Arbitrary Field Research Articles

Integral points on a del Pezzo surface over imaginary quadratic fields

Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. In this paper, we generalise the torsor method for integral points from Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{A}}_3$$\\end{document} over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.

Point neuron learning: a new physics-informed neural network architecture

Machine learning and neural networks have advanced numerous research domains, but challenges such as large training data requirements and inconsistent model performance hinder their application in certain scientific problems. To overcome these challenges, researchers have investigated integrating physics principles into machine learning models, mainly through (i) physics-guided loss functions, generally termed as physics-informed neural networks, and (ii) physics-guided architectural design. While both approaches have demonstrated success across multiple scientific disciplines, they have limitations including being trapped to a local minimum, poor interpretability, and restricted generalizability beyond sampled data range. This paper proposes a new physics-informed neural network (PINN) architecture that combines the strengths of both approaches by embedding the fundamental solution of the wave equation into the network architecture, enabling the learned model to strictly satisfy the wave equation. The proposed point neuron learning method can model an arbitrary sound field based on microphone observations without any dataset. Compared to other PINN methods, our approach directly processes complex numbers, offers better interpretability, and can be generalized to out-of-sample scenarios. We evaluate the versatility of the proposed architecture by a sound field reconstruction problem in a reverberant environment. Results indicate that the point neuron method outperforms two competing methods and can efficiently handle noisy environments with sparse microphone observations.

The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

AbstractWe compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.

Medium Amplitude Field Susceptometry (MAFS) for magnetic nanoparticles

The linear dynamic susceptibility is arguably the most important property to specify the magnetization dynamics of a given system. Therefore, this quantity has been studied in great detail and the corresponding measurements known as susceptometry are well-established. Notwithstanding its relevance, the linear susceptibility is inherently limited to describe the magnetization response to weak external fields only. Here, we suggest the framework of Medium Amplitude Field Susceptometry (MAFS) to study the non-linear response which complements the linear susceptibility and provides additional information on the magnetic properties of the system and is applicable to magnetic fields of medium amplitudes. In particular, we introduce the general third-order nonlinear susceptibility χˆ3 as a central quantity that completely specifies the lowest-order non-linear response to arbitrary time-dependent magnetic fields. We show that response functions in medium amplitude oscillatory magnetic fields and parallel superposition susceptometry are contained in χˆ3 as special cases. Also included in χˆ3 are interesting intermodulation effects when the system is probed by a superposition of oscillating magnetic fields with different frequencies. We work out the explicit form of χˆ3 for several model systems for the dynamics of magnetic nanoparticles (MNPs). We expect this unifying framework to be not only of theoretical interest, but also useful for a deeper characterization of MNP systems, giving additional information on their suitability for various applications.

SPINEPS—automatic whole spine segmentation of T2-weighted MR images using a two-phase approach to multi-class semantic and instance segmentation

Abstract Objectives Introducing SPINEPS, a deep learning method for semantic and instance segmentation of 14 spinal structures (ten vertebra substructures, intervertebral discs, spinal cord, spinal canal, and sacrum) in whole-body sagittal T2-weighted turbo spin echo images. Material and methods This local ethics committee-approved study utilized a public dataset (train/test 179/39 subjects, 137 female), a German National Cohort (NAKO) subset (train/test 1412/65 subjects, mean age 53, 694 female), and an in-house dataset (test 10 subjects, mean age 70, 5 female). SPINEPS is a semantic segmentation model, followed by a sliding window approach utilizing a second model to create instance masks from the semantic ones. Segmentation evaluation metrics included the Dice score and average symmetrical surface distance (ASSD). Statistical significance was assessed using the Wilcoxon signed-rank test. Results On the public dataset, SPINEPS outperformed a nnUNet baseline on every structure and metric (e.g., an average over vertebra instances: dice 0.933 vs 0.911, p < 0.001, ASSD 0.21 vs 0.435, p < 0.001). SPINEPS trained on automated annotations of the NAKO achieves an average global Dice score of 0.918 on the combined NAKO and in-house test split. Adding the training data from the public dataset outperforms this (average instance-wise Dice score over the vertebra substructures 0.803 vs 0.778, average global Dice score 0.931 vs 0.918). Conclusion SPINEPS offers segmentation of 14 spinal structures in T2w sagittal images. It provides a semantic mask and an instance mask separating the vertebrae and intervertebral discs. This is the first publicly available algorithm to enable this segmentation. Key Points QuestionNo publicly available automatic approach can yield semantic and instance segmentation masks for the whole spine (including posterior elements) in T2-weighted sagittal TSE images. FindingsSegmenting semantically first and then instance-wise outperforms a baseline trained directly on instance segmentation. The developed model produces high-resolution MRI segmentations for the whole spine. Clinical relevanceThis study introduces an automatic approach to whole spine segmentation, including posterior elements, in arbitrary fields of view T2w sagittal MR images, enabling easy biomarker extraction, automatic localization of pathologies and degenerative diseases, and quantifying analyses as downstream research.

