scholarly journals Poincaré conjecture: A problem solved after a century of new ideas and continued work

2018 ◽  
pp. 59
Author(s):  
María Teresa Lozano Imízcoz
Keyword(s):  
Fundamental Group ◽  
Algebraic Topology ◽  
Three Dimensional ◽  
Henri Poincaré ◽  
French Mathematician ◽  
New Ideas ◽  
Poincare Conjecture

The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. It characterises three-dimensional spheres in a very simple way. It uses only the first invariant of algebraic topology – the fundamental group – which was also defined and studied by Poincaré. The conjecture implies that if a space does not have essential holes, then it is a sphere. This problem was directly solved between 2002 and 2003 by Grigori Perelman, and as a consequence of his demonstration of the Thurston geometrisation conjecture, which culminated in the path proposed by Richard Hamilton.

5. Flavours of topology

2019 ◽  
pp. 90-114
Author(s):  
Richard Earl
Keyword(s):  
19Th Century ◽  
Set Theory ◽  
Algebraic Topology ◽  
General Topology ◽  
Major Theme ◽  
Henri Poincaré ◽  
Combinatorial Topology ◽  
Differential Topology ◽  
French Mathematician ◽  
Modern Mathematics

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

FROM FLATLAND TO FRACTALAND: NEW GEOMETRIES IN RELATIONSHIP TO ARTISTIC AND SCIENTIFIC REVOLUTIONS

Fractals ◽  
1995 ◽  
Vol 03 (03) ◽  
pp. 617-625 ◽  
Author(s):  
RHONDA ROLAND SHEARER
Keyword(s):  
Social Values ◽  
Modern Art ◽  
Three Dimensional ◽  
Direct Access ◽  
New Paradigm ◽  
Social Commentary ◽  
Scientific Revolutions ◽  
History Of ◽  
New Ideas ◽  
The Universe

Abbott’s 19th century book, Flatland, continues to be popularly interpreted as both a social commentary and a way of visualizing the 4th-dimension by analogy. I attempt here to integrate these two seemingly disparate readings. Flatland is better interpreted as a story with a central theme that social, perceptual, and conceptual innovations are linked to changes in geometry. In such cases as the shift from the two-dimensional world of Flatland to a three-dimensional Spaceland, the taxonomic restructuring of human importance from Linnaeaus to Darwin, or the part/whole proportional shift from Ptolemy’s earth as the center of the universe to Copernicus’s sun, new geometries have changed our thinking, seeing, and social values, and lie at the heart of innovations in both art and science. For example, the two greatest innovations in art — the Renaissance with geometric perspective, and the birth of modern art at the beginning of this century with n-dimensional and non-Euclidean geometries — were developed by artists who were thinking within new geometries. When we view the history of scientific revolutions as new geometries, rather than only as new ideas, we gain direct access to potential manipulations of the structures of human innovation itself. I will discuss the seven historical markers of scientific revolutions (suggested by Kuhn, Cohen, and Popper), and how these seven traits correlate and can now be seen within the new paradigm of fractals and nonlinear sciences.

Preparation of Microporous Supercapacitor Electrode Based on the Triple Networks of Disposable Sheet Mask

NANO ◽  
2019 ◽  
Vol 14 (12) ◽  
pp. 1950157
Author(s):  
Shasha Jiao ◽  
Tiehu Li ◽  
Chuanyin Xiong ◽  
Chen Tang ◽  
Alei Dang ◽  
...  
Keyword(s):  
Specific Capacity ◽  
Three Dimensional ◽  
Cost Savings ◽  
High Specific Capacitance ◽  
Electrochemical Performances ◽  
Energy Storage Devices ◽  
Supercapacitor Electrode ◽  
High Specific Capacity ◽  
Storage Devices ◽  
New Ideas

