Papers

Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.

Two-band superconductors maintain a relative interband phase which can carry winding soliton excitations in a superconducting ring, supported by independent winding numbers in the two bands. In rings of superconductors obeying the inversion symmetry the interband phase solitons are metastable states, separated from the uniform ground state by the energy of screening currents. In this work we find that, breaking the inversion symmetry strongly enough, one can make the soliton a true ground state. In that case, a magneto-electric coupling, absent in centrosymmetric materials, contributes critically above a certain threshold, a relevant, free-energy term, odd with respect to the winding number, which biases the energy balance in favor of a particular winding sign. Once the bias outweighs the energy cost that originally made the soliton metastable, a phase soliton with a finite winding number becomes the ground state, with chirality set by the applied field. In current–flux measurements performed in equilibrium states, the effect is demonstrated by field-odd soliton branches, that replace the metastable ones existing in mesoscopic rings, built by two-component superconductors, realizing a magneto-electric analog of the Little–Parks fluxoid-branch physics in the interband relative-phase sector.

We demonstrate a physical mechanism for causal information filtering in a physical reservoir computing (PRC) by exploiting asymmetric coupling anisotropy. Using a network of coupled Duffing oscillators, we show that the directionality of internal coupling induces a spatial gradient in the effective potential, establishing a deterministic upstream-to-downstream information flow. This anisotropy allows for the selective amplification of semantic drifts, triggering a macroscopic saddle-node bifurcation as a physical interlock before global computational failure. Through spatiotemporal analysis of a 50-node system under traveling wave inputs, we confirm that local phase transitions effectively purge anomalous information while preserving the computational integrity of the remaining nodes. The results suggest that the intrinsic causality of the reservoir's topology provides a robust framework for autonomous reliability and fault-tolerant physical intelligence.

Folding changes the linearized aeroelastic spectrum and can switch which mode becomes unstable first, with consequences for local postflutter interactions. This study formulates a three-component Z-fold wing by assigning every aerodynamic station to a structural component and a material coordinate. The same attachment map generates surface motion and returns pressure loads through virtual work. Geometrically exact component dynamics, an explicit-wake unsteady vortex-lattice model, and block-structured descriptor assembly preserve the physical paths of configuration actions. A two-parameter flutter analysis shows that a smooth flutter-speed envelope conceals a high-low-high sequence of controlling neutral branches, expressed as a critical-frequency valley and a redistribution of component deformation. Numerical continuation locates a near-1:2 double-Hopf point. Within a local model retaining quadratic and cubic structural restoring forces with aerodynamic and inertial operators fixed at the scheduling point, the cubic normal form captures selected 26-state observations and admits high-frequency-dominant and mixed phase-locked periodic solutions. The formulation links configuration-dependent flutter-mode identity to local resonant dynamics and supports blockwise sensitivity and design reasoning.

In swollen elastomers, elasticity can arrest macroscopic demixing between the polymer network and the solvent, producing stable domains of finite size. While previous theories have focused on isotropically swollen elastomers, anisotropy strongly influences the phase behavior of many related systems, such as gels and crosslinked polymer blends. Building on the approach developed in Part I of this two-part paper, we investigate the effects of two experimentally induced sources of anisotropy: uniaxial swelling and stiffness gradients. We show that uniaxial swelling can lead to the formation of stable lamellar microphases and modify phase behavior, while stiffness gradients can cause spatial variations in the characteristic microphase size.

Forecast-error growth has been studied through complementary approaches: scalar models describe the evolution of bulk error, whereas spectral theories follow decorrelation and error transfer across scales. Their connection matters when a fitted growth curve is used to infer a physical mechanism or a limit of predictability. We establish a conditional link between these approaches. Within a bilinear spectral exchange model, assume a stationary reference spectrum and a remainder that vanishes on the proposed common-shape evolution. For a common initial decorrelation fraction strictly between zero and one, the spectral shape is preserved and its amplitude grows logistically exactly when every participating scale shares one positive growth coefficient. The converse inference fails: a logistic aggregate need not imply proportional decorrelation or a common coefficient. An exact two-shell construction demonstrates this failure with identical aggregate curves and different spectral dynamics. For heterogeneous amplitudes, an exact averaging identity separates the contributions of growth-rate mismatch, shape variance, rate–shape covariance, evolving spectral weights, and residual tendencies. A finite-time bound relates their combined effect to departure from a reference logistic curve. Forced surface-quasigeostrophic simulations illustrate how similar retrospective agreement with a logistic reference can coexist with heterogeneous effective shell rates and substantial instantaneous tendency discrepancies. Thus a scalar growth curve alone cannot identify how uncertainty is distributed across scales or justify interpreting a fitted rate as a common spectral growth coefficient. The framework specifies the additional spectral and budget evidence needed to give scalar predictability models that interpretation.

