Papers

In robust variational physics-informed neural networks (RVPINNs), the loss function is formulated in terms of the Riesz representative of the variational residual within a discrete test space. This approach guarantees that the loss function is robust with respect to the true error in the energy norm up to a remainder term that depends on both the neural network approximation and the discrete space configuration. However, in problems with localized singularities, steep gradients, or interface layers, a fixed coarse test space may fail to resolve the continuous Riesz representative of the residual during training. Although this can be avoided by using a sufficiently fine test space from the start, doing so may be computationally inefficient. We therefore propose an adaptive algorithm that enriches the test space only where the error between the discrete and continuous Riesz representatives is pronounced. We establish theoretical adaptive strategies within the RVPINN framework and derive their error bounds. Furthermore, we propose a computable refinement indicator and prove that, under the saturation assumption, it serves as a reliable and efficient error estimator for the non-computable discrepancy between the discrete and continuous Riesz representatives. Finally, we propose a practical adaptive algorithm and demonstrate its effectiveness through numerical experiments on elliptic Dirichlet problems.

We study long-range discrete Dirac operators with Stark potential, extending the theory of Stark localization from scalar lattice models to systems with internal spinorial structure. We initially investigate the local setting, where two distinct localization mechanisms arise. The standard local Dirac-Stark operator yields two Stark-type spectral ladders and exponentially localized spinorial eigenfunctions. Conversely, a related pure-shift local model exhibits an invariant block structure that leads to explicitly computable eigenvalues and exact localization, with eigenfunctions compactly supported on only two spinorial sites. This extreme confinement surpasses the factorial decay characteristic of the classical scalar Stark model. For the general long-range Dirac model, we observe that the eigenvalues remain asymptotically close to the Stark ladder and prove that the corresponding eigenfunctions satisfy power-law localization estimates. Consequently, we establish power-law localization in the sense of finite moments of the position operator for the spinorial evolution. Our results demonstrate that deterministic Stark localization is robust and persists in genuinely matrix-valued lattice systems.

We study a piecewise constant function $\eta:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $\xi:\mathbb R\to\mathbb R$ satisfying $\xi'(x)=\eta(x)$, pinned by $\xi(0)=0$, has finite length excursions over past minima. This is the continuous generalization of an initial ball configuration in the discrete Box Ball System introduced by Takahashi and Satsuma, where solitons of integer sizes $k\ge1$ are identified. We extend the slot decomposition developed by Ferrari, Nguyen, Rolla and Wang in the discrete setting to the continuous case. Each soliton of $\xi$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $\xi$ to a point configuration. We consider a distribution on walks given by a product measure on the decomposition of the path into excursions over past minima. Excursions are distributed as products of their solitons weights, which are determined by the soliton heights. We show that when the weight function is in $L^1$ the slot decomposition of $\xi$ is a Poisson process. This extends to the continuous case an approach of Ferrari and Gabrielli. As an example, we compute the intensity measure of the Poisson process associated to the asymmetric telegraph process introduced by Kac. In a forthcoming paper we discuss the dynamic properties.

Why do unrelated approaches to quantum gravity nearly all find spacetime two-dimensional at the shortest scales? Each theory answers only within its own dynamics; we highlight a single kinematic route to the same value, one assuming no field equation and living in the geometry of the space of states alone. That route is concentration of measure on the Fubini-Study geometry of pure states, which forces the pairwise distances of a random sample to equalize as the dimension grows, so any finite sample collapses to an equidistant dust whose thresholded metric graph is the complete graph. Handed this dust, a diffusion probe reads it as two-dimensional in the large-sample limit, the value the running spectral dimension takes at the dust's single relaxation scale, a property of the measurement rather than the structure; this convergence on two is not, by itself, evidence that spacetime is two-dimensional. Whether a given two is such an artifact is governed by the Laplacian spectrum near zero, and whether that reading carries across an emergence map is the condition we call spectral faithfulness; a single relaxation scale encodes no spectral dimension that tells one structure from another. The collapse, the probe value, and the eigenvalue-density criterion are machine-checked in Lean 4 against Mathlib, resting on the standard Beta law of overlaps; a power-law tail of small eigenvalues reads a genuine dimension, a single scale above a gap reads two at its own clock, and a gapped two-scale band reads off the universal line. These classes are run on graph-Laplacian proxies, and whether a link-graph reading carries to the physical nonlocal operator is left open. The spectral test reads the eigenvalue density near zero and separates, on a given structure, a measurement artifact from a dimension the structure genuinely expresses.

