Mathematics
https://www.researchmatters.in/taxonomy/term/1777/all
enBridged: Gaps between mathematical methods of understanding nature
https://www.researchmatters.in/news/bridged-gaps-between-mathematical-methods-understanding-nature
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/%E0%A6%A6%E0%A7%87%E0%A6%AC%E0%A6%A6%E0%A6%A4%E0%A7%8D%E0%A6%A4-%E0%A6%AA%E0%A6%BE%E0%A6%B2%E0%A5%A4-debdutta-paul">দেবদত্ত পাল। Debdutta Paul</a></div></div></div><span class="read-time">Read time: 8 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/feynman_diagrams-debdutta1.jpg?itok=4UszQ4V2" width="800" height="450" alt="Bridged: Gaps between mathematical methods of understanding nature" title="A Feynman diagram in quantum electrodynamics, a quantum field theory. [Image Credits: Wikimedia Commons / CC BY-SA 4.0]" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: center;"><sup><span style="color:#a9a9a9;">A Feynman diagram in quantum electrodynamics, a quantum field theory. [Image Credits: <a href="https://commons.wikimedia.org/" target="_blank">Wikimedia Commons</a> / CC BY-SA 4.0]</span></sup></p>
<p style="text-align: justify;">In the first two decades of the twentieth century, Albert Einstein developed the special theory of relativity, which unifies the previously separate entities ‘space’ and ‘time’. Later, he developed the general theory of relativity, which provides a mathematical and conceptual framework of gravitation. In the next decade, theoretical physicists had developed a mathematical framework that describes the world of small things –– electrons, protons, atoms, molecules –– called ‘quantum mechanics’. Despite tremendous successes of special relativity and quantum mechanics in describing a plethora of natural phenomena, they remained incompatible in mathematical terms. In the next few years emerged the field of ‘quantum field theory’, a general mathematical framework used to describe multiple physical theories that unify the salient features of special relativity and quantum mechanics.</p>
<p style="text-align: justify;">Over the next few decades, theoretical physicists developed multiple quantum field theories with the help of advanced mathematics. These theories successfully plugged the mathematical loopholes in physical theories and predicted the existence of new particles in nature –– the ‘neutrons’ which along with protons make up the nuclei of atoms, ‘positrons’ or positively charged electrons, and the list continued. Meanwhile, experimental physicists were busy hunting for these particles, and surprisingly, finding them. Due to the interplay of theoretical and experimental physics, one discovery led to another, and ‘particle physics’ became the cool thing to be investigating. The ‘Conseil Européen pour la Recherche Nucléaire’ (CERN), or European Council for Nuclear Research, came into existence, and so did numerous other particle physics detectors around the world. In 2012, CERN finally discovered the ‘Higgs particle’ –– the final building block of the now-famous ‘Standard Model of Particle Physics’, a model that successfully accommodates multiple quantum field theories into one mathematical framework.</p>
<p style="text-align: justify;">The Standard Model of Particle Physics comes with multiple limitations –– for example, it cannot explain the well-established experimental fact that neutrinos, a particle that travels close to the speed of light, possesses mass. Gravitation, the phenomenon that Einstein explained with his general theory of relativity, does not feature in the Standard Model either. In the last few decades, theoretical physicists have worked extensively on alternate theories of the Universe that provides a quantum field theory for gravitation, the most famous among them being ‘string theory’. In the search for such theories, one aspect got largely forgotten in the last 50 years –– that quantum field theories encounter infinities.</p>
<p style="text-align: justify;">While trying to calculate the probabilities of events using quantum field theories, physicists encountered mathematical infinities that made no sense. After all, infinities do not correspond to anything tangible in nature. While a group of physicists concluded that quantum field theories were inadequate in describing nature, another group devised an alternative explanation. According to the second group, while some terms in the equations had positive infinities, others had negative infinities of the same magnitude. These infinities cancelled, they said –– there was no logical fallacy. The infinities arose because of the limitation of the mathematical method they used to calculate probabilities of physical events. However, they were not ready to divorce from the method. Called the ‘Feynman diagrams’, the mathematical method helped physicists picturise physical processes. An alternative mathematical method, called the ‘Bootstrap approach’, was neater. It had no pesky infinities nor cancellation of positive and negative infinities. However, the mathematical steps were complex, and physicists could not clearly understand the probabilities they calculated via physical processes. The Bootstrap approach quickly fell out of favour and was soon forgotten.</p>
<p style="text-align: justify;">In a new <a href="https://doi.org/10.1103/PhysRevLett.126.181601" target="_blank">study</a>, two researchers from the Indian Institute of Science (IISc), Bengaluru, have used the ‘Bootstrap approach’ of quantum field theories to explain ‘Feynman diagrams’. Professor Aninda Sinha and PhD scholar Ahmadullah Zahed carried out the research in the Centre for High Energy Physics of IISc. Published in the journal <em>Physical Review Letters</em>, the study was supported by the Department of Science and Technology, Government of India.</p>
<p style="text-align: justify;">Aninda explains why physicists did not pursue the Bootstrap approach after its formulation. “They did not know how exactly to use the complicated equations that arise from the Bootstrap approach,” he says. But the scenario has changed with the advent of better computers and improved computational algorithms. “Since 2008, new numerical methods have led to new insights into using the Bootstrap equations,” says Aninda. “From 2015, my collaborators and I have been trying to make sense of the correct way to take the Bootstrap approach,” he adds. When they investigated the technical difficulties of this approach, they found that it did not consider an important factor –– certain symmetries of nature.</p>
<p style="text-align: justify;">Symmetries are very common in nature. When a particular object is subjected to a specific change, for example, rotation –– but remains similar to before the change, it is symmetric under that change. Mathematical equations describing physical theories also exhibit symmetries, that is, they remain unchanged when subject to mathematical operations. The mathematician Amelie Emmy Noether discovered that a particular physical quantity remains conserved whenever there are symmetries in physical laws. For example, the conservation of mass-energy is related to symmetries of the mathematical equations describing nature with respect to time. The Feynman diagrams also exhibit a special kind of symmetry, the ‘crossing symmetry’, which has interesting consequences. For example, physical processes involving a couple of electrons and a couple of positrons have the same probabilities even when an electron exchanges with a positron.</p>