Mixed-mode oscillations (MMOs), characterized by the alternation of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and bursting oscillations are common forms of complex oscillatory dynamics observed in systems with multiple timescales and have been widely studied across scientific disciplines. Mixed-mode bursting oscillations (MMBOs) combine features of MMOs and bursting, with LAOs organized into burst events. Most existing studies treat MMBOs as two-timescale phenomena, identifying distinct geometric mechanisms depending on how the timescales are grouped. In this work, we use a three-timescale biophysical cortical neuronal oscillator model to demonstrate that a three-timescale implementation of geometric singular perturbation theory (GSPT) provides stronger predictive insight into MMBO dynamics. This perspective unifies mechanisms previously identified from fast-slow analysis, while revealing the canard-delayed-Hopf (CDH) singularity as an organizing center for MMBOs near the singular limit. We perform a detailed bifurcation analysis of the full eight-dimensional model to determine how MMBOs are organized along families of isolas. We then combine GSPT with full-system bifurcation analysis to show how tuning the relative timescales induces transitions among MMOs, MMBOs, and bursting dynamics. Our results highlight the importance of studying MMBOs from a three-timescale perspective and demonstrate that combining GSPT with full-system bifurcation analysis can reveal organizing structures and mechanisms that are not apparent from either approach alone.

Large natural earthquakes are typically mediated by frictional pulse-like rupture, which features a finite slipping zone. Recently, a comprehensive two-dimensional theory of frictional slip pulses has been developed. It predicts that pulses are categorically unstable rupture modes, whose evolution is intrinsically slow. Unsteady pulses satisfy an equation of motion expressed in terms of discrete observables, which is inherently related to their steady-state counterparts. The theory also predicts a transition from decaying to slowly growing pulses. Here, we perform extensive lab earthquake experiments to test the theory. The experiments confirm the theoretical predictions for pulse-like lab earthquakes over a range of prestress levels, rupture nucleation conditions and small-scale fault roughness amplitudes. Specifically, the predicted time-dependent dynamics are tested in a plane defined by the evolving pulse size and peak slip rate, along with measurements of the dimensionless growth rate of pulses, demonstrating their intrinsically slow unsteady nature. The predicted transition between decaying and growing pulses is also experimentally demonstrated. These results constitute major progress in understanding a dominant earthquake rupture mode.

We introduce a numerical direct scattering transform scheme for dark solitons of the nonlinear Schrodinger equation, enabling the identification and complete characterization of nonlinear coherent structures in defocusing media with a continuous-wave (CW) background. Our scheme is based on numerically solving the auxiliary Zakharov-Shabat scattering problem with CW boundary conditions and on analytically derived expressions that relate the elements of the transfer matrix to the scattering data for dark solitons and continuous-spectrum waves. To test our approach, we consider two analytically solvable cases of the scattering problem: i) rectangular, and ii) hyperbolic tangent hollows in the CW background, which can contain an arbitrary number of dark solitons, with known scattering data. We revisit the analytical derivations and obtain a complete set of soliton parameters represented by discrete eigenvalues and norming constants. By supplementing the direct scattering transform algorithm with high-precision arithmetic to accurately recover soliton norming constants, we provide a robust method to analyze data from numerical or natural experiments on complex wave fields in optical, hydrodynamical, and other physical systems.

In the Derrida–Higgs model of sympatric speciation, a sexually reproducing population with finite binary genomes can mate only when the genetic overlap between two individuals exceeds a threshold $q_{\min}$. Depending on the genome length $L$, the population either remains genetically connected or fragments into reproductively isolated species. A central problem is therefore to predict the critical genome length $L_c$ at which fragmentation begins. At finite $L$, fluctuations broaden the overlap distribution and allow genetically distant regions of the population to remain connected. A previously proposed transient variance criterion captures this effect, but its evaluation requires numerical iteration of coupled moment equations. Here we first correct the unrestricted moment equations by removing a previously implicit assumption and then derive an explicit closed form expression for $L_c$. The resulting formula shows that the critical genome length is determined, at the time the mean overlap reaches $q_{\min}$, by the competition between deterministic separation from the unrestricted equilibrium and the transient genealogical variance of the overlap distribution. This expression permits a systematic asymptotic analysis. When the deterministic contribution dominates, $L_c$ becomes essentially independent of the population size $M$ and scales as $\mu^{-2}$. When the transient genealogical variance dominates, $L_c$ grows as $M^{3/2}$ or as $\sqrt{M}/\mu$, depending on how $M$ and the mutation rate $\mu$ jointly vary. Simulations support all predicted behaviors. Our results identify transient genealogical variance as the principal mechanism linking finite genome fluctuations to the onset of reproductive fragmentation and provide a practical analytical prediction for the critical genome length.

