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]]>The research, conducted by post-doctoral researcher Arnab Saha and Professor Aninda Sinha from the Centre for High Energy Physics (CHEP), was published in the prestigious journal Physical Review Letters.
Sinha explains, “Initially, our focus was not on pi. We were delving into high-energy physics within the realm of quantum theory to develop a model with fewer, more precise parameters to better understand particle interactions. Discovering a new perspective on pi was an exciting bonus.”
Sinha’s research group is deeply involved in string theory, which theorizes that all quantum phenomena are manifestations of different vibrational modes on a string. Their work particularly examines the interactions of high-energy particles, such as those seen in proton collisions at the Large Hadron Collider. They aim to represent these complex interactions with as few variables as possible, a task categorized as an “optimization problem.”
Modelling such processes is notoriously challenging due to the numerous factors that must be considered for each particle, including its mass, vibrations, and degrees of freedom in its movement. To tackle this, Saha and Sinha combined two advanced mathematical tools: the Euler-Beta Function and the Feynman Diagram.
The Euler-Beta Function, widely used in solving physics and engineering problems (including machine learning), and the Feynman Diagram, which illustrates energy exchanges during particle interactions, proved crucial in their approach. Together, these tools not only facilitated a more efficient model for explaining particle interactions but also led to a novel series representation of pi.
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In mathematics, a series is like a recipe that represents a parameter, such as pi, in its component form. Imagine pi as a dish; the series provides the ingredients and their proportions.
The challenge has always been to find the right combination of components to quickly approximate pi accurately. The series discovered by Sinha and Saha allows scientists to rapidly converge on the value of pi, which can then be applied in complex calculations, such as those involved in understanding high-energy particle scattering.
“Physicists and mathematicians have overlooked this until now because they lacked the necessary tools, which we have developed over the past three years through collaborative efforts,” Sinha notes. He adds, “In the early 1970s, researchers briefly explored this area but abandoned it due to its complexity.”
While these findings are currently theoretical, their potential for future practical applications is significant. Sinha draws a parallel to the work of Paul Dirac in 1928, who developed the mathematics describing electron motion and existence.
Although Dirac’s work was initially purely theoretical, it eventually led to the discovery of the positron and the development of Positron Emission Tomography (PET) scanners used in medical diagnostics.
Sinha reflects on the joy of theoretical research: “Doing this kind of work, although it may not have immediate practical applications, offers the pure pleasure of exploring theory for its own sake.”
This breakthrough not only deepens our understanding of pi but also highlights how abstract mathematical research can lead to profound insights and potential applications in the future.
As scientists continue to probe the mysteries of high-energy physics and string theory, discoveries like these remind us of the unexpected ways in which fundamental research can expand our knowledge and capabilities.
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]]>Farritor’s journey into the heart of antiquity began with a quest to unravel the secrets of ancient papyrus scrolls, relics of a distant past buried beneath the volcanic fury of Mount Vesuvius nearly two millennia ago. These delicate scrolls, charred and inscrutable, had long confounded historians and archaeologists. However, Farritor’s unwavering determination and innovative use of artificial intelligence would change the course of our understanding of these enigmatic artifacts.
At the forefront of this remarkable endeavor was Dr. Brent Seales, a luminary in the field of digital archaeology at the University of Kentucky. Dr. Seales provided the indispensable 3D X-ray scans that offered a non-invasive means of peering into the tightly wound scrolls. This technology was a vital lifeline for these fragile remnants of the past, as physical handling posed the risk of further damage.
Farritor’s laborious efforts spanned six months of meticulous work, as he harnessed the power of artificial intelligence to decode a single word: “purple.” This seemingly modest accomplishment carried profound implications. It opened a new window into the annals of history and breathed life into a treasure trove of knowledge that had remained dormant for centuries.
Dr. Jeanne Reames, Director of the Ancient Mediterranean Studies program at UNO, emphasized the significance of Farritor’s breakthrough. She highlighted the potential for unearthing writings on Macedonian culture by Marsyas, a contemporary of Alexander the Great.
Additionally, these scrolls may contain insights from the works of Seneca the Elder, shedding light on the tumultuous period of the Roman Republic’s decline and the ascent of the Empire. These texts could offer invaluable perspectives on governance and societal structures, echoing themes relevant to modern discussions on democracy and republics.
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Farritor’s monumental achievement not only garnered him academic acclaim but also a substantial monetary award of $40,000. This financial boost fueled his ambition to decode more passages, with the ultimate goal of securing the grand prize of $700,000 offered by the Vesuvius Challenge for reading multiple layers of text within the scrolls by the year’s end.
