We show that in Greenberg (1989)'s coalitional repeated game situation, every nondiscriminating Conservative Stable Standard of Behavior is a subset of the set of Perfect Coalitional Equilibrium (Ali and Liu 2026) paths. Moreover, the set of Perfect Coalitional Equilibrium paths itself is a nondiscriminating Conservative Stable Standard of Behavior. The set of Perfect Coalitional Equilibrium paths is therefore the maximal nondiscriminating Conservative Stable Standard of Behavior.

Nonlocal games provide a unified framework for studying the distinction between classical, quantum, and more general no-signaling correlations. In this work, we develop this perspective by connecting the Bell-locality framework to several complementary mathematical representations of nonlocal games and quantum strategies. We begin with local hidden-variable models, the CHSH inequality, and the role of Bell nonlocality as a device-independent witness of entanglement, and then introduce nonlocal games through the standard predicate/verifier formalism. We next examine a set of representative examples, including XOR games, the GHZ game, graph-based coloring games, the Mermin-Peres magic square game, and Hardy's paradox as a related logical manifestation of nonlocality. Building on this foundation, we compare four closely related representation frameworks: conditional-probability and correlation descriptions, Bell-functional formulations, entangled-value optimization, and the quantum-operator approach together with the Navascues-Pironio-Acin (NPA) hierarchy. These viewpoints are then instantiated for the CHSH, magic square, and GHZ games, showing how each representation emphasizes a different aspect of the same underlying task. Taken together, these examples show that nonlocal games can be studied simultaneously as geometric objects in correlation space, optimization problems over entangled resources, and operator-theoretic constructions. This multi-representation viewpoint clarifies the relation between Bell inequality violations, perfect quantum strategies, pseudo-telepathy, and semidefinite relaxations of quantum correlations.

Altermagnetism generates exchange-type spin splitting without net magnetization and, in its $\it d$-wave form, resembles the angular symmetry of unconventional $\it d$-wave superconductivity. Whether this correspondence bears directly on superconducting instabilities in real correlated materials remains open. Here we study the quasi-two-dimensional vanadium oxychalcogenide CsV$_2$Se$_2$O (CVSO), a square-net $\it d$-wave altermagnet candidate, through combined experimental and theoretical investigation of its lattice structure, electronic structure and transport properties. At ambient pressure, CVSO is a weakly insulating parent state with a density-wave-like anomaly near 100 K, and its bulk properties are most consistent with a G-type compensated antiferromagnetic background. Under compression, the density-wave-like feature is suppressed, the magnetoresistance evolves from predominantly negative to positive, and a superconducting-like resistive downturn emerges below about 3 K. This low-temperature anomaly is reproducible across samples and pressure media, and is suppressed by magnetic field. Room-temperature X-ray diffraction reveals no symmetry lowering, whereas does show a pronounced compressibility anomaly over the same pressure range. CVSO thus reveals a pressure-tuned phase diagram in which a reconstructed weakly insulating parent state gives way to strange-metal-like transport and superconducting-like behavior, echoing broader phenomenology associated with unconventional superconductors, including cuprates and nickelates.

We compute the quantum radiation emitted by wavepackets of relativistic single electrons, both at rest and undergoing uniform acceleration in the Minkowski vacuum of the electromagnetic field. We find that the cubic terms in the original nonlinear action of electrodynamics should be considered in obtaining the quantum radiation to the leading order. We show that the quantum radiation from a single-electron wavepacket at rest vanishes exactly. For a uniformly accelerated electron, the quantum radiated power has secular growth in the long-time regime. We demonstrate that this secular growth has a classical interpretation, and argue that the resummed quantum radiation at late times would not diverge. Regarding experimental proposals for the detection of the Unruh effect from the quantum radiation in the `blind spots' of classical radiation we ascertain that quantum corrections in the two blind spots are fully contributed by the transverse deviation correlators, where the dominant contributions are irrelevant to the Unruh effect in electron microscopes.

