Tag: mathematics

  • The mathematical theory that made the internet possible

    The mathematical theory that made the internet possible

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    The internet could not exist without information theory

    Mehmet BAYSAN / Alamy

    The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time here.

    Beginning in the mid-20th century, the digital revolution ushered in the information age, radically changing the way information is processed and transmitted. Audio and video, which traditionally had been analogue, became digital. Phones gained more computational power than the room-filling machines…

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  • The surprising connections between maths and poetry

    The surprising connections between maths and poetry

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    December 2, 2011, New York, New York, USA: December 2, 2011 - New York, New York USA:

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    People like to position maths as cold, hard logic, quite distinct from creative pursuits. Actually, maths often involves a great deal of creativity. As mathematician Sofya Kovalevskaya wrote, “It is impossible to be a mathematician without being a poet in soul.” Poetry is often constrained by rules, and these add to, rather than detract from, its creativity.

    Rhyming poems generally follow a scheme formed by giving each line a letter, so that lines with matching letters rhyme. This verse from a poem by A. A. Milne uses an ABAB scheme:

    What shall I call
    My dear little dormouse?
    His eyes…

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  • Speed of decision-making reflects our biases

    Speed of decision-making reflects our biases

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    Voters cast ballots

    The speed with which we reach a decision can reveal the strength of our biases

    David Williams/Bloomberg via Getty Images

    In a large group of independent decision-makers, those who reach a decision most quickly also tend to be those with the strongest initial biases. The finding comes from a mathematical model of decision-making that could be applied broadly across biology.

    “Many decision-making models in economics assume that people make a decision based on one or two pieces of information, but I think these models have to be expanded,” says Krešimir Josić at the University of Houston in Texas.…

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  • Is the universe a game?

    Is the universe a game?

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    If the universe is a game, then who’s playing it?

    Bruce Rolff/Shutterstock

    The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time for free here.

    Is the universe a game? Famed physicist Richard Feynman certainly thought so: “‘The world’ is something like a great chess game being played by the gods, and we are observers of the game.” As we observe, it is our task…

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  • How mathematics can help you divide anything up fairly

    How mathematics can help you divide anything up fairly

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    People hands taking slices of chocolate pie, above view on a green background. Unequal sharing the cake, concept for financial pie chart

    One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

    But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

    When splitting between two parties, you might…

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  • What Came Before the Big Bang?

    What Came Before the Big Bang?

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    Robert Brandenberger, a physicist at McGill University who was not involved with the study, said the new paper “sets a new standard of rigor for the analysis” of the mathematics of the beginning of time. In some cases, what appears at first to be a singularity—a point in space-time where mathematical descriptions lose their meaning—may in fact be an illusion.

    A Taxonomy of Singularities

    The central issue confronting Geshnizjani, Ling, and Quintin is whether there is a point prior to inflation at which the laws of gravity break down in a singularity. The simplest example of a mathematical singularity is what happens to the function 1/x as x approaches zero. The function takes a number x as an input, and outputs another number. As x gets smaller and smaller, 1/x gets larger and larger, approaching infinity. If x is zero, the function is no longer well defined: It can’t be relied upon as a description of reality.

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    “We mathematically showed that there might be a way to see beyond our universe,” said Eric Ling of the University of Copenhagen.

    Photograph: Annachiara Piubello

    Sometimes, however, mathematicians can get around a singularity. For example, consider the prime meridian, which passes through Greenwich, England, at longitude zero. If you had a function of 1/longitude, it would go berserk in Greenwich. But there’s not actually anything physically special about suburban London: You could easily redefine zero longitude to pass through some other place on Earth, and then your function would behave perfectly normally when approaching the Royal Observatory in Greenwich.

    Something similar happens at the boundary of mathematical models of black holes. The equations that describe spherical nonrotating black holes, worked out by the physicist Karl Schwarzschild in 1916, have a term whose denominator goes to zero at the event horizon of the black hole—the surface surrounding a black hole beyond which nothing can escape. That led physicists to believe that the event horizon was a physical singularity. But eight years later the astronomer Arthur Eddington showed that if a different set of coordinates is used, the singularity disappears. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity, which only arises because of the choice of coordinates.