The greatest common divisor (GCD) of $\binom{2n}{2k}$ for $1\leq k<n$ is known to be some power of 2 times the product of all odd primes p such that $2n=p^i+p^j$. We complete the analysis of this GCD by showing that this power of 2 is either 1 or 0 and relates it to Mersenne primes. We also show how to efficiently compute $GCD \{\binom{mn}{mk}: 1\leq k<n\}$ when n and m satisfy certain conditions.

In this work, the derivation and multiscale analysis of a mathematical model for plant tissue biomechanics is considered. The microscopic model consists of a coupled systems of equations of linear elasticity, describing mechanical deformation, and reaction-diffusion and ordinary differential equations modelling the dynamics of load-bearing cross-links in cell walls and middle lamella, connecting individual cells in a plant tissue. It takes into account a two-way coupling between mechanical deformation and chemical processes in cell walls and middle lamella, where elastic properties depend on the density of cross-links and reactions terms, defining the stretching and breakage of the cross-links, depend on the deformation gradient. The nonlinear dependence of solutions of the ordinary differential equation on the displacement gradient and of the boundary conditions in reaction-diffusion problem on the displacement induces novel approaches in the derivation of a priori estimates. Using homogenization techniques of locally periodic (l-p) two-scale convergence, a macroscopic model for plant tissue biomechanics is derived, representing the first result in the multiscale analysis of a biomechanical model for tissues with non-periodic fibrous microstructure. For multiscale analysis and derivation of the macroscopic model, the non-periodic microstructure of plant cell walls, characterised by rotated planes of parallel aligned microfibrils, is approximated by the corresponding locally periodic microstructure. The two-way coupling between equations of linear elasticity and reaction-diffusion and ordinary differential equations requires the proof of the strong l-p two-scale convergence for the deformation gradient and for the density of stretched cross-links, in order to pass to the limit in the nonlinear functions.

We demonstrate a method to prove that a theta characteristic on Riemann surface $S$ invariant under the action of $G \leq \mathrm{Aut}(S)$ has even parity by considering the representation theory of the double cover $\tilde{G} = 2 \cdot G$. We apply this to certain Hurwitz curves, and prove a recent conjecture of Broughton & Disney-Hogg.

This paper considers a partial linear regression model with scalar response missing at random, one finite-dimensional covariate (a vector, $X$) and one infinite-dimensional covariate (a functional variable, $\mathcal{X}$). While the effect of $X$ on the response is linear, the effect of $\mathcal{X}$ is nonparametric. Three $k$NN-based estimators are proposed for both the vector parameter and the nonparametric operator, and some first asymptotic results are obtained.

Quantum low-density parity-check (QLDPC) codes are strong candidates for fault-tolerant quantum computation, but efficient decoding remains a major challenge due to short cycles, degeneracy, and the poor convergence of standard belief-propagation (BP) decoders. We propose a reinforcement learning-based list sequential (RL-LS) BP decoder for QLDPC codes by extending the reinforcement-learning-based sequential variable-node scheduling (RL-S) framework with list-based search. At each step, the learned policy selects the next variable node to update; the decoder then retains the ordinary RL-S trajectory while also exploring a competing branch obtained by softly biasing the post-update LLR pair toward the second-most likely Pauli symbol, recomputing the incident local BP messages, and setting the visited variable node to that second-best symbol. Candidate trajectories are ranked and pruned using our proposed cumulative path metric. The resulting decoder extends the learned decoder by combining the improved convergence of learned sequential scheduling with list exploration. Numerical results on representative QLDPC benchmark codes over the depolarizing channel show that our proposed method improves the decoding performance of the underlying decoder and compares favorably with existing BP-based decoding methods.

We study the predictable compensators of stochastic processes in a single jump filtration augmented with initial information represented by a sub-$\sigma$-algebra $\mathcal{H}$. We consider adapted càdlàg processes of finite variation and give an explicit construction of their predictable compensators. The main difficulty arises when the jump size has a heavy tail and lacks integrability, so that the classical Doob-Meyer decomposition does not apply. To overcome this, we use the theory of $\sigma$-martingales. We establish necessary and sufficient conditions for a process to be a $\sigma$-martingale and explicitly compute the compensator of a suitably weighted process. This yields an explicit relation between the continuous drift and the expected jump component of the original process.

