Vimeo / Experimental mathematicshttps://vimeo.com/channels/636621/videosRutgers experimental mathematics seminarMon, 22 May 2017 20:06:00 -0400VimeoQuantum A-polynomials for knots using the q-Zeilberger algorithm (Part 2)Mon, 24 Feb 2014 21:00:24 -0500https://vimeo.com/channels/636621/87267455<p><iframe src="https://player.vimeo.com/video/87267455" width="640" height="360" frameborder="0" title="Quantum A-polynomials for knots using the q-Zeilberger algorithm (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Semeon Artamonov, Rutgers Experimental Mathematics Seminar, February 20, 2014</p> <p>See part 1 at <a href="https://vimeo.com/87227355">vimeo.com/87227355</a>.</p> <p>Abstract: My talk is devoted to application of the q-Zeilberger algorithm to knot theory and my recent research in that area. I will start with a basic introduction to knot invariants for graduate students and define braid group and Hecke algebras. Using the Turaev R-matrix approach I will explain how to evaluate one of the most powerful knot invariants, namely the colored HOMFLY polynomial. However, the naive application of the Turaev approach quickly makes evaluations impossible on today's computers. To overcome that issue I will explain the recent developments on cabling techniques and tricks I used in my programs.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-02-24:clip87267455Quantum A-polynomials for knots using the q-Zeilberger algorithm (Part 2)Quantum A-polynomials for knots using the q-Zeilberger algorithm (Part 1)Mon, 24 Feb 2014 21:00:24 -0500https://vimeo.com/channels/636621/87227355<p><iframe src="https://player.vimeo.com/video/87227355" width="640" height="360" frameborder="0" title="Quantum A-polynomials for knots using the q-Zeilberger algorithm (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Semeon Artamonov, Rutgers Experimental Mathematics Seminar, February 20, 2014</p> <p>See part 2 at <a href="https://vimeo.com/87267455">vimeo.com/87267455</a>.</p> <p>Abstract: My talk is devoted to application of the q-Zeilberger algorithm to knot theory and my recent research in that area. I will start with a basic introduction to knot invariants for graduate students and define braid group and Hecke algebras. Using the Turaev R-matrix approach I will explain how to evaluate one of the most powerful knot invariants, namely the colored HOMFLY polynomial. However, the naive application of the Turaev approach quickly makes evaluations impossible on today's computers. To overcome that issue I will explain the recent developments on cabling techniques and tricks I used in my programs.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-02-24:clip87227355Quantum A-polynomials for knots using the q-Zeilberger algorithm (Part 1)Fourier analysis of word maps (Part 1)Mon, 24 Feb 2014 21:00:24 -0500https://vimeo.com/channels/636621/86106466<p><iframe src="https://player.vimeo.com/video/86106466" width="640" height="360" frameborder="0" title="Fourier analysis of word maps (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Ori Parzanchevski, Rutgers Experimental Mathematics Seminar, February 6, 2014</p> <p>See part 2 at <a href="https://vimeo.com/86106849">vimeo.com/86106849</a>.</p> <p>Abstract: Let G be a finite group. How many times is an element g obtained as a commutator in G? Namely, how many solutions are there to the equation x*y*x^-1*y^-1=g ? In 1886 Frobenius gave a striking answer to this question in terms of the character theory of the G. But for a general word w replacing the commutator word x*y*x^-1*y^-1, surprisingly little is known. I will show some examples and survey old and recent results, including recent joint works with Doron Puder and Gili Schul.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-02-24:clip86106466Fourier analysis of word maps (Part 1)Fourier analysis of word maps (Part 2)Mon, 24 Feb 2014 21:00:24 -0500https://vimeo.com/channels/636621/86106849<p><iframe src="https://player.vimeo.com/video/86106849" width="640" height="360" frameborder="0" title="Fourier analysis of word maps (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Ori Parzanchevski, Rutgers Experimental Mathematics Seminar, February 6, 2014</p> <p>See part 1 at <a href="https://vimeo.com/86106466">vimeo.com/86106466</a>.</p> <p>Abstract: Let G be a finite group. How many times is an element g obtained as a commutator in G? Namely, how many solutions are there to the equation x*y*x^-1*y^-1=g ? In 1886 Frobenius gave a striking answer to this question in terms of the character theory of the G. But for a general word w replacing the commutator word x*y*x^-1*y^-1, surprisingly little is known. I will show some examples and survey old and recent results, including recent joint works with Doron Puder and Gili Schul.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-02-24:clip86106849Fourier analysis of word maps (Part 2)Methods of Computer Algebra for Orthogonal Polynomials (Part 2)Fri, 31 Jan 2014 19:12:59 -0500https://vimeo.com/channels/636621/85573712<p><iframe src="https://player.vimeo.com/video/85573712" width="640" height="360" frameborder="0" title="Methods of Computer Algebra for Orthogonal Polynomials (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Wolfram Koepf, Rutgers Experimental Mathematics Seminar, January 30, 2014</p> <p>See part 1 at <a href="https://vimeo.com/85573338">vimeo.com/85573338</a>.