We develop the Sato-theoretic dressing framework for the anti-self-dual Yang-Mills (ASDYM) hierarchy based on a normalized Riemann-Hilbert decomposition. A four-sector expansion of the generating function yields a bi-infinite matrix array of relative coordinates, extending the affine coordinates on the big cell of the Sato Grassmannian. We derive the Sato-Wilson equations, continuous coordinate flows, and their discrete analogues. Several classical integrable hierarchies are recovered under dimensional reduction constraints, with their nonlinear variables identified as specific relative coordinates.

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.

Using monthly Niño-3.4 anomalies through July 2026, we investigate how much predictive information is contained in delayed observations of the index. Ridge regression identifies informative delays, while multilayer perceptron and sparse identification of nonlinear dynamics (SINDy) models test whether nonlinear complexity provides additional direct forecast skill; gated recurrent unit (GRU) and long short-term memory (LSTM) networks provide a complementary test in which the temporal representation is learned internally. Delayed observations substantially improve forecasts over persistence and climatology at leads of up to six months, but increasing model complexity provides no systematic improvement. Historical recursive experiments favor a simple explicit SINDy recurrence and select shallow recurrent architectures, with no appreciable gain from learning the temporal representation internally. These results support a compact predictive representation of Niño-3.4 evolution in which the representation of past information is more consequential than model complexity. As a prospective application, the selected models are used to forecast the developing 2026 event beyond the last available observation and to compare its predicted evolution with completed historical El Niño events.

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlevé V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlevé III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlevé III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

In this paper, we investigate the modified Toda (mToda) hierarchy, which can be regarded as the 2-component first modified Kadomtsev-Petviashvili (mKP) hierarchy. We first investigate the connection between the Toda and mToda tau functions. Based on this, we construct the transformations for the mToda tau functions and Lax operators. Furthermore, we present the mToda squared eigenfunction symmetries and derive the Adler-Shiota-van Moerbeke (ASvM) formula, which plays a crucial role by connecting the actions of the additional symmetries on the wave functions with the Sato–Bäcklund transformations of the tau functions. Finally, by establishing the equivalence between the actions of vertex operators on the mToda tau functions and the multi-step mToda transformations, we derive the mToda addition formulas, also known as the generalized Fay identities.

We predict a new type of stable three-dimensional (3D) vortex quantum droplets in binary Bose-Einstein condensate arranged into ring clusters that per-sistently rotate in the external potential. In contrast to clusters composed from localized quantum droplets, the states introduced here represent interacting vortex lines with identical topological charges nested in common 3D envelope and existing in much broader parameter range in comparison with local-ized quantum droplets. The intricate interplay between mean-field nonlinearity of Bose-Einstein condensate, quantum fluctuations described by Lee-Huang-Yang correction, Coriolis force arising due to rotation, and external potential leads to substantial variations of cluster shape upon increase of its rotation frequency. Rotating vortex clusters bifurcate from single-vortex quantum droplets and exist only above critical value of the rotation frequency that decreases with increase of chemical potential and depends on the number of vortex lines in the cluster. The radius of the cluster decreases with increase of rotation frequency and weakly varies with chemical potential, which determines mostly localization of the individual vortices in cluster and overall width of its envelope. Vortex droplet clusters are very robust objects existing in stable form in wide intervals of the number of particles for any number (odd or even) of vortex lines forming them. Our results may open the route to observation of stable arrays of vortex lines of different configurations per-forming regular collective motion in condensate.

