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Journal articles on the topic 'Algebraic stack'

Author: Grafiati

Published: 14 March 2023

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1

Halpern-Leistner, Daniel, and Anatoly Preygel. "Mapping stacks and categorical notions of properness." Compositio Mathematica 159, no. 3 (March 2023): 530–89. http://dx.doi.org/10.1112/s0010437x22007667.

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One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong $h$ -descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.

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2

Shentu, Junchao, and Dong Wang. "Notes on algebraic log stack." International Journal of Mathematics 27, no. 10 (September 2016): 1650081. http://dx.doi.org/10.1142/s0129167x16500816.

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Let [Formula: see text] be a stack over the category of fine log schemes. If [Formula: see text] has a representable fppf covering, then, it has enough compatible minimal objects. As a consequence, we prove the equivalence between two notions of log moduli stacks which appear in literatures. Also, we obtain several fundamental results of algebraic log stacks.

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3

Hall, Jack, and David Rydh. "Perfect complexes on algebraic stacks." Compositio Mathematica 153, no. 11 (August 17, 2017): 2318–67. http://dx.doi.org/10.1112/s0010437x17007394.

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We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau–Gepner’s results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne–Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

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Shentu, Junchao, and Dong Wang. "Erratum: "Notes on algebraic log stack"." International Journal of Mathematics 28, no. 10 (September 2017): 1792002. http://dx.doi.org/10.1142/s0129167x17920021.

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5

Koppensteiner, Clemens. "Hochschild Cohomology of Torus Equivariant D-modules." International Mathematics Research Notices 2020, no. 19 (August 29, 2018): 6391–420. http://dx.doi.org/10.1093/imrn/rny206.

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Abstract We discuss the Hochschild cohomology of the category of D-modules associated to an algebraic stack. In particular we describe the Hochschild cohomology of the category of torus-equivariant D-modules as the cohomology of a D-module on the loop space of the quotient stack. Finally, we give an approach for understanding the Hochschild cohomology of D-modules on general stacks via a relative compactification of the diagonal. This work is motivated by a desire to understand the support theory (in the sense of [6]) of D-modules on stacks.

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6

Alper, Jarod, Maksym Fedorchuk, and David Ishii Smyth. "Second flip in the Hassett–Keel program: existence of good moduli spaces." Compositio Mathematica 153, no. 8 (May 15, 2017): 1584–609. http://dx.doi.org/10.1112/s0010437x16008289.

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We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

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7

Martino, Ivan. "Introduction to the Ekedahl Invariants." MATHEMATICA SCANDINAVICA 120, no. 2 (May 27, 2017): 211. http://dx.doi.org/10.7146/math.scand.a-25693.

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In 2009, T. Ekedahl introduced certain cohomological invariants for finite groups. In this work we present these invariants and we give an equivalent definition that does not involve the notion of algebraic stacks. Moreover we show certain properties for the class of the classifying stack of a finite group in the Kontsevich value ring.

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8

NITSURE, NITIN. "SCHEMATIC HARDER–NARASIMHAN STRATIFICATION." International Journal of Mathematics 22, no. 10 (October 2011): 1365–73. http://dx.doi.org/10.1142/s0129167x11007264.

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For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder–Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder–Narasimhan stratification. In this paper, we show how to endow each Harder–Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder–Narasimhan filtration with a given Harder–Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder–Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder–Narasimhan type form an algebraic stack in the sense of Artin.

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9

Annala, Toni, and Ryomei Iwasa. "Cohomology of the moduli stack of algebraic vector bundles." Advances in Mathematics 409 (November 2022): 108638. http://dx.doi.org/10.1016/j.aim.2022.108638.

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10

Sakellaridis, Yiannis. "The Schwartz space of a smooth semi-algebraic stack." Selecta Mathematica 22, no. 4 (October 2016): 2401–90. http://dx.doi.org/10.1007/s00029-016-0285-3.

