Normal Crossings Research Articles

Performance of the Earth Explorer 11 SeaSTAR Mission Candidate for Simultaneous Retrieval of Total Surface Current and Wind Vectors

Interactions between ocean surface currents, winds and waves at the atmosphere-ocean interface are key controls of lateral and vertical exchanges of water, heat, carbon, gases and nutrients in the global Earth System. The SeaSTAR satellite mission concept proposes to better quantify and understand these important dynamic processes by measuring two-dimensional fields of total surface current and wind vectors with unparalleled spatial and temporal resolution (1 × 1 km2 or finer, 1 day) and unmatched precision over one continuous wide swath (100 km or more). This paper presents a comprehensive numerical analysis of the expected performance of the Earth Explorer 11 (EE11) SeaSTAR mission candidate in the case of idealised and realistic 2D ocean currents and wind fields. A Bayesian framework derived from satellite scatterometry is adapted and applied to SeaSTAR’s bespoke inversion scheme that simultaneously retrieves total surface current vectors (TSCV) and ocean surface vector winds (OSVW). The results confirm the excellent performance of the EE11 SeaSTAR concept, with Root Mean Square Errors (RMSE) for TSCV and OSVW at 1 × 1 km2 resolution consistently better than 0.1 m/s and 0.4 m/s, respectively. The analyses highlight some performance degradation in some relative wind directions, particularly marked at near range and low wind speeds. Retrieval uncertainties are also reported for several variations around the SeaSTAR baseline three-azimuth configuration, indicating that RMSEs improve only marginally (by ∼0.01 m/s for TSCV) when including broadside Radial Surface Velocity or broadside dual-polarisation data in the inversion. In contrast, our results underscore a) the critical need to include broadside Normalised Radar Cross Section data in the inversion; b) the rapid performance degradation when broadside incidence angles become steeper than 20° from nadir; and c) the benefits of maintaining ground squint angle separation between fore and aft lines-of-sight close to 90°. The numerical results are consistent with experimental performance estimates from airborne data and confirm that the EE11 SeaSTAR concept satisfies the requirements of the mission objectives.

High-precision modelling of deformation of sandwich structures under bilateral symmetric and oblique-symmetric loading

The analysis and assessment of the stress-strain state (STS) of multilayer plates with orthotropic layers under the action of stationary transverse and tangential loads is an urgent task. It includes calculations on the strength and deformability of various homogeneous and multi-layered plates with layers of constant thickness but arbitrary structure according to the thickness of the plate. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexural components, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate. This significantly simplifies its modeling. For unflexural and exclusively flexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic, transverse-isotropic and orthotropic layers are constructed in an elastic formulation. That models takes full account deformation of transverse shear and transverse compression at transverse and the tangential loading of a plate. The model is continuous, that is, the number of equations and the order of differentiation of the calculation system of equations does not depend on the number of layers in the slab. This order of differentiation and the number of calculation equations can depend only on the order of iterative approximation of the model. A way of precise satisfaction of all defining correlations of the layers of material under keeping the conditions of their contact is found, while in the known continual models the dependence between the cross normal stress and cross deformation is only integral.The results of the analytical solution of the problem of deformation of a rectangular plate under boundary conditions of the Navier type under the action of transverse and tangential loads are given. By solving the test problems of the deformation of two-layer plates with transversally isotropic layers and three-layer plates with orthotropic layers and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, an assessment of the accuracy of the proposed refined models is given. The limits of admissible parameters of elastic characteristics of transversely isotropic and orthotropic plates for application of the offered models are established.

Models and integral differentials of hyperelliptic curves

Abstract Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$ , defined over a complete discretely valued field $K$ , with ring of integers $O_K$ . Under certain conditions on $C$ , mild when residue characteristic is not $2$ , we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$ . In the same setting we determine a basis of integral differentials of $C$ , that is an $O_K$ -basis for the global sections of the relative dualising sheaf $\omega _{\mathcal{C}/O_K}$ .

Logarithmic quasimaps

We construct a proper moduli space which is a Deligne–Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella–Nabijou where the latter is defined.

On the weight zero compactly supported cohomology of

Abstract For $g\ge 2$ and $n\ge 0$ , let $\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$ denote the complex moduli stack of n-marked smooth hyperelliptic curves of genus g. A normal crossings compactification of this space is provided by the theory of pointed admissible $\mathbb {Z}/2\mathbb {Z}$ -covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of $\mathcal {H}_{g, n}$ . Using this graph complex, we give a sum-over-graphs formula for the $S_n$ -equivariant weight zero compactly supported Euler characteristic of $\mathcal {H}_{g, n}$ . This formula allows for the computer-aided calculation, for each $g\le 7$ , of the generating function $\mathsf {h}_g$ for these equivariant Euler characteristics for all n. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible G-covers of genus zero curves, when G is abelian, as a symmetric $\Delta $ -complex. We use these complexes to generalize our formula for $\mathsf {h}_g$ to moduli spaces of n-pointed smooth abelian covers of genus zero curves.

Liquid Metal‐Based High‐Density Interconnect Technology for Stretchable Printed Circuits

AbstractHigh‐density interconnect (HDI) technology for liquid metal (LM)‐based stretchable printed circuit boards is crucial for expanding their applicability. HDI technology provides high‐resolution multilayer circuits with a high density of components, as required for next generation of neuroprobes and ultrasonic and sensor arrays. This study presents a HDI technique utilizing laser engraved micro grooves in silicone with a protective lift‐off mask of polyvinyl alcohol (PVA) and subsequent microscale LM particle spray deposition. This approach achieves high‐resolution LM patterns and enables multilayer connectivity and high‐density integration of components simultaneously, that is, realizing HDI technology. It is demonstrated using a stretchable 0201 LED display with a density of six leads per mm2 and a cochlear implant (CI) electrode array. The demonstrated manufacturing of the CI has the potential to enable fully automated printed circuit board manufacturing of such implants with increased precision and throughput. Implants in guinea pigs show that the CI is capable of activating the auditory neuron with electrical auditory brainstem response (eABR) and electrical compound action potential (eCAP) of high quality. Moreover, the U‐shaped cross section of the LM interconnects provides a higher mechanical shock resistance of the circuit than that of normal rectangular cross sections.

