identify a parameter regime in which fully collapsed configurations are linearly unstable. We further characterize asymptotic equilibria through a constrained optimality condition on the subspace retained by the limiting ensemble covariance and derive an exponential decay estimate. Numerical experiments illustrate the effect of inertia and repulsion on the ensemble dynamics and compare the proposed second-order method with first-order EKI-type

Ferroelectric nematic liquid crystals (FNLCs) combine fluidity with spontaneous polarization, offering promising avenues for flexible electromechanical systems. Here, we demonstrate that mechano-electrical conversion in FNLCs can be enhanced by mechanically programming a robust macroscopic polarization alignment. Using hybrid liquid crystal cells composed of rigid glass and flexible substrates, we show that deformation in the ferroelectric nematic phase suppresses polarization domains and produces long-range ordered polarization alignment over millimeter-scale areas. This geometry-driven alignment originates from coupling between the FNLC's spontaneous splay deformation and the deformation-imposed cell geometry, and we further find that the selected polarization direction exhibits clear material dependence. Leveraging this deformation-enabled alignment, we develop an FNLC-based energy harvester that converts mechanical deformation into an output of approximately 1 V. These findings establish geometry-driven alignment as a practical design strategy for boosting FNLC mechano-electrical conversion while providing polarization control for soft electronic devices.

The influence of centrifugal effects in rotating Rayleigh-Benard convection is investigated using direct numerical simulations. We find that the Nusselt number decreases beyond a critical Froude number, Fr_c*. This critical value depends on both the Rayleigh number Ra and the aspect ratio Gamma, following power-law scalings with each parameter. We interpret Fr_c* as the onset of centrifugal effects within the thermal boundary layers. This interpretation is supported by the thickening of the boundary layers and a reduction in the planar heat flux. We compare Fr_c* with two previously proposed critical Froude numbers. The first, Fr_Hu, marks the onset of centrifugal effects in the bulk, as evidenced by changes in local heat flux and radial vortex motion. For Fr_Hu < Fr < Fr_c*, centrifugal effects primarily redistribute heat within the bulk and have little influence on the global heat transfer. The second, Fr_Horn, is based on a global force-balance argument. The similar dependence of Fr_c* and Fr_Horn on the aspect ratio suggests a close connection between the global force balance and the onset of centrifugal effects in the thermal boundary layers. These results demonstrate that centrifugal forcing influences the bulk flow and the thermal boundary layers differently in rotating Rayleigh-Benard convection. While relatively weak centrifugal forcing modifies the bulk dynamics, substantially stronger forcing is required to alter boundary-layer properties and global heat transport.

We introduce a microlocal phase-space-filtered physics-informed neural network (PINN--TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.

The present Reply addresses the Comment as posted on arXiv (arXiv:2606.04137 [nucl-th], June 2026).

We study day-ahead transmission topology control for high-voltage grid operation under $N-1$ security constraints. The operational task is to select, over a 24-hour horizon, a sequence of substation topologies obtained via busbar-coupler switching to relieve line overloads while limiting switching effort and topological complexity. We formulate this task as a sequential multi-objective optimization problem with four objectives used in TSO decision making: worst-case $N-1$ line loading, maximum topological depth, number of topology changes, and time spent outside the reference topology. We propose an exact block algorithm that exploits the temporal structure of topology plans: consecutive hours with the same topology are represented as blocks, enabling enumeration of the complete Pareto front over the admissible set of topologies under fixed operational bounds on depth and switching. We also develop a tailored NSGA-III-based evolutionary heuristic and evaluate it against the exact front. Using real operational data from the Dutch high-voltage transmission grid operated by TenneT, the block algorithm computes the exact front for a highly congested day in under three minutes after topology-level load-flow preprocessing. The exact front reveals low-switching plans with no DC $N-1$ thermal overloads that the tested evolutionary search fails to find. The proposed method, therefore, provides both a practical day-ahead decision-support tool for transmission operators and a benchmark for heuristic and learning-based topology-control methods.