</p> <p>Abstract: In my talk I will give an introduction into the theory of classical orthogonal polynomials and their properties, and I will show, how in this theory computer algebra algorithms are useful. Some techniques that are treated will be demonstrated by live Maple examples.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-01-31:clip85573712Methods of Computer Algebra for Orthogonal Polynomials (Part 2)Methods of Computer Algebra for Orthogonal Polynomials (Part 1)Fri, 31 Jan 2014 19:12:59 -0500https://vimeo.com/channels/636621/85573338<p><iframe src="https://player.vimeo.com/video/85573338" width="640" height="360" frameborder="0" title="Methods of Computer Algebra for Orthogonal Polynomials (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Wolfram Koepf, Rutgers Experimental Mathematics Seminar, January 30, 2014</p> <p>See part 2 at <a href="https://vimeo.com/85573712">vimeo.com/85573712</a>.</p> <p>Abstract: In my talk I will give an introduction into the theory of classical orthogonal polynomials and their properties, and I will show, how in this theory computer algebra algorithms are useful. Some techniques that are treated will be demonstrated by live Maple examples.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2014-01-31:clip85573338Methods of Computer Algebra for Orthogonal Polynomials (Part 1)George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 2)Thu, 12 Dec 2013 19:23:35 -0500https://vimeo.com/channels/636621/81410565<p><iframe src="https://player.vimeo.com/video/81410565" width="640" height="360" frameborder="0" title="George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Doron Zeilberger, Rutgers Experimental Mathematics Seminar, December 5, 2013</p> <p>See part 1 at <a href="https://vimeo.com/81396551">vimeo.com/81396551</a>.</p> <p>Abstract: One of the greatest combinatorialists and number theorists of our time is George Andrews, who made partitions an active and hot mathematical area, and helped make Ramanujan a household name. But all this dwarfs compared with his PIONEERING use of Symbolic computation, starting with very creative use of the early computer algebra system SCRATCHPAD, and continuing to this day with the still-going-strong saga, joint with the RISC-Linz gang, on implementing and beautifully applying MacMahon's Omega calculus. Alas, George, being a tie-and-jacket wearing traditionalist, does not like revolutions, and hence even he does not realize the full impact of his mathematical legacy, that is FAR larger than the sum of its many impressive parts.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-12:clip81410565George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 2)George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 1)Thu, 12 Dec 2013 19:23:34 -0500https://vimeo.com/channels/636621/81396551<p><iframe src="https://player.vimeo.com/video/81396551" width="640" height="360" frameborder="0" title="George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Doron Zeilberger, Rutgers Experimental Mathematics Seminar, December 5, 2013</p> <p>See part 2 at <a href="https://vimeo.com/81410565">vimeo.com/81410565</a>.</p> <p>Abstract: One of the greatest combinatorialists and number theorists of our time is George Andrews, who made partitions an active and hot mathematical area, and helped make Ramanujan a household name. But all this dwarfs compared with his PIONEERING use of Symbolic computation, starting with very creative use of the early computer algebra system SCRATCHPAD, and continuing to this day with the still-going-strong saga, joint with the RISC-Linz gang, on implementing and beautifully applying MacMahon's Omega calculus. Alas, George, being a tie-and-jacket wearing traditionalist, does not like revolutions, and hence even he does not realize the full impact of his mathematical legacy, that is FAR larger than the sum of its many impressive parts.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-12:clip81396551George Eyre Andrews (b. Dec. 4, 1938): A Reluctant REVOLUTIONARY (Part 1)Rutgers Talk on the 290-TheoremMon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/80152913<p><iframe src="https://player.vimeo.com/video/80152913" width="640" height="360" frameborder="0" title="Rutgers Talk on the 290-Theorem" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Rutgers Experimental Math Seminar Talk<br> November 14, 2013 -- 5-5:48pm<br> -------------------------------------------------<br> Title: "The 290-Theorem and Representing Numbers by Quadratic Forms"</p> <p>Abstract: This talk will describe several finiteness theorems for quadratic<br> forms, and progress on the question: "Which positive definite<br> integer-valued quadratic forms represent all positive integers?". The<br> answer to this question depends on settling the related question<br> "Which integers are represented by a given quadratic form?" for<br> finitely many forms. The answer to this question can involve both<br> arithmetic and analytic techniques, though only recently with the <br> use of computers has the analytic approach become practical.