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11

Brochard, Sylvain. "Finiteness theorems for the Picard objects of an algebraic stack." Advances in Mathematics 229, no. 3 (February 2012): 1555–85. http://dx.doi.org/10.1016/j.aim.2011.12.011.

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12

Scherotzke, Sarah, Nicolò Sibilla, and Mattia Talpo. "On a logarithmic version of the derived McKay correspondence." Compositio Mathematica 154, no. 12 (November 8, 2018): 2534–85. http://dx.doi.org/10.1112/s0010437x18007431.

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We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

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13

Alarfaj, Maryam, and Chris Sangwin. "Updating STACK Potential Response Trees Based on Separated Concerns." International Journal of Emerging Technologies in Learning (iJET) 17, no. 23 (December 8, 2022): 94–102. http://dx.doi.org/10.3991/ijet.v17i23.35929.

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The STACK system is a computer aided assessment package for mathematics. STACK questions include full algebraic input with validation. Feedback is provided to students using "potential response trees", written by the teacher for each question. In this research, we are updating STACK potential response tree based on students' concerns in learning induction proofs. We have identified a specific misconception in learning mathematical induction from prior educational research, designed STACK question to test if students exhibit this misconception, and then illustrated how we updated STACK potential response tree based on this misconception. One main goal of separating concerns is to understand students’ responses and evaluate the question itself from an academic prospective and ultimately to improve the questions for future years. In addition, one of the contributions of this study is to improve our general understanding of how to design and use STACK potential response trees.

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14

Sakellaridis, Yiannis. "Correction to: The Schwartz space of a smooth semi-algebraic stack." Selecta Mathematica 24, no. 5 (October 3, 2018): 4961–65. http://dx.doi.org/10.1007/s00029-018-0445-8.

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15

Sala, Francesco. "On the Chow ring of the classifying stack of algebraic tori." Documenta Mathematica 27 (2022): 917–32. http://dx.doi.org/10.4171/dm/888.

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16

Dhillon, Ajneet. "On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn." Canadian Journal of Mathematics 58, no. 5 (October 1, 2006): 1000–1025. http://dx.doi.org/10.4153/cjm-2006-038-8.

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AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

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17

Molcho, Samouil, and Jonathan Wise. "The logarithmic Picard group and its tropicalization." Compositio Mathematica 158, no. 7 (July 2022): 1477–562. http://dx.doi.org/10.1112/s0010437x22007527.

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We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.

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18

Yu, Tony Yue. "Gromov compactness in non-archimedean analytic geometry." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 179–210. http://dx.doi.org/10.1515/crelle-2015-0077.

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Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

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19

Walker, I. S., and D. J. Wilson. "Field Validation of Algebraic Equations for Stack and Wind Driven Air Infiltration Calculations." HVAC&R Research 4, no. 2 (April 1, 1998): 119–39. http://dx.doi.org/10.1080/10789669.1998.10391395.

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20

Frăţilă, Dragoş, Sam Gunningham, and Penghui Li. "The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves." Representation Theory of the American Mathematical Society 26, no. 39 (December 21, 2022): 1268–323. http://dx.doi.org/10.1090/ert/631.

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We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H H -bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.

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21

Schürg, Timo, Bertrand Toën, and Gabriele Vezzosi. "Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 702 (January 1, 2015): 1–40. http://dx.doi.org/10.1515/crelle-2013-0037.

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AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

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22

Lei, Ke, Ali B. Syed, Xucheng Zhu, John M. Pauly, and Shreyas V. Vasanawala. "Automated MRI Field of View Prescription from Region of Interest Prediction by Intra-Stack Attention Neural Network." Bioengineering 10, no. 1 (January 10, 2023): 92. http://dx.doi.org/10.3390/bioengineering10010092.

