Papers

We construct a category of vector space-like objects called linear spaces over perfect idylls $k$ (an algebraic structure generalizing fields and hyperfields), which is a specialization of modules over $k$. Previous authors have noted that naive linear algebra in the product $k^n$ fails because linear independence does not satisfy matroid independence axioms. We give a sufficient axiom so that linear independence in $k$-linear spaces does satisfy matroid independence axioms. We also examine basic categorical properties of $k$-linear spaces. In particular, the category of $k$-linear spaces has no products, explaining the failure of naive linear algebra in $k^n$. Furthermore, we explore the categorical relationship between linear spaces over a perfect idyll, ordinary matroids, and matroids over a perfect idyll, connecting the existing theories of matroids and modules.

We prove $\textit{a priori}$ bounds and MLC (local connectivity of the Mandelbrot set $\mathcal{M}$) for a class of infinitely renormalizable parameters whose renormalization type is primitive but can approach the cusp of $\mathcal{M}$. To this end we develop and refine a variety of tools that allow us to control degeneration of renormalizations. They include the Thin-Thick Decomposition, the Value Calculus, the Wanderers Theorem, and the Wave Lemma.

Loop-erased random walks (LERW), the $O(n)$-model at ${n=-2}$ and Laplacian random walks (LRW) are three realizations of the same random process. While this equivalence holds on any graph, renormalization is possible only via the $O(-2)$-model. To generalize LRWs to $b$-LRWs or to Diffusion Limited Aggregation (DLA), a field theory directly on the Laplacian growth process is necessary. Here we construct an exact lattice action for LRWs and show that its perturbative expansion equals that of LERWs. We then generalize this approach to $b$-LRWs and DLA.

We establish the explicit leading order asymptotics, with a quantitative error bound, of tail probabilities of collision times for a class of integrable space-time inhomogeneous Markov chains, in discrete and continuous time. The corresponding process conditioned not to intersect arises in interacting particle systems with local push-block interactions thereby confirming a recent prediction. The generic discrete nature of the spatial inhomogeneities rules out powerful coupling-with-Brownian-motion techniques, so our proof strategy proceeds instead via a novel steepest-descent analysis combined with a Karlin–McGregor semigroup expansion in terms of dominant-index contributions.

The $w$-optimization problem is a mathematical problem abstracted from a useful tool for general data fusion rooted in various engineering tasks. Purely from the perspective of practical applications for this useful tool namely the split covariance intersection filter (Split CIF), it is sufficient to know that the $w$-optimization problem enjoys not only the second-order convexity (namely convexity in conventional sense) but also the fourth-order convexity, thanks to which a guaranteed fast implementation of the Split CIF can be realized. On the other hand, based on certain observations and analysis, the author proposes a conjecture concerning the $w$-optimization problem for further mathematical study. The conjecture is that the $w$-optimization problem has arbitrary even-order convexity, or in other words, it has the second-order convexity, the fourth-order convexity, the sixth-order convexity, and so on – As mentioned above, whether the $w$-optimization problem has higher-order convexity or not might be of little interest itself for practical applications, yet the author believes that the conjecture would stimulate study on systematic mathematical techniques that are potentially interesting and valuable – This paper presents the conjecture in details.

A sticky diffusion is a process that can stick to and detach from a lower-dimensional boundary. A challenge in simulating such a process is in capturing the change in dimension in a dynamically consistent way. We introduce a numerical algorithm to simulate a one-dimensional sticky diffusion, which sticks to and detaches from a point. Our method is a simple modification of the standard Euler-Maruyama scheme, which chooses with some probability between a reflected Euler-Maruyama update and a jump to the sticky point. We show how to choose this probability to be consistent with the generator of the desired dynamics, and we prove that our scheme converges weakly to a sticky diffusion with order 1.