We investigate eddy-viscosity distributions in pressure-driven wall turbulence for three canonical configurations: plane closed-channel flow, open-channel flow with a free-slip surface, and pipe flow. Using direct numerical simulation (DNS) databases spanning friction Reynolds numbers $Re_τ=$ 2000--12000, we infer the eddy viscosity from one-point statistics through the Boussinesq relation. The DNS-inferred eddy viscosity displays configuration-dependent behavior in the outer region, indicating that a single full-depth expression is not uniformly accurate for all three configurations. Building on the interpretation of eddy viscosity as the product of a velocity scale and a length scale, we extend the log-law scaling into the outer region. Specifically, we adopt a stress-based velocity scale and introduce an outer correction function to capture the remaining dependence on the outer coordinate. We then embed a compact parametric form of this correction into a Cess-type framework with van Driest near-wall damping, yielding a full-depth eddy-viscosity model. We assess the model using eddy-viscosity profiles, the log-law indicator function, and skin friction. The results show that the proposed model yields noticeable improvement for open-channel flow while remaining comparable to the classical Cess model for closed-channel flow and pipe flow. These findings underscore the role of outer boundary conditions in shaping the outer-region eddy viscosity and, consequently, mean-flow predictions.

We present a coarse-graining method applicable to dry scalar active matter with motility regulation. Our approach, based on a multiscale perturbative expansion of the backward Kolmogorov equation, does not rely on any specific microscopic dynamics for the particles' orientations. Its generality allows us to address different forms of motility regulation, from space-dependent self-propulsion speed to taxis, and to extend the analysis to a class of non-Markovian orientational dynamics. Furthermore, we identify general conditions on the microscopic dynamics that ensure the existence of an effective large-scale equilibrium regime. When the latter are violated, our theoretical framework is able to quantitatively capture the emergence of large-scale particle currents. We directly apply our coarse-grained theory to several models of self-propelled agents, ranging from single particles to active polymers, and test our analytical predictions with numerical simulations. Finally, we show that our theory naturally extends to active matter with density-mediated interactions, such as quorum sensing, with potential applications to self-organizing soft materials.

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that can be made colour-balanced by recolouring $O(k^2)$ edges. More generally, we obtain analogous bounds for arbitrary bounded-degree spanning subgraphs of edge-coloured complete graphs and for perfect matchings in edge-coloured $r$-uniform complete hypergraphs in a more general vector-label setting. The former result answers a question recently posed by Banerjee and Hollom, and significantly improves earlier bounds for all previously studied classes of subgraph. Our proofs reduce each of these problems to a setting in which we can apply a bound for perfect matchings in the complete bipartite graph, established via a linear relaxation and a necklace-splitting argument.

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.

Understanding how microscopic motility shapes emergent collective behaviors is a challenging task in active matter, especially when self-propulsion is regulated by external cues or via quorum-sensing interactions. To address this problem, we derive a closed hydrodynamics for scalar active matter with spatially-regulated motility, under general hypotheses for the microscopic dynamics of the particles' orientations. We show that, at large scales, the contribution of the latter is entirely captured by the autocorrelation tensor of the orientations. This allows us to establish a macroscopic equivalence within a broad class of motility-regulated active systems, from single particles to active polymers. Our formalism allows us to reveal a new form of motility-induced phase separation for quorum-sensing active polymers, which we term anti-MIPS, where dense phases exhibit enhanced activity relative to dilute regions. Our theory shows that anti-MIPS generically arises for motility-regulated agents with internal structure, uncovering the existence of several distinct transition pathways.

Large Language Models (LLMs) are widely used by students, yet their tendency to provide fast and complete answers may discourage reflection and foster overconfidence. We examined how alternative LLM interaction designs support deeper thinking without excessively increasing cognitive burden. We conducted a two-phase mixed-methods study. In Phase 1, interviews with 16 Gen Z students informed the design of Deep3, a web-based system with three interaction modes: \emph{a)} future-self explanations, \emph{b)} contrastive learning, and \emph{c)} guided hints. In Phase 2, we evaluated Deep3 with 85 participants across two learning tasks. We found that a standard single-agent baseline produced high perceived understanding despite the lowest objective learning. In contrast, future-self explanations imposed higher cognitive workload yet yielded the closest alignment between perceived and actual understanding, while guided hints achieved the largest learning gains without a proportional increase in frustration. These findings show that effort, confidence, and learning systematically diverge in LLM-supported work.