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Manual prescription of the field of view (FOV) by MRI technologists is variable and prolongs the scanning process. Often, the FOV is too large or crops critical anatomy. We propose a deep learning framework, trained by radiologists’ supervision, for automating FOV prescription. An intra-stack shared feature extraction network and an attention network are used to process a stack of 2D image inputs to generate scalars defining the location of a rectangular region of interest (ROI). The attention mechanism is used to make the model focus on a small number of informative slices in a stack. Then, the smallest FOV that makes the neural network predicted ROI free of aliasing is calculated by an algebraic operation derived from MR sampling theory. The framework’s performance is examined quantitatively with intersection over union (IoU) and pixel error on position and qualitatively with a reader study. The proposed model achieves an average IoU of 0.867 and an average ROI position error of 9.06 out of 512 pixels on 80 test cases, significantly better than two baseline models and not significantly different from a radiologist. Finally, the FOV given by the proposed framework achieves an acceptance rate of 92% from an experienced radiologist.

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23

Nikolić, Dragan D. "Parallelisation of equation-based simulation programs on heterogeneous computing systems." PeerJ Computer Science 4 (August 13, 2018): e160. http://dx.doi.org/10.7717/peerj-cs.160.

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Numerical solutions of equation-based simulations require computationally intensive tasks such as evaluation of model equations, linear algebra operations and solution of systems of linear equations. The focus in this work is on parallel evaluation of model equations on shared memory systems such as general purpose processors (multi-core CPUs and manycore devices), streaming processors (Graphics Processing Units and Field Programmable Gate Arrays) and heterogeneous systems. The current approaches for evaluation of model equations are reviewed and their capabilities and shortcomings analysed. Since stream computing differs from traditional computing in that the system processes a sequential stream of elements, equations must be transformed into a data structure suitable for both types. The postfix notation expression stacks are recognised as a platform and programming language independent method to describe, store in computer memory and evaluate general systems of differential and algebraic equations of any size. Each mathematical operation and its operands are described by a specially designed data structure, and every equation is transformed into an array of these structures (a Compute Stack). Compute Stacks are evaluated by a stack machine using a Last In First Out queue. The stack machine is implemented in the DAE Tools modelling software in the C99 language using two Application Programming Interface (APIs)/frameworks for parallelism. The Open Multi-Processing (OpenMP) API is used for parallelisation on general purpose processors, and the Open Computing Language (OpenCL) framework is used for parallelisation on streaming processors and heterogeneous systems. The performance of the sequential Compute Stack approach is compared to the direct C++ implementation and to the previous approach that uses evaluation trees. The new approach is 45% slower than the C++ implementation and more than five times faster than the previous one. The OpenMP and OpenCL implementations are tested on three medium-scale models using a multi-core CPU, a discrete GPU, an integrated GPU and heterogeneous computing setups. Execution times are compared and analysed and the advantages of the OpenCL implementation running on a discrete GPU and heterogeneous systems are discussed. It is found that the evaluation of model equations using the parallel OpenCL implementation running on a discrete GPU is up to twelve times faster than the sequential version while the overall simulation speed-up gained is more than three times.

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24

Chen, Tsao-Hsien, and Xinwen Zhu. "Geometric Langlands in prime characteristic." Compositio Mathematica 153, no. 2 (February 2017): 395–452. http://dx.doi.org/10.1112/s0010437x16008113.

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Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve{G}$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$-bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$-local systems on $C$. Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$. In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$. This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.

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25

Blanc, Anthony. "Topological K-theory of complex noncommutative spaces." Compositio Mathematica 152, no. 3 (September 22, 2015): 489–555. http://dx.doi.org/10.1112/s0010437x15007617.

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The purpose of this work is to give a definition of a topological K-theory for dg-categories over$\mathbb{C}$and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum$\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with$\mathbb{C}$is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

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26

Kim, Joshua, Huaiqun Guan, David Gersten, and Tiezhi Zhang. "Evaluation of Algebraic Iterative Image Reconstruction Methods for Tetrahedron Beam Computed Tomography Systems." International Journal of Biomedical Imaging 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/609704.

