We study five classical $t$-norms on the unit interval from the viewpoint of ultrafilter concentration. For a fixed ultrafilter $\mathcal U$ on $[0,1]$, we introduce an equivalence relation identifying two operations whenever they coincide on $A\times A$ for some $A\in\mathcal U$. We show that their asymptotic behavior is governed by two concentration regimes. In the near-$1$ regime, the five operations determine four distinct ultrafilter-equivalence classes. In the low-value regime, the Łukasiewicz, nilpotent minimum, and drastic $t$-norms collapse to the zero operation. We encode these reductions in a discrete quotient category and record simple ultrametric models for the two regimes. We further interpret the classification inside classical ultrapowers: the near-$1$ and near-$0$ regimes become exact algebraic phenomena on infinitesimal monads, and saturation yields a compactness principle for countable systems of asymptotic identities. Finally, we indicate how the same viewpoint interacts with residual fuzzy implications generated by $t$-norms.

The interaction between turbulent flows and bubbles is a complex phenomenon ubiquitous in natural and industrial settings. In this work, we experimentally investigate, from a fundamental perspective, the interaction between a rising bubble and a vortex ring in counterflow. Using time-resolved three-dimensional Lagrangian Particle Tracking (4D-LPT) coupled with shadowgraphy, we obtain simultaneous measurements of the bubble motion and the surrounding liquid flow. This approach enables detailed observation of bubble dynamics, deformation, and eventual breakup, as well as the fluid motion. We examine several flow configurations by varying the vortex circulation and the Weber number while maintaining a comparable vortex-to-bubble size ratio. Based on these measurements, we classify the interaction events into three categories according to their impact on bubble dynamics and vortex stability over time. Through experiments, we address for the first time the three-dimensional effects of these interactions, which had not been considered in previous studies. The analysed experiments comprise: Case I, corresponding to a weak interaction in which neither the bubble nor the vortex is significantly affected; Case II, where the bubble is captured and advected by the vortex, leading to a strong distortion of the vortex due to the presence of the bubble within its core; and Case III, involving a stronger vortex capable of capturing the bubble and breaking it into two fragments without a severe loss of energy in the vortex core. The analysis of these results provides insight into the bubble breakup process and the mechanisms responsible for the destabilisation of the vortex ring.

We study conserved worldsheet charges associated with spacetime supersymmetry in heterotic pure-spinor superstrings on curved ten-dimensional superspace backgrounds. Requiring BRST invariance and worldsheet conservation gives a covariant set of superspace constraints. In flat superspace these conditions reproduce the standard ten-dimensional supersymmetry generator. In curved superspace, they organize the requirements for global supersymmetry in terms of a normalizable spinor superfield.

Pure entangled states are already steerable by suitable projective measurements within their Schmidt supports, whereas rank two is the first genuinely mixed rank at which entanglement and Einstein--Podolsky--Rosen steering could bifurcate. We prove that this bifurcation does not occur even under the restricted measurement class of projective measurements: every rank-two bipartite entangled state in arbitrary finite local dimensions is projectively steerable in at least one direction, and is two-way projectively steerable when the effective local dimensions are equal. The proof is a boundary theorem rather than a steering-inequality optimization or a two-qubit reduction. A dimension--rank obstruction forces a projective outcome on the larger effective party to hit a pure boundary point of the trusted state cone. At such a contact, a nonzero support--kernel tangent block is simultaneously an NPT minor and a projective-steering certificate; if the first contact is degenerate, the rank-two constraint closes that branch and forces a second, active contact. Thus the entire first mixed-state stratum is a complete projective-steering sector, and the same support--kernel geometry gives a practical low-rank certificate whenever the relevant boundary contact is nondegenerate.

We prove an isoperimetric inequality for compact bodies bounded by surfaces embedded into a connected, contact, non-unimodular 3-dimensional Lie group.

In this article, we show that any morphism $f$ from a smooth projective variety $X$ to a simple abelian variety $A$ is smooth, if and only if there exists a holomorphic 1-form $ω$ on $A$ such that $f^*ω$ has no zero. As the key ingredient in the proof, we show any $\mathbb{Z}$-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah. Furthermore, we investigate the structure of the spaces of holomorphic 1-forms with zeros, and show that they are linear for large classes of varieties. Also, we construct a delicate example of a smooth projective subvariety of an abelian variety for which the holomorphic 1-forms with positive dimensional zero loci do not form a linear subset. Finally, we study algebraic surfaces admitting holomorphic 1-forms that have zeros and do not arise from cohomology jump loci.

