The anomalous Hall and Nernst effects refer to the perpendicular voltage drop generated by a magnetic material's magnetization in response to an applied current and temperature gradient. These effects can be harnessed to determine the Berry curvature and hold potential for future applications in electronic devices and thermoelectric energy conversion. We investigate the anomalous Hall and Nernst effects in the heavy-fermion ferromagnet CeCrGe$_3$ and its non-4f analog ferromagnet LaCrGe$_3$. We find that CeCrGe$_3$ exhibits a giant anomalous Hall angle and an anomalous Nernst coefficient, reaching values as high as 33% and ~ 10 $\mathrm{μV\ K}^{-1}$, respectively, among the largest reported for topological magnets. Based on electronic band-structure calculations, we identify a series of topological flat bands carrying strong Berry curvature with a pronounced Ce 4f orbital character in CeCrGe$_3$, which are absent in LaCrGe$_3$, highlighting the crucial role of Kondo flat bands in generating large anomalous transport responses. Furthermore, we identify a breakdown of the anomalous Hall scaling relation and the nonlinear anomalous Mott relation, which we attribute to the break of the topological Kondo flat bands at finite temperatures.

Recurrence time quantifies the duration required for a physical system to return to its initial state, playing a pivotal role in understanding the predictability of complex systems. In quantum systems with subspace measurements, recurrence times are governed by Anandan-Aharonov phases, yielding fractionally quantized recurrence times. However, the fractional quantization phenomenon in interacting quantum systems remains unexplored. Here, we address this gap by establishing universal lower and upper bounds for recurrence times in interacting many-body spin systems. Notably, we investigate scenarios where these bounds are approached, shedding light on the speed of quantum processes under monitoring. In specific cases, our findings reveal that the complex many-body system can be effectively mapped onto a dynamical system with a single quasi-particle, leading to integer-quantized recurrence times. Our work demonstrates a valuable link between recurrence times and the number of dark states in the system, thus providing a deeper understanding of the intricate interplay between Hilbert-space fragmentation, ergodicity breaking, measurements, and interaction effects. Finally, our findings have been implemented on an IBM quantum computer, revealing resonances and fractional quantization in agreement with theoretical predictions. This demonstrates the resilience of non-equilibrium topological fractional quantization to noise and highlights its potential use for benchmarking quantum devices and probing dark states.

Inspired by the halfspace theorem for minimal surfaces in $\mathbb{R}^3$ of Hoffman-Meeks, the halfspace theorem of Rodriguez-Rosenberg, and the cone theorem of Omori, we derive new non-existence results for proper harmonic maps into perturbed cones in $\mathbb{R}^n$, horospheres in $\mathbb{H}^n$ and also into perturbed Riemannian cones. The technical tool in use is an extension of the foliated maximum principle appearing in Assimos-Jost to the non-compact setting.

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness, including an application to the Best Linear Unbiased Estimator regression problem.

Individual modules of programmable matter participate in their system's collective behavior by expending energy to perform actions. However, not all modules may have access to the external energy source powering the system, necessitating a local and distributed strategy for supplying energy to modules. In this work, we present a general energy distribution framework for the canonical amoebot model of programmable matter that transforms energy-agnostic algorithms into energy-constrained ones with equivalent behavior and an $\mathcal{O}(n^2)$-round runtime overhead -- even under an unfair adversary -- provided the original algorithms satisfy certain conventions. We then prove that existing amoebot algorithms for leader election (ICDCN 2023) and shape formation (Distributed Computing, 2023) are compatible with this framework and show simulations of their energy-constrained counterparts, demonstrating how other unfair algorithms can be generalized to the energy-constrained setting with relatively little effort. Finally, we show that our energy distribution framework can be composed with the concurrency control framework for amoebot algorithms (Distributed Computing, 2023), allowing algorithm designers to focus on the simpler energy-agnostic, sequential setting but gain the general applicability of energy-constrained, asynchronous correctness.

We show that except for one exceptional case, a lamination on the boundary of a 3-dimensional handlebody H is a Hausdorff limit of meridians if and only if it is commensurable to a lamination with a 'homoclinic leaf'. This is a precise version of a philosophy called Casson's Criterion, which appeared in unpublished notes of A. Casson. Applications include a characterization of when a non-minimal lamination is a Hausdorff limit of meridians, in terms of properties of its minimal components, and a related characterization of which reducible self-homeomorphisms of the boundary of H have powers that extend to subcompressionbodies of H.