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Tetrahedron beam computed tomography (TBCT) performs volumetric imaging using a stack of fan beams generated by a multiple pixel X-ray source. While the TBCT system was designed to overcome the scatter and detector issues faced by cone beam computed tomography (CBCT), it still suffers the same large cone angle artifacts as CBCT due to the use of approximate reconstruction algorithms. It has been shown that iterative reconstruction algorithms are better able to model irregular system geometries and that algebraic iterative algorithms in particular have been able to reduce cone artifacts appearing at large cone angles. In this paper, the SART algorithm is modified for the use with the different TBCT geometries and is tested using both simulated projection data and data acquired using the TBCT benchtop system. The modified SART reconstruction algorithms were able to mitigate the effects of using data generated at large cone angles and were also able to reconstruct CT images without the introduction of artifacts due to either the longitudinal or transverse truncation in the data sets. Algebraic iterative reconstruction can be especially useful for dual-source dual-detector TBCT, wherein the cone angle is the largest in the center of the field of view.

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27

Chaudouard, Pierre-Henri, and Gérard Laumon. "Le lemme fondamental pondéré. I. Constructions géométriques." Compositio Mathematica 146, no. 6 (May 19, 2010): 1416–506. http://dx.doi.org/10.1112/s0010437x10004756.

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AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

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28

JARVIS, TYLER J. "GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES." International Journal of Mathematics 11, no. 05 (July 2000): 637–63. http://dx.doi.org/10.1142/s0129167x00000325.

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This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

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GAVIOLI, FRANCESCA. "THETA FUNCTIONS ON THE MODULI SPACE OF PARABOLIC BUNDLES." International Journal of Mathematics 15, no. 03 (May 2004): 259–87. http://dx.doi.org/10.1142/s0129167x04002272.

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In this paper we extend the result on base point freeness of the powers of the determinant bundle on the moduli space of vector bundles on a curve. We describe the parabolic analogues of parabolic theta functions, then we determine a uniform bound depending only on the rank of the parabolic bundles. In order to get this bound, we construct a parabolic analogue of Grothendieck's scheme of quotients, which parametrizes quotient bundles of a parabolic bundle, of fixed parabolic Hilbert polynomial. We prove an estimate for its dimension, which extends the result of Popa and Roth on the dimension of the Quot scheme. As an application of the theorem on base point freeness, we characterize parabolic semistability on the algebraic stack of quasi-parabolic bundles as the base locus of the linear system of the parabolic determinant bundle.

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30

Hooper, Curtis, and Ian Jones. "Conceptual Statistical Assessment Using JSXGraph." International Journal of Emerging Technologies in Learning (iJET) 18, no. 01 (January 10, 2023): 269–78. http://dx.doi.org/10.3991/ijet.v18i01.36529.

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Traditionally online assessments tend to focus on topics that require students to input algebraic and numeric responses. As such there is a paucity of questions that test students' knowledge of statistics, and what questions there are in our experience focus on computing specific values (mean, standard deviation, and so on). Through making use of a technology called JSXGraph that is supported within the STACK environment for online assessment of mathematical knowledge, we have developed statistics questions that aim to test conceptual knowledge. For example, by requiring students to adjust the bars in a graph in order to produce a dataset that has a required mean, median, mode and range. With careful design this approach enables open-ended questions that have more than one correct answer. In this paper we describe the questions we have designed, and report responses from a sample of students.

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31

Xian, Lu, Henry Adams, Chad M. Topaz, and Lori Ziegelmeier. "Capturing dynamics of time-varying data via topology." Foundations of Data Science 4, no. 1 (2022): 1. http://dx.doi.org/10.3934/fods.2021033.

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<p style='text-indent:20px;'>One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [<xref ref-type="bibr" rid="b19">19</xref>], crocker plots [<xref ref-type="bibr" rid="b55">55</xref>], and multiparameter rank functions [<xref ref-type="bibr" rid="b37">37</xref>]. We then introduce a new tool to summarize time-varying metric spaces: a <i>crocker stack</i>. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable continuity property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [<xref ref-type="bibr" rid="b57">57</xref>]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.</p>

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Iwanari, Isamu. "The category of toric stacks." Compositio Mathematica 145, no. 03 (May 2009): 718–46. http://dx.doi.org/10.1112/s0010437x09003911.

