This document provides an overview of topological quantum computing using anyons. It first discusses motivations for alternative computing approaches due to limitations of traditional computing. It then introduces braid theory, including definitions of braids and their equivalence. Braids form a braid group that can represent any braid as a word. Anyons are quasiparticles that emerge in 2D systems and can represent quantum bits in a way that is robust to environmental perturbations. Braids representing the trajectories of anyons over time can encode quantum information and computations in a topological manner.
This document discusses two approaches for recognizing non-rigid 3D objects using Laplace Beltrami eigensystems. Approach 1 projects point clouds into one dimension, computes moments, and builds a distance matrix to classify objects. Approach 2 directly computes multi-dimensional moments as feature vectors and also builds a distance matrix. Both approaches classify most objects correctly but have some errors, such as splitting the cat class in Approach 1 and splitting the gorilla and seahorse classes in Approach 2.
The document discusses the shapes of chains and arches under gravity. It explains that a catenary is the curve formed by a chain hanging from two fixed points, with the tension in the chain causing it to form a smooth curve. It then explains two methods for analyzing these shapes: using mechanics and forces, and using the calculus of variations to find the curve that minimizes potential energy. The document notes that both methods produce the catenary curve as the solution. It then provides an example analysis of the forces in a tight horizontal string and in a freestanding arch.
1) The document is a student's analysis of a comic page they were assigned to analyze for class.
2) The student found the assignment challenging but enjoyable, as it required a more critical way of thinking than their usual blog posts and discussions.
3) Through an analysis of design choices like emphasis on black areas, hand gestures, and speech bubble construction, the student concludes the comic effectively conveys complex relationships and mature humor to attract an older audience.
El documento trata sobre la doble militancia política en Colombia. Explica que la doble militancia está prohibida por la Constitución y se ha regulado a través de leyes. Define la doble militancia como pertenecer simultáneamente a más de un partido político. Describe las cinco modalidades en que puede ocurrir y establece que la consecuencia jurídica es la revocatoria de la inscripción del candidato.
This document discusses permutation puzzles and their connection to group theory. It specifically examines three puzzles: the Rubik's Cube, Pyraminx, and Megaminx. For the Rubik's Cube, it provides the history, establishes notation for the sides, cubies (pieces), and basic moves, and discusses how the cube's moves form a non-abelian group with specific structure and properties. The Pyraminx and Megaminx are similarly introduced, with notation and an overview of how their moves relate to group theory.
This document introduces a freshman student at Miami University who is majoring in zoology with a desire to minor in marketing or interior design. She is involved in several campus activities including being a member of the dressage team and community engagement chair of her residence hall. She has experience with graphic design through designing posters and flyers for her residence hall and sorority. She is interested in the graphic designer position with Panhellenic because she wants to help inspire and spread positive messages about sororities. While she works well under tight deadlines, she prefers having a few days to create designs to make them as high quality as possible.
El documento describe la cadena de producción de fibra de algodón en un país. Se indica que la producción primaria es de 140 mil toneladas, la industria procesa 170 mil toneladas y se exportan 30 mil toneladas por un valor de USD 306 millones. La cadena no es competitiva debido a falta de integración, altos costos de producción y riesgos de precios. Se proponen medidas como tipo de cambio competitivo, integración de la cadena y eliminar la informalidad para mejorar la competitividad.
The document summarizes and analyzes the graphic novel adaptation of The 9/11 Commission Report titled "The 9/11 Report: A Graphic Depiction" by Sid Jacobson and Ernie Colon. The summary highlights that the graphic novel closely follows the original report, capturing the fear and confusion of 9/11 as well as exposing the government's failure to prevent the attacks despite warnings. Several panels are discussed in detail for how they illustrate key events and reactions accurately while maintaining a respectful tone. The analysis praises the work for making this important history more accessible and raising questions about what the government knew but did not share with the public.
This document discusses two approaches for recognizing non-rigid 3D objects using Laplace Beltrami eigensystems. Approach 1 projects point clouds into one dimension, computes moments, and builds a distance matrix to classify objects. Approach 2 directly computes multi-dimensional moments as feature vectors and also builds a distance matrix. Both approaches classify most objects correctly but have some errors, such as splitting the cat class in Approach 1 and splitting the gorilla and seahorse classes in Approach 2.
The document discusses the shapes of chains and arches under gravity. It explains that a catenary is the curve formed by a chain hanging from two fixed points, with the tension in the chain causing it to form a smooth curve. It then explains two methods for analyzing these shapes: using mechanics and forces, and using the calculus of variations to find the curve that minimizes potential energy. The document notes that both methods produce the catenary curve as the solution. It then provides an example analysis of the forces in a tight horizontal string and in a freestanding arch.
1) The document is a student's analysis of a comic page they were assigned to analyze for class.
2) The student found the assignment challenging but enjoyable, as it required a more critical way of thinking than their usual blog posts and discussions.
3) Through an analysis of design choices like emphasis on black areas, hand gestures, and speech bubble construction, the student concludes the comic effectively conveys complex relationships and mature humor to attract an older audience.