We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard Hénon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line.

We analyze magnetic field profiles of vortices in thin-film superconductors, shedding new light on this old and presumed settled problem. In sufficiently thin films with realistic Ginzburg-Landau parameter $κ= 1/\sqrt{2}$, the magnetic screening around a vortex core is neither exponential -- as is expected in bulk -- nor the power-law that was predicted by Pearl. Instead, a universal curve for the magnetic field variation appears that scales with the sample thickness. The thickness dependence is consistent with the seminal Pearl length, and serves as an indication of the reduced magnetic field screening present in two-dimensional superconductor. Finally, we quantify the crossover from bulk-like to thin superconductors, and establish different screening length-scales relevant for the analysis of experimental data.

Opportunistic photo capture (e.g., slides, exhibits, or artifacts) is a common strategy for preserving information encountered in information-rich environments for later revisitation. While fast and minimally disruptive, such photo collections rarely become meaningful notes. Existing automatic note-generation approaches provide some support but often produce generic summaries that fail to reflect what users intended to capture. We introduce Intent Lenses, a conceptual primitive for intent-mediated note generation and sensemaking. Intent Lenses reify users' capture-time intent inferred from captured information into reusable interactive objects that encode the function to perform, the information sources to focus on, and how results are represented at an appropriate level of detail. These lenses are dynamically generated using the reasoning capabilities of large language models. To investigate this concept, we instantiate Intent Lenses in the context of academic conference photos and present an interactive system that infers lenses from presentation captures to generate structured visual notes on a spatial canvas. Users can further add, link, and arrange lenses across captures to support exploration and sensemaking. A study with nine academics showed that intent-mediated notes aligned with users' expectations, providing effective overviews of their captures while facilitating deeper sensemaking.

The discovery of circumbinary planets (CBPs) has advanced our understanding of planet formation and dynamical evolution in complex environments. However, the population of such planets remains small, leading their underlying physical properties to be loosely constrained. In this work, we have developed a semi-automated framework to identify planetary transit events in light curves of eclipsing binaries observed by the Transiting Exoplanet Survey Satellite (TESS). Our search method, ${\tt mono-cbp}$, removes stellar eclipses and applies a custom detrending procedure, searching for individual transit events and applying automated vetting procedures to filter false positive signals. We searched a sample of binaries from the TESS Eclipsing Binary Catalogue, yielding one candidate transit event. ${\tt mono-cbp}$ was also tested on the known population of transiting CBPs, using the Kepler long-cadence photometry for the Kepler transiting CBPs and the TESS Full Frame Image photometry for the TESS CBPs. Excluding transits that are shallower than the intrinsic noise of the Kepler/TESS data, ${\tt mono-cbp}$ achieved a recovery rate of $\geq50$ per cent for each planet, reaching >75 per cent for 9 of the 14 planets. To test the limits of our framework, we injected simulated transit profiles with varying depth and duration into our sample of TESS light curves, finding that our recovery rate is a strong function of transit duration and the metrics used to filter false positive signals. This framework may be applied to large samples of TESS eclipsing binaries with little computational burden and to photometry from future space-based photometric surveys.

Recent XRISM observations of active galactic nuclei such as PDS 456 have revealed ``forests'' of absorption lines best modeled by five distinct absorption zones with varying large blueshifts. We propose a model in which these relativistic blueshifts originate from the motion of the accretion disc itself, rather than from a clumpy super-Eddington outflow at hundreds of gravitational radii $r_g\equiv GM/c^2$. We demonstrate that thin rings of absorbing material lying just above the accretion disc at varying radii can produce the observed energy shifts and separations of the absorption zones. In this model, the PDS 456 transmission spectrum is well reproduced by rings with widths $Δr\lesssim1r_g$ at locations between the black hole's innermost stable circular orbit (ISCO) and $\approx15r_g$. This model suggests that the absorption forests seen in XRISM observations can probe the surface structure of the innermost ($\lesssim15r_g$) regions of quasar accretion discs.