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AbstractIn this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

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Bittanti, Sergio, Silvia Canevese, Antonio De Marco, Giorgio Giuffrida, Antonio Errigo, and Valter Prandoni. "Molten Carbonate Fuel Cell Dynamical Modeling." Journal of Fuel Cell Science and Technology 4, no. 3 (April 7, 2006): 283–93. http://dx.doi.org/10.1115/1.2743074.

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The aim of this work is to build up a complete dynamical model of a molten carbonate fuel cell (MCFC) stack, describing both the thermo-fluid-dynamical and the electrochemical phenomena involved, i.e., both slow and (relatively) fast dynamics. Following a first-principle approach, a set of differential and algebraic equations is written, based on mass, momentum, energy, and charge balance referred to as small control volumes inside a cell. The outlined two-three-dimensional description takes into account the strong point-to-point anode and cathode reaction coupling due to gas crossflow. Simulations (carried out after suitable thermodynamical and electrochemical parameter tuning) highlight, for instance, the presence of dynamics, linked to the electrochemical behavior, with time constants on the order of a second; besides, rather fair matching to data which can be found in the literature is achieved, in terms of external potential difference and of electric power production. The obtained numerical results, therefore, support model correctness and reliability. This is useful in view of model-based cell operation analysis and control, both in stationary and in transient conditions.

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Jin, Jianhua, Qingguo Li, and Chunquan Li. "On Intuitionistic Fuzzy Context-Free Languages." Journal of Applied Mathematics 2013 (2013): 1–16. http://dx.doi.org/10.1155/2013/825249.

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Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.

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Beraldo, Dario. "Sheaves of categories with local actions of Hochschild cochains." Compositio Mathematica 155, no. 08 (July 4, 2019): 1521–67. http://dx.doi.org/10.1112/s0010437x19007413.

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The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$ , affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ , where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$ . In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$ . Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$ -modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$ , its descent properties and the notion of $\mathbb{H}$ -affineness. We then prove the $\mathbb{H}$ -affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$ -category of modules for $\mathbb{H}({\mathcal{Y}})$ , the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$ , then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$ , where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$ . As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$ , thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$ .

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Naidu, Jaideep T., and John F. Sanford. "Some Comments On: A Historical Note On The Proof Of The Area Of A Circle." American Journal of Business Education (AJBE) 4, no. 12 (November 22, 2011): 45–50. http://dx.doi.org/10.19030/ajbe.v4i12.6613.

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In a recent paper by Wilamowsky et al. [6], an intuitive proof of the area of the circle dating back to the twelfth century was presented. They discuss challenges made to this proof and offer simple rebuttals to these challenges. The alternative solution presented by them is simple and elegant and can be explained rather easily to non-mathematics majors. As business school faculty ourselves, we are in agreement with the authors. Our article is motivated by them and we present yet another alternative method. While we do not make an argument that our proposed method is any simpler, we do feel it may be easier to communicate to business school students. In addition, we present a solution using a rectangle which could be left as an exercise for the student after a brief explanation in class. Finding the area of a stack of rectangles with a rectangle as a starting point may seem redundant at first. However, we show that it is actually an excellent algebraic exercise for students since it offers a certain challenge which a square does not. We also solve this exercise using the quicker triangular approach and feel it can be appreciated by students in an Introduction to Calculus course. We also provide two interesting links that demonstrate the work of the ancient mathematicians for this well known problem.

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Hasegawa, Shigeki, Yoshihiro Ikogi, Sanghong Kim, Miho Kageyama, and Motoaki Kawase. "Modeling of the dynamic behavior of an integrated fuel cell system including fuel cell stack, air system, hydrogen system, and cooling system." ECS Meeting Abstracts MA2022-02, no. 39 (October 9, 2022): 1368. http://dx.doi.org/10.1149/ma2022-02391368mtgabs.