We propose and analyze an adaptive iterative numerical homogenization method to approximate the solution of a class of quasilinear nonmonotone elliptic problems that is of multiscale nature. The method is based on the technique of the Localized Orthogonal Decomposition (LOD) applied to the linear problems in each step of a Kačanov iteration. In this approach, the multiscale basis is recomputed adaptively in each iteration and a linear problem is solved with this updated multiscale space, where in both steps the nonlinearity is evaluated using the approximation of the previous iteration. As a key component of the proposed approach, we present a locally computable error indicator, which at each iteration identifies the basis functions requiring updates, while previously computed basis functions are retained whenever possible. We provide a priori error estimates and show convergence of the method, requiring only higher integrability of the right-hand side, but no higher differentiability of the solution itself. Furthermore, we discuss how to adapt the proposed adaptive iterative LOD in the context of Newton's method as iteration scheme. Numerical experiments illustrate the theory and validate the applicability of the proposed method.

Shortest-path counts describe how many minimal routes connect a pair of graph vertices, complementing distance alone. We study their heterogeneity within graph-distance shells using C_log(p, r_g), the cube root of the third absolute central moment of log shortest-path multiplicities. This permutation-invariant statistic measures variation across shell targets and does not encode direction by itself. In a Flamm–Schwarzschild case study, its radial profile differs from that of a matched-flat control. The binned statistic has a Pearson correlation of approximately 0.912 with a logarithmic Schwarzschild reference profile, indicating radial-reference agreement rather than reconstruction of continuum curvature. Comparisons across sample sizes and neighbor parameters further describe behavior under the tested graph constructions. Preserved summaries and plotting records support the numerical evidence, although original graph realizations and the definitive execution state were not recovered. These results motivate multiplicity heterogeneity as a graph diagnostic for radial organization, with interpretation conditional on sampling, connectivity and boundary effects.

We construct explicit approximations to the solution of a second-order parabolic partial differential equation on the real line with variable coefficients. The method is based on Chernoff's product formula and uses a new operator-valued function defined through proper Riemann integrals over a bounded interval, which makes the approach readily usable in numerical practice. For sufficiently smooth initial data and coefficients, we prove that the resulting Chernoff approximations converge uniformly in space and time with a quadratic rate, improving the standard first-order estimate. The construction yields a new class of quasi-Feynman formulas that are neither grid-based nor Galerkin-type, but instead rely on semigroup theory and multiple bounded integrals. The theoretical findings are validated by symbolic computation, and the paper contributes both refined error bounds and a practical analytical tool for parabolic problems.

We study $p$-approximation properties of $\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For bounded geometry metric spaces, property A implies $p$-nuclearity of $B_u^p(X)$ for $1<p<\infty$, while $B_u^1(X)$ is always $1$-nuclear. Property A also yields coefficient characterizations and $p$-complete exactness results for $B_u^p(X,-)$; at $p=1$ these coefficient results hold without property A. We introduce the $p$-invariant translation approximation property and its operator space version. The $p$-AP implies a coefficient fixed-point identity for arbitrary $SQ_p$ coefficients, and for exact groups it is equivalent to the $p$-operator ITAP. At $p=1$, every discrete group satisfies the full coefficient fixed-point identity. We also establish coefficient permanence under subgroups, finite-index supergroups, and group extensions with $p$-AP quotient, including direct and semidirect products. In particular, the class of exact groups with the $p$-AP is closed under group extensions. We also obtain higher-rank lattices as examples failing the $p$-operator ITAP.