We establish a Remez inequality for functions in quasianalytic Denjoy-Carleman classes $\mathcal C_M$ on a large family of fat compact sets $K \subseteq \mathbb R^n$ with tame geometry, including all fat compact sets definable in the o-minimal expansion of $\mathbb R$ by restricted $\mathcal C_M$ functions. The inequality generalizes the classical Remez inequality for polynomials and its quasianalytic versions on convex bodies, replacing the polynomial degree by the Bang degree, an integer associated with the weight $M$ and the size of the function. The constants depend explicitly on the Bang degree and the geometry of $K$. We derive a range of quantitative consequences: estimates for the volume of sublevel sets, comparison of $L^p$-norms, effective inequalities of Lojasiewicz, Harnack, and Markov type with explicit constants, and decay estimates for oscillatory integrals.

Triple-layer cuprate superconductor $\mathrm{HgBa_2Ca_2Cu_3O_8}$ (Hg1223) keeps the record of the highest superconducting (SC) critical temperature $T_{c}\sim 134$K among all the existing materials at ambient pressure. $T_{c}$ further increases under pressure up to $T_{c}\sim 160$K. However, its microscopic mechanism remains to be elucidated. We solve {\it ab initio} Hamiltonians for Hg1223 using a variational solver supplemented by a neural network. The pressure dependence of the $d$-wave SC order parameter and estimated $T_{c}$ show a dome-like structure in essential agreement with the experimental indications. The origin of the strong SC amplitude at ambient pressure is identified as strong local Coulomb repulsion $U$ attributed to poor screening. Further increase in $T_{c}$ under pressure is understood from interplay of three elements, namely increased electron hopping $t$, decreased $U$ and more importantly, strongly reduced offsite Coulomb repulsion $V$ with increasing pressure. Pairing mechanism is identified as the emergent local attraction counterintuitively generated from the originally strong local repulsion $U$. The emergent attraction is interpreted from ``attraction from reduced repulsion'', originating from the release of the fluctuating doubly-occupied sites characterized from the ``false vacuum'' in the Mott insulator to the double-occupation-free $d$-wave SC states upon carrier doping. This instantaneous attraction is in contrast with the conventional BCS SC mediated by bosonic glues. The local attraction is consistent with the electron fractionalization supported in experimental analyses. The coexistence of the SC and antiferromagnetic order is also demonstrated as a characteristic feature of the multi-layer system. The microscopic understanding of Hg1223 offers a new route explicitly using this emergent attraction to design and optimize SC materials.

We present pre-perihelion observations of methanol, carbon monoxide, and hydrogen cyanide in the Oort cloud comet C/2017 K2 (PANSTARRS), performed with the APEX 12-m and Onsala 20-m telescopes from April to July 2022. As the comets heliocentric distance decreased from 3.4 to 2.7 AU, CH3OH line intensities increased substantially (by factors of 1.1-4.0), with the most pronounced enhancement in lines with the upper-level energies Eu > 40K. In contrast, the brightness of the CO and HCN lines remained constant. We estimate the best-fit gas kinetic temperatures Tgas > 100 K and water production rate of Q(H2O)=(3-10)*10(28) 1/s. The derived methanol-to-water abundance ratio approx. 0.01-0.04, depending on the observed period. Our results demonstrate that non-LTE effects are dominant in the coma and must be accounted for to accurately derive molecular production rates. We also report weak non-thermal excitation, including potential maser activity in the CH3OH 8(0)-7(1) line.

Liquid-gas equilibrium for a simple molecular fluid is considered in view of the existence of the order parameter, in terms of which the symmetry of the binodal is restored not only in the vicinity of the critical point (critical isomorphism) but also globally in the whole coexistence region. This leads to the mapping between fluid and lattice gas (Ising model). We test this approach against the data on the liquid-gas binodal of a two-dimensional Lennard-Jones fluid and monolayers of molecular fluids. The obtained results allow us to speculate about the analog of the Kramers-Wannier duality in such systems and provide the theoretical estimate for $dp/dT$ on the saturation curve at the critical point. The microscopic grounds of the proposed approach are also discussed, and the transition from the continuous fluid model Hamiltonian to the effective quasi-spin lattice model is outlined.