In oscillator networks sustained by finite resources, synchronization can depend on both the total resource and its spatial distribution. We study a duplex system whose activity layer consists of chaotic Hindmarsh–Rose oscillators and whose transport layer redistributes a conserved resource through a degree-biased Markov process. The stationary resource field is characterized analytically, and its existence, uniqueness, and convergence are established. By embedding this field into a local adaptive feedback law, the available resource is converted into node-dependent dissipation, for which Lyapunov analysis guarantees convergence to the synchronization manifold. Numerical results show that topology bias reorganizes the transient route to synchronization. Weak bias produces an almost collective contraction, whereas intermediate bias creates a hub-initiated recruitment hierarchy that extends toward middle-degree and peripheral nodes. Under stronger bias, the degree hierarchy becomes more pronounced while peripheral recruitment slows because adaptive dissipation is concentrated on structurally privileged nodes. Across the explored parameter range, this localization–coverage tradeoff is accompanied by a non-monotonic synchronization response at fixed total resource. The largest Lyapunov exponent remains positive after synchronization and the correlation dimension changes only modestly, consistent with suppression of transverse deviations while chaotic motion is retained on the synchronization manifold.

Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.

The Atlantic Meridional Overturning Circulation (AMOC) carries multi-decadal memory that ensemble-mean risk projections do not capture. We quantify this memory and its consequences for tail risk with a Volterra integral framework applied to ensemble-mean-subtracted variability in 14 CMIP6 models (1850-2100, three SSP scenarios), so that results reflect intrinsic thermohaline memory rather than the common anthropogenic forcing trend. Four results follow from leave-one-out cross-validation and an annealed-quenched tail decomposition. First, the ensemble-mean DFA1 Hurst exponent is H = 0.781 +/- 0.219, with 12 of 14 models showing long-range dependence (H > 0.5), consistent with thermohaline adjustment timescales near 33 yr. Second, a first-order Volterra model reduces out-of-sample RMSE by 16.8% over the best autoregressive baseline and by 12.3% over an unconstrained 20-lag distributed baseline, both robust to 200 circular-shift placebos (p < 0.05): the physically motivated kernel shape carries genuine predictive advantage. Third, the rolling 30-year lower-tail frequency rises 1.9-3.1x above the historical baseline depending on scenario, and the memory amplification index exceeds 1 in 8 of 14 models under SSP5-8.5 (median 1.15): persistence-driven clustering of weak-AMOC states amplifies tail frequency beyond forcing-only projections. Fourth, model-specific optimal memory horizons (15-52 yr) correlate with thermohaline regime (r = 0.74, p < 0.01), and the ensemble-mean Hurst exponent already exceeds a detection threshold H* = 0.70 in 9 of 14 models, with theoretical lead times to near-tipping conditions of 10-35 yr depending on scenario. These results support trajectory-specific, memory-aware AMOC risk assessment under moderate-to-high forcing and a physically grounded, scenario-conditional early-warning framework.

We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form $S$-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang–Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik–Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational $SU(N)$, trigonometric $U_q(\widehat{\mathfrak{sl}}_2)$, and rational $O(N)$ scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.

Bright dissipative Kerr solitons generated in anomalous-dispersion microresonators underpin integrated optical frequency combs that have enabled applications including precision metrology, spectroscopy, coherent communications, and optical frequency synthesis. In contrast, Kerr resonators operating in the normal-dispersion regime can support dark solitons and platicons, but even the initiation of frequency comb generation typically requires auxiliary mechanisms, such as avoided mode crossings or pulsed pumping, to satisfy the phase-matching condition for four-wave mixing. Here we show that topological edge states in two-dimensional Kerr resonator lattices intrinsically satisfy this phase-matching condition, enabling the generation of optical frequency combs and novel coherent dissipative structures in the normal-dispersion regime without any auxiliary mechanisms. The resulting frequency combs self-organize into collective dark solitons that exhibit large-scale spatiotemporal synchronization across multiple resonators at the entire lattice edge. By tuning the pump detuning, we demonstrate the continuous evolution of topological dark solitons into topological platicons while preserving collective synchronization. Remarkably, we show that the topological lattice also supports collective bright-like soliton states despite all constituent resonators operating in the normal-dispersion regime. Our results establish topological Kerr resonator arrays as a versatile route for engineering a broad family of coherent dissipative structures, ranging from dark solitons to bright-like soliton states, within a single platform. They also provide a route toward integrated frequency-comb generation in material platforms and wavelength ranges where normal material dispersion has traditionally constrained comb generation in conventional single-resonator systems.

This work presents a new two-dimensional quantifier plane that combines normalized permutation entropy and normalized permutation mutual information, both derived from Shannon's information theory and normalized consistently through the embedding dimension of the Bandt and Pompe method. Each point on the plane simultaneously characterizes the intrinsic uncertainty of a variable and the amount of information it shares with another. We further show that a third quantity, equivalent to the conditional entropy of one variable given the other, introduces a notion of informational independence and, when both projections of a variable pair are considered jointly, reveals directionality between them. We analyze regular, chaotic, and stochastic dynamics to illustrate the relevance of the informational plane and show how these regimes evolve as a function of a coupling factor.