Characterizing non-Gaussian posterior distributions in partially observed high-dimensional nonlinear systems remains a fundamental challenge in data assimilation. Ensemble Kalman filters rely on Gaussian approximations that can be inaccurate for strongly non-Gaussian posteriors, whereas particle filters suffer from severe scalability limitations. Recent score-based generative approaches improve posterior characterization but typically require supervised training with ground-truth posterior samples, which are unavailable in most practical applications. We introduce $\Omega$ (Operator-based Mixture Ensemble for Generative Assimilation), a scalable framework that integrates conditional Gaussian surrogate modeling, unsupervised score learning, and generative sampling. The conditional Gaussian surrogate provides a nonlinear non-Gaussian baseline approximation while admitting closed-form conditional posterior distributions for the unresolved variables. First, $\Omega$ exploits these closed-form conditional distributions to analytically recover the high-dimensional unobserved component, reducing computational cost and mitigating the curse of dimensionality. Second, $\Omega$ learns only the residual discrepancy beyond an analytical baseline through denoising score matching using ensemble trajectories alone, eliminating the need for ground-truth posterior samples and substantially reducing the learning burden. Third, $\Omega$ reconstructs the full non-Gaussian posterior distribution of both observed and unobserved variables via a Gaussian mixture representation, capturing multimodal, skewed, and heavy-tailed statistics. Finally, $\Omega$ employs annealed Langevin sampling to iteratively refine ensemble members from the baseline toward the target posterior. $\Omega$ is validated on several turbulent models with intermittency and extreme events, consistently improving posterior accuracy.

Modern high performance computing (HPC) applications must target heterogeneous hardware. This requires significant work to ensure domain specific implementations translate to highly performant kernels across a range hardware types and vendors, each requiring bespoke optimization to make use of the specific target architecture. Through the development of a domain specific compiler built with the multi-level intermediate representations (MLIR) project, one can express a high-level, close to the specific domain, abstraction that is progressively lowered to a low, close to metal, abstraction. At each intermediate representation (IR), appropriate optimizations can be applied without costly analysis due to the knowledge embedded in the domain specific IRs. We apply this method to the construction of discrete differential operators for use in spectral/hp element method solvers for computational fluid dynamics (CFD). Here, the performance is driven by a small set of common finite element operators that are composed to create kernels for the discrete differential operators used to solve weak partial differential equations. We create our own MLIR dialect to represent these operators and implement a bespoke lowering pipeline to facilitate the just-in-time compilation of these kernels for both CPU and GPU architecture and illustrate performance comparisons with the Nektar++ spectral/hp element framework.

In this paper, we explore a new strategy towards an Itô-Stratonovich type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously cumbersome. The emergence of mixed differential terms in 2D leads to overlapping iterated integrals, which previously required the construction of exhaustive combinatorial structures. As an example of this kind of structure, let us mention the massive 36-element planar signature introduced by K. Chouk and M. Gubinelli in their influential paper 'Rough sheets'. In this work we propose a simplified setting for rough calculus in the plane, which relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick allows us to significantly streamline the complexity of planar algebraic integration. We illustrate this methodology for a specialized Rough-Young framework: we extend the classical planar change-of-variable formula to paths possessing an asymmetric Hölder regularity: $\gamma_1 > 1/3$ in the first direction and $\gamma_2 > 1/2$ in the second direction. Relying on a structured controlled path expansions, we rigorously minimize the set of iterated integrals needed in the signature, and express the resulting planar change-of-variable formula as the explicit limit of Riemann sums.

