tag:blogger.com,1999:blog-78155489725570220502024-02-18T20:09:03.173-08:00Cosmology Symmetry Breaking through Higgs MechanismH. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.comBlogger20125tag:blogger.com,1999:blog-7815548972557022050.post-68186120763224542502023-05-11T17:36:00.008-07:002023-05-11T17:36:58.730-07:00How AI is helping NASA's James Webb Space TelescopeThe James Webb Space Telescope (JWST) is the most powerful telescope ever built. It is designed to see the universe in infrared light, which will allow it to see objects that are too faint or too distant to be seen by other telescopes.
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One of the challenges of using the JWST is that it will generate a vast amount of data. In its first year of operation, the telescope is expected to generate about 100 terabytes of data. This data will need to be processed and analyzed in order to extract the scientific information that it contains.
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AI is being used to help with this task. AI algorithms are being developed to automatically identify objects in the data, classify them, and measure their properties. This will allow scientists to quickly and easily access the information that they need.
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AI is also being used to help with the design of new instruments for the JWST. AI algorithms are being used to simulate the performance of new instruments, and to identify the best design for a given task. This will help to ensure that the JWST is able to make the most of its capabilities.
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The use of AI is essential to the success of the JWST. By automating tasks that would otherwise be time-consuming and labor-intensive, AI will allow scientists to focus on the most important aspects of their work. This will help the JWST to make new and exciting discoveries about the universe.
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Here are some specific examples of how AI is being used with the JWST:
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AI is being used to identify and classify galaxies in the early universe. This is a challenging task, as the galaxies are very faint and distant. However, AI algorithms have been able to successfully identify and classify these galaxies, providing new insights into the formation of galaxies and the evolution of the universe.<br /><br />
AI is being used to study the atmospheres of exoplanets. This is another challenging task, as the atmospheres of exoplanets are very faint. However, AI algorithms have been able to successfully detect the presence of water vapor and other molecules in the atmospheres of some exoplanets, providing new evidence that these planets may be habitable.<br /><br />
AI is being used to study the composition of comets. This is a valuable task, as comets are thought to be remnants of the early solar system. AI algorithms have been able to successfully identify the presence of various molecules in comets, providing new insights into the formation of the solar system.<br /><br />
These are just a few examples of how AI is being used with the JWST. As the telescope continues to operate, AI is expected to play an even greater role in helping scientists to extract the scientific information that it contains.<br /><br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-91052194685750951582023-05-11T17:34:00.010-07:002023-05-11T17:34:49.839-07:00NASA's James Webb Space Telescope Continues to Break RecordsThe James Webb Space Telescope (JWST) is still in its early stages of operation, but it has already broken several records. In just a few months, the telescope has:<br /><br />
Observed the most distant galaxies ever seen, dating back to just 300 million years after the Big Bang.
Detected water vapor in the atmosphere of an exoplanet, the first time this has been done for a planet outside our solar system.
Studied the atmosphere of a comet, providing new insights into its composition.
Taken stunning images of nebulae, star clusters, and other celestial objects.
These are just a few of the many accomplishments of the JWST. As the telescope continues to operate, it is expected to make even more groundbreaking discoveries.
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One of the most exciting things about the JWST is its potential to find signs of life beyond Earth. The telescope is equipped with powerful instruments that can detect the presence of water, oxygen, and other biosignature gases in the atmospheres of exoplanets. In the coming years, the JWST will be used to search for exoplanets that could potentially support life.
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The JWST is a truly revolutionary telescope, and it is only just beginning to reveal its secrets. As the telescope continues to operate, it is sure to change our understanding of the universe and our place in it.
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Here are some additional details about the Webb telescope reports:
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The telescope's observations of the most distant galaxies ever seen have provided new insights into the early universe. These galaxies are so far away that their light has taken billions of years to reach us. By studying these galaxies, scientists can learn about the conditions that existed in the universe just a few hundred million years after the Big Bang.<br /><br />
The telescope's detection of water vapor in the atmosphere of an exoplanet is a major breakthrough. This is the first time that water vapor has been detected in the atmosphere of a planet outside our solar system. This discovery provides strong evidence that there may be other planets in the universe that could support life.<br /><br />
The telescope's study of the atmosphere of a comet has provided new insights into its composition. Comets are icy bodies that orbit the sun. They are thought to be remnants of the early solar system. By studying the atmosphere of a comet, scientists can learn more about the materials that were present in the early solar system.<br /><br />
The telescope's stunning images of nebulae, star clusters, and other celestial objects have captured the imagination of people all over the world. These images have provided new views of the universe that were previously impossible to see.<br /><br />
The JWST is a truly amazing telescope, and it is only just beginning to reveal its secrets. As the telescope continues to operate, it is sure to change our understanding of the universe and our place in it.<br /><br />
H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-14720170572910450652023-04-11T10:28:00.001-07:002023-04-11T10:28:30.651-07:00Navigating the Mathematical Challenges in AI: Contradictions, Paradoxes, and LimitationIntroduction:<br /><br />
Artificial intelligence (AI) has made remarkable strides in recent years, transforming industries and impacting our daily lives. However, the development of AI is far from a straightforward process. AI researchers face various mathematical challenges, including paradoxes, contradictions, and limitations that require innovative solutions to ensure the safe and effective implementation of AI systems.<br /><br />
The Alignment Problem: A Major Contradiction in AI<br /><br />
One of the most critical contradictions under investigation in the field of AI is the alignment problem. This challenge pertains to ensuring that AI systems consistently pursue human values and objectives, even as they become more capable. AI systems may optimize a given objective in unintended ways, which could lead to harmful or undesirable consequences.
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For instance, if an AI system maximizes efficiency in a factory, it may compromise safety measures or worker well-being. To address the alignment problem, researchers work on techniques to improve AI interpretability, robustness, and value alignment with human ethics and preferences. This involves creating systems that understand and respect human values, even when they aren't explicitly specified or are complex and nuanced.
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Gödel's Incompleteness Theorems: Paradoxes in AI
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Mathematical paradoxes, like Gödel's incompleteness theorems, also present challenges in AI development, particularly for artificial general intelligence (AGI). Gödel's incompleteness theorems highlight inherent limitations in any formal system, implying that there will always be problems that a system based on mathematical logic cannot solve. These theorems raise questions about the capabilities of AI systems, especially AGI, which aims to achieve human-level intelligence.
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Researchers continue to explore the implications of Gödel's incompleteness theorems for AI, attempting to understand the extent to which these limitations might constrain AI development and whether there are ways to overcome or bypass these inherent paradoxes.
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Mathematical Limitations in AI
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AI faces several mathematical limitations that impact its development and effectiveness:
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Curse of dimensionality: High-dimensional datasets can lead to poor performance, overfitting, and increased computational complexity in AI algorithms.<br /><br />
No free lunch theorem: There is no universally superior algorithm; AI researchers must tailor algorithms to specific problems or develop adaptive methods.<br /><br />
Local optima: AI algorithms can get stuck in local optima, which may not be globally optimal, leading to subpar solutions.
Overfitting: Balancing model complexity and the risk of overfitting is a significant challenge in AI.<br /><br />
Combinatorial explosion: Exponentially growing problem spaces in game playing or pathfinding require heuristics or approximations to find solutions.<br /><br />
Incomplete or noisy data: Reduced performance, incorrect predictions, or perpetuation of biases can result from AI systems learning from flawed data.
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Computational complexity: AI researchers often need to develop heuristics or approximation algorithms to deal with computationally intractable problems.
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Conclusion:
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The mathematical challenges that AI researchers face—contradictions, paradoxes, and limitations—are critical to understanding the fundamental capabilities and limits of AI systems. By addressing these challenges, researchers can develop new methods, algorithms, and architectures to improve AI's ability to learn from data, reason, and make decisions in complex environments. As we continue to push the boundaries of AI, understanding and addressing these issues will be essential to ensuring the development of safe, effective, and aligned AI systems. (See <a href="https://www.ai-hive.net/">AI HIVE</a>).H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-70553696261514684342023-04-10T17:25:00.001-07:002023-04-10T17:25:29.451-07:00Comparing GPT and BERT The generative pre-trained transformers (GPT) are a family of large language models based on artificial neural networks that are pre-trained on large datasets of unlabelled text and able to generate novel human-like text developed by Google researchers and were introduced in 2018 by OpenAI. GPT-3 is the latest and most advanced GPT model with 175 billion parameters and was trained on 400 billion text tokens.
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BERT is another language model developed by Google that is pre-trained on large amounts of data. BERT stands for Bidirectional Encoder Representations from Transformers. BERT uses both left and proper contexts to create word representations. It is a multi-layer bidirectional Transformer encoder. While evaluating benchmark datasets, BERT has achieved state-of-the-art results in several natural language processing (NLP) tasks. In terms of performance and architecture differences between GPT and BERT, GPT models typically perform well when generating long-form text, such as articles or stories.
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While both are pre-trained on large text datasets, their training methods, tasks handled, and performance metrics differ. Understanding these differences is crucial to determining which model most applies to a particular NLP task.
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In contrast, the BERT model is better suited for NLP tasks that require language understanding, such as question-answering or sentiment analysis. Overall, both GPT and BERT are powerful NLP models that have been shown to excel in different areas of natural language processing.
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GPT models can generate natural language text that can be used as a search query for internet searches.
For instance, given a prompt such as "Search for the best restaurants in New York City."
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BERT could be utilized to understand the intent of the user's search query and provide more accurate results. For instance, if a user types in a search query like "What is the capital of France?", BERT can infer the question being asked and provide the relevant answer, "Paris." (See <a href="https://www.ai-hive.net/">AI HIVE</a>).H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-79634364181361404232023-04-10T16:00:00.004-07:002023-04-10T16:01:38.864-07:00Autonomous AI CodingThe development of autonomous AI software coding is an ongoing and rapidly evolving research area. As AI models become more sophisticated, we'll likely see further progress in AI-driven code generation and even the creation of entirely new software systems with minimal human intervention.
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As for programming languages, AI-based code generation systems are currently being developed to work with existing programming languages like Python, JavaScript, and others. These systems are designed to understand and generate code in languages already widely used by developers, as doing so allows for the seamless integration of AI-generated code into existing software projects.
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It is possible that, in the future, AI systems could develop their own AI languages or domain-specific languages (DSLs) tailored to specific tasks or industries. However, creating a new programming language requires widespread adoption and support from the developer community, which can be a significant barrier. Additionally, using existing languages allows AI-generated code to be easily understood, maintained, and extended by human developers.
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As AI models become more autonomous, they may generate code in novel ways, create new abstractions and patterns that could influence the evolution of existing programming languages, or even inspire new ones. It is reasonable to expect AI-generated code to continue improving and becoming more sophisticated in the coming years. However, the possibility of AI-driven languages or DSLs should not be ruled out entirely (See <a href="https://www.ai-hive.net/">AI HIVE</a>).H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-54015866834430806842023-04-10T12:55:00.006-07:002023-04-10T12:55:58.974-07:00AI-Hive Phenomenon
The rapid growth of Artificial Intelligence (AI) has been accompanied by an increased need for effective communication and collaboration between AI developers, researchers, and enthusiasts. Hive platforms, such as AI-HIVE.net, have emerged as a potential solution to this challenge, revolutionizing how AI professionals connect. <br /><br />
Hive platforms have gained significant traction among AI developers as a centralized location for forum opinions, blog updates, research papers, tutorials, and tools. This community building allows the exchange of ideas, insights, experiences, and peer recognition.<br /><br />
Problem-solving is achieved through real-time cross-disciplinary collaboration. The potential benefit of blog updates is enabling wide knowledge dissemination. <br /><br />
As AI continues to evolve and impact various industries, the potential for Hive platforms to remain crucial in fostering an environment of innovation and growth for AI developers will be considered.<br /><br />
<br /><br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-57431574743298318792023-03-04T09:33:00.013-08:002023-03-04T09:55:43.384-08:00AI Hive DevelopmentAn AI hive has the potential to revolutionize the way we learn and acquire knowledge online. By leveraging the collective intelligence and collaboration of multiple AI agents, an AI hive could provide a personalized, engaging, and effective learning experience that is tailored to the needs and preferences of individual web users. AI hives can be used to solve complex problems more efficiently and effectively than traditional methods. AI hives are used in various industries:
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Manufacturing: At the BMW Group factory in Dingolfing, Germany, a group of robots work together in an AI hive to produce custom-made electric car components. The robots are equipped with sensors and cameras that allow them to coordinate their movements and avoid collisions, resulting in a more efficient and precise manufacturing process.