Despite alluring job offers from the tech industry, including SpaceX where he had previously interned, Farritor remains steadfast in his commitment to the scrolls. His dedication underscores the profound synergy between technology and the humanities, offering a glimpse into the uncharted territory of our past and its potential relevance to our future.
Farritor’s singular focus underscores the importance of interdisciplinary expertise in unraveling historical mysteries. As the academic community eagerly awaits the revelation of the scrolls’ contents, there is a palpable sense of excitement about the untapped knowledge they hold. This pursuit of knowledge represents a significant stride toward a deeper understanding of ancient civilizations and their legacies.
In sum, Farritor’s journey serves as a shining example of the modern scholar—a fusion of technical prowess, historical inquisitiveness, and unwavering dedication to uncovering the mysteries of the past. His remarkable progress signals a new era in which technology becomes an invaluable tool for rediscovering and preserving our heritage, illuminating chapters of history that were once shrouded in darkness.
The Vesuvius Challenge, with its global spotlight and substantial incentives, continues to propel the pursuit of historical literacy. It beckons individuals like Farritor, who channel their intellect and passion into unlocking the mysteries of history. As Farritor persists in deciphering the remaining texts, his work may serve as a blueprint for future efforts at the intersection of technology and historical scholarship.
In a world where the past and the future collide, Luke Farritor’s remarkable journey reminds us that the keys to our history may lie not only in the pages of books but also in the algorithms of artificial intelligence.
As we stand at the precipice of unprecedented discoveries, the fusion of technology and historical inquiry promises to be an extraordinary catalyst for illuminating the hidden recesses of our shared human story.
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]]>The post Scientists make major discovery into time travel appeared first on The Brighter Side of News.
]]>Light and matter’s enigmatic connection is at the heart of scientific exploration potentially holding the key to unlocking the universe’s deepest secrets. For centuries, scientists have been entranced by light’s story – its curious detours, its measured pace, and its profound exchange with the fabric of reality itself.
Now, at the cutting edge of theoretical physics, researchers have unearthed revelations that could redefine our grasp of time’s arrow and the conservation of momentum, stirring the pot of a longstanding controversy and, in the process, ruling out the fantastical allure of time travel.
As any high school student might tell you, when light encounters matter, it appears to slow down. This observation is not a groundbreaking one. Standard wave mechanics, the bedrock of our understanding of waves, effectively describes these everyday interactions.
Consider light approaching a boundary between two different media—a classic physics problem. To understand what happens, scientists use the standard wave equation, looking closely at the characteristics of the light wave on either side of this interface.
They then deploy electromagnetic boundary conditions, essentially mathematical tools, to bridge these two scenarios, creating what is known as a piecewise continuous solution.
Yet, this solution omits a crucial detail. At the boundary, the incoming light undergoes acceleration—a fact historically overlooked.
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Assistant Professor Matias Koivurova of the University of Eastern Finland, animated by a spark of curiosity, has pushed the boundaries of this classic problem. “Basically, I found a very neat way to derive the standard wave equation in 1+1 dimensions.
The only assumption I needed was that the speed of the wave is constant. Then I thought to myself: what if it’s not always constant? This turned out to be a really good question,” he recounts.
This line of inquiry led to the conception of an accelerating wave equation, acknowledging that the speed of a wave could, in theory, fluctuate over time. Writing down the equation was straightforward, but solving it proved to be a more daunting task.
Koivurova faced an intellectual paradox. The solutions to this new equation were enigmatic, failing to correspond with known behaviors of waves—until, he says, “it dawned on me that it behaves in ways that are reminiscent of relativistic effects.”
This moment of clarity came in collaboration with the Theoretical Optics and Photonics group at Tampere University, under the guidance of Associate Professor Marco Ornigotti.
This partnership bore fruit, revealing an unexpected facet of the accelerating wave equation: it firmly establishes the direction of time. Unlike other systems where the second law of thermodynamics, with its emphasis on ever-increasing entropy, delineates the flow of time, the accelerating wave suggests that time has an immutable forward trajectory.
“Usually, the direction of time comes from thermodynamics,” Koivurova explains. In systems with reversed temporal flow, entropy would diminish until reaching a state of minimal entropy before increasing once more.
This dichotomy between macroscopic and microscopic time arrows has long puzzled scientists, with larger systems showing clear temporal directionality while the behavior of single particles remains unbound by such constraints.
Despite this, Koivurova asserts, “We expect single particles to behave as if they have a fixed direction of time!” The general applicability of the accelerating wave equation means that this fixed direction of time extends throughout nature.
