Bundle Theory Research Articles

A constitutive relation for the tissue composed of type-I collagen fibers under uniaxial tension

Mechanical tests for various collagen tissues showed that the stress-strain curves are composed of two regimes with different deformation mechanisms. In this paper, the constitutive relations are first studied separately for the two regimes and then merged together. For the unfurling regime, a phenomenological power-law relation is utilized to characterize the stress-strain curve, and, for the stretching regime which is modeled as a fiber bundle material, a nonlinear constitutive relation is proposed based on the Weibull distribution and Daniels’ fiber bundle theory. A technique is suggested to continuously merge the two relations for the two regimes of collagen tissues. A strategy is finally devised to quantitatively determine the critical strain for dividing the two regimes and to identify the constitutive parameters for experimental curves. The constitutive relation is first validated by comparing the identified parameters with the references, and then applied to predict the experimental stress-strain curves of collagen tissues. The present work shows that the critical strain plays an essential role in establishing the constitutive relation if experimental data are adopted.

Bundle Theory with Kinds

Abstract Is it possible to get by with just one ontological category? We evaluate L.A. Paul's attempt to do so: the mereological bundle theory. The upshot is that Paul's attempt to construct a one category ontology may be challenged with some of her own arguments. In the positive part of the paper we outline a two category ontology with property universals and kind universals. We will also examine Paul's arguments against a version of universal bundle theory that takes spatiotemporal co-location instead of compresence or coinstantiation as the feature by which we can identify genuine bundles. We compare this novel theory, bundle theory with kinds, and Paul's mereological bundle theory and apply them to a case study concerning entangled fermions and co-located bosons.

Cartan subalgebras in C*-algebras. Existence and uniqueness

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analyzed systematically using the theory of fiber bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear uniform Roe algebras.

Dynamical Systems and Sheaves

A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

The topological chiral homology of the spherical category

We consider the spherical DG category $Sph_G$ attached to an affine algebraic group $G$. By definition, $Sph_G := IndCoh(LS_G(S^2))$ consists of ind-coherent sheaves of the stack of $G$-local systems on the $2$-sphere $S^2$. The $3$-dimensional version of the pair of pants endows $Sph_G$ with an $E_3$-monoidal structure. More generally, for an algebraic stack $Y$ (satisfying some mild conditions) and $n \geq -1$, we can look at the $E_{n+1}$-monoidal DG category $Sph(Y,n) := IndCoh_0((Y^{S^n})^\wedge_Y)$, where $IndCoh_0$ is the sheaf theory introduced in [AG2] and [centerH]. % The case of $Sph_G$ is recovered by setting $Y =BG$ and $n=2$. The cobordism hypothesis associates to $Sph(Y,n)$ an $(n+1)$-dimensional TQFT, whose value of a manifold $M^d$ of dimension $d \leq n+1$ (possibly with boundary) is given by the {topological chiral homology} $\int_{M^d} Sph(Y,n)$. % In this paper, we compute such homology (in virtually all cases): we have the Stokes style formula $$ \int_{M^d} Sph(Y,n) \simeq IndCoh_0 ( (Y^{\partial(M^d \times D^{n+1-d})})^\wedge_{Y^M} ) , $$ where the formal completion is constructed using the obvious projection $\partial(M^d \times D^{n+1-d}) \to M^d$. The most interesting instance of this formula is for $Sph_G \simeq Sph(BG,2)$, the original spherical category, and $X$ a Riemann surface. In this case, we obtain a monoidal equivalence $\int_X Sph_G \simeq H(LS_G^{Betti}(X))$, where $LS_G^{Betti}(X)$ is the stack of $G$-local systems on the topological space underlying $X$ and $H$ is the sheaf theory introduced in [centerH].

Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

Aharonov-Bohm Effect vs. Dirac Monopole: A-B ⇔ D

In the context of fiber bundle theory, we show that the existence of the Aharonov-Bohm connection implies the existence and uniqueness of the Dirac connection.