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Modeling method of an integrated fuel cell system: In this study, the actual FC-system in commercial fuel cell electric vehicle (FCEV), 2nd-generation MIRAI [1], is selected as the target of the validation and verification of the models. It consists of the FC-stack and sub-systems of air, hydrogen (H2), and cooling systems as shown in Fig. 1. The concepts of modelling methods of the FC-system are shown in Fig. 2, where the system configuration is depicted as the function-block diagram. In each function block, the entire system is broken-down to the component level and physical models of the individual components of FC-stack [2], air compressor [3], throttle valve [4], intercooler [5], and piping elements [5] are implemented. The state variables such as total pressure, flowrate, temperature, and gas composition are defined only at the centre of each component and the distribution of the state variables inside each component is not considered for high speed computation of the dynamic system behaviour. The differential equations of mass balance, mole balance, and energy balance across the fluid circuit shown in Fig. 2 are solved by finite difference method. The differentiated linear algebraic equation of mass balance is expressed by Eq. 1. The air, H2, and cooling system models are integrated with FC-stack mass transport and electrochemical models to build an integrated FC-system model in the Simulink block diagram. Results and discussion: The integrated FC-system model was validated and verified with the actual data collected with 2nd-generation MIRAI by comparing the simulation outputs and the system response data in same input and boundary conditions and the acceptable accuracy was confirmed. The relationships among a FC-stack material properties of PEM, CL, and GDL, FC-stack performance, and the system performance of the system efficiency, acceleration response, and heat generation rate are investigated. An example where cathode GDL substrate is removed and only MPL is remained is shown in Figs. 3–6. Fig. 3 is the simulation input conditions of target FC-system power and ambient wind velocity in front of radiator, where the steady state and dynamic transient in low to high load are combined and the constant wind velocity is assumed. Fig. 4 is the simulation results of I–V and I–R performance comparison. The removal of the GDL substrate improved oxygen transport and reduced FC-resistance by 10 mΩ and raised FC-voltage by 20 mV at 2.0 A/cm2. The break-down of voltage improvement of 20 mV consists of 5.5 mV reduction of resistance overpotential and 14.5 mV by concentration overpotential as far as MPL can deliver O2 under the ribs of the flow channels. Fig. 5 is the simulation results of the dynamic FC-system responses of overall heat generation rate, FC-coolant outlet temperature, and overall H2 consumption amount during an entire pattern. It is confirmed that the peak FC-coolant outlet temperature is lowered from 85 to 80 °C following the peak heat generation rate is reduced by 16 %, which also contributes to prevent the PEM and CL from drying. H2 consumption was reduced 4.3 %. Conclusions: The modeling method for an integrated FC-system which consists of the FC-stack, air system, H2 system, and cooling system is investigated. This model was validated and verified by the actual FC-system data collected with the FC-system implemented in the commercial FCEV and an acceptable accuracy was confirmed. The impact of oxygen transport resistance of cathode GDL substrate on the FC-system performance was investigated. It demonstrates impacts of cell configuration including choice of components and materials on the system performance can be investigated by numerical simulation. It is beneficial for an entire FC-system development process. Acknowledgement : This work was supported by the FC-Platform Program: Development of design-for-purpose numerical simulators for attaining long life and high performance project (FY 2020–2022) conducted by the New Energy and Industrial Technology Development Organization (NEDO), Japan. References: Takahashi, and Y. Kakeno, “Development of New Fuel Cell System”, EVTeC 2021 Proceeding, No. B1.2. Hasegawa, M. Kimata, Y. Ikogi, M. Kageyama, S. Kim, and M. Kawase, “Modeling of Fuel Cell Stack for High-Speed Computation and Implementation to Integrated System Model”, ECS Transactions, 104 (8) 3-26 (2021). Moraal and Ilya Kolmanovsky, ” Turbocharger modeling for automotive control applications”, SAE Technical Paper, No. 1999-01-0908 (1999). Bordjane and D. Chalet, “Numerical Investigation Of Throttle Valve Flow Characteristics For Internal Combustion Engines”, J. multidiscip. eng. sci. technol., Vol. 2, Issue 12 (2015). B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Revised Second Edition, (John Wiley & Sons, New York, 2006). Figure 1

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38

Gómez, Tomás L. "Algebraic stacks." Proceedings Mathematical Sciences 111, no. 1 (February 2001): 1–31. http://dx.doi.org/10.1007/bf02829538.