In this paper, we provide the resolution of the degree one resonant case in all dimensions. Our main result reads as follows: for any dimension $D\geq 1$ set $\mathbf{X}(\mathbf{t})=(\mathbf{t},|\mathbf{t}|^2),\; \mathbf{t}\in\mathbb{R}^D$, and let $K(\mathbf{t})$ be any suitable translation invariant Calderón–Zygmund kernel. If $\mathbb{V}\leq\mathbb{R}^{D+1}$ is any linear subspace such that $ \exists\:\:\mathbf{v}_0\in\mathbb{R}^D\times\{0\}$ nontrivial with $\mathbf{v}_0\perp\mathbb{V}$ then the following (maximal) Carleson-Radon transform $CR^\ast_{\mathbb{V}}$ is $L^p(\mathbb{R}^{D+1})-$bounded in the maximal range $1<p<\infty$, where $$CR^\ast_{\mathbb{V}} f(\mathbf{x}):= \sup_{\begin{array}{c} \scriptstyle 0<r<R<\infty \cr \scriptstyle \mathbf{a}\in\mathbb{V} \end{array}} \left| \int_{r<|\mathbf{t}|\leq R} f\left(\mathbf{x}-\mathbf{X}(\mathbf{t})\right) e\left(\mathbf{a}\cdot \mathbf{X}(\mathbf{t})\right) K(\mathbf{t}) d \mathbf{t} \right|.$$ The above choice for $\mathbb{V}$ creates a maximal linear subspace of $\mathbb{R}^{D+1}$ closed under parabolic scaling for which - $CR^\ast_{\mathbb{V}}$ is degree one resonant, and - $CR^\ast_{\mathbb{V}}$ is not degree two (or higher) resonant. The proof of the above result unravels several new manifestations and ideas meant to capture the remarkable features of the resonant Carleson-Radon behavior.

Metaheuristic-based global optimization with embedded, long-running simulations is a computationally expensive process. To support various stages of development and execution, a seamless transition from personal computers to distributed clusters is desired, enabling execution across all computational scales. However, existing tool chains are often characterized by rigidity and hardware-bound constraints, which impede scalability and the integration of complex simulations. Bridging this gap, we present a containerized HPC scalable microservice-based framework for genetic algorithms with embedded simulations (CHAMB-GA). The deployment of the framework scales consistently across cloud infrastructure via container orchestration and HPC clusters via batch-scheduled parallel execution. Users provide the GA operators and simulation backend separately. The framework is designed to run these components in a distributed and decoupled manner, mapped to separate hardware. This approach ensures that the fitness evaluation and genetic operations are not managed within the same process and are utilizing distinct parts of the compute infrastructure. A central message broker coordinates asynchronous manager-worker communication between microservices, thereby parallelizing evolutionary operations and fitness evaluations. We demonstrate CHAMB-GA's scalability, portability, and reproducibility, while facilitating the integration of external tools and complex simulations on benchmark and powerflow problems. The capabilities of CHAMB-GA are validated in a two-part approach: (i) a benchmark study demonstrating minimal overhead while scaling to over 3,500 CPU cores, and (ii) a dispatch optimization of High Voltage Direct Current (HVDC) lines in the German transmission grid, showing seamless migration from Kubernetes to SLURM, combined horizontal and vertical scaling, and integration of multi-stage workflows.

Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines a local matrix-function map rather than a convergent approximation to the full-matrix update: spectral interactions are preserved within tiles and discarded across tile boundaries. For fixed finite $T$, the leading Newton-Schulz work decreases to $O(H W T K)$, and the computation decomposes into independent small dense matrix operations. This structure enables tile-size-dependent GPU kernels, cross-layer batching, memory-bounded chunking, and runtime tile-size schedules. Experiments on transformer training and controlled matrix-function diagnostics show that HiMuon improves optimizer-step efficiency while keeping training behavior close to full-matrix Muon in the tested regimes.

In this paper, we give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen, using differential operators. More precisely, we realize their functions $\omega_{k+1,D}$ as images of the quadratic form Poincaré series $f_{k,D}$ under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue prop erties of $\omega_{k+1,D}$. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.

We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-$Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-$Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\mathbb F_q$ with transversal $\mathsf{C}^m\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\mathbb F_{q^2}$ supporting a transversal $\mathsf{C}^m\mathsf Z$ gate as well as addressable and parallelizable $\mathsf{C}^{m-1}\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.

