scholarly journals Variation of Mixed Hodge Structures Associated to an Equisingular One-dimensional Family of Calabi-Yau Threefolds

2020 ◽  
pp. 1-24
Author(s):  
Isidro Nieto-Baños ◽  
Pedro Luis del Angel-Rodriguez
Keyword(s):  
General Position ◽  
Singular Points ◽  
Good Position ◽  
Hodge Structures ◽  
One Dimensional ◽  
Double Points ◽  
Hodge Filtration ◽  
Starting Point ◽  
Mixed Hodge Structures

Abstract We study the variations of mixed Hodge structures (VMHS) associated with a pencil ${\mathcal{X}}$ of equisingular hypersurfaces of degree $d$ in $\mathbb{P}^{4}$ with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family $\widetilde{{\mathcal{X}}}$ . The notion of a set of singular points being in homologically good position is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths’ description of the $F^{2}$ -term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil ${\mathcal{X}}$ of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention.

Mumford-Tate Groups

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Hodge Structure ◽  
Algebraic Groups ◽  
Hodge Decomposition ◽  
Hodge Structures ◽  
The Third ◽  
Direct Sum ◽  
Hodge Filtration ◽  
Mixed Hodge Structures

This chapter provides an introduction to the basic definitions and properties of Mumford-Tate groups in both the case of Hodge structures and of mixed Hodge structures. Hodge structures of weight n are sometimes called pure Hodge structures, and the term “Hodge structure” then refers to a direct sum of pure Hodge structures. The chapter presents three definitions of a Hodge structure of weight n, given in historical order. In the first definition, a Hodge structure of weight n is given by a Hodge decomposition; in the second, it is given by a Hodge filtration; in the third, it is given by a homomorphism of ℝ-algebraic groups. In the first two definitions, n is assumed to be positive and the p,q's in the definitions are non-negative. In the third definition, n and p,q are arbitrary. For the third definition, the Deligne torus integers are used.

scholarly journals Recognition of Abnormal Chest Compression Depth Using One-Dimensional Convolutional Neural Networks

Sensors ◽  
2021 ◽  
Vol 21 (3) ◽  
pp. 846
Author(s):  
Liang Zhao ◽  
Yu Bao ◽  
Yu Zhang ◽  
Ruidong Ye ◽  
Aijuan Zhang
Keyword(s):  
Chest Compression ◽  
Sensor Data ◽  
Evaluation Procedure ◽  
Support Vector ◽  
Portable Devices ◽  
Sensor Signal ◽  
Distance Measurements ◽  
Compression Depth ◽  
One Dimensional ◽  
Starting Point

When the displacement of an object is evaluated using sensor data, its movement back to the starting point can be used to correct the measurement error of the sensor. In medicine, the movements of chest compressions also involve a reciprocating movement back to the starting point. The traditional method of evaluating the effects of chest compression depth (CCD) is to use an acceleration sensor or gyroscope to obtain chest compression movement data; from these data, the displacement value can be calculated and the CCD effect evaluated. However, this evaluation procedure suffers from sensor errors and environmental interference, limiting its applicability. Our objective is to reduce the auxiliary computing devices employed for CCD effectiveness evaluation and improve the accuracy of the evaluation results. To this end, we propose a one-dimensional convolutional neural network (1D-CNN) classification method. First, we use the chest compression evaluation criterion to classify the pre-collected sensor signal data, from which the proposed 1D-CNN model learns classification features. After training, the model is used to classify and evaluate sensor signal data instead of distance measurements; this effectively avoids the influence of pressure occlusion and electromagnetic waves. We collect and label 937 valid CCD results from an emergency care simulator. In addition, the proposed 1D-CNN structure is experimentally evaluated and compared against other CNN models and support vector machines. The results show that after sufficient training, the proposed 1D-CNN model can recognize the CCD results with an accuracy rate of more than 95%. The execution time suggests that the model balances accuracy and hardware requirements and can be embedded in portable devices.

scholarly journals Hodge correlators

2019 ◽  
Vol 2019 (748) ◽  
pp. 1-138
Author(s):  
Alexander B. Goncharov
Keyword(s):  
Fundamental Group ◽  
Hodge Structure ◽  
Euler System ◽  
Feynman Integral ◽  
Mixed Hodge Structure ◽  
Modular Curves ◽  
Hodge Structures ◽  
Motivic Cohomology ◽  
Linear Differential ◽  
Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

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