As we have entered Exascale computing, the faults in high-performance systems are expected to increase considerably. To compensate for a higher failure rate, the standard checkpoint/restart technique would need to create checkpoints at a much higher frequency resulting in an excessive amount of overhead which would not be sustainable for many scientific applications. Replication allows for fast recovery from failures by simply dropping the failed processes and using their replicas to continue the regular operation of the application. In this paper, we have implemented PartRePer-MPI, a novel fault-tolerant MPI library that adopts partial replication of some of the launched MPI processes in order to provide resilience from failures. The novelty of our work is that it combines both fault tolerance, due to the use of the User Level Failure Mitigation (ULFM) framework in the Open MPI library, and high performance, due to the use of communication protocols in the native MPI library that is generally fine-tuned for specific HPC platforms. We have implemented efficient and parallel communication strategies with computational and replica processes, and our library can seamlessly provide fault tolerance support to an existing MPI application. Our experiments using seven NAS Parallel Benchmarks and two scientific applications show that the failure-free overheads in PartRePer-MPI when compared to the baseline MVAPICH2, are only up to 6.4% for the NAS parallel benchmarks and up to 9.7% for the scientific applications.

In this paper, we propose a new class of local differential privacy (LDP) schemes based on combinatorial block designs for discrete distribution estimation. This class not only recovers many known LDP schemes in a unified framework of combinatorial block design, but also suggests a novel way of finding new schemes achieving the exactly optimal (or near-optimal) privacy-utility trade-off with lower communication costs. Indeed, we find many new LDP schemes that achieve the exactly optimal privacy-utility trade-off, with the minimum communication cost among all the unbiased or consistent schemes, for a certain set of input data size and LDP constraint. Furthermore, to partially solve the sparse existence issue of block design schemes, we consider a broader class of LDP schemes based on regular and pairwise-balanced designs, called RPBD schemes, which relax one of the symmetry requirements on block designs. By considering this broader class of RPBD schemes, we can find LDP schemes achieving near-optimal privacy-utility trade-off with reasonably low communication costs for a much larger set of input data size and LDP constraint.

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\mathbb{R}^N$, where $N=2$ or $3$. We construct axially symmetric normalized concentrating solutions as the parameter $a>0$ approaches $a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to a constant phase (and a rotational transformation for $N=2$), normalized concentrating solutions are unique and axially symmetric as $a\to a_*(N)$. When $N=3$, we also prove that the corresponding unique normalized concentrating solution is free of vortices as $a\to a_*(3)$, even if the anharmonic potential is non-radially symmetric.

Leader-based Byzantine-fault-tolerant (BFT) protocols provide low latency and simple communication structure, but they give the leader short-term control over transaction inclusion. A malicious leader can keep the protocol live while delaying or excluding time-sensitive transactions such as auction bids, oracle updates, liquidations, and bridge messages. Existing responses often build a fixed censorship-resistance, hiding, or ordering mechanism into the protocol path, forcing all transactions to pay for the same protection level. name follows the end-to-end principle: the consensus layer exposes inclusion primitives rather than hardcoding stronger policies. Higher-layer protocols can then choose their own submission strategies and resources, whether through replication, erasure coding, or other mechanisms, to obtain the censorship-resistance, hiding, ordering, or execution guarantees they need. At the core of BigDipper is censorship-resistant data availability, or DA-CR, which certifies available replica-contributed mini-blocks for use by leader-based consensus. A central design goal is that data remains sharded on the consensus critical path: validators do not reconstruct or execute the full payload before voting, but instead check commitments, availability evidence, and the DA-CR inclusion rule. We define DA-CR guarantees for data-tampering resistance, honest mini-block inclusion, and residual leader influence. We then give concrete constructions based on erasure coding and linear commitments, analyze client-tunable transaction submission, and instantiate BigDipper inside HotStuff-2.

A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.

Granger causal inference is a contentious but widespread method used in fields ranging from economics to neuroscience. The original definition addresses the notion of causality in time series by establishing functional dependence conditional on a specified model. Adaptation of Granger causality to nonlinear data remains challenging, and many methods apply in-sample tests that do not incorporate out-of-sample predictability, leading to concerns of model overfitting. To allow for out-of-sample comparison, a measure of functional connectivity is explicitly defined using permutations of the covariate set. Artificial neural networks serve as featurizers of the data to approximate any arbitrary, nonlinear relationship, and consistent estimation of the variance for each permutation is shown under certain conditions on the featurization process and the model residuals. Performance of the permutation method is compared to penalized variable selection, naive replacement, and omission techniques via simulation, and it is applied to neuronal responses of acoustic stimuli in the auditory cortex of anesthetized rats. Targeted use of the Granger causal framework, when prior knowledge of the causal mechanisms in a dataset are limited, can help to reveal potential predictive relationships between sets of variables that warrant further study.