We study the $R$-sequenceability of finite groups of odd order. Building on the classical theory of $R^*$-sequences and orthomorphisms, we explore two tools: the notion of $R^{**}$-sequenceability, a strengthening of $R^*$-sequenceability tailored for inductive arguments over normal subgroups with cyclic quotients, and the odd cycle index $\tau(G)$, which measures how many orthomorphisms are required to generate a full cycle together with an involution. Our main result is a Quotient-Normal Gadget theorem, which shows that if $G$ has a normal subgroup $N$ such that $G/N$ is $R^{**}$-sequenceable and $\tau(N) \leq |G/N| - 3$, then $G$ itself is $R^{**}$-sequenceable. We prove that $\tau(G) = 2$ for cyclic groups of order coprime with $3$, and establish an inductive bound $\tau(G) \leq \max\{\tau(N), \tau(G/N)\}$ for odd ordered groups with a normal subgroup $N$. As consequences, we show that every group whose order is coprime with $30$ is $R$-sequenceable, and that every nilpotent group whose order is coprime with $6$ and not a power of $5$ is $R$-sequenceable. These results extend prior work on abelian groups to broad families of non-abelian groups.

The Bogomolov property \B for an algebraic extension of \(\QQ\) asserts the existence of a uniform positive lower bound for the absolute logarithmic Weil height outside the group of roots of unity. Originally introduced as a weakening of Northcott's property and closely related to Lehmer's conjecture, it has been established for several natural classes of infinite extensions, including abelian extensions and fields generated by torsion points of elliptic curves defined over the rationals. Given a Galois representation of an absolute Galois group, one can associate with it the algebraic extension fixed by its kernel and ask whether this extension has property \B. This point of view allows one to reinterpret classical results, and to generalize them to other Galois representations, both of geometric and non-geometric origin. This expository paper gives a survey of the techniques and results in this framework. We present some results on modular representations, with a particular focus on the role of local \(p\)-adic information. We explain how Sen's theorem on totally ramified \(p\)-adic Lie extensions enters the proof of new criteria for the Bogomolov property, and how these criteria apply to representations with large local image. This contribution is based on joint work with Francesco Amoroso, Andrea Conti, and Pietro Piras.

For a discrete amenable group $G$ with a two-cocycle $\sigma$, we study when the twisted group $C^*$-algebra $C^*_r(G,\sigma)$ is selfless, in the sense of Robert. For a countably infinite abelian group $G$, this holds exactly when the commutator bicharacter of $\sigma$ is nondegenerate, that is, under Kleppner's condition. The nuclear dimension is then zero if $G$ is torsion and one otherwise. We show that root-of-unity-valued cocycles can give selfless algebras of nuclear dimension one, using slow dimension growth. The characterization extends to countably infinite virtually abelian or locally finite FC-hypercentral groups and to infinite finitely generated virtually nilpotent groups. Further, if $G$ is countable and $H\trianglelefteq G$ is infinite, the inclusion $C^*_r(H,\sigma')\subseteq C^*_r(G,\sigma)$ is selfless precisely when $C^*_r(H,\sigma')$ is selfless and $(H\leq G,\sigma)$ satisfies the relative Kleppner condition. As applications, we obtain selfless twisted group $C^*$-algebras of the lamplighter group and of the wreath product $\mathbb{Z}\wr\mathbb{Z}$.

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra over $\mathbb{C}$ and let $L(\rho)$ be the irreducible integrable $\mathfrak{g}$-module with highest weight $\rho$. Let $I$ be a subgraph of the Dynkin diagram of $\mathfrak{g}$ which has only simple bonds and no cycle of length $\geq 3$. For every subset $D$ of $I$, denote by $\beta_D$ the sum of the simple roots corresponding to $D$. To every $D \subset I$ such that $\lambda_{D,I} = 2\rho - \beta_I - \beta_D$ is dominant, we associate certain elements $\pi_{D,I}$ of weight $\lambda_{D,I} {-} \rho$ in the crystal $B(\rho)$, which depend on the choice of a root vertex in each connected component of $I$. Then we prove that our elements are $\rho$-dominant elements of $B(\rho)$, hence provide new families of components of the tensor product $L(\rho)\otimes L(\rho)$.