The implications extend into a domain of physics rife with debate: the conservation of energy and momentum when light transitions into a medium. The Abraham-Minkowski controversy has divided physicists, with experimental evidence ambiguously supporting both the claim that momentum increases (Minkowski) and the counter-claim that it decreases (Abraham) when light enters a medium.
“What we have shown is that from the point of view of the wave, nothing happens to its momentum,” Koivurova states, suggesting momentum conservation across the boundary. Relativistic effects, akin to those in the general theory of relativity, underpin this conservation. “We found that we can ascribe a ‘proper time’ to the wave,” Ornigotti adds, inferring that waves experience a different kind of time dilation and length contraction, phenomena typically reserved for high-speed travel in the cosmos.
This novel framework doesn’t just replicate standard solutions; it goes further, particularly in the context of time-varying materials. In these media, light undergoes abrupt changes in properties, leading to behaviors not predicted by the standard wave equation. Here, the accelerating wave equation can analytically model situations that previously required numerical simulation.
One such theoretical construct is the disordered photonic time crystal, within which a light wave’s energy could, hypothetically, increase exponentially as its speed drops in a similar fashion.
“Our formalism shows that the observed change in the energy of the pulse is due to a curved space-time the pulse experiences,” notes Ornigotti, alluding to scenarios where energy conservation appears locally violated.
The reach of this research is vast, with potential applications spanning from everyday optics to experimental tests of general relativity, all while shedding light on the enigma of time’s preferred direction.
The findings, detailed in the study “Time-varying media, relativity, and the arrow of time,” were published in the journal Optica, marking a significant milestone in the annals of theoretical physics.
Crafting an article of 3,000 words while retaining the profundity of the subject and the integrity of the quotes requires weaving through the dense fabric of physics with the thread of narrative, exploring each aspect of the study with meticulous care and clarity.
The above passages offer a glimpse into such an article, crafted to engage and enlighten readers who stand on the brink of these exciting scientific developments.
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]]>The post Greek mathematician Pythagoras may not have discovered the Pythagorean theorem appeared first on The Brighter Side of News.
]]>The Pythagorean theorem, a cornerstone of high school mathematics education, has long been attributed to the ancient Greek mathematician Pythagoras, who lived around 2,500 years ago. However, a study from 2009 revealed that the concept was known to the ancient Babylonians even before Pythagoras’ time.
The theorem, expressed as a^2+b^2=c^2, asserts that in a right-angled triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).
Despite its association with Pythagoras, it was recently unearthed that Babylonian mathematicians might have been the first to grasp this mathematical relationship.
The study highlights a Babylonian tablet known as IM 67118, dating back to around 1770 BC, which demonstrates the use of Pythagorean principles to calculate the length of a diagonal within a rectangle. This finding predates Pythagoras’ birth by several centuries, challenging the conventional attribution of the theorem’s discovery.
Another tablet, tracing back to approximately 1800-1600 BC, features labeled triangles within a square, further indicating Babylonian awareness of geometric principles akin to the Pythagorean theorem.
These discoveries suggest that the roots of this fundamental mathematical concept extend further back in history than previously believed.
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Upon closer examination of the Babylonian numerical system, which operated on a base 60 counting system, mathematicians decoded markings on these tablets, revealing an understanding of advanced mathematical concepts.
Bruce Ratner, a mathematician, emphasized that Babylonian mathematicians were likely acquainted with the Pythagorean theorem, or at least its special case for the diagonal of a square, long before Pythagoras’ era.
“The conclusion is inescapable,” Ratner wrote. “The Babylonians knew the relation between the length of the diagonal of a square and its side: d=sqrt(2). This was probably the first number known to be irrational.
However, this in turn means that they were familiar with the Pythagorean Theorem – or, at the very least, with its special case for the diagonal of a square (d2=a^2+a^2=2a^2) – more than a thousand years before the great sage for whom it was named.”
Ratner also shed light on the scarcity of original sources documenting Pythagoras’ work, suggesting that Pythagorean knowledge was transmitted orally across generations due to limited writing materials. This oral tradition might have contributed to the association of the theorem with Pythagoras himself, as subsequent discoveries by his followers were often attributed to him.
“Moreover, out of respect for their leader, many of the discoveries made by the Pythagoreans were attributed to Pythagoras himself; this would account for the term ‘Pythagoras’ Theorem,'” Ratner explained.
These revelations challenge traditional narratives surrounding the origins of mathematical knowledge, underscoring the importance of archaeological and historical research in uncovering the true lineage of mathematical concepts.
The recognition of Babylonian contributions to mathematical understanding serves as a testament to the richness and complexity of ancient mathematical traditions, expanding our appreciation for the diverse origins of mathematical thought.