We present a review of theoretical studies of the structure and reactions of N=7 and N=8 nuclei in the vicinity of 11Li, carried out within a framework based on Nuclear Field Theory. The coupling of valence nucleons to low-lying surface vibrations of the spherical core plays a central role, giving rise to self-energy processes that renormalize single-particle states and transfer form factors, as well as to an induced pairing interaction arising from the exchange of collective vibrations, which renormalizes the bare pairing force. Excitation spectra and cross sections for one- and two-nucleon transfer reactions populating states in the quasi-continuum are calculated and compared with available experimental data. Collective excitations in the particle-particle channel are investigated, with particular emphasis on Giant Pairing Vibrations and on their damping mechanisms arising from coupling to more complex configurations and continuum states. Comparisons with other theoretical schemes are also presented. We conclude that a coherent understanding of experimental data requires the detailed consideration of particle-vibration coupling effects.

Swap algebras generalise Bishop's complemented powerset as Boolean algebras generalise the powerset. Actually, all Boolean algebras are swap algebras. We prove constructively a Stone representation theorem for separated swap algebras of type (II), where the notion of a separated swap algebra generalises the corresponding notion of a separated Boolean algebra. Moreover, we prove a Stone-Cech theorem for swap algebras of type (II), showing that the restriction to separated swap algebras is not a loss of generality from the point of view of the theory of swap characters. A constructive Stone representation theorem and a Stone-Cech theorem for Boolean algebras follow as special cases. We introduce sets with a Boolean inequality, that is sets with an internal falsum. These sets allow a book-keeping of the use of the Ex falso principle in constructive mathematics. If we restrict to swap algebras with a Boolean inequality, then the proof of the Stone representation theorem for swap algebras of type (II) is within minimal logic.

Low-rank approximation based on selected rows and columns is a useful alternative to singular value decompositions when the goal is an interpretable and compact matrix representation. A standard way to choose these rows and columns is the maximum-volume principle: it selects submatrices with large volume, which usually leads to stable interpolation coefficients and accurate CUR-type approximations. In this paper, we study this idea for quaternion matrices. This setting is natural for color images, three-dimensional motion data, and multi-channel signals, but requires care because quaternion multiplication is noncommutative. We define quaternion maximum-volume submatrix selection using quaternion singular values and the Study determinant. We then derive quaternion rank-one update formulas and use them to build two selection procedures: a greedy square-core method for row and column replacement, and a rectangular method that enlarges a selected row set until the interpolation coefficients are controlled. We prove that successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used. We also connect the stopping criterion with quasi-dominance, prove an exact quaternion CUR identity in the full-rank case, and derive an interpolation stability bound. For the rectangular case, we derive an append-row pseudoinverse update and show how it gives a natural right preconditioner for overdetermined quaternion least-squares problems. Finally, we illustrate the methods on three applications: quaternion CUR approximation of RGB images, RectMaxVol-based preconditioning for ill-conditioned quaternion least-squares systems, and row selection in quaternion motion-capture data. The experiments show that the proposed quaternion MaxVol and RectMaxVol methods provide stable and efficient selection routines.

The existing Generative Fixed-Filter Active Noise Control (GFANC) method generates a suitable control filter based on the current noise frame. This reactive design aims to estimate a control filter that is optimal for the present frame rather than the upcoming one. Consequently, it suffers from an inherent tracking lag and lacks the predictive capability to handle rapidly varying noises. To address this limitation, we propose the Predictive Fixed-Filter Active Noise Control (PFANC) method with a proactive control paradigm in this paper. In the PFANC method, multiple consecutive noise frames are processed by a Convolutional Recurrent Neural Network (CRNN) to predict the next-frame control filter. By utilizing temporal correlations across noise frames to anticipate the control filter in advance, the PFANC method can effectively track dynamic noise changes. Furthermore, the theoretical analysis based on a high-order Markov chain shows that incorporating multiple noise frames enhances the prediction of the control filter. Numerical simulations with linear and logarithmic chirp signals, as well as real-world dynamic noises, validate the effectiveness of the PFANC method and its superiority over GFANC and its variations. The PFANC method also exhibits good transferability across different acoustic paths.

This expository paper is a companion to \cite{Rav:gjmcyc}, in which we discuss cyclotomic spectra. Both papers are intended to shed light on the recent resolution of the telescope conjecture by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank (hereafter referred to as BHLS) in \cite{BHLS}. Their proof involves both cyclotomic spectra, the subject of \cite{Rav:gjmcyc}, and cyclotomic extensions of spectra, the subject of this paper. Higher cyclotomic extensions of commutative ring spectra are analogous to Galois extensions of $p$-adic number fields (or rings of integers thereof) obtained by adjoining roots of unity.

