Papers

Let $G = (V,E)$ be a finite directed graph with a non-negative real length $\mu_e$ assigned to every directed edge $e \in E$. We assume that $\mu_e = +\infty$ for every non-edge $e \not\in E$. Fix any two distinct vertices $a, b \in V$. A directed path from $a$ to $b$ is called an $(a,b)$-path. An edge $e$ is called an $(a,b)$-bridge if it belongs to all $(a,b)$-paths. Furthermore, it is not difficult to show that all $(a,b)$-paths pass all $(a,b)$-bridges in the same order. Define the distance $\mu(a,b)$ from $a$ to $b$ as the sum of lengths of all $(a,b)$-bridges. Furthermore, $\mu(a,b) = \infty$ if there are no $(a,b)$-paths and $\mu(a,b) = 0$ if $(a,b)$-paths exist but there are no $(a,b)$-bridges. It is easily seen that $\mu(a,b)$ can be computed in polynomial time and the metric inequality $\mu(a,b) \leq \mu(a,c) + \mu(c,b)$ holds for every $a,b,c \in V$. Furthermore, equality holds if and only if each $(a,b)$-bridge is either an $(a,c)$- or a $(c,b)$-bridge. \newline We will show that this is a special limit case $r=s \rightarrow 0$ of the inequality $\mu(a,b)^{s/r} \leq \mu(a,c)^{s/r} + \mu(c,b)^{s/r}$ obtained for all positive real parameters $r$ and $s$ in the paper “Metric and ultrametric inequalities for directed graphs”, Discrete Appl. Math. 314 (2022) 93–104, along with 3 other limit cases $r=s \rightarrow \infty$, $r=1, s \rightarrow \infty$, and $s = 1, r \rightarrow 0$, considered in that paper.

Swarmalators are mobile generalizations of phase oscillators. Introduced to model systems in which sync and self-assembly interact, they remain poorly understood theoretically. Unlike the Kuramoto model for coupled oscillators, existing swarmalator models lack a normal-form foundation, and their basic stabilities and bifurcations remain largely unsolved. Here we address both problems. Building on Tanaka's reduction of chemotactic oscillators, we show that the canonical one-dimensional swarmalator model – previously introduced as an ad hoc toy model – is recovered in the first-harmonic, zero-lag limit, implying its behavior is generic. We then derive the stability boundaries organizing its four collective states, show they meet at a single cusp, correct a previously published order-parameter formula, and uncover a non-monotonic sync response absent in the Kuramoto model.

Current electroencephalography (EEG)-based dream detection relies on power spectral density (PSD) and statistical moment features, achieving a state-of-the-art area under the receiver operating characteristic curve (AUC) of approximately 0.70 on the DREAM database (Wong et al., 2025, Nature Communications). We introduce PHINN-EEG (Persistent Homology Inspired Neural Network for EEG), the first topological time-series framework for dream mentation analysis. Using sliding-window Takens delay embeddings and Vietoris-Rips filtrations on multichannel pre-awakening EEG epochs, we extract Dynamic Betti Curves that characterize the geometric architecture of neural activity, not merely its energy. These topological invariants, combined with topology-conditioned flow matching, are analytically projected to outperform existing PSD and catch22 benchmarks, targeting AUC = 0.82-0.90 on the 1,462-awakening open-access subset of the DREAM database (drawn from a full registry of 3,191 total awakenings from 263 participants across 20 independent laboratories). We further introduce a topology-conditioned rectified flow model for dream-state EEG synthesis-with a spectral-conditioned flow model of comparable feature dimensionality as an additional ablation baseline to isolate the value of topological conditioning specifically-and propose a set of candidate Betti transition archetypes linking topology to phenomenological dream report categories, presented as an exploratory hypothesis space pending empirical validation. If validated, this work represents a paradigm shift from spectral energy to phase-space geometry in neural rare-event detection, with potential future implications for wearable BCI dream monitoring.