<p style="text-align: justify;">The crossing symmetry is an inherent property of the Feynman diagram approach of quantum field theories. However, it is a restriction that needs to be imposed mathematically on the Bootstrap method. Aninda and Ahmadullah did just that. In doing so, the calculations of the Bootstrap approach became simpler. Their equations, which relate the probabilities of different processes, started looking similar to the Feynman diagram method. They calculated some of these mathematical steps with pen on paper. For others, they used an advanced analytical software meant for automating such complicated calculations.</p>
<blockquote><p style="text-align: justify;">“We had to reinterpret conceptually as well as mathematically an older work from the 1970s as well as connect it up with current attempts over the last two years by other groups. It was quite a challenge!” shares Aninda.</p>
</blockquote>
<p style="text-align: justify;">They credit their work to calculations documented in a 1972 study by two physicists Auberson and Khuri. When they came across this paper, they found that it was hardly ever cited by other researchers. “No one knew about this paper, which is evidence that there are hidden treasures in the past,” remarks Aninda. Two or three other groups, one involving S. M. Roy, an Indian physicist at the Tata Institute of Fundamental Physics, Mumbai, had followed up on Auberson and Khuri. However, these efforts remained largely forgotten. Delays caused in the pre-email communication also contributed to a lack of coherent communication. Today, Aninda, Ahmadullah, and their colleagues from various institutions spread across India are perennially connected online.</p>
<p style="text-align: center;"><img alt="" src="/sites/default/files/feynman_diagrams-debdutta2.jpg" style="width: 100%; height:auto;" /><br /><span style="color:#a9a9a9;"><sup>The magic of complex variables enables us to see Feynman diagrams emerge “locally”. [Image Credits: Ahmadullah Zahed, an author of the study.]</sup></span></p>
<p style="text-align: justify;">The study has provided a bridge between the seemingly different approaches to quantum field theories. But, there is more to it. The Feynman diagram approach is also useful in predicting things that happen around us –– atoms do not crumble, radioactive elements decay with time, particles collide to give rise to other particles. Nobel Laureate Kenneth Wilson and his collaborators used it to study physical quantities like specific heat, the amount of heat a kilogram of water requires to be heated per unit degree Celsius rise in temperature. He had shown that it is possible to calculate how the specific heat changes with temperature, specifically at temperatures above which water cannot be liquified even after applying tremendous pressure.</p>
<p style="text-align: justify;">In an earlier work <a href="https://researchmatters.in/article/redrawing-feynman-diagrams-scientists-develop-new-tool-solve-equations-quantum-realm" target="_blank">conducted in 2017</a>, the researchers, in collaboration with Professor Rajesh Gopakumar, director of the International Centre for Theoretical Sciences (ICTS), Bengaluru, had used the Bootstrap approach to study the dependence of specific heat on temperature. “There were a couple of mathematical gaps in the broad scheme of calculation we had proposed in 2017,” says Rajesh. By invoking the crossing symmetries, they have now fixed those gaps, and it has opened up a whole new range of questions both theoretically and experimentally. This study has also been <a href="https://doi.org/10.1103/PhysRevLett.126.211602" target="_blank">published</a> in the journal <em>Physical Review Letters</em>.</p>
<p style="text-align: justify;">Changes between different phases of matter, like solids and liquids, happen primarily in two ways, explains Rajesh. How physical quantities like temperature, pressure change as the transition occurs –– determine the kind of transition. In one type, the changes are steady, while in the other, sudden. Hence, studying these physical properties becomes essential for understanding the properties of the transition. The Bootstrap approach makes mathematical predictions of these physical properties for materials in which the phase transitions are continuous. It is now up to the experimentalists to verify these predictions. Given the tremendous progress of methods in experimental physics, it might take only a few years, opines Rajesh.</p>
<blockquote><p style="text-align: justify;">“Our study shows how ideas inspired by a theory of particle physics and gravitation can play a role in explaining ordinary phenomena. It is a remarkable example of how one field of physics can influence another,” he signs off.</p>
</blockquote>
<hr />
<p style="text-align: justify;"><em>This article has been run past the researchers, whose work is covered, to ensure accuracy</em></p>
<p style="text-align: justify;"><em>Editor's Note: The aticle was edited to include a couple of hyperlinks. </em></p>
</div></div></div><div class="field field-name-field-source field-type-link-field field-label-inline clearfix"><div class="field-label">Source: </div><div class="field-items"><div class="field-item even"><a href="https://doi.org/10.1103/PhysRevLett.126.181601" target="_blank">Crossing Symmetric Dispersion Relations in Quantum Field Theories</a></div><div class="field-item odd"><a href="https://doi.org/10.1103/PhysRevLett.126.211602" target="_blank">Crossing Symmetric Dispersion Relations for Mellin Amplitudes</a></div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/iisc-bangalore" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IISc Bangalore</a></div><div class="field-item odd"><a href="/tags/iisc-bengaluru" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IISc Bengaluru</a></div><div class="field-item even"><a href="/tags/bootstrap-approach" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bootstrap approach</a></div><div class="field-item odd"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div><div class="field-item even"><a href="/tags/feynman-diagrams" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Feynman Diagrams</a></div></div></div><span property="dc:title" content="Bridged: Gaps between mathematical methods of understanding nature" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first last"><span>20397 reads</span></li>
</ul>Fri, 28 May 2021 05:10:05 +0000Research Matters2429 at https://www.researchmatters.inNature's parenting paradox: Males, not females, may gain more by caring for their young
https://www.researchmatters.in/news/natures-parenting-paradox-males-not-females-may-gain-more-caring-their-young
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/swathi-ramesh">Swathi Ramesh</a></div></div></div><span class="read-time">Read time: 6 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/bird_n_egg.jpg?itok=0BAYYix0" width="800" height="533" alt="Nature's parenting paradox. Males, not females, may gain more by caring for their young" title="A pair of Crested treeswift with its egg [Image credits: Aditya Pal CC BY-SA 4.0]" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: center;"><span style="color:#808080;"><sup>A pair of Crested treeswift with its egg [Image credits: Aditya Pal / CC BY-SA 4.0]</sup></span></p>
<p style="text-align: justify;"><em>Mathematical models show that males should be selected to care more for their offspring rather than desert them.</em></p>
