Papers

Complex systems, from gene regulatory networks to neural circuits and ecological food webs, exhibit rich functional behaviour that topology alone does not capture. Yet functional complexity remains difficult to quantify independently of structural organisation. Here we introduce a thermodynamic framework in which functional complexity is characterised through the hierarchical organisation of functional memory, quantifying how the influence of past interactions is distributed and progressively compressed across scales. Across thirty-four empirical networks spanning biological, ecological, social, technological, and biophysical systems and several orders of magnitude in size and density, real interaction strengths organise functional memory at greater hierarchical depth than random weight assignment on the same topology in every domain studied. The framework further reveals that functional memory occupies a remarkably low-dimensional space, collapsing onto four recurrent dynamical organisations. Comparisons with null models that selectively perturb weighted transport geometry, mesoscale wiring, and directionality show that these structural ingredients play distinct roles: weighted transport geometry systematically governs memory depth, whereas mesoscale wiring organises memory across scales and directionality modulates the response of the cascade to structural perturbation. The same comparisons provide an operational criterion for determining whether network weights encode functionally meaningful structure beyond topology. These results establish weighted transport geometry as a primary organiser of functional memory and provide a quantitative framework for studying functional complexity in directed weighted networks.

It is well known that $F(x)=\prod_{n=0}^{\infty}(1-x^{2^n})$ is the generating function of the Prouhet-Thue-Morse sequence $\{(-1)^{\sigma_2(n)}\}_{n=0}^\infty$, where $\sigma_2(n)$ is the sum of (binary) digits of $n$. Let $m$ be an integer. In 2018, Gawron, Miska and Ulas initiated the study of arithmetic properties of power series expansion of the function $$F_m(x)=F(x)^m=\sum_{n=0}^{\infty}t_m(n) x^n,$$ and proposed a conjecture stating that for any given integer $m\ge 2$, the sequence $\{t_m(n)\}_{n=0}^{\infty}$ is unbounded. In this paper, we introduce a new method to investigate this conjecture. In fact, by making use of algebraic, $p$-adic and analytic methods, we show that the Gawron-Miska-Ulas conjecture is true.

A method for testing the hypothesis of an ultrametric organization of the energy landscape of RNA secondary structures is proposed, based on the analysis of the kinetics of transitions between basins of attraction. The method is based on a kinetic metric constructed from the spectral decomposition of the symmetrized Kramers transition rate matrix and the Mahalanobis distance, and it is not an ultrametric by construction. A computational scheme has been developed that includes automatic filtering of noise eigenmodes and a procedure for analyzing disconnected structure graphs. The performance of the method is demonstrated on a sample of reference and random RNAs.

Motivated by sequence reconstruction and reconstruction codes, we study extremal intersections of deletion balls over a fixed $q$-ary alphabet. Let $\Sigma_q^n$ be the set of sequences of length $n$ over $\Sigma_q$, and let $D_t(x)$ denote the set of all sequences obtained from $x\in\Sigma_q^n$ by deleting exactly $t$ symbols. Our first result gives a finite upper bound under a lower-order deletion-correction constraint. We prove that if $x,y\in\Sigma_q^n$ satisfy $D_{s-1}(x)\cap D_{s-1}(y)=\varnothing$, then \[ |D_t(x)\cap D_t(y)| \le \binom{2s}{s}\binom{n-s}{t-s}. \] For binary alphabets, this strengthens a recent asymptotic upper bound of Pham, Goyal, and Kiah (2025, JCTA). We then investigate deletion-ball intersections under simultaneous constraints on run counts and lower-order deletion-ball intersections. For fixed $0<\gamma\le1$, integers $1\le s\le t$, and $m\ge1$, we show that if $x,y\in\Sigma_q^n$ have at most $\gamma n$ runs and satisfy $|D_s(x)\cap D_s(y)|\le m$, then \[ |D_t(x)\cap D_t(y)|\le \frac{m\gamma^{t-s}}{(t-s)!}n^{t-s}+O_{s,t,m}(n^{t-s-1}). \] Moreover, the leading term can be attainable whenever $m$ is realized by a fixed finite-length seed pair. As a consequence, we obtain a direct lifting theorem for deletion reconstruction codes, transferring reconstruction properties from radius $s$ to larger radii $t$. Finally, we establish a parallel insertion theory and derive corresponding results for insertion-ball intersections and insertion reconstruction codes.