The propagation of heat and thermal signals in the form of travelling waves is investigated for a nonlinear Maxwell-Cattaneo-Vernotte equation. The exact wave solutions are derived by expressing the thermal conductivity and the relaxation time as polynomial functions of the temperature. This approach enables the identification of suitable degrees of nonlinearity that give rise to soliton solutions. Finally, exact solutions are shown through plots for the values of the selected parameters.

We develop a framework to construct worldvolume gauge theories on D2-branes probing compound Du Val (cDV) Calabi-Yau threefold singularities. By viewing these singularities as ADE surface fibrations over the complex $w$-plane, we encode their geometry in a Higgs field $Φ(w)$. A D2-brane probe perceives $Φ(w)$ as an $\mathcal{N}=2$ deformation of its 3d $\mathcal{N}=4$ affine Dynkin quiver gauge theory via polynomial and monopole superpotential terms. By exploiting 3d mirror symmetry, we obtain an effective theory that correctly reproduces the quiver-collapsing mechanism known in the mathematical literature. We present several examples, including non-toric and non-resolvable cases.

The Koopman operator is a powerful approach to global stability analysis of nonlinear systems, which provides a systematic procedure for Lyapunov function design. In this framework, Lyapunov functions are obtained through the eigenfunctions of the Koopman operator associated with the eigenvalues of the Jacobian matrix at the equilibrium. In practice, the eigenfunctions are approximated via a finite-dimensional representation of the operator, and there is no guarantee that the approximated spectrum accurately matches the true one. In this paper, we develop a kernel-based method to compute Koopman eigenfunctions and preserve the spectrum of the Jacobian matrix. This approach is suitable for stability analysis of high-dimensional systems thanks to the kernel trick. Moreover, the Lyapunov function candidate is validated through a scenario-based optimization technique that provides a reliable estimation of the region of attraction of the system.

We study the periodic properties of sequences of quantum channels sampled from an ergodic stochastic process satisfying a natural irreducibility condition. We relate these periodic properties to certain global spectral data defined by the sequence of quantum channels, proving a general Perron-Frobenius-type theorem. We give examples to motivate the theory and conclude with some open problems and conjectures.

Artificial spin ice (ASI) are metamaterials composed of interacting nanomagnets. Although ASI hold promise for low-power computing, the ability to transmit information through these two-dimensional systems has been limited. Inspired by non-reciprocal transport in nature, we develop a framework for non-reciprocal influence between nanomagnets. Using the framework we discover a family of ASI geometries with inherent directionality. Directional ASI have the property that, when driven by an external field protocol, domains grow and reverse in the same direction, illustrating an emergent non-reciprocity of the system. Combining growth and reversal results in unidirectional domain movement through the metamaterial. We focus on one member of the directional ASI family, and demonstrate unidirectional domain growth experimentally. Furthermore, we show that the direction of growth is reconfigurable by tuning the external field strengths. Finally, we demonstrate how the directionality of the system significantly improves memory capabilities in a reservoir computing framework. Our work is the first demonstration of an ASI with inherent directionality, offering a magnetic computing platform that combines memory and computation within a single neuromorphic substrate.

This paper studies the KL-Schubert classes defined by Kazhdan-Lusztig bases in $K$-theory and hyperbolic cohomology of flag varieties. We first establish Poincaré dualities of these classes. We then focus on Grassmannians, and establish the Billey-type formula for KL-Schubert classes in hyperbolic cohomology.