<p style="text-align: justify;">Building nests, incubating eggs, feeding the young ones, protecting them from predators, teaching the required skills for life—parenting is hard work! Nevertheless, many animals do it to ensure their progeny continues through their offspring. When it comes to parenting, not all animals follow the same playbook. Among most birds, both parents are involved in caring for the young ones, while most mammals rely on the females. Then there are many fish and amphibians, where the females are not too keen on any care—they often just lay their eggs and hope for the best. In some of these species, only the males provide care, while in others both parents abandon the eggs.</p>
<p style="text-align: justify;">Why is parental care so diverse? What decides these patterns? Science has shown that parental care patterns depend on the number of offspring an animal has and their probability of surviving until they are adults. In a recent theoretical <a href="https://onlinelibrary.wiley.com/doi/abs/10.1111/evo.13969#.XuuDMr5QsW4.twitter" target="_blank">study</a>, researchers at the Indian Institute of Science, Bengaluru, the University of British Columbia, Vancouver and Indian Institute of Science Education and Research, Mohali have found some new insights on how parental care may be expected to evolve. They show that when costs and benefits are considered and the only difference assumed between the sexes is the size of the gametes they produce, male-biased parental care should be more likely. This result is in contrast with the wide-spread prevalence of female-biased care in nature. The study, published in the journal <em>Evolution</em> was funded by the Department of Science and Technology (DST).</p>
<p style="text-align: justify;">Robert Trivers, an American biologist, proposed a theory on how parental care depends on the gamete size differences between the sexes. He argued that since females produce a limited number of large eggs in their lifetime, as compared to unlimited tiny sperms produced by the males, they lose more if the offspring do not survive. Hence, females tend to invest more in parental care. Males, on the other hand, fiercely compete with each other to woo the females—a tremendous task in itself—and hence have a minimal role in raising their young ones.</p>
<blockquote><p style="text-align: justify;">"Trivers's parental investment hypothesis has been very influential, not just in the field of animal behaviour, but also in shaping the understanding of human behaviour and psychology. People have pointed out flaws in these arguments before. Still, it continues to be invoked," says Dr Priya Iyer, the lead author of the study.</p>
</blockquote>
<p style="text-align: justify;">Enthused by this hypothesis, several mathematicians tried to construct models to evaluate it. One such model was by John Maynard Smith, a British evolutionary biologist. He designed a framework based on game theory, where each player had a particular strategy that decided their actions. In his framework, mating males and females could choose to care for the young ones or desert them. Females who abandoned could lay more eggs, while abandoning males got to mate with other females. Solving this game would result in four patterns of parental care—male-biased, female-biased, biparental, or no care. However, a 2002 <a href="http://www2.nau.edu/~shuster/isopod/Pubs/WadeandShuster2002.pdf" target="_blank">study</a> found a fundamental flaw in this model—it was not self-consistent.</p>
<blockquote><p style="text-align: justify;">“Self-consistency is where each offspring has exactly one mother and one father. In a model, this means that the total number of offspring of males in a population has to be equal to that of females in the population”, points out Priya.</p>
</blockquote>
<p style="text-align: justify;">The researchers from IISc built on Maynard Smith’s model by developing many self-consistent versions that incorporate the consequences of the gamete size difference, and the trade-off of gamete production with parental care. “We wanted to explicitly incorporate the consequences of egg production being expensive and sperm production being cheap to re-examine the parental investment hypothesis,” explains Priya.</p>
<p style="text-align: justify;">They developed two sets of models, one assuming that all receptive females become available to mate at the same time (synchronous model). The other model considered that receptive females become available to mate sequentially (asynchronous model). Thus, a male deserting one clutch has the possibility to mate with an unmated female in the asynchronous model, but not in the synchronous model. Each of these models had three game frameworks.</p>
<p style="text-align: justify;">The first game of the synchronous model considered that deserting males could mate again. Since the number of females is finite, more deserting males within the population would mean fewer chances for males to mate. On the other hand, when only a few males desert, the males benefit more by mating with many females.</p>
<p style="text-align: justify;">Deserting females get to lay a second clutch of eggs, increasing the number of their offspring. Hence, the two coevolving male and female strategies led to males and females to care equally for their offspring, resulting in biparental care in the first game.</p>
<p style="text-align: justify;">The second game was similar to the first, except that, males who cared could also get to mate again, especially as sperm production is assumed to be cheap, and hence need not trade-off with male parental care. However, it can be assumed that deserting males have an advantage with these rematings. The third game assumed that females who abandon could lay more eggs in the first clutch itself and that only males who deserted could mate again. These two games gave rise to either similar results as the first model or male-biased care. It would be more beneficial for females to abandon if males cared for their offspring.</p>
<p style="text-align: justify;">In nature, seldom does mating occur synchronously. When the researchers included asynchronous mating, the first game resulted in all four patterns of parental care. The second game showed either male-biased or biparental care, and the third game resulted in more male care than the first game. Though these results differ from the synchronous models, the assumptions that go into games 2 and 3 select for more male care.</p>
<p style="text-align: justify;">The results of this study contradicts Trivers' hypothesis in that, incorporating the logical consequences of gamete size differences between sexes, male-biased parental care is the only biased pattern of care in the synchronous model. The asynchronous model also selects for more male care when these consequences are incorporated. This result is seen despite the differing costs of sperm and egg production and their trade-off with parental care. The researchers insist on the need for a better understanding of the implications of differential gamete size between sexes, coevolution of parental strategies and competition for mates to interpret the patterns of parental care observed in nature.</p>
<blockquote><p style="text-align: justify;">"We are working on two follow-up studies to this, where we also incorporate parentage and sexual selection considerations. In all of these studies, the intent is to question and extend some of the existing theories of evolution of sex roles, and attempt to explain the diversity of sex roles seen in nature," signs off Priya.</p>
