tag:blogger.com,1999:blog-102740842024-03-07T21:15:45.278-08:00GBGaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comBlogger129125tag:blogger.com,1999:blog-10274084.post-59568746901138419462022-06-19T09:11:00.009-07:002022-06-24T18:44:23.173-07:00Dr. Sneh Raj - Sneh Mausi - Bua - Daadi - is no more. <p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbqPB_glRJcJ0UIOzhyoxsgTqVqpSBimikd1wj__rkNTE8Dt7mLmkoJ_KxbfWzAyeydu1o9-cVuBtNzkayAohaBOa72jjaZqBm7UPfZ2Y3LowTQmpYLjZGOTg5aZxL_-zyXLwtcOcajQ93DB0bBetxUHSMHDuiJFKMCT8kb9SceUQO7PGEhA/s714/Sneh-Mausi.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="714" data-original-width="570" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbqPB_glRJcJ0UIOzhyoxsgTqVqpSBimikd1wj__rkNTE8Dt7mLmkoJ_KxbfWzAyeydu1o9-cVuBtNzkayAohaBOa72jjaZqBm7UPfZ2Y3LowTQmpYLjZGOTg5aZxL_-zyXLwtcOcajQ93DB0bBetxUHSMHDuiJFKMCT8kb9SceUQO7PGEhA/w319-h400/Sneh-Mausi.jpg" width="319" /></a></div><br /><p>My Mausi, Dr. Sneh Raj is no more. She died at age 91. She went on her own terms. She said she did not wish any tubes or artificial means of respiration (and such) and her family respected her wishes. All her life she has battled the circumstances she found herself in cheerfully and with enthusiasm. She decided when it was time not to battle any more. </p><p>She made a family and community across three countries -- India, US and Canada. </p><p>When I first landed in Columbus, she was there to pick me up. When Tejasi was born, she was the Daadi available to look after us and to teach me how to wrap Tejasi in a blanket and hold her properly. And was available in every visit to the US and every marriage in India. Her empathy, sympathy and good humor was legendary -- she shared herself generously with all of us. </p><p>On her 90th birthday, her daughter, Madhulika Raj, together with Madhulika Agarwal, made a book of wishes for her. The picture above is the cover. Click here to <a href="https://drive.google.com/file/d/1IxMenLzicWAJWxFP_6yz42QNbftXfYy6/view?usp=sharing" target="_blank">download </a>a copy. I had written there what I wanted to say to her. </p><p>Here is an <a href="https://www.beaconjournal.com/obituaries/pwoo0236234" target="_blank">obituary</a> published in the Akron Beacon Journal.</p><p>Mausi was the youngest of three sisters. After them three brothers followed. It is difficult to accept that all three sisters -- Kusum Mausi, <a href="https://www.hemaunty.in/p/hem-aunty.html" target="_blank">Mummy</a> and Sneh Mausi, left so soon after each other. All three sisters in their own way were forces of nature. Their force multiplied because of each other. The three Shaktis of this family. </p><p>Their passing is the end of an era. </p>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-85327075739977349722022-04-06T02:14:00.012-07:002022-06-19T03:58:01.396-07:00Telescoping continued fractions<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZSqNF_GscPc7SCxIogPbpN0uwpSCrpNCnRZ4GHlDsZiT3w-IRKMDOZDKce50ljFMN1r-L5wFn_QQTzrApWKbfBMJn2r7_NNsL1yUN8aEgerGAxy3LN-PnHy0pBNT3rmeQWUVhCnPwifwWZrxmtD8mESdV6t0STBvKMceZfYx8mN-bhUtIIA/s4000/Kishnan-June2022.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3000" data-original-width="4000" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZSqNF_GscPc7SCxIogPbpN0uwpSCrpNCnRZ4GHlDsZiT3w-IRKMDOZDKce50ljFMN1r-L5wFn_QQTzrApWKbfBMJn2r7_NNsL1yUN8aEgerGAxy3LN-PnHy0pBNT3rmeQWUVhCnPwifwWZrxmtD8mESdV6t0STBvKMceZfYx8mN-bhUtIIA/w400-h300/Kishnan-June2022.jpg" width="400" /></a></div><p>Krishnan Rajkumar (JNU) and I have a new preprint -- on telescoping continued fractions. I have written previously on telescoping and several times on continued fractions, but this one is unique. I don't think anyone has tried to combine the two ideas. We apply it to find lower bounds for the remainder term for Stirling's formula. Ultimately, we discovered a nice new technique, proved several things required to make it work, but were unable to take it to its natural conclusion (so far!). The preprint (now available at arxiv) has several conjectures. </p><p>The pic is taken on June 9, 2022, in the Blue Door Cafe, Khan Market where we revised the paper. (That day I showed symptoms of COVID, my second time.)</p><p>Here is a <a href="https://arxiv.org/abs/2204.00962" target="_blank">link to the preprint.</a></p><p>Here is the abstract:</p><p><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">Title: Telescoping continued fractions for the error term in Stirling's formula</span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">Authors: Gaurav Bhatnagar, Krishnan Rajkumar</span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">Categories: math.CA math.NT</span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">In this paper, we introduce telescoping continued fractions to find lower</span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">bounds for the error term $r_n$ in Stirling's approximation \[ n! =\sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.\] This improves lower bounds given earlier by<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">(2017). The expression is in terms of a continued fraction, together with an<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">algorithm to find successive terms of this continued fraction. The technique we<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">introduce allows us to experimentally obtain upper and lower bounds for a<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">sequence of convergents of a continued fraction in terms of a difference of two<span face="Arial, Helvetica, sans-serif"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">continued fractions.</span><br /></p><p>Here is a talk introducing the method. I presented in the Ashoka Math Colloquium on November 2, 2021. It has an overview of the technique. The talk was made keeping undergraduate students in mind, so there is something here which is quite accessible. In particular I have outlined Robbins' approach at the beginning of the talk.</p><p><br /></p>
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<div><br /></div><div><p style="color: #222222; font-family: helvetica; font-size: 18px; line-height: normal; margin: 0px; text-size-adjust: auto;"><br /></p><p style="color: #222222; line-height: normal; margin: 0px; text-size-adjust: auto;"><span style="font-family: inherit; font-size: x-large;"><b>Abstract</b></span></p><p style="color: #222222; line-height: normal; margin: 0px; text-size-adjust: auto;"><span style="font-family: inherit;"><br /></span></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-family: inherit;">The two Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter in 1913. As an example, here is the first Rogers-Ramanujan identity:</span></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">$$1+\frac{q}{(1-q)}+\frac{q^4}{(1-q)(1-q^2)}+\frac{q^9}{(1-q)(1-q^2)(1-q^3)}+\cdots $$</p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;">$$=\frac{1}{(1-q)(1-q^6)(1-q^{11})\dots}\times \frac{1}{(1-q^4)(1-q^{9})(1-q^{14})\dots}$$</p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><br /></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-family: inherit;">They look less forbidding when interpreted in terms of partitions, which is how MacMahon considered them. A partition of a number is a way of writing it as an unordered sum of other numbers. Unordered means that<span class="Apple-converted-space"> </span>$2+3$ and $3+2$ are considered the same. For example, </span></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-family: inherit;">$$5 = 4+1 = 3+2 = 3+1+1 = 2+1+1+1$$ </span></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-family: inherit;">are partitions<span class="Apple-converted-space"> </span>of $5$. (Two partitions of $5$ are missing in this list; can you find them?) The theory of partitions is an<span class="Apple-converted-space"> </span>attractive area of mathematics, where many complicated formulas are rendered completely obvious by<span class="Apple-converted-space"> </span>making the `right picture'. However, while each side of the Rogers--Ramanujan identities are represented<span class="Apple-converted-space"> </span>naturally in terms of partitions, they are still far from obvious.</span></p><p class="p2" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; min-height: 22px;"><span style="font-family: inherit;"><br /></span></p><p class="p1" style="font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px;"><span style="font-family: inherit;">In this talk, we will introduce partitions, explain how to enumerate them systematically, represent them graphically, and write their generating functions.<span class="Apple-converted-space"> </span>We present an experimental approach<span class="Apple-converted-space"> </span>to discover the Rogers-Ramanujan identities. This approach is due to Professor George Andrews of Penn State University.</span></p></div><div><br /></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-633013180621705272020-04-30T01:16:00.000-07:002020-05-01T00:12:52.082-07:00Lest we forget them (Edited) - By P. K. Ghosh<span style="color: #38761d; font-size: large;">An email Asoke and I sent with a link to P K Ghosh's article "Lest we forget them". I have previously written a book review of PKG's novel: <a href="https://www.gbhatnagar.com/2003/04/on-returning-to-desh.html" target="_blank">On returning to desh.</a></span><br />