In this study, a three-dimensional hybrid was synthesized via depositing of carbon nanotubes (CNTs) and ferroferric oxide (Fe3O4) particles on the abandoned disposable sheet mask fabric, followed by the polymerization of polypyrrole (PPY). The as-prepared nanocomposite shows superior electrochemical performances when it was used for the material for the flexible supercapacitor electrode. Benefiting from the synergistic effect of CNTs, Fe3O4 and PPY in such a porous structure, cyclic voltammetry and galvanostatic charge/discharge measurements indicated that the as-prepared hybrid possessed a good reversibility and high specific capacity at various scanning rates. It turned out that the as-prepared electrode demonstrated a high specific capacitance of 221.7[Formula: see text]F/g at the scanning rate of 50[Formula: see text]mV/s and long-life cycling stability of 88.2% after 10[Formula: see text]000 cycles. Besides, the electrode composite had good flexibility after repeated bending times of 3000. With the exception of improved electrochemical properties, this hybrid electrode material also showed many advantages, including facile preparation, flexibility and cost savings. These results will provide new ideas and solutions to design and fabricate the flexible supercapacitors, which has great prospect in the development of energy storage devices.

scholarly journals Partial hyperbolicity and classification: a survey

2016 ◽  
Vol 38 (2) ◽  
pp. 401-443 ◽  
Author(s):  
ANDY HAMMERLINDL ◽  
RAFAEL POTRIE
Keyword(s):  
Fundamental Group ◽  
Three Dimensional ◽  
Group Classification ◽  
Partial Hyperbolicity ◽  
Hyperbolic Diffeomorphisms ◽  
Higher Dimensional ◽  
Partially Hyperbolic ◽  
Anosov Flows ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Solvable Fundamental Group

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

Three Dimensional Creativity: Three Navigations to Extend our Thoughts

2013 ◽  
Vol 17 (2) ◽  
pp. 157-160
Author(s):  
Kwang H. Lee ◽  
Keyword(s):  
Three Dimensional ◽  
Time Space ◽  
New Ideas ◽  
Asking Questions ◽  
To Come ◽  
The Brain

Creativity is an ability to come up with a new idea. In many cases, getting out of reality can bring forth a new idea. Since asking questions stimulates the brain to release us from reality, repeating such questions forms the habit of asking many questions that increases creativity. A framework consisting of three kinds of questions is provided. The three kinds of questions are on axes of “time,” “space,” and “field” and the framework is called as three dimensional creativity. Traveling along the three axes allows escaping from a fixed idea, and thus helps us to raise new ideas.

ADVANCED CONCEPTS FOR FLOATING-BODY MEMORIES

2012 ◽  
Vol 21 (01) ◽  
pp. 1250002
Author(s):  
FRANCISCO GÁMIZ ◽  
NOEL RODRIGUEZ ◽  
SORIN CRISTOLOVEANU
Keyword(s):  
Three Dimensional ◽  
Primary Source ◽  
Physical Structure ◽  
Floating Body ◽  
Memory Cells ◽  
State Discrimination ◽  
Floating Body Effect ◽  
New Concepts ◽  
Dram Industry ◽  
New Ideas

With 30nm-class memory cells in production and 20nm-class (20-29nm feature-size) memory targeted for next year, the standard 1-Transistor + 1-Capacitor (1T+1C) DRAM industry is making prominent efforts to improve the scalability of the cell capacitor while maintaining the minimum capacitance requirements for state discrimination, immune to noise (C~25fF/cell). To achieve the capacitance requirement, the DRAM cell has evolved from its initial planar implementation to complex three-dimensional structures. The increment in complexity and the large difference in size between the transistor and capacitor of each cell have motivated the search for Floating-Body Single-Transistor DRAM (1T-DRAM). The underlying idea behind 1T-DRAMs is the development of single-device memory cells with a pronounced hysteresis effect and fast operation. This chapter is focused on the floating-body effect as a primary source of hysteresis. We present new concepts able to deal with the basic limitations of 1T-DRAM while maintaining its simplicity. The floating-body 1T-DRAMs can be reconciled with the aggressive scaling constrains by considering new ideas which make possible the coexistence of electron and holes in the same ultrathin transistor. The best approach is to isolate each type of carrier in an specific potential well which is not created specifically by the bias conditions (unlike standard 1T-DRAMs) but by the physical structure of the device.

scholarly journals A COVID-19 Drug Repurposing Strategy Through Quantitative Homological Similarities by using a Topological Data Analysis Based Formalism

2020 ◽  
Author(s):  
Raul Pérez-Moraga ◽  
Jaume Forés-Martos ◽  
Beatriz Suay ◽  
Jean-Louis Duval ◽  
Antonio Falcó ◽  
...  
Keyword(s):  
Data Analysis ◽  
Algebraic Topology ◽  
Protein Structures ◽  
Three Dimensional ◽  
Drug Repurposing ◽  
Topological Data Analysis ◽  
Drug Identification ◽  
Global Pandemic ◽  
Candidate Drug ◽  
Topological Data

