Singular Surface Research Articles

On Bruhat-Tits theory over a higher dimensional base

Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies certain tameness assumptions \eqref{tameness}. Let $\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ and set $K_{_n} := \text{Fract}~\cO_{_n}$. Let $G$ be an almost-simple, simply-connected affine Chevalley group scheme with a maximal torus $T$ and a Borel subgroup $B$. Given a $n$-tuple ${\bf f} = (f_{_1}, \ldots, f_{_n})$ of concave functions on the root system of $G$ as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we define {\it {\tt n}-bounded subgroups ${\tt P}_{_{\bf f}}\subset G(K_{_n})$} as a direct generalization of Bruhat-Tits groups for the case $n=1$. We show that these groups are {\it schematic}, i.e. they are valued points of smooth {\em quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it adapted to the divisor with normal crossing $z_1 \cdots z_n =0$} in the sense that the restriction to the generic point of the divisor $z_i=0$ is given by $f_i$ (resp. sums of concave functions given by points of the apartment). This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In \S\ref{mixedstuff}, under suitable assumptions on $k$ \S \ref{charassum}, we extend all these results for a $n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the root system of $G$ replacing $\mathcal O_{_n}$ by ${\cO} \llbracket x_{_1},\cdots,x_{_n} \rrbracket$ where $\cO$ is a complete discrete valuation ring with a perfect residue field $k$ of characteristic $p$. In the last part of the paper, we give applications in char zero to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of {\tt 2-parahoric} group schemes on minimal resolutions of surface singularities that arose in \cite{balaproc}.

Steenbrink‐type vanishing for surfaces in positive characteristic

AbstractLet be a pair of a normal surface over a perfect field of characteristic and an effective ‐divisor on . We prove that Steenbrink‐type vanishing holds for if it is log canonical and , or it is ‐pure. We also show that rational surface singularities satisfying the vanishing are ‐injective.

Universal phase-field mixture representation of thermodynamics and shock-wave mechanics in porous soft biologic continua.

A continuum mixture theory is formulated for large deformations, thermal effects, phase interactions, and degradation of soft biologic tissues suitable at high pressures and low to very high strain rates. Tissues consist of one or more solid and fluid phases and can demonstrate nonlinear anisotropic elastic, viscoelastic, thermoelastic, and poroelastic physics. Under extreme deformations or shock loading, tissues may fracture, tear, or rupture. Existing models do not account for all physics simultaneously, and most poromechanics and soft-tissue models assume incompressibility of some or all constituents, generally inappropriate for modeling shock waves or extreme compressions. Motivated by these prior limitations, a thermodynamically consistent formulation that combines a continuum theory of mixtures, compressible nonlinear anisotropic thermoelasticity, viscoelasticity, and phase-field mechanics of fracture is constructed to resolve the pertinent physics. A metric tensor of generalized Finsler space supplies geometric insight on effects of rearrangements of microstructure, for example degradation, growth, and remodeling. Shocks are modeled as singular surfaces. Hugoniot states and shock decay are analyzed: Solutions account for concurrent viscoelasticity, fracture, and interphase momentum and energy exchange not all contained in previous analyses. Suitability of the framework for representing blood, skeletal muscle, and liver is demonstrated by agreement with experimental data and observations across a range of loading rates and pressures. Insight into previously unresolved physics is obtained, for example importance of rate sensitivity of damage and quantification of effects of dissipation from viscoelasticity and phase interactions on shock decay.

K-Unstable Singular del Pezzo Surfaces Without Anticanonical Polar Cylinder

Abstract We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.

Categorification of the plurigenera of Gorenstein normal surface singularities

Consider a complex normal surface singularity and its three plurigenera, the m-th L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}–plurigenus of Watanabe, the m-th plurigenus of Knöller and the m-th log-plurigenus of Morales. For any of these invariants we construct a double graded Z[U]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}[U]$$\\end{document}–module, whose Euler characteristic is the chosen plurigenus. The three outputs are compared with the analytic lattice cohomology of the germ, whose Euler characteristic is the classical geometric genus.

Nanosecond laser-assisted fabrication of Ti6Al4V surfaces with gradient wettability and robust cross-linked microstructures

Abstract Surfaces with gradient wettability outperform singular superhydrophobic surfaces in terms of self-cleaning efficiency, anti-contamination properties, and fluid manipulation. These attributes offer extensive application potential across various industrial and scientific domains. This study introduces and employs nanosecond pulsed laser ablation to create wettability gradient surfaces on Ti6Al4V alloys, featuring robust cross-linked frame microstructures. Experimental results demonstrate that by varying the ridge width (w), associated with the liquid-solid contact fraction, we can achieve varying wettability and mechanical durability in these cross-linked frame microstructures. Wettability tests indicate static contact angles ranging from 150.7° to 105.45°. Furthermore, the sandpaper linear abrasion test illustrates a decrease in material wear rate with an increase in w. Under similar test conditions, the proposed surfaces demonstrate superior mechanical durability compared to two other prevalent wettability surface structures. The proposed surfaces, efficiently and eco-friendly produced through nanosecond laser fabrication, hold tremendous potential for diverse applications.

