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Category: mathematics – Page 76

Do you want to know whether a very large integer is a prime number or not? Or if it is a “lucky number”? A new study by SISSA, carried out in collaboration with the University of Trieste and the University of Saint Andrews, suggests an innovative method that could help answer such questions through physics, using some sort of “quantum abacus.”

By combining theoretical and , scientists were able to reproduce a quantum potential with corresponding to the first 15 and the first 10 lucky numbers using holographic laser techniques. This result, published in PNAS Nexus, opens the door to obtaining potentials with finite sequences of integers as arbitrary quantum energies, and to addressing mathematical questions related to with quantum mechanical experiments.

“Every physical system is characterized by a certain set of energy levels, which basically make up its ID,” explains Giuseppe Mussardo, at SISSA—International School for Advanced Studies. “In this work, we have reversed this line of reasoning: is it possible—starting from an arithmetic sequence, for example that of prime numbers—to obtain a quantum system with those very numbers as energy levels?”

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“All things are numbers,” avowed Pythagoras. Today, 25 centuries later, algebra and mathematics are everywhere in our lives, whether we see them or not. The Cambrian-like explosion of artificial intelligence (AI) brought numbers even closer to us all, since technological evolution allows for parallel processing of a vast amounts of operations.

Progressively, operations between scalars (numbers) were parallelized into operations between vectors, and subsequently, matrices. Multiplication between matrices now trends as the most time-and energy-demanding operation of contemporary AI computational systems. A technique called “tiled matrix multiplication” (TMM) helps to speed computation by decomposing matrix operations into smaller tiles to be computed by the same system in consecutive time slots. But modern electronic AI engines, employing transistors, are approaching their intrinsic limits and can hardly compute at clock-frequencies higher than ~2 GHz.

The compelling credentials of light—ultrahigh speeds and significant energy and footprint savings—offer a solution. Recently a team of photonic researchers of the WinPhos Research group, led by Prof. Nikos Pleros from the Aristotle University of Thessaloniki, harnessed the power of light to develop a compact silicon photonic computer engine capable of computing TMMs at a record-high 50 GHz clock frequency.

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A recent study from researchers at the University of California, Irvine found that the removal of cilia from the striatum region of the brain negatively impacted time perception and judgement, opening the possibility for new therapeutic targets for mental and neurological conditions such as schizophrenia, Parkinson’s and Huntington’s diseases, autism spectrum disorder.

Autism Spectrum Disorder (ASD) is a complex developmental disorder that affects how a person communicates and interacts with others. It is characterized by difficulty with social communication and interaction, as well as repetitive behaviors and interests. ASD can range from mild to severe, and individuals with ASD may have a wide range of abilities and challenges. It is a spectrum disorder because the symptoms and characteristics of ASD can vary widely from person to person. Some people with ASD are highly skilled in certain areas, such as music or math, while others may have significant learning disabilities.

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Connectome harmonic decomposition (CHD) generalises the mathematics of the Fourier transform to the network structure of the human brain. The traditional Fourier transform operates in the temporal domain (Fig. 1a): decomposition into temporal harmonics quantifies to what extent the signal varies slowly (low-frequency temporal harmonics) or quickly (high-frequency temporal harmonics) over time (Fig. 1b). Analogously, CHD re-represents a spatial signal in terms of harmonic modes of the human connectome, so that the spatial frequency (granularity) of each connectome harmonic quantifies to what extent the organization of functional brain signals deviates from the organization of the underlying structural network (Fig. 1c, d). Therefore, CHD is fundamentally different from, and complementary to, traditional approaches to functional MRI data analysis. This is because CHD does not view functional brain activity as composed of signals from discrete spatial locations, but rather as composed of contributions from distinct spatial frequencies: each connectome harmonic is a whole-brain pattern with a characteristic spatial scale (granularity)—from an entire hemisphere to just a few millimetres.

