tag:blogger.com,1999:blog-72625148356471367912024-03-23T23:56:49.352-04:00MATHEMATICAL TRIPSInvitations, in an educational vein, to problem solving, ideas and their history, biographical snapshots, visuals, classroom capsules, undergraduate research ideas, recreations, experiments, arts and poetry, philosophical and foundational matters and connections.Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comBlogger140125tag:blogger.com,1999:blog-7262514835647136791.post-44551420632207676712024-03-23T23:56:00.000-04:002024-03-23T23:56:18.655-04:00Paper on subprime tribonacci sequences<p><span style="font-size: large;"> "<a href="https://www.tandfonline.com/doi/abs/10.1080/07468342.2023.2263109" target="_blank"><i>Three’s a crowd: an exploration of subprime tribonacci sequences</i></a>" - <i>College Mathematics Journal</i> 54, No. 5, 464-475 (2023) - by Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright.</span></p><p><span style="font-size: large;"> </span><br /></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-3357300471421836272024-02-22T14:59:00.004-05:002024-02-22T14:59:34.529-05:00Iwasawa decomposition in a nutshell<iframe width="560" height="315" src="https://www.youtube.com/embed/TwCkMtzQeMo?si=apX4IutQd54ZyEKC" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-21334395133802813672024-02-07T17:33:00.004-05:002024-02-07T17:33:35.700-05:00130 years of mind games and quantum challenges<p> <span style="font-size: large;">A wonderful article by Laurie Wurth-Pressel (<a href="https://www.onu.edu/news/130-years-mind-games-and-quantum-challenges" target="_blank"><b>source</b></a>)</span><br /></p><div class="block block-core block-page-title-block" id="block-pagetitle">
<h1 class="heading-style-2 text-orange"><span class="field field--name-title field--type-string field--label-hidden">130 years of mind games and quantum challenges</span>
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<time class="datetime" datetime="2024-01-22T19:11:00Z">January 22, 2024</time></div><div class="article--date mb-3 mt-2 field field--name-field-article-date field--type-datetime field--label-hidden field__items"><time class="datetime" datetime="2024-01-22T19:11:00Z"> </time>
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<img alt="graphic showing Benjamin Finkel, a mathematic trailblazer" src="https://www.onu.edu/sites/default/files/styles/max_325x325/public/benjaminfinkel_news.jpg?itok=TvjTPWEY" />
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<p>2024 marks the 130th anniversary of The American Mathematical Monthly. </p>
<p>The trailblazer who launched this premier academic journal in 1894
and inspired the establishment of the American Mathematical Association
(AMA) graduated from Ohio Northern University in the late 1800s—<b>Benjamin Franklin Finkel,</b> BS 1888, BA 1896.</p>
<p>Mathematical problems fascinated Finkel from a young age, and his
eagerness to solve them impelled his ambition to “publish a journal
devoted solely to mathematics and suitable to the needs of teachers of
mathematics.”</p>
<p>He once stated: “Many dormant minds have been aroused into activity through the mastery of a single problem.” </p>
<p>Over a century later, Finkel’s statement still rings true. </p>
<p>While much has changed at Ohio Northern since Finkel’s days, the
University continues to attract brilliant problem-solvers molded in
Finkel’s character who graduate to become leading mathematicians and
physicists. </p>
<p>Read about three outstanding alumni from ONU’s School of Science,
Technology, and Mathematics, housed in the Getty College of Arts &
Sciences, who are making a mark in the world of physics and mathematics.</p>
<h3><img alt="Photo of Tommy Steinberger" class="pull-right" src="https://my.onu.edu/sites/default/files/styles/medium/public/images/news/tommysteinberger.jpg?itok=Du36zGzj" />From algebra to plasma physics</h3>
<p><b>Thomas (Tommy) Steinberger</b>, BS ’14, Ph.D., is breaking new ground in experimental plasma physics at West Virginia University in Morgantown, W.V.</p>
<p>A research assistant professor in the Department of Physics and
Astronomy, he explores charged gas systems (plasmas), investigating gas
particle motion and temperature. His work aims to enhance electronics
manufacturing, space travel, and our understanding of space phenomena,
such as magnetic reconnection.</p>
<p>“I come into the lab and can work on 10 different projects before
lunch,” he said. “I have a lot of ongoing projects that are unique. Most
of my effort is helping all these projects move forward.”</p>
<p>Steinberger’s journey began in freshman Algebra class in high school.
While his classmates bellyached about having to solve 50 of the 100
algebra problems listed in the final exam, he felt a burst of
excitement.</p>
<p> “I completed nearly all 100 in a couple of days, almost obsessively,” he recalled. </p>
<p>His ONU experience further ignited his passion for problem solving.