We present an operator learning framework based on a coarse data-driven multiscale space for nonlinear flow in random heterogeneous porous media. The multiscale space is constructed from local representative fine-scale solution snapshots, yielding an accurate low-dimensional representation of the solution manifold. This multiscale basis serves as the trunk of a neural operator, while a branch network predicts the corresponding reduced coefficients from the input permeability field. Unlike Galerkin projection methods, the neural operator learns a global nonlinear mapping from permeability fields to solution coefficients, providing greater flexibility, improved accuracy, and eliminating the need for online nonlinear coarse-grid solves and coefficient evaluations. Numerical results show that the proposed approach achieves good accuracy and substantially lower computational cost than projection-based methods for nonlinear flow in high-contrast heterogeneous media.

Let Q_n = B_n + J_n be the quadratic covariation matrix of a high-dimensional semimartingale, where J_n is the jump component and B_n is the diffusion component. We prove that spectral collapse occurs – meaning the ratio of the leading eigenvalue to the trace converges to 1 and the effective rank converges to 1 – if and only if the jump directions are geometrically aligned in a weighted sense and the background diffusion is asymptotically negligible. The proof separates into two steps: geometric alignment of jump directions forces spectral concentration of J_n; background negligibility then propagates this to the full system. We extend to the stochastic setting and prove convergence in probability under natural conditions on the jump process. The framework gives a scalar diagnostic for detecting when a high-dimensional system is dominated by extreme events.

We propose a predict-optimize-explain framework that uses gradient-based sample generation to interpret various portfolio models by identifying macroeconomic conditions that induce specified portfolio outcomes. Unlike traditional feature-importance methods, this approach directly probes decision pipelines (predictive models coupled with portfolio optimization) by constructing economically meaningful what-if questions. We focus on four such questions: under what macroeconomic conditions a predict-then-optimize pipeline closes or reverses its return gap with a predict-and-optimize pipeline; what conditions lead a pipeline to diversify rather than concentrate its allocation; when a pipeline trained on calm markets overtakes one trained through crises; and what conditions would let a pipeline match a benchmark return. These examples illustrate how our framework uncovers key behavioral differences between various decision pipelines. Beyond these cases, the proposed framework is flexible and can support a wide range of probing questions tailored to specific portfolio objectives. Our findings highlight the value of integrating prediction, optimization, and explanation to produce more robust and transparent portfolio strategies.

We establish global-in-time Strichartz estimates without loss of derivatives for wave equations with time-dependent Lipschitz coefficients, which satisfy an additional structural assumption. Our approach is based on a parametrix construction through the Phillips functional calculus. We furthermore obtain the well-posedness of such wave equations with Lipschitz coefficients in $H^1$.

We study an implicit iteration for approximating common fixed points of strongly continuous semigroups on closed convex subsets of real Banach spaces. Under a stepwise Lipschitz restriction and hemi-contractivity relative to the common fixed point set, we establish well-definedness and Fej% ér monotonicity of the iteration. In reflexive Banach spaces satisfying Opial's condition, we obtain a sufficient condition for weak convergence based on a normalized residual estimate, uniform continuity at zero along the generated sequence, and demiclosedness at fixed positive times. In general real Banach spaces, we prove strong convergence under a time-dependent directional contraction condition and a divergence condition coupling the iteration weights with the contraction strength. A linear example shows that vanishing time steps and a divergent sum of iteration weights alone do not ensure convergence to a common fixed point. A nonlinear planar evolution semigroup illustrates the applicability of the strong convergence theorem beyond the nonexpansive class.