Sculpturing metal nanoparticles by controlled massive deformation

Controlling the shape and orientation of metal nanoparticles constitutes fundamental strategies employed to tune their distinctive properties. However, singular high-energy surfaces are notably missing from nanoparticles fabricated using existing synthesis techniques. In this work, we fabricated supported (111)-oriented Pt nanoparticles that expose low-energy surfaces and subsequently reshaped them using a metal forming technique. We found that the orientations of deformed particles span a continuum encompassing (111), (110) and (112) and observed a linear dependence of this re-orientation on the plastic strain. We proposed a model describing particle rotation during deformation in terms of synergistic interplay between slip and diffusion.

G-torsors and universal torsors over nonsplit del Pezzo surfaces

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of SL in PicSL, or a form of it containing the Néron–Severi torus. Let G be the G-torsor over SL obtained by extension of structure group from a universal torsor T over SL. We prove that G does not descend to S unless T does. This is in contrast to a result of Friedman and Morgan that such G always descend to singular del Pezzo surfaces over C from their desingularizations.

Non-contact discharge estimation at a river site by using only the maximum surface flow velocity

The study proposes a novel method of computing river discharge based on the maximum surface velocity recorded using a non-contact-based measurement at a singular water surface point. This location, generally, coincides with the maximum flow depth of the cross-section and accounts for the dip phenomena, where the maximum instream velocity occurs below the water surface. The method is based on information entropy theory developed by Shannon (1948) and applied to river hydraulics. In this study an alternate form of entropy is used to compute discharge as a function of the cross-sectional mean velocity, maximum velocity and shear velocity (Keulegan,1938) by minimizing the error of the state equilibrium constant, ΦM, which is the ratio between the mean and maximum flow velocity, and that estimated using the Keulegan-based relationship. To test the accuracy of the proposed method, the maximum surface flow velocities measured at two gauging stations, each located on two different Italian rivers were studied. The estimated discharges by the proposed method were found to be comparable with the existing non-contact discharge method advocated by Moramarco et al. (2017) and, the traditional velocity-area method, using, e.g., the mean-section approach, based on the following metrics: the Nash-sutcliffe Efficiency (NSE), the coefficient of correlation and the percent bias (PBIAS). The mean velocity error emulates a Gaussian distribution for both the gauging stations and was within 95% and 5% confidance levels. Further, the entropy-based velocity profiles generated by the proposed method at the y-axis are consistent with those of the depth-based velocity profiles observed by the mechanical-current meter, thus, proving the appropriateness of the proposed discharge estimation method.

Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli

In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems it is crucial to establish several isomorphisms between different deformation functors, that we expect that will be useful in further work as well.

Joshi–Malafarina–Narayan singularity in weak magnetic field

The importance and significance of magnetic fields in the astrophysical scenario is well known. Many domains of astrophysical black hole physics such as polarized shadow image, high energy emitting processes and jet formation are dependent on the behavior of the magnetic fields in the vicinity of the compact objects. In light of this, we determine the master equation and master differential equation that determine the spatial behavior of the magnetic field inside a matter distribution or vacuum region, of general spherically symmetric metric, which is immersed in a test magnetic field. We also investigate here the case of JMN-1 singularity immersed in a uniform weak magnetic field and determine the behavior of magnetic fields by defining electromagnetic four potential vector. We find that the tangential component of the magnetic field is discontinuous at the matching surface of the JMN-1 singularity with the external Schwarzschild metric, resulting in surface currents. We define the covariant expression of surface current density in this scenario. We also analyze the behavior of center-of-mass energy of two oppositely charged particles in the geometry of the magnetized JMN-1 singularity. We briefly discuss the possible scenarios which would possess a discontinuous magnetic field and implications of the same and future possibilities in the realm of astrophysics are indicated.

On Yau sequence over complete intersection surface singularities of Brieskorn type

On Yau sequence over complete intersection surface singularities of Brieskorn type

On the application of Krylov subspace spectral methodologies to poroacoustic shock formation in an exponential class of inhomogeneous gases

In this communication, the propagation of poroacoustic acceleration waves in a class of inhomogeneous gases whose ambient mass density varies exponentially is considered. Employing the tools of singular surface theory, we first determine the evolution of both the jump amplitudes and the locations/velocities of their associated wave-fronts, along with a variety of related analytical results. We then turn to what have become known as Krylov subspace spectral (KSS) methods to numerically simulate the evolution of the full waveforms under consideration. These simulations are not only performed quite efficiently, since KSS allows the use of “large” CFL numbers, but also quite accurately, in the sense of capturing theoretically-predicted features of the solution profiles, since KSS customizes the computation of the components (of a solution) corresponding to the different frequencies involved. The presentation concludes with a discussion of possible acoustics-related follow-on studies.