He joined ONU-SOLVE, a problem-solving group of students that tackle the
challenging problems found in leading math magazines such as Fibonacci
Quarterly, Mathematics Magazine, The College Mathematics Journal, and
Finkel’s own The American Mathematical Monthly.</p>
<p>ONU-SOLVE has been recognized multiple times in recent years for
submitting correct solutions, and several have been published in leading
journals as the most well-written solutions, according to Mihai
Caragiu, Ph.D., professor of mathematics.</p>
<p>According to Steinberger, several solutions he worked on with the
assistance of ONU professors received honorable mentions in academic
magazines.</p>
<p>“My time at ONU really fostered my interest in math and its
application to other fields,” he said. “I received fantastic instruction
from professors who truly cared about their students.” </p>
<p>After graduating from ONU with a double major in <a href="https://www.onu.edu/academics/mathematics" target="_blank">mathematics</a> and <a href="https://www.onu.edu/academics/physics" target="_blank">physics</a>, and minor in <a href="https://www.onu.edu/academics/astronomy" target="_blank">astronomy</a>,
Steinberger received a master’s degree in physics and astronomy, and a
Ph.D. in experimental plasma physics from West Virginia University. </p>
<p>“The challenge of the subjects (math and physics) resonates with my
stubborn nature,” he added. “It leads me to be ever more persistent in
my studies.”</p>
<h3><img alt="Photo of Ashley Ernst" class="pull-left" src="https://my.onu.edu/sites/default/files/styles/medium/public/images/news/ashleyernst.jpg?itok=Lz-SftE_" />From jigsaw mastery to national defense</h3>
<p><b>Ashley Ernst,</b> BS ’15, Ph.D., is a senior physicist
at Arcfield in Colorado Springs, Colo, where she helps to solve complex
challenges in support of the United States’ most critical national
security missions.</p>
<p>Arcfield is a leading provider of mission-focused systems engineering
and integration capabilities to the U.S. Intelligence Community,
Department of Defense, and other agencies.</p>
<p>Ernst is currently working primarily with simulation and modeling of
radiation in both vacuum and in atmosphere or material. She develops
technical documentation, presents work to clients, and performs
calculations with the aid of software.</p>
<p>“There is always another problem on the horizon,” she says. “The
quest for the solution to the next problem is what drives me every day.”</p>
<p>Growing up, Ernst loved to solve jigsaw puzzles with her mom. As her skill increased, so did her hunger for harder challenges.</p>
<p>“They became larger in puzzle number, smaller in puzzle piece size,
and more complex in shape and design,” she said. “When that wasn’t
enough, I started solving puzzles without guide pictures.”</p>
<p>When she first arrived at ONU, however, she lacked focus and felt
extremely homesick. Her ONU professors noticed her struggles and made
special efforts to engage her in math conferences and ONU-SOLVE.
Professor Caragiu spurred her mind into action by exposing her to Graph
Theory and Discrete mathematics.</p>
<p>“Having a direction for my mental energy definitely helped me,” she said.</p>
<p>Within a short time, she says, she was on a better path forward. “I
would not be where I am today without their help. The math and physics
programs at ONU nurture the search for knowledge. No matter the level,
the program meets the student at that level and pushes them to the next
level.”</p>
<p>After graduating from ONU with a double major in <a href="https://www.onu.edu/academics/physics" target="_blank">physics</a> and <a href="https://www.onu.edu/academics/applied-mathematics" target="_blank">applied mathematics</a>,
Ernst earned a master’s degree in physics and a Ph.D. in hadronic
physics from Florida State University. Her second year of graduate
school, she received a highly-competitive National Science Foundation
(NSF) Graduate Research Fellowship, which she credits to the quality of
education and one-on-one mentoring she received at ONU.</p>
<p>“Each student that passes through the math program at ONU is
instilled with a sense of excitement regarding a problem. While that
problem may change, that excitement stays,” she said. </p>
<h3><img alt="Photo of Matthew Golden" class="pull-right" src="https://my.onu.edu/sites/default/files/styles/medium/public/images/news/matthewgolden.jpg?itok=j9FBbgSO" />From twisty puzzles to cosmic enigmas</h3>
<p><b>Matthew Golden,</b> BS ’17, Ph.D., is a postdoctoral
fellow at the Georgia Institute of Technology in the Xtreme Astrophysics
group. The group is led by two founding members of the Event Horizon
Telescope Collaboration, which released the first image of a black hole
in 2019.</p>
<p>“I am a full-time researcher,” he said. “My research focuses on the
interface of machine learning and physics. Specifically, I work on using
machine learning to accelerate human learning. Our goal is to produce
physics equations directly from complex data with minimal human
intervention.” His <a href="https://www.science.org/doi/10.1126/sciadv.abq6120" target="_blank">recent publication</a>
in Science Advances showed how machine learning learned the equations
of a living fluid directly from a video of the experiment.</p>
<p>In high school, Golden became enthralled with solving “twisty
puzzles”—think Rubik’s cube, only the more complicated versions. He had
puzzles of every platonic solid and with hundreds of pieces. Some would
take him mere minutes to solve, others weeks.</p>
<p>“I eventually went on to solve the four-dimensional 3x3x3x3 Rubik’s
cube,” he said. “You can find my name in the 4D Hall of Fame as solver
#196.”</p>
<p>At ONU, his obsession switched to General Relativity. He spent many
late nights in the Mathile Center for the Natural Sciences working
through derivations. Then, he’d head to the third floor of Heterick
Memorial Library, pull a random math or physics book from the shelf, and
read until he was “hopelessly confused.” </p>
<p>“My schooling was significantly accelerated compared to the usual
undergraduate,” he said. “That’s because the physics department was
small and the teachers eager to teach at any pace.”</p>
<p>His professors allowed him to take courses in any order he desired.
