In the field of statistical learning and data analysis, estimating precision matrices (i.e., the inverse of covariance matrices) is a critical task, particularly for understanding dependency structures among variables. However, traditional methods often fall short when dealing with high-dimensional interval-valued data, where each observation is represented as an interval rather than a single point. This paper proposes a novel framework for estimating precision matrices in such contexts, addressing the unique challenges posed by the interval nature of the data. Specifically, we assume that the upper and lower bounds of the intervals share the same conditional dependency structure, and then formulate the interval graphical lasso optimization objective to estimate the precision matrix. At the optimization level, we provide an efficient computational approach, while at the theoretical level, we prove the sparsity and consistency of the estimator. Experimental results on simulated studies and real data applications demonstrate the superiority of the proposed method in terms of estimation precision and interpretability.

In the paper, we study spatially distributed particle systems whose time evolution is governed by vanishing diffusion in space $\mathbb{R}^d$, $d\ge 1$, and by size-continuous fragmentation and coagulation processes with unbounded rates. We show that for a large class of coefficients, such systems are classically locally well-posed, provided the diffusion and the coagulation processes are suitably dominated by the fragmentation. In the special case of power rates, we demonstrate existence of global in time classical solutions in all spatial dimensions $d\ge 1$ and without any restrictions on the size of input data.

The inherent randomness of communication symbols creates a fundamental tension in Integrated Sensing and Communications (ISAC). On the one hand, they enable data transmission while allowing sensing to fully reuse communication resources. On the other hand, their randomness induces waveform-dependent fluctuations that directly affect sensing accuracy. This paper investigates a foundational question arising from this tradeoff: \textit{How does the modulation waveform affect the ranging Cramér--Rao Bound (CRB) when sensing reuses random data symbols?} We address this question by revealing a structural factorization of the Fisher information matrix (FIM) for joint delay-amplitude estimation, which separates the deterministic Jacobian of the target geometry from the random frequency-domain signal power induced by the data symbols. This structure yields a Jensen-type universal lower bound on the CRB, which is exactly attained by CP-OFDM under PSK constellations. For QAM and broader sub-Gaussian constellations, we develop an asymptotic perturbation analysis of the inverse FIM and prove that, when the number of transmitted symbols $N$ grows large, CP-OFDM achieves a lower ranging CRB than any frequency-spread orthogonal waveform over the almost-sure event where the random FIM is invertible. This superiority is further extended to amplitude estimation and full joint delay-amplitude estimation. We also characterize the local geometry of the stochastic CRB minimization problem over the unitary group. The analysis reveals that CP-OFDM is a stationary point for finite $N$, and its Riemannian Hessian is positive semidefinite for sufficiently large $N$, establishing its asymptotic local optimality. Numerical results confirm that OFDM outperforms representative waveforms including SC, OTFS, and AFDM.

Listwise reranking utilizing Large Language Models (LLMs) has achieved state-of-the-art retrieval effectiveness. Recently, reasoning-enhanced models have further pushed these boundaries by employing Chain-of-Thought (CoT) to perform deep comparative analysis of candidate documents. However, this performance gain comes at a prohibitive computational cost, as models often generate thousands of reasoning tokens before producing a final ranking. In this work, we investigate the relationship between reasoning length and ranking quality, revealing an overthinking phenomenon where extended reasoning yields diminishing returns. To address this, we propose a Length-Regularized Self-Distillation framework. We synthesize a dataset by sampling diverse reasoning traces from a teacher model (Rank-K) and applying a Pareto-inspired filter to select traces that achieve high ranking performance with minimal token usage. By fine-tuning on these concise, high-quality rationales, the student model learns to internalize efficient reasoning patterns, effectively pruning redundant deliberation. Experiments on TREC Deep Learning and NeuCLIR benchmarks demonstrate that our method maintains the teacher's effectiveness while reducing inference token consumption by 34%-37% across different retrieval settings, offering a practical solution for deploying reasoning-enhanced rerankers in latency-sensitive applications.

