Obstruction Theory Research Articles

Liberal Citizenship Through the Prism of Shia Jurisprudence: Embracing Fundamental over Partial Solutions

This article explores the compatibility of liberal citizenship with Twelver Shia jurisprudence, a topic previously analyzed from the perspective of Sunni schools, most notably in the extensive research of Andrew F. March. This study confronts the challenges of reconciling liberal citizenship with Islamic jurisprudence, as highlighted in March’s work, through the lens of Shia legal thought. Rather than aiming to critique or review March’s research, this article considers his work solely as a representative example addressing the topic from the perspective of Sunni jurisprudence. This approach provides readers with a fundamental contrast, illuminating the unique insights that emerge from examining the subject within the framework of Shia jurisprudence. Unlike Sunni jurisprudence, which addresses these issues case-by-case by reviewing relevant Quranic and narrational sources, Twelver Shia jurisprudence offers a more foundational resolution. Owing to the belief in the occultation of the twelfth Imam and its implications for the implementation of Islamic law, Shia scholars have advanced theories such as the theory of obstruction (insidād) and the suspension of the social and political dimensions of Sharia. These theories effectively narrow the scope of Sharia, allowing for the acceptance of laws from non-Islamic states and circumventing potential conflicts with liberal citizenship in the absence of the twelfth Imam.

Importance of adsorption compared with complexation for retarding C3S hydration via adding sodium gluconate

Adsorption effect on particle surfaces and complexation effect with free Ca2+ mostly determine the retarding performance of organic admixtures on cement hydration. However, it is difficult to identify which effect plays a more important role in retarding hydration by experimental methods. Here, a theoretical model was developed to investigate the retarding mechanisms of sodium gluconate (SG) on hydration of tricalcium silicate (C3S). Based on obstruction theory and complexation reaction kinetics, effects of adsorption and complexation were simulated to examine the retarding performance of C3S hydration with addition of SG. The proposed model well predicted the effect of additional dosing of SG on the retarding performance of C3S hydration. Theoretical parameter studies demonstrated that adsorption ratio contributed much largely to the delays in C3S hydration, compared with rate constant of complex generation. Therefore, it is confirmed that adsorption plays a more important role in regulating the retarding mechanism of C3S hydration.

Obstruction Theory for the $$\mathbb {Z}_2$$-Index of 4-Manifolds

Obstruction Theory for the $$\mathbb {Z}_2$$-Index of 4-Manifolds

Versality for pairs

For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs.

Circle-valued angle structures and obstruction theory

We study spaces of circle-valued angle structures, introduced by Feng Luo, on ideal triangulations of 3-manifolds. We prove that the connected components of these spaces are enumerated by certain cohomology groups of the 3-manifold with [Formula: see text]-coefficients. Our main theorem shows that this establishes a geometrically natural bijection between the connected components of the spaces of circle-valued angle structures and the obstruction classes to lifting boundary-parabolic PSL[Formula: see text]-representations of the fundamental group of the 3-manifold to boundary-unipotent representations into SL[Formula: see text]. In particular, these connected components have topological and algebraic significance independent of the ideal triangulations chosen to construct them. The motivation and main application of this study is to understand the domain of the state-integral defining the meromorphic 3D-index of Garoufalidis and Kashaev, necessary in order to classify the boundary-parabolic representations contributing to its asymptotics.

Semiregularity as a consequence of Goodwillie’s theorem

Abstract We realise Buchweitz and Flenner’s semiregularity map (and hence a fortiori Bloch’s semiregularity map) for a smooth variety X as the tangent of a generalised Abel–Jacobi map on the derived moduli stack of perfect complexes on X. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie’s theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex). Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if X is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov–Witten and Pandharipande–Thomas invariants. We also give generalisations allowing X to be singular, and even a derived stack.

The Atiyah class on algebraic stacks

Abstract We generalize Illusie’s definition of the Atiyah class to complexes with quasi-coherent cohomology on arbitrary algebraic stacks. We show that this gives a global obstruction theory for moduli stacks of complexes in algebraic geometry without derived methods. We give a similar generalization of the reduced Atiyah class, and we show various useful properties for working with Atiyah classes, such as compatibilities between the reduced and ordinary Atiyah class, and compatibility with tensor products and determinants.