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IITs of India are known worldwide as excellent academic institutions. For several decades, IIT Kanpur was the leader in this group. Those who were fortunate to have studied there, would fondly remember the ambience, the Halls of Residence, the Central Library, the canteens, the lecture hall complex...and certainly the teachers. They were great men, and sought to ignite the spark in kindred spirits among younger souls. Among them, Prof. P. K. Ghosh stands out as one who always maintained a high standard in teaching and research, and demanded the same from his students. The attached booklet gives a glimpse of his time in the Chemistry Department at IIT Kanpur, where he and his students created one of the highest resolution optical spectrometers in the country, an unusual programmable microprocessor for training, data acquisition and display, wrote several monographs which have been praised internationally, among other things. Prof. Ghosh has recounted those days in a delightful booklet "Lest We Forget Them."</div>
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Who are the people whom he doesn't want us to forget? What else does he not wish to forget? The mood. The vision. The bravado. The students who will go on to achieve great success and renown.</div>
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For details, you must read the booklet.</div>
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Here is a link: </div>
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Best wishes,</div>
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Asoke and Gaurav </div>
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PS. We would appreciate it if viewers and readers can identify themselves and comment below. Any messages to PKG will be forwarded to him along with your contact details. I will have to `approve' the comment. This is to avoid spam.<br />
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PPS. This is an edited version. Some pictures and letters have been edited from here.If you know PKG personally, he may share with you the complete article. Please send a request for the same to Asoke. </div>
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Asoke Chattopadhyay</div>
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Professor</div>
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Department of Chemistry</div>
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University of Kalyani</div>
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Kalyani 741235</div>
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(MSc integrated, IIT Kanpur) </div>
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asoke dot chattopadhyay at gmail<span style="color: #888888;"><br /></span></div>
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Gaurav Bhatnagar<br />
bhatnagarg at gmail com</div>
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Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-92202294451298472602019-10-26T22:47:00.002-07:002019-12-19T04:41:02.279-08:00Mummy's 90th birthday<br />
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My mother, Dr. Hem Bhatnagar, completed 90 years on October 28, 2019. We asked her family and friends if they wished to say something in tribute or regarding their association with her. Madhulika and I have collated them as a book, which was presented to mummy on her birthday. It was meant to be a book of wishes, but it ended up presenting a very multidimensional view of her as a person. It is also inspirational, in parts, and has some interesting historical photographs too.<br />
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<a href="https://drive.google.com/file/d/1a188PozQB_mqPOeTemqea6eKvmqrBjsn/view?usp=sharing" target="_blank">Click on this sentence to download the book</a>. It will open up as a Google Document, you can save it.<br />
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<span style="background-color: white; font-family: "roboto" , "robotodraft" , "helvetica" , "arial" , sans-serif; font-size: 14px;">The Sanskar calendar was also released on this occasion. It is made up of her paintings and writings. This will also be available for sale as the annual sanskara calendar. You can <a href="https://drive.google.com/file/d/1UgPwMzlgl0FgYDJ-BosukvLpYqcxz4aq/view?usp=sharing" target="_blank">download it here</a>. </span><br />
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As a PDF this book is available for free. We have got a small number of copies printed for a very limited distribution, only for mom's sisters and brothers. If you want a printed copy for yourself, Sanskara Society will make it available at cost. If you want a copy for yourself, please write to <span style="background-color: white; font-family: "roboto" , "robotodraft" , "helvetica" , "arial" , sans-serif; font-size: 14px;">samskar_culp@hotmail.com with your request. The cost is 700 (USD 10) plus S/H. </span><br />
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-21986123481029342382019-09-10T23:53:00.000-07:002019-10-11T10:08:57.801-07:00Thank you, Dick<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-DPsTHMhTdLk/XYCBHZDT2xI/AAAAAAAAR4Q/be5RNxfiYs01WZZWF5ffMG03H4wO33P6QCNcBGAsYHQ/s1600/DSC00035.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-DPsTHMhTdLk/XYCBHZDT2xI/AAAAAAAAR4Q/be5RNxfiYs01WZZWF5ffMG03H4wO33P6QCNcBGAsYHQ/s400/DSC00035.JPG" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">From R to L (facing camera): Ae Ja Yee, Bruce Berndt, Dick Askey, Shaun Cooper, Michael Schlosser, and me<br />
At Alladi 60 conference at a conference reception at the Alladi residence</td></tr>
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<span style="font-family: "trebuchet ms" , sans-serif;">Howard Cohl and Mourad Ismail created a <span style="background-color: white;">Liber Amicorum (Friendship Book) to present to Richard Askey. Askey is not well, and he and his wife Liz have moved into a Hospice in Wisconsin. Askey is our leader, the leader of our field, and of the people in the field. </span></span><br />
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<span style="font-family: "trebuchet ms" , sans-serif;"><span style="background-color: yellow;">UPDATE (October 9, 2019): Alex Berkovich and Howard Cohl informed that Dick is no more.</span></span><br />
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<span style="font-family: "helvetica neue" , "arial" , "helvetica" , sans-serif;"><span style="background-color: white; font-family: "trebuchet ms" , sans-serif;">My entry for his book is here: <a href="https://drive.google.com/file/d/1zTENhtugc6SJHQi3mTfM8tgxAmFbflSz/view?usp=sharing" target="_blank">Thank you, Dick</a></span></span><br />
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<span style="font-family: "helvetica neue" , "arial" , "helvetica" , sans-serif;"><span style="background-color: white; font-family: "trebuchet ms" , sans-serif;">The title is appropriate. When Dick autographed my copy of the book, Special Functions by Andrews, Askey and Roy, he wrote "Thank you for your work, early and now". (Here is my <a href="https://www.gbhatnagar.com/2010/06/q-disease.html" target="_blank">book review of this book</a>.) He gave extensive comments on receiving a draft copy of my unpublished book "<a href="http://www.gbhatnagar.com/2010/08/experience-mathematics.html" target="_blank">Experience Mathematics</a>" and tried to help me get it published. My paper "<a href="https://drive.google.com/file/d/0B1Tsyq_cbXz9TFdoaDRPRWVsZUE/view?usp=sharing" target="_blank">How to discover the Rogers-Ramanujan Identities</a>" is essentially an expansion of something that took Dick a couple of paragraphs. </span></span><br />
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<span style="font-family: "helvetica neue" , "arial" , "helvetica" , sans-serif;"><span style="background-color: white; font-family: "trebuchet ms" , sans-serif;">When I returned to mathematics, I hung out with him in many conferences, and he was very supportive, coming for my talks, making remarks. In general, he was very welcoming. I have missed him the last couple of years. </span></span>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-25280804394320578532019-08-11T21:37:00.001-07:002019-10-30T08:52:32.866-07:00On Entry II.16.12 of Ramanujan<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-mrBn6JpUeOQ/XVqezbfX3KI/AAAAAAAARzg/77EYTfv0ZX8KwvNAVelTbr2O6c017-RSgCLcBGAs/s1600/Bendt-Mug.jpeg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="710" data-original-width="737" height="308" src="https://1.bp.blogspot.com/-mrBn6JpUeOQ/XVqezbfX3KI/AAAAAAAARzg/77EYTfv0ZX8KwvNAVelTbr2O6c017-RSgCLcBGAs/s320/Bendt-Mug.jpeg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Bruce's Return gift: A daily reminder of Ramanujan</td></tr>
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In June, 2019, I attended a very inspiring conference, held at the University of Illinois at Urbana-Champaign. This was to <a href="https://math.illinois.edu/nt2019" target="_blank">celebrate Bruce Berndt's 80th birthday</a> and his retirement. Bruce is the greatest living expert on Ramanujan and one of the nicest people in the world. He is scholarly and an inspiration. His retirement means nothing, he is already traveling around the world and he said he is working on another book with Ae Ja Yee on partitions. My wishes to him were more for myself than for him: <span style="font-family: inherit;"><span style="background-color: white;">Wishing us many, many books and papers from you for many, many years!</span></span><br />