We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.

A characteristic particle method for the simulation of first order macroscopic traffic models on road networks is presented. The approach is based on the method "particleclaw", which solves scalar one dimensional hyperbolic conservations laws exactly, except for a small error right around shocks. The method is generalized to nonlinear network flows, where particle approximations on the edges are suitably coupled together at the network nodes. It is demonstrated in numerical examples that the resulting particle method can approximate traffic jams accurately, while only devoting a few degrees of freedom to each edge of the network.

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the efficiency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. Moreover, the performance of various schemes in tracking solution contours is investigated. It is demonstrated that jet schemes possess the simplicity and speed of WENO schemes, while showing several of the advantages as well as the accuracy of DG methods.

We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through the tracking of characteristics and the use of suitable Hermite interpolations, high order is achieved in an optimally local fashion, i.e. the update for the data at any grid point uses information from a single grid cell only. We show that jet schemes can be interpreted as advect-and-project processes in function spaces, where the projection step minimizes a stability functional. Furthermore, this function space framework makes it possible to systematically inherit update rules for the higher derivatives from the ODE solver for the characteristics. Jet schemes of orders up to five are applied in numerical benchmark tests, and systematically compared with classical WENO finite difference schemes. It is observed that jet schemes tend to possess a higher accuracy than WENO schemes of the same order.

An "exact" method for scalar one-dimensional hyperbolic conservation laws is presented. The approach is based on the evolution of shock particles, separated by local similarity solutions. The numerical solution is defined everywhere, and is as accurate as the applied ODE solver. Furthermore, the method is extended to stiff balance laws. A special correction approach yields a method that evolves detonation waves at correct velocities, without resolving their internal dynamics. The particle approach is compared to a classical finite volume method in terms of numerical accuracy, both for conservation laws and for an application in reaction kinetics.

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an AMLI type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity.

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.

This study explores the transfer of nonclassical correlations from an ultra-cold atom system to a pair of pulsed laser beams. Through nondestructive local probe measurements, we introduce an alternative to destructive techniques for mapping BEC entanglement. Operating at ultralow temperatures, the setup emulates a relativistic vacuum field. We show that lasers can serve as Unruh-DeWitt detectors for BEC vacuum phonons. A quantum vacuum holds intrinsic entanglement, transferable to distant probes briefly interacting with it - a phenomenon termed `entanglement harvesting'. Our study accomplishes two primary objectives: first, establishing a mathematical equivalence between a pair of pulsed laser probes interacting with an effective relativistic field and the entanglement harvesting protocol; and second, to closely examine the potential and persisting obstacles for realising this protocol in an ultra-cold atom experiment.

We compute the first two k-invariants of the Picard spectra of $KU$ and $KO$ by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the E_\infty-structures of their truncations Pic(KU)[0,3] and Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the Brauer groups of Z/2-graded algebra bundles of Donovan-Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed, Hopkins and Teleman. Our results also imply that that these spaces represent twists of String and Spin structures on manifolds and can be used to twist tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a cotruncation of the Anderson dual of the sphere spectrum.

In this article, we investigate the explicit formula for the uniformizers of the false-Tate curve extension of $\mathbb{Q}_p$. More precisely, we establish the formula for the fields ${\mathbb{K}}_p^{m,1}={\mathbb{Q}}_p(ζ_{p^m}, p^{1/p})$ with $m\geq 1$ and for general $n\geq 2$, we prove the existence of the recurrence polynomials ${\mathcal{R}}_p^{m,n}$ for general field extensions ${\mathbb{K}}_p^{m, n}$ of ${\mathbb{Q}}_p$, which shows the possibility to construct the uniformizers systematically.