</p> <p>We will describe the theory of quadratic forms as it relates to<br> answering these questions, its connections with the theory of modular<br> forms, and give an idea of how one can obtain explicit bounds to<br> describe which numbers are represented by a given quadratic form.<br> --------------------------------<br> Comments/Corrections: (Here [minute:second] describes the time stamp referenced)</p> <p>[5:58] Legendre's result on the sum of 4 squares was proved around 1770 (not in the 1800s), though Jacobi's result on the sum of 4 squares was proved around 1834.</p> <p>[12:14] "representing something in this set" should be "representing everything in this set'.</p> <p>[16:57] The escalator construction first appears in William Schneeberger's 1997 Princeton PhD thesis, and was later used by Bhargava to prove the finiteness theorems described in the talk. The escalator form approach is due to some combination of Conway and Schneeberger.</p> <p>[17:20] As an example of the "something else" one needs to know, one needs to get lucky that no ternary forms could appear as leaves of the escalator tree. If ternary forms do appear, then the analytic theory presented becomes ineffective without assuming there are no Siegel zeros (or the stronger statement of GRH).</p> <p>[20:55] The form in the genus of x^2 + 55y^2 representing 5 is 5x^2 + 11y^2.</p> <p>[21:52] There are only finitely many classes of positive-definite quadratic forms with class number one. In the indefinite case this not true.</p> <p>[22:27] The notion of spinor genus allows one to inderstand what numbers are represented by integer-valued indefinite quadratic forms in n >= 3 variables, however the given example of numbers represented by x^2 - 2y^2 cannot be easily understood in this way. This comment appears in the lower left corner of the transparency.</p> <p>[23:59] The real local densities are not exactly given by the surface area of the level set ellipsoid, but it is something very similar to this.</p> <p>[26:28] The stated quadratic form only has four variables, instead of the requisite five for the example, so I should also have said something like "+ 13 v^2" at the end.</p> <p>[32:23] An Eisenstein series (in our sense of being in the subspace of modular forms orthogonal to all cusp forms) is non-zero at at least one of the cusps, though perhaps not at all of them at once.</p> <p>[33:09] Actually the Eisenstein series grow more like m^{k-1}, but only when they are non-zero.</p> <p>[35:30] Again, the lower bound only applies to the non-zero coefficients a_E(m).</p> <p>[35:55] The number r_Q(m) is always non-negative, not always positive. (The whole point to for us is to understand exactly when it is positive, and when it is zero!)</p> <p>[36:06] This comment is premature (i.e. doesn't follow from the previous theorem as stated), though it is stated appropriately in the next transparency at [36:16].</p> <p>[37:10] I should have said what appears on the transparency, where one divides by p^{n-1} with n being the number of variables of the quadratic form (e.g. n=4).</p> <p>[37:55] Again, this effective lower bound is valid only when a_E(m) is not zero, which follows from the condition stated there that m is locally represented.</p> <p>[39:48] TO DO: Check this reference about the Ramanujan bound!</p> <p>[41:18] Here "quadratic form" means positive-definite integer-valued quadratic form in at least four variables.</p> <p>[43:21] My answer here that the the numerical methods always have bounds that worked is accurate, though with the caveat that for safety/comfort we consistently added one to the numerical bounds, and the exact answers always were within this "+1" tolerance that we used for further checking. Without this tolerance the numerical answers were not always sufficient, so it's very important to be aware of possible roundoff error!</p> <p>[46:22] There is a subtlety here that in (3) one may need to check a few more primes than those not meeting the bound individually in (1) when assembling the eligible squarefree numbers in (2) which strictly speaking I have glossed over, but it is very minor.</p> <p>[46:34] I meant that one doesn't want to give this to a graduate student not using a computer. I did actually give this to my graduate student as an exercise to check her understanding! =)</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/jonhanke">Jonathan Hanke</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Math">Math</a>, <a href="https://vimeo.com/tag:Quadratic">Quadratic</a>, <a href="https://vimeo.com/tag:Forms">Forms</a> and <a href="https://vimeo.com/tag:290-Theorem">290-Theorem</a></p>tag:vimeo,2013-12-02:clip80152913Rutgers Talk on the 290-TheoremGyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/74422863<p><iframe src="https://player.vimeo.com/video/74422863" width="640" height="360" frameborder="0" title="Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Doron Zeilberger, Rutgers Experimental Mathematics Seminar, September 12, 2013</p> <p>See part 2 at <a href="https://vimeo.com/75107518">vimeo.com/75107518</a>.