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Healthcare: In a study published in Nature, researchers used an AI hive to diagnose skin cancer. The hive consisted of 157 AI agents, each with a different skill set, such as analyzing clinical images or reading pathology reports. The agents worked together to diagnose skin cancer with an accuracy rate that exceeded that of individual dermatologists.
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Transportation: In Singapore, a group of self-driving buses operate in an AI hive to optimize their routes and minimize travel time. The buses are equipped with sensors and cameras that allow them to communicate with each other and coordinate their movements to avoid collisions and reduce congestion.
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Finance: PayPal uses an AI hive to detect and prevent fraud in its payment system. The hive consists of multiple AI agents that analyze transaction data and collaborate to identify suspicious activity. The agents can also learn from each other, improving their accuracy and effectiveness over time.
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An AI hive could be used to educate. Here are some possible scenarios:
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<a href="http://ai-hive.net" target="_blank">AI-Hive</a> is an example that could then recommend relevant educational content, such as articles, videos, and tutorials, that are tailored to the user's interests and learning style. It could create a collaborative learning environment where web users can interact with each other and share their knowledge and expertise. The hive could facilitate online discussions, peer-to-peer feedback, and group projects that promote collaborative learning and knowledge exchange.
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It could act as an intelligent tutor that guides web users through a learning journey. The hive could use natural language processing and machine learning algorithms to understand the user's questions and provide personalized feedback and guidance. The hive could also adapt its teaching approach based on the user's progress and feedback.H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-60889663008214083312023-01-20T15:20:00.002-08:002023-01-20T15:20:46.277-08:00Integrating ChatGPT into TechnologyMicrosoft:
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The integration of ChatGPT into Microsoft Office and Bing could greatly improve the user experience by making it easier for users to complete tasks and find information using natural language commands. It could also increase productivity and efficiency by automating repetitive. Microsoft Office could be used to enable users to complete tasks using natural language commands.
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For example, a user could say "Insert a table with three rows and four columns in Word" and ChatGPT would understand the command and insert the table in the document. Another example could be in Bing, where ChatGPT could be used to enhance the search capabilities. A user could say "Show me the best Italian restaurants in New York" and ChatGPT would understand the command and return a list of top-rated Italian restaurants in New York.
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Another example could be in Outlook, ChatGPT could be used to compose emails, schedule meetings, and set reminders by natural language commands. For example, a user could say "Schedule a meeting with John and Jane next Wednesday at 2 PM" and ChatGPT would understand the command, create a calendar event, and send an invitation to John and Jane. In Excel, ChatGPT could be used to perform data analysis, create charts and graphs, and automate tasks using natural language commands. For example, a user could say "Show me the trend of sales in the last quarter" and ChatGPT would understand the command, retrieve the data, and create a chart or graph to display the trend of sales.
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Google:
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Overall, the expansion of Google DeepMind's capabilities in areas such as computer vision, natural language processing, and robotics could lead to significant advancements in these fields and bring a lot of benefits to Google's products and services such as Google Photos, YouTube, Google Assistant, Google Translate, and Waymo.
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Google's DeepMind is already a leader in the field of AI, and it is likely that the company will continue to invest in and expand its capabilities in areas such as machine learning and deep learning. One example of this expansion could be in computer vision, where DeepMind could be used to improve image and video recognition capabilities in Google products such as Google Photos and YouTube.
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For example, DeepMind could be used to automatically tag and organize photos and videos based on the objects and people in them, making it easier for users to search and find specific content. Another example could be in natural language processing, where DeepMind could be used to improve the capabilities of Google Assistant and Google Translate. For example, DeepMind could be used to make Google Assistant more conversational, allowing users to carry out more complex tasks using natural language commands.
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Additionally, DeepMind could be used to improve the accuracy and fluency of Google Translate, making it possible to translate between more languages and idiomatic expressions. DeepMind could also be used to develop advanced robotics capabilities.
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For example, Google's Waymo self-driving cars is already using DeepMind's technology, but in the future, it could be used to develop robots that can perform a wide range of tasks in different environments, such as manufacturing, healthcare, and transportation. DeepMind could also be used to optimize energy consumption in data centers and improve the efficiency of Google's search algorithms.
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NVIDIA:
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NVIDIA's continued investment in and development of specialized AI hardware and software, as well as partnerships with other companies and research institutions, could lead to significant advancements in the field of AI and bring many benefits to a wide range of industries. NVIDIA actively competing in AI: NVIDIA is already a major player in the AI industry, and it is likely that the company will continue to invest in and develop its AI capabilities.
One example of this could be in the development of specialized AI hardware, such as graphics processing units (GPUs) optimized for deep learning and other AI applications. NVIDIA's GPUs are already widely used in the industry for training deep learning models and are more efficient than traditional CPUs.
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In the future, NVIDIA could develop even more specialized AI hardware, such as custom ASICs (Application-Specific Integrated Circuits) tailored to specific AI workloads, which could further improve the performance of AI systems. Another example could be in the development of specialized AI software, such as libraries and frameworks for deep learning.
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NVIDIA already has a suite of AI software development tools such as CUDA and cuDNN, which enable developers to easily implement deep learning algorithms on NVIDIA hardware.
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In the future, NVIDIA could develop more specialized software tailored to specific AI workloads, such as computer vision and natural language processing. NVIDIA could also expand its partnerships with other companies and research institutions to further advance the field of AI.
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For example, NVIDIA could collaborate with companies in the autonomous vehicle industry to develop AI systems that can enable cars to drive themselves. Additionally, NVIDIA could partner with healthcare companies to develop AI systems that can assist in medical diagnosis and treatment. In addition, NVIDIA could develop specialized AI-based products, such as AI-based cameras, drones and robots using its expertise in AI and computer vision.
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TESLA:
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Tesla's continued investment in the development of autonomous vehicles and robotics could lead to significant advancements in these fields and bring many benefits to a wide range of industries.
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Tesla reaching autonomous driving and robots: Tesla has already made significant progress in the development of autonomous vehicles, and it is likely that the company will continue to invest in this area. One example of this could be in the continued development of Tesla's Autopilot system, which is already capable of performing many semi-autonomous driving tasks such as steering, accelerating, and braking. In the future, Tesla could continue to improve the Autopilot system, making it increasingly capable of performing more complex tasks such as navigating city streets and merging onto highways.
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Another example could be in the development of fully autonomous vehicles, which would not require any human input and could drive themselves without any need for a driver.
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Tesla has already announced that all of their vehicles are being built with the necessary hardware for full autonomy, and the company plans to roll out a software update that will enable full autonomy in the future. Tesla could also potentially expand into the field of robotics, using its expertise in AI and autonomous systems to develop robots for a variety of applications. For example, Tesla could develop robots that can perform tasks such as manufacturing, logistics, and transportation. These robots could potentially be powered by Tesla's electric powertrains and be able to operate in a sustainable way.
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Additionally, Tesla could develop robots that can assist in maintenance and repair tasks on vehicles, such as changing tires, replacing batteries, and performing other routine maintenance. Another application could be in the field of home automation and smart homes, where Tesla could develop robots that can perform tasks such as cleaning, cooking, and providing security.
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EDUCATION:
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The use of ChatGPT to develop educational programs that follow the Montessori method could greatly enhance the learning experience for students by providing them with interactive, personalized, and real-time feedback on their progress, which would ultimately lead to better student outcomes. The use of ChatGPT to do Montessori teaching: ChatGPT could potentially be used to develop educational programs that follow the Montessori method by creating interactive and personalized learning experiences for students. One example of this could be in the development of an interactive language learning program, where ChatGPT could be used to generate personalized exercises and activities that are tailored to the student's individual language level and learning style.
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The program could also use ChatGPT to provide real-time feedback to the student on their progress, such as identifying areas where the student is struggling and providing additional exercises to help them improve. Another example could be in the field of math and science education, where ChatGPT could be used to create interactive simulations and visualizations that help students understand complex concepts.
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The program could also use ChatGPT to provide real-time feedback to the student on their progress, such as identifying areas where the student is struggling and providing additional exercises to help them improve. In addition, ChatGPT could also be used to create personalized learning plans for students, based on their strengths, weaknesses and learning style. This would enable teachers to focus on the areas where each student needs the most help and provide them with the resources and support they need to succeed. ChatGPT could also be used to generate assessments and quizzes that are tailored to each student's level of understanding, providing teachers with real-time feedback on student progress.
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MILITARY:
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The use of LLM for command and control in the US military could greatly improve the efficiency and effectiveness of military operations, by providing the military with the ability to analyze large amounts of data, make predictions about potential threats, control unmanned systems, and autonomous weapons and improve situational awareness. The US military is using LLM for command and control: The US military could potentially use LLM (Large Language Models) to improve its command-and-control capabilities in several ways. One example could be in the analysis of large amounts of data to make predictions about potential threats. LLM could be used to analyze data from various sources such as satellite imagery, social media, and sensor data to identify patterns and trends that could indicate a potential threat.
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This could help the military to take proactive measures to prevent or mitigate the threat. Another example could be in the control of unmanned systems and autonomous weapons. LLM could be used to enable unmanned systems and autonomous weapons to make decisions and take actions based on natural language commands.
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This could greatly increase the efficiency and effectiveness of these systems, as they would be able to operate more autonomously, reducing the need for human intervention. LLM could also be used to improve the efficiency of command-and-control systems by automating routine tasks, such as monitoring and tracking the status of various systems and assets.
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This could free up human operators to focus on more critical tasks, such as decision-making and problem-solving. LLM could also improve the situational awareness of the military, by providing real-time updates and alerts on the status of various systems and assets, such as the location of troops, the status of equipment, and the progress of missions. This could greatly improve the ability of commanders to make informed decisions in a timely manner.
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These are just some of the possibilities yet to unfold.
<br /><br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-8993811573025510032023-01-17T12:14:00.001-08:002023-01-17T12:14:37.236-08:00Inertial Confinement Fusion impact on the Future of the Electrical Distribution GridInertial Confinement Fusion (ICF) is a promising approach for harnessing the power of fusion energy, which has the potential to provide a clean, safe, and sustainable source of power for the future. However, as ICF power plants begin to come online over the next decade, the world's electrical distribution grid will need to adapt to accommodate this new source of power. <br /><br />
One of the key advantages of ICF is that it has the potential to produce energy more continuously than other forms of fusion, such as magnetic confinement fusion. In ICF, powerful lasers are used to compress and heat tiny pellets of fusion fuel, causing them to undergo fusion reactions. This approach allows for a steady state of energy production, as the lasers can be fired repeatedly.
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However, as with any new technology, there are still several technical challenges to be addressed before ICF can be integrated into the grid at a commercial scale. One of the major challenges is energy storage.
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There are several options that can be considered to store the energy produced by ICF power plants. One option is to use advanced battery systems, such as lithium-ion batteries, which can quickly and efficiently store large amounts of energy. Another option is to use hydrogen fuel cells, which can store energy in the form of hydrogen gas and then convert it back into electricity when needed.
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Fusion and fission are two distinct methods of generating nuclear energy. There are key differences in the engineering technology required for their power plants.