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<a href="https://arxiv.org/pdf/1908.03333.pdf" style="background-color: transparent;" target="_blank">Mourad Ismail and I have written a paper</a><span style="background-color: transparent;"> for the occasion. I presented it in Illinois in front of a small audience, which included Bruce. This is the second paper with Mourad on the subject of Ramanujan's continued fractions, using techniques he has taught me. I gave a talk on this continued fraction in 2016, in SLC 77. The talk is available here: </span><a data-saferedirecturl="https://www.google.com/url?q=https://www.mat.univie.ac.at/~slc//wpapers/s77vortrag/bhatnagar.pdf&source=gmail&ust=1572518426860000&usg=AFQjCNF8zJHqLizBknwxKuWEv850xF1WWA" href="https://www.mat.univie.ac.at/~slc//wpapers/s77vortrag/bhatnagar.pdf" style="color: #1155cc; font-family: Arial, Helvetica, sans-serif;" target="_blank">https://www.mat.univie.ac.at/~<wbr></wbr>slc//wpapers/s77vortrag/<wbr></wbr>bhatnagar.pdf</a>. Some of this material is included in the paper with Mourad. </div>
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Here is a picture from the conference. (More pictures below)<br />
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<a href="https://1.bp.blogspot.com/-GhOyphpPIyc/XVEhmYQRIkI/AAAAAAAARvU/36WBzjNCL_484oOLqZFdx4RS1G2fxvgIgCLcBGAs/s1600/Berndt-group-photo2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-GhOyphpPIyc/XVEhmYQRIkI/AAAAAAAARvU/36WBzjNCL_484oOLqZFdx4RS1G2fxvgIgCLcBGAs/s400/Berndt-group-photo2.jpg" width="400" /></a></div>
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Picture in front of Altgeld Hall, the iconic department of mathematics, UIUC</div>
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Here is the announcement of the paper from ArXiv. Click on the link in the title to get to the preprint on ArXiv.<br />
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<span style="font-family: inherit;"><span style="background-color: white; color: #222222;">Title: <a href="https://arxiv.org/pdf/1908.03333.pdf" target="_blank">On Entry II.16.12: A continued fraction of Ramanujan</a></span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Authors: Gaurav Bhatnagar and Mourad E. H. Ismail</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Categories: math.CA</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Comments: 15 Pages</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">MSC-class: 33D45 (Primary), 30B70 (Secondary)</span></span></blockquote>
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Abstract:</blockquote>
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<span style="font-family: inherit;"><span style="background-color: white; color: #222222;">We study a continued fraction due to Ramanujan, that he recorded as Entry 12</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">in Chapter 16 of his second notebook. It is presented in Part III of Berndt's</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">volumes on Ramanujan's notebooks. </span><span style="background-color: white; color: #222222;">We give two alternate approaches to proving Ramanujan's Entry 12, one using a </span><span style="background-color: white; color: #222222;">method of Euler, and another using the theory of orthogonal polynomials. We </span><span style="background-color: white; color: #222222;">consider a natural generalization of Entry 12 suggested by the theory of </span><span style="background-color: white; color: #222222;">orthogonal polynom</span></span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">ials.</span></span></blockquote>
Here is a picture taken at the Banquet.<br />
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<a href="https://1.bp.blogspot.com/-pKtK3mD_FqE/XVqbBQOl9sI/AAAAAAAARzU/NZxbybBGeHI7CGKZX6Y00xe_IsVebgvNwCLcBGAs/s1600/Berndt80-MS-GB-BCB.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-pKtK3mD_FqE/XVqbBQOl9sI/AAAAAAAARzU/NZxbybBGeHI7CGKZX6Y00xe_IsVebgvNwCLcBGAs/s400/Berndt80-MS-GB-BCB.jpg" width="400" /></a></div>
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With Bruce Berndt and Michael Schlosser in the Banquet Hall</div>
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Here is a picture from Bruce's office, which I saw courtesy Atul Dixit who had the keys.<br />
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<a href="https://1.bp.blogspot.com/-CnWueAjHzoQ/XVqgIuCYZ-I/AAAAAAAARzs/I3QG9_pDz60KFhzhA7DhZmyC1sGs-_bbQCLcBGAs/s1600/Berndt-Office.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1200" height="320" src="https://1.bp.blogspot.com/-CnWueAjHzoQ/XVqgIuCYZ-I/AAAAAAAARzs/I3QG9_pDz60KFhzhA7DhZmyC1sGs-_bbQCLcBGAs/s320/Berndt-Office.jpg" width="240" /></a></div>
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A picture of some of the pictures in Bruce's office</div>
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I presented this work in the OPSFA2019 conference in Hagenberg, Austria. Here are the <a href="https://www3.risc.jku.at/conferences/opsfa2019/talk/bhat.pdf" target="_blank">slides</a> from the talk in Hagenberg. OPSFA2019 was an amazing conference. I got a chance to hear Christian's concert on the church organ, saw and heard Chihara, met Alan Sokal (among others), hung out with Michael and Hjalmar and worked with Mourad on this paper and on our next joint paper. Here is a picture from Hagenberg.<br />
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<a href="https://1.bp.blogspot.com/-Uf8dvGaCgTg/XVEii6nAvjI/AAAAAAAARvg/XIlMLGbj2FIKTmnlp5mHJ0Z4TRIj9NmCQCLcBGAs/s1600/OPSFA_group-photo_small.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="474" data-original-width="980" height="191" src="https://1.bp.blogspot.com/-Uf8dvGaCgTg/XVEii6nAvjI/AAAAAAAARvg/XIlMLGbj2FIKTmnlp5mHJ0Z4TRIj9NmCQCLcBGAs/s400/OPSFA_group-photo_small.jpg" width="400" /></a></div>
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The group photo from OPSFA</div>
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Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-91917187879404109502019-03-24T19:59:00.003-07:002021-07-10T19:11:41.163-07:00Prime number conjectures from the Shapiro class structureThe first of hopefully many joint projects with my childhood friend Hartosh Singh Bal. For many reasons this has been a most exciting collaboration. For one thing, Hartosh and I have been discussing mathematical ideas since Class 11 in Modern School. So it was good to work on something which will lead to something new. For another, Shapiro was Hartosh's number theory Professor at NYU. And for three more reasons, you will have to look at the last section of this paper.<br />
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Here is the abstract:<br />
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The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest positive integer $i$ such that the $i$th iterate of Euler's totient function at $n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all numbers at a height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function. </blockquote>
Here is a link to a reprint of the paper.<br />
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<a href="http://math.colgate.edu/~integers/u11/u11.pdf" style="color: #3d85c6; font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 13.2px; text-decoration: none;" target="_blank">Prime number conjectures from the Shapiro class structure</a><span face=""trebuchet ms" , "trebuchet" , sans-serif" style="color: #222222; font-size: 13.2px;"> (with Hartosh Singh Bal), </span><br />
<span face=""trebuchet ms" , "trebuchet" , sans-serif" style="color: #222222; font-size: 13.2px;"><span style="background-color: yellow;">UPDATE (Feb 14, 2020)</span>: The paper has appeared in <a href="http://math.colgate.edu/~integers/" target="_blank">INTEGERS: Electronic Journal of Combinatorial Number Theory</a> (Volume 20), #A11, 23pp.</span><br />
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<a href="https://3.bp.blogspot.com/-DQkINV35uMA/XJhD6r5IGzI/AAAAAAAAQqQ/9-C2TdV9oz0qTcnFTAujyYYxOPYAHO78gCLcBGAs/s1600/HartoshPunyaSonitGB.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="960" data-original-width="660" height="640" src="https://3.bp.blogspot.com/-DQkINV35uMA/XJhD6r5IGzI/AAAAAAAAQqQ/9-C2TdV9oz0qTcnFTAujyYYxOPYAHO78gCLcBGAs/s640/HartoshPunyaSonitGB.jpg" width="440" /></a></div>
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From left to right: Sonit, Hartosh, me, Punya in 1983 or so</div>
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See here for <a href="https://www.gbhatnagar.com/2014/01/of-art-and-math-new-series-of-articles.html" target="_blank">my collaboration with Punya.</a></div>
<span face=""trebuchet ms" , "trebuchet" , sans-serif" style="color: #222222; font-size: 13.2px;"><br /></span>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-37509080865549290862019-02-12T13:04:00.002-08:002019-09-16T09:24:58.026-07:00An infinite family of Borwein-type + - - conjecturesAnother collaboration with Michael Schlosser written to celebrate the 80th birthday of Professor George Andrews. The paper's opening paragraph talks about a very interesting event that took place in a conference celebrating George's birthday in June 2018.<br />
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The so-called Borwein conjectures, due to Peter Borwein (circa 1990), were popularized by Andrews. The first of these concerns the expansion of finite products of the form<br />
$$(1-q)(1-q^2)(1-q^4)(1-q^5)(1-q^7)(1-q^8)\cdots$$<br />
into a power series in $q$ and the sign pattern displayed by the coefficients. In June 2018, in a conference at Penn State celebrating Andrews' 80th birthday, Chen Wang, a young Ph.D. student studying at the University of Vienna, announced that he has vanquished the first of the Borwein conjectures. In this paper, we propose another set of Borwein-type conjectures. The conjectures here are consistent with the first two Borwein conjectures as well as what is known about their refinement proposed by Andrews. At the same time, they do not appear to be very far from these conjectures in form and content.</blockquote>
