Van der Waerden’s conjecture mystified mathematicians for 85 years. Its solution shows how polynomial roots relate to one another.
Category: mathematics – Page 101
Circa 2018 face_with_colon_three
For millennia mathematicians have struggled to unify arithmetic and geometry. Now one young genius could have brought them in sight of the ultimate prize.
Circa 2016 face_with_colon_three
A subset C of infinite-dimensional binary cube is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.
A paradigm shift away from the 3D mathematical description developed by Schrödinger and others to describe how we see color could result in more vibrant computer displays, TVs, textiles, printed materials, and more.
New research corrects a significant error in the 3D mathematical space developed by the Nobel Prize-winning physicist Erwin Schrödinger and others to describe how your eye distinguishes one color from another. This incorrect model has been used by scientists and industry for more than 100 years. The study has the potential to boost scientific data visualizations, improve televisions, and recalibrate the textile and paint industries.
“The assumed shape of color space requires a paradigm shift,” said Roxana Bujack, a computer scientist with a background in mathematics who creates scientific visualizations at Los Alamos National Laboratory. Bujack is lead author of the paper on the mathematics of color perception by a Los Alamos team. It was published in the Proceedings of the National Academy of Sciences.
In recent decades, the scientific study of consciousness has significantly increased our understanding of this elusive phenomenon. Yet, despite critical development in our understanding of the functional side of consciousness, we still lack a fundamental theory regarding its phenomenal aspect. There is an “explanatory gap” between our scientific knowledge of functional consciousness and its “subjective,” phenomenal aspects, referred to as the “hard problem” of consciousness. The phenomenal aspect of consciousness is the first-person answer to “what it’s like” question, and it has thus far proved recalcitrant to direct scientific investigation. Naturalistic dualists argue that it is composed of a primitive, private, non-reductive element of reality that is independent from the functional and physical aspects of consciousness. Illusionists, on the other hand, argue that it is merely a cognitive illusion, and that all that exists are ultimately physical, non-phenomenal properties. We contend that both the dualist and illusionist positions are flawed because they tacitly assume consciousness to be an absolute property that doesn’t depend on the observer. We develop a conceptual and a mathematical argument for a relativistic theory of consciousness in which a system either has or doesn’t have phenomenal consciousness with respect to some observer. Phenomenal consciousness is neither private nor delusional, just relativistic. In the frame of reference of the cognitive system, it will be observable (first-person perspective) and in other frame of reference it will not (third-person perspective). These two cognitive frames of reference are both correct, just as in the case of an observer that claims to be at rest while another will claim that the observer has constant velocity. Given that consciousness is a relativistic phenomenon, neither observer position can be privileged, as they both describe the same underlying reality. Based on relativistic phenomena in physics we developed a mathematical formalization for consciousness which bridges the explanatory gap and dissolves the hard problem. Given that the first-person cognitive frame of reference also offers legitimate observations on consciousness, we conclude by arguing that philosophers can usefully contribute to the science of consciousness by collaborating with neuroscientists to explore the neural basis of phenomenal structures.
As one of the most complex structures we know of nature, the brain poses a great challenge to us in understanding how higher functions like perception, cognition, and the self arise from it. One of its most baffling abilities is its capacity for conscious experience (van Gulick, 2014). Thomas Nagel (1974) suggests a now widely accepted definition of consciousness: a being is conscious just if there is “something that it is like” to be that creature, i.e., some subjective way the world seems or appears from the creature’s point of view. For example, if bats are conscious, that means there is something it is like for a bat to experience its world through its echolocational senses. On the other hand, under deep sleep (with no dreams) humans are unconscious because there is nothing it is like for humans to experience their world in that state.
In the last several decades, consciousness has transformed from an elusive metaphysical problem into an empirical research topic. Nevertheless, it remains a puzzling and thorny issue for science. At the heart of the problem lies the question of the brute phenomena that we experience from a first-person perspective—e.g., what it is like to feel redness, happiness, or a thought. These qualitative states, or qualia, compose much of the phenomenal side of consciousness. These qualia are arranged into spatial and temporal patterns and formal structures in phenomenal experience, called eidetic or transcendental structures1. For example, while qualia pick out how a specific note sounds, eidetic structures refer to the temporal form of the whole melody. Hence, our inventory of the elusive properties of phenomenal consciousness includes both qualia and eidetic structures.
On June 27th 2022, Yann LeCun, one of the godfathers of artificial intelligence and Head of AI at Meta released his vision on how to build autonomous AI systems. Here is the link to the paper.
First of all, I really suggest you to read this paper. As mentioned in the prologue, the text is written with as little jargon as possible. It uses as little mathematical prior knowledge as possible to appeal to readers with various backgrounds. It’s essentially a vision of what might direct the research efforts at Meta and elsewhere in the industry.
When you start reading the paper, quite quickly, you realize that this vision is very ambitious and futuristic. After all, Yann is describing an autonomous and polyvalent AI system.
Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.
Agility Robotics recently raised $150 million USD in part from Amazon to further develop its humanoid robot called “Digit” for logistics automation. New wearable, bendable, stretchable neuromorphic AI chip monitors health in real time. New machine learning model from MIT does college level math at a human level.
AI News Timestamps:
0:00 Humanoid Robot Worker For Amazon Automation.
3:00 New Wearable AI Chip.
5:08 New Machine Learning Math Model.
👉 Crypto AI News: https://www.youtube.com/c/CryptoAINews/videos.
#ai #robot #news
Electrons inhabit a strange and topsy-turvy world. These infinitesimally small particles have never ceased to amaze and mystify despite the more than a century that scientists have studied them. Now, in an even more amazing twist, physicists have discovered that, under certain conditions, interacting electrons can create what are called ‘topological quantum states.’ This finding, which was recently published in the journal Nature, has implications for many technological fields of study, especially information technology.
Topological states of matter are particularly intriguing classes of quantum phenomena. Their study combines quantum physics with topology, which is the branch of theoretical mathematics that studies geometric properties that can be deformed but not intrinsically changed. Topological quantum states first came to the public’s attention in 2016 when three scientists—Princeton’s Duncan Haldane, who is Princeton’s Thomas D. Jones Professor of Mathematical Physics and Sherman Fairchild University Professor of Physics, together with David Thouless and Michael Kosterlitz—were awarded the Nobel Prize for their work in uncovering the role of topology in electronic materials.
“The last decade has seen quite a lot of excitement about new topological quantum states of electrons,” said Ali Yazdani, the Class of 1909 Professor of Physics at Princeton and the senior author of the study. “Most of what we have uncovered in the last decade has been focused on how electrons get these topological properties, without thinking about them interacting with one another.”
A hidden link between two seemingly unrelated particle collision outcomes shows a mysterious web of mathematical connections between disparate theories.