Inferring thermal fields and thermophysical properties from limited measurements is a fundamental challenge in micro- and nanoscale heat conduction, where the classical Fourier law breaks down and the phonon Boltzmann transport equation (BTE) is needed to capture non-diffusive transport effects. In this work, we extend Monte Carlo physics-informed neural networks (MC-PINNs), originally developed for forward phonon BTE problems [J. Comput. Phys. 542, 114364, 2025], to inverse multiscale heat conduction problems. Two representative classes of inverse problems are considered: (i) reconstructing the full thermal field from sparse interior temperature measurements when boundary conditions are unknown, and (ii) simultaneously inferring the unknown relaxation time together with the thermal field. Problem-specific MC-PINN architectures and training strategies are designed for each class. The mesh-free Monte Carlo sampling strategy enables a unified treatment across diffusive, transitional, and ballistic transport regimes without requiring a priori knowledge of the relaxation time. The proposed method is evaluated on quasi-one-dimensional, quasi-two-dimensional, and three-dimensional benchmark problems covering a wide range of Knudsen numbers, as well as on a realistic 3D fin field-effect transistor (FinFET) structure. Results demonstrate that MC-PINNs consistently outperform purely data-driven deep neural networks, particularly in the sparse-data regime, and can accurately infer spatially uniform relaxation times. For spatially varying relaxation times, the inferred distributions capture the dominant thermal response, and numerical simulations using the recovered parameters reproduce the macroscopic fields with good accuracy. These findings establish MC-PINNs as an effective and physically consistent framework for inverse thermal analysis at micro- and nanoscales.

In this paper, we show that every biharmonic Wintgen ideal submanifold of dimension at least three or dimension two with codimension at most two in a Riemannian manifold of nonpositive constant sectional curvature is minimal. Under the same dimensional assumption, we also prove that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature on each connected component. This gives partial affirmative answers to Chen's conjecture, to the generalized Chen's conjecture in hyperbolic spaces, and to the Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.

Many statistical models have natural symmetries described by a group action. We study how such symmetries affect the comparison of two distributions. We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every $f$-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher–Rao distance.

For graphs $H$ and $F$, the generalized Turán number $\operatorname{ex}(n,H,F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We prove a general lifting principle for the case $H=K_3$ and the forbidden graph is a vertex-disjoint union of several copies of a graph. The key hypothesis is a local neighborhood-forcing condition: there is a graph $R$ with $\operatorname{ex}(n,R)=o(n^2)$ such that $F\subseteq K_1\nabla R$. Under this condition, the corresponding single-forbidden-graph asymptotics, together with a construction attaining the relevant extremal triangle and edge densities simultaneously, lift to an asymptotic value for \(\operatorname{ex}(n,K_3,(s+1)F)\) for every integer \(s \). We also prove an exact version in terms of the maximum value of a weighted expression over all graphs of a given size that avoid the forbidden graph. As applications, we obtain exact or asymptotic results for disjoint unions of suspensions of paths and stars. We also recover known exact results for disjoint odd cycles.

A cross-diffusion system with advection is considered on the whole line, describing the dynamics of two segregating population species. Starting from two initial densities supported on the half-lines $x\leq 0$ and $x\geq 0$, respectively, we construct two distinct solutions of the system: one pair of densities remains confined to their initial supports and stay completely segregated, while the second pair of densities begins to invade the opposite half-line after a finite time (mixing). To the best of our knowledge, this provides one of the first examples of non-uniqueness for this class of equations, together with an explicit demonstration of mixing phenomena. The construction produces infinitely many mixing solutions and applies throughout the full range of pressure exponents. For a certain range of exponents, we additionally obtain quantitative estimates on the mixing process.