We study the impact of symmetry energy ($S$), hyperons, and dark matter (DM) on structural and oscillatory properties of neutron stars (NSs). Uncertainty from hadronic equation of state for NSs is considered with 15 relativistic mean field models having slope parameter ($L_0$) of $S$ in range $40-120$ MeV. DM admixed NSs (DMANSs) are described with feeble interaction between light DM fermions ($χ$) with hadronic matter in the presence of hyperons via scalar ($η$) and vector ($ξ$) dark mediators. The masses $m_χ$, $m_η$ and $m_ξ$ are related by self-interaction constraints from bullet cluster. DM self-interaction couplings are related to $m_χ$ by relic density constraint. The DM density is taken as an exponential function of baryon density with a free parameter $α$. Uncertainty from DM model is incorporated by exploring the dependence on $m_χ$ and $α$. Several DM search experiments have almost ruled out the existence of massive DM ($\gtrsim$ GeV). Lately, pursuit for sub-GeV DM has attracted significant attention. Therefore, we consider $m_χ<$ 1 GeV and $α\leq$ 0.1 such that the contribution of DM to the total mass of the DMANSs is $<10\%$. Comparing our results with various astrophysical constraints, we find that the HESS J1731-347 and GW170817 data are very important in determining the presence of light DM in NSs in moderate amount, relevant in the range $L_0\lesssim$ 58 MeV. Employing models of DMANSs that satisfy several observational data, we infer with Bayesian analysis, the likely ranges of $m_χ$ and $α$ are almost independent of the underlying hadronic model within 40 MeV $\lesssim$ $L_0$ $<$ 58 MeV. In the absence of DM and with the most probable values of $m_χ$ and $α$ obtained from the Bayesian inference, we calculate the frequencies of non-radial $f$- and $p_1$-modes oscillation of NSs/DMANSs.

In the early 1920s, Hardy and Littlewood considered the number of integer points in the right-angled triangles with irrational inclines of the diagonal. We extend their results to higher dimensions.

Robust point-set registration in the presence of noise and outliers is challenging because the matched points (inliers) must be identified before reliable alignment can be performed. Existing robust registration methods typically optimize over the transformation space and are often designed for regimes with a nonvanishing fraction of inliers. In this paper, we study the inlier recovery problem arising in robust registration by comparing two datasets through the Hadamard product of their Gram matrices. This formulation converts the inlier identification into a structured recovery problem and avoids direct optimization over the rotation group. Based on this idea, we develop two methods: an eigenvector matching method based on the leading eigenvector of the Gram-matrix overlap, and a row-sum matching method based on aggregated entrywise comparison. We show that the eigenvector method achieves weak recovery when the dimension and sample size are of the same order, while the row-sum method achieves exact recovery under a broader range of dimensional scalings. In particular, when the dimension is comparable to the sample size, exact recovery is possible even when the inlier fraction vanishes, with the number of inliers as small as order $\sqrt{n}$, up to logarithmic factors. We also discuss a parallel implementation for large-scale settings. Numerical experiments on brain imaging data and image examples demonstrate that the proposed methods effectively identify matched structure under substantial corruption.

Characterizing the physiological life boundaries of microbial strains, including viable temperature, pH, salinity, substrate utilization, and morphology, is central to biotechnology and ecology, yet traditionally requires exhaustive in vitro screening. Existing computational approaches either treat physiological traits as isolated supervised targets or repurpose biological foundation models as static encoders, leaving the genotype-to-physiology gap largely unbridged. We formulate microbial life-boundary prediction as a unified genome-to-physiology task and address it with a genome-conditioned, tool-augmented LLM agent. To support this task, we curate a strain-centric benchmark from IJSEM, NCBI, and BacDive covering 1,525 strains and 6,448 instances across viability intervals, environmental optima, substrate utilization, categorical traits, and morphology. Architecturally, the agent injects frozen LucaOne genome embeddings into a Qwen backbone via lightweight token fusion, and reasons over a similarity-based RAG module and a Genome-scale Metabolic Model (GEM) perturbation tool. We optimize the agent through a three-stage pipeline of gene-text alignment, agentic SFT on distilled trajectories, and GRPO with a novel counterfactual gene-grounding reward that reinforces the policy only when the authentic genome embedding causally improves correct-token generation relative to a zero-gene ablation. The resulting 4B-parameter agent matches or surpasses substantially larger frontier LLMs, with ablations confirming that genome-token fusion, dynamic tool use, and the counterfactual reward each yield distinct, significant gains.