El documento trata sobre la doble militancia política en Colombia. Explica que la doble militancia está prohibida por la Constitución y se ha regulado a través de leyes. Define la doble militancia como pertenecer simultáneamente a más de un partido político. Describe las cinco modalidades en que puede ocurrir y establece que la consecuencia jurídica es la revocatoria de la inscripción del candidato.
This document discusses permutation puzzles and their connection to group theory. It specifically examines three puzzles: the Rubik's Cube, Pyraminx, and Megaminx. For the Rubik's Cube, it provides the history, establishes notation for the sides, cubies (pieces), and basic moves, and discusses how the cube's moves form a non-abelian group with specific structure and properties. The Pyraminx and Megaminx are similarly introduced, with notation and an overview of how their moves relate to group theory.
This document introduces a freshman student at Miami University who is majoring in zoology with a desire to minor in marketing or interior design. She is involved in several campus activities including being a member of the dressage team and community engagement chair of her residence hall. She has experience with graphic design through designing posters and flyers for her residence hall and sorority. She is interested in the graphic designer position with Panhellenic because she wants to help inspire and spread positive messages about sororities. While she works well under tight deadlines, she prefers having a few days to create designs to make them as high quality as possible.
El documento describe la cadena de producción de fibra de algodón en un país. Se indica que la producción primaria es de 140 mil toneladas, la industria procesa 170 mil toneladas y se exportan 30 mil toneladas por un valor de USD 306 millones. La cadena no es competitiva debido a falta de integración, altos costos de producción y riesgos de precios. Se proponen medidas como tipo de cambio competitivo, integración de la cadena y eliminar la informalidad para mejorar la competitividad.
The document summarizes and analyzes the graphic novel adaptation of The 9/11 Commission Report titled "The 9/11 Report: A Graphic Depiction" by Sid Jacobson and Ernie Colon. The summary highlights that the graphic novel closely follows the original report, capturing the fear and confusion of 9/11 as well as exposing the government's failure to prevent the attacks despite warnings. Several panels are discussed in detail for how they illustrate key events and reactions accurately while maintaining a respectful tone. The analysis praises the work for making this important history more accessible and raising questions about what the government knew but did not share with the public.
This document discusses geometric modeling and representations. It introduces polygonal and polyhedral models for representing obstacle regions as the union or intersection of half-plane or half-space primitives. Logical predicates can also represent these models using logical ANDs and ORs. Transformations of geometric bodies are also introduced to enable modeling motion.
Chapter 10Matching MarketsFrom the book Networks, Crow.docxcravennichole326
Chapter 10
Matching Markets
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World.
By David Easley and Jon Kleinberg. Cambridge University Press, 2010.
Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/
We have now seen a number of ways of thinking both about network structure and
about the behavior of agents as they interact with each other. A few of our examples have
brought these together directly — such as the issue of tra!c in a network, including Braess’s
Paradox — and in the next few chapters we explore this convergence of network structure
and strategic interaction more fully, and in a range of di"erent settings.
First, we think about markets as a prime example of network-structured interaction
between many agents. When we consider markets creating opportunities for interaction
among buyers and sellers, there is an implicit network encoding the access between these
buyers and sellers. In fact, there are a number of ways of using networks to model interactions
among market participants, and we will discuss several of these models. Later, in Chapter 12
on network exchange theory, we will discuss how market-style interactions become a metaphor
for the broad notion of social exchange, in which the social dynamics within a group can be
modeled by the power imbalances of the interactions within the group’s social network.
10.1 Bipartite Graphs and Perfect Matchings
Matching markets form the first class of models we consider, as the focus of the current
chapter. Matching markets have a long history of study in economics, operations research,
and other areas because they embody, in a very clean and stylized way, a number of basic
principles: the way in which people may have di"erent preferences for di"erent kinds of
goods, the way in which prices can decentralize the allocation of goods to people, and the
way in which such prices can in fact lead to allocations that are socially optimal.
We will introduce these various ingredients gradually, by progressing through a succession
of increasingly rich models. We begin with a setting in which goods will be allocated to people
Draft version: June 10, 2010
277
278 CHAPTER 10. MATCHING MARKETS
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(a) Bipartite Graph
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(b) A Perfect Matching
Figure 10.1: (a) An example of a bipartite graph. (b) A perfect matching in this graph,
indicated via the dark edges.
based on preferences, and these preferences will be expressed in network form, but there is
no explicit buying, selling, or price-setting. This first setting will also be a crucial component
of the more complex ones that follow.
Bipartite Graphs. The model we start with is called the bipartite matching problem, and
we can motivate it via the following scenario. Suppose that the administrators of a college
dormitory are ...
Quantum computing uses quantum mechanics phenomena like superposition, entanglement, and interference to perform computation. Quantum computers are improving at an exponential rate according to Neven's Law, doubling their processing power exponentially faster than classical computers. The basic unit of quantum information is the qubit, which can exist in superposition and represent a '1' and '0' simultaneously. This allows quantum computers to explore all computational paths at once, greatly increasing their processing speed over classical computers for certain problems.