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39

Hollander, Sharon. "Characterizing algebraic stacks." Proceedings of the American Mathematical Society 136, no. 04 (December 6, 2007): 1465–76. http://dx.doi.org/10.1090/s0002-9939-07-08832-6.

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40

Toën, Bertrand. "Descente fidèlement plate pour les n-champs d’Artin." Compositio Mathematica 147, no. 5 (July 25, 2011): 1382–412. http://dx.doi.org/10.1112/s0010437x10005245.

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AbstractWe prove two flat descent results in the setting of Artin n-stacks. First of all, a stack for the etale topology which is an Artin n-stack (in the sense of Simpson and Toën–Vezzosi) is also a stack for the flat (fppf) topology. Moreover, an n-stack, for the fppf topology, which admits a flat (fppf) n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these two results a comparison between etale and fppf cohomologies (with coefficients in non-smooth group schemes and also non-abelian). This work is written in the setting of the derived stacks of Toën and Vezzosi, and all of these results are therefore also valid for derived Artin n-stacks.

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41

Aoki, Masao. "Deformation theory of algebraic stacks." Compositio Mathematica 141, no. 01 (December 1, 2004): 19–34. http://dx.doi.org/10.1112/s0010437x04000806.

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42

Noohi, B. "FUNDAMENTAL GROUPS OF ALGEBRAIC STACKS." Journal of the Institute of Mathematics of Jussieu 3, no. 1 (January 2004): 69–103. http://dx.doi.org/10.1017/s1474748004000039.

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43

Kamgarpour, Masoud. "Stacky abelianization of algebraic groups." Transformation Groups 14, no. 4 (October 31, 2009): 825–46. http://dx.doi.org/10.1007/s00031-009-9067-8.

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44

Iwanari, Isamu. "Stable points on algebraic stacks." Advances in Mathematics 223, no. 1 (January 2010): 257–99. http://dx.doi.org/10.1016/j.aim.2009.08.007.

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OLSSON, M. "Logarithmic geometry and algebraic stacks." Annales Scientifiques de l’École Normale Supérieure 36, no. 5 (September 2003): 747–91. http://dx.doi.org/10.1016/j.ansens.2002.11.001.

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46

Pirisi, Roberto. "Cohomological invariants of algebraic stacks." Transactions of the American Mathematical Society 370, no. 3 (November 22, 2017): 1885–906. http://dx.doi.org/10.1090/tran/7006.

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Kinjo, Tasuki. "Dimensional reduction in cohomological Donaldson–Thomas theory." Compositio Mathematica 158, no. 1 (January 2022): 123–67. http://dx.doi.org/10.1112/s0010437x21007740.

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For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel–Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson–Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.

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Krishna, Amalendu, and Charanya Ravi. "Algebraic K-theory of quotient stacks." Annals of K-Theory 3, no. 2 (March 24, 2018): 207–33. http://dx.doi.org/10.2140/akt.2018.3.207.

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Huber, Annette. "M. Olsson: “Algebraic Spaces and Stacks”." Jahresbericht der Deutschen Mathematiker-Vereinigung 120, no. 2 (September 13, 2017): 143–46. http://dx.doi.org/10.1365/s13291-017-0172-7.

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Hall, Jack, and David Rydh. "The telescope conjecture for algebraic stacks." Journal of Topology 10, no. 3 (July 4, 2017): 776–94. http://dx.doi.org/10.1112/topo.12021.

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