Quasimaps to moduli spaces of sheaves on a surface

Abstract In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$ , if $g(C)\leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$ .

The Physiological Occlusion of the Central Canal May Be a Prerequisite for Syringomyelia Formation.

Syringomyelia is a common central nervous system disease characterized by the dilation of the central canal (CC). Regarding the pathogenesis of syringomyelia, cerebrospinal fluid (CSF) circulation obstruction in the subarachnoid space (SAS) of the spinal cord has been widely accepted. However, clinical and animal studies on obstructing the CSF in SAS failed to form syringomyelia, challenging the theory of SAS obstruction. The precise pathogenesis remains unknown. We utilized an extradural compression rat model to investigate the pathogenesis underlying syringomyelia. Magnetic resonance imaging enabled detection of syringomyelia formation. To assess CSF flow within the SAS, Evans blue was infused into the cisterna magna. Histological analysis allowed morphological examination of the CC. Furthermore, CSF flow through the CC was traced using Ovalbumin Alexa-Flour 647 conjugate (OAF-647). Scanning electron microscopy (SEM) enabled visualization of ependymal cilia. The findings showed that the dura mater below the compression segment exhibited lighter coloration relative to the region above the compression, indicative of partial obstruction within the SAS. However, the degree of SAS occlusion did not significantly differ between syringomyelia (SM-Y group) and those without (SM-N group). Intriguingly, hematoxylin and eosin staining and CSF tracing revealed occlusion of the CC accompanied by reduced CSF flow in the SM-Y group compared to SM-N and control groups. SEM images uncovered impairment of ependymal cilia inside the syringomyelia. CC occlusion may represent a physiological prerequisite for syringomyelia formation, while SAS obstruction serves to initiate disease onset. The impairment of ependymal cilia appears to facilitate progression of syringomyelia.

A robust theoretical approach to analyze the diffusion of natural and synthetic wound antimicrobials through polyethylene glycol hydrogel using obstruction theory

A robust theoretical approach to analyze the diffusion of natural and synthetic wound antimicrobials through polyethylene glycol hydrogel using obstruction theory

Mackey's obstruction map for discrete graded algebras

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles.Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules.The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules.The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.

The d-critical structure on the Quot scheme of points of a Calabi–Yau 3-fold

The Artin stack [Formula: see text] of zero-dimensional sheaves of length [Formula: see text] on [Formula: see text] carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack [Formula: see text], another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points [Formula: see text], which is a global critical locus for every [Formula: see text], and also carries a derived-in-flavor d-critical structure besides the one induced by the potential [Formula: see text]. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on [Formula: see text], where [Formula: see text] is a locally free sheaf of rank [Formula: see text] on a projective Calabi–Yau [Formula: see text]-fold [Formula: see text]. Finally, we prove that the perfect obstruction theory on [Formula: see text] induced by the Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the Hessian of the potential [Formula: see text].

Single-Particle Tracking in Poly(Ethylene Glycol) Diacrylate: Probe Size Effect on the Diffusion Behaviors of Nanoparticles in Unentangled Polymer Solutions.

A thorough understanding of the relevant factors governing the transport of nanoparticles in poly(ethylene glycol) diacrylate (PEGDA) is crucial for many applications utilizing this polymer. Here, single-particle tracking (SPT) was used to systematically investigate the role of the probe size (3-200 nm) on the diffusion behaviors of individual fluorescent nanoparticles in semidilute and unentangled PEGDA solutions. The quantitative assessment of the SPT data via the recorded single-particle trajectories and diffusion coefficients (D) not only showed that the observed probe dynamics in PEGDA were temporally and spatially heterogeneous, but more importantly that the measured D were observed to be significantly reduced (vs in solvent) and strongly size-dependent. We explained these results based on a modified multiscale model for particle diffusion, built upon well-established hydrodynamics and obstruction theories. We furthermore showed that the presence of steric interactions and probe confinement effects in highly crowded, unentangled PEGDA microstructures can lead to deviations in the single-particle displacements from the expected Gaussian behavior, as revealed by the van Hove displacement distributions and the associated non-Gaussian parameters. This study has demonstrated the power of SPT methods in offering an advanced characterization of the transport characteristics in complex polymer structures, overcoming challenges posed by traditional characterization techniques.