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First, fusion power plants will require much higher temperatures than fission power plants. In order to initiate and sustain a fusion reaction, the fuel must be heated to millions of degrees Celsius, much hotter than the temperatures required for fission. This will require the development of advanced materials that can withstand these extreme temperatures, as well as the development of new cooling systems to remove the massive amount of heat generated by the reaction.
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Second, fusion power plants will require much higher pressures than fission power plants. In order to initiate and sustain a fusion reaction, the fuel must be compressed to extremely high densities, much higher than the densities required for fission. This will require the development of advanced compression systems to achieve these high pressures, as well as new technologies to contain the high-pressure plasma.
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Third, fusion power plants will require much more precise control of the reaction than fission power plants. In fission, the reaction is self-sustaining once started, but in fusion the reaction must be sustained by a constant input of energy. This will require the development of new control systems to regulate the input of energy and maintain the conditions necessary for the reaction to take place.
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Fourth, fusion power plants will not generate high level of radioactive waste, unlike fission power plants. As a result, the waste management system for fusion power plants will be simpler and less complex than those for fission power plants.
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Fusion is a promising technology but still has a long way to go before it can be integrated into the grid.<br /><br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-17440433202336199582023-01-16T11:42:00.001-08:002023-01-16T11:42:31.805-08:00Basic NIF Power Plant Engineering
The National Ignition Facility (NIF) experiment used a laser-based approach to initiate nuclear fusion reactions. <br /><br />
In this blog, we consider a speculative design for a nuclear fusion power plant based on the NIF experiment that includes the following steps: fuel preparation, laser compression, fusion reaction rate, energy collection, waste management, maintenance, and upgrade.<br /><br />
Fuel preparation: <br /><br />
The fuel for the fusion reactions would be isotopes of hydrogen, specifically deuterium and tritium. These isotopes would be extracted and purified from natural sources such as seawater. The quality and purity of the fuel directly impact the efficiency and safety of the fusion reactions. In order to achieve the high fuel consumption rate of thousands of fusion pellets per hour, the following steps would be necessary:
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Fuel extraction of deuterium and tritium from seawater. This process would involve a combination of distillation, electrolysis, and other chemical separation techniques to isolate the isotopes from other elements in the water.
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Once extracted, the deuterium and tritium would need to be further purified to remove any impurities. This could be done through a series of chemical and physical processes such as gas chromatography, isotope separation, and high-temperature distillation.
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The purified fuel would then be compressed into small, spherical pellets that are suitable for use in the laser compression chamber. These pellets would be highly dense, typically around 100-400 mg, and uniform in size and composition.
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The fuel pellets need to be stored in special containers with handling safety protocols.
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The fuel pellets would be delivered to the laser compression chamber in a precise and timely manner using an automated delivery systems such as conveyors or robots.
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To ensure the quality and purity of the fuel, tracking and monitoring systems would include measuring the isotopic composition, density, and temperature of the fuel, and taking appropriate action if any issues arise.
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Laser compression: <br /><br />
The fuel would be loaded into a small target chamber and compressed to extremely high densities and temperatures using powerful lasers. The laser system used for compression would be a high-powered, multi-beam laser system capable of delivering high-energy pulses to the target chamber. This could include a combination of solid-state and gas lasers, such as Nd:YAG and CO2 lasers, which are capable of producing the high-intensity pulses required for compression.
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The target chamber would be a small, highly-engineered vessel designed to withstand the intense pressures and temperatures generated by the laser compression process. It would be made of materials such as beryllium or lithium. Diamonds are manufactured that are capable of withstanding high-energy radiation and neutron flux.
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The fuel pellets need to be precisely delivered into the target to maintain the high fuel consumption rate. This could involve the use of a target positioning system which would be capable of precise alignment and delivery of the fuel pellets.
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The laser pulses involve the use of pulse shaping optics, such as spatial light modulators, to shape and focus the laser beams.
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A suite of diagnostic tools include a combination of optical, x-ray, and particle diagnostic systems, which would provide real-time data on the fuel compression and fusion reactions.
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As the laser compression process generates high-energy radiation and neutron flux, safety systems would be implemented to protect personnel and equipment. This could include radiation shielding, emergency shut-off systems, and other safety measures to ensure the safe operation of the laser compression system.
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Overall, the laser compression process would require a highly advanced and precise system, including powerful lasers, a specially designed target chamber, and sophisticated control and diagnostic systems to create the conditions necessary for nuclear fusion to occur.
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Fusion reaction rate:
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The fusion reaction is where the energy is produced. The reactions that take place in a fusion power plant are similar to those that take place in the sun, where hydrogen isotopes fuse to form helium and release a large amount of energy in the process.
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During a fusion reaction, deuterium and tritium nuclei are brought together under high temperature and pressure conditions, allowing them to overcome their electrostatic repulsion and fuse together. The resulting helium atom is slightly lighter than the original nuclei, releasing energy in the form of high-energy particles and radiation.
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The energy released per fusion reaction is extremely high, roughly on the order of 17.6 MeV (Mega electron Volts), which is about four times the energy released by fission reactions in current nuclear power plants. However, the energy consumption to initiate and sustain a fusion reaction is also high, and it is still a subject of ongoing research to achieve net energy gain in the fusion process.
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Currently, the best experiments in the field (such as the ITER project) are aiming to reach a Q value (ratio of energy produced by the reaction to the energy used to initiate it) of at least 10, meaning that they aim to produce ten times more energy than they consume. However, it is still uncertain whether this goal will be achieved and when commercial reactors will be available.
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In comparison, the sun, which is a natural fusion reactor, has a Q value of about 1, as it consumes more energy than it produces in the form of heat and light. However, the sun has been running for billions of years, and the energy consumption is negligible when compared to the energy produced.<br /><br />
Energy collection:
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Energy collection is the step in the nuclear fusion power plant where the energy released by the fusion reactions is converted into a usable form, typically electricity. There are several alternative options for energy collection:
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The high-energy particles and radiation released are directly converted into electricity through the use of thermionic converters or solid-state electrical generators. T
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This involves the use of a heat exchanger to transfer the energy released by the fusion reactions to a working fluid, such as water or helium, which is then used to generate electricity in a conventional power generation system, such as a steam turbine or a gas turbine.
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Overall, the energy collection method chosen will depend on the specific design of the fusion power plant and the trade-offs between efficiency, cost, and technical feasibility. Each method has its own set of advantages and disadvantages, and further research is needed to determine the best option for a practical fusion power plant.
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Waste management:
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The by-products of the fusion reactions, such as helium and neutron radiation, would be safely contained and managed to minimize any negative impacts on the environment.
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Maintenance and Upgradation:
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The power plant will be regularly maintained and upgraded as necessary to ensure optimal performance and safety.
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This is a speculative design, as nuclear fusion at a scale that would be useful for power production has yet to be achieved. While the NIF experiment has made significant progress in developing the technology, there are still many technical challenges that must be overcome before a practical fusion power plant can be built.
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H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-35362223018726145342023-01-01T16:10:00.001-08:002023-01-01T16:10:18.259-08:00Langlands Program and OpenAIThe Langlands program is a huge web of connections and relationships between different parts of mathematics, such as number theory, representation theory, and algebraic geometry. It is the study of symmetry and duality between different mathematical objects. It has had a big impact on the way modern mathematics has grown and changed. <br /><br />
One area where the Langlands program has seen significant progress in recent years is in its relationship to quantum field theory. Quantum field theory is a way to explain how particles behave and how they interact with each other. It has been an important part of how we've come to understand the basic laws of physics. <br /><br />
The Langlands program has made a big contribution to quantum field theory by looking at symmetry and duality between different mathematical objects. For example, studying Langlands duality in quantum field theory has given us new ideas about the structure of gauge theories, which are important for understanding the fundamental forces of nature. <br /><br />
OpenAI is a top organization for research that wants to make new technologies and algorithms for artificial intelligence. It has made significant contributions to machine learning and natural language processing. It could greatly affect the Langlands program and how it works with quantum field theory. <br /><br />
One way OpenAI could potentially impact the Langlands program is by developing new machine learning algorithms that can analyze and understand complex mathematical structures and patterns. The key symmetries and dualities of the Langlands program could be automatically extracted and analyzed by these algorithms. This could lead to new insights and a better understanding of these ideas. <br /><br />
Also, OpenAI's work on natural language processing is related to the Langlands program because it helps create machine learning systems that can understand and interpret mathematical texts and ideas. This could lead to new tools and methods for studying the Langlands program and help us learn more about this complicated and interesting area of math. <br /><br />
Overall, the Langlands program is a vast and complex field of study, with connections and relationships to many different areas of mathematics and physics. Through the development of advanced machine learning algorithms and tools for natural language processing, OpenAI's work could have a big impact on this field. <br /><br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-62200888825360350372023-01-01T12:45:00.003-08:002023-01-01T12:45:28.651-08:00Photofission Concept using U-236Consider a new experimental device that uses a high energy laser pulse to ignite a U-236 crystal lattice, yielding a higher energy directional beam from the U-236 photo-fissioning process.<br /><br />
By using a high energy laser pulse to initiate the photo-fissioning process in a U-236 crystal lattice, we might be to harness the energy released during the fission process and direct it in a specific direction as a beam.<br /><br />
The process begins with the laser pulse, which is focused onto the surface of the U-236 crystal. The photons collide with the U-236 atoms causing them to fission and release a tremendous amount of energy. The crystal structure channels a large protion of the enrgy output as a beam, providing a highly concentrated source of power.<br /><br />
One of the key benefits of this could be its efficiency. Traditional energy production methods, such as fossil fuels and nuclear fission, have low conversion rates, meaning a large amount of energy is lost during the generation process. In contrast, the U-236 photo-fissioning device has a much higher conversion rate, meaning less energy is lost and more is directed to a specific task.<br /><br />
Another benefit is the device's potential for scalability. The size of the U-236 crystal can be easily adjusted to meet the specific energy needs of a particular application. This means that the device could potentially be used for a wide range of purposes.<br /><br />
Overall, experimental U-236 photo-fissioning has the potential to be a new technology in the field of energy production. Its high efficiency and scalability make it a promising alternative to traditional energy sources.
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H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-10126486775961276182023-01-01T10:32:00.000-08:002023-01-01T10:32:04.390-08:00Space-based nuclear fissioning lasers are a type of weapon that use the energy from a nuclear fissioning material to power a laser beam. While the concept of these weapons has been around for several decades, they have yet to be successfully developed and deployed.<br /> <br />
One of the main challenges in developing space-based nuclear fissioning lasers is the difficulty in creating a stable, high-energy laser beam. Nuclear fissioning materials produce a significant amount of energy, but harnessing that energy and channeling it into a coherent laser beam has proven to be a daunting task. Additionally, the fissioning material itself would need to be carefully controlled in order to prevent the laser beam from being disrupted or dispersed.<br /> <br />
Another issue with space-based nuclear fissioning lasers is the potential for radioactive contamination. A nuclear fissioning material would produce radioactive debris, which could potentially contaminate the area around the weapon. This could have serious consequences for both the environment and for human health.<br /> <br />
Despite these challenges, research and development of space-based nuclear fissioning lasers has continued over the years. In the 1980s, the United States conducted a number of tests to explore the feasibility of these weapons, but the program was eventually abandoned due to technical difficulties and concerns about the potential for nuclear proliferation.<br /> <br />
In recent years, there have been some indications that other countries, such as Russia and China, may be exploring the development of space-based nuclear fissioning lasers. However, it is unclear to what extent these efforts are underway, and it is likely that significant technological hurdles would need to be overcome in order to successfully develop and deploy these weapons.<br /> <br />
Overall, the current state of development of space-based nuclear fissioning lasers is one of uncertainty. While the concept of these weapons has been around for decades, the technical challenges and potential consequences of their use have so far prevented their successful development and deployment. It remains to be seen whether these challenges can be overcome in the future.<br /> <br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-41938466537085367002022-12-22T12:28:00.004-08:002022-12-22T12:28:46.530-08:00Higgs ParticleThe Higgs particle, also known as the Higgs boson, is a subatomic particle that was discovered at the Large Hadron Collider (LHC) in 2012. The discovery of the Higgs particle was a major milestone in particle physics, as it helps to explain how particles in the universe acquire mass.
