We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $α$-stable offspring tails for some $α\in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $α$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).

``Little Red Dots'' (LRDs) are an abundant high-redshift population newly discovered by the James Webb Space Telescope (JWST) and considered to be an early growth phase of supermassive black holes (SMBHs). Using a method of pixel-by-pixel color selection and relaxing the compactness criteria, we identify four dual LRD candidates in the COSMOS-Web survey with projected separations of $0.\!\!^{\prime\prime}2$-$1.\!\!^{\prime\prime}2$. A comparison between existing LRD samples and mock data reveals that the projected separations of these dual LRD candidates are unlikely to result from chance projections of objects at different redshifts. Furthermore, two of the four systems are covered by COSMOS-3D slitless spectroscopy, and a single-line detection at the same observed wavelength for each LRD in a pair strongly supports that they are at identical redshifts. Assuming that the detected lines are H$α$ based on their high equivalent width and broad profile, the spectroscopic redshifts of $z=5.822$ and $5.464$ for the two pairs are consistent with their photometric redshifts, yielding projected separations of $1.64$ and $7.36\,{\rm kpc}$. These discoveries suggest that the angular auto-correlation function (ACF) of LRDs exhibits an excess ($\sim20$-$30$ times) on sub-arcsec (kilo-parsec) separations compared to an extrapolation of a power-law ACF of JWST-found AGNs measured over $10^{\prime\prime}$-$100^{\prime\prime}$. Our sample is likely to represent precursors of mergers between LRDs, and such mergers may be one of the mechanisms that can drive the rapid growth of SMBHs in their early evolutionary stages.

Entity resolution (ER) is a fundamental task in data integration that enables insights from heterogeneous data sources. The primary challenge of ER lies in classifying record pairs as matches or nonmatches, which in multi-source ER (MS-ER) scenarios can become complicated due to data source heterogeneity and scalability issues. Existing methods for MS-ER generally require labeled record pairs, and such methods fail to effectively reuse models across multiple ER tasks. We propose MoRER (Model Repositories for Entity Resolution), a novel method for building a model repository consisting of classification models that solve ER problems. By leveraging feature distribution analysis, MoRER clusters similar ER tasks, thereby enabling the effective initialization of a model repository with a moderate labeling effort. Experimental results on three multi-source datasets demonstrate that MoRER achieves comparable or better results to methods that have label-limited budgets, such as active learning and transfer learning approaches, while outperforming self-supervised approaches that utilize large pre-trained language models. When compared to supervised transformer-based methods, MoRER achieves comparable or better results, depending on the size of the training data set used.

Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPS) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPS, as well as further relevant examples of MPS including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of ansätze simpler than tensor networks in systems whose structure is invariant under permutations.

With the increasing demand for quantum hardware, shared and multi-tenant environments have been proposed to optimize resource utilization. However, the multi-tenancy paradigm in quantum computing inherently introduces security threats. This paper examines crosstalk-mediated attacks targeting three-qubit Grover's search algorithm and explores two fundamental mitigation strategies: gate-based dynamical decoupling and the use of a buffer qubit. We evaluate the effectiveness of each method individually and in combination, finding that while both strategies offer some level of attack mitigation, their combined application yields the most significant performance improvement. Beyond security vulnerabilities, our work also has implications for unintentional circuit interference that can occur when multiple quantum circuits are executed in close proximity.

This article investigates Dehornoy's monomial representations for structure groups and Coxeter-like groups associated with a set-theoretic solution to the Yang--Baxter equation. Using the brace structure of these groups and the language of cycle sets, we prove that the irreducibility of the associated monomial representations is equivalent to the indecomposability of the underlying solutions, except when the Dehornoy class is two. For indecomposable solutions, we show that these representations are induced from certain explicitly constructed one-dimensional representations.

Polymer mixtures fractionate between phases depending on their molecular weight. Consequently, by varying solvent conditions, a polydisperse polymer sample can be separated between phases so as to achieve a particular molecular weight distribution in each phase. In principle, predictive physics-based theories can help guide separation design and interpret experimental phase-diagram and fractionation measurements. Even so, applying the standard Flory-Huggins model can require numerical computations that hamper the predictions considering the full molecular weight distribution. Here, we apply a recently-derived exact analytical solution of multi-component Flory-Huggins theory for polydisperse homopolymers to understand the principles of polymer fractionation for common molecular weight distributions. Consistent with previous studies, the method highlights the sensitivity of polymer fractionation to the shape, and in particular the tails, of this distribution. Our results provide a systematic approach to evaluate the full molecular weight distribution in phase coexistence calculations over the possible composition space.

We study a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the edges of the graph, with a scattering process at each vertex governed by the scattering matrix assigned to it. We show that Scattering Quantum Walks encompass several known Quantum Walks. Additionally, we introduce two classes of Open Scattering Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one class defined on the edges and the other on the vertices of the graph. We show that these walks give rise to proper Quantum Channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.