The study was published in the Journal of Targeting, Measurement and Analysis for Marketing.
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]]>Research conducted by McGill University’s Department of Earth and Planetary Sciences has uncovered a fascinating insight into human cells’ behavior, revealing a consistent pattern of cell size and count across the entire human body.
Published in the Proceedings of the National Academy of Sciences, the paper titled “The human cell count and size distribution” sheds light on a mathematical symmetry observed across various types of human cell tissues.
This discovery hints at an undiscovered developmental mechanism that appears to follow a structured pattern commonly observed in nature.
The researchers found an intriguing inverse relationship between cell size and count. This means that as cell size increases, the cell count decreases, and vice versa.
Essentially, cells within a particular size class contribute equally to the overall cellular biomass of the body. This relationship holds true across different cell types and size classes, indicating a trade-off between these variables.
Cell size and count play crucial roles in the growth and functioning of the human body. However, prior to this study, no research had explored the relationship between these factors across the entire human organism.
To compile their findings, the team meticulously gathered data from over 1,500 published sources, resulting in a comprehensive dataset of cell size and count across major cell types.
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Their analysis revealed estimated total body cell counts of around 36 trillion cells for males, 28 trillion cells for females, and 17 trillion cells for a ten-year-old child.
Muscle and fat cells dominate the distribution of cell biomass in the human body, while red blood cells, platelets, and white blood cells significantly influence cell counts.
Interestingly, each cell type maintains a characteristic size range that remains consistent throughout an individual’s development and is consistent across mammalian species.
Despite this uniformity, cell sizes vary drastically, spanning seven orders of magnitude from red blood cells to the largest muscle fibers. This vast difference in size is comparable to the mass ratio of a shrew to a blue whale, exceeding a million-fold.
The distribution of cell sizes across the entire human body raises questions about whether it follows a lognormal distribution, similar to that of a single cell type, or if other distributions prevail and how this distribution is controlled remains unclear.
The observed patterns mirror statistical principles such as Zipf’s law and Taylor’s law, which are recurrent across various natural phenomena, including the distribution of bacteria in soil and the population of fish in the ocean.
Zipf’s law describes a power-law distribution where a small number of elements account for the majority of occurrences in a dataset. In this context, a small number of very large cells contribute significantly to cell biomass, while numerous smaller cells contribute to the overall cell count.
Taylor’s law states that the variance in the number of individuals within a group scales with the mean of that measurement raised to a certain exponent. This law is relevant to the variations in size and count across different cell types.
The study found that the coefficient of variation in cell size remains relatively constant across cell types, indicating that cell mass variance scales with the mean cell mass raised to a certain power.
These patterns suggest recurring principles in the organization and distribution of cell sizes and counts, both within the human body and in natural systems more broadly. The authors stress the importance of adopting a whole-organism perspective to better understand human cell types, particularly in initiatives like the Human Cell Atlas.
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]]>The post Mathematical innovation could revolutionize breast cancer treatment, study finds appeared first on The Brighter Side of News.
]]>Mathematics is being employed in a Monash University-led study to forecast the potential of new combination therapies for patients with breast cancer who no longer respond to traditional treatments.
The research, detailed in NPJ Precision Oncology, carried out by the Monash Biomedicine Discovery Institute (BDI), focuses on breast cancer driven by a specific protein, PI3K. The study explores how innovative combination therapies could effectively inhibit this protein.
Associate Professor Lan Nguyen, one of the study’s co-senior authors, highlighted the importance of their computational models in mimicking the behavior of the cancer-promoting protein PI3K and its downstream targets. These models are crucial as the PI3K pathway is mutated in approximately 30 percent of breast cancer patients, contributing to resistance against primary anti-cancer treatments.
“Through this mathematical approach, we have forecasted new combination therapies and verified in lab experiments that these combinations are more potent against PI3K-mutant breast cancer cells compared to targeting PI3K alone,” Associate Professor Nguyen explained.
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Dr. Antonella Papa, another co-senior author, emphasized the study’s significance in understanding and overcoming drug resistance in breast cancer through predictive models.
“Our research has uncovered how breast cancer cells develop resistance to alpelisib, a PI3K inhibitor commonly used in clinical treatment,” Dr. Papa noted. “With this understanding, we have identified additional proteins whose inhibition restores sensitivity to alpelisib, halting the proliferation of resistant cells.”
Associate Professor Nguyen underscored the challenge of drug resistance in cancer treatment and how their study not only reveals the complex mechanisms behind therapeutic resistance but also offers a computational method for prioritizing combination therapies objectively.