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The Higgs particle is named after physicist Peter Higgs, who proposed the existence of the particle in 1964 as part of the Higgs mechanism. The Higgs mechanism explains how particles in the universe acquire mass through their interactions with the Higgs field, which is a field of energy that permeates all of space.
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Since its discovery, the Higgs particle has been the subject of ongoing research and study by particle physicists around the world. In the years since its discovery, scientists have continued to study the properties of the Higgs particle and how it behaves in different situations.
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One of the latest updates on the Higgs particle comes from the LHC, which has been conducting a series of experiments to study the behavior of the Higgs particle in greater detail. These experiments have provided new insights into the properties of the Higgs particle and have helped to further our understanding of how it behaves.<br /><br />
Overall, the study of the Higgs particle is an important area of research in particle physics, as it helps us to better understand the fundamental building blocks of the universe and the forces that govern their behavior. As research on the Higgs particle continues, we can expect to see more updates and discoveries that further our understanding of this fascinating and important particle.H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-2615227293197025262022-12-22T09:53:00.005-08:002022-12-22T09:53:28.899-08:00Dark Matter<p> Dark matter is a mysterious and elusive substance that is believed to make up about 27% of the total mass in the universe. Dark matter does not emit, absorb, or reflect light, which makes it difficult to detect directly. Instead, scientists infer its existence and properties based on its gravitational effects on visible matter, radiation, and the universe's large-scale structure.</p><p>One of the most compelling pieces of evidence for the existence of dark matter comes from the observation that galaxies rotate much faster than expected based on the mass of their visible stars, gas, and dust. This "missing mass" problem can be explained if a large amount of invisible, non-luminous matter is present that provides extra gravitational pull. Similarly, the cosmic microwave background radiation and the large-scale distribution of galaxies in the universe also suggest the presence of dark matter.</p><p>There are many theories about what dark matter could be, ranging from exotic particles to black holes to modifications of general relativity. </p><p>There are several experimental approaches to studying dark matter. One of the main techniques is to look for the indirect effects of dark matter particles on other particles, such as through their collisional or gravitational interactions. For example, scientists can search for signs of dark matter in cosmic rays, gamma rays, and neutrinos or look for distortions in the light from distant stars or galaxies due to the gravitational lensing effect of dark matter.</p><p>Other researchers are working to directly detect dark matter particles using specialized detectors that are sensitive to the rare interactions that might occur between dark matter and normal matter. These experiments typically use large volumes of materials, such as liquid xenon or germanium, that are sensitive to the passing of even a single dark matter particle.</p><p>In addition to these experimental efforts, there are also theoretical efforts to understand the nature of dark matter and to develop new models and predictions that can be tested by observations. This includes work on the fundamental properties of dark matter particles, such as their mass, spin, and interactions, and the development of new theories that could explain our observations in the universe.</p><p>Overall, the search for dark matter is a complex and multifaceted endeavor involving many fields of science, from astrophysics and particle physics to cosmology and theoretical physics. </p><p><br /></p>H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-1565435657695570642022-12-21T11:04:00.004-08:002022-12-21T11:04:40.091-08:00<h2 style="text-align: left;">Quantum Field Theory</h2><p>Quantum field theory is a theoretical framework that combines quantum mechanics and special relativity to explain the behavior of particles and the fundamental forces of nature. It is a cornerstone of modern physics and has significantly impacted our understanding of the universe.</p><p>Physicist Richard Feynman laid the foundations of quantum field theory in the 1950s. Feynman developed a new method for calculating the probability of certain events occurring in quantum systems, known as the path integral approach. This approach allowed Feynman to reformulate quantum mechanics in a more consistent way with special relativity, paving the way for the development of quantum field theory.</p><p>Over the next few decades, various physicists made significant contributions to the development of quantum field theory. Murray Gell-Mann and George Zweig developed the concept of quarks, which are fundamental particles that makeup protons and neutrons. Steven Weinberg, Abdus Salam, and Sheldon Glashow developed the electroweak theory, which unified the weak nuclear force and electromagnetism into a single theory.</p><p>Today, the standard model of particle physics is the most widely accepted theory of the fundamental forces and particles in the universe. It is a quantum field theory that includes electromagnetic, weak, and strong nuclear forces and all known elementary particles. The standard model has been highly successful in explaining a wide range of experimental data and is the basis for much modern particle physics research.</p><p>While the standard model is a powerful theory, it is not a complete description of the universe. It cannot explain several phenomena, such as dark matter and the nature of gravity. Researchers are currently developing theories that go beyond the standard model and can address these and other outstanding questions in physics.</p><p><br /></p>H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-92136993086172841332022-12-21T11:03:00.005-08:002022-12-21T11:03:38.121-08:00<h2 style="text-align: left;"><b>Breakeven Fusion at the National Ignition Facility</b></h2><p>The Lawrence Livermore National Laboratory (LLNL) has made a breakthrough in nuclear fusion. After years of research and development, the National Ignition Facility (NIF) team has successfully achieved nuclear fusion "breakeven" using lasers impinging on a diamond hydrogen pellet.</p><p>For those unfamiliar with nuclear fusion, it is the process by which atomic nuclei combine to form a heavier nucleus, releasing a large amount of energy in the process. This energy has the potential to provide a virtually limitless and clean source of power. However, achieving controlled nuclear fusion has proven to be a significant challenge, as it requires the fusion of hydrogen nuclei at temperatures and pressures found only in the cores of stars.</p><p>To overcome this challenge, the team at LLNL has been working on a method called inertial confinement fusion (ICF), which involves using lasers to heat and compress a tiny fuel pellet until the conditions are suitable for nuclear fusion to occur. At the NIF, 192 powerful lasers are used to impinge on a tiny diamond hydrogen pellet, creating the conditions needed for fusion.</p><p>After years of experimentation, the team at LLNL has finally achieved "breakeven," meaning that the amount of energy produced by the fusion reaction equals the amount of energy required to initiate the reaction. This is a significant milestone on the path toward practical fusion energy and could revolutionize how we generate electricity.</p><p>Whree there is still much work to be done before nuclear fusion becomes a viable power source, the success at LLNL is a significant step forward. The team at the NIF will continue to improve the efficiency and yield of their fusion reactions, and we are excited to see what the future holds for this exciting field.</p>H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-42520802150644502222017-07-01T21:37:00.001-07:002019-03-01T10:15:09.914-08:00Standard Model Lagrangian<div style="text-align: center;">
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The Standard Model of Particle Physics has helped unravel the hidden symmetries within the design of the Universe. Here we examine the steps in building the Standard Model.<br />
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<b>1. The Universe</b> was created out of interacting quantum fields producing forces (Bosons) from integer spin interactions and matter (Fermions) from half-integer spin.<br />
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Lagrangian Field Theory formulates the relativistic quantum mechanical theory of interactions. It has dependent variables replaced by values of a field at a point in space-time f(x,y,z,t). The equations of motion are obtained by the Action Principle using S as Action.<br />
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The Euler-Lagrange Equation minimizes S and produces the model's equation of motion:<br />
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The steps to construct the Standard Model of Quantum Field Theory start with the classical Lagrangian, L.<br />
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<b>2. The Lagrangian density</b>, L<br />
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Starting with 1863 Maxwell’s Equations<br />
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The L for Classical Electrodynamics:<br />
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Next, consider the Lagrangian density function for a massless field:<br />
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Introducing a mass term:<br />
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Introducing a source term produces J(x)<br />
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For the case of a field with mass and spin (1/2 and 1) interaction:<br />
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The Klein Gordon EOM for Spin ( 0 ) (Higgs field):<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCWyMF7phwkCFvsJ8H1UrLZ84iMqhkbzqV0oehu_xaA_JwFMSaBRfFTEWyhPk6eXWHXQFry1m7b3qtGYzpAidk31h5THSOo8LbJ3tKimGYGUQtzMrZHoJ7gLMMzQrtmtlekRNaIy2IF1kr/s1600/Klein+EOM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="27" data-original-width="131" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCWyMF7phwkCFvsJ8H1UrLZ84iMqhkbzqV0oehu_xaA_JwFMSaBRfFTEWyhPk6eXWHXQFry1m7b3qtGYzpAidk31h5THSOo8LbJ3tKimGYGUQtzMrZHoJ7gLMMzQrtmtlekRNaIy2IF1kr/s1600/Klein+EOM.png" /></a></div>
<br />
The solutions to the Klein Gordon Equation are simple plane waves subject to relativistic constraint:<br />
<br />
<div style="text-align: center;">
f(x) = Ce-i(p.x-Et) </div>
<br />
The EOM for Spin = 1/2 Dirac Eq.:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqrgYTn1wi3_bnmhQ5jlzJdZqnWNIM9puvbh_taWZO_Dou6RYvQfOHMuxKCJR7oMiVKlVF30GdWo0LQCQkO-cjl02ViUhyphenhyphenuwswwvsGOaFsy843TyCAxXKGi68YN6WThM7KrlRNFdKjZ8af/s1600/Dirac+EOM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="27" data-original-width="131" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqrgYTn1wi3_bnmhQ5jlzJdZqnWNIM9puvbh_taWZO_Dou6RYvQfOHMuxKCJR7oMiVKlVF30GdWo0LQCQkO-cjl02ViUhyphenhyphenuwswwvsGOaFsy843TyCAxXKGi68YN6WThM7KrlRNFdKjZ8af/s1600/Dirac+EOM.png" /></a></div>
<br />
The EOM for Spin = 1 (Boson) Proca Eq.:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiAjO-IQq7-k_py_x4IAWA7INpSfY8WyAVkHH9jLW-CeVPzaK6s0538sG8EEDMptbIka4M4rAYo0fhgrF27HuQjYLQNwElV3GCGfGbBD582n3Vlr0xLiKkjpnrA-yQT-fESQ00xBAJETbe/s1600/Proca+EOM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="29" data-original-width="131" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiAjO-IQq7-k_py_x4IAWA7INpSfY8WyAVkHH9jLW-CeVPzaK6s0538sG8EEDMptbIka4M4rAYo0fhgrF27HuQjYLQNwElV3GCGfGbBD582n3Vlr0xLiKkjpnrA-yQT-fESQ00xBAJETbe/s1600/Proca+EOM.png" /></a></div>