Thermal behavior in subsystems of closed quantum systems is commonly attributed to dynamical chaos, quantum ergodicity, canonical typicality, or the eigenstate thermalization hypothesis, suggesting a fundamentally statistical origin of thermalization. Here, we propose a potential alternative mechanism in which thermal structures emerge deterministically from symmetry considerations alone, without recourse to statistical arguments. We prove a finite de Finetti-type theorem for quantum states invariant under energy-preserving unitaries, establishing that the reduced marginals of any such invariant $N$-qudit state are close (both in trace distance and relative entropy) to convex mixtures of thermal product states, with explicit error bounds vanishing as $N \to \infty$. We further present an example of energy-conserving Lindblad dynamics whose long-time limit is invariant under energy-preserving unitaries, providing a dynamical realization of the desired symmetry class. These results imply that invariance under energy-preserving unitaries suffices as a sole fundamental, deterministic principle to enforce thermal structures.

Software logging is essential for maintaining and debugging complex systems, yet it remains unclear how AI coding agents handle this non-functional requirement. While prior work characterizes human logging practices, the behaviors of AI coding agents and the efficacy of natural language instructions in governing them are unexplored. To address this gap, we conduct an empirical study of 4,550 agentic pull requests across 81 open-source repositories. We compare agent logging patterns against human baselines and analyze the impact of explicit logging instructions. We find that agents change logging less often than humans in 58.4% of repositories, though they exhibit higher log density when they do. Furthermore, explicit logging instructions are rare (4.7%) and ineffective, as agents fail to comply with constructive requests 67% of the time. Finally, we observe that humans perform 72.5% of post-generation log repairs, acting as "silent janitors" who fix logging and observability issues without explicit review feedback. These findings indicate a dual failure in natural language instruction (i.e., scarcity of logging instructions and low agent compliance), suggesting that deterministic guardrails might be necessary to ensure consistent logging practices.

The 1D flux power spectrum ($P_{\mathrm{1D}}$) of the Ly$α$ forest provides an exceptionally high-resolution probe of structure formation down to small scales ($k\approx1-10~\text{$h~$Mpc$^{-1}$}$). These scales carry the imprints of massive neutrinos, warm dark matter, and the running of the primordial power spectrum spectral index. The effective field theory (EFT) is a promising perturbative approach to systematically and efficiently describe the Ly$α$ forest, but it faces challenges in its application to $P_{\mathrm{1D}}$, as many EFT parameters become degenerate when projected along the line of sight. In addition, this projection generates new stochastic terms from the integration over small-scale modes. In this work, we address these issues by compressing the EFT model space using the Fisher matrix formalism and linearizing the resulting compression directions, enabling analytic template marginalization and significantly reducing the computational cost of likelihood evaluation. We use hydrodynamical simulations to obtain a baseline estimate of EFT parameters, and use the DESI DR1 $P_{\mathrm{1D}}$ measurements to derive compression directions. We then marginalize over deviations from the baseline using these compression directions and forecast the constraining power of our formalism. We find that even in conservative scenarios where each data redshift bin requires its own set of EFT parameters, the cosmological constraints saturate with the linear bias, two leading-order 1D stochastic terms, and three principal combinations of the remaining EFT templates. In this case, our forecasted precision of the amplitude ($Δ^2_p$) and the logarithmic slope ($n_p$) of the linear matter power spectrum at the pivot scale ($k_p=0.7~\text{Mpc}^{-1}$) is $10\%$ and $2.0\%$, respectively, which is similar to emulator-based analyses that include observational data systematics.

We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our strategy is to reduce this classification to determining the orbits of the parity and conjugation functors on irreducible modules of the complexifications of the aforementioned algebras. Then we provide explicit results for the computation of these orbits. For Lie superalgebras of basic type or of type $\mathbf Q(n)$, our classification applies to any highest-weight parametrization of irreducible complex modules with respect to an arbitrary Borel subalgebra. As a consequence, in the special case of real simple Lie algebras we obtain a new perspective on the classification of real simple modules and establish a conceptual connection with Kostant's cascade of strongly orthogonal roots.