This paper develops a reinforcement-learning approach to continuous-time risk-sensitive benchmarked asset allocation in a partly model-based setting. The benchmarked problem does not directly fit the standard Markovian stochastic-control template: the state is uncontrolled, whereas the terminal reward contains a controlled Itô integral. We use free energy-entropy duality to reformulate the problem as a linear-quadratic-Gaussian stochastic differential game under an equivalent probability measure, yielding explicit finite- and infinite-horizon saddle-point solutions. This structure guides a continuous-time $q$-learning actor-critic method: the quadratic value function motivates the critic, while the affine saddle-point controls motivate deterministic actors for the portfolio allocation and adversarial control. The learned allocation admits an economic interpretation through fractional Kelly decompositions. A proof-of-concept implementation calibrated to U.S. equity data shows that the actors learn the optimal policy with high accuracy and reveals a favorable asymmetry: the portfolio actor receives a cleaner learning signal than the auxiliary adversarial actor.

We develop a finite-window recursive audit framework for Navier–Stokes-generated packages. On a fixed window, the underlying anti-phantom certificate asserts that a baseline-visible package is either detected locally or charged to an explicit residual ledger. The first component of the framework is a broad one-step admissibility criterion for pressure-admissible finite-energy Navier–Stokes-generated audit packages under an explicit ledger of synchronization, localization, projection, harmonic-tail, chart, clean-gap, gate/slack, and detector mismatches. The second component is a finite-chain propagation theorem: once one-step admissibility supplies a finite renormalized chain of packages and a static finite-window audit certificate is available at each scale, variable-coefficient error recursion and weighted summation yield a recursive finite-window anti-phantom lower bound. We also give compact/effective pressure projection criteria, reduced quotient chart visibility, clean detector gaps, a pressure–flux–energy matrix kernel condition, a smooth reduced NS-generated verification class, and a conditional Caffarelli–Kohn–Nirenberg compatible defect-extraction criterion.

A fundamental identity for the number of vacillating tableaux was originally obtained through the representation theory of partition algebras. We extend this identity to arbitrary differential posets and show that it, together with several analogous identities, follows directly from the structural properties of differential posets. Specializing to Young's lattice and its Cartesian powers, we further obtain new bijective proofs via a simple deletion–insertion process.

In this paper we present new differential addition and doubling formulas for Jacobi quartic curves. Several differential addition and doubling formulas are presented with cost of 5M+4S+1D,3M+7S+1D,3M+6S+3D when the given difference point is in affine form. Here M,S,D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively.

Holographic beamforming enabled by metamaterial antennas has been proposed to facilitate spatial multiplexing at low hardware cost and low power consumption. However, existing holographic beamforming schemes are mainly developed for conventional bit-communication systems, which have not considered semantic-level importance and thus cannot be directly applied to support semantic communication. Specifically, in conventional bit communication, all bits are treated as equally important. In contrast, in semantic communication, different semantic information contribute unequally to task completion and therefore has different degrees of importance, with more important information requiring higher transmission quality. Ignoring semantic importance in holographic beamforming causes mismatches between importance of semantic information and its received SNR, thus degrading performances. In this paper, we propose a semantic-importance-aware holographic beamforming scheme enabled by metamaterial antennas with tunable radiated amplitudes to support semantic communication. It is challenging to design semantic-aware holographic beamforming schemes due to non-trivial modeling of the impact of semantic importance and unique amplitude-controlled structures of holographic beamforming. To address this, we characterize the dependence of semantic communication performance on semantic importance and received SNR via data fitting, and design a semantic-aware holographic beamforming algorithm to ensure reliable delivery of highly important semantic information. Simulation results validate effectiveness of the proposed method.

We consider the Poisson Problem for the stability operator of the helicoid on a vertical strip, under the assumption that the source term is supported on a strip of fixed height. We prove that solutions decay at a definite rate away from the support assuming natural orthogonality conditions on the source term.

The paper establishes integrated local energy decay (ILED) estimates for the damped wave equation on certain non-stationary spacetimes. The main technical result is a high frequency estimate that holds in great generality, provided that null geodesics trapped in a compact region are sufficiently damped.

We study an inverse initial data problem for a convection-diffusion equation with memory, where the goal is to recover the unknown initial condition from final-time data. The model includes convection, an instantaneous Laplacian term, and a nonlocal-in-time memory term involving the Laplacian of the past states, which leads to a severely ill-posed backward problem. We prove uniqueness in a spatially independent coefficient setting by applying the Fourier transform and using an analyticity argument for a scalar Volterra equation. For the variable-coefficient case, we develop a computational method based on Legendre spatial dimensional reduction and Tikhonov regularization. The solution is approximated by a finite tensor-product Legendre expansion, thereby reducing the inverse problem to a finite-dimensional terminal-value system for the time-dependent coefficients. We solve the reduced problem by a Tikhonov-regularized least-squares method with an $H^2$ penalty. For a fixed truncation order, we prove that the regularized minimizers converge to the finite-dimensional minimum-norm solution as the noise level and the regularization parameter vanish, under a suitable choice of the regularization parameter. Some two-dimensional numerical examples are presented to illustrate the performance of the proposed method.