He completed Quantum Mechanics his first semester, then continued to
grow his knowledge in leaps and bounds. He’ll never forget being the
only student in Dr. Khristo Boyadzhiev’s Real Analysis class. Dr.
Boyadzhiev, who obtained <a href="https://www.youtube.com/shorts/IWk2PYMkj1U" target="_blank">YouTube fame</a> for his consistent classroom outfit and greeting, passed away in June 2023.</p>
<p>“It feels like half the people my age know Dr. Khristo Boyadzhiev
because of his lovable appearance on Vine,” said Golden. “He was a micro
celebrity, and I still remember him laughing about it in the hall.”</p>
<p>Golden also recalls leading a small group of students to first place
in the 2017 Ohio MAA Leo Schneider Team Math Competition, dethroning
Case Western Reserve University for the first time in many years. </p>
<p>With accelerated learning and one-on-one attention, Golden says he was way ahead of his peers in graduate school. </p>
<p>“The education I received was passionate and tailored to me,” he
said. “There is no math program in the Midwest that could compete with
the personal attention I received at ONU, and it paid off.”</p>
<p>He’s ecstatic that his professional career is centered on gravity research.</p>
<p>“I love that every day I get to think about machine learning and
extreme astrophysical environments,” he said. “I get to interact with
some of the greatest minds in physics.”</p>
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</div>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-39367727518405844602024-01-31T22:57:00.000-05:002024-01-31T22:57:02.591-05:00"Model Theory (Ax's Theorem)" by Piotr Kowalski<p> </p><iframe width="560" height="315" src="https://www.youtube.com/embed/t5NTBa75CoM?si=EWfyGa-yy8f57B2n" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-81738413161177637902024-01-23T14:13:00.000-05:002024-01-23T14:13:09.205-05:00A Fall 2023 full messy chalkboard - random graphs<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCn57bBBBZMvC6p6BvZeaJIdCjJiL8RBTUWwHVkir1V_WI9zvu3_t95y-nA6MjmhBwFgZk-qk9mC4BVHjjEjFFXu70_pH3l2I7G9ZwiA3cL1G8O3Lo5ZIqtWJBwDLoTdxMgAdMDHfzLqq8etbCKPADItEXffGWBQZIslEtIv0h_KJ6KU1RFHmOVruvf7g/s2929/GIANT.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="693" data-original-width="2929" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCn57bBBBZMvC6p6BvZeaJIdCjJiL8RBTUWwHVkir1V_WI9zvu3_t95y-nA6MjmhBwFgZk-qk9mC4BVHjjEjFFXu70_pH3l2I7G9ZwiA3cL1G8O3Lo5ZIqtWJBwDLoTdxMgAdMDHfzLqq8etbCKPADItEXffGWBQZIslEtIv0h_KJ6KU1RFHmOVruvf7g/s16000/GIANT.jpg" /></a></div><br /><p></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-41180932443190070832024-01-19T23:39:00.003-05:002024-01-19T23:39:54.429-05:00Quanta Magazine: 2023's Biggest Breakthroughs in Physics<p> </p><iframe width="560" height="315" src="https://www.youtube.com/embed/580V0wRl1Lo?si=z4Avk6d1r_GFPMht" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-31219152347113310432024-01-08T21:39:00.004-05:002024-01-08T21:39:55.939-05:00Quanta Magazine: 2023's Biggest Breakthroughs in Math<p><br /></p> <iframe width="560" height="315" src="https://www.youtube.com/embed/4HHUGnHcDQw?si=_B45f7OOzQP1q0Vy" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-53180513089805563182023-12-17T22:50:00.004-05:002023-12-17T22:50:37.207-05:00Florian Richter: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions<iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/2n0elpGbvYU?si=NowYakW_hzAHZmAx" title="YouTube video player" width="560"></iframe><p><span class="yt-core-attributed-string yt-core-attributed-string--white-space-pre-wrap" style="font-size: large;"><span class="yt-core-attributed-string--link-inherit-color" style="color: #131313;">Topic: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions </span></span></p><p><span class="yt-core-attributed-string yt-core-attributed-string--white-space-pre-wrap" style="font-size: large;"><span class="yt-core-attributed-string--link-inherit-color" style="color: #131313;">Speaker: Florian Richter, Northwestern University</span></span></p><p><span class="yt-core-attributed-string yt-core-attributed-string--white-space-pre-wrap" style="font-size: large;"><span class="yt-core-attributed-string--link-inherit-color" style="color: #131313;">June 4, 2020</span></span></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-3461503612940455222023-11-13T20:51:00.000-05:002023-11-13T20:51:34.209-05:00Ramsey Numbers asymptotics - R(4,t)<p><span style="font-size: large;">An amazing <a href="https://arxiv.org/abs/2306.04007" target="_blank">new result</a></span></p><p><span style="font-size: large;"><b>The asymptotics of r(4,t)</b> - by Sam Mattheus and Jacques Verstraete</span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_FElPghu5JpLWrGMI3hdZi8ThKzKjJ4yt0HTnUqAhuQ42dBGmbUp6U7QRRtL77LeeC_C8CQ3mb0XSErCfLLJ0zaL_6TXOWy6XXy4EwLIaymPudJhmI9AEl9iIed-jUxULzl16ZPAzwpwXgtx6u2k2QMkEvPHscRKD6QQH2d_royZr-jSQz_DOEJMLKpM/s756/Ramsey.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="264" data-original-width="756" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_FElPghu5JpLWrGMI3hdZi8ThKzKjJ4yt0HTnUqAhuQ42dBGmbUp6U7QRRtL77LeeC_C8CQ3mb0XSErCfLLJ0zaL_6TXOWy6XXy4EwLIaymPudJhmI9AEl9iIed-jUxULzl16ZPAzwpwXgtx6u2k2QMkEvPHscRKD6QQH2d_royZr-jSQz_DOEJMLKpM/w640-h224/Ramsey.png" width="640" /></a></div><br /><p><br /></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-17503272861835202602023-11-07T23:03:00.000-05:002023-11-07T23:03:04.925-05:00Random Graphs<i><span style="font-size: large;">Math circle session on random graphs led by Alon Amit at the 2012 Summer BACT