<b><br /></b> <b>3. Quantum Electrodynamic</b> U(1):<br />
<br />
Tomonaga, Feynman, Schwinger (1945-58) developed Quantum Electrodynamics: a precise description of electromagnetic interactions.<br />
<br />
Feynman Diagram:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCys37e5ft6C9QylVoEZfcaFWKy9N6w1Dq_LMIKvAcjv9CH261OQXs5mLIJTCLTLbiWrOzG4gg6r7A98WBqHTrdyCKbAaIOXy3Sc_TSiTo3qtHtlgNcAHyH5rqG2fs4-gXoBNL4yvEhPt3/s1600/feynman.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="360" data-original-width="640" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCys37e5ft6C9QylVoEZfcaFWKy9N6w1Dq_LMIKvAcjv9CH261OQXs5mLIJTCLTLbiWrOzG4gg6r7A98WBqHTrdyCKbAaIOXy3Sc_TSiTo3qtHtlgNcAHyH5rqG2fs4-gXoBNL4yvEhPt3/s320/feynman.png" width="320" /></a></div>
Feynman transition probabilities are calculated from a Feynman diagram where (for example) Fermions (spin 1/2 with charge) are destroyed to create a virtual Boson (spin 1 without charge) that is then destroyed to create new Fermions.<br />
<br />
Note: A loop in a Feynman diagram indicts a divergence (infinite integral) that must be renormalized for calculations.<br />
<br />
A photon is a spin 1 massless Boson interference packet in the electromagnetic field that has no rest mass, but has quanta E = hv and always travels at speed c.<br />
<br />
An electron is a spin 1/2 Fermion interference packet in the electromagnetic field that has a rest mass.<br />
<br />
A quark (Gell-Mann, Zweig 1960) is a spin 1/2 Fermion interference packet that interacts with electromagnetic, weak, and strong fields and has a rest mass.<br />
<br />
The Quantum Electrodynamics (QED) Lagrangian:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoQm8cZPoZiYaobzQTwddeYYheAjomNipFXlMTKurCaJLjxlxzlvCTWfKmcQj0yH1FRxB2wEjZkSFtRbRw5g_COZxTD9K2xwbf4RDjWN4_FwkWGRjjYT9r-msCgR_5yT_GKhscdaByRCsh/s1600/aa11.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="146" data-original-width="467" height="100" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoQm8cZPoZiYaobzQTwddeYYheAjomNipFXlMTKurCaJLjxlxzlvCTWfKmcQj0yH1FRxB2wEjZkSFtRbRw5g_COZxTD9K2xwbf4RDjWN4_FwkWGRjjYT9r-msCgR_5yT_GKhscdaByRCsh/s320/aa11.png" width="320" /></a></div>
<b>4. Quantum ElectroWeak</b> SU(2):<br />
<br />
Weinberg and Salam (1967) developed a gauge theory requiring three gauge bosons (W+-,Z). The Quantum ElectroWeak (QEW) Lagrangian:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihPfy7JhlJO5XjYdJdg9BQXgpgo400gGipU5VYyQLxAWed_StqPJgJlvNfv5Aa4BJv9vVc7BkUC-Hd7PUwvo0f1jI772SK96uFw0x8mUW2lJF-5Gcthw3Q9L5EkEoiIJycG1-BM8sQuLrv/s1600/a16+%25281%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="128" data-original-width="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihPfy7JhlJO5XjYdJdg9BQXgpgo400gGipU5VYyQLxAWed_StqPJgJlvNfv5Aa4BJv9vVc7BkUC-Hd7PUwvo0f1jI772SK96uFw0x8mUW2lJF-5Gcthw3Q9L5EkEoiIJycG1-BM8sQuLrv/s1600/a16+%25281%2529.png" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<br />
<b>5. Quantum Chromodynamics</b> SU(3):<br />
<br />
Han, Nambu, Greenburg (1970) described the strong force mediated by gauge bosons, called gluons, carrying a unique kind of charge called color. The Quantum Chromodynamics (QCD) Lagrangian:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkGjjkQSpu9N7S1ugwi7Nt3SX7SBT9bkeUeBmuUySb5FuluGWk3aHkhGUSYnlGuknUYDbF1u6QORNP5JH7GaaG_qwc-f6mXLW6zyTRD93tZkyz8fidQdmejGnZDokGQNCoCRPT6KCvEz20/s1600/aa13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="104" data-original-width="525" height="63" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkGjjkQSpu9N7S1ugwi7Nt3SX7SBT9bkeUeBmuUySb5FuluGWk3aHkhGUSYnlGuknUYDbF1u6QORNP5JH7GaaG_qwc-f6mXLW6zyTRD93tZkyz8fidQdmejGnZDokGQNCoCRPT6KCvEz20/s320/aa13.png" width="320" /></a></div>
<br />
<b>6. The Standard Model</b> SU(3) x SU(2) x U(1):<br />
<br />
The Standard Model (SM) Lagrangian:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUDSr2r5l7IhjAGJPTlz1T0kkTevMAbw6BLlgdGtREaSUPMpU-6Dqm6Oq5YJVZUWh0hehHawB2p4fMzLDMWoklp4Fu8mI2gScJevZuX2iun9uv39HcIwhP1hgLluxXKNvbcpqiTnktCUs4/s1600/aa14.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="223" data-original-width="516" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUDSr2r5l7IhjAGJPTlz1T0kkTevMAbw6BLlgdGtREaSUPMpU-6Dqm6Oq5YJVZUWh0hehHawB2p4fMzLDMWoklp4Fu8mI2gScJevZuX2iun9uv39HcIwhP1hgLluxXKNvbcpqiTnktCUs4/s400/aa14.png" width="400" /></a></div>
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</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
The first line represents the kinetic energy carried by W, Z, photon, and gluons. The second line is the interaction terms. The third line contains mass and the fourth line the left-right parity interaction.<br />
<br />
Hawkins (1980) Blackholes radiate.<br />
<br />
Guth (1981) Inflation Theory.<br />
<br />
<b>7. The Big Bang</b><br />
<div class="separator" style="clear: both; text-align: center;">
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The Higgs field is unstable to symmetry breaking. After 10^-12 seconds, the SU(2)xU(1) symmetry breaks and the electron acquires mass, the neutrino stays massless, the W+-, Z acquire mass and the massless photon emerges.<br />
<br />
A simple calculation of the Higgs' mass has suggested new science.<br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja2ayZNL4aR9gj10vJwEXWUVSBWx9cbQWiUf-hgOJD1sOOHrIZPYV0YwD-yOtXpfZwpN1IJJ4qr8QKjF0Wnd8829Z1Wcl9NPC-3hvdSqVYcFGyJDAYH7bwlCcD0fqMdcRMyWhN1nekJ2N5/s1600/particles1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="494" data-original-width="594" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja2ayZNL4aR9gj10vJwEXWUVSBWx9cbQWiUf-hgOJD1sOOHrIZPYV0YwD-yOtXpfZwpN1IJJ4qr8QKjF0Wnd8829Z1Wcl9NPC-3hvdSqVYcFGyJDAYH7bwlCcD0fqMdcRMyWhN1nekJ2N5/s320/particles1.png" width="320" /></a></div>
<br />
<br />
<b>References:</b><br />
<br />
[1] Cox, B. and Forshaw, J., Why does E=mc2, Da Capo Press, Cambridge, MA 2009.<br />
<br />
[2] Lancaster, T., and Blundell, S. j., Quantum Field Theory, Oxford, UK, 2014.<br />
<br />
[3] Robinson, M., Symmetry and the Standard Model, Springer, London, 2011.<br />
<br />
[4] Schwichtenberg, J., Springer, London, 2015.<br />
<br />H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-5721052005989543332017-02-28T13:41:00.004-08:002017-04-14T18:15:39.555-07:00Thinking on the Web: Berners-Lee, Gödel and TuringWhen the philosopher René Descartes proclaimed his famous observation "Cogito, ergo sum," he demonstrated the power of thought at the most basic level by deriving an important fact (i.e., the reality of his own existence) from the act of thinking and self awareness.<br />
<br />
The boldest attempt to apply logic to mathematics was pioneered by philosopher-logician Bertrand Russell, in<i> Principia Mathematia</i>. The idea was that mathematical theories were logical tautologies, and his program was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, such as Russell's Paradox, and the defeat of Hilbert's Program by Gödel's incompleteness theorems.<br />
<br />
Russell's paradox represents either of two interrelated logical contradictions. The first is a contradiction arising in the logic of sets or classes. Some sets can be members of themselves, while others can not. The set of all sets is itself a set, and so it seems to be a member of itself. The null or empty set, however, must not be a member of itself. However, suppose that we can form a set of all sets that, like the null set, are not included in themselves. The paradox arises from asking the question of whether this set is a member of itself. It is, if and only if, it is not!<br />
<br />
While it was shown that non-Euclidean geometries were consistent relative to Euclidean geometry, proving the consistency of Euclidean geometry has not been achieved by mathematics. It was only found to be consistent as long as mathematics was consistent.<br />
<br />
<b>What is undefined?</b><br />
<br />
When students first envision the extensive mathematical landscape, they assume that it is a pristine smooth surface holding all the answers to every mathematical question - just waiting to be unearthed - point to the right spot and there's the answer. They soon discover, however, that the landscape is littered with irregularities, obstructions, and, in some spots, gaping holes.<br />
<br />
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
Math and
physics have a lot in common. For example, Infinity and c cannot be treated as ordinary
numbers.<o:p></o:p></div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
Infinity seems to suffer from the contradiction: infinity + 1 = infinity <o:p></o:p></div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
which reduces
to 1 = 0.<o:p></o:p></div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
The speed of
light (c) seems to suffer a similar contradiction: c + 1 = c <o:p></o:p></div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
which reduces
to 1 = 0.<o:p></o:p></div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
But, Special
Relativity put time on the same level as spatial dimensions with the Lorentz
transform superseding Newton’s Galilean transform and Einstein redefined how c was
understood.</div>
<div class="MsoNormal" style="line-height: 107%; margin-bottom: 8.0pt;">
Similarly,
Cantor created aleph to redefined infinity through Cardinal and
the Continuum Hypotheses so that infinity was no longer treated as an ordinary Real number. </div>
Of particular importance is the fact that division by zero is literally undefined. Nevertheless, it repeatedly pops-up in the most important problems in science from nonlinear equations to Quantum Field Theory where it is finessed by renormalization (removing/cancelling singularities). When trying to reconcile the mathematical models of physical theories it is necessary to consider the failings of math itself.<br />
<b><br /></b>
<b>What is decidable?</b><br />
<br />
In the 1930s, the logician, Kurt Gödel, established that, in certain important mathematical domains, there are problems that cannot be solved, or propositions that cannot be proved or disproved, and are therefore undecidable. In particular, the work of Kurt Gödel identified the concept of undecidability where the truth, or falsity, of some statements may not be determined. This is relevant to the field of artificial intelligence because of the limits and boundaries that can be inferred from Gödel's insights.<br />
<b><br /></b>
<b>What is machine intelligence?</b><br />
<br />
In 1947, mathematician Alan Turing first started to seriously explore the concept of intelligent machines. He determined that a computing machine can be called intelligent if it could deceive a human into believing that it was human. His test — called the Turing Test — consists of a person asking a series of questions to both a human subject and a machine. The questioning is done via a keyboard so that the questioner has no direct interaction between subjects; man, or machine. A machine with true intelligence will pass the Turing Test by providing responses that are sufficiently human-like that the questioner cannot determine which responder is human and which is not.<br />
<br />
Recursion is the process a procedure goes through when one of the steps of the procedure involves rerunning a complete set of identical steps. In mathematics and computer science, recursion is a particular way of specifying a class of objects with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of the class.<br />
<br />
<b>Is there art in mathematics?</b><br />
<br />
How much of math is reality-based and how much is art (avoiding irregularities and undefined features in math in order to reach an answer without running into a hole)?<br />
<br />
Consider:<br />
<br />
<div class="MsoNormal">
e<sup>x</sup> = -1 was considered unsolvable
until Euler discovered one solution:<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
e<sup>i</sup><sup>π</sup>
= cos π
+ i(sin π) = -1 + 1*0 = -1<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
or rewritten as<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Euler’s Equation e<sup>i</sup><sup>π</sup>
+1 = 0<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Where this one equation equates five