The Erdős–Pósa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erdős–Pósa property also holds for $H$-minor models of any planar graph $H$. Equivalently, if $G$ has no $k$ minor models of $H$ pairwise at distance at least 1 (i.e. disjoint), then one can remove $f(k,H)$ balls of radius 0 (i.e. vertices) to make the graph $H$-minor free. We show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős–Pósa property for planar minors. Namely, every graph $G$ which does not contain $k$ pairwise non-adjacent minor models of a planar graph $H$ (we say that $G$ is $kH$-free) can be made $H$-minor free by removing $f(k,H)$ neighborhoods. The proof relies on the fact that sparse $kH$-free graphs have linearly many independent large protrusions. The same method gives that sparse $kH$-free graphs can be made $H$-minor free by deleting $O(\log n)$ vertices (and thus have logarithmic tree-width). This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for $kH$-free graphs.

Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \citep{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theory and methodology at scale and under DAG misspecification. We then model a latent-collider DAG, a protein signalling network, and a multi-fidelity \mbox{heavy-ion} collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability over the simulator hierarchy.

We investigate the relationship between the weak-type regularity of the Bergman projection, $\Pi_{\Omega}$, of a simply connected domain $\Omega \subset \mathbb{C}$ and the boundary geometry of $\Omega$ in terms of a conformal map $\psi\colon\mathbb{D}\rightarrow\Omega$. We show that $\Pi_{\Omega}$ is of weak-type $(1,1)$ whenever $|\psi'|$ is in the Bekollé-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of $\Pi_{\Omega}$ when $1\leq p<\infty$, and establish sharpened sufficient conditions for the weak-type bounds when $p>1$. Our results follow from a reformulation in terms of mixed-weighted weak-type inequalities for $\Pi_{\mathbb{D}}$. We provide several applications.

Partial differential equations are a promising application area for fault-tolerant quantum algorithms, but the subject lies between two communities with different languages: numerical analysis and quantum computation. These lecture notes provide a numerically grounded introduction for readers entering from either field. Block encoding is the organizing principle: once a discretized differential operator is embedded in a unitary, primitives such as quantum singular value transformation, Hamiltonian simulation, linear combinations of unitaries, amplitude amplification, and measurement can be assembled into algorithms for elliptic, hyperbolic, and parabolic PDEs. Each chapter begins with a standard finite difference or finite element discretization and follows the full pipeline from the continuous PDE to quantum encoding, transformation, and extraction of a quantity of interest. Particular attention is paid to the factors governing end-to-end performance, including discretization error, state preparation, normalization, postselection, and measurement cost. A final chapter introduces nonlinear problems through Carleman and Koopman-von Neumann linearizations. The aim is not a comprehensive survey or a claim of universal quantum advantage, but a mathematically transparent entry point and a shared vocabulary for researchers in both communities.

On $\mathbb{X}$, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension $n \ge 3$, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions lying outside the $L^2$-spectrum of $\Delta$ and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of $\mathbb{X}$.

In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations $\rho_{p,g,\underline{\lambda}}\colon\mathrm{PMod} (\Sigma_{g,n})\longrightarrow \mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$ of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface $\Sigma_{g,n}$ of genus $g\geq 3$, any number $n\geq 0$ of punctures, and any coloration $\underline{\lambda}$ of the punctures, $\rho_{p,g,\underline{\lambda}}$ has dense image in the projective unitary group $\mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$, extending a landmark result of Larsen and Wang. Moreover, we show that the representations $\rho_{p,g,\underline{\lambda}}$ are surjective modulo any unramified maximal ideal of $\mathbb{Z}[\zeta_p]$, establishing an effective version of strong approximation for these representations. We also give several applications of our main results to residual finite simpleness of $\mathrm{PMod}(\Sigma_{g,n})$ (answering a question of Masbaum and Reid); to subnormal cores of some subgroups of $\mathrm{PMod}(\Sigma_{g,n})$; to realizability of congruence classes of quantum invariants; to embedding obstructions between $3$-manifolds; and to homological stability for mapping class groups with coefficients in $\mathrm{SO}(3)$-quantum representations.

We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.