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Our first conjecture considers products of the form<br />
$$<br />
\prod_{i=0}^{n-1} (1-q^{3i+1}) (1-q^{3i+2})<br />
\prod_{j=1}^m \prod_{i=-n}^{n-1} (1-p^jq^{3i+1})(1- p^jq^{3i+2})<br />
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These are motivated by theta products. </blockquote>
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Here is a link to a preprint of the paper.<br />
<a href="https://arxiv.org/pdf/1902.04447.pdf" target="_blank">A partial theta function Borwein conjecture</a>, by Gaurav Bhatnagar and Michael Schlosser.<br />
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<span style="background-color: yellow;">UPDATE (September 16, 2019). </span>The paper has been accepted to appear in the Andrews 80 Special Issue in the Annals of Combinatorics.<br />
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Here is a picture from a trip to Hong Kong for an OPSF meeting in June 2017. From left to right: Heng Huat Chan (Singapore), Michael Schlosser (Vienna), Hjalmar Rosengren (Sweden), Shaun Cooper (New Zealand), me. A special team of Special Functions people from around the world!<br />
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<a href="https://2.bp.blogspot.com/-RroysoWhPME/XGM0LQtD8iI/AAAAAAAAQg4/tJNKm8WbXsUymlAeuOjTDQE6Amm7OSDmACLcBGAs/s1600/q-gang2-HK2017.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://2.bp.blogspot.com/-RroysoWhPME/XGM0LQtD8iI/AAAAAAAAQg4/tJNKm8WbXsUymlAeuOjTDQE6Amm7OSDmACLcBGAs/s400/q-gang2-HK2017.jpg" width="400" /></a></div>
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-69821118547878183942019-01-31T07:38:00.001-08:002019-07-15T10:34:03.899-07:00Orthogonal polynomials associated with continued fractionsMy first joint paper with Professor Mourad Ismail. This has been a most interesting collaboration. Mourad taught me what to do on the sidelines of a series of meetings around the world. Most of these meetings were organized by the OPSF activity group of SIAM, one of the most interesting and diverse group of mathematicians and physicists. It began with a couple of meetings in Maryland in July 2016. The next one was in Hong Kong in July 2017, followed by a week long visit of Mourad to Austria (Oct 2017), where he gave me an exclusive, one-on-one, tutorial. Next we met in a summer school on $q$-series in Tianjin university in July-Aug 2018. Finally, we finished up things when I met him in Baltimore at the joint AMS meetings (Jan 2019), followed by a weekend trip to Orlando, right after visiting UF in Gainesville. This work was presented in Baltimore in a special session on continued fractions.<br />
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I hope it is the first in a series on Orthogonal Polynomials. There is much to learn and much to do.<br />
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<a href="https://arxiv.org/pdf/1901.09985.pdf" target="_blank">Here is a link to the preprint on ArXiv.</a><br />
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<b>Orthogonal polynomials associated with a continued fraction of Hirschhorn</b></div>
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Gaurav Bhatnagar and Mourad E. H. Ismail</div>
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<b>Abstract</b><br />
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We study orthogonal polynomials associated with a continued fraction due to Hirschhorn.<br />
Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of<br />
polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan's continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.<br />
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Here is a picture of Mourad with me in Tianjin (July-Aug 2018). <br />
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<a href="https://1.bp.blogspot.com/-Tlv4d60v7lQ/XIykB7Q0ddI/AAAAAAAAQoU/_y-0NNV1qQsFbN6mfqANsJd1-APftb6iwCEwYBhgL/s1600/GB-Mourad.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="" border="0" data-original-height="1068" data-original-width="1600" height="266" src="https://1.bp.blogspot.com/-Tlv4d60v7lQ/XIykB7Q0ddI/AAAAAAAAQoU/_y-0NNV1qQsFbN6mfqANsJd1-APftb6iwCEwYBhgL/s400/GB-Mourad.JPG" title="" width="400" /></a></div>
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The picture below is the conference group photo from Hong Kong (July 2017).<br />
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Mourad is seated in the front row second from the left. Many of the leading lights of the OPSF world are in this picture.<br />
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-39348484414207618232018-11-16T12:37:00.000-08:002019-03-05T22:08:25.305-08:00A bibasic Heine transformation formula<div style="text-align: justify;">
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While studying chapter 1 of Andrews and Berndt's Lost Notebook, Part II, I stumbled upon a bibasic Heine's transformation. A special case is Heine's 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I'm happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan's own $_2\phi_1$ transformations.</div>
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And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.</div>
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Here is a <a href="http://arxiv.org/abs/1606.05460" target="_blank">preprint</a>.<br />
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Here is a <a href="https://www.youtube.com/watch?v=E7vQ34ERh2E" target="_blank">video</a> of a talk I presented at the <a href="http://www.qseries.org/fgarvan/alladi60.html" target="_blank">Alladi 60 Conference</a>. March 17-21, 2016.</div>
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<span style="background-color: #ffe599;">Update (November 10, 2018).</span> The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine's method and $A_n$ to $A_m$ transformation formulas.<br />
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<span style="font-family: inherit;"><span style="color: #222222; white-space: nowrap;">Here is a </span><a href="https://drive.google.com/file/d/1yBDPJBgJULRt7fFecJON5i7ewCKWZSZ4/view?usp=sharing" target="_blank">reprint.</a></span><br />
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<span style="background-color: #ffe599;">UPDATE </span>(Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations, <i>in </i>Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, Mar 2016, G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)</div>
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The book is available <a href="https://link.springer.com/book/10.1007%2F978-3-319-68376-8" target="_blank">here.</a> The <a href="https://link.springer.com/content/pdf/bfm%3A978-3-319-68376-8%2F1.pdf" target="_blank">front matter</a> from the Springer site.</div>
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<span style="background-color: white;">UPDATE (June 16, 2016). </span><span style="background-color: white;"> The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.)</span></div>
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-48083812865312072312018-08-27T01:06:00.000-07:002018-08-29T06:22:08.700-07:00Mathematics and Life: A Speech<div style="text-align: left;">
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<a href="https://3.bp.blogspot.com/-kRJlpBFFVvk/W4acFislvdI/AAAAAAAAPfs/5hZ7MkCf61cg3Juu0rCe4JJwAMSXMYlBACLcBGAs/s1600/Millennium-26-8-2018.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="960" height="80%" src="https://3.bp.blogspot.com/-kRJlpBFFVvk/W4acFislvdI/AAAAAAAAPfs/5hZ7MkCf61cg3Juu0rCe4JJwAMSXMYlBACLcBGAs/s320/Millennium-26-8-2018.jpg" width="80%" /></a></div>
<br />
<br />
On August 27, 2018, I was invited by the Millennium School, Noida to their investiture ceremony. I have previously taught mathematics to Class 11 students in another campus of the school. At that time, I instituted a "Mathguru Prize" for one or two students who did well in mathematics in Grade 10. (The first winner was my student, Ayush Tripathi, who was in the first graduating batch of the school). Every year I buy some books to be given to the winner, with a copy for the school library. </div>
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Here is a speech I gave at the occasion (with some editing). </div>
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***</div>
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<br /></div>
<div style="text-align: left;">
I am a mathematician, so I speak here only in terms of mathematics. Today we will be awarding the Mathguru prize to two very bright students. The first thing I wish to tell you is something they know very well.</div>
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<br /></div>
<div style="text-align: left;">
To get 100% in math, you have to do two things. </div>
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</div>
<ol>
<li style="text-align: left;">Solve all the problems in the book</li>
<li style="text-align: left;">Write the solutions in a way that others can understand it. Even you should be able to understand what you have written if you read the solution after 6 months. </li>
</ol>
<div style="text-align: left;">
The second thing I wish to tell you is something which Professor Littlewood said. Littlewood was a famous mathematician, who played a big part in Ramanujan's life. He said that if you are trying to solve a really hard problem, then you may not make much progress in a year or two. But you will certainly make a lot of progress in 10 years or so.</div>
<br />