</blockquote>
<hr />
<p style="text-align: justify;"><em>This article has been run past the researchers, whose work is covered, to ensure accuracy. </em></p>
</div></div></div><div class="field field-name-field-source field-type-link-field field-label-inline clearfix"><div class="field-label">Source: </div><div class="field-items"><div class="field-item even"><a href="https://onlinelibrary.wiley.com/doi/abs/10.1111/evo.13969" target="_blank">Anisogamy selects for male-biased care in self-consistent games with synchronous matings</a></div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/parenting" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Parenting</a></div><div class="field-item odd"><a href="/tags/parental-care" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">parental care</a></div><div class="field-item even"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div><div class="field-item odd"><a href="/tags/modelling" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Modelling</a></div></div></div><span property="dc:title" content="Nature's parenting paradox: Males, not females, may gain more by caring for their young" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first last"><span>8114 reads</span></li>
</ul>Thu, 10 Sep 2020 14:35:50 +0000Research Matters2187 at https://www.researchmatters.inPoets and Quants
https://www.researchmatters.in/news/poets-and-quants
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/srikanth-desikan">Srikanth Desikan</a></div></div></div><span class="read-time">Read time: 5 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/untitled_design_6.jpg?itok=ynlErkWZ" width="800" height="450" alt="Poets and Quants" title="Srinivasa Ramanujam | Wikimedia Commons | Public Domain" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;"><em>100th death anniversary of the great mathematician Srinivasa Ramanujan</em></p>
<p style="text-align: justify;">Genius is when someone’s works are so profound that they not only stand the test of time but test the truths of time. It has been 100 years today since Srinivasa Ramanujan was cruelly snatched away at a young age of 32. His works are yet to be fully deciphered.</p>
<blockquote><p style="text-align: justify;">“An equation for me has no meaning unless it expresses a thought of God”, Ramanujan once said.</p>
</blockquote>
<p style="text-align: justify;">That this devotee of Goddess Namagiri, a goddess who he claimed wrote numbers on his tongue, reached out to an avowed atheist mathematician who eventually peeled the layers of his genius is irony of the century.</p>
<p style="text-align: justify;">We use the term visionary loosely today but here was a man who had visions of the infinity, imbibed in him at birth and reinforced by his dreams and his deity.</p>
<p style="text-align: justify;">Much has been written by this South Indian genius. Ken Uno of Emory University calls him a <em>Poet</em>. Pages and pages of notations sing and dance to those who can read his mathematical music. Most humans with celebrated talent start with a problem to solve and work with proof towards the pinnacle of an answer. The genius of Ramanujan was that he had the answers not only for the now but the next as well. Many of his notes and notations point to potential solutions for which there are no questions yet.</p>
<p style="text-align: justify;">Labels of genius stick to someone like Einstein but what do you call someone who wrote about entire fields of mathematical theory that were invented much after he passed away. I say invented because one has to work from an answer to see why that answer is relevant to us mere mortals.</p>
<p style="text-align: justify;">Does all this beg the question as to how do we find and nurture the Ramanujan’s of the world? Is it nature or nurture that produces such path-breaking talent? Ramanujan would not even have been a footnote in the annals of mathematics if not for Hardy. So what makes diamonds out of coal? Is such talent a rare mutation, a once in a thousand-year marvel that cannot be produced in the labs of our local universities? Something alien?</p>
<blockquote><p style="text-align: justify;">“Ramanujan’s math was almost from outer space”, says Dr. Manjul Bhargava, a fields medal winner the equivalent of a Nobel laureate in Math. Manjul learned the Tabla at a young age and talks about the source of music as a motivation for math. He ascribes the choices of notes in music to the choice of frequencies that sound good. “…two notes sound good together (i.e., are resonant) if the ratio of their frequencies is a simple whole-number ratio, like 2:1 (which is the distance of one saptak, from sa to sa) or 3:2 (which is the distance from sa to pa)”, says Bhargava.</p>
</blockquote>
<p style="text-align: justify;">The science of music and mathematics had long been documented in Hindu culture. India or Bharat is the land of Bharatha munivar. His Natya Sastra contains an elaborate set of rules and discusses among other things the harmonic scale, a unit known as Śruti. The melodies, transformed by Bhakti or devotion, manifested in the art form of Carnatic music and is to this day celebrated and revered all over south India.</p>
<p style="text-align: justify;">It is in this tradition that Ramanujan was born into. He was surrounded by the songs of devotion in the temple town of Kumbakonam. He was a scholar and well versed in Vedic thought. He was known to keep his agraharam audiences spellbound with his commentary on the Shastras, bridging God and Math, Zero and Infinity.</p>
<p style="text-align: justify;">Could it be that these vibrations of prose and poetry stoked a latent mind to conjure miracles in math?</p>
<p style="text-align: justify;">Robert Kanigel in his epic biography on Ramanujan - <em>“The man who knew infinity”</em>, talks about a sullen, sick Ramanujan, a few months before his death, in a letter to Hardy.</p>
<p style="text-align: justify;"><em>“ I am extremely sorry for not writing you a single letter up to now”.</em></p>
<p style="text-align: justify;">He goes on to say <em>“….discovered some very interesting functions which I call “Mock” theta functions…”</em></p>
<p style="text-align: justify;">Mathematicians 80 years later would go on to apply them to our understanding about Black Holes and much is still an open book.</p>
<p style="text-align: justify;">Ramanujan’s final years of sickness made him testy to the point of erupting in anger at people around him. But he produced his best work as he suffered through his illness. Finally, on April 26, 1920, he slipped into unconsciousness and passed away soon after.</p>
<p style="text-align: justify;">The very same shastras, followed to the rule by his agraharam neighbors that Ramanujan grew up with, kept his brahmin orthodox relatives away. He had crossed the ocean and that had caused his Brahmin status to elapse.</p>
<p style="text-align: justify;">A mathematical athlete, a sprinter whose strides were cruelly cut short before the glory of a gold medal, Ramanujan was almost forgotten. 56 years into the date of his death a young student made a trip to France from Wisconsin. With spare time to visit Cambridge, he dug into papers left behind by G.N Watson. He hit gold with 140 pages of what would eventually be called the <em>“The Lost Notebook”.</em></p>