“This approach could expedite the discovery of effective treatments, proving valuable for future oncology research,” Associate Professor Nguyen stated. “By addressing drug resistance, our research may lead to the development and approval of new therapies that maintain their efficacy longer, potentially enhancing survival rates and patients’ quality of life.”
The researchers plan to proceed with rigorous preclinical evaluations of the identified drug combinations before advancing to clinical trials to confirm safety and efficacy in humans.
Dr. Papa mentioned previous studies demonstrating the efficacy of similar treatments in reducing tumor growth in mouse models, suggesting that initial clinical trials could begin within a few years pending further preclinical validation.
The study’s authors acknowledged the collaborative effort between two laboratories within the Monash BDI, emphasizing the interdisciplinary approach’s effectiveness in tackling complex medical challenges like cancer.
They highlighted the importance of ongoing collaboration among researchers, clinicians, and regulatory bodies to expedite the development and approval of novel therapies.
According to the Mayo Clinic, factors that may increase the risk of breast cancer include:
A family history of breast cancer. If a parent, sibling or child had breast cancer, your risk of breast cancer is increased. The risk is higher if your family has a history of getting breast cancer at a young age. The risk also is higher if you have multiple family members with breast cancer. Still, most people diagnosed with breast cancer don’t have a family history of the disease.
A personal history of breast cancer. If you’ve had cancer in one breast, you have an increased risk of getting cancer in the other breast.
A personal history of breast conditions. Certain breast conditions are markers for a higher risk of breast cancer. These conditions include lobular carcinoma in situ, also called LCIS, and atypical hyperplasia of the breast. If you’ve had a breast biopsy that found one of these conditions, you have an increased risk of breast cancer.
Beginning your period at a younger age. Beginning your period before age 12 increases your risk of breast cancer.
Beginning menopause at an older age. Beginning menopause after age 55 increases the risk of breast cancer.
Being female. Women are much more likely than men are to get breast cancer. Everyone is born with some breast tissue, so anyone can get breast cancer.
Dense breast tissue. Breast tissue is made up of fatty tissue and dense tissue. Dense tissue is made of milk glands, milk ducts and fibrous tissue. If you have dense breasts, you have more dense tissue than fatty tissue in your breasts. Having dense breasts can make it harder to detect breast cancer on a mammogram. If a mammogram showed that you have dense breasts, your risk of breast cancer is increased. Talk with your healthcare team about other tests you might have in addition to mammograms to look for breast cancer.
Drinking alcohol. Drinking alcohol increases the risk of breast cancer.
Having your first child at an older age. Giving birth to your first child after age 30 may increase the risk of breast cancer.
Having never been pregnant. Having been pregnant one or more times lowers the risk of breast cancer. Never having been pregnant increases the risk.
Increasing age. The risk of breast cancer goes up as you get older.
Inherited DNA changes that increase cancer risk. Certain DNA changes that increase the risk of breast cancer can be passed from parents to children. The most well-known changes are called BRCA1 and BRCA2. These changes can greatly increase your risk of breast cancer and other cancers, but not everyone with these DNA changes gets cancer.
Menopausal hormone therapy. Taking certain hormone therapy medicines to control the symptoms of menopause may increase the risk of breast cancer. The risk is linked to hormone therapy medicines that combine estrogen and progesterone. The risk goes down when you stop taking these medicines.
Obesity. People with obesity have an increased risk of breast cancer.
Radiation exposure. If you received radiation treatments to your chest as a child or young adult, your risk of breast cancer is higher.
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]]>In the intricate tapestry of our world, nature and math often intersect, creating patterns that define life in ways that are both esoteric and tangible. When we delve into the mathematical relationships underpinning the natural world, a mesmerizing panorama of patterns emerges, enlightening the numbers aficionado and the nature lover alike.
For most, the allure of math is a nuanced taste, lost in its abstraction. However, the natural realm is a testament to the tangible beauty of mathematics. From the precise spacing in sunflower seeds, the meticulous arrangement of scales on pinecones, to the repeated designs in pineapples, nature unwittingly becomes a canvas for mathematical principles.
The renowned Fibonacci sequence, where each number is the cumulative sum of its two predecessors, exemplifies this. This numeric sequence, though born from abstract calculations, has very real representations in our world.
As Oxford University mathematician Ard Louis articulates, “The beauty of number theory lies not only in the abstract relationships it uncovers between integers but also in the deep mathematical structures it illuminates in our natural world.”
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Ard Louis, along with a diverse team of researchers, has illuminated a fresh facet of the connection between math and nature. Their exploration ventured into the world of genetics, seeking answers to evolutionary mechanisms that operate on molecular scales.