This paper investigates two problems unified by the study of the uniform boundedness of the dilation operators (UBD) T_r f(z)=f(rz), 0<r<1, acting on weighted Bergman spaces A^p_omega with not necessarily radial weights. We first characterize the random symbol space for A^p_omega under a mild admissible condition (Theorem 1.2). This extends the main result of [7] from radial weights to non-radial weights. We then introduce two new notions, namely non-radial mixed norm spaces M(p,q;omega) and analytic tent spaces A(p,q;omega), and we characterize their corresponding symbol spaces as well (Theorem 1.8, Theorem 1.12). The novelty here is to employ a measure-disintegration framework to prove a general Littlewood-type theorem (M(p,q;omega))* = H(2,q;omega_r), from which the Bergman space result (A^p_omega)* = H(2,p;omega_r) follows as the special case p=q. Among other things, UBD plays a pivotal role in the proofs of the preceding theorems. The second main problem addressed in this paper is to establish a local-to-global criterion for UBD, which remains largely unexplored for non-radial weights. Our principle result in this part (Theorem 1.16) asserts that UBD is guaranteed by two local geometric conditions: bounded hyperbolic oscillation (BHO) and a reverse-Carleson tail condition (RC). This is the technical heart of the paper. In the course of our investigation, three new types of problems arise naturally, each of independent interest: a two-weight top-maximal operator (Theorem 1.17); a truncated maximal operator over hyperbolic balls (Theorem 1.18); and a single-testing Carleson embedding problem (Theorem 1.19).

Large Language Model (LLM) agent systems suffer from failures that occur without external triggers -- no injection, no adversarial input, no resource exhaustion. These silent failures -- unexpected deviations from intended behavior under normal conditions -- are routinely misattributed to bugs or configuration errors. Through systematic analysis of over 40,000 controlled trials and long-term production observations spanning 100,000+ agent interactions, we identify a common structural logic underlying these failures. Building on patterns observed in our experiments, we survey the global research literature on autonomous agent reliability and synthesize 22 intrinsic properties of LLM agent systems across six lifecycle layers: foundation semantics, inter-agent transmission, memory persistence, task execution, feedback correction, and systemic evolution. We demonstrate that whenever a sufficient subset of these properties co-exist, system entropy -- the measurable accumulation of disorder: loss of output consistency, task accuracy, and cross-session coherence -- increases monotonically with interaction rounds. We formalize this as the Entropy Principle: S(t) = S0 * e^(alpha * t), with alpha measured empirically across multiple architectures. We propose the PIG (Physical Integrity Gate) Engine with the ADE (Agent Delivery Engineering) protocol suite as an engineering countermeasure to entropy-driven disorder. Our findings establish silent failure not as a bug to be fixed but as a manifestation of Intelligence Entropy -- a physical constraint to be managed through deterministic governance. We argue that any engineering effort stabilizing the structure and order of agent systems participates in a unified mission: keeping intelligent systems reliable as they grow in scale and complexity.

In this article, the surface tension and stability of quark matter modified by the vector interaction in a strong magnetic field are investigated in the quasiparticle model with the multiple reflection expansion. The self-consistent thermodynamic treatment of the chemical-potential-dependent quark mass is maintained by the effective bag function, which depends on both the chemical potential and the magnetic field. It is found that the vector interaction could enlarge the surface tension in both the parallel and transverse directions with respect to the magnetic field. In a stronger magnetic field region, the presence of the vector repulsive interaction leads to an increase in transverse surface tension with the magnetic field strength, which is opposite to the vanishing value without repulsive interaction in the previous work. Consequently, it is concluded that a moderate-intensity magnetic field is required for the formation of a quark matter bubble with the vector interaction. Finally, it is demonstrated that the vector interaction slightly reduces the stability of quark matter.

In this paper, we introduce and investigate the $\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_1,\ldots,k_n)\in \mathbb{N}^n$ satisfying $1\leq k_i\leq \textrm{deg}_G(i)+1$ for all $i$, the $\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\textrm{NI}_{\textbf{k}}(G)=\sum_{i=1}^n\, (\textbf{x}_W:\, W\subseteq N_G[i],\, |W|=k_i)$$ of $S$, where $\textbf{x}_W=\prod_{i\in W} x_i$. We study homological invariants and properties of $\textrm{NI}_{\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\textrm{NI}_{\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\textrm{NI}_{\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.