Let $\widehat{\mathcal{C}}_\mathbb{Z}$ be the group consists of manifold-knot pairs $(Y,K)$ modulo homology concordance, where $Y$ is an integer homology sphere bounding an integer homology ball, and let $\mathcal{C}_\mathbb{Z}$ be the subgroup consisting of pairs $(S^3,K)$. Dai-Hom-Stoffregen-Truong show that the quotient group ${\widehat{\mathcal{C}}_\mathbb{Z}}/{\mathcal{C}_\mathbb{Z}}$ admits a $\mathbb{Z}^\infty$-summand. In this paper, we improve the result by showing that there exists a family $\{(Y,K_m)\}_{m>1 }$ generating the $\mathbb{Z}^\infty$-summand where $Y$ is the boundary of a smooth contractible $4$-manifold. In fact, we give a $\mathbb{Z}$-count of such families. The examples are constructed using a family of knots obtained by blowing down a component of a two-bridge link. They are studied in Jonathan Hales's thesis. Using the algorithm due to Ozsváth, Szabó and Hales we give a classification of the knot Floer homology of a larger family of such knots, that might be of independent interest.

In this paper, we propose a novel, computationally efficient reduced order method to solve linear parabolic inverse source problems. Our approach provides accurate numerical solutions without relying on specific training data. The forward solution is constructed using a Krylov sequence, while the source term is recovered via the conjugate gradient (CG) method. Under a weak regularity assumption on the solution of the parabolic partial differential equations (PDEs), we establish convergence of the forward solution and provide a rigorous error estimate for our method. Numerical results demonstrate that our approach offers substantial computational savings compared to the traditional finite element method (FEM) and retains equivalent accuracy.

We examine what is an efficient and scalable nonlinear solver, with low work and memory complexity, for many classes of discretized partial differential equations (PDEs) - matrix-free Full multigrid (FMG) with a Full Approximation Storage (FAS) - in the context of current trends in computer architectures. Brandt proposed an extremely low memory FMG-FAS algorithm over 25 years ago that has several attractive properties for reducing costs on modern - memory centric -- machines and has not been developed to our knowledge. This method, segmental refinement (SR), has very low memory requirements because the finest grids need not be held in memory at any one time but can be "swept" through, computing coarse grid correction and any quantities of interest, allowing for orders of magnitude reduction in memory usage. This algorithm has two useful ideas for effectively exploiting future architectures: improved data locality and reuse via "vertical" processing of the multigrid algorithms and the method of $τ$-corrections, which allows for not storing the entire fine grids at any one time. This report develops this algorithm for a model problem and a parallel generalization of the original sweeping technique. We show that FMG-FAS-SR can work as originally predicted, solving systems accurately enough to maintain the convergence rate of the discretization with one FMG iteration, and that the parallel algorithm provides a natural approach to fully exploiting the available parallelism of FMG.

In this work we provide a model-independent notion of local fibrations of $(\infty,2)$-categories which generalises the well-known theory of locally coCartesian fibrations of $(\infty,1)$-categories. Based on previous work, we construct a model category which serves as a specific combinatorial model for this type of fibrations. Our main result is a generalisation of the locally coCartesian straightening and unstraightening construction of Lurie, which yields for any scaled simplicial set $S$ an equivalence of $(\infty,2)$-categories between the $(\infty,2)$-category of $(0,1)$-fibrations over $S$ (also known as inner coCartesian fibrations) and the $(\infty,2)$-category of functors $S \to \mathbb{C}\!\operatorname{at}_{(\infty,2)}$ with values in $(\infty,2)$-categories. Given an $(\infty,2)$-category $\mathbb{B}$, our Grothendieck construction can be specialised to produce an equivalence between the $(\infty,2)$-category of local fibrations over $\mathbb{B}$ and the $(\infty,2)$-category of oplax unital functors with values in $\mathbb{C}\!\operatorname{at}_{(\infty,2)}$. Finally, as an application of our results we provide a version of the Yoneda lemma for $(\infty,2)$-categories.

In this paper we introduce a novel block-based regression strategy for image denoising based on edge-aware Steered-Mixture-of-Experts (SMoE) models. SMoEs provide very sparse image representations, able to model sharp edges as well as smooth transitions in images efficiently with few parameters. A multi-model inference strategy is developed that improves significantly the denoising capacity of single SMoE models. We show that the important edge reconstruction properties of SMoEs are well preserved, even when many models are fused under severe noise. We investigate model-inference from local neighborhood blocks as well as from distant blocks using block-matching as in BM3D. Our initial results indicate that SMoE multi-model regression can provide promising results compared to state-of-the-art BM3D with excellent edge quality.

The symmetric case of the Szemerédi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemerédi Trotter is densely related to a successful instance of the recipe. We obtain this result by getting structural information on cell decompositions for extremal Szemerédi-Trotter examples. We obtain analogous results for unit circles.

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.