We study a one-dimensional cross-diffusion system of two populations. Their densities are diffused with a common pressure that depends on the total density, but are transported by two independent external potentials. Starting from segregated initial data with a single interface, we show that the way the two densities meet is not given by the equation alone: it depends on the notion of solution one chooses. When the drifts push the two phases towards each other at the interface, the vanishing-viscosity solution creates an overlap, whereas a segregated weak solution also exists. The two solutions are distinct, so the Cauchy problem is not well posed in the class of weak solutions. We first prove the result for an explicit stationary segregated solution, and then extend the non-uniqueness to general segregated data.

In this paper, we present the construction of equilibrium states for a gas of weakly interacting non-relativistic bosons, focusing on the case of a non-trivial background field in infinitely extended space. Building upon a method introduced by Araki and further developed by Fredenhagen and Lindner, we derive the generating function of the correlation functions of the theory as a suitable series. By applying a Hubbard-Stratonovich transformation, we rewrite this quantity into a more mathematically tractable form, allowing us to establish the convergence of the corresponding loop vertex expansion in certain intermediate regimes. Furthermore, we isolate the tree diagrams that produce the scattering length in the dispersion relations of the two-point function of the state within the Gross-Pitaevskii regime. Finally, we use this scattering length to renormalise the background and the two-point function of the fluctuations and we discuss convergence of the generating function of the connected correlation functions of the renormalised theory in the limit of vanishing temperature.

We introduce an estimator for the pointwise, time-varying Hölder exponent (Hurst parameter) of a stochastic process, based on the geometry accumulation integral G_Lambda(t) = integral from Lambda to 1 of |eth_s X(t)| s^{-1} ds, where eth_s X(t) = (X(t+s)-X(t))/s is the scale derivative at resolution s. We prove consistency, noise robustness with explicit threshold Lambda* = sigma^{1/H}, and a CLT at rate (log Lambda)^{-1/2}. The estimator is pointwise in time, defined at finite resolution, and eliminates microstructure noise by scale separation. Existing estimators give a global H from integrated variance; this gives a time-varying H(t) directly from the price path.

In the three-rotor problem, three equally massive point particles move on a circle interacting via attractive pairwise cosine potentials. Rotors can represent superconducting phases of metallic segments in a chain of coupled Josephson junctions. We propose a digitization of the classical dynamics that records successive pair and triple coincidences of rotors using four symbols. Rotor coincidences correspond to boundaries in a disjoint partition of the configuration torus into cells where the rotors are ordered clockwise and anticlockwise. It is shown that isolated rotor coincidences must be crossings. Despite being a rather coarse digitization, we find that replacing trajectories by coincidence symbol sequences captures significant qualitative features of the dynamics through word statistics. Word-complexity $C_n$ measures the diversity of $n$-letter words in the symbol sequence while topological entropy governs asymptotic exponential growth of $C_n$. Sequences from periodic orbits have a word-complexity that saturates at the period. Ultra-high-energy trajectories with irrational `slope' are quasiperiodic. We show that they have zero entropy and $C_n = n+3$ by examining limiting slopes and by a mapping to Sturmian sequences. We examine their grammar rules and propose how their right-special words may be identified. On the other hand, numerical investigation of sequences from globally chaotic orbits leads us to conjecture an exponentially growing word-complexity $C_n = 3 \times 2^{n-1}$ with a topological entropy $\log 2$. We identify grammar rules that imply this conjectured word-complexity and model the resulting sequences as a subshift of finite type, unlike those at ultra-high energies which are not topological Markov chains.

We extend the notion of an inexact first-order oracle from the Euclidean setting to Riemannian optimization, and conduct a convergence analysis for the Riemannian proximal gradient method equipped with this oracle, which we refer to as RPG-IO. Under mild conditions on the oracle errors, we establish the global convergence of RPG-IO. Specifically, we prove that (i) the norm of the search direction converges to zero; (ii) the sequence of function values converges to the function value at any accumulation point of the iterates; and (iii) every accumulation point of the iterates is a stationary point. Under the additional assumption of the Riemannian Kurdyka–Łojasiewicz (KL) property, we prove that the full sequence of iterates generated by RPG-IO converges to a single stationary point. Moreover, we derive explicit convergence rates when the KL exponent is specified. Finally, when a strong inexact oracle is employed, we establish the convergence rate of the sequence of function values to the optimal value.

