Residual Field Research Articles

EXTENDING TORSORS UNDER QUASI-FINITE FLAT GROUP SCHEMES

Abstract Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$ . In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

On general fiber product rings, Poincaré series and their structure

The present paper deals with the investigation of the structure of general fiber product rings R × T S , where R, S and T are local rings with common residue field. We show that the Poincaré series of any R-module over the fiber product ring R × T S is bounded by a rational function. In addition, we give a description of depth ( R × T S ) , which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R × T S obtained, we provide certain cases of the Cohen-Macaulayness of R × T S and, in particular, we show that R × T S is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R × T S are also established.

Clinical study of immune-targeting combined with attenuated chemotherapy in the treatment of children with classic Hodgkin lymphoma

Objective: To evaluate the efficacy and safety of brentuximab vedotin (BV) combined with rituximab and attenuated chemotherapy in the treatment of children with classic Hodgkin lymphoma (cHL). Methods: A prospective, non-randomized, risk-assigned study. Clinical data (including age, gender, B symptoms, bulky disease, CD30 and Epstein-Barr virus-encoded RNA(EBER) expression, clinical stage, risk stratification, etc.) of 28 intermediate to high-risk cHL children diagnosed and treated at Beijing Children's Hospital Affiliated to Capital Medical University from October 2022 to May 2024 were collected. Immuno-targeted combined with attenuated chemotherapy was administered based on risk stratification and early treatment response. The patients were followed up until May 1st, 2024. The infusion reactions and adverse reactions after treatment were recorded. Results: In all 28 patients, there were 22 males and 6 females, the age was 12 (5,16) years, 16 cases (57%) presented with bulky disease and 10 cases (36%) with B symptoms. The most common pathological type was nodular sclerosis (14 cases, 50%). There were 7 cases of stage Ⅱ, 14 cases of stage Ⅲ and 7 cases of stage Ⅳ according to the Ann Arbor staging system. There were 5 cases in the intermediate-risk group and 23 cases in the high-risk group. EBER was positive in 20 cases (71%) and negative in 6 cases (21%), and CD30 antigen was expressed in tumor cells of all enrolled children. Treatment duration: 5 cases (18%) received 4 courses of treatment, 21 cases (75%) received 6 courses of treatment, and 2 cases (7%) received 8 courses of treatment, 25 cases (89%) achieved complete metabolism response (CMR) through early assessment after 2 courses of chemotherapy. The CMR rates were 100% in intermediate-risk group and 87% in high-risk group, respectively. Four patients (14%) finally received residual field radiotherapy. Toxicities included grade Ⅰ-Ⅱ myelosuppression, early infusion reaction and mild peripheral neuropathy, only one case of grade 3 adverse events was recorded and did not affect sequential treatment. At the end of treatment and 3 months of follow-up, the levels of IgA, IgG and IgM were all decreased compared with the baseline before chemotherapy, and the total B cell count began to be lower than the level before chemotherapy at the early stage of treatment (after 2 courses). The total B cell count monitored during treatment was 50 (0, 101)×106/L and was 12 (0, 25)×106/L at the end of treatment. The follow-up time was 6 (3, 13) months, all 28 children had event-free survival and all achieved complete remission. At 6 and 9 months of follow-up, IgA, IgG, IgM and total B cell counts returned to pre-chemotherapy baseline levels, respectively. Conclusion: BV combined with rituximab attenuated chemotherapy has demonstrated efficacy and a tolerable safety profile in the treatment of cHL in children, and significantly reduce radiation rate.