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart–Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

Cooperation is ubiquitous in both natural and human societies, yet its evolutionary basis remains a major challenge. A long-standing puzzle is whether having more information leads to better decision-making and thus a higher level of cooperation. To address this question, we adopt a recently developed reinforcement learning framework in which individuals learn through trial and error to maximize cumulative rewards - a paradigm that has successfully explained diverse emergent patterns in human behaviors. Specifically, we equip a structured population with the Q-learning algorithm and systematically vary the size of the interactive neighborhood, which serves as a proxy for perceived information. Interestingly, we observe a non-monotonic relationship between cooperation prevalence and neighborhood size in both two-dimensional square lattices and Barabasi-Albert scale-free networks. This inverted U-shaped dependence reveals that an optimal amount of information exists, yielding the highest level of cooperation. Mechanistic analyses show that a moderate neighborhood size enables individuals to strike an optimal balance between information sufficiency and decision-making tractability. This balance allows them to detect reciprocal opportunities while avoiding the deterioration of decision quality due to information overload. Our findings challenge everyday intuition, suggesting that a proper amount of information - not more - is optimal for the emergence of cooperation.

This work develops a discrete-time framework for identifying probability localization dynamics through finite stochastic representations adapted in space, time, memory, and state information. A compact dynamically relevant set is localized by a finite measurable partition, producing an observable probability state and a relational graph of admissible transitions. Structured stochastic liftings derived from Stochastically Structured Reservoir Computing (SSRC) give lossless polynomial representations of the observable state, while stochastic delay liftings add finite observable memory. These are distinguished from dynamically informed state-space enrichment: refinement of observational fibers containing states with the same present observation but different observable futures, yielding an exact obstruction-to-closure criterion. A route-network toy problem gives a minimal obstruction example, while four numerical laboratories (rotational phase dynamics, the chaotic logistic map, the Van der Pol oscillator, and a synthetic cyclic inventory system) show how spatial scale, temporal scale, polynomial degree, and delay depth interact. The logistic map isolates representation-induced memory in an otherwise Markovian chaotic system, using its exact invariant law as an ergodic benchmark and its zero-mass pseudospectrum to separate relaxation from transient amplification. An exact rotational cycle calibrates pseudospectra as a robustness diagnostic rather than a closure certificate. The inventory example gives a closure-driven enrichment procedure: residence-age hazards trigger age-refined states that improve predictive scores. These results motivate a minimal adequate representation: the least complex representation meeting predictive, structural, and identifiability requirements.

Zhou and Yu [Appl. Math. Lett. 178 (2026) 109937] studied the space-time shifted nonlocal nonlinear Schrödinger (NLS) equation $iq_t(x,t)=q_{xx}(x,t)-2\sigma q^2(x,t) \bar{q}(x_0-x,t_0+\lambda t),\,\, \lambda=\pm1$, and claimed that for both choices of $\lambda$ the equation is integrable. Here we examine this claim from the perspective of reductions of the standard integrable NLS system. We find that the compatibility of the equation with the NLS system is true only when there is no time reversal in the argument. In other words, for this equation it requires $\lambda=1$ and $t_0=0$. The case with $\lambda=-1$ is a valid nonlocal equation, but cannot be derived from the standard NLS system by the corresponding complex conjugate shifted nonlocal reduction, and its integrability therefore does not follow from that integrable NLS system. We also discuss the equivalence between shifted (with real shifts) and unshifted nonlocal reductions and give the adjusted form of the Hirota bilinear equation.

In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.

The Bandt-Pompe permutation entropy framework, alongside the complexity-entropy causality plane, has become a standard tool for characterizing the dynamical properties of time series. However, observational noise distorts ordinal pattern probability distributions in ways that can systematically misplace time series within the causality plane, compromising dynamical classification. This effect is particularly relevant for geophysical signals, which are typically poorly and irregularly sampled, and have a low signal-to-noise level. In this work, we characterize the distortions on ordinal pattern statistics using the formalism of majorization. We provide theoretical results and propose corrective strategies that restore discriminability under realistic measurement conditions. To achieve this, we introduce methodology that allows the characterization of noisy dynamical series and further allows the quantification of observational noise without the need of a fitting procedure. Finally, we illustrate our methodology by analyzing paleomagnetic records to determine if the geological evolution of the Earth dipole is better described by a stochastic or chaotic system.