When gradient information is unavailable, black-box optimization (BBO) methods provide a practical alternative. While Evolution Strategies (ES), Consensus-Based Optimization (CBO), Optimization via Integration (OVI), and related methods have each been studied independently, their connections remain underexplored. We unify these approaches within a common theoretical framework, revealing that they differ primarily in two design choices: fitness aggregation (controlling sharpness preference) and consensus scope (controlling modality). Leveraging these insights, we introduce hybrid optimizers that interpolate between existing methods. Our ES-OVI hybrid allows explicit control over the preference for flat minima, enabling a trade-off between performance and robustness in continuous control tasks. Our CBO-OVI hybrids combine the higher-dimensional efficiency of parametric methods with the multimodal capabilities of particle-based approaches, achieving competitive results on language model merging under limited evaluation budgets. We validate our methods on standard BBO benchmarks and higher-dimensional locomotion tasks, demonstrating that the hybrid methods can outperform their constituent algorithms.

We adapt Sbierski's proof of $C^0$-inextendibility of the maximal analytic Schwarzschild spacetime to a broad class of warped-product black hole spacetimes with a static exterior region. These spacetimes are globally hyperbolic, have a codimension-two Riemannian fibre and a radial coordinate $(r)$, which serves as the warping function of the fibre. They admit a spacetime singularity as $r \to 0$, characterised by the divergence of the Kretschmann scalar. This class encompasses nonvacuum black hole models and geometries beyond spherical symmetry. Under suitable assumptions, including that the fibre is closed (compact without boundary), connected, homogeneous, and orientable, we establish future $C^0$-inextendibility for spacetimes in this class. The result further extends to spacetimes possessing more than one regular black hole horizon.

Gradient-based inverse lithography technology~(ILT) for extreme ultraviolet~(EUV) masks is presented. A novel framework treats the differentiable waveguide method and the recently proposed waveguide neural operator~(WGNO) as end-to-end physics engines, recovering the permittivity of the absorber of the mask through automatic differentiation of the full forward diffraction model. Numerical experiments on realistic 2D and 3D absorbers of the mask (TaBN, La, U) at $\lambda{=}11.2$~nm show that the considered ILT methods make it possible to obtain a mask structure that achieves the desired field on the wafer.

A 2+1-dimensional extension of the solitonic Dym equation is shown to admit a Painlevé II symmetry reduction which permits the exact solution of a class of Stefan-type moving boundary problems.

In this paper, we initiate Iwasawa theory for K3 surfaces over finite fields. First, using the Artin-Tate conjecture, which is known to hold for K3 surfaces, we prove an analogue of Mazur's control theorem for elliptic curves over number fields. Second, we prove an analogue of Iwasawa's class number formula for Brauer groups in two different ways. We also give explicit examples in the case of Kummer surfaces. Finally, we establish an analogue of the Iwasawa main conjecture for Brauer groups.

We present a Pontryagin-based numerical solver for deterministic optimal control problems in Bolza form. The solver regularizes the generally nonsmooth Hamiltonian using a log-sum-exp smoothing of a concave piecewise-affine bundle surrogate, yielding a smoothed Hamiltonian $H_\delta$ that is $C^\infty$ and concave in the costate and $C^{1,1}$ in the state. The resulting two-point boundary value problem is discretized by a symplectic Euler scheme and solved by damped Newton iteration on the full-space nonlinear system in the discrete state and costate variables. A unified adaptive outer loop jointly controls the time-step distribution $\Delta t$, the number of planes in the bundle surrogate, and the smoothing parameter $\delta$. The time-adaptation strategy follows the error-density framework of Karlsson, Larsson, Sandberg, and Tempone (2015). Related approaches include Pontryagin shooting, Hamilton–Jacobi–Bellman methods, occupation-measure relaxations, max-plus approximations, and direct collocation. Numerical examples, including nonsmooth, singular, hypersensitive, and quadratic-programming-oracle benchmarks, demonstrate the roles of time refinement, bundle enrichment, and smoothing continuation. A linear–quadratic regulator with a closed-form Riccati solution is used as a calibration check.