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Keep this in mind when choosing problems to solve. Know that even if the problem is very tough, if you keep at it for years, you will make a lot of progress.</div>
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<br /></div>
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Finally, the third thing I wish to share are some words of George Polya, another famous mathematician. Polya said:</div>
<blockquote class="tr_bq" style="text-align: left;">
<span style="font-family: inherit;">Beauty in mathematics is seeing the truth without effort.</span></blockquote>
<div style="text-align: left;">
So one must aspire to understand things so well, that we can see the beauty of it without any effort. The same goes when we are presenting something that we have understood. </div>
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I speak in terms of mathematics, but I speak not only of mathematics. Much of what I said is applicable in other domains of life.</div>
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Best wishes and good luck to all of you, as you pursue your aspirations.<br />
<br />
<div style="text-align: center;">
***</div>
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PS. I may have been influenced a bit in the way I phrased certain things by a book I just finished reading for the nth time. The book is called Shibumi, written by Trevanian. </div>
Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-70539861669830713672018-07-08T00:01:00.002-07:002019-03-05T22:10:59.140-08:00The determinant of an elliptic, Sylvesteresque matrixMy second determinant project with Christian Krattenthaler.<br />
<br />
The determinant of the Sylvester matrix corresponding to the polynomials <br />
\[<br />
x^2+2s_1x+s_1^2 = (x+s_1)^2<br />
\]<br />
and<br />
\[<br />
x^3+3 s_2 x^2 +3s_2^2 x + s_2^3 = (x+s_2)^3<br />
\]<br />
is given by<br />
\[<br />
\det<br />
\begin{pmatrix}<br />
1 & 2s_1 & s_1^2 & 0 & 0\\<br />
0 & 1& 2s_1 & s_1^2 & 0\\<br />
0 & 0 &1 & 2s_1 & s_1^2 \\<br />
1 & 3s_2 & 3s_2^2 & s_2^3 & 0\\<br />
0& 1 & 3s_2 & 3s_2^2 & s_2^3 \\<br />
\end{pmatrix}<br />
= (s_1-s_2)^6.<br />
\]<br />
<br />
The determinant is $0$ when $s_1$ and $s_2$ are both $1$. In general, if the determinant of a Sylvester matrix is $0$, then this indicates that the two polynomials have a common root. <br />
<br />
Here is an abstract of our paper.<br />
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<blockquote class="tr_bq">
<div style="text-align: justify;">
We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. In particular, it generalizes the determinant evaluation above. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaev's $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson. </div>
</blockquote>
<div>
<br /></div>
This paper has been published in Sigma. Here is a link:<br />
<a href="https://www.emis.de/journals/SIGMA/2018/052/" style="color: #3d85c6; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13.199999809265137px; text-decoration: none;" target="_blank">The determinant of an elliptic Sylvesteresque matrix</a><span style="color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13.199999809265137px;"> (with Christian Krattenthaler), SIGMA, </span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13.199999809265137px;">14</b><span style="color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13.199999809265137px;"> (2018), 052, 15pp.</span><br />
<br />
I presented this paper in <a href="http://personal.psu.edu/jxs23/gea80/" target="_blank">Combinatory Analysis 2018</a>, a conference in honor of George Andrews' 80th birthday conference. Here is a picture from Andrews' talk. (The picture inside the picture is of Freeman J. Dyson.)<br />
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<a href="https://1.bp.blogspot.com/-TlbBEU--jos/W0G0p9S7-sI/AAAAAAAAPUM/IetRkxnsNCkLpvup74JicSZiNtFIepDfwCLcBGAs/s1600/P_20180624_112208_vHDR_On.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="200" src="https://1.bp.blogspot.com/-TlbBEU--jos/W0G0p9S7-sI/AAAAAAAAPUM/IetRkxnsNCkLpvup74JicSZiNtFIepDfwCLcBGAs/s400/P_20180624_112208_vHDR_On.jpg" width="400" /></a></div>
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Next I expect to present the same paper in a Summer Research Institute on $q$-series in the University of Tianjin, China.</div>
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A long version (with lots of background information) was presented in our "Arbeitsgemeinschaft "Diskrete Mathematik" (working group in Discrete Mathematics) Seminar, TU-Wien and Uni Wien, on Tuesday, June 5, 2018. </div>
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-59439341974311671362017-06-15T08:00:00.000-07:002017-12-05T22:53:13.515-08:00How to discover the exponential function<script type="text/x-mathjax-config">
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Another article on the "How to discover/guess/prove/..." series written for a high school audience. The basic idea is to find a function whose derivative is itself, and to find the power series which satisfies this. Then messing with it to guess it must be the exponential function. No proofs, in fact, it is outrageously un-rigourous. I hope the editor allows it.<br />
<br />
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
I try to include only the most beautiful items, and state facts which I feel every high school student should know, even if they doesn't appear formally in the syllabus. </div>
<div>
<br />
<span style="background-color: yellow;">Update (Nov 2017).</span> The article was published in the <a href="http://azimpremjiuniversity.edu.in/SitePages/resources-at-right-angles-november-2017.aspx" target="_blank">November issue of At Right Angles.</a> A nice surprise was Shailesh Shirali's companion <a href="http://azimpremjiuniversity.edu.in/SitePages/resources-ara-november-2017-exponential-series-addendum.aspx" target="_blank">article</a> which gives some graphical intuition to complement the algebraic computations in my article. Here is the link to a<a href="https://drive.google.com/file/d/1g9rIKQaasYJuD1Obq3uHWl3Miuj8kmZ-/view?usp=sharing" target="_blank"> reprint</a>. </div>
<br />
<b>Abstract</b><br />
<b><br /></b>
If a function is such that its derivative is the function itself, then what would it be? Some interesting mathematical objects appear while trying to answer this question, including a power series, the irrational number $e$ and the exponential function $e^x$. The article ends with a beautiful formula that connects $e$, $\pi$, the complex number $i=\sqrt{-1}$, $1$ and $0$.<br />
<br />
<span style="background-color: #ffd966;">Update: 15/June/2017.</span> I was wondering what happened to this article, and the editor said he had sent some comments from the referee which were yet to be incorporated. I resent the article after incorporating the referee's comments, and now this article is slated to appear in the November issue of At Right Angles. Time to think about the next article in the series.<br />
<br />
Here is a link to the updated <a href="https://drive.google.com/file/d/0B1Tsyq_cbXz9ZktsMUtuMTFlVFE/edit?usp=sharing" target="_blank">preprint</a>. Please do give comments.Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-23631335389910783952017-05-09T09:53:00.003-07:002019-03-05T22:09:55.728-08:00WP Bailey Lemmas (Elliptic, multivariable)After many many years, Michael Schlosser and I wrote another joint paper. We first collaborated in 1995-96 when both of us were Ph.D. students or shortly thereafter. Our joint work was part of his thesis, and published in Constructive Approximation. This time around, I was his post-doc in Vienna from Feb 1, 2016 to Feb 28, 2017.<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-znTGNOT6aB8/WRHwf-VTw-I/AAAAAAAALBI/MveVHHpr5do1ofqo-YL4qsYr7ODYwU9fgCLcB/s1600/MS-GB-Strobl-Nov2016.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-znTGNOT6aB8/WRHwf-VTw-I/AAAAAAAALBI/MveVHHpr5do1ofqo-YL4qsYr7ODYwU9fgCLcB/s640/MS-GB-Strobl-Nov2016.jpg" width="90%" /></a></div>
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The picture was taken in Strobl, a favorite place for small meetings and conferences for Krattenthaler's group in the University of Vienna.<br />
<br />
In this paper, we give multivariable extensions (over root systems) of the elliptic well-poised (WP) Bailey Transform and Lemma. In the classical (i.e. dimension = 1) case, this work was done by Spiridonov, who in turn extended the work of Andrews and Bailey. It is Andrews' exposition which we found very useful while finding generalizations. We used previous $q$-Dougall summations due to Rosengren, and Rosengren and Schlosser, and found a few of our own along with some new elliptic Bailey $_{10}\phi_9$ transformation formulas, extending some fundamental formulas given in the classical case by Frenkel and Turaev in 1997. Along the way, we discovered a nice trick to generalize the theorem of my advisor, Steve Milne, that I had named "Fundamental Theorem of $U(n)$ series" in my thesis.<br />
<br />
<span style="font-family: inherit;">Hopefully, there will be many more collaborative ventures in the near future.</span><br />
<br />
<span style="font-family: inherit;"><span style="background-color: yellow; color: #222222;"><b>Update (Mar 22, 2018):</b></span><span style="color: #222222;"> <span style="font-family: inherit;">The paper has been published. Here is the reference and Link</span></span><span style="color: #222222; font-family: inherit;">:</span></span><br />
<span style="font-family: inherit;">G. Bhatnagar and M.J. Schlosser, <a href="https://www.emis.de/journals/SIGMA/2018/025/" target="_blank">Elliptic well-poised Bailey transforms and lemmas on root systems</a>, SIGMA, <b>14</b> (2018), 025, 44pp.</span><br />