This work conceives a unified channel estimation and beamforming framework, formulated within the principles of variational Bayesian inference. Recognizing the limitations imposed by hardware constraints, frequency-dependent propagation effects, and the structural restrictions of partially connected architectures in the Terahertz (THz) band, we formulate a dual-wideband channel model incorporating root raised cosine (RRC) pulse shape to account its band-limited nature. To further address the nonlinear distortions introduced by low-resolution ADCs, Bussgang decomposition is employed, enabling a tractable linearized inference process. Unlike conventional techniques, the proposed method accommodates both on-grid and off-grid angular domains, capturing spatial sparsity with improved resolution and robustness. The multi-user (MU) Bayesian Cramér-Rao lower bound is also derived to benchmark the performance of the proposed estimator. Moreover, the framework incorporates a true time delay (TTD)-based hybrid transceiver design that inherently compensates for the beam-squint effect; a frequency-dependent angular deviation that arises due to the fixedphase nature of the conventional beamformer in wideband systems, thereby ensuring accurate directional alignment across all subcarriers. Extensive simulation results validate the effectiveness of the proposed variational Bayesian inference-based estimator and the TTD-enabled beamforming architecture, highlighting their robustness and performance gains under practical wideband THz system.

We present a galaxy evolution model that incorporates grain-size evolution in a multiphase interstellar medium (ISM) to investigate dust attenuation in galaxies at $z \geq 5$. Our fiducial setup assumes a low dust yield of $y_{\rm d} = 10^{-4}~\rm M_\odot$ and a small characteristic size of stellar dust of $a_0 = 0.01~μ$m, motivated by efficient dust destruction by reverse shocks in dense ISM environments. Our model demonstrates that, even with such low dust yields, massive galaxies with $M_\ast > 10^9~\rm M_\odot$ reach high dust-to-stellar mass ratios of $M_{\rm d}/M_\ast \sim 10^{-2}$ by $z \sim 7$ because small grains supplied by SNe efficiently serve as seeds for metal accretion in the ISM. Because dust growth significantly lags behind star formation, the outer regions beyond the half-star-formation-rate radius remain relatively dust poor, allowing a non-negligible fraction of UV photons to escape without strong attenuation. We further find that dust growth becomes most efficient when the ISM is dominated by cold dense gas but still contains a modest warm component, as the former promotes metal accretion while the latter supplies additional small grains through shattering, thereby further enhancing subsequent grain growth. In particular, with a cold dense gas fraction of $\sim 90~\%$, our model predictions become broadly consistent with the dust-to-stellar mass ratios inferred for dust-rich galaxies at $z \sim 7$, as well as the upper limits for blue galaxies at $z \gtrsim 10$. Self-consistently, the model successfully reproduces the UV luminosity functions observed at both $z = 7$ and $z = 12$. Overall, this study demonstrates that a physically motivated treatment of grain growth in a multiphase ISM is essential for linking the dust content of high-redshift galaxies to their radiative properties during cosmic dawn.

Internal link optimization is a recurring task in search engine optimization, yet many production workflows rely on manual judgment, fixed page templates, or generic tool recommendations. Practitioners need ways to evaluate candidate links before deployment because link changes can redistribute authority and affect semantic coherence in ways that are difficult to isolate after release. We present WebKnoGraph, an open-source framework for evaluating internal linking strategies on website crawls. The framework models a website as a directed graph, represents pages by embeddings, scores candidate links with GraphSAGE, and evaluates interventions by embedding the site into larger host environments. We instantiate WebKnoGraph on a production crawl of Kalicube.com and compare automatic with expert-assisted link selection in an empirical FineWeb-based host graph and a synthetic Barabási-Albert host graph, using PageRank-based authority metrics and semantic coherence. The results show that automatic selection generally produces stronger authority redistribution, with higher Authority Yield, but also larger semantic coherence costs. Expert-assisted selection better preserves semantic coherence and, when targeting low-PageRank pages, achieves the highest Authority Yield, although with the least favorable loss-gain balance. Authority Volatility provides an additional stability perspective, but is interpreted cautiously because the two regimes use different numbers of intervention sets. These findings support a practical workflow in which candidate intervention sets are generated at scale, evaluated jointly across authority gain, volatility, loss-gain balance, and semantic coherence, and then reviewed for editorial deployability before implementation.

The bulk viscosity $ζ$ is a transport coefficient which is of central importance for various areas of modern physics. In particular, its determination for a mixture of more than one fluid is challenging, since it involves a complex interplay of multiple microscopic processes that operate on different time scales. Within the Chapman-Enskog framework, based on a series expansion of the Boltzmann distribution function, many previous works have derived the 1$^{\text{st}}$ order result for the $ζ$ of a mixture. However, such a result fails to reproduce relevant physical features of the system, especially when the masses of the two components are similar. In this work we improve the 1$^{\text{st}}$ order Chapman-Enskog result by deriving the $ζ$ at the 2$^{\text{nd}}$ order in the expansion. We show that this improved formula encodes many physical properties that the 1$^{\text{st}}$ order result misses: under specific conditions, the 2$^{\text{nd}}$ order result can be qualitatively and quantitatively very different from the 1$^{\text{st}}$ order one. Moreover, this result is compared against the $ζ$ evaluated within the Green-Kubo formalism, by means of a numerical solution of the Relativistic Boltzmann equation. The agreement with respect to this benchmark is significantly improved when moving from the 1$^{\text{st}}$ to the 2$^{\text{nd}}$ order CE result.