Studying Deformations of Fuchsian Representations with Higgs Bundles

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichm\"uller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747].

The Chimeric Self: A Neo Naturalist Bundle Theory of the Self.

In the contemporary cognitive science and philosophy of mind debate the definition of the ontology of the Self is difficult if not downright dubious. Thus, different theories aim to provide an account, especially where further neuroscientific research could be implemented. To this extent, the identity of the Self is suggested to be pinpointed by virtue of its specific set of mechanical features or brain functions, or it is considered the product of cognitive and conceptual capacities that build representations and narratives about ourselves. In this paper, I propose an alternative approach, based on the Spread Mind Theory (Manzotti, 2017a). Starting from the idea that the Self isn’t just an aprioristic or transcendental form, I claim, endorsing a Neo Naturalist approach, that our first-personal experience is identical to the external objects which, due to a physical relation, constitute the same experience. Thanks to an externalist explanation of the experience of the Self it is possible to avoid multiple ontologies, causal foundationalism, naïve materialism and questions begging about what we should explain. Clarifying the concept of the Self, as a bundle of relative objects which are identical with the experiences themselves, allows us to draw a physical ontology, based on the neutral (and natural) idea of relative existence rather than any posited concept of subjectivity.

УПРАВЛЕНИЕ НЕЛИНЕЙНОСТЬЮ ДЕФОРМИРОВАНИЯ КОМПОЗИТА С ПОМОЩЬЮ ЛОКАЛЬНОГО ИСКРИВЛЕНИЯ НИТЕЙ

Using the refined theory of dry bundles, we analyzed the curvature (length diversity) of yarns in the fabric composite for obtaining the maximum effect of pseudo-ductility (constant yield stresses during the deformation). Because of length diversity, yarns destruction happens consequently, starting with the straight yarns and ending with the most curved ones. The polymer matrix ensures a monolithic structure and high damping of the oscillations due to the intensification of matrix shears during tension/compression of the composite. To simulate an ideal elastoplastic stress-strain diagram of the composite, we formulated the law of changes in fiber lengths, the mechanical properties of which were considered to be the same. To study the technological possibilities of the local curvatures of yarns in fabric preforms, we performed experiments using the indenting method and arrays of conical and flat needles. We showed that conical needles allow us to obtain simultaneous curvature of warp and weft yarns to the same extent. The needle spacing controls the ratio of straight and curved yarns; so, by changing the spacing we can bring this ratio closer to the value required for getting the needed length of the yield plateau. In the case of flat needles with various orientation to the warp yarns, we can obtain anisotropic yarns curvature (of only one yarns family, for example). For experimental deformation of the fabrics with locally curved yarns we used aramid fabrics of plain weave and carbon fabrics of twill weave.

Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif dans le cas rationnel

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in ${\mathbb C}/\pi i {\mathbb Z}$. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that the holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.

THE CONORMAL TORUS IS A COMPLETE KNOT INVARIANT

We use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.

The identity of experiences and the identity of the subject

Barry Dainton has developed a sophisticated version of the bundle theory of the subject of experiences. I shall focus on three claims Dainton makes: the identity-conditions of subjects can be specified in terms of capacities to produce experiences; the identity-conditions of token capacities are not determined by their subjects; and a subject is nothing over and above a bundle of such capacities. I shall argue that Dainton’s key notion of co-consciousness, a primitive relation of experienced togetherness, presupposes a subject common to each of the experiences which are experienced together. Therefore, co-consciousness cannot be used to state the identity-conditions of subjects in a non-circular manner. I shall also argue that none of the different options Dainton offers for specifying the identity-conditions of token experiential capacities independently of their subjects are successful. I shall then outline a way in which this can be done, but argue that it undermines the claim that a subject is nothing over and above a bundle of such capacities.

Metric Properties of Parabolic Ample Bundles

AbstractWe introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.

Going Beyond the Data as the Patching (Sheaving) of Local Knowledge.