<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-24832111425255061662017-04-07T01:21:00.000-07:002019-11-10T17:58:11.584-08:00Spiral Determinants<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-dyudLHkjnJM/WOdPUWI1nHI/AAAAAAAAKls/HSCg7qgSO0kTBHlOxdEC8FREIUJL8UDTwCLcB/s1600/krattenthaler-lecture-2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-dyudLHkjnJM/WOdPUWI1nHI/AAAAAAAAKls/HSCg7qgSO0kTBHlOxdEC8FREIUJL8UDTwCLcB/s400/krattenthaler-lecture-2.jpg" width="90%" /></a></div>
<br />
<br />
<br />
We consider Spiral Determinants of the kind<br />
$$\text{det}\left(<br />
\begin{matrix}<br />
{16}&{15}&{14}&{13}\\<br />
{5}&{4}&{3}&{12}\\<br />
{6}&1&{2}&{11}\\<br />
{7}&{8}&{9}&{10}<br />
\end{matrix}<br />
\right)<br />
$$<br />
and<br />
$$\text{det}<br />
\left(<br />
\begin{matrix}<br />
{17}&{16}&{15}&{14}&{13}\\<br />
{18}&{5}&{4}&{3}&{12}\\<br />
{19}&{6}&1&{2}&{11}\\<br />
{20}&{7}&{8}&{9}&{10}\\<br />
{21}&{22}&{23}&{24}&{25}<br />
\end{matrix}<br />
\right)<br />
$$<br />
<div>
where the entries spiral out from the center. Christian Krattenthaler, who is one of the greatest experts on determinants, tells the story of how he came across such determinants and how he went about discovering the formulas for such determinants. The preprint is available on <a href="https://arxiv.org/pdf/1704.02859.pdf" target="_blank">arxiv</a>.<br />
<br />
I have wanted to work with Christian ever since my Ph.D. days, when I tried to generalize a matrix inversion due to him. Finally, we have a joint paper. This also means that my Erdos number has come down from 4 to 3. <br />
<br />
<div>
The picture above is from Christian's course on "Bijections" which I had an opportunity to attend in the University of Vienna during the period October 2016 to January 2017.<br />
<br />
<span style="background-color: yellow;">Update: April 26, 2017 </span><span style="background-color: white;"> The paper has been accepted and will appear in Linear Algebra and its Applications. Here is a preprint</span> on <a href="https://arxiv.org/pdf/1704.02859.pdf" target="_blank">arxiv</a>.<br />
<span style="background-color: yellow;"><span style="background-color: yellow;">Update: May 10, 2017</span><span style="background-color: white;">.</span><span style="background-color: white;"> </span></span><span style="background-color: white;">The paper is published online. The reference is:</span><br />
<span style="background-color: white;">G. Bhatnagar and C. Krattenthaler, Spiral Determinants, Linear Algebra Appl., <b>529</b> (2017) 374-390.</span><br />
Here is a link to the publisher's site: <a href="https://www.sciencedirect.com/science/article/pii/S0024379517302719">https://www.sciencedirect.com/science/article/pii/S0024379517302719</a><br />
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Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-32776291299109646082016-02-14T08:57:00.001-08:002019-03-05T22:10:59.114-08:00Analogues of a Fibonacci-Lucas IdentityRecently, in 2014, Sury published a Fibonacci-Lucas identity in the Monthly. It turned out that the identity had appeared earlier (as Identity 236 in Benjamin and Quinn's book: <i>Proofs that count: The art of combinatorial proof</i>). When I tried to prove it using my usual telescoping method, I found its connection with one of the oldest Fibonacci identities due to Lucas in 1876. I also found many generalizations and analogous identities for other Fibonacci type sequences and polynomials. This small paper has been accepted in the Fibonacci Quarterly.<br />
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Here is a link to a preprint: <a href="http://arxiv.org/pdf/1510.03159v3.pdf" target="_blank">Analogues of a Fibonacci-Lucas Identity</a><br />
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Update: Its has appeared. The ref is: <span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px;">Analogues of a Fibonacci-Lucas identity, Fibonacci Quart., </span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px;">54 (no. 2)</b><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px;">, </span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px;"> </b><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px;">166-171</span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px;">,</b><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px;"> (2016)</span><br />
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I use the approach of my earlier paper on Telescoping: <a href="http://www.combinatorics.org/Volume_18/PDF/v18i2p13.pdf" target="_blank">In praise of an elementary identity of Euler.</a><br />
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I am pleased, because I have thought of getting a paper in the Fibonacci Quarterly since I was in high school, and feel lucky I found something they found acceptable!Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-69964990164241897602015-05-26T00:49:00.000-07:002019-03-05T22:10:59.135-08:00How to Discover the Rogers-Ramanujan Identities<div class="separator" style="clear: both; text-align: left;">
<b>Dec 22, 2012:</b> It is Ramanujan's 125th birthday, but how many of his famous identities do you know? Here we examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.</div>
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Here is a link: <a href="https://drive.google.com/file/d/0B1Tsyq_cbXz9TFdoaDRPRWVsZUE/view?usp=sharing" target="_blank"> How to Discover the Rogers-Ramanujan Identities</a>.<br />
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This was presented to a some high school math teachers <a href="http://theindianschool.in/conference-for-math-teachers-a-report/" target="_blank">in a conference</a>. I tried to write it in a way that it could be understood by a motivated high school student.<br />
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<b><span style="background-color: yellow;">Update (May 26, 2015):</span> </b>The article has been published. Here is a reference. <span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px; line-height: 18px;">Resonance, </span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px; line-height: 18px;">20 (no. 5)</b><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px; line-height: 18px;">, 416-430, (May 2015).</span><br />
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<b>Update (January 18, 2014): </b>This article has been accepted for publication in Resonance, a popular science magazine aimed at the undergraduate level.<br />
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<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-78346640827837189472015-03-31T23:53:00.000-07:002015-12-08T00:22:02.050-08:00Of Art and Math: A series of articles with Punya Mishra<br />
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<span style="font-size: x-small;">Right Angle: One of the many ambigrams made by Punya Mishra that appear in this series of articles appearing in "At Right Angles". </span><span style="font-size: x-small;">All ambigrams are copyright Punya Mishra and cannot be used without permission. </span></div>
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<a href="http://punya.educ.msu.edu/" target="_blank">Punya</a> and I are writing a series of articles on the subject of ambigrams. All the ambigrams are made by Punya. For this series, he has been making many new ambigrams, which communicate mathematical ideas. Already, in the space of working on a few articles, it looks like he has made the largest number of mathematical ambigrams.<br />
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<a href="http://punya.educ.msu.edu/2013/12/10/of-math-and-ambigrams-a-new-series-of-articles/#more-2863" target="_blank">Here is a longer blog</a> entry from Punya's blog, about this series of articles. His blog has further links to his amazing ambigrams.<br />
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<b>Updates</b><br />
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<b><span style="background-color: yellow;">Dec, 2015.</span> </b>I presented Punya's and my work in <a href="http://www.math.iitb.ac.in/TIME2015/index.html" target="_blank">TIME 2015</a>, in Baramati, Maharashtra in my talk: <i>On Punya Mishra's Mathematical Ambigrams</i>. This was the seventh edition of TIME, which stands for 'Technology and innovation in Math education'.<br />
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<b style="background-color: yellow;">July, 2015.</b><b style="background-color: white;"> </b><span style="background-color: white;">The fifth article is Part 2 of 2 on the subject of <a href="https://drive.google.com/file/d/0B1Tsyq_cbXz9TDZZRDBKTnZJU1k/view?usp=sharing" target="_blank">paradoxes</a>. It covers self-reference, Russell's Paradox and visual paradoxes. This article includes a 'new paradox', a version of Jourdain's card paradox by Punya. </span><br />
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<span style="background-color: yellow;"><b>Mar, 2015.</b> </span>The fourth article is on <a href="https://drive.google.com/file/d/0B1Tsyq_cbXz9R1VDLWIwNDdYOHc/view?usp=sharing" target="_blank">Paradoxes</a>. It is part 1 of 2 articles on this topic. Here we consider what TRUE and FALSE mean in the context of mathematics. Its an introduction to math philosophy. Again, it has many interesting ambigrams.<br />
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<b style="background-color: white;"><span style="background-color: yellow;">Feb 2015.</span> </b>The Michigan State Museum has launched an exhibit entitled "Deep Play: Creativity in Math and Art through Visual Wordplay." <a href="http://deep-play.com/creative/exhibition/" target="_blank">Check out: the exhibitions web-page</a></div>