We investigate the different oscillatory modes, namely, complete synchronization, inphase synchronization, antiphase synchronization and desynchronization in a one-dimensional anisotropic Heisenberg ferromagnetic spin chain consisting of a large number of spins. By solving the associated Landau-Lifshitz-Gilbert-Slonczewski equation for the spins we show the simultaneous existence of the above mentioned oscillatory modes in the spins. We observe that when the number of the spins is large the synchronization is lost between the spins; however, we identify that the field-like torque is able to induce synchronous oscillations of the spins in the chain again. We also confirm the agreement of the numerically obtained values of the frequency of the inphase synchronized oscillations with the analytically obtained values.

This work develops an iterative computational framework to obtain saddle-point solutions for infinite-horizon two-person mean-field linear-quadratic zero-sum stochastic differential games. By generalizing classical iterative framework, we construct a monotonically increasing matrix sequence to decouple the strongly coupled, analytically intractable original problem into a set of tractable subproblems. By sequentially computing the stabilizing solutions of the coupled algebraic Riccati equations for these subproblems, we can further derive the solution to the original problem. Rigorous convergence analysis is established to validate the proposed iterative algorithm. Different from existing algorithms limited to specific simplified setups, this framework proposes the first universal computational paradigm applicable to a broad class of game-driven Riccati equations.

A field theory in $3+1$ dimensions contains two fermions, two Abelian gauge fields, and one complex scalar, with dynamics fixed by a BRST symmetry. For a specific parameter configuration, the fermion mass matrix becomes non-Hermitian, and the propagators exhibit complex conjugate poles. We evaluate the one-loop fermion contribution to the two-point function of the composite operator $ϕ^{\dagger}ϕ$. Once the factor $i$ of the $e^{iS}$ normalization is removed, the contribution is real for real external momentum, which follows from the pairing of complex conjugate terms in the loop integral. The BRST construction of the action sets up a proof of renormalizability.

Particle physics models beyond the Standard Model often contain global symmetries to address various unanswered questions. However, a common criticism of theories based on global symmetries is that such symmetries are generally expected to be explicitly violated by gravitational effects at the Planck scale. In the case of a global $U(1)$ symmetry, this explicit breaking can reduce the symmetry to a discrete subgroup of $U(1)$, leading to the formation of cosmic strings attached to multiple domain walls (DWs). These DWs are usually cosmologically problematic, since their slow scaling behavior can eventually dominate the energy density of the Universe, giving rise to the well-known cosmological DW problem, which is strongly constrained by Big Bang Nucleosynthesis. In this letter, we propose a new annihilation mechanism of such DWs in theories with the simplest continuous global symmetry, $U(1)$, in the presence of gravitational effects. The mechanism is as follows: if a fermion coupled to the symmetry-breaking scalar possesses a small bare mass term, radiative corrections can generate a temperature-dependent bias for triggering DW annihilation. As a representative example, we study a majoron framework containing right-handed neutrinos with small bare mass terms, in which a wall-string network can arise once gravitational effects are taken into account. Within this setup, we show that the small bare masses of the right-handed neutrinos provide the origin of the bias responsible for triggering the annihilation of the DW network.

This paper proposes a Structural Matrix Autoregressive (SMAR) model for the joint analysis of asset returns, realized volatility, and trading volume in a large-dimensional setting. This framework simultaneously captures dynamic spillovers across financial variables and cross-sectional dependence across assets while preserving a parsimonious parameterization relative to conventional vector autoregressive models. The model is estimated on daily data for the constituents of the Dow Jones Industrial Average over the period 2021-2025 and is structurally identified through restrictions consistent with the Mixture of Distributions Hypothesis and efficient market theory. The empirical findings indicate that volatility is the primary driver of trading activity, suggesting that informational shocks are predominantly incorporated into markets through price variability. Forecast error variance decompositions further reveal that, although internal shocks dominate short-term volume dynamics, cross-asset spillovers account for more than 50% of trading volume variation at longer horizons. Finally, an event-study analysis around FOMC announcements supports the proposed decomposition by identifying significant increases in the informative component of trading activity on announcement days followed by rapid mean reversion.

We study charge transport in a magnetized relativistic plasma using kinetic theory within the relaxation-time approximation. By exactly solving the linearized Boltzmann equation in a uniform magnetic field, we obtain an analytic solution for the distribution function in terms of Bessel functions. Using this solution, we compute the full set of retarded current-current correlators and verify the Ward identities. In the hydrodynamic limit, we extract the charge diffusion modes, demonstrating that the transverse diffusion coefficient is strongly suppressed by the magnetic field, scaling as $1/B_0^2$ in the strong-field regime, while the longitudinal diffusion remains unaffected. Furthermore, we analyze the non-hydrodynamic branch cuts in the complex frequency plane, determining their kinematic thresholds and identifying the underlying wave-particle interactions as longitudinal Landau damping and transverse cyclotron damping.