Let $\mathcal{S}$ be the class of analytic univalent functions defined in the open unit disc $\mathbb{D}$ of the complex plane with the normalizations $f(0)=0$ and $f'(0)=1$. For $A\in (1,2]$, let $Co(A)$ denote the class of concave univalent functions defined in $\mathbb{D}$ with the opening angle $\pi A$ at infinity. In this article, by applying certain convolution techniques, we investigate the radius of convexity for the class $Co(A)\ast\mathcal{S}t(1/2)$, where $\mathcal{S}t(1/2)\subsetneq \mathcal{S}$ denotes the class of starlike functions of order $1/2$. Furthermore, we establish that the radius of convexity of the class $\mathcal{S}\ast\mathcal{S}t(1/2)$ is at least $0.19191$ (approximately). Here, `$\ast$' denotes the convolution (or Hadamard product) of two classes of functions.

This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass $8\pi$, solutions concentrate in infinite time around the stationary state with a universal logarithmic rate. The determination of this behaviour was the last missing step for achieving a complete radial classification, and we prove it in a companion paper (in fact without radial assumption). For a supercritical mass, solutions undergo type II finite-time blow-up with an explicit universal asymptotic rate and the stationary state as profile. This trichotomy holds for all radial initial data with finite second momentum. For non-radial data or infinite second momentum, it is known that other dynamics can be possible; for each of these three universal regimes we review the known results showing how they persist.

We consider the parabolic-elliptic Keller-Segel equation in two dimensions on the whole space. We prove that for arbitrary initial data with critical mass $8\pi$ and finite second momentum, all solutions have the same universal behaviour. They are globally defined and, asymptotically for large times, they converge to a renormalized stationary state of the equation which concentrates around the center of mass of the solution at a universal logarithmic-in-time scale. Our result holds for general solutions without symmetry assumptions, and we furthermore provide explicit convergence rates. In the radial case, by combining our result with previous ones (by Blanchet-Dolbeault-Perthame, Mizoguchi, and related works), this achieves a complete classification of the possible dynamics: for subcritical masses solutions converge to a self-similar expander, at the critical mass they concentrate a stationary state in infinite time at the aforementioned universal scale, and for supercritical masses they blow up in finite time by type II concentration of a stationary state at another universal scale. Our proof starts with soliton resolution, then controls the motion of the stationary state and the remainder in new spaces, and devises new techniques for the multi-scale linearized analysis, which eventually enables the stability and modulation analysis around an approximate solution.

In many applications, the precision by the available hardware arithmetic is insufficient to guarantee accurate results. Multiword arithmetic is a special type of multiprecision arithmetic where a multiple double is an unevaluated sum of 64-bit doubles, or where a multiple integer is an unevaluated sum of 64-bit integers. Parallel computing is applied to compensate for the cost overhead of multiword arithmetic. This type of arithmetic exploits naturally the optimized hardware, allows for efficient type conversions, memory layouts, all favorable for parallel computing. For example, storing a multiword in registers rather than arrays is beneficial to parallel computing by tasking and acceleration by graphics processing units. Code for multiword arithmetic is available in the software PHCpack, written mainly in Ada, publicly available at github, and as an Alire crate, released under the GNU GPL v3.0 license.

Let $\mathcal{S}$ be the set of all analytic univalent functions $f$ defined in the open unit disc $\mathbb{D}$, with $f(0)=0=f'(0)-1$. For $\alpha\in[0,1)$, let $\mathcal{S}t(\alpha)$ be the set of all starlike functions of order $\alpha$ in $\mathcal{S}$. In this article, by applying duality technique we obtain the radius of a disc that is mapped onto a starlike domain with respect to the origin by the functions in the set $\mathcal{S}\ast \mathcal {S}t(\alpha):=\{f\ast g :f\in\mathcal{S},~g\in\mathcal{S}t(\alpha)\}$. Here, `$\ast$' denotes the convolution (or Hadamard product) of two analytic functions in $\mathbb{D}$.