Necessary optimality conditions in Lagrangian form and the sequential minimization framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, a characterization of local minimizers and critical points is given for problems including also nonlinear constraints. This approach lays the foundations for developing affordable sequential minimization algorithms with convergence guarantees to critical points from arbitrary initializations. A primal-dual perspective, a local saddle point property, and the dual relationships with the proximal point algorithm are also advanced in the presence of integer variables. Preliminary numerical results are presented for an augmented Lagrangian and an interior point method.

Advanced atomic magnetometers have made it possible to acquire nuclear magnetic resonance spectra in zero to ultralow magnetic fields. This regime carries the benefit of compact, low-cost instrumentation with reduced spin relaxation effects and the ability to probe phenomena that are inaccessible in conventional high-field experiments. A tradeoff is that the resulting spectra must be interpreted using simulations that are taxing for classical computation. Working by example for small-molecule and protein spectroscopy, we demonstrate that these simulations are a promising target for fault-tolerant quantum computation. Our holistic analysis spans from input selection to the construction of explicit circuits for qubitized quantum dynamics. By maintaining parity with experimental requirements, we demonstrate how certain cases might be especially promising for early fault-tolerant architectures.

In this work, an extension of the 1D Klausmeier model that accounts for the toxicity compounds is considered and the occurrence of travelling stripes is investigated. Numerical simulations are firstly conducted to capture the qualitative behaviours of the pulse-type solutions and, then, geometric singular perturbation theory is used to prove the existence of such travelling pulses by constructing the corresponding homoclinic orbits in the associated 4-dimensional system. A scaling analysis on the investigated model is performed to identify the asymptotic scaling regime in which travelling pulses can be constructed. Biological observations are extracted from the analytical results and the role of autotoxicity in travelling patterns is emphasized. Finally, the analytically constructed solutions are compared with the numerical ones, leading to a good agreement that confirms the validity of the conducted analysis. Numerical investigations are also carried out in order to gain additional information on vegetation dynamics.

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

The hybrid-high order (HHO) scheme has many successful applications including linear elasticity as the first step towards computational solid mechanics. The striking advantage is the simplicity among other higher-order nonconforming schemes and its geometric flexibility as a polytopal method on the expanse of a parameter-free refined stabilization. This paper utilizes just one reconstruction operator for the linear Green strain and therefore does not rely on a split in deviatoric and spherical behaviour as in the classical HHO discretization. The a priori error analysis provides quasi-best approximation with $λ$-independent equivalence constants. The reliable and (up to data oscillations) efficient a posteriori error estimates are stabilization-free and $λ$-robust. The error analysis is carried out on simplicial meshes to allow conforming piecewise polynomials finite elements in the kernel of the stabilization terms. Numerical benchmarks provide empirical evidence for optimal convergence rates of the a posteriori error estimator in some associated adaptive mesh-refining algorithm also in the incompressible limit, where this paper provides corresponding assertions for the Stokes problem.

Carrier graphs of groups representing subgroups of a given relatively hyperbolic groups are introduced and a combination theorem for relatively quasi-convex subgroups is proven. Subsequently a theory of folds for such carrier graphs is introduced and finiteness results for subgroups of locally relatively quasiconvex relatively hyperbolic groups and Kleinian groups are established.

We introduce a synthetic Mach-Zehnder interferometer for digitized quantum computing devices to probe fractional exchange statistics of anyonic excitations that appear in quantum spin liquids. Employing an IonQ quantum computer, we apply this scheme to the toric ladder, a quasi-one-dimensional reduction of the toric code. We observe interference patterns resulting from the movement of `electric' excitations in the presence and absence of `magnetic' ones. We model the noise in IonQ via depolarizing Lindbladian dynamics, and find quantitative agreement with the measurements obtained from the quantum device. The synthetic Mach-Zehnder interferometer can thus also serve as an effective means to probe the coherence length and time scales of multi-qubit noisy quantum devices.

In this work, we show that several problems naturally represented as Nonlinear Absolute Value Equations (NAVE) can be reformulated as Nonlinear Complementarity Problems (NCP) and efficiently solved using smoothing regularization techniques under mild assumptions. As far as we know, this is the first numerical approach that directly deals with NAVE. We also identify a technical assumption commonly utilized in smoothing techniques and prove its equivalence to a classical __ojasiewicz inequality at infinity, validating its non-restrictive nature. Furthermore, we extend established error estimates for NCP solvers to derive error bounds for NAVE problems under weaker assumptions. We illustrate the effectiveness of our approach through applications including asymmetric ridge optimization and nonlinear ordinary differential equations.

In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $I\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $I\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the product metric. Other related results for arbitrary dimensions are also discussed.

Notes on algebraic stacks, prepared for an 11-lecture course at the NCTS, Taipei, during the fall of 2022.