    At a black hole’s center, by contrast, the density and curvature go to infinity in a way that can’t be eliminated by using a different coordinate system. The laws of general relativity start spewing out gibberish. This is called a curvature singularity. It implies that something is taking place that’s beyond the ability of current physical and mathematical theories to describe.

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  • The Monty Hall problem shows how tricky judging the odds can be

    The Monty Hall problem shows how tricky judging the odds can be

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    shutterstock/Jose Luis Stephens

    Calculating probabilities can be tricky, with subtle changes in context giving quite different results. I was reminded of this recently after setting BrainTwister #10 for New Scientist readers, which was about the odds of seating two pairs of people adjacently in a row of 22 chairs.

    Several readers wrote to say my solution was wrong. I had figured out all the possible seating arrangements and counted the ones that had the two groups adjacent. The readers, meanwhile, seated one pair first and then counted the ways of seating the second pair adjacently. Neither approach was wrong, depending on how you…

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  • What are fractals and how can they help us understand the world?

    What are fractals and how can they help us understand the world?

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    Fractal geometry is common in nature

    Shutterstock/Sabine Hortebusch

    You have almost certainly seen computer-generated fractals – beautiful, trippy images in which colourful, intricate structures repeat ad infinitum as you fall ever further down the rabbit hole. Formally speaking, fractals are infinitely complex patterns that are self-similar across different scales. But, in an echo of their geometry, fractals can help us better understand the world on many levels.

    Let’s start with the familiar: fractals in nature. “They are all around us – in trees, mountain ranges, river deltas and so on,” says Dave Feldman at the College of the Atlantic in Bar Harbour, Maine. Such ubiquity makes sense because of the way fractals are made: “a simple iterative process – repeated folding or branching – can produce fractals”, he says.

    These forms aren’t just for gawping at, though. The inside of your lungs is fractal for a reason: such arrangements cram a huge surface area into a small volume of space. This is how evolution solved the problem of maximising the area of tissue that can absorb oxygen.

    Where fractals get baffling, however, is in the reasons they captivate mathematicians, not least as a playground for exploring numbers. This is where we encounter fractal dimensions. In our everyday world, a straight line is one-dimensional, a square or rectangle two-dimensional and a cube or sphere three-dimensional. But fractal structures have dimensions in between these values. Though largely impossible to visualise, you can think of these as a measure of how much complexity a fractal contains…

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  • Game theory shows we can never learn perfectly from our mistakes

    Game theory shows we can never learn perfectly from our mistakes

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    traders in stock market

    When people trade stocks, they don’t always learn from experience

    Bill Ross/Getty Images

    Even when we learn from past mistakes, we may never become optimal decision-makers. The finding comes from an analysis of a mathematical game that simulates a large economy, and suggests we may need to rethink some of the common assumptions built into existing economic theories.

    In such theories, people are typically represented as rational agents who learn from past experiences to optimise their performance, eventually reaching a stable state in which they know how to maximise their earnings. This assumption…

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  • Unveiling Platinum’s Role in Clean Energy Catalysts

    Unveiling Platinum’s Role in Clean Energy Catalysts

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    Oxide Formation Mechanism on Platinum Surface

    Interfacial modification using hydrophilic/hydrophobic cations to control both electrochemical activity and stability of platinum electrode. Credit: Masashi Nakamura from Chiba University, edited

    Researchers elucidate mechanisms for controlling the surface oxidation processes that affect the performance of platinum catalysts in alkaline media.

    Platinum (Pt) electrodes are crucial for clean power technologies like hydrogen fuel cells and electrolysis. However, the surface oxidation that occurs during such processes degrades catalyst performance and stability. To address this, researchers investigated the mechanisms of surface oxidation on Pt surface in alkaline media, a previously unexplored avenue of research. Their experiments revealed crucial insights that can aid in the development of next-generation catalysts, paving the way for a carbon-neutral society.

    Hydrogen Fuel Cells and Electrolysis

    The pursuit of carbon neutrality drives the exploration of clean energy sources, with hydrogen fuel cells emerging as a promising avenue. In these cells, hydrogen undergoes an electrochemical reaction with oxygen to produce electricity and water. Also, the reverse of this process, called electrolysis, can be used to split the abundantly available water to produce hydrogen and oxygen. These two technologies can work in tandem to provide a clean and renewable source of energy. A pivotal element in these two technologies is the platinum (Pt) electrode.