This document discusses network science and graph theory. It begins by introducing the Human Disease Network, which connects diseases that share a common genetic origin. It then crossed disciplinary boundaries and was featured in various publications and exhibitions. The rest of the document discusses network representations as graphs, using the example of the Bridges of Königsberg problem solved by Euler in 1735. It introduces basic graph concepts like nodes, links, directed and undirected networks, and discusses how different systems can be represented by the same graph structure.
This document summarizes a student paper about using the shortest path algorithm to interpolate contours in images. It discusses how the human visual system perceives 3D representations from 2D images and how extracting meaningful contours is challenging due to noise and discontinuities. The paper proposes using a modified Dijkstra's algorithm to find the shortest path in log-polar space, which maps circles in images to straight lines. This approach aims to identify simple, closed curves representing object contours while ignoring irrelevant edges.
Is Abstraction the Key to Artificial Intelligence? - Lorenza SaittaWithTheBest
With this comprehensive breakdown of abstraction's multiple layers and components, we can understand and answer the question if abstraction is essential to artificial intelligence.
Lorenza Saitta, Università del Piemonte Orientale
1. The document proposes a new experiment involving entanglement and gravitational decoherence using a dual Mach-Zehnder interferometer setup.
2. In the proposed experiment, one interferometer would be placed in a gravitational field, while the other would not. This would result in non-unitary evolution and allow for nonlocal signaling between the two locations.
3. By moving his interferometer in and out of the gravitational field, one experimenter could encode and transmit binary messages to the other, who could decode the messages by observing changes in interference patterns resulting from the non-unitary evolution.
This document summarizes the key differences between classical and quantum computing. Classical computing uses binary bits that are either 1 or 0, while quantum computing uses quantum bits (qubits) that can be 1, 0, or both at the same time due to quantum superposition. The document explains how qubits are based on properties of electrons and their spin, and how quantum gates manipulate qubit states. It discusses how quantum entanglement allows qubits to influence each other in a way that could solve complex problems more efficiently than classical computing. However, the document notes that quantum computing is still in development and some dispute claims about its current capabilities.
This document discusses solving the inverse kinematics problem for a 6 degree of freedom robot using numerical methods and algorithms. It begins by introducing the robot and describing the inverse kinematics problem. It then outlines the process used to determine the lengths of each link by analyzing the robot configuration. Equations are developed relating the joint angles to cylindrical coordinates. Initial conditions are established indicating how the links contribute to the radial and length components at different joint angles.
Methods of Preventing Decoherence in Quantum BitsDurham Abric
The document discusses various methods for preventing decoherence in quantum bits (qubits) in order to improve quantum computing capabilities. It analyzes constructing qubits within diamond (natural or synthetic), continuous-wave driving fields, optical superlattices, fault-tolerant quantum codes. Based on criteria of practicality, simplicity, and development expediency, constructing qubits within diamond is identified as the most ideal method, as the diamond structure both creates and shields qubits from decoherence, though improved fabrication is still needed.
This document discusses various topics in computer vision including affinity measures for image segmentation, normalized cuts, human stereopsis, epipolar geometry, and trinocular stereo. It also discusses tracking applications such as motion capture, recognition from motion, surveillance, and targeting. Vehicle tracking is discussed in detail for applications in predicting traffic flow using video from fixed cameras to initiate tracks automatically by constructing regions of interest that span each lane.
This document discusses structured knowledge representation using semantic nets and frames. It covers key concepts like semantic nets, frames, slots, exceptions, probabilistic reasoning, and fuzzy logic. Specifically, it explains how semantic nets can be used to represent relationships between nodes and inheritance of properties, and how frames allow for default values and inheritance of attributes from superclasses.
This document discusses social networks and methods for analyzing them. It begins by defining social networks and their essential characteristics. It then provides examples of different types of social networks including telephone networks, email networks, collaboration networks, and Wikipedia. It discusses techniques for clustering social networks including betweenness and the Girvan-Newman algorithm. It also covers partitioning graphs, neighborhood properties of graphs, directed graphs, the diameter of graphs, transitive closure, and reachability. Finally, it explains the concept of simrank, which measures node similarity in networks with multiple node types based on the probability of random walkers starting at one node ending up at another.
The document discusses key concepts for quantifying and modeling social networks. It covers the following network properties:
1. Degree distribution - The distribution of the number of connections for each node. Real-world networks often have skewed degree distributions.
2. Path length and diameter - The shortest and longest distances between node pairs, averaged over all pairs. Real-world networks tend to have small path lengths.
3. Clustering coefficient - The likelihood that two neighbors of a node are also neighbors, quantifying local clustering. Social networks exhibit high clustering.
4. Connected components - The size of the largest subset of nodes that are all reachable from each other by paths. Real-world networks often have
1) Biased Normalized Cuts presents a modification of Normalized Cuts that incorporates priors to allow for constrained image segmentation.
2) It seeks solutions that are sufficiently "correlated" with noisy top-down priors, like an object detector, and can be computed quickly given the unconstrained solution.
3) The algorithm constructs a "biased normalized cut vector" that linearly combines eigenvectors such that those correlated with a user-specified seed vector are upweighted while inversely correlated ones have their sign flipped.