</span></i><p></p><p></p><p><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/kXfca4CqSo0?si=S6LHAMGC12RCkSVn" title="YouTube video player" width="560"></iframe></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-61602429204549830072023-10-09T22:36:00.007-04:002023-10-09T22:41:50.736-04:00Cellular Automata: Rule 110 + Conway’s Game of Life<p><span style="font-size: large;"><span class="yt-core-attributed-string yt-core-attributed-string--white-space-pre-wrap"><span class="yt-core-attributed-string--link-inherit-color" style="color: #131313;"></span></span></span></p><blockquote><span style="font-size: large;"><span class="yt-core-attributed-string yt-core-attributed-string--white-space-pre-wrap"><span class="yt-core-attributed-string--link-inherit-color" style="color: #131313;">"<i>A </i>1D<i> cellular automaton, </i>Rule 110<i> (bottom), being fed as input to a 2D cellular automaton, </i>Conway’s Game of Life<i> (top)</i>"</span></span></span></blockquote><p></p><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/P2uhhAXd7PI?si=WtaSCbi-6sWQrBz0" title="YouTube video player" width="560"></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-65018506704995578672023-07-20T22:29:00.003-04:002023-07-20T22:29:44.668-04:00California Approves Revised Math Framework as a Step Forward for Equity and Excellence - July 12, 2023<p><span style="font-size: large;"><b><a href="https://www.cde.ca.gov/nr/ne/yr23/yr23rel54.asp" target="_blank">https://www.cde.ca.gov/nr/ne/yr23/yr23rel54.asp </a></b></span><br /><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"></span></p><div class="col-md-9">
<div class=" hidden-xs" id="newsreleaseimage"><img alt="California Department of Education News Release" class="img-responsive" src="https://www.cde.ca.gov/images/tthurmondnewsrelease840.gif" /></div>
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CDE/SBE Joint Release: #23-54<br />
July 12, 2023
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Contact: Communications<br />E-mail: <a href="mailto:communications@cde.ca.gov">communications@cde.ca.gov</a><br />Phone: 916-319-0818
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<h1 class="centeredText" style="padding-bottom: 20px; padding-top: 20px;">
California Approves Revised Math Framework as a Step Forward for Equity and Excellence
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<p>SACRAMENTO—The California State Board of Education today approved the <em>2023</em> <em>Mathematics Framework for California Public Schools, </em>instructional
guidance for educators that affirms California’s commitment to
ensuring equity and excellence in math learning for all students.</p>
<p>“I’m thankful for everyone who worked tirelessly to develop this
framework to ensure California’s students have equitable access to
rigorous and high-quality math instruction that will prepare them for
the future. The framework has struck a great balance in new ways to
engage students in developing a love for math while supporting those on
an accelerated path,” said Mary Nicely, Chief Deputy Superintendent of
Public Instruction. “Our State Superintendent is a champion of equity
and excellence, and it is our core mission that every child—regardless
of race, ZIP code, or background—has access to a quality education. The
approval of the revised Math Framework is one more step forward to
meeting the needs of all California’s students.”</p>
<p>The vote today concludes four years of work to update math instructional guidance aligned with the <a href="https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf">California Common Core State Standards for Mathematics<span class="LinkNotation"> (PDF)</span></a>,
which are rigorous learning standards that detail what every student
should know and be able to do at every grade level. The framework
approved today is the third iteration and reflects revisions responsive
to thousands of public comments fielded over two 60-day public comment
periods and two public hearings.</p>
<p>The draft was presented by Dr. Mike Torres, Executive Director of
the Instructional Quality Commission and a former high school math
teacher. Others who participated in the presentation include Dr.