<p style="text-align: justify;">Progress in civilized societies does not appear to be linear. Less change is brought about by cultivated curriculum than by leaps of the untrained mind. The world should celebrate the native genius. Raw talent untainted by formal education should have the opportunity to blossom without the trappings of topics not essential to the core of that genius. Of what use are dates and History, the periodic table, or even particle physics to the mind that only sees numbers and patterns? A system that only recognizes nurtured talent, one that systematically climbs the ladder of elitist institutions is bound to inhibit leaps in true progress. The progress that India and the world desperately needs.</p>
<p style="text-align: justify;">We need more Poets and fewer Quants to compose a better future.</p>
<hr />
<p>Editor's note - This article has originally been published <a href="https://ohzone.substack.com/p/poets-and-quants" target="_blank">here</a>. Republished here with the permission from the <a href="https://ohzone.substack.com/people/1869291" target="_blank">author</a>. </p>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/srinivasa-ramanujam" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Srinivasa Ramanujam</a></div><div class="field-item odd"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div><div class="field-item even"><a href="/tags/100th-death-anniversary" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">100th death anniversary</a></div></div></div><span property="dc:title" content="Poets and Quants" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first last"><span>6261 reads</span></li>
</ul>Wed, 29 Apr 2020 03:51:57 +0000Research Matters2051 at https://www.researchmatters.inProf Nitin Saxena from IIT Kanpur awarded the Shanti Swarup Bhatnagar Prize 2018 for his work on algebraic circuits
https://www.researchmatters.in/news/prof-nitin-saxena-iit-kanpur-awarded-shanti-swarup-bhatnagar-prize-2018-his-work-algebraic
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/arati-halbe-%E0%A4%86%E0%A4%B0%E0%A4%A4%E0%A5%80-%E0%A4%B9%E0%A4%B3%E0%A4%AC%E0%A5%87">Arati Halbe (आरती हळबे) </a></div></div></div><span class="read-time">Read time: 4 mins | <br><br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/ssb.png?itok=Oha2Q9R7" width="800" height="450" alt="Image: Prof. Nitin Saxena, Photo credit: Mr. Girish Pant" title="Image: Prof. Nitin Saxena, Photo credit: Mr. Girish Pant" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">Prof. Nitin Saxena, Professor at the Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, has been awarded the 2018 Shanti Swarup Bhatnagar Prize for his work in Algebraic Complexity Theory. One of the youngest awardees, Prof. Saxena’s research interests include Computational Complexity and Algebraic Geometry.</p>
<p style="text-align: justify;">The Shanti Swarup Bhatnagar Prize, named after the founder of the Council of Scientific and Industrial Research (CSIR), is awarded for outstanding and notable research in the field of science and engineering. It includes Rs 5,00,000 prize money, a citation plaque and a fellowship till the age of 65.</p>
<blockquote><p style="text-align: justify;">“The award, with a long history of 61 years, has witnessed great scientists and engineers. I feel very fortunate to have become part of this illustrious gathering. It inspires me to continue working on hard problems in my field and to mentor the next generation of complexity theorists,” says Prof. Saxena about this honour. He is thankful to his family for their love and support; and teachers and friends for the motivation. “I would like to dedicate all my wins to them,” he adds.</p>
</blockquote>
<p style="text-align: justify;">Prof. Saxena’s work deals with a topic in Mathematics called Polynomial Identity Testing and certain mathematical objects called ‘algebraic circuits’. Some mathematical problems, like solving a sudoku puzzle, could have fast and accurate approaches. However, a problem like finding all possible ways to solve a sudoku puzzle of any size becomes an extremely complex one. Such complex problems can be represented by polynomials—expressions that involve variables and operations like addition, subtraction, multiplication and exponents.</p>
<p style="text-align: justify;">Polynomial Identity Testing explores if two polynomials, involving multiple variables, are equal. Algebraic methods help in determining how complicated it would be to solve this. Theoretical research in the area of Polynomial Identity Testing could potentially help in cryptography, error-correction, optimisation of algorithms and systems, and machine learning.</p>
<p style="text-align: justify;">Algebraic circuits are used to compute polynomial equations and for Polynomial Identity Testing. A problem to be solved is represented as an algebraic circuit—a graph consisting of nodes and gates to represent inputs and operations, with directed connections between them. The number of nodes and edges represent the ‘size’ of the graph and is a representative of how long a computer could take to solve the problem.</p>
<p style="text-align: justify;">Given a polynomial equation, one needs to find some circuit that can compute the polynomial, called upper bound problem, and also prove that a proposed circuit corresponds to the fastest way to solve it, known as the lower bound problem. “Often, this structural 'size' is more convenient to study than the sequential notion of 'time',” explains Prof. Saxena. A lower bound problem can be described as finding the smallest algebraic circuit corresponding to a polynomial. </p>
<p style="text-align: justify;">Simple polynomial equalities, like (X-Y) (X+Y) = X<sup>2</sup> - Y<sup>2</sup>, can be established by solving algebraically. However, when the polynomials of interest contain many variables and higher powers, the complexity increases exponentially. Finding out the exact computational complexity of Polynomial Identity Testing is one of the most important unsolved problems in this subject area. Prof. Saxena took a step forward in solving this problem through establishing that studying a very special type of circuit models would be enough to understand the properties of general circuits.</p>
<p style="text-align: justify;">In a related study, Prof. Saxena has also developed a new framework that helped to solve another unsolved problem. He and his collaborators found a solution for Polynomial Identity Testing for the model of an algebraic circuit that has very few input variables. Here, only the inputs and outputs of a circuit are known, and the exact internal connections are not known. The proposed technique to solve this problem is likely to be useful in addressing other unsolved problems in mathematics.</p>
<p style="text-align: justify;">The study of algebraic circuits also gives rise to new mathematical concepts. For example, Prof. Saxena has proved that the roots or solutions of specific ‘small’ circuits will be ‘small circuits’. The result is not obvious, as some small size circuits can compute very large polynomials, and the solutions of those can be large. He has also developed a new algorithm to find out whether a system of polynomial equations has a root which is very closely approximate but not exact. The new algorithm is orders of magnitude better than previous algorithms regarding computing resources and implementation time.</p>