Optical Skyrmions have recently garnered much interest providing a potential avenue for high capacity, robust topological information transfer. Typically, Skyrmions are derived from the coupling of just two degrees of freedom (DoFs) limiting their versatility. In this work we realize spatio-spectral Skyrmions derived from the non-separability between three DoFs: wavelength, space and polarization. A compact and simple technique is used to generate the spatio-spectral vector beams (SSVB) carrying the desired Skyrmionic structure, offering simple pathways for complex Skyrmionic beam design. The topological structure, witnessed through a map between the spatio-spectral plane and the Poincaré sphere, exhibits an additional tunable $kπ$ parameter thereby enhancing the number of controllable DoFs. Our three DoF construction allows us to propose a novel topological multiplexing strategy that independently encodes different Skyrmion numbers at different radii of the field. We experimentally demonstrate the practicality of this approach by transmitting and receiving three distinct Skyrmion numbers encoded into a single topological field, for the first form of mode division multiplexing with Skyrmion topology. This work opens up new avenues for dense information encoding using multiple topological channels encoded in a single light field.

Early-time flux excesses in Type Ia supernovae (SNe~Ia) offer a unique insight into their progenitor systems and explosion mechanisms. Although individual early-excess events and larger searches have been reported, demographic studies remain limited by sample size. We present a systematic search for early-time excess emission in a volume-limited sample ($z<0.06$) of SNe~Ia based on the Zwicky Transient Facility Data Release 2 (ZTF DR2). Using ZTF $g$- and $r$-band light curves, we identify candidates showing early-excesses shortly after the explosion time, and we apply conservative coverage and quality requirements to build reliable ``excess'' and ``no-excess'' bump and no-bump catalogs. From an initial sample of 1547 SNe~Ia, our final catalogs contain 42 early-excess and 110 no-excess events. We compare the two populations using SN and host environment parameters from ZTF DR2 and quantify the differences using two-sample statistical tests. We find the strongest differences are in SN light-curve properties: early-excess events have larger SALT2 stretch $x_1$ ($7.91σ$) and larger $r$-band secondary-maximum flux $\mathcal{F}_{r_2}$ ($6.25σ$), while differences in SALT2 color $c$ are weak ($0.57σ$). Early-excess events also favor bluer $(g-z)_{\rm local}$ ($3.41σ$) and lower $\log_{10} (M_*/M_\odot)_{\rm local}$ ($2.73σ$). Our results connect early excesses with SNe~Ia diversity, and motivate further analyses of upcoming larger samples.

We derive a way to diagnose band topology for graphene with triangular and/or honeycomb array of nanoholes directly from the lattice constant of superstructure $m\sqrt{3}\times m\sqrt{3}$ with integer $m$. Taking into account the $C_{6v}$ crystalline symmetry respected by nanoholes and their array, we demonstrate that nontrivial topology appears periodically with $m$ with period two (six) for triangular (honeycomb) array. These behaviors are verified by Wyckoff positions of Wannier centers and parity index of valence bands at high-symmetry points in Brillouin zone. The results provide a convenient guide for material design of topological electronic states based on graphene derivatives.

The Invisible Internet Project (I2P) routes data via encrypted, decentralized tunnels. Peer selection can significantly affect security and performance. This empirical study examines whether geographic location systematically influences I2P's routing topology. Consistent with I2P's design principles, which include avoiding multiple peers from the same /16 IP subnet to maximize anonymity, we conducted assortativity analysis, community detection, and permutation testing on data from 327 routers and 254 connections (SWARM-I2P). We found a network-level absence of significant geographic homophily. The assortativity coefficient was r = 0.017 (p = 0.222). Same-country connections (11.1%) are statistically near random expectation (10.91%). Community detection found 110 highly modular groups (Q = 0.972) only moderately aligned geographically (NMI = 0.521). We conclude that aggregate peer selection in I2P leads to a highly heterogeneous, random geographical mixing, providing a foundation for understanding the performance-anonymity tradeoff.