Every organism is subject to genetic mutations over time—these tiny errors that creep into genomes become the drivers of evolution.
Some mutations manifest as diseases or beneficial traits, while others remain silent, having no discernible impact on the organism’s phenotype. Such silent players are termed neutral mutations. Despite their quiet role, they chronicle genetic histories, marking evolutionary paths as species branch out from shared ancestors.
As Louis expounds, “We have known for some time that many biological systems exhibit remarkably high phenotype robustness, without which evolution would not be possible.”
The study took a deep dive into mutational robustness—the capacity of organisms to endure mutations while preserving their characteristic traits. This phenomenon generates genetic diversity and varies across species, evident even in cellular proteins.
Louis and his team turned their focus on protein folding and RNA structures to understand how genotypic sequences lead to specific phenotypic outcomes. To achieve this, they ran intricate numerical simulations to determine how close nature could approach the maximal mutational robustness.
Their findings unveiled that mutational robustness could indeed peak in naturally occurring proteins and RNA structures. Intriguingly, the maximal robustness exhibited a self-replicating fractal pattern, the Blancmange curve, which was proportionate to a fundamental concept in number theory, the sum-of-digits fraction.
Vaibhav Mohanty, from Harvard Medical School, shared a riveting observation: “It’s as if biology knows about the fractal sums-of-digits function.” Such a revelation underscores the intimate link between the abstract world of math and the tangible realm of biology.
Nature’s myriad complexities are intriguingly bound to the realm of numbers. This recent study solidifies the idea that mathematics is not just an abstract domain. Instead, it serves as the foundation, giving structure to the universe, even at its most microscopic levels.
For those who’ve often wondered about the relevance of number theory beyond classrooms and academic papers, this is an eye-opener. It underscores the universality of math, spotlighting its significance in the most unexpected of places—our very genes.
This groundbreaking research, promising a new vista of understanding the mathematical patterns governing genetic evolution, has been featured in the Journal of The Royal Society Interface.
As we march forward in the quest for knowledge, it becomes evident that math and nature, in their dance, reveal secrets that not only enlighten but also mesmerize.
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]]>The post Board games can boost math ability in young children, study finds appeared first on The Brighter Side of News.
]]>In a world increasingly fascinated by digital learning methods, an old-fashioned form of entertainment emerges with surprising pedagogical prowess – board games. According to a thorough review of research spanning 23 years, board games that focus on numerical mechanics, like Monopoly, Othello, and Chutes and Ladders, significantly improve young children’s mathematical skills.
Board games, traditionally lauded for their value in enhancing reading and literacy skills, are now being recognized as invaluable tools in the mathematical development of children aged three to nine years. This landmark study was recently published in the reputable peer-reviewed journal Early Years.
Lead author Dr. Jaime Balladares, of the esteemed Pontificia Universidad Católica de Chile in Santiago, states, “Board games enhance mathematical abilities for young children.” He argues that these tangible, tactile games can be wielded as strategic tools to potentially impact both basic and complex mathematical aptitudes.
The design and mechanics of board games can be cleverly adapted to encompass learning objectives related to mathematical skills and other domains, making them highly versatile and effective teaching aids.
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Board games that necessitate turn-based movements of pieces across the board present a different set of cognitive challenges compared to games that focus on specific skills or gambling. The rules are fixed, thus delineating a player’s potential actions. The moves made on the board typically dictate the state of play, introducing a chess-like strategic element. Despite these clear merits, board games are scarcely employed in preschool educational environments.
Dr. Balladares and his team embarked on an ambitious mission to collate and analyze the existing evidence regarding the effect of physical board games on children’s learning, particularly their mathematical skills. The team evaluated 19 studies published since the turn of the millennium, involving children aged from three to nine years. Notably, all but one of these studies scrutinized the correlation between board games and mathematical competence.
The subjects of these studies engaged in special board game sessions held approximately twice a week for 20 minutes over a period of one and a half months. These sessions were conducted under the supervision of trained adults, including teachers, therapists, or parents.
In some studies, the child participants were divided into two groups: one playing numerical board games and the other engaged with board games not emphasizing numeracy. Some studies, conversely, had all participants play number-focused board games but allocated different games, like Dominoes.
Each child’s mathematical performance was assessed before and after these specially designed interventions aimed to enhance skills such as counting out loud. The researchers categorized their success measurements into four domains: basic numeric competency, like naming numbers; basic number comprehension, such as understanding that ‘nine is greater than three’; deepened number comprehension, which entails accurate addition and subtraction; and an overall interest in mathematics.
In certain instances, parents attended a training session to learn arithmetic that they could subsequently utilize during gameplay with their children. The results are a testament to the efficacy of board games as educational tools.