Modeling of inhaled corticosteroids delivery for topical croup treatment in pediatric upper airways

Croup is the most frequent cause of pediatric upper airway obstruction characterized by spindle-shaped stenosis in the subglottis mucosa. Inhaled corticosteroids (ICSs) serve as the first-line therapy for croup. Traditional ICS particles (1–4 μm in diameter) are primarily designed for trachea and lung diseases and show extremely low larynx deposition. Moreover, the specific correlation between airway minimal cross-sectional area (CSA) and airway resistance has not been fully understood. In this study, three healthy pediatric upper airway models with commercial nebulizer masks attached to their faces were reconstructed from computed tomography (CT) scans. Virtual mild, moderate, and severe croup were incorporated into these healthy models. To enhance the performance of conventional nebulizing drug delivery, the aerodynamic properties of croup with different degrees of stenosis were quantitatively analyzed, and the respiratory transit and deposition of ICS particles sized between 1 to 20 μm in our target area (glottis + subglottis) were modeled utilizing the Computational Fluid Particle Dynamics (CFPD) method. Results showed that in all models, maximum deposition fractions (DF) can be reached when the ICS particle sizes are 7–8 μm, and for particles sized at 8 and 9 μm, all models can achieve effective target area delivery (≥75% of the maximum DF), whereas the majority of traditional nebulizers produce smaller particles than what we recommended. Pediatric upper airway resistance is negatively correlated with the minimum airspace CSA (R ∝ CSA−1), which is in good agreement with the Bernoulli Obstruction Theory. Furthermore, when the constriction of the subglottis reaches a specific level (≥70% obstruction), the upper airway pressure drop abruptly surged and the dyspneic respiration symptoms of patients develop instantly.

Obstruction theories of extensions via track categories

Abelian track categories can be classified via the third Baues–Wirsching cohomology of small categories. This approach is used in this paper to derive the different generalizations of the obstruction theory of non-abelian group extensions, due to Cegarra, Garzón and Grandjean, Cegarra, Garzón and Ortega, and Chen, Du and Wang as particular cases of a general theory of abelian track categories and to give unified proofs of the main properties of these obstruction classes.

Deformation theory of orthogonal and symplectic sheaves

We show that the deformation functor of an orthogonal (resp. symplectic) sheaf over a smooth projective scheme admits a miniversal pro-family, identifying its space of first-order deformations with the first hypercohomology space of a complex which is naturally constructed out of the orthogonal (resp. symplectic) sheaf. We also provide an obstruction theory of these objects whose target is the second hypercohomology space of this complex.

Complete L∞-algebras and their homotopy theory

We analyze a model for the homotopy theory of complete filtered L∞-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E. Getzler which states that the category Lieˆ∞ of such L∞-algebras and filtration-preserving ∞-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Novel applications of our approach include explicit models for homotopy pullbacks, and an analog of Whitehead's Theorem: under some mild conditions, every filtered L∞-quasi-isomorphism in Lieˆ∞ has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer–Cartan functor, which assigns a Kan simplicial set to each L∞-algebra in Lieˆ∞, is an exact functor between the respective CFOs. Finally, we provide an obstruction theory for the general problem of lifting a Maurer-Cartan element through an ∞-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.

Virtual Riemann–Roch theorems for almost perfect obstruction theories

This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.

The inverse function theorem for curved $L$-infinity spaces

In this paper, we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved L_\infty spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for L_\infty algebras. The main ingredients are the obstruction theory for L_\infty homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L_\infty algebras. Both techniques work in the A_\infty case as well.

Rational points of algebraic varieties: a homotopical approach

This article, dedicated to the 100-th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.