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<b>July 2014. </b>The third article on <a href="http://teachersofindia.org/en/article/self-similarity" target="_blank">Self-similarity</a>. This one has some amazing ambigrams, and a graphic of the binary pascal's triangle I made many years ago.<br />
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<b>Mar, 2014.</b> The second article is on <a href="http://teachersofindia.org/en/article/art-math-introducing-symmetry" target="_blank">Introducing Symmetry</a>. I think Punya outdid himself in some of the ambigrams here. The ambigram for sin (which is periodic, a sin wave, an odd function) and inverse (modeled on a hyperbola) and exp-log were my favorites. But this month's puzzle ambigram is mind-blowing too.</div>
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<b>Nov, 2013.</b> The first article has come out. It is: <a href="http://www.teachersofindia.org/en/article/introducing-ambigrams" target="_blank">Introducing Ambigrams</a>. There is a hidden message in the article. See if you can find it.Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-56337196357421404832014-12-22T18:11:00.000-08:002019-03-05T22:10:59.129-08:00How to prove Ramanujan's q-Continued Fractions<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
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<tr><td class="tr-caption" style="text-align: center;">The photograph of a page in Ramanujan's Lost Notebook where he expanded a ratio of two series in terms of three continued fractions. These three are among the continued fractions explained in this paper.</td></tr>
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Its the 125th year of Ramanujan's birth, but how many of his formulas do you know? Here is an opportunity to get familiar with 9 of Ramanujan's continued fraction formulas. These include the three continued fractions that appear in the Lost Notebook in the above photograph. <script type="text/x-mathjax-config">
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Abstract:</div>
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By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.</div>
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A <a href="http://arxiv.org/pdf/1205.5455v3.pdf" target="_blank">Preprint</a> of this expository paper is now available from arXiv. The latest version fixes a typo. The final version appears in <a href="http://www.ams.org/books/conm/627/" target="_blank">this book.</a> You may wish to buy/access the entire volume from the AMS, its really an amazing piece of work.<br />
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<b><span style="background-color: yellow;">Update (Sept 7, 2018):</span> </b>I presented this topic in IISER, Mohali, after adding a few ideas from the recent joint work with Mourad Ismail. Here is the <a href="https://drive.google.com/file/d/17V44MPruLM4OkSjOOZAH0I0cKTB7VEKu/view?usp=sharing" target="_blank">presentation</a>.<br />
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<b style="background-color: white;">Update (December 20, 2014):</b> Published in <i style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px; line-height: 18px;">Contemporary Mathematics: Ramanujan 125</i><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px; line-height: 18px;">, K. Alladi, F. Garvan, A. J. Yee (eds.) </span><b style="color: #222222; font-family: 'Trebuchet MS', Trebuchet, sans-serif; font-size: 13px; line-height: 18px;">627,</b><span style="background-color: white; color: #222222; font-family: "trebuchet ms" , "trebuchet" , sans-serif; font-size: 13px; line-height: 18px;"> 49-68 (2014)</span><br />
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<b>Update (July 17, 2013): </b>Accepted for publication in the proceedings of Ramanujan 125, where I presented this paper.<br />
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<a href="http://www.blogger.com/blogger.g?blogID=10274084">
</a>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-25850154236805366602014-03-31T00:35:00.000-07:002015-05-28T22:48:18.053-07:00How to discover 22/7 and other rational approximations to $\pi$ <div class="separator" style="clear: both; text-align: center;">
<a href="http://teachersofindia.org/sites/default/files/styles/periodicals_thumb/public/periodicals/atria_march_cover.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://teachersofindia.org/sites/default/files/styles/periodicals_thumb/public/periodicals/atria_march_cover.jpg" height="320" width="240" /></a></div>
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A short article written for a magazine that caters to high-school students. The basic idea is to use continued fractions found using Euclid's algorithm, and then to chop off the continued fraction to get rational approximations. Written at a high-school level. Some of the material was already present in my book Maths Concepts.<br />
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Update: <b>March 2014.</b> This article is published in "At Right Angles." Here is a <a href="http://teachersofindia.org/en/article/how-discover-227" target="_blank">link to the published article</a>.Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-9956240319431321182014-02-06T23:08:00.000-08:002015-08-20T10:32:55.171-07:00Interview in Annulus - Hindu College math department magazine <div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 18.399999618530273px;"><i>This is an email interview with Annulus, a magazine taken out by Acuity, the mathematics society run by mathematics students of Hindu College, Delhi University. Most of it was published in the magazine. I thought it was meant as a tribute to Saroj Bala Malik, who taught me 4 classes when I attended Hindu, from 1984-87, but they edited out the question about her in the final article. </i></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b><span style="line-height: 115%;">Tell us about your time
at Hindu College. What made you pursue Mathematics here?</span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">I wanted to become a mathematician. I joined Hindu college
because it was the best college where I managed to get admission to study Math.
It was the best! And I had the most wonderful time. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">Hindu was a most liberal place, where a lot of leeway and
freedom was given to students to figure out their own approach to life. Teachers
did not impose upon us. The students came from all kinds of economic and social
backgrounds, which was great for me, because I had gone to a somewhat elitist
school. The cafeteria those days was really a park in front of the hostel, with
a few chairs, but plenty of sunshine. A lot of people (even from other
colleges) just hung around. A lot of time was spent in ‘Café Hons’. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">The key highlights was the annual ‘trip’ which was 4-5 days
of concentrated fun, followed by discussions of what all happened there for the
rest of the year. Plus, of course, Mecca. One year in Mecca, we took out a
daily magazine called ‘The Quark that Quakes’, consisting of mathematical puzzles,
limericks and bad jokes. It was a big hit.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">What career option were
you looking for when you decided to take up Mathematics? <o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">I wanted to be a research mathematician. I knew that all that
I had studied in school was essentially stuff known to Newton and Archimedes
many centuries ago. I wanted to reach the frontiers, whatever that meant. I had
no idea what it takes to discover and prove your own theorem. I just thought it
would be cool to have one I can call my own! <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">Questions on whether I could become rich, or even survive
financially, didn’t really enter my head. Perhaps the practice—prevalent in
Hindu—of treating our friends’ money and possessions as our own, contributed to
this attitude. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">How hard was it to make
it to IIT Delhi back then? Any tips that you would like to share with the
students?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">I don’t recall having studied at all for the IIT entrance. The
exam was so tricky that it was fair game for anyone. About 250 students took
the test, and 20 were selected. My rank was 2, so I suppose I did quite well! The
test required understanding the basic ideas/definitions rather than extensive
knowledge of the subject. In fact, I recall that one of the questions was to
state and prove your favorite theorem, so they were looking to see if you liked
math and what you liked in it. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">An idea that works for me is to find one book that gives a
historical overview of a particular subject. After going through it in a week
or two, I am able to understand what’s happening for the entire semester. There
are books like this for algebra, analysis, complex analysis, number theory—you just
have to find one that you like.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">If you do this, then you can begin to appreciate the beauty
of the subject, and are able to understand why you are doing what you are
doing. The subject becomes easy, and you will be able to answer the kind of
questions that examiners are looking for. <span style="mso-spacerun: yes;"> </span>You will also be able to slog through the
difficult theorems and proofs, because you have a sense of where you are going.