The origin of hot Jupiters remains a key open question. In the high-eccentricity migration scenario, traditional coreless models predict a strict tidal exclusion zone within $\sim 2.7$ tidal radii $r_\textrm{t}$, in which giant planets are either fully disrupted or ejected. We revisit this limit using three-dimensional hydrodynamic simulations of giant planets with realistic dense cores (10 - 20 $M_\oplus$). We find that even a few-percent-mass core fundamentally changes the outcome: \textbf{no total disruptions} occur within the previously suggested destruction zone ($\lesssim 2.7 \, r_\textrm{t}$). For deep encounters ($\lesssim 1.7 \, r_\textrm{t}$) planets suffer severe envelope stripping and are either progressively downsized to dense remnants or ejected after a few close encounters, possibly contributing to the free-floating planet population. In the intermediate regime ($ \sim 1.7 $--$2.0, r_\mathrm{t}$), planets experience significant partial mass loss over repeated encounters. For wider encounters ($ \gtrsim 2.0\, r_\mathrm{t} $), mass loss is minimal, allowing the planets gradually circularize into hot Jupiters. Furthermore, we show that for highly eccentric orbits ($e\gtrsim 0.9$), the change in specific orbital energy $ ΔE_{\mathrm{orb}} $ depends primarily on periastron distance $ r_\mathrm{p} $ rather than semi-major axis $ a $. This enables us to extrapolate our fixed-$ a $ results across a broad ($a$, $e$) parameter space and identify a well-defined tidal ejection zone whose sharp boundaries converge asymptotically. Our results highlight the crucial role of planetary internal structure in high-eccentricity migration and suggest that the survival and transformation of core-bearing giant planets are far more common than previously thought.

Library fuzzing is essential for hardening the software supply chain, but adopting it at scale remains expensive. Practitioners still spend substantial effort on environment setup, struggle to generate harnesses that respect intricate API constraints, and lack reliable means to tell genuine library bugs from harness-induced crashes. Recent LLM-based systems automate parts of this pipeline, yet they typically operate as one-shot code generators that ignore runtime feedback, which limits both the depth of code they reach and the validity of the bugs they report. We argue that effective library fuzzing is iterative by nature: each campaign exposes new coverage bottlenecks and crashes, and the next campaign should evolve from these signals rather than restart from scratch. Building on this insight, we present FuzzAgent, a multi-agent system that turns library fuzzing into an evolutionary process, in which a team of specialized agents collaborates over the full fuzzing lifecycle and grounds every decision in concrete runtime evidence, so that the harness suite is successively refined toward deeper coverage and higher-fidelity crash analysis across rounds. We evaluate FuzzAgent on 20 real-world C/C++ libraries against four state-of-the-art baselines (OSS-Fuzz, OSS-Fuzz-Gen, PromptFuzz, and PromeFuzz). FuzzAgent completes the full fuzzing lifecycle for all 20 libraries without human intervention and reaches 179619 branches, exceeding OSS-Fuzz, PromptFuzz, PromeFuzz, and OSS-Fuzz-Gen by 45.1%, 73.2%, 92.1%, and 191.2%, respectively. FuzzAgent also identifies 102 genuine library bugs, 78 of which have already been acknowledged and fixed by upstream maintainers.

We study department-level retail space optimization, where limited bay capacity must be allocated among planograms (POGs) under business and operational constraints. The problem is formulated as a linear binary knapsack model, with potential SKUs treated as items characterized by space requirements and weighted value contributions from sales, margin, units, and assortment similarity. Dynamic Programming (DP) is employed to obtain exact and reproducible assortment decisions in O(nc) time, avoiding the variance inherent in heuristic approaches. These decisions are integrated with a second-stage bay optimization model formulated as a mixed-integer program. Evaluated end-to-end across ten optimization runs spanning multiple departments and store clusters, the OPTIMUS framework achieves an average sales lift of 11.8% and an average margin lift of 9.5%. Overall, OPTIMUS provides a scalable, interpretable, and profit-driven solution for enterprise-scale retail space management.

In this paper we use a method based on the Grunsky coefficients to find upper bounds of the modulus of the initial coefficients, difference of the moduli of two consecutive initial coefficients, of the modulus of the initial logarithmic coefficient, and of the second Hankel determinant for the class of normalized bi-univalent functions.