Gravity and magnetic study of the Xitle volcanic field, Basin of Mexico – Inferences on the pre-eruptive topography

Results of a gravity and magnetic study over a lava field in the southern Basin of Mexico are used to investigate on lava emplacement and pre-eruptive topography. The characteristics of lava flows depend on multiple factors, including the lava composition, effusion rate, rheology, fluid content and underlying terrain. The ∼2000 years old volcanic field, formed by seven flow units emplaced by the Xitle eruption, covers an extensive area >80 km2 characterized by topographic relief on the Ajusco volcano flank and surrounding plain. The study is conducted on the northern end of the volcanic field. The gravity and magnetic study is based on regional-residual separation, vertical and horizontal derivatives, reduced to the pole, analytical signal, spectral analysis and forward modeling. The regional gravity field shows two anomaly lows in the central sector, with the residual field showing trends of gravity highs. The residual magnetic anomaly is characterized by trends of magnetic highs associated to the lava field. Depths estimated from spectral analysis of gravity and magnetic anomalies show shallow sources at ∼22 m and ∼46 m, and deep sources at ∼469 m and ∼268 m, respectively. The underground sequence is modeled as a volcanoclastic succession of varying thickness. The first vertical derivatives, gradients and analytical signal enhance the anomalies associated to lava thickness variations. Joint gravity and magnetic analysis constraints the deep/shallow anomaly sources in the volcanic field and relations to pre-eruptive structure.

Non-uniform dynamic magnetic field compensation for a lightweight magnetically shielded room based on distributed differential coils

Abstract Due to the influence of material characteristics and engineering imperfections, the residual magnetic fields including static and dynamic field in a lightweight magnetically shielded room (MSR) present non-uniform distribution characteristic. To effectively reduce the residual static field, a non-uniform field compensation method based on distributed coils has been proposed. This method takes the L2-norm of the magnetic field at the target array points as the objective function, to optimize the compensation current in each of the designed distributed coils, achieving non-uniform field compensation. To further achieve compensation for non-uniform residual dynamic field, a closed-loop control system containing a voltage divider and multi-channel amplifiers is designed in this paper. Additionally, the coil constant of the distributed differential coils and the initial magnetic field distribution vector for dynamic compensation are redefined. In conjunction with a PI controller, dynamic compensation is achieved using a single optically pumped magnetometer (OPM) as the feedback component. The experimental results indicate that the average dynamic field decreased from 376.75 pT to 27.80 pT over the central volume of (300 mm)³ in MSR, achieving a near-zero magnetic environment. This method has universal applicability for compensating non-uniform residual field under different working conditions.

Relativistic beam loading, recoil-reduction, and residual-wake acceleration with a covariant retarded-potential integrator

An algorithm is demonstrated that performs first-principles tracking of relativistic charged-particles. A covariant approach is used which relies on retarded vector potentials for trajectory integration instead of performing electromagnetic field calculations. When accounting for retardation effects, the peak vector potential and corresponding Lorentz force in the direction of travel increase asymptotically for high-β particles. This produces a very strong field distribution at small angles from the particle’s direction of travel, which can result in considerable change in momentum when approaching a conducting or charged object. We study these dynamics using protons and electrons at relativistic energies passing through apertures in conducting surfaces, where substantial energy shifts are observed for particles passing within roughly 10μm of the aperture boundary.We also simulate breaking a test particle’s line of sight with a conductor or other charged body. After this instant, the test particle continues to accelerate due to residual fields, but no longer produces an opposing force on any charged or conducting object; thus any recoil on the enclosing structure is effectively reduced. In this test, a 1% energy gain is observed for an 85MeV electron traversing its reflected wake after having conducting plate in its path screened by a dielectric object.We then incorporate a micro-scale dielectric laser acceleration (DLA) device into our simulations. Compared with a 2 mm DLA on its own, we find a factor of two increase in energy gain when adding a series of conducting-surface choppers.

Admissibility over semi-global fields in the bad characteristic case

A finite group G is said to be admissible over a field F if there exists a division algebra D central over F with a maximal subfield L such that L/F is Galois with group G. In this paper we give a complete characterization of admissible groups over function fields of curves over equicharacteristic complete discretely valued fields with algebraically closed residue fields, such as the field FP‾((t))(x).

On Linear Codes over Local Rings of Order p4

Suppose R is a local ring with invariants p,n,r,m,k and mr=4, that is R of order p4. Then, R=R0+uR0+vR0+wR0 has maximal ideal J=uR0+vR0+wR0 of order p(m−1)r and a residue field F of order pr, where R0=GR(pn,r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p,n,r,m,k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.