</p> <p>Abstract: See <a href="http://math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html" target="_blank" rel="nofollow noopener noreferrer">math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html</a>.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip74422863Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 1)Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/75107518<p><iframe src="https://player.vimeo.com/video/75107518" width="640" height="360" frameborder="0" title="Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Doron Zeilberger, Rutgers Experimental Mathematics Seminar, September 12, 2013</p> <p>See part 1 at <a href="https://vimeo.com/74422863">vimeo.com/74422863</a>.</p> <p>Abstract: See <a href="http://math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html" target="_blank" rel="nofollow noopener noreferrer">math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html</a>.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip75107518Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem (Part 2)Pattern Avoidance in Trees (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/75574041<p><iframe src="https://player.vimeo.com/video/75574041" width="640" height="360" frameborder="0" title="Pattern Avoidance in Trees (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Lara Pudwell, Rutgers Experimental Mathematics Seminar, September 26, 2013.</p> <p>See part 1 at <a href="https://vimeo.com/75964711">vimeo.com/75964711</a>.</p> <p>Abstract: Pattern-avoiding trees have appeared in various computational contexts since at least the 1980s. A more recent topic of interest is the exact enumeration of trees that avoid some other tree pattern. In 2010, Rowland considered this enumeration problem for rooted, ordered binary trees where tree T contains tree pattern t if and only if T contains t as a contiguous rooted, ordered subtree. In this talk we consider Rowland's contiguous tree patterns as well as non-contiguous tree patterns. We also consider implications of tree enumeration results for pattern-avoiding permutations.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip75574041Pattern Avoidance in Trees (Part 2)Congruences for algebraic sequences (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/75612108<p><iframe src="https://player.vimeo.com/video/75612108" width="640" height="360" frameborder="0" title="Congruences for algebraic sequences (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Eric Rowland, Rutgers Mathematics Department Colloquium and Experimental Mathematics Seminar, September 27, 2013.</p> <p>See part 1 at <a href="https://vimeo.com/75965904">vimeo.com/75965904</a>.</p> <p>Abstract: In the past decade there have been several papers studying number theoretic properties of fundamental combinatorial sequences such as the Catalan and Motzkin numbers. For the most part, the proofs use ad hoc methods particular to the combinatorics of each sequence. I will talk about a general method for producing congruences for sequences whose generating functions are algebraic. This method gives completely automatic proofs of existing theorems and has produced many new theorems. Main ingredients include finite automata and diagonals of formal power series. Joint work with Reem Yassawi.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip75612108Congruences for algebraic sequences (Part 2)Pattern Avoidance in Trees (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/75964711<p><iframe src="https://player.vimeo.com/video/75964711" width="640" height="360" frameborder="0" title="Pattern Avoidance in Trees (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Lara Pudwell, Rutgers Experimental Mathematics Seminar, September 26, 2013.</p> <p>See part 2 at <a href="https://vimeo.com/75574041">vimeo.com/75574041</a>.</p> <p>Abstract: Pattern-avoiding trees have appeared in various computational contexts since at least the 1980s. A more recent topic of interest is the exact enumeration of trees that avoid some other tree pattern. In 2010, Rowland considered this enumeration problem for rooted, ordered binary trees where tree T contains tree pattern t if and only if T contains t as a contiguous rooted, ordered subtree. In this talk we consider Rowland's contiguous tree patterns as well as non-contiguous tree patterns. We also consider implications of tree enumeration results for pattern-avoiding permutations.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip75964711Pattern Avoidance in Trees (Part 1)Congruences for algebraic sequences (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/75965904<p><iframe src="https://player.vimeo.com/video/75965904" width="640" height="360" frameborder="0" title="Congruences for algebraic sequences (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Eric Rowland, Rutgers Mathematics Department Colloquium and Experimental Mathematics Seminar, September 27, 2013.</p> <p>See part 2 at <a href="https://vimeo.com/75612108">vimeo.com/75612108</a>.