<blockquote><p style="text-align: justify;">Prof. Saxena wishes to focus on other open problems in the area of algebraic circuits. “I would like to continue working towards strengthening the techniques and increasing their scope,” he concludes.</p>
</blockquote>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/nitin-saxena" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Nitin Saxena</a></div><div class="field-item odd"><a href="/tags/iit-kanpur" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IIT Kanpur</a></div><div class="field-item even"><a href="/tags/shanti-swarup-bhatnagar-prize" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Shanti Swarup Bhatnagar Prize</a></div><div class="field-item odd"><a href="/tags/csir" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">CSIR</a></div><div class="field-item even"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div></div></div><span property="dc:title" content="Prof Nitin Saxena from IIT Kanpur awarded the Shanti Swarup Bhatnagar Prize 2018 for his work on algebraic circuits" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first"><span>8759 reads</span></li>
<li class="translation_hi last"><a href="/hi/news/%E0%A4%86%E0%A4%88%E0%A4%86%E0%A4%88%E0%A4%9F%E0%A5%80-%E0%A4%95%E0%A4%BE%E0%A4%A8%E0%A4%AA%E0%A5%81%E0%A4%B0-%E0%A4%95%E0%A5%87-%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AA%E0%A4%95-%E0%A4%A8%E0%A4%BF%E0%A4%A4%E0%A4%BF%E0%A4%A8-%E0%A4%B8%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A5%87%E0%A4%A8%E0%A4%BE-%E0%A4%95%E0%A5%8B-%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF-%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%BF%E0%A4%9F-%E0%A4%AA%E0%A4%B0-%E0%A4%89%E0%A4%A8%E0%A4%95%E0%A5%87-%E0%A4%95%E0%A4%BE%E0%A4%AE-%E0%A4%95%E0%A5%87-%E0%A4%B2%E0%A4%BF%E0%A4%8F-%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%BF-%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%B0%E0%A5%81%E0%A4%AA" title="आईआईटी कानपुर के प्राध्यापक नितिन सक्सेना को बीजगणितीय सर्किट पर उनके काम के लिए शांति स्वरुप भटनागर पुरस्कार 2018 से सम्मानित किया गया।" class="translation-link" xml:lang="hi">हिन्दी</a></li>
</ul>Thu, 15 Nov 2018 13:49:20 +0000Research Matters1243 at https://www.researchmatters.inकायमाराचे रहस्य
https://www.researchmatters.in/node/1154
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/ashlesha-gore-%E0%A4%86%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%BE-%E0%A4%97%E0%A5%8B%E0%A4%B0%E0%A5%87">Ashlesha Gore (आश्लेषा गोरे) </a></div></div></div><span class="read-time">Read time: १ मिनिट<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/main_15.jpg?itok=mh_BhTiY" width="800" height="608" alt="कायमारा [Public domain], विकिमिडीया कॉमन्स वरून" title="कायमारा [Public domain], विकिमिडीया कॉमन्स वरून" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">संगीत वाद्ये, समुद्राच्या लाटा, गुरुत्वीय तरंग, अॅन्टेना आणि लंबक या सगळ्यांमध्ये सामायिक असलेला एक घटक म्हणजे या सर्वांचाच संबंध दोलनांशी (ऑसीलेशन्स) आहे. विविध दोलक एकत्र आल्यावर काय होते, हा प्रश्न महत्त्वाचा आहे. म्हणजे अनेक दोलक एकमेकांवर परिणाम करतील अशा पद्धतीने एकत्र आले तर काय होईल? भारतीय तंत्रज्ञान संस्था मुंबई येथील श्री तेजस कोतवाल यांनी चीन येथील बेहांग विद्यापीठाचे डॉ. झिन जियांग आणि नॉर्थवेस्टर्न विद्यापीठातील प्राध्यापक डॅनियल अब्राहम्स यांच्या सहकार्याने युग्मित दोलकांच्या कायमारा अवस्थेचे मूळ काय आहे याचे स्पष्टीकरण एका साध्या गणिती समीकरणाच्या आधारे <a href="https://doi.org/10.1103/PhysRevLett.119.264101" target="_blank">दिले</a> आहे.</p>
<p style="text-align: justify;">एकसारखे युग्मित दोलक (कपल्ड ऑसीलेटर्स ) असतील तर त्यांची दोलने एकतर वाट्टेल तशी होतील किंवा एकमेकांशी संपूर्णपणे सुसंगत अशी होतील, अशी एकविसाव्या शतकाच्या सुरुवातीला मोठ्या प्रमाणावर समजूत होती. मात्र नंतर असे दिसले की, काही विशिष्ट परिस्थितीत त्यातले काही दोलक लहानशा गटात एकमेकांशी सुसंगत वागतात तर काहींची दोलने कशीही होतात. जणू काही प्रत्येक गटाची एक स्वतंत्र ओळख असते. २००२ मध्ये लक्षात आलेल्या या विरोधाभासात्मक वर्तनाला “कायमारा अवस्था” असे म्हणतात. कवी होमरच्या इलियड या महाकाव्यात एकापेक्षा अधिक प्राण्यांनी बनलेल्या आणि आग ओकणाऱ्या कायमारा नावाच्या एका राक्षसी प्राण्याचे वर्णन येते. या प्राण्याचे धड सिंहाचे असून त्याच्या पाठीतून बकऱ्याचे डोके उगवल्यासारखे दिसते आणि शेपटी सापासारखी असते. त्याच्यावरूनच “कायमारा अवस्था” हे नाव पडले आहे. फिजिकल रिव्हयू लेटर्स या नियतकालिकात प्रसिद्ध झालेल्या शोधनिबंधात संशोधकांनी असे दाखवून दिले आहे की युग्मित दोलकांसाठी असलेली गणिती समीकरणे (प्रतिरूप) वापरून “कायमारा अवस्थेचे” मूळ समजून घेता येते.</p>
<p style="text-align: justify;">त्यांच्या विश्लेषणाची सुरुवात होते ती कुरामोतो मॉडेल वापरून. “१९७० पासून कुरामोतो मॉडेल अस्तित्त्वात आहे. आत्तापर्यंत हजारो संशोधकांनी त्याचा वापर करून निसर्गातील गोष्टी एकमेकांशी सुसंगत कशा असतात ते समजून घेण्याचा प्रयत्न केला आहे. काजवे एकसाथ चमकतात, रातकिडे सुरात सूर मिसळून किरकिरतात, हृदयातल्या सगळ्या पेशी एकत्र आकुंचन पावून रक्ताला वाट करून देतात इत्यादी काही उदाहरणे. हे मॉडेल आवश्यक तितके क्लिष्ट असले तरी साधे पेन आणि कागद घेऊनही सोडवता येण्यासारखे आहे.” असे या शोधनिबंधाचे सहलेखक असलेले नॉर्थवेस्टर्न विद्यापीठाचे प्राध्यापक डॅनियल अब्राहम्स म्हणतात. “इतक्या अरेषीय आणि निश्चित उकल असलेल्या” समीकरणांचा हा दुर्मिळ संयोग म्हणजे कोणत्याही सैद्धांतिक गणितज्ञाचे स्वप्न सत्यात उतरल्यासारखेच आहे.</p>
<blockquote><p style="text-align: justify;">दोलकांची नैसर्गिक वारंवारिता, त्यांच्यातील युग्मतेची बळकटी आणि त्यांच्य प्रावस्थेत (फेज) असलेला फरक अशा विविध घटकात बदल झाले असता कुरामोतो मॉडेलमध्ये काय बदल घडून येतात याचा संशोधकांनी अभ्यास केला. “या शोधनिबंधातील महत्त्वाचा मुद्दा म्हणजे कुरामोतो सुसंगत अवस्थेतून पिचफोर्क (द्विशूल) द्विभाजानाद्वारे कायमारा अवस्था गाठता येते. कायमारा अवस्था तयार होणे हा निखालसपणे सममिती बिघडण्याचा (सिमेट्री ब्रेकिंग) प्रकार आहे असे आमच्या विश्लेषणातून दिसते.” असे या शोधनिबंधाचे प्रथम लेखक, आयआयटी मुंबईचे तेजस कोतवाल सांगतात.</p>
</blockquote>
<p style="text-align: justify;">एखादी प्रणाली नियंत्रित करणाऱ्या इनपुटपैकी एखादे इनपुट सावकाश बदलत नेले असता त्या प्रणालीत अचानकपणे जो नाट्यमय बदल घडून येतो त्याचे वर्णन करण्यासाठी “द्विभाजन” ही संज्ञा गणिती अर्थाने वापरली जाते. उदाहरणार्थ, पाण्याचे तापमान सावकाश वाढवत नेले असता, ते १०० अंश सेल्सियसला पोहोचल्यावर पाण्याचे रुपांतर अचानक द्रवस्थितीतून वायुस्थितीत होते. पिचफोर्क द्विभाजानात नियंत्रण करणारे इनपुट सावकाश बदलत द्विभाजन उंबरठ्यापलीकडे नेले असता, त्या प्रणालीतील समतोलाची एक अवस्था जाऊन त्याऐवजी तिला दोन समतोल अवस्था प्राप्त होतात आणि या दोन्ही अवस्था मूळ समतोल अवस्थेपेक्षा वेगळ्या असतात. मूळ समतोल अवस्था नाहीशी होते. (म्हणजेच ती अवस्था स्थिर राहत नाही.) “युग्मित दोलकाच्या उदाहरणात, नियंत्रित करणारे घटक सावकाशपणे बदलत नेले असता एका समतोल अवस्थेकडून, म्हणजेच संपूर्णपणे सुसंगत अवस्थेकडून दोन वेगेवेगळ्या स्थिर संतुलित अवस्थांकडे, म्हणजेच दोन प्रकारच्या कायमारा अवस्थेकडे जाता येते.” असे प्राध्यापक अब्राहम्स सांगतात.</p>
<p style="text-align: justify;">‘सममिती बिघडणे’ ही संकल्पनादेखील एका साध्याशा उदाहरणाने समजून घेता येईल. अशी कल्पना करा की, एखादी पेन्सिल तिच्या टोकावर तोल साधून उभी आहे. ती एकदम संतुलित अवस्थेत असली तरीही कोणत्या ना कोणत्या बाजूला पडणारच आहे. निसर्गनियमानुसार ती एखाद्या विशिष्ट बाजूलाच पडेल असे नाही. मात्र एकदा का तिचा तोल ढळला की ती एखाद्या दिशेला पडते आणि सममिती बिघडली असे आपण म्हणतो. कायमारा अवस्था तयार होणे हा सममिती बिघडण्याचाच एक प्रकार आहे का हे याआधीच्या अभ्यासातून स्पष्ट झाले नव्हते. तसेच कायमारा अवस्था आणि संपूर्ण सुसंगत अवस्था यातील ठळक संबंधही दिसला नव्हता. मात्र, वरील अभ्यासातून कायमारा अवस्था जाणून घेणे आणि या अवस्था कुठून निर्माण होतात ते समजणे शक्य होते.</p>
<blockquote><p style="text-align: justify;">या संशोधनाचा वापर कुठे होऊ शकतो याबद्दल बोलताना श्री. कोतवाल म्हणतात, “या मॉडेलच्या मांडणीचा उपयोग रासायनिक आणि जैविक दोलकांच्या किंवा लेझर आणि यांत्रिक लंबकांच्या प्रणाली कशा चालतात हे विशद करण्यासाठी होऊ शकतो. चेतासंस्थाशास्त्रात आणि हृदयाच्या पेशींच्या हालचाली समजून घेण्यासाठी देखील या अभ्यासाचा बराच उपयोग होऊ शकतो.”</p>