Ax–Kochen–Ershov principles for finitely ramified henselian fields

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax–Kochen–Ershov principle as follows. We identify the induced structure on the residue field and show that once the residue field is endowed with this structure, the theory of the valued field is determined by the theories of the enriched residue field and the value group. Similarly, we show that the existential theory of the valued field is determined by the positive existential theory of the enriched residue field. We also prove that an embedding of finitely ramified henselian valued fields is existentially closed as soon as the induced embeddings of value group and residue field are existentially closed. This last result requires no enrichment of the residue field, in analogy to the corresponding result for model completeness, which holds by results of Ershov and Ziegler.

Brightness and contrast corrections for space–time stereocorrelation via proper generalized decomposition

Stereocorrelation (SC) is a flexible, non-contact experimental tool for measuring 3D shape and surface kinematics in complex mechanical tests. The violation of gray level conservation raises the issue of convergence of SC (e.g., when local specular reflections are observed). Brightness and contrast corrections (BCCs) then become necessary. Space–time separation allows modes of the displacement, brightness, and contrast fields to be computed on the fly. The paper introduces a framework for Brightness and Contrast Corrections in Proper Generalized Decomposition stereocorrelation (BCCs-PGD-SC). It is shown that BCCs-PGD-SC improves the faithfulness of measured displacement fields and makes the identification of cracks viable from residual fields. The computational cost to include BCCs in PGD stereocorrelation is shown to be minimal within a PGD framework.

A prismatic approach to $(\varphi, \hat G)$-modules and $F$-crystals

We give a new construction of (\varphi, \hat G) -modules using the theory of prisms developed by Bhatt and Scholze. We give two applications of our results. Firstly, we provide a new proof for the equivalence between the category of prismatic F -crystals in finite locally free \mathcal{O}_{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}} -modules over (\mathcal{O}_{K})_{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}} and the category of lattices in crystalline representations of G_{K} , where K is a complete discretely valued field of mixed characteristic with perfect residue field. Moreover, we generalize this result to semistable representations using the absolute logarithmic prismatic site defined by Koshikawa.

Defining ideals of Cohen–Macaulay fiber cones

Let [Formula: see text] be a commutative Noetherian local ring with maximal ideal [Formula: see text], and let [Formula: see text] be an ideal. The fiber cone is then an image of the polynomial ring over the residue field [Formula: see text]. The kernel of this map is called the defining ideal, and it is natural to ask how to compute it. In this paper, we provide a construction for the defining ideals of Cohen–Macaulay fiber cones.

Experimental study on microscopic production characteristics and influencing factors during dynamic imbibition of shale reservoir with online NMR and fractal theory

Dynamic imbibition and displacement between the matrix and fractures in shale reservoirs can significantly enhance oil recovery (EOR) following initial depletion. However, the microscopic production characteristics and seepage mechanisms at different pore scales during the dynamic imbibition process remain incompletely understood. In this study, we established an online physical simulation experiment method that combines dynamic displacement and imbibition based on nuclear magnetic resonance (NMR) and fractal theory, and a series of online NMR water flooding dynamic imbibition experiments were conducted. Through real-time dynamic monitoring of multiphase flow and migration behavior of crude oil in each stage of dynamic imbibition, the microscopic production characteristics and influencing factors were quantitatively studied from the recovery and residual oil saturation field distribution of different scale pores. Additionally, we developed a novel two-dimensional (2-D) NMR interpretation model to investigate the dynamic development characteristics of shale oil injection-imbibition-soaking-production integration, and the multiphase and multi-scale dynamic imbibition crude oil migration models were discussed. The results show that shale oil occurrence pores can be categorized into two types based on the corresponding production mode (imbibition + displacement). The water flooding dynamic imbibition process in shale reservoirs unfolds in three stages: strong displacement and weak imbibition stage characterized by the rapid production of large pores and fractures under displacement action; weak displacement and strong imbibition stage involving the slow production of small pores under reverse imbibition; and dynamic equilibrium stage characterized by weak displacement and weak imbibition. When viewing the entire water flooding dynamic imbibition process as an organism, it becomes imperative to maximize the recovery of large pores while ensuring the recovery degree of small pores. High permeability contributes to better pore throat connectivity and a greater imbibition and displacement production degree. The contribution of the.two production modes to the recovery of small and large pores increased by 4.67 % and 3.17 %, respectively. A negative correlation was observed between high displacement pressure and imbibition recovery, yet it is conducive to the contribution of displacement to total recovery. Notably, fractures can effectively reduce oil-water seepage resistance, and increase the contribution of displacement to recovery by 21.65 %. Finally.a positive correlation was noted between increased imbibition amount and free oil recovery, while residual oil gradually shifted towards adsorbed and organic matter-dominated forms, leading to decreased recoverability. Effectively utilizing the bridge flow conductivity of fractures, fluid imbibition displacement, and carrying effect is crucial for achieving optimal EOR. The findings provide theoretical support for clarifying the interaction between matrix-fracture imbibition and displacement and for the efficient development of shale oil.