Kyndall Brown, Executive Director of the California Mathematics Project
at the University of California, Los Angeles; Omowale Moses, Founder
and Chief Executive Officer of Math Talks; Dr. Adrian Mims, Founder of
The Calculus Project; Ellen Barger, Chair of the Curricular and
Improvement Support Committee of the California County Superintendents;
and Dr. Linsey Gotanda, Vice Chair of the Instructional Quality
Commission.</p>
<p>“This framework provides strategies to challenge, engage, and
support all students in deep and relevant math learning by building on
successful approaches used in nations that produce high and equitable
achievement in math,” said State Board President Linda Darling-Hammond.
“It also draws on the experiences of educators who have worked for a
decade to develop successful strategies for teaching California’s
rigorous standards, carrying those lessons to others across the state.
This framework provides teachers and schools with a path to greater
excellence with greater equity.”</p>
<p>The guidance includes strategies to:</p>
<ul><li>Structure the teaching of the state’s math standards around
“big ideas” that integrate rather than isolate math concepts—a
best practice in high-performing countries.</li><li>Increase focus on developing student mathematical expertise
as described in California’s Standards for Mathematical Practice,
which include the ability to make sense of problems and persevere
to solve them; to reason abstractly and quantitively; to attend to
precision; and to apply the mathematics they know to solve
problems arising in everyday life, society, and the workplace. </li><li>Connect learning to the “real world” through authentic
examples and use of data, prompting students to ask and answer
meaningful questions. Adding authenticity to lessons helps
teachers answer students’ questions around “why do I need to learn
this?”</li><li>Allow students to “see themselves” in curriculum and in
math-related careers by making math instruction culturally
relevant and empowering.</li><li>Stimulate deep learning by sparking student curiosity through lessons that encourage inquiry and problem-solving.</li><li>Ensure that students develop both appreciation of math
concepts and fluency in using math efficiently through the
productive use of algorithms and mastery of math facts they have
come to understand.</li><li>Integrate and align math concepts taught at the elementary, middle, and high school levels.</li><li>Ensure that all high school math pathways are open to all students.</li><li>Support multiple ways to get more students to higher level
mathematics—ranging from successful acceleration to differentiated
instruction, personalized supports, extra lab sections, and
additional coursework offered at multiple junctures—augmenting
more effective core instruction.</li><li>Expand high school math course options to encourage more students to go beyond minimum course requirements.</li><li>Encourage students across age spans to become proficient at
understanding and using data—a key skill in the 21st century job
market.</li><li>Help students to identify misleading uses of data and use data to make decisions in their roles as global citizens. </li><li>Develop in students a “growth mindset” about mathematics, in
line with the groundbreaking research of Stanford’s Dr. Carol
Dweck, that supports effort and perseverance.</li><li>Instill confidence in learners by dispelling myths about who can and cannot learn math.</li><li>Develop instruction and curriculum that is
“multi-dimensional” and employs the use of visuals, graphics, and
words in addition to numbers and equations. </li></ul>
<p>More information is available on the <a href="https://www.cde.ca.gov/ci/ma/cf/">California Department of Education's Mathematics Frameworks web page</a>, which includes frequently asked questions, an overview, and a timeline of events in the framework’s development.</p>