We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set $\Gamma$ of equivalence classes of $C_0$-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric $d$: the distance between two classes is the convergence exponent of the series $\sum_j|\langle\varphi_j,\psi_j\rangle-1|$ formed from any pair of representatives. We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so $d$ separates points only after passing to the quotient $\widetilde\Gamma$ of $\Gamma$ by the relation $d=0$; the pair $(\widetilde\Gamma,d)$ is then a complete ultrametric space. As an application, we show that a product unitary $\bigotimes_j U$ whose factor $U$ satisfies $\inf_{\|x\|=1}|\langle x,Ux\rangle-1|>0$ (in particular, a unitary on a finite dimensional space with $1\notin\sigma(U)$) displaces every class to the maximal distance $1$. Guided by the intended application – a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds – we also develop a gauge-invariant variant $\tilde d$ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by $\tilde d$, is class dependent and realizes every value in $[0,1]$. We interpret $\tilde d$ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.

We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized planar Turán number $\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex planar graph that contains no graph $F \in \mathcal{F}$ as a subgraph. When only induced copies of $H$ are counted, we denote the corresponding generalized planar Turán number by $\mathrm{ex}_\mathcal{P}(n, H^{\mathrm{ind}}, \mathcal{F})$. Győri and Karim determined $\mathrm{ex}_\mathcal{P}(n, C_{5}, \{C_3\})$. In this paper, we determine the exact value of $\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})$ for every $k \ge 3$. Since all shorter odd cycles are forbidden, every $C_{2k+1}$ is induced. This problem is closely related to the inducibility of odd cycles in planar graphs. Ghosh, Győri, Janzer, Paulos, Salia and Zamora~(and independently Savery) determined the exact value of $\mathrm{ex}_\mathcal{P}(n, C_5^{\mathrm{ind}}, \emptyset)$. Moreover, they established a conjecture for all odd cycles $C_{2k+1}$ with $k \ge 3$. Our result confirms their conjecture under the additional assumption that all shorter odd cycles are forbidden.

In this note, we explore the connection between hyperkähler manifolds with large Picard numbers and abelian varieties. In particular, we are interested in Morrison's solution to the (modified) Oda's conjecture: every K3 surface whose Picard group is large enough (in a certain precise sense) must be related via an algebraic correspondence to an abelian surface. We generalize this theorem to the case of known examples of hyperkähler manifolds such as pointed Hilbert schemes on K3 surfaces.

We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of scaling (the natural parabolic scaling) and rotation about a given axis, at a constant angular speed $\alpha$ in self-similar time. For these so-called rotated self-similar solutions (RSS), we prove that if they satisfy a Type~I upper bound, and if the rotation parameter $\alpha$ is either too small, or too large, then they must be trivial. This Liouville-type result extends the classical works of Nečas-Růžička-Šverák ('96) and Tsai ('98), which only consider $\alpha=0$, to the case of similarity profiles which experience nontrivial rotation. Our results partially answer a question posed by Perelman. For backwards globally self-similar solutions which are invariant under the discrete action of scaling and rotation, the so-called rotated discretely self-similar solutions (RDSS), we obtain similar Liouville-type results under a Type~I upper bound, assuming extreme values of the rotation parameter $\alpha$, and if the scaling factor $\lambda$ is sufficiently close to $1$. We also establish a new regularity criterion for 3D Navier-Stokes which is local in nature: if the solution satisfies a Type~I upper bound in a unit parabolic cylinder, and there is a single time-slice at which the solution is locally approximately self-similar, then the top-center of the parabolic cylinder is a regular point of the Navier-Stokes flow. The proof of all these results rests on the introduction of a robust weighted-$L^2$ framework. In particular, our method is quantitative and is not sensitive to whether the Bernoulli head pressure satisfies a maximum principle, which was a key obstruction in previous works.

In this paper we provide extensions of the Birkhoff-Smale Homoclinic Theorem and the $\lambda$-Lemma for piecewise smooth vector fields free of sliding motion. We prove that if the vector field is $T$-periodic in its independent variable and has a transverse homoclinic point, then its time-$T$-map is conjugated with a Bernoulli Shift and thus the vector field is chaotic. The Bernoulli Shifts considered here may have any finite number of symbols, or countably infinite many.