For an $n$-element matroid $M$ given by an $n \times n$ matrix representation over a finite field $\mathbb F$ and an integer $k$, we present an $(O_{k,\mathbb F}(n^2)+O(n^ω))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $ω< 2.3714$ is the matrix multiplication exponent, and the $O_{k,\mathbb F}(\cdot)$-notation hides factors that depend on $k$ and $\mathbb F$ in a computable manner. All previous algorithms including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $Ω(n^3)$ time. Moreover, if the input matrix representation is given by a standard form, our algorithm runs in $O_{k,\mathbb F}(n^2)$-time, since $O(n^ω)$-time is only needed for finding a standard form of the input matrix. When $M$ is given by an $m \times n$ matrix, the overhead for finding a standard form is $O(mn \min(m,n)^{ω-2})$. As corollaries, we obtain faster algorithms for rank-width of directed graphs and path-width of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is of a standard form and contains a bounded number of distinct values of entries. To suggest that our algorithm is optimal, we observe that for every field $\mathbb F$, deciding whether the branch-width of a matroid represented over $\mathbb F$ is $0$ is as hard as deciding whether a square matrix over $\mathbb F$ is singular. Under the assumption that singularity testing requires $Ω(n^ω)$-time, this implies that the overhead of $O(n^ω)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.

In this paper we consider some properties of the initial logarithmic coefficients for inverse functions of functions univalent in the unit disc. The case of convex functions is treated separately. We give estimate, in some cases sharp, of the modulus of the initial coefficients, as well as the difference of the modulus of two consecutive coefficients.

Nitrogen-bearing polycyclic aromatic hydrocarbons (N-PAHs) are key precursors to complex organic molecules in both the interstellar medium and the nitrogen-rich planetary atmospheres. Despite the recent detections of nitrogen-functionalized astromolecules, their formation pathways remain an open question. The discrepancies between their predicted and observed abundances point to unknown mechanism that govern their evolution in the astrophysical environments. Employing an ion trap technique and electronic structure calculations, we unravel multiple barrier-less reactions between gas-phase pyrimidine cations (C$_4$H$_4$N$_2^+$) and acetylene (C$_2$H$_2$) which form an hitherto unreported endocyclic- N-PAHs (C$_8$H$_7$N$_2^+$). The present measurements on reactions involving a double-nitrogen subsituted aromatic heterocycle have implications to the astrochemistry of both the Titan's atmosphere and interstellar medium.

Sorting is a foundational primitive in modern data processing, influencing the execution speed of high-performance data pipelines. However, the algorithmic landscape is currently bifurcated by a pervasive "Stability Tax": practitioners must sacrifice either order preservation for high throughput or execution speed for stability. To address these limitations, this paper introduces, zSort, an adaptive z-score based distribution sorting algorithm that guarantees stability while avoiding pass complexity that scales with key-width. The performance of the proposed technique is evaluated using Microarchitectural analysis and experimental results. Microarchitectural analysis shows that zSort achieves a lower bad-speculation overhead (19.7%) than both stable baselines and several high-performance unstable algorithms and sustains a competitive IPC of 1.44. Empirical evaluation across diverse input distributions and datasets of up to 10^7 elements (64 bit) demonstrates that zSort consistently outperforms widely used comparison based stable sorting algorithms, achieving up to 3x-4.5x speedups, and a relatively better performance compared to LSD Radix, with larger gains on duplicate heavy and partially ordered inputs. Despite providing stability, zSort achieves comparable throughput as compared to high-performance unstable algorithms such as Skasort. It also maintains this performance on adaptive workloads where methods like Pdqsort typically excel and doesn't exhibit any extreme worst case. These results indicate that zSort substantially narrows the traditional performance gap between stable and unstable sorting and provides an efficient, stable sorting alternative.

We propose a bilayer all-dielectric PhC for controlling Janus bound states in the continuum (BIC) and optical chirality through symmetry-selective perturbations. Starting from a symmetry-protected $Γ$-point BIC, we use interlayer displacement as one geometric control knob to generate different topological charges in the upward radiation and downward radiation channels. A subsequent diagonal in-plane displacement reconstructs the polarization topology around the BIC and generates a Janus-chiral BIC with strong handedness selectivity. In contrast, other in-plane perturbations generate chiral quasi-BICs with finite radiative coupling, for which the circular dichroism (CD) and resonance wavelength can be continuously tuned. We further show that material conductivity provides an additional dissipative degree of freedom for actively modulating the chiral response, with a switchable CD exceeding 0.89. Near-field optical-chirality distributions and multipole decompositions reveal that the chiral response originates from a symmetry-induced imbalance of local optical handedness and a spin-selective magnetic-dipole resonance. These results reveal the topological relationship between Janus radiation, polarization singularities and intrinsic chirality, thus paving a scalable route toward reconfigurable high-$Q$ chiral photonics.