Regular models of hyperelliptic curves

Let K be a complete discretely valued field of residue characteristic not 2 and OK its ring of integers. We explicitly construct a regular model over OK with strict normal crossings of any hyperelliptic curve C/K:y2=f(x). For this purpose, we introduce the new notion of MacLane cluster picture, that aims to be a link between clusters and MacLane valuations.

On the Ohsawa–Takegoshi Extension Theorem

We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold. Some of the examples presented in the appendix show that the bounds we obtain for the extension are sharp.

Features of calculations of normal cross sections of reinforced concrete structures according to SP 63.13330.2018

Introduction. SP 63.13330.2018 provides two approaches to calculating the strength of normal cross-sections of reinforced concrete structures: the nonlinear deformation model NDM (general approach) and the method of ultimate forces UFM for simple elements of rectangular, T-shaped and I-shaped sections with reinforcement located at the edges of the element. As far as the authors are aware, no detailed studies have been conducted to compare the results obtained with both approaches.Aim. Conduct researchе to compare the results obtained in the calculations for NDM and UFM, identify the differences, establish their physical cause, evaluate the economic aspects of the problem, and make the necessary recommendations.Materials and methods. Calculations were carried out according to the program "OM SNiP Reinforced Concrete". Studies were performed for three types of structures. The first type included four cross sections of elements under two stress states: bending (T-shaped and rectangular), off-center compression (rectangular and annular). The second type of structures included beams tested at the NIIZHB named after A.A. Gvozdev. The third type is lintels according to State Standard 948-84. In all types of structures, under the action of the same forces, reinforcement was selected according to NDM and UFM.Results. It turned out that the compressed zone of concrete in UFM participates in the resistance of cross sections to acting forces much more effectively than in NDM. This can lead to a significant difference in reinforcement. In one of the cases, the steel consumption in the calculations for UFM turned out to be 78 % less than for NDM. Conclusions. For the considered structures, the calculation of the strength of normal cross sections by UFM turned out to be more effective than by NDM. At the same time, all the necessary regulatory requirements are met. Further research is needed to amend the existing regulations.

Torus fibers and the weight filtration

We show that if ( X , Y ) (X,Y) is a simple normal crossings log Calabi–Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of Y Y which can be used to construct W 2 k − 1 H k ( X ∖ Y ; Q ) W_{2k-1}\mathrm {H}^k(X \setminus Y;\mathbb {Q}) for all k k . We use this to show that a P=W type result holds for pairs ( X , Y ) (X,Y) consisting of a rational surface X X and a nodal anticanonical divisor Y Y .

Resolution of singularities for C∞ functions and meromorphy of local zeta functions

In this paper, we attempt to resolve the singularities of the zero variety of a C∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.

Measurement of inclusive and differential cross sections for single top quark production in association with a W boson in proton-proton collisions at sqrt{s} = 13 TeV

Measurements of the inclusive and normalised differential cross sections are presented for the production of single top quarks in association with a W boson in proton-proton collisions at a centre-of-mass energy of 13 TeV. The data used were recorded with the CMS detector at the LHC during 2016–2018, and correspond to an integrated luminosity of 138 fb−1. Events containing one electron and one muon in the final state are analysed. For the inclusive measurement, a multivariate discriminant, exploiting the kinematic properties of the events is used to separate the signal from the dominant textrm{t}overline{textrm{t}} background. A cross section of 79.2pm 0.9{left(textrm{stat}right)}_{-8.0}^{+7.7}left(textrm{syst}right)pm 1.2left(textrm{lumi}right) pb is obtained, consistent with the predictions of the standard model. For the differential measurements, a fiducial region is defined according to the detector acceptance, and the requirement of exactly one jet coming from the fragmentation of a bottom quark. The resulting distributions are unfolded to particle level and agree with the predictions at next-to-leading order in perturbative quantum chromodynamics.

Parabolic connections and stack of roots

Given a scheme over a field endowed with a strict normal crossings divisor, we define strongly parabolic connections, consistently with the current terminology for Higgs bundles. When the weights are rational with prescribed denominators, we show that strongly parabolic connections correspond to holomorphic connections on the corresponding stack of roots. We use this correspondence to establish that a holomorphic connection on a stack of roots can be reconstructed from its direct image to the moduli space.

Quadratic relations between periods of connections

We prove the existence of quadratic relations between periods of meromorphic flat bundles on complex manifolds with poles along a divisor with normal crossings under the assumption of "goodness". In dimension one, for which goodness is always satisfied, we provide methods to compute the various pairings involved. In an appendix, we give details on the classical results needed for the proofs. V3: Revised version, various proofs simplified in Section 3, exposition improved.

On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows

Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.

Tame fundamental groups of pure pairs and Abhyankar’s lemma

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the \'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods.

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

Complexes, residues and obstructions for log-symplectic manifolds

We consider compact Kählerian manifolds X of even dimension 4 or more, endowed with a log-symplectic structure Phi , a generically nondegenerate closed 2-form with simple poles on a divisor D with local normal crossings. A simple linear inequality involving the iterated Poincaré residues of Phi at components of the double locus of D ensures that the pair (X, Phi ) has unobstructed deformations and that D deforms locally trivially.

Logarithmic Gromov–Witten theory with expansions

We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.

Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.