We investigate the dissipative dynamics of multilevel quantum systems strongly coupled either to a lossy cavity mode or directly to a bosonic environment. By deriving spectral localization bounds, we establish conditions under which strong system-bath coupling gives rise to a hierarchy of population relaxation timescales. Our approach builds on the reaction-coordinate polaron-transform framework. By mapping the original strong-coupling problem onto an effective weakly dissipative model, we analyze the spectrum of the resulting Liouvillian superoperator through localization bounds. For the generalized V model, we find that strong system-bath coupling gives rise to a bright-dark structure in the effective system-bath coupling operator: a single collective mode remains strongly coupled to the environment, while the remaining modes become progressively dark. Consequently, the dynamics separate into fast and slow sectors and, at finite coupling strengths, develop a hierarchy of population relaxation timescales. Numerical simulations based on both secular and non-secular quantum master equations corroborate the emergence of timescale separation and the pronounced slowing down of dissipative dynamics at strong coupling. Our results reveal a general mechanism underlying anomalously slow relaxation in strongly coupled open quantum systems and provide a route for engineering long-lived states through system-environment interactions.

A fundamental assumption in statistics and machine learning is that “the future looks like the past,” formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.

Conjecture 3 in [A. Aldroubi, C. Cabrelli, I. Krishtal, and U. Molter, Dynamical Sampling: A Survey, La Matematica 5 (2026), Article 37] postulates that for any bounded normal operator $T$ on a Hilbert space $H$ and any vector $g\in H$ the system \[ \left\{\frac{T^k g}{\|T^k g\|}: k=0,1,2,\ldots\right\} \] is not a frame. It was motivated by [A. Aldroubi, C. Cabrelli, A. F. Çakmak, U. Molter, and A. Petrosyan, Iterative actions of normal operators, J. Funct. Anal. 272 (2017), no. 3, 1121–1146], where it was established that such frames do not exist when $T$ is a self adjoint operator. We show, however, that this conjecture is false by presenting a construction of $H$, $T$, and $g$ such that the normalized orbit considered is indeed a frame. The operator is diagonal and is defined via a decomposition of the space into finite blocks rapidly increasing in size. We also provide an $\epsilon$-perturbation $S$ of the operator $T$ such that the system \[ \left\{{S^k g}: k=0,1,2,\ldots\right\} \] is a Carleson frame in the sense of [A. Aldroubi, C. Cabrelli, U. Molter, and S. Tang, Dynamical sampling, Appl. Comput. Harmon. Anal. 42 (2017), no. 3, 378–401] and [O. Christensen, M. Hasannasab, F. M. Philipp, and D. Stoeva, The mystery of Carleson frames, Appl. Comput. Harmon. Anal. 72 (2024), Article 101659]. The constructions were achieved using ChatGPT, whose assistance was also employed in the preparation of this manuscript.

Belief revision is a process in which an agent begins to believe in something she previously did not. I begin the paper by presenting, based on (Gärdenfors, 1998; Hansson, 1999), postulates for belief revision that constitute the basis of the AGM theory. I will then briefly show the semantics of a modal logic introduced in (van Ditmarsch, 2005), which I call `$P$'. This logic formalizes static epistemic states and has greater expressive power than AGM in doing so because it captures the quantitative notion of "degrees of conviction". The third step is to introduce revision operators on $P$ and, mostly following (van Ditmarsch, 2005), obtain the Dynamic Epistemic Logic (DEL) I call `$P*$'. It models processes of belief revision in several ways. Original results are presented in the following two sections. The first one of these sections revolves around a formalization of AGM postulates within $P*$ by proving some theorems related to the satisfaction of those postulates by revisions defined in $P*$. The last section features an analysis of $P*$'s revisions that go beyond the mere satisfaction of postulates. I compare their formal behavior with respect to some philosophical criteria. At last, I conclude that the functions presented in (van Ditmarsch, 2005) are not good formalizations of the philosophical intuition behind AGM. Instead, it is captured by the function $*^0$ originally defined in this paper (but highly inspired by (van Benthem, 2007)). An implementation of this function is also provided.