</p> <p>Abstract: In the past decade there have been several papers studying number theoretic properties of fundamental combinatorial sequences such as the Catalan and Motzkin numbers. For the most part, the proofs use ad hoc methods particular to the combinatorics of each sequence. I will talk about a general method for producing congruences for sequences whose generating functions are algebraic. This method gives completely automatic proofs of existing theorems and has produced many new theorems. Main ingredients include finite automata and diagonals of formal power series. Joint work with Reem Yassawi.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip75965904Congruences for algebraic sequences (Part 1)On configurations in finite projective planes (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/76108328<p><iframe src="https://player.vimeo.com/video/76108328" width="640" height="360" frameborder="0" title="On configurations in finite projective planes (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Felix Lazebnik, Rutgers Experimental Mathematics Seminar, October 3, 2013.</p> <p>See part 2 at <a href="https://vimeo.com/76139900">vimeo.com/76139900</a>.</p> <p>See the paper at <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p24/pdf" target="_blank" rel="nofollow noopener noreferrer">combinatorics.org/ojs/index.php/eljc/article/view/v20i3p24/pdf</a>.</p> <p>Abstract: In this talk I will discuss some old and some new results and open problems on point-line configurations that one can find in finite projective planes.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip76108328On configurations in finite projective planes (Part 1)On configurations in finite projective planes (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/76139900<p><iframe src="https://player.vimeo.com/video/76139900" width="640" height="360" frameborder="0" title="On configurations in finite projective planes (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Felix Lazebnik, Rutgers Experimental Mathematics Seminar, October 3, 2013.</p> <p>See part 1 at <a href="https://vimeo.com/76108328">vimeo.com/76108328</a>.</p> <p>See the paper at <a href="http://combinatorics.org/ojs/index.php/eljc/article/view/v20i3p24/pdf" target="_blank" rel="nofollow noopener noreferrer">combinatorics.org/ojs/index.php/eljc/article/view/v20i3p24/pdf</a>.</p> <p>Abstract: In this talk I will discuss some old and some new results and open problems on point-line configurations that one can find in finite projective planes.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip76139900On configurations in finite projective planes (Part 2)2178 And All That (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/76725343<p><iframe src="https://player.vimeo.com/video/76725343" width="640" height="360" frameborder="0" title="2178 And All That (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Neil J. A. Sloane, Rutgers Experimental Mathematics Seminar, October 10, 2013.</p> <p>See part 2 at <a href="https://vimeo.com/77255410">vimeo.com/77255410</a>.</p> <p>Abstract: For integers g >= 2, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood. The talk will mention William The Conquerer, G. H. Hardy, The Unabomber, Lara Pudwell, and others.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip767253432178 And All That (Part 1)"Magic" numbers in Smale's 7th problemMon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/77192778<p><iframe src="https://player.vimeo.com/video/77192778" width="640" height="360" frameborder="0" title=""Magic" numbers in Smale's 7th problem" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Michael Kiessling, Rutgers Experimental Mathematics Seminar, October 17, 2013.</p> <p>Abstract: Smale's 7th problem concerns N-point configurations on the 2-sphere which minimize the logarithmic pair-energy V(r) = - ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V(r) may be replaced by the standardized Riesz pair-energy. In a recent paper with Brauchart and Nerattini we inquired into the concavity of the map from the integers >1 into the minimal average standardized Riesz pair-energies of the N-point configurations on the sphere. It is known that this map is strictly increasing and in some Riesz parameter range bounded above, hence "overall concave." It is (easily) proved that for the Riesz parameter -2 it is even locally strictly concave. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies for the Riesz parameters -1,0,1,2,3 and N up to 200 we found that the map in question is locally strictly concave for parameter -1, while not always locally strictly concave for the other parameter values. It is found that the empirical map from the Riesz parameter into the set of convex defect-N is set-theoretically increasing; moreover, the percentage of odd numbers in the range is found to increase with the Riesz parameter. They form a curious sequence of numbers for the logarithmic kernel, reminiscent of the "magic numbers" in nuclear physics; it is conjectured that the "magic numbers" in Smale's 7th problem are associated with optimally symmetric optimal-energy configurations. The talk emphasizes the role of computer experiments, in particular also of Maple, in our investigation.