Generalized posteriors replace the likelihood by an exponentiated empirical criterion, but their credible sets generally lack asymptotic justification for frequentist coverage. General posterior calibration selects a scalar learning rate by estimating coverage with the bootstrap. Using Edgeworth expansions under regular fixed-dimensional asymptotics, we derive higher-order coverage expansions and analyze the stochastic approximation step used in the implemented algorithm. For a fixed nominal level, the root of the bootstrap coverage equation is consistent under a uniform coverage approximation and local identification. The higher-order expansions separate two sources of coverage error: the sampling Edgeworth correction for the estimator and the posterior Edgeworth correction for credible set boundaries, centres, and shapes. A scalar learning rate can calibrate all nominal levels in the Gaussian limit only when the posterior covariance and the sampling covariance are proportional. Hence, bootstrap calibration is a level-specific scale correction, not a remedy for general shape misspecification.

Let $K$ be a finite extension of $\mathbb{Q}_p$. We prove that a one-dimensional formal group law over $K$ has integral coefficients if and only if its multiplication-by-$n$ endomorphisms have integral coefficients for all integers $n$, in the height-one case, that is when the multiplication by $p$ has Weierstrass degree $p$. The proof relies on $p$-adic Hodge theory.

We classify all possible finite groups acting faithfully on the superspecial $K3$ surface which contain the maximal symplectic groups, as classified by Ohashi and Schütt.

We propose a task-oriented multiuser wireless communication framework for distributed classification based on a MAP-driven system design under wireless channel impairments. \textcolor{black}{By deriving tractable approximations of the MAP error bound}, the proposed approach enables the design of learning-based feature extraction and precoding strategies. Unlike existing approaches that optimize intermediate reconstruction, information-theoretic, or feature-separability objectives, the proposed formulation directly optimizes objectives derived from the MAP decision error, thereby providing a more direct connection to the final classification performance. Simulation results demonstrate that the proposed schemes achieve higher classification accuracy than their counterparts, with lower or comparable computational complexity.

In applied health research, Markov cohort models are built on a precisely specified transition probability matrix. However, in many applications, the available evidence – transition counts, structural constraints, and treatment-effect data – identifies a set of admissible matrices rather than one uniquely justified matrix. This paper formulates an imprecise-probability extension in which inference yields lower and upper expectations over an evidence-compatible set of precise Markov cohort models. The contribution differs from existing imprecise Markov-chain work by focusing on finite-horizon cohort trajectories, additive accumulated outcomes, and transition matrices constructed from empirical transition counts. Under non-empty compact separately specified outgoing-row sets, the lower and upper accumulated outcomes are computed exactly by Bellman-style lower and upper transition operators. We prove the envelope theorem, reduction to the classical model, coherence properties of the lower transition operator, and algebraic conditions under which a single selected matrix yields a non-robust decision. We then show how multinomial transition counts induce admissible matrix sets through the Imprecise Dirichlet Model. A real-world cost-effectiveness example of patent foramen ovale closure after cryptogenic stroke illustrates the practical consequence: the empirical transition matrix slightly favors closure, whereas the imprecise analysis yields an incremental net monetary benefit interval crossing zero. The method provides both a rigorous lower-expectation formulation and a practical diagnostic for decisions that depend on transition probabilities not fully resolved by the evidence.