A functorial approach to monomorphism categories II: Indecomposables

AbstractWe investigate the (separated) monomorphism category of a quiver over an Artin algebra . We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from to , where is the category of finitely generated ‐modules and and denote the respective injectively stable categories. Furthermore, if has at least one arrow, then we show that this is an equivalence if and only if is hereditary. In general, the epivalence induces a bijection between indecomposable objects in and noninjective indecomposable objects in , and we show that the generalized Mimo‐construction, an explicit minimal right approximation into , gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radical square zero Nakayama algebras, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. The main tool to prove these results is the language of a free monad of an exact endofunctor on an arbitrary abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras that were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the algebras that were introduced by Lu and Wang to realize groups via semiderived Hall algebras.

High-quality lipid suppression and B0 shimming for human brain 1H MRSI

Magnetic Resonance Spectroscopic Imaging (MRSI) is a powerful technique that can map the metabolic profile in the brain non-invasively. Extracranial lipid contamination and insufficient B0 homogeneity however hampers robustness, and as a result has hindered widespread use of MRSI in clinical and research settings. Over the last six years we have developed highly effective extracranial lipid suppression methods with a second order gradient insert (ECLIPSE) utilizing inner volume selection (IVS) and outer volume suppression (OVS) methods. While ECLIPSE provides > 100-fold in lipid suppression with modest radio frequency (RF) power requirements and immunity to B1+ field variations, axial coverage is reduced for non-elliptical head shapes. In this work we detail the design, construction, and utility of MC-ECLIPSE, a pulsed second order gradient coil with Z2 and X2Y2 fields, combined with a 54-channel multi-coil (MC) array. The MC-ECLIPSE platform allows arbitrary region of interest (ROI) shaped OVS for full-axial slice coverage, in addition to MC-based B0 field shimming, for robust human brain proton MRSI.In vivo experiments demonstrate that MC-ECLIPSE allows axial brain coverage of 92–95 % is achieved following arbitrary ROI shaped OVS for various head shapes. The standard deviation (SD) of the residual B0 field following SH2 and MC shimming were 25 ± 9 Hz and 18 ± 8 Hz over a 5 cm slab, and 18 ± 5 Hz and 14 ± 6 Hz over a 1.5 cm slab, respectively. These results demonstrate that B0 magnetic field shimming with the MC array supersedes second order harmonic capabilities available on standard MRI systems for both restricted and large ROIs. Furthermore, MC based B0 shimming provides comparable shimming performance to an unrestricted SH5 shim set for both restricted, and 5-cm slab shim challenges.Phantom experiments demonstrate the high level of localization performance achievable with MC-ECLIPSE, with ROI edge chemical shift displacements ranging from 1–3 mm with a median value of 2 mm, and transition width metrics ranging from 1–2.5 mm throughout the ROI edge. Furthermore, MC based B0 shimming is comparable to performance following a full set of unrestricted spherical harmonic fields up to order 5. Short echo time MRSI and GABA-edited MRSI acquisitions in the human brain following MC-shimming and arbitrary ROI shaping demonstrate full-axial slice coverage and extracranial lipid artifact free spectra. MC-ECLIPSE allows full-axial coverage and robust MRSI acquisitions, while allowing interrogation of cortical tissue proximal to the skull, which has significant value in a wide range of neurological and psychiatric conditions.