This summary provides the key details about the document in 3 sentences:
The document describes an algorithm for performing Constructive Solid Geometry (CSG) operations like union, intersection, and difference on 3D polyhedral objects constructed from convex polygons. The algorithm first subdivides polygons so they do not intersect, then classifies each polygon as inside, outside or on the boundary of the other object. It uses this classification to determine which polygons should be retained or deleted to produce the result of the specified CSG operation.
ICPSR - Complex Systems Models in the Social Sciences - Lecture 3 - Professor...Daniel Katz
This document provides a summary of Stanley Milgram's small world experiment and discussion of complex network models. It discusses how Milgram found that the average path length between individuals in society is around 6 degrees of separation. Later work by Watts and Strogatz showed that networks with a small amount of randomness can display both clustering and small world properties. Degree distributions and other network measures like clustering coefficients and connected components are discussed. Preferential attachment models that generate power law degree distributions are presented.
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
This document discusses geometric modeling and representations. It introduces polygonal and polyhedral models for representing obstacle regions as the union or intersection of half-plane or half-space primitives. Logical predicates can also represent these models using logical ANDs and ORs. Transformations of geometric bodies are also introduced to enable modeling motion.
Chapter 10Matching MarketsFrom the book Networks, Crow.docxcravennichole326
Chapter 10
Matching Markets
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World.
By David Easley and Jon Kleinberg. Cambridge University Press, 2010.
Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/
We have now seen a number of ways of thinking both about network structure and
about the behavior of agents as they interact with each other. A few of our examples have
brought these together directly — such as the issue of tra!c in a network, including Braess’s
Paradox — and in the next few chapters we explore this convergence of network structure
and strategic interaction more fully, and in a range of di"erent settings.
First, we think about markets as a prime example of network-structured interaction
between many agents. When we consider markets creating opportunities for interaction
among buyers and sellers, there is an implicit network encoding the access between these
buyers and sellers. In fact, there are a number of ways of using networks to model interactions
among market participants, and we will discuss several of these models. Later, in Chapter 12
on network exchange theory, we will discuss how market-style interactions become a metaphor
for the broad notion of social exchange, in which the social dynamics within a group can be
modeled by the power imbalances of the interactions within the group’s social network.
10.1 Bipartite Graphs and Perfect Matchings
Matching markets form the first class of models we consider, as the focus of the current
chapter. Matching markets have a long history of study in economics, operations research,
and other areas because they embody, in a very clean and stylized way, a number of basic
principles: the way in which people may have di"erent preferences for di"erent kinds of
goods, the way in which prices can decentralize the allocation of goods to people, and the
way in which such prices can in fact lead to allocations that are socially optimal.
We will introduce these various ingredients gradually, by progressing through a succession
of increasingly rich models. We begin with a setting in which goods will be allocated to people
Draft version: June 10, 2010
277
278 CHAPTER 10. MATCHING MARKETS
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(a) Bipartite Graph
Room1
Room2
Room3
Room4
Room5
Vikram
Wendy
Xin
Yoram
Zoe
(b) A Perfect Matching
Figure 10.1: (a) An example of a bipartite graph. (b) A perfect matching in this graph,
indicated via the dark edges.
based on preferences, and these preferences will be expressed in network form, but there is
no explicit buying, selling, or price-setting. This first setting will also be a crucial component
of the more complex ones that follow.
Bipartite Graphs. The model we start with is called the bipartite matching problem, and
we can motivate it via the following scenario. Suppose that the administrators of a college
dormitory are ...
Quantum computing uses quantum mechanics phenomena like superposition, entanglement, and interference to perform computation. Quantum computers are improving at an exponential rate according to Neven's Law, doubling their processing power exponentially faster than classical computers. The basic unit of quantum information is the qubit, which can exist in superposition and represent a '1' and '0' simultaneously. This allows quantum computers to explore all computational paths at once, greatly increasing their processing speed over classical computers for certain problems.
This document discusses network science and graph theory. It begins by introducing the Human Disease Network, which connects diseases that share a common genetic origin. It then crossed disciplinary boundaries and was featured in various publications and exhibitions. The rest of the document discusses network representations as graphs, using the example of the Bridges of Königsberg problem solved by Euler in 1735. It introduces basic graph concepts like nodes, links, directed and undirected networks, and discusses how different systems can be represented by the same graph structure.
This document summarizes a student paper about using the shortest path algorithm to interpolate contours in images. It discusses how the human visual system perceives 3D representations from 2D images and how extracting meaningful contours is challenging due to noise and discontinuities. The paper proposes using a modified Dijkstra's algorithm to find the shortest path in log-polar space, which maps circles in images to straight lines. This approach aims to identify simple, closed curves representing object contours while ignoring irrelevant edges.
Is Abstraction the Key to Artificial Intelligence? - Lorenza SaittaWithTheBest
With this comprehensive breakdown of abstraction's multiple layers and components, we can understand and answer the question if abstraction is essential to artificial intelligence.
Lorenza Saitta, Università del Piemonte Orientale
1. The document proposes a new experiment involving entanglement and gravitational decoherence using a dual Mach-Zehnder interferometer setup.