On the dualizability of fusion 2-categories

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple tensor 2-categories, separable bimodule 2-categories, and their morphisms. Categorifying a result of Douglas, Schommer-Pries, Snyder [Mem. Amer. Math. Soc. 268 (2021)], we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of Décoppet [Comp. Math.] that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object of the above Morita 4-category. We explain how this can be extended to any field of characteristic zero. Finally, we discuss the field theoretic interpretation of our results.

COLMAR1d: A Web Server for Automated, Quantitative One-Dimensional Nuclear Magnetic Resonance-Based Metabolomics at Arbitrary Magnetic Fields.

The field of metabolomics, which is quintessential in today's omics research, involves the large-scale detection, identification, and quantification of small-molecule metabolites in a wide range of biological samples. Nuclear magnetic resonance spectroscopy (NMR) has emerged as a powerful tool for metabolomics due to its high resolution, reproducibility, and exceptional quantitative nature. One of the key bottlenecks of metabolomics studies, however, remains the accurate and automated analysis of the resulting NMR spectra with good accuracy and minimal human intervention. Here, we present the COLMAR1d platform, consisting of a public web server and an optimized database, for one-dimensional (1D) NMR-based metabolomics analysis to address these challenges. The COLMAR1d database comprises more than 480 metabolites from GISSMO enabling a database query of spectra measured at arbitrary magnetic field strengths, as is demonstrated for spectra acquired between 1H resonance frequencies of 80 MHz and 1.2 GHz of mouse serum, DMEM cell growth medium, and wine. COLMAR1d combines the GISSMO metabolomics database concept with the latest tools for automated processing, spectral deconvolution, database querying, and globally optimized mixture analysis for improved accuracy and efficiency. By leveraging advanced computational algorithms, COLMAR1d offers a user-friendly, automated platform for quantitative 1D NMR-based metabolomics analysis allowing a wide range of applications, including biomarker discovery, metabolic pathway elucidation, and integration with multiomics strategies.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

On affine spaces of rectangular matrices with constant rank

Let F be a field, and n≥p≥r>0 be integers. In a recent article, Rubei has determined, when F is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in F in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than r+1 elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of Ms(F) with dimension (s2). The latter is known to be connected to the classification of nonisotropic quadratic forms over F up to congruence.

Fusion categories over non-algebraically closed fields

Several complications arise when attempting to work with fusion categories over arbitrary fields. Here we describe some of the new phenomena that occur when the field is not algebraically closed, and we adapt tools such as the Frobenius-Perron dimension in order to accommodate these new effects.

On improving sound source localization in the wind tunnel

When traveling in an open-jet wind tunnel, the path of an acoustic wave is affected by the flow causing a shift of source positions in acoustical maps of phased arrays outside the flow. In this paper, we start by comparing several well-known approaches to correct travel times between microphones and assumed sources, used, for example, by beamforming algorithms in such an environment. The methods under consideration include the original 1D-Amiet/Bahr formulation, a standard 2.5D-planar approach that assumes the constancy of angular frequency and wave-number-vector components along a plane, and finally, a 3D-ray-tracing method. Common to the former two of these methods is the assumption that the boundary layer of the flow, the so-called shear layer, is infinitely thin and refracts the acoustical ray, whereas, in principle, the latter algorithm allows for an arbitrary (but sufficiently smooth) vector field modeling the flow. Going through these methods, we discuss travel-time results relative to each other and in terms of differences in source localization using beamforming maps. In particular, we model the flow of an existing wind tunnel and discuss results of real-world measurements comparing the 2.5D-planar approach with the 3D-ray-tracing method.

Torsion homology growth and cheap rebuilding of inner-amenable groups

We prove that virtually torsion-free, residually finite groups that are inner-amenable and non-amenable have the cheap 1-rebuilding property, a notion recently introduced by Abért, Bergeron, Frączyk and Gaboriau. As a consequence, the first \ell^{2} -Betti number with arbitrary field coefficients and log-torsion in degree 1 vanish for these groups. This extends results previously known for amenable groups to inner-amenable groups. We use a structure theorem of Tucker-Drob for inner-amenable groups showing the existence of a chain of q -normal subgroups.

The rank of sparse symmetric matrices over arbitrary fields

AbstractLet be an arbitrary field and be a sequence of sparse weighted Erdős–Rényi random graphs on vertices with edge probability , where weights from are assigned to the edges according to a matrix . We show that the normalized rank of the adjacency matrix of converges to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalized rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse nonweighted Erdős–Rényi matrices over . Our proof, which is purely combinatorial in its nature, is based on an intricate extension of a novel perturbation approach to the symmetric setting.