Eukaryotic DNA replication must remain robust under thermal, chemical, and genotoxic stress despite large fluctuations in replication dynamics. Here, we develop a lattice-based stochastic Monte Carlo framework for whole-genome replication in Saccharomyces cerevisiae at single base-pair resolution, incorporating probabilistic origin firing, replication fork-speed distributions, and a time-dependent limiting factor that governs the availability of cellular replication resources. The model is benchmarked quantitatively against experimental replication profiles before being applied to stress conditions, and reproduces diverse replication stress responses using only two effective parameters. Importantly, the analysis reveals that replication fork-speed heterogeneity underlies the emergence of Erlang-distributed S-phase durations and rare, anomalously prolonged replication events observed experimentally in Escherichia coli and human cell lines, while predicting similar behavior in S. cerevisiae. The framework further predicts non-monotonic thermal behavior, power-law scaling under hydroxyurea stress, and total replication-time dynamics under diverse genotoxic conditions.

Affine formation control offers immense flexibility for coordinating multi-agent maneuvers, but guaranteeing the safety of agents under parametric uncertainties remains an open challenge. This paper proposes a novel safe affine formation control framework for second-order multi-agent systems by integrating Higher-Order Control Barrier Functions (HOCBFs) with Adaptive Dynamic Programming (ADP). We introduce a barrier-modulated control architecture that smoothly attenuates the nominal formation tracking objective when agents approach safety boundaries, preventing conflicting control inputs. Within this architecture, two distinct safety controllers are developed: (1) an analytical barrier-gradient repulsive controller that provides a computationally efficient, rigorous mathematical baseline, and (2) a data-driven optimal safety controller. The data-driven approach utilizes an actor-critic neural network to solve the Hamilton-Jacobi-Bellman (HJB) equation online, enabling optimal collision avoidance even in the presence of unknown system parameters. Using Nagumo's theorem and Lyapunov stability analysis, we formally prove that both controllers guarantee the forward invariance of the safe set ensuring absolute collision avoidance while maintaining Uniformly Ultimately Bounded (UUB) formation tracking errors. Finally, simulations validate the theoretical findings and demonstrate the robustness of the proposed controllers in dynamic obstacle avoidance scenarios.

Repository-level code generation with Large Language Models (LLMs) remains challenging, primarily due to complex dependencies and limited context windows. Recent approaches adopt retrieval-augmented generation (RAG) and the planning mechanism to reuse potential callee functions in the repository. However, these approaches often suffer from two limitations: lack of test-driven behavioral guidance during planning and overlooking the implementation logic embedded in repository code during reuse. As a result, generated plans may not align with expected behaviors, and retrieved functions may not be effectively reused. In this paper, we propose TICoder, a novel repository-level code generation framework that improves both planning and reuse. TICoder introduces a test-driven iterative planning mechanism that leverages test cases as behavioral specifications to refine implementation steps. Furthermore, TICoder employs an implementation-aware code reuse strategy, which retrieves potential callee functions using a dual-view similarity that captures both functional and implementation aspects. We then identify relevant usage patterns through a dual-stage selection strategy, combining structure-based clustering and perplexity-based filtering. We conduct extensive experiments on widely used repository-level code generation benchmarks with various LLMs. Experimental results demonstrate that TICoder outperforms state-of-the-art (SOTA) methods, achieving an average improvement of 11.52%.

We analyze fractional phase-transition energies with periodic hetero\-genei\-ties at the critical $H^{1/2}$ scaling. We prove that the $Γ$-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case $s>1/2$.

This study employs a reflexive thematic analysis to systematically examine the design patterns of 48 first-person Virtual reality (VR) animal avatar games. The research identifies four primary design themes: Animal Biomimicry, Limited Animal Simulation, Hybrid HumanAnimal Features, and Human Behavior with Animal Avatar. The analysis reveals that approximately 77 percent of the games remain grounded in human-centered interaction logic, with animal forms primarily serving as visual representations. The study highlights the core tension between authenticity and usability in current VR animal avatar design, and points toward design opportunities for achieving more authentic animal avatar's interactive experience through directions such as controller innovation, unconventional body mapping, and dynamic feedback. This research provides a thematic classification framework for understanding the representation of non-human perspectives in VR games.