A 85Rb transverse modulation magnetometer in the geophysical range based on atomic alignment states

Abstract We have constructed a transverse modulated magnetometer based on spin alignment in a paraffin-coated 85 Rb cell operated in a geophysical magnetic field of 47.1 µT. When an orthogonal driving magnetic field ( x ^ axis) is resonant on the Larmor frequency ( z ^ axis), we have proposed a new method to zero the static residual magnetic fields in the transverse plane and achieved a sensitivity of 1.8 pT Hz − 1 with bandwidth of 200 Hz. The repump light ( F g = 2 → F e = 2 ) redistributes the populations in the ground state, rendering the state | F g = 2 ⟩ dark. This effect significantly amplifies the optical rotation signals nearly fourfold. The numerical solution of the Liouville equation is in good agreement with the experimental results. By using perturbation treatment and employing appropriate approximations, the derived analytical expressions for optical rotation are deduced to succinctly elucidate the dynamics of atomic alignment under parametric modulation. These outcomes could be extended for the advancement of an alignment magnetometer that has been designed to detect a weak magnetic signal in the geophysical range.

Physics-constrained adaptive kernel interpolation for region-to-region acoustic transfer function: a Bayesian approach

A kernel interpolation method for the acoustic transfer function (ATF) between regions constrained by the physics of sound while being adaptive to the data is proposed. Most ATF interpolation methods aim to model the ATF for fixed source by using techniques that fit the estimation to the measurements while not taking the physics of the problem into consideration. We aim to interpolate the ATF for a region-to-region estimation, meaning we account for variation of both source and receiver positions. By using a very general formulation for the reproducing kernel function, we have created a kernel function that considers both directed and residual fields as two separate kernel functions. The directed field kernel considers a sparse selection of reflective field components with large amplitudes and is formulated as a combination of directional kernels. The residual field is composed of the remaining densely distributed components with lower amplitudes. Its kernel weight is represented by a universal approximator, a neural network, in order to learn patterns from the data freely. These kernel parameters are learned using Bayesian inference both under the assumption of Gaussian priors and by using a Markov chain Monte Carlo simulation method to perform inference in a more directed manner. We compare all established kernel formulations with each other in numerical simulations, showing that the proposed kernel model is capable of properly representing the complexities of the ATF.

Optimized design of cubic grid coil system with high space utilization ratio for weak biomagnetic signal measurement

The uniform field coils with large uniform region are important for improving the resolution in bio-magnetic signal measurements. The traditional coil design methods suffer from small space utilization ratio which limits the mobility and accuracy of bio-magnetic signal measurements. In this paper, a novel cubic grid coil (CGC) design method is presented, with significantly improved space utilization ratio and simple structure. The iterative fastest current drop method (IFCD) is proposed to simplify coil structure, thus ensuring the controllability of the cubic grid coil system. The superiority of CGCs has been demonstrated by finite element simulations and a series of experiments. The space utilization ratio of CGCs reaches 22.4% while maintaining 2% inhomogeneity, improved by about 6 times compared with traditional coils. Furthermore, the CGC-based system for directly eliminating the geomagnetic field achieves a low residual magnetic field noise of 29.7 pT/Hz/1/2 at 1 Hz. The CGCs can provide new ideas for high-resolution, lightweight mobile measurements of weak bio-magnetic signals in the human body.

Branching rules for the restriction of regular representations of GL2(o/℘ r ) to SL2(o/℘ r )

Let o be a compact discrete valuation ring with maximal ideal ℘ such that the finite residue field o / ℘ has characteristic p. For r ≥ 2 and p = 2 , we obtain the branching rules for the restriction of a regular representation of GL 2 ( o / ℘ r ) to SL 2 ( o / ℘ r ) . These results have different behavior than that of the known case of p ≠ 2.

On Waring numbers of henselian rings

AbstractLet be a positive integer. Let be a henselian local ring with residue field of th level . We give some upper and lower bounds for the th Waring number in terms of and . In large number of cases, we are able to compute . Similar results for the th Waring number of the total ring of fractions of are obtained. We then provide applications. In particular, we compute and for and any prime .