<p style="text-align: center;"># # # #</p>
<p style="font-weight: bold; text-align: center;">
<a href="https://www.cde.ca.gov/eo/">Tony Thurmond</a> —
State Superintendent of Public Instruction<br />
Communications Division, Room 5602, 916-319-0818, Fax 916-319-0100</p>
<div class="lastmodified">
Last Reviewed: Wednesday, July 12, 2023
</div>
</div><p><br /><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"></span></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-87333426083667551222023-05-20T12:29:00.004-04:002023-05-20T12:29:26.410-04:00Peter Sarnak on Möbius Disjointness <iframe width="560" height="315" src="https://www.youtube.com/embed/lrEz7y5-VRs" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-35937235642390709182023-05-11T14:20:00.002-04:002023-05-11T14:20:46.614-04:00Senior Capstone presentations - May 10, 2023<p><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span style="font-size: large;">Session
of Senior Capstone presentations today at ONU, featuring topics on
Partial Differential Equations and Number Theory. It was definitely a
success, with a lively attendance. Many thanks to these awesome students!</span></span></p><p><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span style="font-size: large;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXvXbq0h4gquA4qPIiacaiA0GfXaB8zH0YYiVap7CcDPMRuG791MAqDIxS6LwLeq6hB3wlGk7wrK65Pk8-pljjTn0Grq8lUYSV48T7GwwOcPPALwiSYXdDpcCSq3lUvcwIhzmHduenNEYbWjuEvOb1KA8K1nim2qqX6BYnuofDpVLshfowcHAXyrYo/s2048/C1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2048" data-original-width="2048" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXvXbq0h4gquA4qPIiacaiA0GfXaB8zH0YYiVap7CcDPMRuG791MAqDIxS6LwLeq6hB3wlGk7wrK65Pk8-pljjTn0Grq8lUYSV48T7GwwOcPPALwiSYXdDpcCSq3lUvcwIhzmHduenNEYbWjuEvOb1KA8K1nim2qqX6BYnuofDpVLshfowcHAXyrYo/w640-h640/C1.jpg" width="640" /></a></span></div><span style="font-size: large;"><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjEGQqgCfMPVjN0rcogBgcpR5tw3LlyTrx13hlLbz49kvtw7kXI0GR4R-opHMOiGXWNl2gL0GhvQfhwH983abhBUNHuEPHiP1dFfOfvvQcghJxJkKYPFNLKLrpMZCxYcb5RpPn6iT8jpvLXoZBhkMLnOOCKqKDWwll3tdFaF6uPL25XOTuebB4O-ja/s2048/C2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2048" data-original-width="2048" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjEGQqgCfMPVjN0rcogBgcpR5tw3LlyTrx13hlLbz49kvtw7kXI0GR4R-opHMOiGXWNl2gL0GhvQfhwH983abhBUNHuEPHiP1dFfOfvvQcghJxJkKYPFNLKLrpMZCxYcb5RpPn6iT8jpvLXoZBhkMLnOOCKqKDWwll3tdFaF6uPL25XOTuebB4O-ja/w640-h640/C2.jpg" width="640" /></a></div><br /></span><br /><p></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-20115789917571754122023-05-01T21:32:00.005-04:002023-05-01T21:32:59.397-04:00The Patterns in the Primes, with Andrew Granville<p> </p><iframe width="560" height="315" src="https://www.youtube.com/embed/pO7Egc5Dtqs" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-71190907698866436102023-05-01T21:27:00.000-04:002023-05-01T21:27:06.583-04:00The Search for Randomness | Jean Bourgain<p> </p><iframe width="560" height="315" src="https://www.youtube.com/embed/xMWlv_1DqyM" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-28147751035485871712023-03-01T12:07:00.000-05:002023-03-01T12:07:03.199-05:00More teaching chalkboards<p><span style="font-size: large;">Chinese Remainder Theorem...</span><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXwv8EdgxPJNrGaaMFCgeyS6TgpCezfMegMPaLrIXxVBlcDryhDd0XfLIt379ZcG1iG99jfmBYyHQzXKBM5GzF995kYzLxzoDMgWRcO2Lwu0-3rz4fGVSVmmead8O5uUAUNfYp0xM0eJHoZ_aavrtMKemqLvs1696ZpzRuJm5S8EBWOoNilISwLoAX/s2992/NT1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2992" data-original-width="2992" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXwv8EdgxPJNrGaaMFCgeyS6TgpCezfMegMPaLrIXxVBlcDryhDd0XfLIt379ZcG1iG99jfmBYyHQzXKBM5GzF995kYzLxzoDMgWRcO2Lwu0-3rz4fGVSVmmead8O5uUAUNfYp0xM0eJHoZ_aavrtMKemqLvs1696ZpzRuJm5S8EBWOoNilISwLoAX/w640-h640/NT1.jpg" width="640" /></a></div><br /><p></p><p></p><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOSnAk_E0c5xQVWEyZSyKuOa9R0UAuCWWSJ_PX8sfghFQ5-mokJSOE-KcIV3WTrDqG_KveoOzs3jdnwChI1jO_wq2kG0K4RPYfMef2Ar9Khpug8aWPdhKrBK5ZB8KDLNIX1VasI8xva8lu8zF6yVOYP7UPDBxKuITgD5LM0V5uz9mde-g-i74myFcg/s2992/NT2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2992" data-original-width="2992" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOSnAk_E0c5xQVWEyZSyKuOa9R0UAuCWWSJ_PX8sfghFQ5-mokJSOE-KcIV3WTrDqG_KveoOzs3jdnwChI1jO_wq2kG0K4RPYfMef2Ar9Khpug8aWPdhKrBK5ZB8KDLNIX1VasI8xva8lu8zF6yVOYP7UPDBxKuITgD5LM0V5uz9mde-g-i74myFcg/w640-h640/NT2.jpg" width="640" /></a></div><br /><p><br /></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-9677848174937369262023-02-01T10:49:00.001-05:002023-02-01T10:50:42.586-05:00Number theory and cryptography class - chalkboard pictures<p><span style="font-size: large;">Today, the topic was the Euclidean algorithm...</span></p><p><span style="font-size: large;">Just before erasing this number theoretic Tibetan mandala :) <br /></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikr29RVacNMFjBXy3VMwQIR5ISX6ZQDn1VDj3ELP761arkTeM8_J2w7T38DMEv7rLy7zyvLODHD6-qrlz4FLqW3iklY-vPhaPBwbbuIuOm0Y_9ixc4uabny8MG-Yc5RWgLPsF3aLloV8NjBQ_DYFDR37klG0YF74SMFgYxY_RVHI5SYYfm0qVJYkA8/s2992/N1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2992" data-original-width="2992" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikr29RVacNMFjBXy3VMwQIR5ISX6ZQDn1VDj3ELP761arkTeM8_J2w7T38DMEv7rLy7zyvLODHD6-qrlz4FLqW3iklY-vPhaPBwbbuIuOm0Y_9ixc4uabny8MG-Yc5RWgLPsF3aLloV8NjBQ_DYFDR37klG0YF74SMFgYxY_RVHI5SYYfm0qVJYkA8/w640-h640/N1.jpg" width="640" /></a></span></div><span style="font-size: large;"><br /></span><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAL_ta7og71Vpy90gJapuCkHSoPcthUYR20Fpc5FLpr9Dney2CgdlDBw4RR8b8Jl9vfBZSDhYzfAjngfNOyse83WL13wNDCowVq3aNSDtGfizqBTumpX4PhK8Xpfj0i-DHgjgOjW4FkwfoobuYiH8_Bxpjt02_BWnTPEfp33yjN40sHRYkC5FWwPfD/s2992/N2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2992" data-original-width="2992" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAL_ta7og71Vpy90gJapuCkHSoPcthUYR20Fpc5FLpr9Dney2CgdlDBw4RR8b8Jl9vfBZSDhYzfAjngfNOyse83WL13wNDCowVq3aNSDtGfizqBTumpX4PhK8Xpfj0i-DHgjgOjW4FkwfoobuYiH8_Bxpjt02_BWnTPEfp33yjN40sHRYkC5FWwPfD/w640-h640/N2.jpg" width="640" /></a></div><br /><p><br /></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-65302829146637824112022-12-16T21:34:00.002-05:002022-12-16T21:36:34.579-05:00Undergraduate research - a post pandemic restart<p><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto">My
first post-pandemic faculty-student <a href="http://www.pphmj.com/abstract/14613.htm" target="_blank">paper</a> and the 14-th overall
(written with Rachael Harbaugh, ONU '23, a talented Mathematics
Education major). Glad for this "restart". Students need confidence,
need to be exposed to interesting math topics, and then we hope for the
best in their future timelines.</span></span></p><p><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisbXIazkw9_EfNA60EvsGhCrZnStZ9oFJM6C4rqtGPifIg4ELuEfzdIl2So8yti40FCWnc-ffbWTdb8K5oBrE-pObaGFhgCK5NMOxoSVrx-fT_4xtNocAxVZ9c2dBc-gLkIq89jR-v7BIKEyNocDuclmUopFlhMpGNu406lHU0MulCD2dOFwGXLS1f/s1796/JPANTA%20nov%208%202022.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1796" data-original-width="1080" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisbXIazkw9_EfNA60EvsGhCrZnStZ9oFJM6C4rqtGPifIg4ELuEfzdIl2So8yti40FCWnc-ffbWTdb8K5oBrE-pObaGFhgCK5NMOxoSVrx-fT_4xtNocAxVZ9c2dBc-gLkIq89jR-v7BIKEyNocDuclmUopFlhMpGNu406lHU0MulCD2dOFwGXLS1f/w385-h640/JPANTA%20nov%208%202022.jpg" width="385" /></a></div><br /> <p></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-64782819375353087492022-12-09T23:37:00.004-05:002022-12-09T23:38:16.462-05:00Senior Capstone Colloquium - Fall 2022<p><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto">Mathematics
Capstone Colloquium - December 7, 2022 @ohionorthern (from L to R: Dr.