Consistently predicting outcomes in novel situations is colloquially called “going beyond the data,” or “generalization.” Going beyond the data features in spatial and non-spatial cognition, raising the question of whether such features have a common basis—a kind of systematicity of generalization. Here, we conceptualize this ability as the patching of local knowledge to obtain non-local (global) information. Tracking the passage from local to global properties is the purview of sheaf theory, a branch of mathematics at the nexus of algebra and geometry/topology. Two cognitive domains are examined: (1) learning cue-target patterns that conform to an underlying algebraic rule, and (2) visual attention requiring the integration of space-based feature maps. In both cases, going beyond the data is obtained from a (universal) sheaf theory construction called “sheaving,” i.e., the “patching” of local data attached to a topological space to obtain a representation considered as a globally coherent cognitive map. These results are discussed in the context of a previous (category theory) explanation for systematicity, vis-a-vis, categorical universal constructions, along with other cognitive domains where going beyond the data is apparent. Analogous to higher-order function (i.e., a function that takes/returns a function), going beyond the data as a higher-order systematicity property is explained by sheaving, a higher-order (categorical) universal construction.

Hacia una concepción no atomista de la identidad personal

This paper argues that the classical debate on personal identity in analytical philosophy implicitly rests upon the untenable doctrine of philosophical atomism. Both the Cartesian Ego Theory and the Empiricist Bundle Theory are built upon the indefensible epistemological presupposition that the self is a private object of introspective knowledge. It is suggested that Peter Strawson's descriptive metaphysics of the person as a public agent contains the essential preliminaries for a non-atomistic view of personal identity.

How to Be a Pluralist in Substance Ontology

The four principal competing substance ontologies are substratum theory, bundle theory, primitive substance theory, and hylomorphism. Both historically and in the recent literature, most arguments pertaining to these four theories have been developed under the assumption that only one of them can be true. (E.g., substratum theorists typically think that substrata must be real if substances are real at all, and that the other three theories are impossible.) However there is room in this debate for various forms of pluralism: mild pluralism here refers to the view that while only one of these four theories is true of our world, there is at least one other possible world in which a different substance ontology obtains; moderate pluralism refers to the view that while only one of these four theories is true of our world, there is at least one other possible world in which multiple substance ontologies obtain (e.g., a possible world in which some substances are compounds of substrata and properties and other substances consist solely of compresent properties); and extreme pluralism refers to the view that our own actual world contains substances belonging to different substance ontologies. In this paper I lay out a novel argument for a version of extreme pluralism.

Persistent homology and microlocal sheaf theory

We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using the operation of convolution, we introduce a pseudo-distance on this category and prove in particular a stability result for direct images. Then we assume that V is endowed with a closed convex proper cone $\gamma$ with non empty interior and study $\gamma$-sheaves, that is, constructible sheaves with microsupport contained in the antipodal to the polar cone (equivalently, constructible sheaves for the $\gamma$-topology). We prove that such sheaves may be approximated (for the pseudo-distance) by "piecewise linear" $\gamma$-sheaves. Finally we show that these last sheaves are constant on stratifications by $\gamma$-locally closed sets, an analogue of barcodes in higher dimension.

Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

We discuss the Aharonov-Bohm ($A-B$) effect and the Dirac ($D$) monopole of magnetic charge $g={{1}\over{2}}$ in the context of bundle theory, exhibiting a purely geometric relation between them. If $\xi_{A-B}$ and $\xi_D$ are the respective $U(1)$-bundles, we show that $\xi_{A-B}$ is isomorphic to the pull-back of $\xi_D$ induced by the inclusion of the corresponding base spaces $\iota:(D_0^2)^*\to S^2$}. The fact that the $A-B$ effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux $\Phi_0={{2\pi}\over{\vert e\vert}}$ associated with the electric charge $\vert e\vert$, reflects here as a consequence of the pull-back by $\iota$ of the Dirac connection in $\xi_D$ to $\xi_{A-B}$, and the Dirac quantization condition. We also show the necessary vanishing in $\xi_{A-B}$ of the pull-back of the Chern class $c_1$ in $\xi_D$.