</blockquote>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/iitb" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IITB</a></div><div class="field-item odd"><a href="/tags/chimera" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chimera</a></div><div class="field-item even"><a href="/tags/coupled-oscillations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Coupled Oscillations</a></div><div class="field-item odd"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div></div></div><span property="dc:title" content="कायमाराचे रहस्य" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first"><span>2934 reads</span></li>
<li class="translation_en last"><a href="/news/unravelling-secret-behind-identities-chimera" title="Unravelling the Secret behind the Identities of Chimera" class="translation-link" xml:lang="en">English</a></li>
</ul>Thu, 04 Oct 2018 02:05:40 +0000Research Matters1154 at https://www.researchmatters.inUnravelling the Secret behind the Identities of Chimera
https://www.researchmatters.in/news/unravelling-secret-behind-identities-chimera
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/vimal-simha">Vimal Simha</a></div></div></div><span class="read-time">Read time: 4 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/main_15.jpg?itok=mh_BhTiY" width="800" height="608" alt="Photo : Chimera, [Public domain], via Wikimedia Commons" title="Photo : Chimera, [Public domain], via Wikimedia Commons" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">If there is one aspect that is common to musical instruments, ocean waves, gravitational waves, antennae, and pendula, it is that all of these are related to oscillations. An important question to ask is what happens when several oscillators are coupled, i.e., they are put together in a situation where one influences another. A mathematical <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.264101" target="_blank">study</a> by Mr Tejas Kotwal, from the Indian Institute of Technology Bombay, collaborating with Dr Xin Jiang at Beihang University in China and Prof. Daniel Abrams at the Northwestern University, has found a simple mathematical explanation of the origin of a special situation known as the Chimera State of these coupled oscillators.</p>
<p style="text-align: justify;">Before the turn of the twenty-first century, it was widely believed that a collection of identical coupled oscillators, would either oscillate randomly or in complete synchrony with one another. However, it was later found that under some conditions some of them synchronise in little groups, while others swing at random, as though each of these groups has a different identity. This paradoxical behaviour, discovered in 2002, has been dubbed the “chimera state”, named after a fire-breathing monster, composed of parts of more than one animal, in Homer’s Iliad. In their study published in the journal <em>Physical Review Letters</em>, researchers have shown how we can understand the origin of the “chimera state” using known mathematical equations (model) of coupled oscillators.</p>
<blockquote><p style="text-align: justify;">The starting point of their analysis is the Kuramoto model. “The Kuramoto model has been around since the 1970s and has been used by thousands of researchers to understand why things in nature tend to synchronise with each other, such as fireflies may flash in unison, crickets may chirp in sync, cells in the heart contract together to pump blood, etc. While the model has the necessary complexity, it can also be solved using pen-and-paper”, explains Prof. Daniel Abrams from Northwestern University, who is an author of the study. Such a rare combination of “strongly nonlinear and exactly solvable” equations are a dream come true for theoretical mathematicians.”</p>
</blockquote>
<p style="text-align: justify;">The researchers studied how the Kuramoto model behaved when different parameters like the natural frequencies of the oscillators, the coupling strength between them and the phase lag between them were varied. "The key highlight of this paper is that the chimera state can be achieved via a pitchfork bifurcation off of the well-understood Kuramoto synchronised state. Our analysis reveals that the formation of the chimera state is indeed a case of symmetry breaking,” says lead author Tejas Kotwal from IIT Bombay.</p>
<p style="text-align: justify;">The term ‘bifurcation’ is used in a mathematical sense to describe a sudden, dramatic change in a system when some controlling input is gradually changed. For example, if you slowly change the temperature of water, it will suddenly go from liquid to gas when that temperature passes 100 degrees Celsius. In a pitchfork bifurcation, as you slowly change the control input beyond the bifurcation threshold, the system goes from having one equilibrium state to having two equilibria, both different from the original one. The original equilibrium behaviour disappears (it is no longer stable). “In our coupled oscillator example, as we slowly change a control parameter, we go from a single equilibrium--the fully synchronised state--to having two different stable equilibria--two types of chimera states,” explains Prof. Abrams.</p>
<p style="text-align: justify;">One can also understand ‘symmetry-breaking’ using a simple example. Imagine a pencil balanced on its tip. It is perfectly symmetrical but bound to topple one way or the other although the laws of nature do not prescribe a particular direction. But once it topples, it does so in a particular direction and we say that the symmetry has been broken. In previous works, it was unclear whether the formation of a chimera state is a case of symmetry breaking and there was no explicit connection found between the chimera state and the fully synchronised state. However, this study presents an intuitive understanding of chimera states and where they originate from.</p>
<blockquote><p style="text-align: justify;">Discussing the potential applications of this paper, Mr Kotwal says, “The formulation of this model can be used to describe the behaviour of systems of chemical and biological oscillators, lasers and mechanical pendula. It is also of widespread use in neuroscience and heart cell dynamics.”</p>
</blockquote>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/iitb" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IITB</a></div><div class="field-item odd"><a href="/tags/chimera" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chimera</a></div><div class="field-item even"><a href="/tags/coupled-oscillations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Coupled Oscillations</a></div><div class="field-item odd"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div></div></div><span property="dc:title" content="Unravelling the Secret behind the Identities of Chimera" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first"><span>4039 reads</span></li>
<li class="translation_mr last"><a href="/mr/news/%E0%A4%95%E0%A4%BE%E0%A4%AF%E0%A4%AE%E0%A4%BE%E0%A4%B0%E0%A4%BE%E0%A4%9A%E0%A5%87-%E0%A4%B0%E0%A4%B9%E0%A4%B8%E0%A5%8D%E0%A4%AF" title="कायमाराचे रहस्य" class="translation-link" xml:lang="mr">मराठी</a></li>
</ul>Mon, 06 Aug 2018 02:55:01 +0000Research Matters1049 at https://www.researchmatters.inEnsuring Enough Electricity for All
https://www.researchmatters.in/news/ensuring-enough-electricity-all
