Archive for the ‘mathematics’ category: Page 3

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years.

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

This essay addresses Cartesian duality and how its implicit dialectic might be repaired using physics and information theory. Our agenda is to describe a key distinction in the physical sciences that may provide a foundation for the distinction between mind and matter, and between sentient and intentional systems. From this perspective, it becomes tenable to talk about the physics of sentience and ‘forces’ that underwrite our beliefs (in the sense of probability distributions represented by our internal states), which may ground our mental states and consciousness. We will refer to this view as Markovian monism, which entails two claims: fundamentally, there is only one type of thing and only one type of irreducible property (hence monism). All systems possessing a Markov blanket have properties that are relevant for understanding the mind and consciousness: if such systems have mental properties, then they have them partly by virtue of possessing a Markov blanket (hence Markovian). Markovian monism rests upon the information geometry of random dynamic systems. In brief, the information geometry induced in any system—whose internal states can be distinguished from external states—must acquire a dual aspect. This dual aspect concerns the (intrinsic) information geometry of the probabilistic evolution of internal states and a separate (extrinsic) information geometry of probabilistic beliefs about external states that are parameterised by internal states. We call these intrinsic (i.e., mechanical, or state-based) and extrinsic (i.e., Markovian, or belief-based) information geometries, respectively. Although these mathematical notions may sound complicated, they are fairly straightforward to handle, and may offer a means through which to frame the origins of consciousness.

Keywords: consciousness, information geometry, Markovian monism.

Ontic structural realism argues that structure is all there is. In (French, 2014) I argued for an ‘eliminativist’ version of this view, according to which the world should be conceived, metaphysically, as structure, and objects, at both the fundamental and ‘everyday’ levels, should be eliminated. This paper is a response to a number of profound concerns that have been raised, such as how we might distinguish between the kind of structure invoked by this view and mathematical structure in general, how we should choose between eliminativist ontic structural realism and alternative metaphysical accounts such as dispositionalism, and how we should capture, in metaphysical terms, the relationship between structures and particles. In developing my response I shall touch on a number of broad issues, including the applicability of mathematics, the nature of representation and the relationship between metaphysics and science in general.

Keywords: Causation; Dependence; Disposition; Metaphysics; Object; Representation; Structure.

Most of us are familiar with the classic example of a liquid-gas moving contact line on a solid surface: a raindrop, sheared by the wind, creeps along a glass windscreen. The contact line’s movements depend on the interplay between viscous and surface tension forces—a relationship that has been thoroughly investigated in experimental fluid mechanics.

In a study published in The European Physical Journal Special Topics, Harish Dixit, of the Indian Institute of Technology Hyderabad, and his colleagues now examine the movements of a contact line formed at the interface between two immiscible liquids and a solid. The experiments fill a gap in and suggest a mechanism for an imposed boundary condition that eludes mathematical description.

According to theory, the movement of a liquid-liquid contact line should be governed entirely by the liquids’ viscosity ratio and the angle at which the liquid interface meets the solid. To examine this in a real-world system, Dixit and his colleagues filled a rectangular tank with two liquid layers—silicone oil atop sugar water—with similar densities but significantly different viscosities. The researchers placed a glass slide at the edge of the tank, which they could slide vertically to create a moving contact line.

Mathematics suggested that time travel is physically possible – and Kurt Gödel proved it. Mathematician Karl Sigmund explains how the polymath did it.

Computers have come so far in terms of their power and potential, rivaling and even eclipsing human brains in their ability to store and crunch data, make predictions and communicate. But there is one domain where human brains continue to dominate: energy efficiency.

“The most efficient computers are still approximately four orders of magnitude — that’s 10,000 times — higher in energy requirements compared to the human brain for specific tasks such as image processing and recognition, although they outperform the brain in tasks like mathematical calculations,” said UC Santa Barbara electrical and computer engineering Professor Kaustav Banerjee, a world expert in the realm of nanoelectronics. “Making computers more energy efficient is crucial because the worldwide energy consumption by on-chip electronics stands at #4 in the global rankings of nation-wise energy consumption, and it is increasing exponentially each year, fueled by applications such as artificial intelligence.” Additionally, he said, the problem of energy inefficient computing is particularly pressing in the context of global warming, “highlighting the urgent need to develop more energy-efficient computing technologies.”

Neuromorphic computing has emerged as a promising way to bridge the energy efficiency gap. By mimicking the structure and operations of the human brain, where processing occurs in parallel across an array of low power-consuming neurons, it may be possible to approach brain-like energy efficiency.

Schizophrenia, a neurodevelopmental disorder that features psychosis among its symptoms, is thought to arise from disorganization in brain connectivity and functional integration. Now, a recent study in Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, finds differences in functional brain connectivity in people with and without psychosis and schizophrenia that could help researchers understand the neural underpinnings of this disease.

The brain’s cortex is organized in a hierarchical fashion, anchored by the sensorimotor cortex at one end and by multimodal association areas at the other, with the task of integrating incoming sensory information with internal and external sensory signals. The loss of executive control in schizophrenia may stem from disruption of this hierarchical signaling.

Alexander Holmes, a Ph.D. candidate at Monash University who led the study, said, “We used brain imaging and novel mathematical techniques to investigate the hierarchical organization of the brains of individuals with early psychosis and established schizophrenia. This organization is important for brain health, as it regulates how we can effectively respond to and process stimuli from the external world.”

Euler’s Identity:

The most beautiful equation in mathematics that combines five of the most important constants of nature: 0, 1, π, e and i, with the three fundamental operations: addition, multiplication and exponentiation.

It’s mystical.

A research team is studying how light moves through special circuits called optical waveguides, using a concept called topology. They’ve made an important discovery that combines stable light paths with light particle interactions, which could make quantum computers more reliable and lead to new technological advancements.

Scientific innovation often arises as synthesis from seemingly unrelated concepts. For instance, the reciprocity of electricity and magnetism paved the way for Maxwell’s theory of light, which, up until now, is continually being refined and extended with ideas from quantum mechanics.

Similarly, the research group of Professor Alexander Szameit at the Institute of Physics at the University of Rostock explores light evolution in optical waveguide circuits in the presence of topology. This abstract mathematical concept was initially developed to classify solid geometries according to their global properties. Szameit explains: “In topological systems, light only follows the global characteristics of the waveguide system. Local perturbations to the waveguides such as defects, vacancies, and disorder cannot divert its path.”

Our favorite mathematical constant, pi (π), describing the ratio between a circle’s circumference and its diameter, has taken on new meaning.

The new representation was borne out of the twists and turns of string theory, and two mathematicians’ attempts to better describe particle collisions.

“Our efforts, initially, were never to find a way to look at pi,” says Aninda Sinha of the Indian Institute of Science (IISc) who co-authored the new work with fellow IISc mathematician Arnab Priya Saha.

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