    Challenges in Fuel Cell Technology

    Hydrogen fuel cells consist of two electrodes: an anode and a cathode, with an electrolyte between them. Pt serves as a fundamental catalyst in low-temperature fuel cells, such as alkaline fuel cells and polymer electrolyte fuel cells (PEFCs). Pt has a high activity for the oxygen reduction reaction (ORR), which is crucial for fuel cells, in alkaline and acidic conditions at the operating voltage of PEFC cathodes. However, this also leads to oxide formation on the surface, which roughens and dissolves the Pt layer, ultimately degrading the cathodes and affecting performance and stability.

    Understanding surface oxide formation mechanisms is thus crucial for developing Pt cathode catalysts that work well in alkaline conditions. Studies have shown that the oxide formation on the Pt surface depends on the electrode potential, the electrolyte, and the electrical double layer (EDL). While studies have investigated the oxide formation and reduction on the Pt surface in acidic media, few of them have addressed the same in alkaline media, present in fuel cells and electrolyzers with anion exchange membranes.

    Advancements in Alkaline Media Research

    To address this gap, a team of researchers led by Professor Masashi Nakamura from the Graduate School of Engineering, Chiba University, Japan, dug deep into the oxide formation mechanisms on Pt surfaces in alkaline media.

    “In a previous study, we reported that interfacial hydrophobic ions with long alkyl chains can enhance ORR. This suggests that it is possible to construct an interfacial reaction field that not only activates the ORR but also improves the durability of Pt electrodes by using optimal interfacial ions,” explains Prof. Nakamura.

    The study also included contributions from Dr. Tomoaki Kumeda and Professor Nagahiro Hoshi, both from the Graduate School of Engineering at Chiba University, along with Dr. Osami Sakata from the Center for Synchrotron Radiation Research at Japan Synchrotron Radiation Research Institute. Their findings have been published in the Journal of the American Chemical Society.

    Innovative Techniques and Findings

    The team investigated the oxide formation on the Pt (111) surface in alkaline aqueous solutions containing different cations, namely Lithium cation (Li+), Potassium (K+) cation and Tetramethylammonium cation (TMA+), using advanced methods like X-ray crystal truncation rod (CTR) scattering, gold nanoparticle-based surface-enhanced Raman spectroscopy (GNP-SERS), and infrared reflection absorption spectroscopy (IRAS).

    “Studies have shown that a combination of vibrational spectroscopy and X-ray diffraction is effective for elucidating surface oxidation processes,” adds Prof. Nakamura.

    X-ray CTR revealed that oxide formation results in surface buckling and Pt extraction. SERS and IRAS measurements revealed the potential and cation-dependent formation of three oxide species, namely infrared (IR)-active adsorbed hydroxide OH (OHad), Raman active adsorbed water (H2O)ad, and Raman-active oxygen (Oad). The team found that hydrophilic cations like Li+ stabilize IR-active OHad, thus preventing harmful oxide formation, while moderate hydrophilicity of K+ has no protective effect.

    Interestingly, bulky hydrophobic cations such as TMA+ also reduce irreversible oxidation, similar to Li+. Notably, the team also found that the electrostatic repulsion between Raman-active (H2O)ad and neighboring Raman-active Oad facilitates Pt extraction.

    Conclusion and Implications for Clean Energy

    These results suggest that interfacial cations play an essential role in oxide formation on Pt surfaces, which can be controlled by selecting appropriate cations.

    Elaborating on these results, Prof. Nakamura remarks: “These insights are crucial for understanding the surface oxidation mechanisms and the EDL structure, which can be beneficial for achieving high-performance and stable Pt electrocatalysts for use in next-generation electrochemical devices.”

    Overall, this study takes us a step further in achieving a zero-carbon future powered by abundant and clean hydrogen.

    Reference: “Surface Extraction Process During Initial Oxidation of Pt(111): Effect of Hydrophilic/Hydrophobic Cations in Alkaline Media” by Tomoaki Kumeda, Kenshin Kondo, Syunnosuke Tanaka, Osami Sakata, Nagahiro Hoshi and Masashi Nakamura, 20 March 2024, Journal of the American Chemical Society.
    DOI: 10.1021/jacs.3c11334



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