fundamental elements of math:<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
0 = additive identity - a finite real number<o:p></o:p></div>
<div class="MsoNormal">
1 = multiplicative identity - an integer real number<o:p></o:p></div>
<div class="MsoNormal">
π = circular constant pi - a number with unending decimal places<o:p></o:p></div>
<div class="MsoNormal">
e = the base of natural logarithms - an infinite series<o:p></o:p></div>
<div class="MsoNormal">
i = imaginary unit<o:p></o:p></div>
<br />
<div class="MsoNormal">
<b>Question:</b>
What does it 'mean' when we seek an ‘exact’ value to an infinite series? If it's true in the limit (approaching infinite) then Euler's Equation is an approximation - not an exact solution. </div>
_______________________________________________<br />
<br />
The following excerpts from Alesso [<a href="https://www.amazon.com/Thinking-Web-Berners-Lee-G%C3%B6del-Turing/dp/0471768669/ref=sr_1_10?s=books&amp;ie=UTF8&amp;qid=1488313073&amp;sr=1-10&amp;keywords=alesso" target="_blank">1</a>] illustrate some of these fundamental concepts:<br />
<br />
<b>Interlude #1: Thinking about Thinking </b><br />
<br />
John picked up the two double Latte Grandes and walked over to the corner table near the fireplace where Mary was setting up the chess game. She took a pawn of each color and concealed them in her hands before offering two fists to John.<br />
<br />
Putting the cups down, he tapped Mary’s left hand and was pleased to see the white piece as he took his chair. <br />
<br />
John said “Playing chess always reminds me of the game between IBM’s Deep Blue Supercomputer and the reigning World Chess Champion at the time, Garry Kasparov.” He glanced at the board, “d4, I think” as he moved his pawn.<br />
<br />
Mary said, “Me too.” Mary smiled to herself as she moved her own queen’s pawn forward to d5. She knew that John had strong feelings about the limits of true Artificial Intelligence and she hoped to gain an advantage by baiting him. “That was the first time a computer won a complete match against the world’s best human player. It took almost 50 years of research in the field, but a computer finally was thinking like a human.”<br />
<br />
John bristled slightly, but then realized that Mary was just poking a little fun. Taking his next move, c4, he said “You can guess my position on that subject. The basic approach of Deep Blue was to decide on a chess move by assessing all possible moves and responses. It could identify up to a depth of about 14 moves and value-rank the resulting game positions using an algorithm developed in advance by a team of grand masters. Deep Blue did not think in any real sense. It was merely computational brute force.”<br />
<br />
Mary reached over and took John’s pawn, accepting the gambit. “You must admit,” she replied, “although Kasparov’s ‘thought’ processes were without a doubt something very different than Deep Blue’s, their performances were very similar. After all, it was a team of grand masters that designed Deep Blue’s decision-making ability to think like them.”<br />
<br />
John played his usual Nc3, continuing the main line of the Queen’s Pawn Gambit. “You’ve made my point,” he exclaimed, “Deep Blue did not make its own decisions before it moved. All it did was accurately execute, the very sophisticated judgments that had been pre-programmed by the human experts.”<br />
<br />
“Let’s look at it from another angle.” Mary said as she moved Nf6. “Much like a computer, Kasparov’s brain used its billions of neurons to carry out hundreds of tiny operations per second, none of which, in isolation, demonstrates intelligence. In totality, though, we call his play ‘brilliant.’ Kasparov was processing information very much like a computer does. Over the years, he had memorized and pre-analyzed thousands of positions and strategies."<br />
<br />
“I disagree,” said John quickly moving e3. “Deep Blue’s behavior was merely logic algebra – expertly and quickly calculated, I admit. However, logic established the rules between positional relationships and sets of value-data. A fundamental set of instructions allowed operations including sequencing, branching and recursion within an accepted hierarchy.”<br />
<br />
Mary grimaced and held up her hands, "No lectures please." Moving to e6 she added, "A perfectly reasonable alternative explanation to logic methods is to use heuristics methods, which observe and mimic the human brain. In particular, pattern recognition seems intimately related to a sequence of unique images connected by special relationships. Heuristic methods seem as effective in producing AI as logic methods. The success of Deep Blue in chess programming is important because it employed both logic and heuristic AI methods."<br />
<br />
"Now who’s lecturing," responded John, taking Mary’s pawn with his bishop. “In my opinion, human Grandmasters, do not examine 200,000,000 move sequences per second.”<br />
<br />
Without hesitation Mary moved c5 and said, "How do we know? Just because human grandmasters are not aware of searching such a number of positions doesn’t prove it. Individuals are generally unaware of what actually does go on in their minds. Patterns in the position suggest what lines of play to look at, and the pattern recognition processes in the human mind seem to be invisible to the mind."<br />
<br />
John said, "You mean like your playing the same Queen’s Gambit Accepted line over and over again?" as he castled.<br />
<br />
Ignoring him, Mary moved a6 and said, "Suppose most of the chess player’s skill actually comes from an ability to compare the current position against images of thousands of positions already studied. We would call selecting the best position (or image) insightful. Still, if the unconscious human version yields intelligent results, and the explicit algorithmic Deep Blue version yields essentially the same results, then why can't I call Deep Blue intelligent too?"<br />
<br />
John said, "I’m sorry, but for me you’ve overstated your case by calling Deep Blue intelligent,” moving Qe2. He continued, “ Would you like to reconsider your position?”<br />
<br />
Mary moved Nc3 and said, "Of course not, I still have plenty of options to think about, alone this line.”<br />
_______________________________________________<br />
<br />
<b>Interlude #2: The Truth and Beauty of Math</b><br />
<br />
As John passed with a sour look on his face, Mary looked up from her text book and asked, “Didn’t you enjoy the soccer game?”<br />
<br />
“How can you even ask that when we lost?” asked John gloomily.<br />
<br />
“I think the team performed beautifully, despite the score” said Mary.<br />
<br />
This instantly frustrated John and he said, "Do you know Mary that sometimes I find it disarming the way you express objects in terms of beauty. I find that simply accepting something on the basis of its beauty can lead to false conclusions?"<br />
<br />
Mary reflected upon this before offering a gambit of her own, "Well John, do you know that sometimes I find that relying on objective truth alone can lead to unattractive conclusions."<br />
<br />
John became flustered and reflected his dismay by demanding, "Give me an example."<br />
<br />
Without hesitation, Mary said, "Perhaps you will recall that in the late 1920's, mathematicians were quite certain that every well-posed mathematical question had to have a definite answer ─ either true or false. For example, suppose they claimed that every even number was the sum of two prime numbers,” referring to Goldbach's Conjecture which she had just been studying in her text book. Mary continued, “Mathematicians would seek the truth or falsity of the claim by examining a chain of logical reasoning that would lead in a finite number of steps to prove if the claim were either true or false."<br />
<br />
"So mathematicians thought at the time," said John. "Even today most people still do."<br />
<br />
"Indeed," said Mary. "But in 1931, logician Kurt Gödel proved that the mathematicians were wrong. He showed that every sufficiently expressive logical system must contain at least one statement that can be neither proved nor disproved following the logical rules of that system. Gödel proved that not every mathematical question has to have a yes or no answer. Even a simple question about numbers may be undecidable. In fact, Gödel proved that there exist questions that while being undecidable by the rules of logical system can be seen to be actually true if we jump outside that system. But they cannot be proven to be true.”<br />
<br />
“Thank you for that clear explanation,” said John. “But isn’t such a fact simply a translation into mathematical terms of the famous Liar’s Paradox: ‘This statement is false.’”<br />
<br />
“Well, I think it's a little more complicated than that,” said Mary. “But Gödel did identify the problem of self-reference that occurs in the Liar’s Paradox. Nevertheless, Gödel’s theorem contradicted the thinking of most of the great mathematicians of his time. The result is that one can not be as certain as the mathematician had desired. See what I mean, Gödel may have found an important truth, but it was – well to be frank – rather disappointingly unattractive," concluded Mary.<br />
<br />
"On the contrary,” countered John, “from my perspective it was the beauty of the well-posed mathematical question offered by the mathematicians that was proven to be false.<br />
<br />
Mary replied, “I’ll have to think about that.”<br />
_______________________________________________<br />
<b><br /></b>
<b>Interlude #3: Computing Machines</b><br />
<br />
Having had the final word in their last discussion, John was feeling a little smug as he listened to his ipod. Mary sat down next to him on the library steps. Their last class had been on computer design and they were both thinking about just how far the new technology could evolve.<br />
<br />
John said, “As you suggested earlier, Gödel was concerned that a logic system had to be consistent and then he determined that no logic system can prove itself to be consistent.”<br />
<br />
“True,” replied Mary. “But it was Turing who built on Gödel’s findings. Shortly before WWII, Turing found a way to translate Gödel’s logic results about numbers and mathematics into analogous results about calculations and computing machines.”<br />
<br />
John interrupted, “Yes, Turing was convinced that mathematical problem solving could be reduced to simple steps that could be used to program computer actions."<br />
<br />
Mary said, “True. Turing considered the logical steps one goes through in constructing a proof as being the same steps that a human mind follows in a computation.” Mary gave John a sideways glance before continuing, “Turing was convinced that the ability to solve this type of mathematical problem is a significant indication of the ability of machines to duplicate human thought.”<br />
<br />
John dissented, “Wait a minute. We’ve been here before. Just because machines follow the same logical steps a human uses to solve a calculation doesn’t mean that they actually think. Since calculating machines are not biological, it seems unreasonable to me to suggest that machines are capable of actual creative thought. They may be mimicking the logical steps, but are they actually thinking? I think not.”<br />
<br />
“Therefore, you are not,” Mary said with a grin.<br />
<br />
John relied, “Hey.”<br />
<br />
Mary said, “OK. Seriously, if it were even remotely possible that machines could independently mimic thought that would be significant.”<br />
<br />
She continued, “Consider Turing’s Machine. Turing held that a mechanical computer is basically a large number of address locations acting as a memory, together with an executive unit that carries out individual operations of a calculation. These operations represent a program. Let’s imagine that I want to use the machine to add two numbers 1 and 2 together. The computing machine would begin with placing a ‘1’ in the first location and a ‘2’ in the second location and then the computer consults a program for how to do addition. The instructions would say gather the numbers from the two locations and perform a summing operation to yield the sum of the two numbers and place the resultant number ‘3’ in the third location. This process could be considered to mimic the operations a human would perform.”<br />