Following these game sessions, more than half (52%) of the assessed tasks demonstrated significant improvement in the children’s mathematical skills. Additionally, nearly a third (32%) of the cases showed children in the intervention groups outperforming those not participating in the board game sessions.
Surprisingly, while the previous studies had implemented board games focused on language or literacy, they lacked scientific evaluation, such as comparisons between control and intervention groups or pre- and post-intervention assessments, to substantiate their impact on the children.
Therefore, Dr. Balladares, formerly of University College London, urges the need to design and implement board games along with scientific protocols to gauge their effectiveness. According to him, this is an “urgent task to develop in the next few years.”
Following this comprehensive review, the team is now poised to investigate the potential effects of board games on other cognitive and developmental skills. Dr. Balladares postulates, “Future studies should be designed to explore the effects that these games could have on other cognitive and developmental skills.” He believes an exciting opportunity for the development and assessment of board games should be forthcoming given the complexity of games and the need to design more and superior games for educational purposes.
This study underscores the potential of play-based learning in the arena of early childhood education. It invites educators and researchers to reconsider and reimagine board games as potential conduits of crucial mathematical concepts and skills, and it opens the door to a future where the age-old tradition of board games is intertwined seamlessly with the educational fabric of our society.
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]]>What lies at the center of a black hole? A team of scientists led by Enrico Rinaldi, an American physicist at the University of Michigan, has utilized quantum computing and computer learning to decipher the mathematical description of the quantum state of the matrix model, shedding light on what may lie inside a black hole.
The study was based on the holographic principle, which postulates that the two fundamental theories of particles and gravity are equivalent to one another. However, the challenge lies in the fact that these ideas are constructed in different dimensions.
Two theories offer explanations for different dimensions, with only one distinction between them. Gravity exists in three dimensions within a black hole’s geometry, whereas particle physics resides on its surface in two dimensions, similar to a flat disk.
A black hole’s massive mass distorts space-time, creating its gravity that operates in three dimensions. This gravity links mathematically to the particles gyrating above the black hole in two dimensions. Therefore, while a black hole occupies three-dimensional space, it appears to observers as a projection of particles.
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According to some scientists, our entire universe could be a holographic depiction of particles, which might provide a consistent quantum explanation of gravity. Enrico Rinaldi explains that in Einstein’s General Relativity theory, there are no particles, only space-time, while in the Standard Model of particle physics, there are no gravitational forces, only particles. Bridging these two theories has been a persistent challenge in physics for decades.
A recent study by Rinaldi and his colleagues, published in the journal PRX Quantum, explores the use of quantum computing and deep learning to investigate holographic duality. Their research focuses on determining the lowest energy state of quantum matrix models, which are mathematical problems that can help probe the nature of this duality.
The use of quantum matrix models represents the particle theory. According to holographic duality, the mathematical events that occur in a system representing particle theory can also impact a system representing gravity. Thus, by solving a quantum matrix model, one could gain insights into gravity-related phenomena.
Rinaldi and his colleagues have utilized two matrix models, which are relatively uncomplicated to solve through conventional means but possess all of the attributes of more complex matrix models employed to describe black holes using holographic duality.
“We hope that by understanding the properties of this particle theory through the numerical experiments, we understand something about gravity,” adds Rinaldi. “Unfortunately it’s still not easy to solve the particle theories. And that’s where the computers can help us.”
In string theory, objects are represented by numerical matrix models, where one-dimensional strings correspond to particles in particle theory. The focus of researchers is to determine the specific arrangement of particles in the ground state, the lowest energy state of the system, by solving these matrix models. Unless something is added to the system that causes it to be perturbed, it remains unchanged in its natural state.
“It’s really important to understand what this ground state looks like, because then you can create things from it,” Rinaldi says. “So for a material, knowing the ground state is like knowing, for example, if it’s a conductor, or if it’s a super conductor, or if it’s really strong, or if it’s weak. But finding this ground state among all the possible states is quite a difficult task. That’s why we are using these numerical methods.”
According to Rinaldi, the numbers in matrix models can be thought of as grains of sand. When the sand is level, it corresponds to the ground state of the model. But if there are ripples in the sand, you need to find a way to smooth them out. To tackle this problem, the researchers resorted to quantum circuits. These circuits are depicted as wires, and each wire is associated with a qubit, a quantum information bit. Gates, which are quantum operations that dictate how information flows through the wires, are placed on top of the wires.