The skew-gradient embedding (SGE) framework~\cite{GuWangSGE2025} reformulates a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution in a skew-symmetric operator. In a time-discrete scheme, the profiles defining this operator may be evaluated at previous time levels. The resulting operator remains skew-symmetric, so its contribution to the discrete energy balance vanishes; this explicit treatment often decouples multiphysics systems. We show that this operator is not unique: the admissible gauges form an affine space, and we call the resulting family generalized skew-gradient embeddings (GSGE). For any positive definite metric, least squares selects a unique minimum-Hilbert–Schmidt gauge, and the native metric recovers SGE. This construction also gives regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants. For rank-two gauges, we use a necessary and sufficient Jacobi criterion. Applying this criterion to a compatible MAC discretization of the incompressible Navier–Stokes equations gives a finite-dimensional rank-two Poisson–GENERIC formulation at the semi-discrete level; the implicit midpoint rule preserves this rank-two GENERIC structure at the fully discrete level and satisfies the exact discrete energy law. For the Cahn–Hilliard–Navier–Stokes system, the regularized GSGE–BDF2 scheme preserves mass, dissipates the discrete energy unconditionally, and admits a decoupled implementation.

A classical theorem of category theory says that given an adjunction between monoidal categories with lax monoidal right adjoint, there is an induced adjunction between categories of enriched categories. We extend this result to monoidal model categories (with some model categorical assumptions), improving on previous theorems which required the derived left adjoint to be strong monoidal. As an application, we consider the category of chain complexes over a varying groupoid of operators, equipped with two different monoidal structures: the standard tensor product, and the cartesian product. We show that both of these products yield monoidal model categories, and use our theorems to reproduce the square-zero extensions adjunction for augmented dg-categories.

Reed-Muller (RM) codes have been shown to achieve capacity over a range of channels, and recently proposed projection-aggregation (PA) decoding has been experimentally shown to achieve near-maximum-likelihood decoding performance. These recent achievements motivate theoretical research on PA decoding. In this work, we analyze the density function of the soft output from collapsed projection-aggregation (CPA) decoding for RM codes over the binary-input additive white Gaussian noise (BIAWGN) channel. We prove that soft-decision CPA decoding returns an exact marginal probability and is symmetric. Based on the analysis, we build a density evolution model for CPA decoding. To simplify the density evolution, we approximate the projection and the fast Hadamard transform decoding using hard-decision decoding. Simulation results over the BIAWGN channel show that our proposed density evolution model captures the fast reduction in the mean and the variance of the soft information returned from the CPA decoding, which qualitatively explains the decoding mechanism and the fast convergence speed of the CPA decoding. We perform an asymptotic analysis based on the proposed density evolution, and we show that CPA decoding can achieve a vanishing error probability for RM codes with a vanishing code rate.

We develop a new framework for quantum invariants of $3$-manifolds by extending to cobordisms a homological construction of mapping class group representations. More specifically, we construct a $(2+1)$-dimensional topological quantum field theory (TQFT) that assigns to each surface the twisted homology of its unordered configuration space. The construction requires a choice of local systems on configuration spaces together with additional data. We formulate sufficient conditions on these data that guarantee the TQFT axioms, and we show that there are at least two useful examples satisfying them. One of them yields a homological construction of the projective Kerler–Lyubashenko TQFT, while the other recovers the Frohman–Nicas–Donaldson TQFT. In contrast to the classical algebraic constructions of quantum invariants, our approach is purely topological and relies on multi-trajectory spaces of cobordisms.

We develop a monadic approach to actions of internal categories. Given an internal category, we construct a monad on a slice whose algebras are the actions of that internal category. Then, we give a complete characterization of the monads that arise in this way; we call them generalized action monads. Finally, we prove that these monads yield a well-behaved notion of generalized descent.