On one hand, this means that CHD is unsuitable to address questions pertaining to spatial localisation and the involvement of specific neuroanatomical regions; such questions have been extensively investigated within the traditional framework of viewing brain activity in terms of spatially discrete regions, and several previous studies have implicated specific neuroanatomical regions in supporting consciousness33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49. On the other hand, CHD enables us to consider how brain activity across states of consciousness is shaped by the brain’s distributed network of structural connections, reflecting the contribution of global patterns at different spatial scales—each arising from the network topology of the human connectome. We emphasise that neither approach is inherently superior, but rather they each provide a unique perspective on brain function: one localised, the other distributed.

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There are two aspects to a computer’s power: the number of operations its hardware can execute per second and the efficiency of the algorithms it runs. The hardware speed is limited by the laws of physics. Algorithms—basically sets of instructions —are written by humans and translated into a sequence of operations that computer hardware can execute. Even if a computer’s speed could reach the physical limit, computational hurdles remain due to the limits of algorithms.

These hurdles include problems that are impossible for computers to solve and problems that are theoretically solvable but in practice are beyond the capabilities of even the most powerful versions of today’s computers imaginable. Mathematicians and computer scientists attempt to determine whether a problem is solvable by trying them out on an imaginary machine.

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Mc.openredstone.org.

Instruction Set: https://1drv.ms/x/s!AkiZre7Tutskiw5SEwrIS90RQEK9?e=YhXXzR
Assembler: https://1drv.ms/u/s!AkiZre7TutskjCaMNSG6SuvbnN-B?e=kdPKtb.

Music used in this video:
Aria Math Space Remix: https://youtu.be/ajlXyZQp9N4
I do NOT own any of the music used in this video!

Links:

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‘Spin’ is a fundamental quality of fundamental particles like the electron, invoking images of a tiny sphere revolving rapidly on its axis like a planet in a shrunken solar system.

Only it isn’t. It can’t. For one thing, electrons aren’t spheres of matter but points described by the mathematics of probability.

But California Institute of Technology philosopher of physics Charles T. Sebens argues such a particle-based approach to one of the most accurate theories in physics might be misleading us.

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Year 2017 face_with_colon_three

A basic question [1] in the study of the gauge-gravity duality is this: which field theories have a gravity dual? In the case of applications to actual strongly coupled systems such as the Quark–Gluon Plasma [2], [3], [4], [5], [6], this question becomes: does every realistic strongly coupled system have such a dual? To settle this, one needs to examine the most extreme cases. The most extreme strongly-coupled systems currently accessible to experiment are probably (see below) the plasmas produced by collisions of heavy ions at the LHC [7], [8] ; so one needs to consider whether holography works in this case.

In [9] we adduced evidence suggesting that it does not. The problem is a very fundamental one: it appears that the purported gravity dual in some cases does not exist when one attempts to interpret it (as one ultimately must [10]) as a string-theoretic system.

The situation may be briefly explained as follows. Ferrari and co-workers have shown [11], [12], [13], [14] that, simply for reasons of internal mathematical consistency, a string-theoretic bulk spacetime with a holographic dual must satisfy certain fundamental relations between the Euclidean spacetime action and the action of probes (such as branes). This has been explicitly confirmed in a large number of concrete cases [14].

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A model for information storage in the brain reveals how memories decay with age.

Theoretical constructs called attractor networks provide a model for memory in the brain. A new study of such networks traces the route by which memories are stored and ultimately forgotten [1]. The mathematical model and simulations show that, as they age, memories recorded in patterns of neural activity become chaotic—impossible to predict—before disintegrating into random noise. Whether this behavior occurs in real brains remains to be seen, but the researchers propose looking for it by monitoring how neural activity changes over time in memory-retrieval tasks.

Memories in both artificial and biological neural networks are stored and retrieved as patterns in the way signals are passed among many nodes (neurons) in a network. In an artificial neural network, each node’s output value at any time is determined by the inputs it receives from the other nodes to which it’s connected. Analogously, the likelihood of a biological neuron “firing” (sending out an electrical pulse), as well as the frequency of firing, depends on its inputs. In another analogy with neurons, the links between nodes, which represent synapses, have “weights” that can amplify or reduce the signals they transmit. The weight of a given link is determined by the degree of synchronization of the two nodes that it connects and may be altered as new memories are stored.

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