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip77192778"Magic" numbers in Smale's 7th problem2178 And All That (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/77255410<p><iframe src="https://player.vimeo.com/video/77255410" width="640" height="360" frameborder="0" title="2178 And All That (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Neil J. A. Sloane, Rutgers Experimental Mathematics Seminar, October 10, 2013.</p> <p>See part 1 at <a href="https://vimeo.com/76725343">vimeo.com/76725343</a>.</p> <p>Abstract: For integers g >= 2, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood. The talk will mention William The Conquerer, G. H. Hardy, The Unabomber, Lara Pudwell, and others.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip772554102178 And All That (Part 2)Computer-assisted bijectification of algebraic proofsMon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/77743227<p><iframe src="https://player.vimeo.com/video/77743227" width="640" height="360" frameborder="0" title="Computer-assisted bijectification of algebraic proofs" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Nathaniel Shar, Rutgers Experimental Mathematics Seminar, October 24, 2013.</p> <p>Abstract: If a(n) is the sum of the cubes of the entries on the nth row of Pascal's triangle, then (n+1)^2 a(n) = (7n^2 - 7n + 2)a(n-1) + 8(n-1)^2a(n-2). It seems challenging for a human to find a bijective proof of this, but a computer can do it, with a little help. I'll show you a real live bijection, implemented of course by the computer, that proves this identity, and describe a method that might help computers bijectify other difficult identities.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip77743227Computer-assisted bijectification of algebraic proofsZeta Function-Like Sums Over Lucas Numbers (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/78868802<p><iframe src="https://player.vimeo.com/video/78868802" width="640" height="360" frameborder="0" title="Zeta Function-Like Sums Over Lucas Numbers (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">William Kang, Rutgers Experimental Mathematics Seminar, November 7, 2013.</p> <p>See part 2 at <a href="https://vimeo.com/78869277">vimeo.com/78869277</a>.</p> <p>Abstract: The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if F(n) is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F(0) = 0, F(1) =1,F(n) =F(n-1)+F(n-2) and L0 =2,L1 =1,Ln =L(n-1)+L(n-2), respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip78868802Zeta Function-Like Sums Over Lucas Numbers (Part 1)Zeta Function-Like Sums Over Lucas Numbers (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/78869277<p><iframe src="https://player.vimeo.com/video/78869277" width="640" height="360" frameborder="0" title="Zeta Function-Like Sums Over Lucas Numbers (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">William Kang, Rutgers Experimental Mathematics Seminar, November 7, 2013.</p> <p>See part 1 at <a href="https://vimeo.com/78868802">vimeo.com/78868802</a>.</p> <p>Abstract: The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if F(n) is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F(0) = 0, F(1) =1,F(n) =F(n-1)+F(n-2) and L0 =2,L1 =1,Ln =L(n-1)+L(n-2), respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip78869277Zeta Function-Like Sums Over Lucas Numbers (Part 2)Noncommutative recursions and the Laurent phenomenon (Part 1)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/80043138<p><iframe src="https://player.vimeo.com/video/80043138" width="640" height="360" frameborder="0" title="Noncommutative recursions and the Laurent phenomenon (Part 1)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Matthew Russell, Rutgers Experimental Mathematics Seminar, November 21, 2013</p> <p>See part 2 at <a href="https://vimeo.com/80043139">vimeo.com/80043139</a>.</p> <p>Abstract: We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip80043138Noncommutative recursions and the Laurent phenomenon (Part 1)Noncommutative recursions and the Laurent phenomenon (Part 2)Mon, 02 Dec 2013 19:03:59 -0500https://vimeo.com/channels/636621/80043139<p><iframe src="https://player.vimeo.com/video/80043139" width="640" height="360" frameborder="0" title="Noncommutative recursions and the Laurent phenomenon (Part 2)" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe></p><p><p class="first">Matthew Russell, Rutgers Experimental Mathematics Seminar, November 21, 2013</p> <p>See part 1 at <a href="https://vimeo.com/80043138">vimeo.com/80043138</a>.</p> <p>Abstract: We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.</p></p><p><strong>Cast:</strong> <a href="https://vimeo.com/experimentalmathematics">Experimental Mathematics</a></p><p><strong>Tags:</strong> <a href="https://vimeo.com/tag:Experimental+mathematics">Experimental mathematics</a> and <a href="https://vimeo.com/tag:Rutgers">Rutgers</a></p>tag:vimeo,2013-12-02:clip80043139Noncommutative recursions and the Laurent phenomenon (Part 2)