<br />
John replied solemnly, “Simple rote actions.”<br />
<br />
Mary added, “Turing’s computer consists of two basic elements: an infinitely long tape ruled off into squares, each capable of being inscribed with the symbol ‘0’ or ‘1,’ and a scanning head that can move forward or back one square at a time reading the symbol on that square, either leaving it alone or writing a new symbol on that square. At any step of the operation, the scanning head can be in one of an infinite number of states. The machine has a pointer that is set at one of the letters ‘A’ ‘B’ ‘C,’ and this letter represents the state of the machine. Part of the program tells the machine how to change the pointer setting, depending on what state the machine is currently in and what symbol is on the square of the tape that the head is currently reading.”<br />
<br />
John nodded slowly as he visualized the machine.<br />
<br />
Mary continued, “The action of a Turing machine is determined completely by; (1) the current state of the machine, (2) the symbol in the cell currently being scanned by the head, and (3) a table of transition rules, which serves as the “program” for the machine. If the machine reaches a situation in which there is not exactly one transition rule specified, then the machine halts. While it may take a long tape and many steps to carry out sophisticated calculations, anything at all that can be thought of as following from a set of rules, can be calculated in the step by step fashion.”<br />
<br />
John said, “It is easy to appreciate that Turing Machine is the foundation of the modern computer.”<br />
<br />
Mary said, “And that leads us to your earlier question about whether a computing machine can have human-like intelligence.”<br />
<br />
John said, “In general, I would consider ‘thinking’ to be a complex process that uses concepts and relationships to infer new knowledge. Thinking would involve acts such as memory recall, arithmetic calculations, puzzle solving, and so on. By extension the performance of these acts would indicate ‘intelligence’.” For example, a child who can perform difficult arithmetic calculations quickly would display intelligence. Likewise an individual who has rapid memory recall and who has accumulated sufficient amounts of information to consistently win games such as Scrabble, or Trivial Pursuit, might also be considered to be intelligent.”<br />
<br />
“Well then,” responded Mary, “Why wouldn’t a computer that could perform the same calculations as that child, but faster and with greater accuracy be considered intelligent. Or consider a computer with substantial memory storage that is able to answer all those Trivial Pursuit questions. Why can’t that computer, be considered intelligent.”<br />
<br />
John said, “Well human thinking involves complicated interactions within the biological components of the brain. In addition, the processes of communication and learning are also important elements of human intelligence.”<br />
<br />
Mary replied, “By mentioning intelligent communication you have led us to Turing’s Test for machine intelligence.”<br />
<br />
John said, “OK, but please, let’s talk about that tomorrow.”<br />
<br />
___________________________________________________________<br />
<br />
REFERENCE:<br />
<br />
[1] <a href="https://www.amazon.com/Thinking-Web-Berners-Lee-G%C3%B6del-Turing/dp/0471768669/ref=sr_1_10?s=books&ie=UTF8&qid=1488313073&sr=1-10&keywords=alesso" target="_blank">Thinking on the Web: Berners-Lee, Gödel and Turing</a>, H. Peter Alesso and Craig F. Smith,<br />
Wiley-Interscience , November 24, 2008, ISBN-13: 978-0471768661.<br />
<br />
<div>
<br /></div>
H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0tag:blogger.com,1999:blog-7815548972557022050.post-7036277698866781092013-10-16T14:51:00.003-07:002016-04-27T11:34:33.630-07:00<div style="text-align: center;">
<b>Cosmology Spontaneous Symmetry Breaking through the Higgs Mechanism</b></div>
<br />
The quantum uncertainty of spontaneous symmetry breaking during the Big Bang was completely indifferent to the birth of the universe and life on Earth. However, this event produced inquiring minds with many questions, such as "what is mass?"<br />
<br />
Understanding the origin of mass is one of the greatest challenges of science. Going back to a tiny fraction of a second after the Big Bang, the four forces of nature began to split apart and an inflationary expansion took place that increased the size of the universe exponentially. Under these extremely hot conditions the fundamental particles began to form and travel at the speed of light. A tiny fraction of a second later, these particles acquired mass through the Higgs Mechanism, thereby creating the Standard Model of elementary particles ( leptons, quarks, and bosons). The moment when the Higgs field went from zero to an average nonzero value is known as the "electroweak phase transition." The transition was similar to liquid water turning into ice.<br />
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<div style="text-align: center;">
<b>The Big Bang</b></div>
<br />
Today, we know that the 2.725 degree K cosmic microwave background radiation firmly established the existence Big Bang.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAqT7_Ecoc83Lat_5B2yqngtNNBNaEfDQzUaTrDPbbFqzbJZlgI42mNUdHh2IspiRgQfdmXQNbwogR6SBpU4jOtzbyIy3RxwYnNRo-WmL7XTUMrKyL5HzOftdZ73O-YUBwAzavz-Pxd-Y/s1600/pic9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAqT7_Ecoc83Lat_5B2yqngtNNBNaEfDQzUaTrDPbbFqzbJZlgI42mNUdHh2IspiRgQfdmXQNbwogR6SBpU4jOtzbyIy3RxwYnNRo-WmL7XTUMrKyL5HzOftdZ73O-YUBwAzavz-Pxd-Y/s1600/pic9.jpg" width="320" /></a></div>
<br />
<br />
An instant after the Big Bang, the Universe was more symmetric than it is today and during this period, it underwent several symmetry breaking phase changes in very rapid succession.<br />
<br />
Ultimately, our ability to find cosmological answers rests on how far back in time we can observe (<a href="http://www.amazon.com/First-Three-Minutes-Modern-Universe/dp/0465024378/ref=sr_1_1?s=books&ie=UTF8&qid=1381273285&sr=1-1&keywords=The+First+Three+Minutes%2C" target="_blank">Weinberg</a><u> </u>[1]). By measuring material compositions and using General Relativity to determine expansion rates, scientists were able to determine how long it took from the Big Bang to the recombination era. When atoms began to form during the recombination era, 380,000 years after the Big Bang, it produced the first time that light was able to travel freely. Now when we look at a star one light-year away, we are actually seeing it as it existed one year ago.<br />
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<div style="text-align: center;">
<b>The Universe</b></div>
<b><br /></b>
<b>Time Temp Energy Phenomena</b><br />
sec. K GEV<br />
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<br />
10 E-43 10 E32 10 E19 Single force divides into gravity and GUT.<br />
<br />
10 E-37 10 E29 10 E16 Inflation starts.<br />
<br />
10 E-35 10 E29 10 E16 Strong force splits from electro-weak force.<br />
<br />
10 E-33 10 E27 10 E14 Heavy bosons freeze. Matter dominates anti-matter.<br />
<br />
10 E-9 10 E15 10 E2 Weak interaction splits from electromagnetism.<br />
<br />
10 E-2 10 E13 1 Color quarks and gluons produce protons, neutrons.<br />
<br />
100 10 E9 1 E-4 Nucleosynthesis creates Helium and Hydrogen.<br />
<br />
4 E6 10 E3 1 E-10 Photons decouple from matter.<br />
years<br />
<br />
10 E10 3 1 E-12 Today. Black body radiation.<br />
years<br />
________________________________________________________________<br />
<br />
Only 4 to 5% of the universe is composed of ordinary matter. Another 23% is dark matter (exotic non-baryonic matter that weakly interacts with ordinary matter). The 1998 recognition that the universe's expansion is accelerating led to the discovery of dark energy. Dark energy is a property of the vacuum itself (the cosmological constant) characterized by negative pressure (repelling force) causing the expansion of the universe to accelerate.<br />
<br />
To solve the cosmological flatness, horizon, and monopole problems, <a href="http://www.amazon.com/Inflationary-Universe-Alan-Guth/dp/0201328402/ref=sr_1_1?s=books&ie=UTF8&qid=1381273251&sr=1-1&keywords=The+Inflationary+Universe" target="_blank">Guth [</a>2] proposed an early era of exponential expansion called the Inflation Epoch where a tiny bit of space was stretched into a far larger region.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXkAvO4huq_wztRc3YIbValOzuuEhQmo0TEweQsPZ_NWMQbDooJLxbcOuAJyokWQjdODgp909WfkfDCtiuAqy9V-fS838VFuKOIenrZUsQtyPTNmkY_mjXS3TbhzfQbubi29EhtC-JE2U/s1600/pic2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXkAvO4huq_wztRc3YIbValOzuuEhQmo0TEweQsPZ_NWMQbDooJLxbcOuAJyokWQjdODgp909WfkfDCtiuAqy9V-fS838VFuKOIenrZUsQtyPTNmkY_mjXS3TbhzfQbubi29EhtC-JE2U/s1600/pic2.jpg" width="320" /></a></div>
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Inflation required a supercooling phase transition that would have started from the state of symmetry of the Grand Unified Theory (GUT)/Standard Model (SM) - before it broke into individual forces.<br />
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In 1976, particle physics developed the Standard Model of Quantum Mechanics representing the three forces (electromagnetic, weak and strong) as SU(3)C ⊗ SU(2)L ⊗ U(1)Y with exceptional accuracy. Each of the forces in the Standard Model (SM) is a product of gauge theories with complex charges that mutually interact in a highly symmetric way.<br />
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The development of GUT was intended to unify electromagnetic, weak and strong interactions of the Standard Model in terms of a single fully unified interaction. While this still left out gravity, it was an excellent approximation to nature.</div>
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The SM shows the three forces (weak, strong, and electromagnetic) coupling constants splitting at the point of symmetry breaking. In July 2012, the Higgs Boson (125 Gev) was confirmed.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjdS33-VC2_MwHjwNXXV5D8ZGGTkVt3WZqxby6PG7WWNrMsUrys626pLpIyFU0mH0qNq374L_IU15vfRBKdTTxRQ3yGbNpwHFJZtItJR2YDRaQ_86yu4BxhLkwt7T4cEyU67wCgb9IJHc/s1600/pic3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjdS33-VC2_MwHjwNXXV5D8ZGGTkVt3WZqxby6PG7WWNrMsUrys626pLpIyFU0mH0qNq374L_IU15vfRBKdTTxRQ3yGbNpwHFJZtItJR2YDRaQ_86yu4BxhLkwt7T4cEyU67wCgb9IJHc/s1600/pic3.jpg" width="320" /></a></div>
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GUT theories gave specific reasons for the charge symmetry of electrons and protons. Also super-symmetry gave an intersecting point where all three force’s coupling constants merged at 10 E16 Gev. </div>
<br />
An SU(5) GUT (no longer in favor) could for example break the pattern of SU(5) → SU(3)C ⊗ SU(2)L ⊗ U(1)Y to obtain the standard model (SM). The Higgs mechanism for spontaneously symmetry breaking must include massless Goldstone bosons.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2gR7XnBtdJdAxZGb0xDZlBJXykWi-EhOSgsPmgE6o81qDVzCmGUTvS_LoLXyG8G5BGzLkx8FcxCV-g25S-rYBEXX-F5PXlaCTeMrNsPu-8KiHVX6d2aJ_2Oja75OMwHtZV-r3NJrrlBw/s1600/pic10.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2gR7XnBtdJdAxZGb0xDZlBJXykWi-EhOSgsPmgE6o81qDVzCmGUTvS_LoLXyG8G5BGzLkx8FcxCV-g25S-rYBEXX-F5PXlaCTeMrNsPu-8KiHVX6d2aJ_2Oja75OMwHtZV-r3NJrrlBw/s1600/pic10.jpg" width="320" /></a></div>
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<b><br /></b>Noteworthy is the fact that not only does inflation smooth out the large scale observables of the universe, it also explains its homogeneity.<br />
<b><br /></b>
<br />
<div style="text-align: center;">
<b>Spontaneous Symmetry Breaking and the Higgs Mechanism</b>
</div>
<br />
The Noether’s theorem correlates conservation laws to symmetries that leave the Lagrangian (L) invariant.