Since its emergence in March 2020, the SARS-CoV-2 global pandemic has produced more than 65 million cases and one point five million deaths worldwide. Despite the enormous efforts carried out by the scientific community, no effective treatments have been developed to date. We created a novel computational pipeline aimed to speed up the process of repurposable candidate drug identification. Compared with current drug repurposing methodologies, our strategy is centered on filtering the best candidate among all selected targets focused on the introduction of a mathematical formalism motivated by recent advances in the fields of algebraic topology and topological data analysis (TDA). This formalism allows us to compare three-dimensional protein structures. Its use in conjunction with two in silico validation strategies (molecular docking and transcriptomic analyses) allowed us to identify a set of potential drug repurposing candidates targeting three viral proteins (3CL viral protease, NSP15 endoribonuclease, and NSP12 RNA-dependent RNA polymerase), which included rutin, dexamethasone, and vemurafenib among others. To our knowledge, it is the first time that a TDA based strategy has been used to compare a massive amount of protein structures with the final objective of performing drug repurposing

scholarly journals Tissue Engineering and Three-Dimensional Printing in Periodontal Regeneration: A Literature Review

2020 ◽  
Vol 9 (12) ◽  
pp. 4008
Author(s):  
Simon Raveau ◽  
Fabienne Jordana
Keyword(s):  
Selective Laser Sintering ◽  
Periodontal Regeneration ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Natural Polymers ◽  
Three Dimensional Printing ◽  
Computer Assisted Design ◽  
New Ideas ◽  
3D Printed ◽  
Dimensional Printing

The three-dimensional printing of scaffolds is an interesting alternative to the traditional techniques of periodontal regeneration. This technique uses computer assisted design and manufacturing after CT scan. After 3D modelling, individualized scaffolds are printed by extrusion, selective laser sintering, stereolithography, or powder bed inkjet printing. These scaffolds can be made of one or several materials such as natural polymers, synthetic polymers, or bioceramics. They can be monophasic or multiphasic and tend to recreate the architectural structure of the periodontal tissue. In order to enhance the bioactivity and have a higher regeneration, the scaffolds can be embedded with stem cells and/or growth factors. This new technique could enhance a complete periodontal regeneration. This review summarizes the application of 3D printed scaffolds in periodontal regeneration. The process, the materials and designs, the key advantages and prospects of 3D bioprinting are highlighted, providing new ideas for tissue regeneration.

Physics and Geometry

2017 ◽  
Author(s):  
Hanoch Gutfreund ◽  
Jürgen Renn
Keyword(s):  
General Relativity ◽  
Special Relativity ◽  
Symmetry Property ◽  
Maxwell's Equations ◽  
Theory Of Relativity ◽  
Relativistic Invariance ◽  
Henri Poincaré ◽  
Ongoing Debate ◽  
French Mathematician

This chapter concerns the ongoing debate about the meaning of Einstein's theory in the formative years, with particular attention to the relation between physics and geometry. It also compares Einstein's thinking on this issue with that of the French mathematician and philosopher Henri Poincaré and deals with the role of symmetry in the theory of relativity—one of Einstein's enduring legacies. The role of symmetry becomes evident, for instance, in the lecture on special relativity, in which it is shown how relativistic invariance, a symmetry property of the spacetime continuum, shapes Maxwell's equations and other laws of physics. In the period under consideration, the understanding of symmetry is deepened by the emergence of Emmy Noether's famous theorems, for which the theory of general relativity was an important source of inspiration.

Path Components and the Fundamental Group

2019 ◽  
pp. 1-126
Author(s):  
Graham Ellis
Keyword(s):  
Fundamental Group ◽  
Algebraic Topology ◽  
High Dimensional ◽  
Discrete Groups ◽  
Cloud Data ◽  
Cw Complex ◽  
Random Simplicial Complexes ◽  
Knot Complements ◽  
Protein Backbones ◽  
Basic Concepts

This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, non-regular CW-complex, simplicial complex, cubical complex, permutahedral complex, simple homotopy, set of path-components, fundamental group, van Kampen’s theorem, knot quandle, Alexander polynomial of a knot, covering space. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.

Sign in / Sign up

Export Citation Format

Share Document