2. In the proposed experiment, one interferometer would be placed in a gravitational field, while the other would not. This would result in non-unitary evolution and allow for nonlocal signaling between the two locations.
3. By moving his interferometer in and out of the gravitational field, one experimenter could encode and transmit binary messages to the other, who could decode the messages by observing changes in interference patterns resulting from the non-unitary evolution.
This document summarizes the key differences between classical and quantum computing. Classical computing uses binary bits that are either 1 or 0, while quantum computing uses quantum bits (qubits) that can be 1, 0, or both at the same time due to quantum superposition. The document explains how qubits are based on properties of electrons and their spin, and how quantum gates manipulate qubit states. It discusses how quantum entanglement allows qubits to influence each other in a way that could solve complex problems more efficiently than classical computing. However, the document notes that quantum computing is still in development and some dispute claims about its current capabilities.
This document discusses solving the inverse kinematics problem for a 6 degree of freedom robot using numerical methods and algorithms. It begins by introducing the robot and describing the inverse kinematics problem. It then outlines the process used to determine the lengths of each link by analyzing the robot configuration. Equations are developed relating the joint angles to cylindrical coordinates. Initial conditions are established indicating how the links contribute to the radial and length components at different joint angles.
Methods of Preventing Decoherence in Quantum BitsDurham Abric
The document discusses various methods for preventing decoherence in quantum bits (qubits) in order to improve quantum computing capabilities. It analyzes constructing qubits within diamond (natural or synthetic), continuous-wave driving fields, optical superlattices, fault-tolerant quantum codes. Based on criteria of practicality, simplicity, and development expediency, constructing qubits within diamond is identified as the most ideal method, as the diamond structure both creates and shields qubits from decoherence, though improved fabrication is still needed.
This document discusses various topics in computer vision including affinity measures for image segmentation, normalized cuts, human stereopsis, epipolar geometry, and trinocular stereo. It also discusses tracking applications such as motion capture, recognition from motion, surveillance, and targeting. Vehicle tracking is discussed in detail for applications in predicting traffic flow using video from fixed cameras to initiate tracks automatically by constructing regions of interest that span each lane.
This document discusses structured knowledge representation using semantic nets and frames. It covers key concepts like semantic nets, frames, slots, exceptions, probabilistic reasoning, and fuzzy logic. Specifically, it explains how semantic nets can be used to represent relationships between nodes and inheritance of properties, and how frames allow for default values and inheritance of attributes from superclasses.
This document discusses social networks and methods for analyzing them. It begins by defining social networks and their essential characteristics. It then provides examples of different types of social networks including telephone networks, email networks, collaboration networks, and Wikipedia. It discusses techniques for clustering social networks including betweenness and the Girvan-Newman algorithm. It also covers partitioning graphs, neighborhood properties of graphs, directed graphs, the diameter of graphs, transitive closure, and reachability. Finally, it explains the concept of simrank, which measures node similarity in networks with multiple node types based on the probability of random walkers starting at one node ending up at another.
The document discusses key concepts for quantifying and modeling social networks. It covers the following network properties:
1. Degree distribution - The distribution of the number of connections for each node. Real-world networks often have skewed degree distributions.
2. Path length and diameter - The shortest and longest distances between node pairs, averaged over all pairs. Real-world networks tend to have small path lengths.
3. Clustering coefficient - The likelihood that two neighbors of a node are also neighbors, quantifying local clustering. Social networks exhibit high clustering.
4. Connected components - The size of the largest subset of nodes that are all reachable from each other by paths. Real-world networks often have
1) Biased Normalized Cuts presents a modification of Normalized Cuts that incorporates priors to allow for constrained image segmentation.
2) It seeks solutions that are sufficiently "correlated" with noisy top-down priors, like an object detector, and can be computed quickly given the unconstrained solution.
3) The algorithm constructs a "biased normalized cut vector" that linearly combines eigenvectors such that those correlated with a user-specified seed vector are upweighted while inversely correlated ones have their sign flipped.
This summary provides the key details about the document in 3 sentences:
The document describes an algorithm for performing Constructive Solid Geometry (CSG) operations like union, intersection, and difference on 3D polyhedral objects constructed from convex polygons. The algorithm first subdivides polygons so they do not intersect, then classifies each polygon as inside, outside or on the boundary of the other object. It uses this classification to determine which polygons should be retained or deleted to produce the result of the specified CSG operation.
ICPSR - Complex Systems Models in the Social Sciences - Lecture 3 - Professor...Daniel Katz
This document provides a summary of Stanley Milgram's small world experiment and discussion of complex network models. It discusses how Milgram found that the average path length between individuals in society is around 6 degrees of separation. Later work by Watts and Strogatz showed that networks with a small amount of randomness can display both clustering and small world properties. Degree distributions and other network measures like clustering coefficients and connected components are discussed. Preferential attachment models that generate power law degree distributions are presented.
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
3. 1 Motivations
The free lunch is over. This is a sort of slogan which has become quite com-
mon in the world of computing during recent years, but what does it mean?