It has recently been revealed that dyonic black-hole spacetimes of a quasitopological non-linear electrodynamic field theory may be characterized by discrete radial regions with the property $dg_{tt}(r)/dr=0$ in which spherically symmetric massive {\it test} shells (Dyson shells with negligible self-gravity) can be supported in static equilibrium states. In the present paper we prove that the dyonic spacetimes of the non-linear electrodynamic field theory may also be characterized by the presence of radial regions with the dimensionless property $d[r\cdot g_{tt}(r)]/dr\to0^+$ in which massive {\it self-gravitating} thin shells that are on the verge of becoming black holes can be supported in static equilibrium states. Intriguingly, it is proved that the discrete radii of these self-gravitating nearly-black Dyson shells are universal in the sense that they are independent of the masses of the central supporting dyonic compact objects.

In this paper, we study the regularity assumptions commonly adopted in bilevel optimization with constrained lower-level problems, including the linear independence constraint qualification, the strict complementary slackness condition, and the second-order sufficient condition. These conditions are typically required to hold for the lower-level problem at every upper-level variable $x$. We first show that the requirement that these conditions hold at every upper-level variable $x$ is strong, in the sense that it is non-prevalent: there exist problems for which no sufficiently small perturbation of the lower-level objective and constraints can make the conditions hold at every $x$. To establish the result, we prove rigidity theorems showing that certain structural quantities of the lower-level problem must remain invariant across all $x$ whenever these conditions hold everywhere. We then construct explicit counterexamples in which these invariants differ between two values of $x$. In contrast, we show that the weaker requirement, that these conditions hold at almost every $x$, is a weak assumption, in the sense that it is prevalent: with probability one over a random perturbation of the lower-level objective and constraints, each condition holds at almost every $x$. We further analyze the gap between the two requirements. Although the ``every $x$'' and ``almost every $x$'' versions differ only on a measure-zero set, we show that this difference introduces fundamental difficulties in both theory and computation for bilevel optimization.

First-principles density functional theory (DFT) is employed to evaluate the structural, electronic, optical, thermoelectric, and electrocatalytic properties of monolayer CrSi2N4. Its symmetric N-Si-N-Cr-N-Si-N septuple-layer structure exhibits dynamic, thermal (300 K), and mechanical stability, supported by a -8.76 eV/atom cohesive energy. PBE and HSE06 functionals reveal an indirect bandgap of 0.58 eV and 2.16 eV, respectively, driven by localized Cr-3d and N-2p states. The monolayer features 15.57 static dielectric constant and maximum absorption coefficients of 0.9 X 10^6 cm-1 (visible) and 1.4 X 10^6 cm-1 (deep-UV). Semiclassical Boltzmann calculations predict an outstanding room-temperature n-type thermoelectric power factor of 3.5 x mW/mK2. For hydrogen evolution (HER), the basal plane yields a baseline hydrogen adsorption free energy (ΔGH) of 1.05 eV at the N-site. Applying +5% expansive biaxial strain improves HER kinetics, reducing ΔGH to 0.46 eV. Thus, CrSi2N4 is a resilient, tuneable candidate for waste-heat recovery, photodetectors, and sustainable electrocatalysis.

In this study, we obtain the following two families of circulant graphs each has Type-2 isomorphic circulant graphs w.r.t. $m$ such that $m$ has more than one value. (i) Family of circulant graphs $C_{432}(R)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3; and (ii) Family of circulant graphs $C_{6750}(S)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3 as well as $m$ = 5. This study is the $8^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}.