Singularities of Flat Dual Surfaces of Cuspidal Edges in the Three-Sphere from Duality Viewpoint

Singularities of Flat Dual Surfaces of Cuspidal Edges in the Three-Sphere from Duality Viewpoint

Approximations of Singular Surfaces with Standard Cuspidal Edges

We consider a function from the Euclidean three space whose zero set is the image of the standard cuspidal edge. The composition of a parametrized singular surface in the three space with this function provides an approximation of the surface by the standard cuspidal edge. Taking a look at singularities of this composition, we study various approximations of singular surfaces like the cross cap, the generalized cuspidal edge and the swallowtail by standard cuspidal edges.

Nodal quintic surfaces and lines on cubic fourfolds (with an appendix by John Christian Ottem)

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16 -nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville (1979), Murre (1972), and Voisin (1986), Results due to Shen (2012, 2014) are particularly close to some of the statements. In this sense, the text is mostly expository (but with complete proofs), although our arguments often differ substantially from the original sources.

Long-lasting low fluorinated stainless steel hierarchical surfaces for omniphobic, anti-fouling and anti-icing applications

Stainless steel (SS) alloys are prevalent in many industries, household appliances or other commodities, where a strict control of surface properties is required to tailor their interaction with the environment. In this work we report a new procedure of stainless steel surface processing that provides a multifunctional response including superhydrophobicity, omniphobicity, self-cleaning, anti-fouling and effective anti-icing capacity, while still preserving a corrosion resistance similar to that of this material in compact form. The method consists of a first nanostructuration step followed by a low fluorination. The nanostructured surfaces presented a dual scale roughness of hierarchical character. The liquid free approach developed in this work to get this singular surface nanostructuration entails a first laser treatment of stainless steel flat substrates, followed by the deposition of a nanostructured thin layer of this material by electron beam evaporation in an oblique angle configuration. The resulting hierarchical surfaces were subjected to fluorination by: (i) the plasma-assisted deposition of a thin Teflon-like coating or (ii) the grafting of fluorinated molecules. The self-cleanable, anti-adherent and ice repellent character of the resulting low fluorinated surfaces outperformed the behaviour of classical slippery surfaces obtained by the infusion of high amounts of fluorinated liquids. These hierarchical SS surfaces withstood mild abrasion tests and the effect of water jets. Moreover, the corrosion behaviour of the fluorinated surfaces determined through their potentiodynamic analysis revealed a similar corrosion resistance than the flat SS substrates. Outstandingly, after these corrosion tests, the fluorinated samples obtained by grafting preserved their surface functionalities without significant degradation. The high mechanical and chemical stability of these low fluorinated samples support their usage for a large variety of applications.

Cn -symmetric quasi-periodic Chern insulators

Chern insulators have been recently extended to quasicrystal systems, namely the quasi-periodic Chern insulators (QCIs). Here we study the topological properties and topological response of QCIs on two-dimensional singular surfaces. Such singular QCIs with arbitrary n− fold rotational symmetry (i.e. C n -symmetric QCIs) can be constructed by ‘cutting and gluing’ unit sectors on the Dürer’s pentagonal tiling. Chiral edge states and real-space Chern number can well characterize the topological properties of C n -symmetric QCIs. Intriguingly, we numerically identify the emergence of charge fractionalization in unit of e/10 around the singular center in Cn -symmetric QCIs though their bulk densities are inhomogeneous. In addition, we explore the phase transitions of these C n -symmetric QCIs.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Monte Carlo simulation of planar GaAs nanowire growth

The Monte Carlo model for the planar GaAs nanowire growth via the vapor–liquid-solid mechanism on GaAs substrate using a gallium droplet as a catalyst is realized. The effect of temperature, Ga and As2 deposition rates, substrate orientation and surface properties on the morphology of planar GaAs nanowires is considered. It is shown, that at the initial stages of nanowire growth, a 3D GaAs crystal is formed under a gallium droplet, which sets the nanowire growth direction. The range of temperatures and deposition rates of gallium and arsenic, which ensure the planar nanowire formation on the GaAs(111)A surface, is determined. The investigation of surface properties influence on the nanowire morphology showed that the arsenic diffusion influx into a gallium droplet is of critical importance for the planar GaAs nanowire growth. Ways for achieving a unidirectional growth of planar nanowires on singular and vicinal GaAs surfaces are proposed.