<div class="field field-name-field-op-author field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/people/arul-ganesh-s-s">Arul Ganesh S S </a></div></div></div><span class="read-time">Read time: 4 mins<br /></span><span class="submitted-by"></span><div class="field field-name-field-graphic field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img typeof="foaf:Image" src="https://www.researchmatters.in/sites/default/files/styles/large_800w_scale/public/main_14.jpg?itok=Fabh2jW2" width="800" height="500" alt="Photo : Karsog (17) by Travelling Slacker" title="Photo : Karsog (17) by Travelling Slacker" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even" property="content:encoded"><p style="text-align: justify;">Ever wondered how is it that a bulb lights the moment you switch it on? Every time we switch on an appliance, some power station burns a little more fuel to supply the energy demand from the appliance. Any imbalance in the supply and demand can lead to to a power cut or power loss. In a recent survey article published in the <em>Journal of Philosophical transactions</em>, Dr Ankur Kulkarni of Indian Institute of Technology Bombay has put together a novel theoretical framework to handle this problem. In the <a href="http://rsta.royalsocietypublishing.org/content/375/2100/20160302" target="_blank">article</a>, he has collated relevant results from game theory, a field of mathematics that studies ‘games’ as the name suggests.</p>
<p style="text-align: justify;">The power grid is a network of interconnected power lines and substations and used for distribution and transmission of electricity. Effective operation of the power grid is the key to minimise energy loss, making power grid management one of the most researched topics in energy production and distribution. A new dimension is added to these studies with the increasing deployment of renewable sources like solar and wind energy. The energy generated from these sources is unpredictable, prompting the design of new models where demand is adjusted based on supply. It is well established that renewable energy sources are the ones for the future. But effectively tapping on to their potential is no easy task. One of the hurdles is the unreliability of the sources themselves. For instance, we need not have the wind at a desired speed or sunlight at the desired intensity at a certain time of the day. A way to overcome this challenge is to adjust the consumption based on production. That is to produce and use electricity when sunlight or wind is available.</p>
<p style="text-align: justify;">Game Theory emerged during the early part of 20th century as an attempt to mathematize social sciences. In a game, players try to effectively use available resources to their advantage. Often the strategies of a player should take into consideration the fact that other players must also be devising counter strategies. Game theory is widely used to model problems involving many players, especially when each of them tries to maximise their benefit, independently or otherwise. Nash equilibrium, named after Prof. John Nash (seen the movie A Beautiful Mind?), is an important property of games. It is a condition in which no player can extract any advantage by changing their strategies unilaterally. Knowing a Nash equilibrium of a game can help us design the rules in such a way that the game leads to scenarios which are desirable for all those involved. For instance, in the case of power grid management, a desirable scenario will be characterised by optimal profit for the supplier, minimised cost for the consumers and reduced wastage of power. A Nash equilibrium is difficult to compute, especially if the game has a large number of players and possible moves and may exist only under certain conditions.</p>
<p style="text-align: justify;">In the article, citing known results in the field of game theory Dr Kulkarni demonstrates that the reversed problem of adjusting demand of power as the supply changes can be fit into a game-theoretic framework. The problem can be modelled as a game where each player (the consumer, at the demand end) is bound by constraints like the total energy available and price per unit of power dictated by the supply side. What is remarkable is that a Nash equilibrium is possible for such a system even under fairly generalised assumptions regarding the number of consumers, price, the total power available etc. Once these parameters are set in such a way that the game is in equilibrium, a steady state of ‘consumption based on supply’ emerges. This steady state makes an optimal and integrated operation of next-generation power distribution systems possible. So, in other words, it is possible to design a distribution system in which the optimal strategy for the end users would be to adjust their usage based on supply or information provided by the supplier. Any attempt at deviating from it would result in a loss for the consumer.</p>
<p style="text-align: justify;">This article proposes a theoretical framework for designing next-generation power distribution systems. This is an instance of how abstract mathematical theories can help the development of technology and accelerate progress. But putting it into practice would require fine tuning of the mathematical results and also development and design of necessary technology including one which will ensure two-way communication between demand side and supply side.</p>
</div></div></div><div class="field field-name-field-tags field-type-taxonomy-term-reference field-label-inline clearfix"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/tags/iitb" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">IITB</a></div><div class="field-item odd"><a href="/tags/dst" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">DST</a></div><div class="field-item even"><a href="/tags/electricity" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Electricity</a></div><div class="field-item odd"><a href="/tags/game-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">game theory</a></div><div class="field-item even"><a href="/tags/mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics</a></div></div></div><span property="dc:title" content="Ensuring Enough Electricity for All" class="rdf-meta element-hidden"></span><ul class="links inline"><li class="statistics_counter first"><span>3005 reads</span></li>
<li class="translation_mr"><a href="/mr/news/%E0%A4%B8%E0%A4%97%E0%A4%B3%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%B8%E0%A4%BE%E0%A4%A0%E0%A5%80-%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AA%E0%A5%8D%E0%A4%A4-%E0%A4%B5%E0%A4%BF%E0%A4%9C%E0%A5%87%E0%A4%9A%E0%A5%80-%E0%A4%A4%E0%A4%9C%E0%A4%B5%E0%A5%80%E0%A4%9C-%E0%A4%95%E0%A4%B6%E0%A5%80-%E0%A4%95%E0%A4%B0%E0%A4%BE%E0%A4%B5%E0%A5%80" title="सगळ्यांसाठी पर्याप्त विजेची तजवीज कशी करावी" class="translation-link" xml:lang="mr">मराठी</a></li>
<li class="translation_hi last"><a href="/hi/news/%E0%A4%B8%E0%A4%AD%E0%A5%80-%E0%A4%95%E0%A5%87-%E0%A4%B2%E0%A4%BF%E0%A4%8F-%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AA%E0%A5%8D%E0%A4%A4-%E0%A4%AC%E0%A4%BF%E0%A4%9C%E0%A4%B2%E0%A5%80-%E0%A4%9C%E0%A5%81%E0%A4%9F%E0%A4%BE%E0%A4%A8%E0%A4%BE-%E0%A4%8F%E0%A4%95-%E0%A4%AA%E0%A4%B9%E0%A4%B2" title="सभी के लिए पर्याप्त बिजली जुटाना - एक पहल " class="translation-link" xml:lang="hi">हिन्दी</a></li>
</ul>Tue, 31 Jul 2018 02:27:47 +0000Research Matters1040 at https://www.researchmatters.in