What it does not mean is that Moore’s Law is dead. It does mean that, to
take advantage of emerging information processing technology, programmers
must learn to be increasingly clever. Hardware designers have been banging
their heads against a few physical “walls” for over a decade now. Heat is-
sues, power constraints, and a shortage of low hanging fruit with respect to
instruction level processing optimizations have stopped the advance of the
single-processor machine dead in its tracks [7].
Figure 1: Processing power no longer increases at the same rate as transistor
count [7].
Designers have since turned to parallel computing in order to maintain the
expected annual exponential increases in processing power. However, due to
the sequential nature of many (perhaps most) programming problems, dou-
bling the number of cores or processors does not double the processing power.
In response, we are seeing the slow death of the personal computer. Many
are investing in massive distributed networks, to which small, lightweight
personal computers can connect and unload most of their work. This is an
interesting reversal of the move from mainframes and timeshare computing
to personal computers in the 1980’s, but it is certainly not an ideal solution.
Many believe that computing power must be kept in the hands of individuals.
3
4. With this in mind, one of the most hyped technologies on the horizon seems
to be quantum computing.
Although some “large” strides (many of which are of debatable significance)
have been made by researchers of quantum computing techniques, many ma-
jor issues continue to persist. One of these is the incredible fragility of a
quantum system. Quantum computers make use of the fact that quantum
bits may hold both of their two possible states simultaneously, with each
state’s complex amplitude expressing its probability of being observed. We
can call these two states 0 and 1, as we do when discussing classical comput-
ing bits. A simpler way of visualizing this is to imagine that a qubit may hold
only one state at each instant, but that that state is a superposition of both 0
and 1. In other words, we can imagine the states as points along the complex
sphere, where the north pole maps to the state 1, the south pole maps to the
state 0, and everything else is a superposition of the two, each weighted by
its amplitude [3]. These superpositions are written as α|0 ą `β|1 ą, where
α and β are complex numbers (the amplitudes of 0 and 1).
Figure 2: A spherical representation of a quantum bit.
The issue is that minute fluctuations in temperature, magnetic field, etc.
can easily disrupt the states of the qubits [3]. This requires incredibly well-
controlled environments to be created, making the personal/portable quan-
tum computer quite impractical. However, a topological approach to quan-
tum computing yields a possible solution.
4
5. 2 Braid Theory
In order to understand how the qubits in a topological quantum computer
can be impervious to environmental perturbations, we must first discuss the
topological structure that they represent. We will therefore give a brief intro-
duction of a subfield of topology called braid theory, a close cousin of knot
theory.
2.0.1 What is a Braid?
One common way to define a braid is by viewing it as a collection of non-
intersecting paths in R3
, each of which connects a point in tpx,0,1q|x P Zu
with a point in tpx,0,0q|x P Zu [4]. Recall that a path from x to y, where
x,y P X is a continuous function f : r0,1s Ñ X, such that fp0q “ x and
fp1q “ y. It may be helpful to point out that, since the paths may not inter-
sect, neither may their endpoints intersect. Figure 3 depicts a braid. Notice
that the endpoints are lined up and evenly spaced, since they are fixed in a
vertical, two-dimensional plane. Conversely, the paths connecting them are
free to travel anywhere in R3
, with two restrictions. First, no point along any
path may have a z-coordinate greater than 1, or less than 0 (i.e. the paths
must stay between the endpoints). Second, the paths must move in the neg-
ative z-direction at all times [1]. Figure 4 shows an example of a braid which
breaks the second rule.
Figure 3: A braid.
5
6. Figure 4: Not a braid.
Instead of imagining braids as a series of paths embedded within R3
, it is
often convenient to project them onto the plane. This is typically done in such
a way as to guarantee that there are finitely many points on the projection
which correspond to two points on the braid, and none which correspond to
more than two points on the braid. This is called a regular projection, and
guarantees that we do not lose information by projecting the braid into a
lower dimensional space [2]. The method of projection is obvious, as we have
already done it by displaying Figure 3 on a two-dimensional sheet of paper.
To project a braid onto the x-z plane, we simply remove the y-coordinate
from every point in path. In order to avoid losing important topological
information, when two paths intersect, we draw the intersection in such a
way that it is obvious which path was at a higher y-value (which path crossed
in front of the other).
Of course, one might imagine that two paths could run for a while with
the same position in the x-z plane (the y-values must differ), so that one
would be hidden behind the other after projection, thus violating our rules.
It turns out that we can always manipulate our paths in such a way that,
after projection, they only intersect at a finite number of points, and that at
most two paths intersect at any point. This deformation does not change the
braid topologically, which brings us to our next section.
2.0.2 Equivalence of Braids
Topological spaces are identified, not by any specific representation, nor by
their appearance when viewed by any specific perspective. Instead, they are
identified by a special topological properties which are preserved by home-
omorphisms. Such topological properties are aptly named invariants [2]. We
give an example with knots before discussing braids, so as to form a connec-
tion with a more familiar subject. The connections will continue to emerge
as we discuss ambient isotopies and words. Such connections arise because
every knot is actually the closure of some braid, formed by connecting the
endpoints of the braid as shown in Figure 5.
6
7. Figure 5: A Knot as the Closure of a Braid.
Imagine knotting up a piece of string, and then fusing the ends together.