This work introduces an optimization model to support decision-making in the design of integrated transit systems, with the aim of improving urban mobility. We assume that daily user trips consist of two types of segments: short segments where users apply their own transport between their homes or workplaces and the public transit access points, and a longer segment on the transportation network. Users' accessibility is ensured by imposing an upper limit on the travel times of the short segments. We assume that only users within a given coverage radius will employ the system. On the other hand, system effectiveness is ensured by limiting travel time on the transportation network. Two mixed-integer linear programming formulations and a matheuristic procedure are proposed, and their efficiency is evaluated through computational experiments. The results allow us to study the structure of the solutions obtained and analyze the effects of the parameters considered.

The aim of this paper is to introduce a generalization of Sobolev spaces based on the Dobrakov integral. More precisely, we consider the setting of Banach-valued functions and Fomin differentiable Borel operator-valued measures on a finite-dimensional space. To build the necessary rigorous foundation, we establish analogs of several key results from the theory of differentiable real-valued measures, including the Leibniz rule and the integration by parts formula, all within the context of Dobrakov integration. These results are then embedded into the general scheme of vector-valued distribution theory. In particular, we describe the configuration of test spaces that yields an appropriate definition of a generalized derivative with respect to a differentiable operator-valued measure.

For Borel sets $A\subset \mathbb{R}^n\times \mathbb{R}$ we prove lower bounds for the parabolic Hausdorff dimension of the orthogonal projections of $A$ on generic $m$-dimensional linear subspaces of $\mathbb{R}^n\times \mathbb{R}$.

We investigate the distribution of primes satisfying arithmetic inequalities involving the Fourier coefficients of two non-CM newforms at prime powers. More precisely, we establish asymptotic formulas for the number of primes for which the differences, products, and ratios of the Fourier coefficients satisfy prescribed inequalities, together with explicit estimates for the corresponding densities. The proofs combine an effective joint Sato–Tate theorem with a geometric analysis of the associated semi-algebraic regions. As applications, we obtain new multiplicity one criteria, improve a theorem of Matomäki on small differences between Fourier coefficients, establish density-one analogues in the spirit of the Atkin–Serre conjecture, and derive a new characterization of twist-equivalence through the distribution of ratios of Fourier coefficients.

In this work, we study the optimization landscape of homogeneous quadratic programs with orthogonality constraints (QPOC) and apply the resulting theory to heteroscedastic probabilistic PCA (HePPCA). For QPOC, we establish a complete characterization of the benign optimization landscape by showing that every critical point is either a global maximizer or a strict saddle point. Our analysis builds on a closed-form characterization of the critical point set, from which we derive a necessary and sufficient condition for global optimality and show that every non-optimal critical point has a direction of positive curvature. As an application, we show that the population version of HePPCA is a special instance of QPOC and therefore has a benign optimization landscape; moreover, it satisfies local geodesic strong concavity near every global maximizer. We furthermore prove that, when the sample size is sufficiently large, the sample version of HePPCA inherits these favorable properties with high probability. Together with existing theory on avoiding strict saddle points, our results provide a theoretical justification for the observed local linear convergence of retraction-based optimization methods to global solutions for both QPOC and HePPCA. Finally, we present numerical experiments to corroborate our theoretical results.

We study the macroscopic fluctuation theory (MFT) of a multispecies generalization of the symmetric simple exclusion process (mSSEP) on the infinite line, in which particles of $N+1$ species – including, possibly, a vacancy species – exchange positions at unit rate. Working with the full, redundant set of coarse-grained densities $\bfrho=\{\rho_0,\dots,\rho_N\}$ keeps the relabelling symmetry of the model manifest throughout. We first extend the argument of Derrida and Gerschenfeld to the multispecies setting, showing that the cumulant generating function of the multispecies current between two regions of an arbitrary graph depends on the boundary densities $ \bfrho_L,\bfrho_R$ and on the fugacities $\bflambda$ only through a single scalar variable $\omega$. We then formulate the MFT saddle-point equations for the mSSEP on the infinite line and show that they define an integrable system: they are of Landau–Lifshitz type, and a Zakharov–Takhtajan gauge transformation recasts them in AKNS form. Solving the resulting linear scattering problem by the inverse scattering method, we recover the cumulant generating function $F(\omega)$ for the multispecies current, as well as the initial and final density profiles conditioned on a prescribed current fluctuation. In particular, we show that $F(\omega)$ coincides with the function obtained by Derrida and Gerschenfeld for the single-species SSEP, now derived for an arbitrary number of species directly from the integrable structure of the multispecies MFT equations.