In this paper, we consider some variations on Mann's definition $\infty$-categorical definition of abstract six-functor formalisms. We consider Nagata six-functor formalisms, that have the additional requirement of having Grothendieck and Wirthmüller contexts. We also consider local six-functor formalisms, which in addition to this, take values in presentable stable $\infty$-categories, and have recollements. Using Nagata's compactification theorem, we show that Nagata six-functor formalism on varieties can be given by just specifying adjoint triples for open immersions and for proper morphisms, satisfying certain compatibilities. The existence of recollements is (almost) equivalent to a hypersheaf condition for a Grothendieck topology on the category of ``varieties and spans consisting of an open immersion and a proper map''. Using this characterisation, we show that the category of local six-functor formalisms embeds faithfully into the category of lax symmetric monoidal functors from the category of smooth and complete varieties to the category of presentable stable $\infty$-categories and adjoint triples. We characterise which lax symmetric monoidal functors on complete varieties, taking values in the category of presentable stable $\infty$-categories and adjoint triples, extend to local six-functor formalisms.

This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a~posteriori error estimates on regular triangulations into simplices using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs superiorly compared to uniform mesh refinements.

We present new JWST-NIRSpec IFS data for the luminous infrared galaxy NGC7469: a nearby (70.6Mpc) active galaxy with a Sy 1.5 nucleus that drives a highly ionized gas outflow and a prominent nuclear star-forming ring. Using the superb sensitivity and high spatial resolution of the JWST instrument NIRSpec-IFS, we investigate the role of the Seyfert nucleus in the excitation and dynamics of the circumnuclear gas. Our analysis focuses on the [Fe ii], H2, and hydrogen recombination lines that trace the radiation/shocked-excited molecular and ionized ISM around the AGN. We investigate the gas excitation through H2/Brγ and [Fe ii]/Pa\b{eta} emission line ratios and find that photoionization by the AGN dominates within the central 300 pc of the galaxy and together with a small region show ing signatures of shock-heated gas; these shock-heated regions are likely associated with a compact radio jet. In addition, the velocity field and velocity dispersion maps reveal complex gas kinematics. Rotation is the dominant feature, but we also identify non-circular motions consistent with gas inflows as traced by the velocity residuals and the spiral pattern in the Paα velocity dispersion map. The inflow is consistent with the mass outflow rate and two orders of magnitude higher than the AGN accretion rate. The compact nuclear radio jet has enough power to drive the highly ionized outflow. This scenario suggests that the inflow and outflow are in a self-regulating feeding-feedback process, with a contribution from the radio jet helping to drive the outflow.

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code.

This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overlineΩ) \cap W^{2,n}_\mathrm{loc}(Ω)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm.

In this paper we investigate the stationary profiles of a nonlinear Fokker-Planck equation with small diffusion and nonlinear in- and outflow boundary conditions. We consider corridors with a bottleneck whose width has a global nondegenerate minimum in the interior. In the small diffusion limit the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the in- and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.

We compute the geometric Sen operator for arbitrary Shimura varieties in terms of equivariant vector bundles of flag varieties and the Hodge-Tate period map. As an application, we obtain the rational vanishing of completed cohomology in the Calegari-Emerton conjectures.

A fundamental example of reaction-diffusion system exhibiting Turing type pattern formation is the Gierer-Meinhardt system, which reduces to the shadow Gierer-Meinhardt problem in a suitable singular limit. Thanks to its applicability in a large range of biological applications, this singularly perturbed problem has been widely studied in the last few decades via rigorous, asymptotic, and numerical methods. However, standard matched asymptotics methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions for single spike solutions are generally lacking. By introducing an ansatz based on generalized hyperbolic functions, we determine exact radially symmetric solutions to the one-dimensional shadow Gierer-Meinhardt problem for any $1 < p < \infty$, representing both inner and boundary spike solutions depending on the location of the peak. Our approach not only confirms numerical results existing in literature, but also provides guidance for tackling extensions of the shadow Gierer-Meinhardt problem based on different boundary conditions (e.g. mixed) and/or $n$-dimensional domains.

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $SL_n(\mathbb{Z}, p^m)$ in degrees $* < p-1$, for all large enough $n$, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $SL_n(\mathbb{Z}/p)$ with certain twisted coefficients.

In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

We study distributed agreement in microbial distributed systems under stochastic population dynamics and competitive interactions. Motivated by recent applications in synthetic biology, we examine how the presence and absence of direct competition among microbial species influences their ability to reach majority consensus. In this problem, two species are designated as input species, and the goal is to guarantee that eventually only the input species which had the highest initial count prevails. We show that direct competition dynamics reach majority consensus with high probability even when the initial gap between the species is small, i.e., $Ω(\sqrt{n\log n})$, where $n$ is the initial population size. In contrast, we show that absence of direct competition is not robust: solving majority consensus with constant probability requires a large initial gap of $Ω(n)$. To corroborate our analytical results, we use simulations to show that these consensus dynamics occur within practical biological time scales.