Invariants between knots may be (extremely) loosely defined as loops, coils,
ties, etc. which cannot be removed from the knot without cutting it at some
point, unraveling it, and gluing the severed ends back together. Any features
of a knot which are not invariants, may be removed from the knot, so that two
knots with the same invariant features, but with any number of non-invariant
features, may be uncoiled back into identical states. These knots would then
be considered equivalent. Figure 6 shows two such equivalent knots.
Figure 6: Equivalent Knots.
Braids, being very closely related to knots, have a very similar equivalence
relation. Here, it helps to imagine that your braid consists of a series of
strings, tied to two boards. The endpoints of the strings cannot move and
you cannot spin the boards around, but you can otherwise wiggle the strings
around to your heart’s content without changing the identity of the braid.
Figure 7 shows pairs of equivalent braids.
7
8. Figure 7: Equivalent Braids.
A much more formal definition of this equivalence relation is that any two
braids which are ambient isotopic are equivalent. An isotopy is a path in a
space of continuous functions mapping X to Y , connecting two such functions
f : X Ñ Y and g : X Ñ Y in such a way that every point on the path is
a homeomorphism from X to Y . An isotopy may also be thought of as a
continuous deformation of the aforementioned space X. This deformation
does not change the topology on X. An ambient isotopy is an isotopy on
the space of continuous functions from X to itself, acting as a path from the
identity function on X to a function linking two embeddings of another space
in X [2]. What this means is that two braids are equivalent if they can be
continuously deformed to one another. In addition, this definition shows why
the strings of a braid cannot pass through each other during this deformation
(if they could, all braids would be equivalent). At the point of intersection
during such a deformation, our isotopy would not be a homeomorphism.
Now we know that the set of all braids can actually be compressed into a set
of equivalence classes, but finding ambient isotopies between braids would
not be a very practical way to establish equivalence. Before finishing our
discussion of braid equivalence, we will need to discuss the braid group.
8
9. 2.0.3 The Braid Group
It turns out that braids form a very simple group, which will help us to
understand braid equivalence. Recall that a group is a set of elements, along
with an associative binary operation ‹, for which an identity element exists,
along with an inverse for every element in the group. In the braid group, the
binary operation is simply concatenation of braids, as shown in Figure 8.
Figure 8: Braid Concatenation.
It is clear that concatenation is associative, and that it results in another
element of the braid group. It should also be clear that the identity element
is the trivial braid, represented by a series of vertical lines (i.e. none of the
lines are coiled around each other).
Suppose that we are working with the group of all braids with n paths/strings.
To see that every braid has an inverse, we should first point out that the
entire group is generated from a set of basic braids, and so is denoted
ă σ1,σ2,...,σn´1 ą. If we assign our paths the numbers 1 to n in order,
each σi represents path i crossing over path i ` 1. The inverse of σi, denoted
σ´1
i , represents path i crossing under path i ` 1. It is not difficult to con-
vince ourselves that all possible n-braids may be formed from combinations
of these elements and their inverses, and it follows that every such braid has
an inverse composed of the inverses of the generators from which it is made
(in reverse sequence) [1]. Figure 9 depicts the generators (and their inverses)
of the 3-braid group.
9
10. Figure 9: 3-Braid Group Generators and their Inverses.
Although we have described each σi or σ´1
i as a distinct braid, all of which are
concatenated together by the group operation to form more complex braids,
it is just as easy to imagine that we have only one braid, and that the σi’s
and their inverses are operations that we apply to this braid to modify it.
Whichever visualization style you prefer, the point is that we can represent
any braid with a string of these crossings. Such a string is formally called a
word, and we will use them to easily establish equivalence between braids [1].
In order to compare the word representations of braids, we must compile a
list of relations, and moves which do not change a braid’s identity. The list
is as follows:
1. Adding or removing σiσ´1
i or σ´1
i σi does not change that word.
2. σi`1σiσi`1 “ σiσi`1σi
3. σiσj “ σjσi when |i ´ j| ě 2
The first rule comes from the existence of an element’s inverse in a group.
The second relation comes from the third Reidemeister move, an ambient
isotopy preserving operation on knots which passes a strand over a crossing
of two other strands. When performed on the closure of a braid (which is
a knot), it actually is a Type III Reidemeister move [1]. Using the 3-braid
group as an example, we can imagine starting with the identity and applying
σ2 (reference Figure 9). Then the application of σ1σ2 has the effect of sliding
strand 1 over the crossing of strands 2 and 3. On the other hand, the sequence
σ1σ2σ1 has the effect of simultaneously crossing strands 2 and 3, and sliding
them underneath strand 1.
10
11. If we imagine that the σis are moves applied sequentially to a single braid,
the third rule simply states that two moves which do not affect the same
paths may be applied in any order. Be careful not to forget that, in general,
the braid group is not abelian. We are now able to determine whether or not
two braids are topologically equivalent algorithmically, by comparing their
word representations. This allows us to finally begin to discuss the physical
manifestation of a braid that can be used to simulate a quantum computer.