This paper extends the incomplete model of Haile and Tamer (2003) from static English auctions to sequential English auctions. Because bidders may wait for future opportunities, the static condition that bidders do not let rivals win at beatable prices need not hold. We replace it with a dynamic opportunity-cost restriction, yielding nonparametric valuation bounds without solving a dynamic equilibrium. Sharp bounds are also characterized. We propose a novel moment-condition inversion estimator that pools auctions with heterogeneous bidder counts, mitigating finite-sample instability of order statistics approaches and admitting analytical standard errors and smooth confidence intervals. Applications to Korean wholesale used-car auctions and Cars and Bids online auctions deliver informative bounds. Counterfactual analyses show that the option to wait lowers first-period revenue by 8--11% in the Korean market, that increasing effective competition from 8 to 20 serious bidders in Cars and Bids raises seller revenue by 40--65%, and that maximin reserve prices vary substantially across vehicle clusters.

We present an implicit, fully-coupled hydro-mechanical solver for the three dimensional simulation of fluid-driven rupture propagation along existing discontinuities. The solver handles simultaneously frictional slip (shear failure) and tensile opening (hydraulic fracture) along arbitrary intersecting fractures and faults in a linearly elastic and impermeable rock matrix. The spatial discretization combines a collocation displacement discontinuity boundary element method for quasi-static elasticity with a Galerkin finite element method for nonlinear pore-fluid diffusion along the discontinuities. Frictional and tensile failure are governed by a poro-elastoplastic cohesive zone like interface law with slip-weakening friction, dilatancy, and tensile strength degradation, integrated via an elastic predictor-plastic corrector scheme. The strong nonlinear coupling between mechanical deformation and fracture permeability is handled via adaptive implicit time-stepping. Efficient block preconditioning of the coupled tangent system, leveraging hierarchical matrix representations of the boundary element operator, is essential to achieve robustness across the full range of fracture behaviors. Accuracy and convergence are demonstrated against a comprehensive suite of analytical and semi-analytical solutions of increasing complexity: fluid-driven frictional ruptures under constant and slip-weakening friction, dilatant ruptures with permeability changes, and penny shaped hydraulic fractures spanning the viscosity-to-toughness transition. The solver is further assessed on two multi-fracture configurations: injection into three intersecting fractures, and a height-confined hydraulic fracture intersecting a strike-slip fault. The proposed framework simultaneously captures frictional slip, dilatancy, permeability evolution, and tensile opening.

Cavity-magnon systems, combining magnons and photons, offer a versatile platform for studying quantum entanglement and advancing quantum information science. In this work, we propose a scheme for generating nonreciprocal magnon-magnon entanglement in a hybrid system consisting of two yttrium iron garnet spheres coupled to a spinning whispering-gallery-mode cavity. By leveraging the magnon Kerr nonlinearity and the Sagnac effect arising from the cavity rotation, we show that the entanglement can be substantially enhanced, and the resulting entanglement exhibits pronounced nonreciprocal characteristics. Furthermore, our scheme demonstrates that the entanglement remains robust against thermal noise and persists at bath temperatures up to 100 mK. This work underscores the potential of spinning cavity-magnon systems as a versatile platform for realizing nonreciprocal quantum devices and facilitating the development of quantum technologies.

It is shown that finite-index extensions and finite-index subgroups of $ω$-stable groups can be model-theoretically wild. More precisely, there exists an $ω$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$.

Although recurrent neural networks (RNNs) trained on cognitive tasks have become a widely used framework for studying neural computation, the internal mechanisms by which RNNs switch between rhythms across multiple frequency bands, and how these mechanisms relate to neuronal time constants, have not been systematically analyzed. We trained leaky integrator RNNs with neuron-specific learnable time constants on a four-band (theta, alpha, beta, gamma) rhythm-switching task and analyzed 20 independently trained networks. Whereas low-frequency rhythms were produced by distributed participation of many neurons, high-frequency rhythms were dominated by a small subpopulation of short-time-constant neurons, and the negative correlation between time constant and matched-mode amplitude strengthened monotonically with frequency. Rhythm switching was supported by multiple coexisting mechanisms: turnover of the active subpopulation, network-wide baseline shifts that reposition the operating point near distinct unstable fixed points, and inter-neuronal phase reorganization that selectively cancels or supports band components in the population output. The mechanism deployed for each mode pair varied across training runs, exposing a degeneracy of learned solutions. These findings parallel the coexistence of rhythm-specific and multi-rhythm interneurons reported in biological circuits and provide a candidate framework for interpreting frequency-band-specific functional differentiation in neural systems.