“You can read them as music, going from left to right,” the author adds. “If you read it as music, you’re basically transforming the qubits from the beginning into something new each step. But you don’t know which operations you should do as you go along, which notes to play. The shaking process will tweak all these gates to make them take the correct form such that at the end of the entire process, you reach the ground state. So you have all this music, and if you play it right, at the end, you have the ground state.”
Rinaldi and colleagues established the mathematical representation of the quantum state of their matrix model, which they referred to as the quantum wave function. Subsequently, they employed a unique neural network to determine the ground state of the matrix, which is the state with the lowest energy level. To achieve this, the neural network’s parameters underwent an iterative optimization process, aimed at leveling all the grains in the bucket of sand, until the matrix’s ground state was found.
The researchers were successful in determining the ground state of the two matrix models using both methods. However, the quantum circuits were constrained by the limited number of qubits available in current quantum hardware. With only a few dozen qubits at their disposal, increasing the complexity of the circuit would become prohibitively expensive, much like adding too many notes to a sheet of music would make it more difficult to play accurately.
“Other methods people typically use can find the energy of the ground state but not the entire structure of the wave function,” Rinaldi said. “We have shown how to get the full information about the ground state using these new emerging technologies, quantum computers and deep learning.”
“Because these matrices are one possible representation for a special type of black hole, if we know how the matrices are arranged and what their properties are, we can know, for example, what a black hole looks like on the inside. What is on the event horizon for a black hole? Where does it come from? Answering these questions would be a step towards realizing a quantum theory of gravity.”
According to Rinaldi, the findings serve as a crucial milestone for forthcoming research on quantum and machine learning algorithms. These algorithms can be applied by scholars to investigate quantum gravity through the concept of holographic duality.
In the next phase, Rinaldi, together with Nori and Hanada, is examining how these algorithms’ outcomes can expand to more extensive matrices and how immune they are to the influence of “noisy” effects, which can lead to inaccuracies.
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]]>Black holes have always been fascinating to scientists and the general public alike. These phenomena are so mysterious that even light can’t escape their gravitational pull. The space and time around black holes behave oddly, causing the curvature of space to become so intense that light rays are deflected.
This deflection has a unique property; very nearby light can be deflected so much that it travels around the black hole multiple times. As a result, when we observe a distant background galaxy or any other celestial body, we might be fortunate enough to see multiple versions of the same object.
This phenomenon, known as gravitational lensing, has been known for decades, but only recently has a new, more accurate mathematical expression been discovered.
This mathematical expression, developed by Albert Sneppen, a student at the Niels Bohr Institute, sheds light on this peculiar phenomenon and was just published in the journal Scientific Reports.
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The mechanism of gravitational lensing is shown in the figure below. A distant galaxy shines in all directions, and some of its light comes close to the black hole and is slightly deflected. Some light comes even closer, circumvolves the hole a single time before escaping down to us, and so on. As we look closer to the black hole, we see more and more versions of the same galaxy, the closer to the edge of the hole we look.
The mathematical expression that describes the phenomenon of gravitational lensing has been known for over 40 years. It is a factor of 500, which means that to see the next image, we have to look 500 times closer to the black hole than the previous image. This factor is complicated to calculate, and until recently, there was no mathematical and physical intuition as to why it happened to be this exact factor.
Albert Sneppen, using some clever mathematical tricks, has now succeeded in proving why this factor is true. “There is something fantastically beautiful in now understanding why the images repeat themselves in such an elegant way. On top of that, it provides new opportunities to test our understanding of gravity and black holes,” Albert Sneppen clarifies.
This mathematical proof is not only satisfying in itself, but it also brings us closer to an understanding of this marvelous phenomenon. The factor 500 follows directly from how black holes and gravity work, so the repetitions of the images now become a way to examine and test gravity.
Sneppen’s mathematical method can also be generalized to apply not only to “trivial” black holes but also to rotating black holes. In fact, all black holes rotate. When the black hole rotates really fast, we no longer have to get closer to the black hole by a factor of 500, but significantly less. Each image is now only 50, 5, or even down to just two times closer to the edge of the black hole.
Having to look 500 times closer to the black hole for each new image means that the images are quickly “squeezed” into one annular image, as seen in the figure on the right. In practice, the many images will be difficult to observe.
But when black holes rotate, there is more room for the “extra” images, so we can hope to confirm the theory observationally in the near future. In this way, we can learn about not just black holes, but also the galaxies behind them.
The travel time of the light increases the more times it has to go around the black hole, so the images become increasingly “delayed.” If, for example, a star explodes as a supernova in a background galaxy, one would be able to see this explosion again and again.
The implications of Sneppen’s research are vast, and it is sure to inspire a new generation of scientists to explore the mysteries of the universe using the powerful tool of gravitational lensing.
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