We study periodic traveling waves in a general class of nonlinear dispersive equations with Burgers dissipation and a monostable Fisher-KPP reaction source. Under natural assumptions on the dispersion symbol, we prove the existence of small-amplitude periodic traveling waves bifurcating from a constant equilibrium by means of a Lyapunov-Schmidt reduction. We then investigate the spectral stability of these waves by combining Floquet-Bloch decomposition with perturbation theory for linear operators. It is shown that the Floquet spectrum of the linearized operator intersects the unstable complex half-plane, implying spectral instability with respect to localized perturbations. The results recover the recently established instability theory for the Korteweg-de Vries-Burgers-Fisher equation as a special case and provide a unified framework for a broad class of local and nonlocal dispersive systems. These results reveal a common instability mechanism generated by the interplay of dispersion, Burgers dissipation, and monostable reaction dynamics.

Path-dependent McKean–Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of $O(h^{1/4})$. Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of $O(h^{1/4}) + O(M^{-\gamma})$ for $M$ particles and an explicitly given $\gamma > 0$. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.

Erasure codes are a core component of most existing large-scale distributed storage systems, ensuring reliability against node failures. Recent work has shown that adapting code parameters to changing node failure rates can lead to significant storage savings. The default approach is to re-encode the data under a new code, which consumes substantial system resources. Code conversion was introduced to reduce this cost. However, existing work has mainly focused on conversions within specific classes of codes. In this paper, we study scalar linear code conversion in the merge regime for arbitrary linear codes. We derive universal lower bounds on the write and read costs in terms of unchanged and read symbols. The bounds are refined using generalized Hamming weights, which capture support-growth properties of subcodes and can give sharper estimates than minimum-distance-only arguments. We show that the framework recovers known bounds for important special cases and can be strictly stronger when the final code has nontrivial jumps in its generalized Hamming weight hierarchy. We then apply the framework to Reed-Muller codes and construct explicit Reed-Muller convertible codes using the Plotkin decomposition. For a natural Reed-Muller parameter regime, the construction attains the derived write-cost lower bound. For the read cost, the generalized-Hamming-weight analysis is sharp for one initial block, while a gap remains for the other block.

In the present paper, we introduce a numerical method for second-kind Fredholm integral equations (FIEs) based on de la Vallée Poussin-type (VP) polynomial approximations at Jacobi zeros. This class of approximations offers several advantages over classical Lagrange interpolation at the same nodes. In particular, it guarantees uniformly bounded Lebesgue constants in suitable weighted function spaces and provides near-best uniform approximation for functions in these spaces, while also significantly mitigating the Gibbs phenomenon. We show how these properties can be exploited in the numerical solution of FIEs. In particular, the proposed approach effectively handles functions with possible algebraic endpoint singularities and kernel functions featuring weak singularities or highly oscillatory behavior. Under suitable assumptions, we prove stability and convergence of the method in weighted uniform spaces. Furthermore, we develop an efficient implementation based on the solution of a well-conditioned linear system. Numerical results confirm the theoretical error estimates and show that the proposed method achieves higher local accuracy than the corresponding Lagrange-based projection method.

In this paper, we consider Fourier phase retrieval from differential intensity measurements, i.e., the problem of determining the phase of a complex-valued function from a series of intensity measurements differing only by slight modulations in Fourier domain. These modulations may be induced by optical elements such as prisms or phase plates, or via spatial-light modulators. Generalizing the principles behind the transport of intensity equation, we show that given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics. Numerical experiments illustrate the usefulness of our proposed approach.