Chowdhury, Joelena Brown, Rachael Harbaugh, and Dr. Caragiu)</span></span></p><p><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0">Joelena Brown: "Rectangular Donut Numbers" (advisor Dr. Chowdhury)<br /></span></span></span></p><p><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto">Rachael Harbaugh: </span></span></span></span></span><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0"><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0">"Extending a Putnam Problem to Fields of Various Characteristics" (advisor Dr. Caragiu)</span></span></span> </span></span></span></p><p><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh49siApntpLM4JkuFnE_FY5N7ownnkEg_0Yfs1tRe82Wfz-uSwI5Yy7fWvayg99suec3APE1Mxy_41iq80tX6BBdzTUCjJ6Gh9Et2UT0GsCP29ELsDfZNnlhvROucw_wGTvsWK9RdjhHiQFYvvxq5sB-Y1sH_XoP9RZ5CE8biIWL70Z2FycBlSYwla/s2048/Capstone.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="2048" data-original-width="2048" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh49siApntpLM4JkuFnE_FY5N7ownnkEg_0Yfs1tRe82Wfz-uSwI5Yy7fWvayg99suec3APE1Mxy_41iq80tX6BBdzTUCjJ6Gh9Et2UT0GsCP29ELsDfZNnlhvROucw_wGTvsWK9RdjhHiQFYvvxq5sB-Y1sH_XoP9RZ5CE8biIWL70Z2FycBlSYwla/w640-h640/Capstone.jpg" width="640" /></a></span></span></div><span style="font-size: large;"><span class="x193iq5w xeuugli x13faqbe x1vvkbs x10flsy6 x1lliihq x1s928wv xhkezso x1gmr53x x1cpjm7i x1fgarty x1943h6x x4zkp8e x41vudc x6prxxf xvq8zen xo1l8bm xzsf02u x1yc453h" dir="auto"><br /> </span></span><p></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-32074414055288173182022-11-07T14:17:00.006-05:002022-11-07T14:17:44.461-05:00Zermelo Fraenkel Choice - with Richard Borcherds<iframe width="560" height="315" src="https://www.youtube.com/embed/pMda7i88xBc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-81436765555074944002022-11-03T21:58:00.005-04:002022-11-03T21:58:38.727-04:00An Invitation to Model Theory - by Jonathan Kirby, UEA<iframe width="560" height="315" src="https://www.youtube.com/embed/fssR2tDwIBc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-18606469480299471412022-06-29T22:16:00.000-04:002022-06-29T22:16:00.716-04:00Mobius Randomness and Dynamics - Peter Sarnak<iframe width="560" height="315" src="https://www.youtube.com/embed/reJ53lFF304" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-57300289550915677132022-06-20T22:51:00.000-04:002022-06-20T22:51:12.708-04:00The random Fibonacci sequence and the Viswanath's constant<p><span style="font-size: large;">Random Fibonacci recursion </span></p><p><span style="font-size: large;"><a href="https://en.wikipedia.org/wiki/Random_Fibonacci_sequence">https://en.wikipedia.org/wiki/Random_Fibonacci_sequence</a><br /></span></p><p>
</p><span style="font-size: large;"><blockquote>
In mathematics, the <b>random Fibonacci sequence</b> is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>±</mo>
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}</annotation>
</semantics>
</math></span><img alt="{\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1e263037c612f118013f1480d0bb42f7c66727" style="height: 2.509ex; vertical-align: -0.671ex; width: 17.213ex;" /></span>, where the signs + or − are chosen at random with equal probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>
</semantics>
</math></span><img alt="{\tfrac {1}{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" style="height: 3.509ex; vertical-align: -1.171ex; width: 1.658ex;" /></span>, independently for different <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img alt="n" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" style="height: 1.676ex; vertical-align: -0.338ex; width: 1.395ex;" /></span>. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943...(sequence <a href="https://oeis.org/A078416" target="_blank">A078416</a> in the OEIS), a mathematical constant that was later named Viswanath's constant.</blockquote></span>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.comtag:blogger.com,1999:blog-7262514835647136791.post-27071170078438930902022-05-31T02:15:00.005-04:002022-05-31T02:15:58.074-04:00Practice test for MATH 3701 - Complex Systems<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU0R08OTSjZwp1JvjDR0KzFS4BFcXYLh7V2rEsI2UylSl7JuWKLjKLmSSNkrr5N_aNO9qxmXwp3ySCx3egloLqj6lI_uQsOxsbyt2RDPjJarKH1uRxkqKJEoKsvnP10BcjU2HYeYpaw1veQtcd3KUjZI_YGyeadXuUBN2gUm2Awrh7a_2ziaCUdZYV/s1600/complex%20systems.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU0R08OTSjZwp1JvjDR0KzFS4BFcXYLh7V2rEsI2UylSl7JuWKLjKLmSSNkrr5N_aNO9qxmXwp3ySCx3egloLqj6lI_uQsOxsbyt2RDPjJarKH1uRxkqKJEoKsvnP10BcjU2HYeYpaw1veQtcd3KUjZI_YGyeadXuUBN2gUm2Awrh7a_2ziaCUdZYV/w640-h360/complex%20systems.jpg" width="640" /></a></div><br /><p></p>Mihai Caragiuhttp://www.blogger.com/profile/17518637740425832023noreply@blogger.com