Identifying dynamically influential nodes in biological networks is a central problem in systems biology, particularly for prioritizing intervention targets in gene regulatory networks. In this paper, we propose a Shapley-value-based framework for assessing the importance of nodes in a Boolean network with respect to a given target node. The framework comprises two complementary measures: the Knock-out and the Knock-in Shapley values. Moreover, we present a propagation-based method that enables their efficient computation. By exploiting the logical structure of the network, the method avoids exhaustive simulations. The approach is exact for acyclic networks and provides good approximations for cyclic networks. Evaluation on benchmark models from the Cell Collective database shows that the propagation method accurately recovers node importance rankings while achieving substantial speed-ups.

Gamma-ray astronomy from hundreds of GeV to PeV is confined to ground-based experiments that detect air showers induced by $γ$-rays entering Earth's atmosphere. While particle detector arrays feature huge detection areas, accurately reconstructing the primary particle properties is difficult due to the sparse sampling of the air shower and its intrinsic fluctuations. In this work, using simulations of a future water-Cherenkov array, we investigate two end-to-end deep learning approaches based on the transformer architecture with different computational complexities that utilize calibrated raw data. We benchmark both methods against well-established methods in the field in terms of $γ$-hadron separation, angular, core, and energy reconstruction. Our results show significant improvements across the whole energy range, particularly at low and intermediate energies. This work is the first to consistently demonstrate improved performance in both event reconstruction and $γ$-hadron separation using a single architecture.

Thin films of Cu-W-O ternary compounds were fabricated via DC magnetron co-sputtering from Cu and W metallic targets under controlled oxygen partial pressures, followed by thermal annealing. Low-oxygen conditions favored the formation of a single CuWO4 phase, whereas higher oxygen levels produced a mixture of CuWO4 and Cu3WO6. Structural and optical properties were investigated by X-ray diffraction (XRD) and spectrophotometry, revealing phase coexistence and changes in preferential orientation depending on the deposition conditions. A detailed and carefully validated X-ray photoelectron spectroscopy (XPS) analysis provides insight into the surface chemical environment of Cu and W, indicating the presence of compositional inhomogeneities and surface-bulk differences associated with Cu migration and segregation. While the W 4f core levels remain remarkably stable across all tested oxygen partial pressures, a systematic shift is observed in the Cu 2p3/2 binding energy. Wagner plot analysis confirms that this displacement is dominated by initial-state effects, reflecting modifications of the Cu ground-state electronic structure and Cu-O-W hybridization rather than changes in final-state screening. Our findings demonstrate that sputtered Cu-W-O films, even when nominally identified as CuWO4, can exhibit substantially different structural and electronic states depending on synthesis conditions, highlighting the need for rigorous characterization to ensure reproducibility in ternary oxide research.

''Little Red Dots'' (LRDs) are broad-line sources at high redshift, initially identified by their compact morphologies, red colours and prominent Balmer breaks. The origin of their optical-to-near-infrared continua is debated, with proposed explanations ranging from direct recombination emission to thermalised blackbodies from stellar-like atmospheres. Here we report evidence for Paschen jumps in a subset of LRDs, consistent with free-bound recombination to hydrogen $n=3$. The Paschen and Brackett continuum shapes across the sample are consistent with minimally reddened emission from low-temperature gas with $T_e\lesssim10\,000$ K, while the presence of Paschen jump signatures limits scenarios in which the emission is thermalised. Further, the extreme H$α$ equivalent widths and the tight observed correlation between H$α$ and the continuum follow naturally if both originate in recombination emission. This provides an observational upper limit on the contribution of any direct AGN accretion component and any stellar-atmosphere-like component, as well as on the fraction of line emission that can be thermalised as it traverses the cocoon. Ultimately, nebular radiative-transfer models provide a self-consistent explanation of the continuum, line strengths and line profiles without requiring multiple separately fitted components.

The calculation of the decay rate of a metastable state in the path-integral formulation of stochastic processes is revisited. Previous derivations of this rate were achieved at the cost of a step that is difficult to justify mathematically. We show that this difficulty can be resolved by working with an extremal solution that arises naturally in the Ito formulation of the path integral. To make the analysis as transparent as possible, we choose a simple potential for which the extremal solution can be written in terms of elementary functions. The mechanism identified here, however, is not restricted to this example and holds more generally.