The problem of multi-snapshot spike deconvolution is studied, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. A variable-projection formulation is adopted, in which the amplitudes are eliminated in closed form, thereby reducing the task to a nonconvex least-squares problem over the spike locations alone. This formulation is referred to as the variable-projection formulation of spike deconvolution (VarProSD). An explicit characterization of the basin of convexity of the VarProSD objective is provided in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation affect the local geometry. Within this basin, consistency of the estimator in the number of snapshots is established under stochastic noise, and a complementary, sharper error bound is derived under adversarial noise through the local Lipschitz property of the inverse map. Local convergence guarantees for gradient descent are further established when initialization is performed within the basin. A central role throughout the analysis is played by Beurling–Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments are presented to corroborate the theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.

Let $H_a=-\Delta+a|x|^{-2}$ be the Friedrichs extension on $L^2(\mathbb{R}^d)$, where $d\ge 3$ and $-(d-2)^2/4\le a<0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\sigma=(d-2-\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\sigma$, then [ ||x|^{-\beta}H_a^{-s/2}f|{L^q} \lesssim ||x|^\alpha f|{L^p} ] holds for $1<p,q<\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. At either origin-critical boundary, the strong estimate and the corresponding weighted $L^p\to L^{q,\infty}$ estimate fail, whereas the Lorentz replacement $L^{p,1}\to L^{q,\infty}$ holds. We also derive weighted Sobolev consequences and treat the Hardy-critical Friedrichs case separately. No new heat-kernel, spectral multiplier, Bernstein, or Littlewood–Paley theorem is claimed.

As almost all models are wrong, a mean model's usefulness is often accepted as sufficient justification for its use. In practice, however, standard statistical theory breaks when the mean model fit is imperfect, limiting this usefulness. This tension between fit and usefulness arises from a dichotomization of model fit: the model is either right or it is wrong. Motivated by the linear regression framework, we propose an alternative viewpoint that leverages the mean model's usefulness without assuming it is right or wrong. We define a new model usefulness index and use it to share information across individual observations through the mean model. The result is an estimator of each outcome's mean that shrinks individualized means towards the shared mean model, with the degree of shrinkage governed by this usefulness index. We draw connections between our estimator and the James-Stein estimator and establish when and how our estimators of the individualized means yield more efficient inference than model-based and non-model-based alternatives. We also propose a data-dependent estimate of the usefulness index that balances statistical intuition with efficiency considerations. We illustrate our method's practical value in an analysis of personal tracker data.

In this note, we establish various probabilistic estimates for an interacting particle system that describes the evolution of the membrane potentials in a network of excitatory integrate-and-fire neurons, which are subject to both idiosyncratic and common noise. These estimates serve to support a separate work that studies the large population limit and the corresponding stochastic Fokker–Planck equation for the membrane potential density.

This study investigates the boundary conditions for fluid-dynamic equations at the interface of a gas and its condensed phase during steady evaporation of a polyatomic gas. Evaporation curves illustrating the dependence of the temperature and pressure ratios on the Mach number are considered for a calorically perfect gas whose molecules behave like rigid rotors. At the condensed phase, complete absorption conditions are assumed. Also an extension to cases in which a part of the molecules is diffusely reflected at the condensed phase is also considered. We revisit the half-space evaporation problem for polyatomic gases based on previous results, applied to a slightly modified entropy functional. The structure of the (modified) maximal total entropy production curves is investigated, and simple functions that accurately fit the numerical results are proposed. Functions of the proposed form, with modified coefficients, also fit the evaporation curves previously obtained by different numerical methods surprisingly well. The approximation is performed for different numbers of internal degrees of freedom or ratios of specific heats. Simple functions that depend on the ratio of specific heats are found to fit the obtained evaporation curves for different ratios of specific heats very well.