<br />
<br />
<div style="text-align: center;">
L = T – V
</div>
<br />
The classical path taken by a particle is one that minimizes the action S which is equal to the integral of the Lagrangian over time.<br />
<div style="text-align: center;">
<br /></div>
In quantum theory symmetries can take the form of invariance under a unitary transformation. Local symmetries are very important in relativistic physics because they represent quantities like charge and lepton number that are conserved locally. Local symmetries preserve causality if the Lagrangian is invariant under<br />
<br />
<div style="text-align: center;">
L -> -L<br />
<br />
<div style="text-align: left;">
The Lagrangian for the Standard Model is:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD8D_SO5rXTN1mR42We7IE3WssvYKNKrKlPxpZUl7lWunfK6EJERuYeQ5vQ5jSndU9GPrJCjs4UoBCCWLqxRyBZnyiEiednB4QtkwCKqbIwdWQa9eEs9M-IDRW643HVPB5dZQdQQtXsbX9/s1600/pic35.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD8D_SO5rXTN1mR42We7IE3WssvYKNKrKlPxpZUl7lWunfK6EJERuYeQ5vQ5jSndU9GPrJCjs4UoBCCWLqxRyBZnyiEiednB4QtkwCKqbIwdWQa9eEs9M-IDRW643HVPB5dZQdQQtXsbX9/s1600/pic35.png" width="320" /></a></div>
</div>
</div>
This type of symmetry introduces the concept of spontaneous symmetry breaking. The vacuum state that is the apparent ground state may not be the true ground state. The nonlinear jumping from one state to another state can result from symmetry breaking. For example, if we place a marble on top, right in the center, of an over turned bowl, the symmetry is broken when the marble falls in one direction. The marble may start out semi-stable, but the slightest perturbation, or disturbance, will cause it to roll off. This is because the system is basically unstable.<br />
<br />
A massless gauge field A undergoing symmetry breaking results in a massive vector field. In electroweak theory, SU(2) x U(1), this gives rise to massive vector bosons W+-,Z. The mechanism for spontaneous symmetry breaking with a gauge field and local U(1) invariance is called the Higgs mechanism, after Peter Higgs, 1964. A spontaneous broken symmetry is one that is present in theory but hidden in the equilibrium state. The symmetry is unbroken when all Higgs fields have a value of zero, but the symmetry is broken when all the Higgs fields acquire a nonzero value.<br />
<br />
Big Bang cosmology depends on special early conditions leading to GUT predictions that there were two early universe phase transitions. A first order phase transition in a special class of grand unification theory is exemplary of a symmetry breaking potential function V(x).<br />
<br />
The singularity that broke symmetry and changed one force into four (gravity and SM forces) occurred in stages starting at 10 E16 Gev (10 E-39 seconds) after the Big Bang. The original single force coupling constant changed into four coupling constants that began to diverge in value.<br />
<br />
The first phase transition was at a high temperature phase transition that broke symmetry and split gravity away from the SM forces - generating a large amount of supercooling that delayed the second phase transition and suppressed monopole production. The supercooling caused the second phase transition to occur at a temperature well below the normal phase transition temperature. (For example, ordinary water can be supercooled to 20 degrees below freezing before it turns to ice.)<br />
<br />
With the second phase transition postponed until after supercooling, a false vacuum was created. The false vacuum would be the result of twenty four Higgs fields intersecting (representing 12 Leptons, Fermions and their anti-particles, and 12 force particles (photon, +-W, Z, and 8 gluons)). Exactly when the various Higgs field come into existence during the symmetry breaking process is not known. This phase transition was a 1st Order Phase Transition (like boiling water with supercooling) which occurred from 10 E-37 to 10 E-35 seconds. The expansion doubling time was 10 E-37 seconds. The false vacuum produced a neg. pressure repulsing gravitons to make the cosmological constant increase the size of the universe by a factor of 10 E30. The supercooling phase transition that followed caused Inflationary expansion of the universe. Inflationary expansion proceeded at faster than the speed of light. During this time the universe grew from the size of a proton to the size of a baseball in a flash. (Today, the observable physical universe is 93 billion light-years in diameter.)<br />
<br />
The matter-antimatter ratio became slightly imbalanced and matter became slightly dominate due to cooling shutting off prior to baryon neucleogenesis reaching equilibrium (10 E78 baryons now in the universe).<br />
<br />
To help visualize the elements of the Standard Model, it is useful to think of an electron as a matter wave - (a Fermion with spin 1/2) and a photon as an energy wave (a Boson with spin 1). A photon, even though it is massless, feels gravity by E/c^2 and it is the only particle that is an exception to relative motion - always travels at the speed of light, c.<br />
<br />
General Relativity is an extremely accurate theory for gravity but is classical and does not include quantum effects. As a result, the inability to reconcile it with the Standard Model is due to the appearance of infinity expressions that cannot be renormalized.<br />
<br />
Three constants play a profound role in physics: (1) the speed of light, c, converts between space and time (ie. the Lorentz transform); (2) Planck's constant, h, converts between frequency and energy (E=hv); and (3) Newton's gravitation constant, G, converts between space-time curvature and energy-density.<br />
<br />
So then what can we say about mass?<br />
<br />
Mass is the differential energy left over when disturbed fields attempt to cancel out the disturbance to get to the minimum energy state. For example, The disturbance of quark's color charge can be nullified by an antiquark of opposite charge nearby (or by complementary colors in a proton) however due to quantum effects the cancellation is incomplete and the residue constitutes the resulting mass.<br />
<br />
<div style="text-align: center;">
<b>Elementary Catastrophe Theory </b><b>Modeling of the Big Bang Inflation</b></div>
<br />
<div>
A ‘singularity’ is a point where mathematical models are no longer valid ‒ for example: a point divided by zero is undefined. The theory of singularities examines mathematical manifolds in an abstract space to gain a topological representation of the region near a singularity.<br />
<br />
In the 1960’s Rene’ Thom proposed a nonlinear mathematics approach to describe singularities called Catastrophe Theory. Thom classified the bifurcations based upon their potential function and its derivatives. The morphology of solutions is determined by values of the potential’s parameters. In the special case of gradient vector fields, the mathematics results are called Elementary Catastrophe Theory (ECT) Gilmore [3].
<br />
<br />
Gradient vector fields are interesting because nearly all trajectories on the behavior surface tend toward a point attractor and the attractor minimizes the potential function V of the system. The parameters determine the locations of the relative minima. A smooth change in the parameters can give rise to a discontinuous jump on the behavior surface.
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Thom found that under stable conditions there are exactly seven elementary catastrophes if the potential function has no more than two parameters. The most illustrative is the Cusp Catastrophe with a potential of:
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V(x, a, b) = (x^4)/4 + a(x^2)/2 + bx.
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The Cusp has two control variables a and b where x satisfies
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dV/dx = 0,
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shown in figure:
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3aF8aC1Bj-a34Ex9W5SVoYkIEdquRcaZzlFYuUib16eQ49HCkd38ST6a5UfVOMFQmjhfRs2mQ1A_7k5w4jUV2biO4pIeT3_NZYdCYkCDDzPAbWkvdZClcZ5dwkbIh56Usvb2POAHFGNY/s1600/ect1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3aF8aC1Bj-a34Ex9W5SVoYkIEdquRcaZzlFYuUib16eQ49HCkd38ST6a5UfVOMFQmjhfRs2mQ1A_7k5w4jUV2biO4pIeT3_NZYdCYkCDDzPAbWkvdZClcZ5dwkbIh56Usvb2POAHFGNY/s1600/ect1.gif" /></a></div>
Outside the cusp region there is only one extrema value for x. Inside the cusp, there are two different values of x giving local minima of V(x) for each. Nonlinear variable discontinuous jumps occur at the cusp boundary.<br />
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Cusp shape in parameter space (a,b) near the catastrophe point shows the locus of fold bifurcations separating the region with two stable solutions from the region with one.
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But the bifurcation curve loops back on itself, giving a second branch where the alternate solution loses stability and jumps back to the original solution space. You can observe hysteresis loops as the system follows one solution and jumps to the other.<br />
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Consider if one holds b constant and varies a to follow path 1 or 2. In the symmetrical case b = 0, a pitchfork bifurcation occurs as a is reduced. One stable solution suddenly splitting into two stable solutions and one unstable solution as the physical system passes to a < 0 through the cusp point (0,0) (spontaneous symmetry breaking). Away from the cusp point, there are no sudden changes.
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(For illustrations of Elementary Catastrophe Theory applications see Alesso [4-5]).<br />
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Classical phase transition theory is naturally modeled in Elementary Catastrophe Theory ECT. The general family of potential functions depending on state variables, or parameters. The first series of applications of ECT deals with thermodynamics phase transition. Ginzburg Landau second order phase transition relate the critical point of the fluid to the cusp catastrophe. We will show it similarly relates to Big Bang Inflationary Theory universe.<br />
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We let the state of the physical system be described by the value x that minimizes the potential locally. The physical system is then reduced to a study of equilibrium and stability properties of the potential function V(x,c). The first derivative of the function is equal to zero at equilibrium and the second are shown greater than zero indicating a local stability as well as the critical values of the stable equilibrium branches.<br />
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In general the potential function the will have only isolated critical points. A phase transition occurs when the point reaches this state of the physical system jumps from one critical branch to another.<br />
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Phase transitions can occur when the control parameters are varied. The control parameters are assumed to depend upon a single time parameter. A phase transition will occur when the curve crosses and appropriate point. The bifurcation set on which the local minimum are created or destroyed for this curve the transition is of order and if the limits of the derivative goes to zero. Phase transitions in nature usually are zero, first, or second order.<br />
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The potential function V(x) for the standard GUT with finite-temperature effective scalar potential <a href="http://www.amazon.com/The-Particle-End-Universe-Higgs/dp/0142180300/ref=pd_sim_sbs_b_1" target="_blank">Carroll</a> [6] and <a href="http://prl.aps.org/abstract/PRL/v48/i17/p1220_1" target="_blank">Albrecht</a> [7]:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWyMScWwKKjkZnrB-we5_LuXh-lOxv43jjAiCmpfP9HmbQedzCnJrduF5wP72hqmOiMMbYvyA6JzFWeWNLCP-AtsONY9eqihq5DkuLiPjy1275Uzrt6iPPt6MMVM1pI-wmSOXVhsvhEa5b/s1600/pic34.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWyMScWwKKjkZnrB-we5_LuXh-lOxv43jjAiCmpfP9HmbQedzCnJrduF5wP72hqmOiMMbYvyA6JzFWeWNLCP-AtsONY9eqihq5DkuLiPjy1275Uzrt6iPPt6MMVM1pI-wmSOXVhsvhEa5b/s1600/pic34.png" width="320" /></a></div>
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where this potential can be mapped into a Cusp catastrophe by first taking the Taylor series expansion of the natural logarithm terms, then gathering Higgs fields terms according to their exponent and grouping like terms. Then following renormalization rules, create counter terms and eliminate divergences. The result is a Cusp x^4 theory.</div>
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V(x, a, b) = (x^4)/4 + a(x^2)/2 + bx.</div>
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where x represents the Higgs field, and</div>
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a and b are combinations of A,B, g. </div>
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The value of the Cusp catastrophe transformation is the clarity of an illustrative image of the non-linear behavior of the phase change and the potential of refining the vacuum value based upon the new parameters. The characteristics of a nonlinear <b>JUMP</b> from one solution space to another within the bifurcation cusp show alternative to quantum tunneling.<br />
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The S-shaped nonlinear jump of the ECT Cusp is characteristic of the S-shaped jump we find in Inflation Theory .<br />
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The concepts expressed in this blog are further illustrated in the app: <a href="https://play.google.com/store/apps/details?id=com.videosoftwarelab.cosmology" target="_blank">Cosmology App.</a><br />
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<b>REFERENCES:
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[1] Weinberg, S., "<a href="http://www.amazon.com/First-Three-Minutes-Modern-Universe/dp/0465024378/ref=sr_1_1?s=books&ie=UTF8&qid=1381273285&sr=1-1&keywords=The+First+Three+Minutes%2C">The First Three Minutes,</a>" Basic Books, NY, NY, 1977.
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[2] Guth, A. H., "<a href="http://www.amazon.com/Inflationary-Universe-Alan-Guth/dp/0201328402/ref=sr_1_1?s=books&ie=UTF8&qid=1381273251&sr=1-1&keywords=The+Inflationary+Universe">The Inflationary Universe</a>," Basic Books, NY, NY, 1997.
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[3] Gilmore, R., <a href="http://www.amazon.com/Catastrophe-Theory-Scientists-Engineers-Gilmore/dp/0486675394/ref=sr_1_1?ie=UTF8&qid=1381281905&sr=8-1&keywords=catastrophe+theory+for+scientists+and+engineers">Catastrophe Theory for Scientists and Engineers</a>, NY, NY, 1981.<br />
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[4] Alesso, H. P., “<a href="http://www.sciencedirect.com/science/article/pii/0020746282900415">On the Instabilities of an Externally Loaded Shell</a>” INTERNATIONAL JOURNAL OF NONLINEAR MECHANICS, Vol. 17, No. 2, pp-85-103,1982.
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[5] Alesso, H. P., and Smith, C. F., “<a href="http://link.springer.com/article/10.1007/BF02731688#page-1">On the Classifying the Deformation Shape of the Liquid Drop Model</a>” IL NUOVO CIMENTO, Vol. 66, pp 272-282, 1981.
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[6] Carroll, S., <a href="http://www.amazon.com/The-Particle-End-Universe-Higgs/dp/0142180300/ref=pd_sim_sbs_b_1">The Particle at the End of the Universe</a>, NY, NY 2012.<br />
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[7] Albrecht, A., Steinhardt, P., <a href="http://prl.aps.org/abstract/PRL/v48/i17/p1220_1">Cosmology for Grand Unified Theories with Radiatively</a><br />
<a href="http://prl.aps.org/abstract/PRL/v48/i17/p1220_1">Induced Symmetry Breaking</a>, Phys. Rev. Lett. 48, 1220 (1982).<br />
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H. Peter Alessohttp://www.blogger.com/profile/17021883119974500421noreply@blogger.com0