We establish the ultimate precision limit for estimating the optical spatial coherence radius (Fried parameter) within a quantum metrological framework. In the weak field regime, we show that spatial-mode decomposition -- originally introduced for superresolution imaging -- enables substantially more precise estimation than conventional direct imaging when the receiver aperture is smaller than the coherence radius.

We study natural automorphisms of the universal virtual braid group $UV_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to $\mathbb{Z}_2^k\times\mathbb{Z}_2$. We then show that the two one-forbidden quotients of $UV_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UUV_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_kVB_n$ is a quotient of $UV_n(k)$, the corresponding results for $M_kVB_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $VB_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.

Laplace transforms and their numerical inverses arise throughout applied mathematics, physics, finance, and probability theory. Numerical inversion, however, quickly becomes intractable in high dimensions because the number of quadrature evaluations grows exponentially with dimension. We develop a tensor train (TT) formulation of the multidimensional inverse Laplace transform. The method constructs a TT approximation of the transformed function on the complex quadrature grid and then performs the inversion through a sequence of tensor contractions. Under suitable low-rank assumptions, this reduces the computational cost from exponential to polynomial in the dimension, provided that the relevant bond dimensions remain bounded. The method has only a small number of tunable parameters and admits error estimations. We demonstrate its performance in numerical experiments, benchmarked against Monte Carlo estimates and exact references, for multivariate normal-inverse Gaussian, Wishart, and correlated Gamma-type distributions.

In this paper, we introduce a notion of Higgs bundles on the Fargues-Fontaine curve. We establish a version of the BNR correspondence, which relates Higgs bundles to line bundles on suitable curves. We then describe an action of a Picard stack on the moduli stack of Higgs bundles and show that, modulo this action, there is a natural injective map of étale-stacks from the product of $B_{dR}^+$-affine Springer fibers to the Hitchin fiber that induces an equivalence of categories on every geometric point. Finally, we discuss connections with number-theoretic objects.

Hyperproperties express, e.g., information-flow properties of systems, which involves the simultaneous reasoning about multiple execution traces of a system. Consequently, HyperLTL, the most important specification logic for hyperproperties, extends LTL with quantification over traces. However, HyperLTL can only express synchronous hyperproperties. Recently, several logics for asynchronous hyperproperties have been proposed. Here, we focus on AHLTL, asynchronous HyperLTL, which extends HyperLTL with quantification over trajectories that control the relative speed at which time progresses on the quantified traces. Model-checking AHLTL is known to be undecidable while satisfiability is known to be $Σ_1^1$-hard, but the precise complexity of both problems is open. Here, we close these gaps and show that model-checking is equivalent to truth in second-order arithmetic while satisfiability is $Σ_1^1$-complete if the trajectory is existentially quantified and $Σ_1^1$-hard and in $Σ_2^1$ if the trajectory is universally quantified.

Actuaries rely primarily on structured numerical data for reserving and ratemaking, while valuable predictive information in unstructured text including medical records, adjuster notes, and call transcripts remains largely unused. Manual processing of these documents is time-consuming, inconsistent across reviewers, and unscalable. We present a proof-of-concept framework using large language models (LLMs) to extract structured actuarial variables from unstructured claims data. We implement a two-stage processing architecture separating document-level extraction (Stage 1) from claim-level synthesis (Stage 2). A modular four-script Python pipeline processes synthetic FHIR-based claims data and real claims documents, extracting 36 actuarial variables across reserving, ratemaking, and claims management categories. We validate 14 core variables using two independent clinical expert reviewers scoring 20 synthetic claims on a five-point Likert rubric, achieving mean scores above 4.0 and a weighted kappa of 0.53. Integration with chain ladder reserving demonstrates practical actuarial value: severity-segmented analysis reduced reserve estimation error from 6.5% to 4.0%. The open-source implementation includes audit trails and confidence scoring, providing a replicable foundation for LLM-based actuarial variable extraction in property-casualty insurance.