<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">Any Dr. Saroj Bala
Malik memories?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">SBM has been one of the most influential teachers in my life.
In our first class, she asked questions and I was one of the two or three
students who answered her. The same day I met her at the bus stop, and she
recognized me. I told her I want to do research in mathematics. And from that
day on, she took it upon herself to help me in whichever way she can.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">Our entire batch was her favorite. It wasn’t that she was all
mushy or soft on us. She practiced what is called ‘tough love’. She worked hard
at her teaching and demanded we work hard at our learning. She asked a lot of
questions. She praised us when we could answer, and, well, took our trip, when we
couldn’t. She went out of her way to fund our activities, and covered for us in
case we got into trouble with other faculty members!<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">However, there were a few rocky moments too. I used to
organize a weekly puzzle contest. Every week, I would post a new puzzle, along
with the answer of the previous puzzle, and the names of those who got it right.
All went well for a few weeks. Until one day, when SBM got (in my view) the
wrong answer! Her view was that the question was wrongly worded. She demanded
that I correct my mistake. We fought long and hard. It wasn’t pretty—but it was
interesting, and kind of fun!<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 18.399999618530273px;"><i>The question above was not included in the printed interview.</i></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">Tell us about your time
at IIT Delhi.<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">I spent only a year in IIT Delhi. IIT was mostly about very
brilliant lecturers and a fun hostel life. But I did not learn much there,
because I did not work very hard. Most of the time I was busy applying abroad.
I got a scholarship, and left without finishing my MSc. But there was one important
aspect of my year at IIT. I met the person whom I eventually married. So
all-in-all it turned out to be a good year!<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">What was the experience
at Ohio State University like? <o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">Ohio State was truly the best educational experience
possible. There were many famous mathematicians who taught me, among the best
people in their area. My Ph.D. advisor was Steve Milne, who had given the first
combinatorial proof of the Rogers-Ramanujan identities, thereby solving a
long-standing problem. My story with him was similar to SBMs. He gave a talk
about his area, and showed how he had extended a famous result of Ramanujan.
Right after his talk I went and told him I wanted to work with him for my Ph.D.
<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">The biggest truth I learnt at Ohio State was that mathematics
is learnt by doing mathematics. Your professor can be the most brilliant lecturer
in the world (or not), but you will learn only if you do all the problems of
the textbook on your own. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">In our department, there were people from all over the world;
plus, I interacted with hundreds of American students as their Teaching
Assistant. Living in the US, with enough money to have some fun, and hanging
out with many people of many different countries—I think that was the most
amazing and enjoyable part of doing a Ph.D. in the US. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">From Modern School to
Ohio State University, how has Mathematics shaped your life? <o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">When I was in class 11, I took a Math Olympiad exam, where I
happened to crack a problem I had never seen before. And I felt wonderful! I
had got an exhilarating high, and it happened because I got a creative idea in
mathematics. I figured that I want to have this feeling again and again, for
the rest of my life. So I decided to become a mathematician. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">From Modern to Hindu and IIT, and on to Ohio State, I stayed
with this for nearly 15 years. <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">But I forgot about this after returning to India after my
Ph.D. After a year in ISI, Delhi, I took up a job in the industry and thought I
cannot pursue math any more. This went on for a few years, and I was totally
miserable, and didn’t know why. Then one fine day I got a project to write a
math book, and got reminded about this exhilarating feeling again! <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">That is when I realized that math is what keeps me happy. Now,
despite a full time job, I look to do something mathematical, whether it is
research, teaching, writing books, articles or papers, or even reading math books.<span style="mso-spacerun: yes;"> </span>The thrill that comes from solving a math
problem—especially a tough math problem—has never gone away. That is what keeps
me happy.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;"><br /></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="line-height: 115%;">The generation of today
is somewhat reluctant to pursue Mathematics as a subject. What will be your
advice to the students who are looking to or currently pursuing Mathematics?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">My advice would be to do as much as you can handle, and then
a little more. If you cannot do math just for the love of it, then consider the
following 5 things that a math education does: <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">#1: <b style="mso-bidi-font-weight: normal;">It teaches you to
question. </b></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">Why prove theorems, when they have been proved a million times
before? Because, as our teachers tell us, you need to see for yourself that the
theorem is true. This is so unlike the real world, where often people tend to
prove things to you by intimidation, or by asserting their authority. However,
unless you question things, you will not get creative ideas. And in math, we
question everything!<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">#2: <b style="mso-bidi-font-weight: normal;">It teaches you to
reason.</b> </span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">We learn to apply logic to prove theorems. In the real world,
people frequently confuse a statement with its converse, and don’t believe that
if ‘A or B’ is true, then both ‘A and B’ could also be true! Your capability to
reason correctly and think clearly will quickly get you noticed.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">#3: <b style="mso-bidi-font-weight: normal;">It teaches you to
communicate clearly. </b></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">The practice of understanding mathematical definitions
and proving theorems teaches us that words have a precise meaning. Being able
to communicate clearly is perhaps the most important requirement for success.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">#4: <b style="mso-bidi-font-weight: normal;">It teaches you to think
abstractly.</b> </span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">As you grow in responsibility in an organization, you need to
deal with a large number of facts. However, the time to deal with them is
finite. At this time the ability to think abstractly becomes hugely important.
Abstraction is a key requirement of any leadership position whether it is in
academia, industry or the government!<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">#5. <b style="mso-bidi-font-weight: normal;">It gives you
confidence.</b> </span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">If you have done well in mathematics, or even reasonably well,
you should take a huge amount of confidence from this. For someone who is a
master of epsilon-delta proofs, point-set topology, or abstract algebra, most
management or technical problems at the workplace are a piece of cake! <o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="line-height: 115%;">In short, a good mathematical education gives you an unfair
advantage in the real world. So if you can handle it, go for it!<o:p></o:p></span></div>
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Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-10677020331216994092013-07-29T00:11:00.001-07:002017-08-17T20:57:08.933-07:00New website: Teaching WebsiteI have made a new website, that collates the math materials I keep creating and with information for my students. It is available on<br />
<a href="http://gb-teaching.blogspot.in/" target="_blank">http://gb-teaching.blogspot.com</a>. If you, or your child, is in high school, there are many materials available that may be useful. Eventually, I hope some of the new materials I am placing there also become a book or perhaps an e-book.Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-48375563875471865372012-06-30T21:58:00.004-07:002015-05-26T19:26:45.708-07:00Get Smart Maths Concepts now available as an e-book<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-wcLquywICxg/ScCrzliWE7I/AAAAAAAAAE4/ecO077qZbJU/s1600/cover_b.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-wcLquywICxg/ScCrzliWE7I/AAAAAAAAAE4/ecO077qZbJU/s1600/cover_b.jpg" /></a></div>
<br />
Get Smart! Maths Concepts, published by Penguin in 2008 is now available as an e-book.<br />
<br />
Check it out!<br />
<a href="http://www.amazon.com/kindle/dp/B008ESLWRK/ref=rdr_kindle_ext_eos_detail">http://www.amazon.com/kindle/dp/B008ESLWRK/ref=rdr_kindle_ext_eos_detail</a><br />
<br />
Or<br />
<br />
<a href="http://www.flipkart.com/get-smart-maths-concepts/p/itmdumyhbgh6dyqh?pid=DGBDGGY4M2SVYDMH">http://www.flipkart.com/get-smart-maths-concepts/p/itmdumyhbgh6dyqh?pid=DGBDGGY4M2SVYDMH</a><br />
<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comtag:blogger.com,1999:blog-10274084.post-4234154385932280042012-04-02T10:27:00.001-07:002012-05-21T05:36:21.183-07:00Arun Bhaiyya -- A limerick<br />
Texas ka famous letter writer, our Arun Bhaiya,<br />
He loves to cook large amounts of lobhia,<br />
Eating those greens,<br />
Keeps him full of beans--<br />
But TT: he thinks its a taste worse than ghia!<br />
<br />
<i>Arun Kumar used to write an interesting "Letter from Austin" which was quite famous in the early days of the Internet. Even now, his letters to his mailing list "dakghar" are quite interesting and varied in the topics they cover. Now these letters are being recorded as his Facebook notes. This piece was inspired by one of his Facebook notes, where he describes his love for cooking (and eating) lobhia. TeeTee (TT for short) is his son. I hear he is quite a mathematician and a hockey player.</i><br />
<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com