3 Quantum Computing with Anyons
The first breakthrough in forming a topological approach to quantum com-
puting was the discovery of anyons. Anyons are quasiparticles with a frac-
tional charge, restricted to movement in two dimensions. Although quantum
particles living in three dimensions are required to be either fermions or
bosons, anyons may take on a complex phase which falls into neither cate-
gory. This property emerges from the two dimensional world in which anyons
live.
In order to produce anyons, we must create an environment which restricts
movement to two dimensions. Incredibly, this can be done by pairing gallium
arsenide semiconductors, creating what is called a two dimensional electron
gas. Essentially, the semiconductors prevent electrons between them from
moving in the third dimension. Such conditions produce what is called the
fractional quantum Hall effect, in which quasiparticles appear to be anyons
[3].
Recall that numbers of the form eiα
are typically taken to represent a rotation
by α in two complex dimensions, because the group formed by the members
of the complex unit circle has the map θ Ñ z “ eiθ
“ cosθ ` isinθ [8]. As
it turns out, identical anyons can only accumulate phase as they are rotated
around each other. Each time two identical anyons are swapped in a clockwise
direction, the phase factor added to the system takes the form eiα
for some
0 ă α ă π. A counterclockwise swapping results in half the phase factor.
Notice that eiˆπ
“ ´1 by Euler’s formula, and that eiˆ0
“ 1. Traditionally,
interchanging bosons can only add a phase factor of 1, and interchanging
fermions adds a phase factor of -1 [5].
What is absolutely amazing about all of this is that the phase of a sys-
11
12. tem where anyons are used to represent qubits cannot be affected by small
deformations in the trajectories of the particles caused by the ambient envi-
ronment. In this way, anyons can be considered to be a topological state of
matter.
Recall that a braid is constructed from a collection of paths. We know that
a path connecting two points, a and b, is typically expressed with the pa-
rameterized formula fptq “ p1 ´ tq ˆ a ` t ˆ b. If we take t to represent
time (fitting, since a braid’s paths must make forward progress at all times,
and cannot loop back on themselves), we can imagine that a braid repre-
sents the movement of a collection of particles over time. This representation
of spatial movement over time is referred to as a particle’s world line. An
anyon’s world line would therefore be imagined as a 2 ` 1 dimensional line
[3]. However, since the two dimensional movement of anyons has no effect
on the total phase of the system unless two anyons are interchanged, such a
representation is actually quite simple. We can imagine that the crossing of
braid path i over path i ` 1 represents a clockwise interchanging of the two
anyons represented by those paths. Likewise, a counterclockwise interchang-
ing of anyons can be represented by path i crossing under path j. This gives
us the relations σj “ eiθ
and σ´1
j “ epiθq{2
, where θ is the phase factor added
by such an interchanging [6].
Figure 10: The interchanging of anyons represented as a braid.
Due to the inherent topological properties of these anyons, we can be sure
that none of the slight deformations which threaten the stability of most
quantum computers will have any effect on this one. Logic gates can be con-
structed by sequentially interchanging the positions of a collection of anyons,
and measuring the resultant phase of the system. This sequence of inter-
changes could be represented by the anyons’ braided world lines, as seen
12
13. in Figure 11. Braids may be distinguished by comparing their word repre-
sentations, and these words are simply generator moves applied sequentially
to the identity braid. Since we have mapped the braid group generators to
phase changes, it follows that we can distinguish our braided anyons by the
final phase, resulting from sequentially applied phase changes. As long as the
world lines of the anyons can be continuously deformed back to their original
state (if they haven’t been coiled up), the total phase of the system at the
end will not be altered.
Figure 11: A CNOT logic gate, built with anyons.
4 Conclusion
In conclusion, it has been shown that two dimensional particles, called anyons,
exist as a physical manifestation of the braid group. Within a system, their
innate topological properties protect the state (or phase) of that system from
alterations caused by the ambient environment. On the other hand, the topo-
logical quantum computer falls victim to new problems, such as the appear-
ance of stray anyons, generated by thermal fluctuations, which can be caught
up in the braiding of the anyons (this might be avoided by spacing out the
anyons). No topological quantum computer has yet been constructed, but
the potential remains. By studying topological braid theory and knot theory,
we might be able to gain a better understanding of such a system, and to
take another small step into the world of reliable quantum computing.
13
14. References
[1] Colin C. Adams. The Knot Book: An Elementary Introduction to the
Mathematical Theory of Knots. American Mathematical Society, 2001.
[2] Colin C. Adams and Robert Franzosa. Introduction to Topology: Pure
and Applied. Pearson Education, Inc., 2008.
[3] Graham P. Collins. Computing with quantum knots. Scientific American,
2006.
[4] Rebecca Hoberg. Knots and braids, 2011.
[5] Sankar D. Sarma, Michael Freedman, and Chetan Nayak. Topological
quantum computing. American Institute of Physics, 2006.
[6] Michael K. Spillane. An introduction to the theory of topological quan-
tum computing.
[7] Herb Sutter. The free lunch is over: A fundamental turn toward concur-
rency in software. Dr. Dobb’s Journal, 30(3), 2005.
[8] Wikipedia. Circle group.
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