In this paper, we prove a generalization of a conjecture of Amdeberhan, Andrews, and Ballantine on double Lambert series. Motivated by a question raised by Cui, Kumar, and Singh concerning the existence of a generalization of this conjecture, we establish an identity in which the coefficients are given by the generalized divisor function $σ_k(n)$. As a special case, our result includes the original conjecture.

We study the revenue of approximate correlated equilibrium in discrete first price auctions - the set of allowable bids is $\mathcal{B} = \{0, 1/k, \dots, 1 - 1/k, 1\}$ for some $k \in \mathbb{N}$. We show that the revenue of any $ε$-approximate correlated equilibrium is at least $v_2 - Θ(1/k)- Θ(εk^2)$, where $v_2 \geq 0$ is the second-highest valuation. Our results establish the first polynomial convergence rates on the revenue generated by no-swap regret bidders in first-price auctions. For instance, if bidders admit the optimal swap regret of $\mathcal{O}(\sqrt{k T})$, then the time-averaged revenue is at least $v_2 - Θ(1/k) - Θ(ε)$ after $\mathcal{O}(k^5/ε^2)$ rounds.

Efficient discrimination of circularly polarized light (CPL) is of great significance in photonics. In this study, we propose a nanophotonic device based on asymmetric V-shaped metasurfaces that exhibits strong discrimination between left- and right-circularly polarized (LCP and RCP) light at the resonant wavelengths of the absorption spectra. The chiral-selective response originates from a quasi-bound state in the continuum (qBIC) mode enabled by controlled symmetry breaking in the V-shaped structure, which exhibits high absorption under LCP illumination while the resonance is strongly suppressed under RCP incidence, leading to a large absorption contrast between the two circular polarizations. This significant absorption difference enables highly efficient chiral discrimination. Furthermore, the resonant wavelength can be broadly tuned from 5200 to 6200 nm by scaling the structural dimensions without compromising the high absorption contrast between LCP and RCP. This work not only enables high-performance chiral detection and separation, but also offers valuable guidance for reconfigurable chiral nanodevices, with promising applications in areas such as enantiomer sensing, chiral imaging, and CPL spectroscopy.

The human-dignity-centric digital social contract grounds personal data in human dignity, data personalism, and data sovereignty, and articulates six dimensions of data governance: technological oversight, automation limits, economic justice, political legitimacy, social cohesion, and legal guarantees. It presupposes, however, that enforcement falls to State regulators, licensed fiduciaries, and multi-stakeholder bodies embedded in existing legal systems. This paper asks whether its normative content can instead be realized not as rules imposed on the owners of the AI stack from without, but as a commons-governed infrastructure that any person, firm, or State may use and fund while its governance stays horizontal, polycentric, and subsidiary. We construct the Dignity Stack, a six-layer architecture mapping each dimension onto a layer of commons-governed AI infrastructure, with protocols drawn from the Liberation Stack framework and from the cooperative, mutualist, and libertarian-municipalist traditions. The commons is State-agnostic rather than anti-State, anarchist in its horizontal means but not in the abolition of the State. Its central device is a decoupling of capital from control, by which the stack functions as a shared civic battery, charged by many contributors yet steered by none in proportion to its charge. We prove that this defeats formal capture through votes or surplus, and show that structural capture, the leverage of a dominant supplier free to withdraw what it provides, is resisted only insofar as operational supply is polycentric and substitutable, a condition demanding at the lower layers and perhaps presently unattainable at chip fabrication. We conclude, with explicit attention to its limits, that commons-governed AI realizes the values the contract proclaims more faithfully than the regulation it presupposes.

In the present paper, we have performed 2.5D resistive magnetohydrodynamic simulations of the interaction of a fast magnetoacoustic wave with a localized coronal magnetic null point. As a result, an Alfvén wave packet is generated by the mode conversion when a fast magnetoacoustic perturbation interacts with the null point. The field-aligned plasma flows are also generated due to the non-linear effects. When the fast mode wavefront interacts with the null, some parts of this wavefront get refracted around it, while some other part is trapped at the null region. Subsequently, the velocity fluctuation out of the plane and in-phase magnetic field fluctuations have evolved and propagated with the local Alfvén speed along the separatrixes at one side of the coronal null region. The resulting disturbance behaves as an incompressible and energetic Alfvén wave packet. A secondary fast magnetoacoustic wave is also produced and propagates. In the synthetic SDO/AIA observations, no intensity fluctuations are evident in the region where the Alfvén wave packet propagates, while the fast magnetoacoustic wave fronts are clearly evident. Our results suggest that given the appropriate physical conditions at the null, when the fast mode wave is incident, Alfvén packets can be excited due to the mode conversion, further carrying substantial momentum and energy flux in the solar corona.