Exact results of the one-dimensional repulsive Hubbard model

We present analytical results of the fundamental properties of the one-dimensional (1D) Hubbard model with a repulsive interaction. The new model results with arbitrary external fields include: (I) using the exact solutions of the Bethe ansatz equations of the Hubbard model, we first rigorously calculate the gapless spin and charge excitations, exhibiting exotic features of fractionalized spinons and holons. We then investigate the gapped excitations in terms of the spin string and thek-Λstring bound states at arbitrary driving fields, showing subtle differences in spin magnons and chargeη-pair excitations. (II) For a high-density and high spin magnetization region, i.e. near the quadruple critical point, we further analytically obtain the thermodynamic properties, dimensionless ratios and scaling functions near quantum phase transitions. (III) Importantly, we give the general scaling functions at quantum criticality for arbitrary filling and interaction strength. These can directly apply to other integrable models. (IV) Based on the fractional excitations and the scaling laws, the spin-incoherent Luttinger liquid (SILL) with only the charge propagation mode is elucidated by the asymptotic of the two-point correlation functions with the help of conformal field theory. We also, for the first time, obtain the analytical results of the thermodynamics for the SILL. (V) Finally, to capture deeper insights into the Mott insulator and interaction-driven criticality, we further study the double occupancy and propose its associated contact and contact susceptibilities, through which an adiabatic cooling scheme based upon quantum criticality is proposed. In this scenario, we build up general relations among arbitrary external- and internal-potential-driven quantum phase transitions, providing a comprehensive understanding of quantum criticality. Our methods offer rich perspectives of quantum integrability and offer promising guidance for future experiments with interacting electrons and ultracold atoms, both with and without a lattice.

Gradings on the algebra of triangular matrices as a Lie algebra: Revisited

We investigate the group gradings on the algebras of upper triangular matrices over an arbitrary field, viewed as Lie algebras. Classification results were obtained in 2017 by the same authors when the base field has characteristic different from 2. In this paper we provide streamlined proofs of these results. Moreover we present a complete classification of isomorphism classes of the group gradings on these algebras over an arbitrary field. Recall that two graded Lie algebras L1 and L2 are practically-isomorphic if there exists an (ungraded) algebra isomorphism L1→L2 that induces a graded-algebra isomorphism L1/z(L1)→L2/z(L2). We provide a classification of the practically-isomorphism classes of the group gradings on the Lie algebra of upper triangular matrices. The latter classification is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

Central limit theorem of overlap for the mean field Ghatak–Sherrington model

The Ghatak–Sherrington spin glass model is a random probability measure defined on the configuration space {0,±1,±2,…,±S}N with system size N and S⩾1 finite. This generalizes the classical Sherrington–Kirkpatrick (SK) model on the boolean cube {−1, +1}N to capture more complex behaviors, including the spontaneous inverse freezing phenomenon. We give a quantitative joint central limit theorem for the overlap and self-overlap array at sufficiently high temperature under arbitrary crystal and external fields. Our proof uses the moment method combined with the cavity approach. Compared to the SK model, the main challenge comes from the non-trivial self-overlap terms that correlate with the standard overlap terms.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Degenerate fourfold Massey products over arbitrary fields

We prove that, for all fields F of characteristic different from 2 and all a,b,c\in F^{\times} , the mod 2 Massey product \langle a,b,c,a\rangle vanishes as soon as it is defined. For every field F_{0} , we construct a field F containing F_{0} and a,b,c,d\in F^{\times} such that \langle a,b,c\rangle and \langle b,c,d\rangle vanish but \langle a,b,c,d\rangle is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod 2 cochain differential graded algebra of the absolute Galois group is not formal.

Seebeck and Nernst effects in the mixed state of strongly fluctuating superconductors

We study Seebeck and Nernst effects in strongly fluctuating superconductors by calculating the Seebeck and Nernst coefficients in the Ginzburg–Landau Lawrence–Doniach model with thermal noise. Strongly thermal fluctuations are represented by the Langevin thermal noise. To capture the interesting fluctuation effects, we use the self-consistent Gaussian approximation to treat cubic term in the equation of motion. Our results including all Landau levels are to treat arbitrary magnetic fields. The Seebeck and Nernst coefficients are calculated for arbitrary temperature and magnetic field. The results are compared to experimental data on Fe-based superconductor Fe1+yTe0.6Se0.4 in both the superconducting state below the critical temperature Tc and the normal state above Tc. Our calculation supports the picture of phase disordering of fluctuations above Tc in Fe-based superconductors.

Physics-informed neural network for volumetric sound field reconstruction of speech signals

Recent developments in acoustic signal processing have seen the integration of deep learning methodologies, alongside the continued prominence of classical wave expansion-based approaches, particularly in sound field reconstruction. Physics-informed neural networks (PINNs) have emerged as a novel framework, bridging the gap between data-driven and model-based techniques for addressing physical phenomena governed by partial differential equations. This paper introduces a PINN-based approach for the recovery of arbitrary volumetric acoustic fields. The network incorporates the wave equation to impose a regularization on signal reconstruction in the time domain. This methodology enables the network to learn the physical law of sound propagation and allows for the complete characterization of the sound field based on a limited set of observations. The proposed method’s efficacy is validated through experiments involving speech signals in a real-world environment, considering varying numbers of available measurements. Moreover, a comparative analysis is undertaken against state-of-the-art frequency domain and time domain reconstruction methods from existing literature, highlighting the increased accuracy across the various measurement configurations.