We present the H-cloud formalism, a cloud-accessible and open-source finite-element framework for electromagnetic modelling of applied superconductors. The proposed method expresses the nonlinear electromagnetic response of type-II superconductors in a curl-conforming discretisation based on Nédélec finite elements, where the tangential applied-field boundary condition, nonlinear E-J power law, and fully implicit time-discrete residual are stated explicitly at the variational level, all within a scripted Python finite-element workflow. The weak form is used as the basis for forward simulation and for extension to adjoint differentiation and PDE-constrained optimisation, while keeping the governing assumptions, boundary conditions, and solver structure fully visible to the user. The implementation is realised in Firedrake with UFL and PETSc-backed nonlinear solvers, allowing the identical script to run on local machines and in browser-accessible environments such as Google Colab without reformulating the problem. The method is verified on the canonical magnetisation benchmark of a cylindrical superconductor under Bean-like penetration conditions and then benchmarked against an independently constructed COMSOL model for a practical high temperature superconducting Bi2212 wire. Across matched mesh studies, the open-source workflow reproduces the commercial-reference magnetisation loops to within approximately \(1\%\) , with relative peak errors below 1.5%, while cloud execution preserves the same numerical solution at rather modest additional runtime considering the use of (freely available) reduced hardware resources. The proposed framework provides a rigorous, reproducible, and portable route for superconducting simulation, benchmarking, and future optimisation-led modelling of applied and functional superconductors, shareable and executable into open cloud environments.

Large-scale traffic assignment requires equilibrium models that are both behaviorally plausible and computationally tractable. This paper develops a perturbed utility Markovian equilibrium (PUME) framework that preserves the scalability of link-based Markovian traffic equilibrium models and extends their applicability to settings with boundary choice probabilities, undiscounted network loading, and general link interactions. As the behavioral basis of PUME, we first develop the perturbed utility Markovian choice model (PUMCM) in which the Bellman optimality operator is defined through a convex surplus function whose gradient directly yields the optimal policy. The model generalizes existing additive random utility (ARUM) Markovian choice models and admits both interior and boundary choice probabilities. Accordingly, unattractive links can receive zero flow without imposing ex ante choice-set restrictions as in existing ARUM models. We establish conditions under which the corresponding Markov decision problem is well posed and yields a proper demand mapping. We then formulate the equilibrium as a variational inequality (VI) problem on the dual cost space and establish its existence and uniqueness. Particularly, the VI formulation of PUME accommodates non-separable and asymmetric cost structures and thus offers a more flexible modeling framework than existing Markovian traffic equilibrium (MTE) models. For computation, we develop a modified policy iteration method for network loading and a safeguarded accelerated meta-algorithm for computing equilibrium. Both algorithms are proven to be globally convergent and have demonstrated satisfactory numerical performances. Experiments on benchmark and synthetic networks further show that the proposed framework is highly scalable and robust towards a wide variety of demand-supply settings.

Decision-making is posing an increasingly formidable challenge to investors because of the growing number of alternatives available in financial markets. A hot area of research over the past few decades has been portfolio optimization that seeks to determine how much an investor should invest in which asset. Introducing real-world conditions to the optimization model turns the problem into an NP-hard one for whose solution exact methods become inefficient; hence, researchers have turned to evolutionary algorithms to approximate solutions. In this paper, strengthening strategies are presented for multi-objective evolutionary algorithms that can provide a faster convergence rate and extensive search ability in the portfolio optimization problem under the cardinality constraint. To implement those features, a unique solution representation, a novel operator, and new repair mechanisms are introduced for solving the aforementioned problem in which lower and upper limits are set on the number of assets in the portfolio. For this purpose, new mating strategies along with the aforesaid package are implemented in well-known multi-objective evolutionary algorithms to solve the problem. The customized algorithms are subsequently tested against traditional ones using well-known market indices as benchmarks. Results indicate that the proposed strategy not only provides better approximations but also converges faster as well at no loss of performance with an increasing number of assets in the market.

This paper computes explicit constants in Bombieri and André's height bound for points at which there are unexpected "global" polymomial relations between values of G-functions. It applies these bounds for G-functions to obtain explicit height bounds for unlikely intersections with lines in tori, making explicit a weak version of the bounded height theorem of Bombieri, Masser and Zannier and, in some cases, improving an explicit bound of Habegger.