Wireless Access and Backhaul (WAB) is emerging as a key enabler for flexible and cost-efficient 5G deployments, offering a modular architecture that decouples access and backhaul while supporting multi-technology and mobile backhaul links. This article introduces the WAB framework standardized in 3GPP Release 19, outlining its architecture and operational principles. A practical implementation built with commercial hardware and open-source software demonstrates the feasibility and efficiency of WAB systems. We further explore four representative application scenarios - ranging from on-demand coverage to mobile Software-Defined Wide Area Network (SD-WAN) connectivity - and discuss the technical challenges that must be addressed for large-scale adoption. These insights highlight WAB as a promising foundation for 5G-Advanced and a stepping stone toward future 6G networks.

Recently, graph-aware Large Language Models (LLMs) have shown promising capabilities in jointly modeling graph-structured data and textual information. Existing approaches typically employ a graph encoder and a frozen LLM to obtain node representations from graph and textual views, followed by node-level alignment to bridge the two modalities. However, such alignment mechanisms primarily focus on node information while overlooking edge-level structures, leading to suboptimal information propagation across views. In this work, we conduct a comprehensive theoretical analysis to uncover why node-level alignment is insufficient for aligning textual and graph representations. Specifically, we prove theoretically for the first time that neglecting edge information leads to suboptimal solutions and negatively curved edges induce bottlenecked information flow, giving rise to the over-squashing phenomenon between graph and textual views. To address the two challenges, we innovatively proposed a CureLLM framework of Curvature-enhanced Graph Representations for Large Language Model whose goal is to inject the signals of edge information into the existing LLMs. Specifically, CureLLM first introduces the training-free textual prompt mechanism to make the LLM model generate the output directly based on the edge-aware prompt without learnable parameter costs. Furthermore, a novel curvature-aware graph representation learning is designed to capture the edge structure information to enhance the downstream tasks, where the message passing between text and graph representations only depends on edges with positive curvature. Finally, we conduct evaluations with 20 different compared methods on 11 real world datasets from various domains and the experiment results demonstrate the superiority of our proposed CureLLM framework.

We prove a version of GAGA theorem for a normal complex analytic variety $X$ equipped with an invertible holomorphic contraction $γ$ with center in $x$. We show that $X$ admits a natural structure of an affine variety, and any $γ$-equivariant complex analytic reflexive coherent sheaf on $X$ admits a natural algebraic structure. We prove a structure theorem for $X_0:=X\backslash x$, showing that it admits a proper action of ${\Bbb C}^*$, and is isomorphic to the space of non-zero vectors in the total space of an ample line bundle over the projective variety $Z:= X_0/{\mathbb C}^*$ equipped with an orbifold structure. We show that the quotient $M:=X_0/γ$ admits a holomorphic embedding to a Hopf manifold, and, conversely, any normal subvariety $M$ in a Hopf manifold is obtained this way. We prove a form of structure theorem, showing that any reflexive coherent sheaf on $M$, $\dim M > 2$, admits a filtration such that its associated graded subquotients, tensored with an appropriate line bundle, are obtained as pullbacks of coherent sheaves on the projective variety $Z=X_0/{\mathbb C}^*$. This is used to show that any reflexive coherent sheaf on $M$ is filtrable, that is, admits a filtration with associated graded quotients of rank $\leq 1$.

This paper studies the weak convergence order of structure-preserving stochastic theta methods for a class of index-$1$ stochastic differential algebraic equations with time-dependent singular matrices. The singular matrix is allowed to vary in time but preserves a fixed differential-algebraic splitting, thereby extending the constant singular-matrix setting while retaining the projector structure required for constraint preservation. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we establish an abstract weak convergence theorem for constraint-preserving one-step approximations and apply it to the stochastic theta method with $θ\in (0,1]$. Under global Lipschitz, linear growth, and suitable smoothness assumptions, the considered method is proved to be well posed, to preserve the algebraic constraints at all time levels, and to converge with weak order one. Numerical experiments are finally presented to confirm the structure-preserving property and the theoretical convergence order.

Controlled time-evolution circuits select forward or backward Hamiltonian time evolution according to the state of an ancilla qubit. They are fundamental building blocks in quantum eigenvalue transformation of unitaries, Hamiltonian filtering, and related quantum algorithms. A direct realization adds ancilla control to the elementary gates of the time-evolution circuit and therefore increases the two-qubit gate count, compiled T depth and CX depth. We develop an efficient recursive algorithm that, for an arbitrary Pauli-sum Hamiltonian, partitions the input Pauli terms into groups and assigns to each group a sign-flip Pauli string that anti-commutes with the in-group terms, thereby removing ancilla control from the grouped time-evolution blocks. Numerical benchmarks on random Hamiltonians and structured spin Hamiltonians show reductions in compiled T depth and compiled CX depth. For a Kagome Hamiltonian with 24 spins under full connectivity, the proposed construction reduces the compiled T depth by 85.2% and the compiled CX depth by 68.9%, compared with a conventional implementation that decomposes the Hamiltonian into individual Pauli terms and implements the controlled time evolution of each term by directly adding the ancilla qubit to the corresponding Pauli-rotation circuit.

This paper is concerned with uniqueness and stability of Lagrange multipliers for constrained optimization problems in abstract spaces. It is well known that validity of the strict Robinson-Zowe-Kurcyusz condition implies the so-called isolated calmness, a one-sided Lipschitz property tailored for set-valued mappings, of some Lagrange multiplier mapping associated with a perturbed version of the original optimization problem, and the latter indeed is enough to guarantee uniqueness of the Lagrange multiplier. The paper studies the isolated calmness of the Lagrange multiplier mapping in detail. Exemplary, it is shown that this condition is sufficient for the Robinson-Zowe-Kurcyusz constraint qualification and, in the presence of additional assumptions, even equivalent to the strict Robinson-Zowe-Kurcyusz condition. Illustrative examples are presented to underline the necessity of postulated assumptions.

Let $a_L(n)$ be the maximum number of regions into which $n$ lollipops divide the plane. Paulsen introduced a second obstruction for this problem, based on pairs of circles meeting at obtuse angle, in addition to the stem-direction obstruction of Cutler-Karlsson-Sloane. We recast Paulsen's argument as a weighted problem for two graphs: a $K_4$-free graph $D$ of non-close stem pairs and a $K_5$-free graph $E$ of non-intriguing circle pairs. For the total number $C$ of pairwise crossings, $$ C\le 4\binom n2+|D|+|E|+|D\cap E|. $$ Paulsen bounds the final term by $|D|$. We keep the overlap term and analyze near-extremal configurations of $D$ and $E$. This closes all of Paulsen's remaining gaps up to $n=17$, and also closes $n=19$: $$ \begin{array}{c} a_L(0),a_L(1),\ldots,a_L(17)\\ =1,2,10,25,45,71,104,142,186,237,294,356,425,500,580,667,761,859, \end{array} $$ and $$ a_L(19)=1076. $$ The same method gives the one-region gaps $$ 964\le a_L(18)\le965,\qquad 1193\le a_L(20)\le1194. $$

When porting high-performance computing (HPC) code from CPU to GPU, CPU-oriented optimizations may obstruct LLM-based CUDA translation. We design and evaluate a Deopt-Reopt workflow that first simplifies the input C++ code and then retranslates and reoptimizes it for CUDA, comparing it against direct translation (Direct) on twelve HPC kernels with two LLMs (gpt-oss-120b (O120) and qwen-3-235b-a22b-instruct-2507 (Q235)) in Single-shot (one pass) and Iterative (repeated refinement) settings. In Single-shot, among 18 testable cases Deopt-Reopt was significantly faster among successful trials (after BH-FDR correction) in five - most clearly for conv2d, where CPU- and GPU-oriented designs diverge - but Direct was faster in three, so removing CPU-specific optimizations is not universally beneficial. An exploratory Direct-3 control that equalizes the LLM-call count left Deopt-Reopt ahead in only four of nineteen testable cases, with Direct-3 ahead in four others. In Iterative, repeated generation and repair narrow the mode gap - markedly so for O120 - while Q235 retains large Deopt-Reopt advantages on conv2d, ddgemm, and bgemm. Deopt-Reopt's effect on feasibility is also mixed - sharply higher for some kernels Direct rarely compiles, lower for others. Because performance is conditioned on successful trials, the benefit is conditional rather than a guaranteed end-to-end gain. Overall, Deopt-Reopt is an effective but non-universal technique for LLM-based GPU porting, with gains that depend on the kernel, the model, the search budget, and the success rate.