tag:blogger.com,1999:blog-7315507820652697532014-10-02T21:53:25.008-07:00Frank Moore's classFrankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.comBlogger29125tag:blogger.com,1999:blog-731550782065269753.post-12680724689596588652010-05-11T09:40:00.000-07:002010-05-11T09:52:42.528-07:00Additional Hint on 3b)Here is an additional hint on 3b: Note that with such a substantial hint, I expect cleaner proofs ;)<br /><br />i) Prove that (I:f) is monomial by induction on the number of terms of f, which we may assume are all *not* in I by part a). To do this, let g be in (I:f). We need to show that every term of g is in (I:f). Consider gf. It is in I, so LT(g)LT(f) is in I.<br /><br />Now consider (I:LT(g)f). Now (I:f) is a subset of (I:LT(g)f). Also, note that (I:LT(g)f), if prime, is monomial by the inductive hypothesis (Why?!?). Now show that if (I:f) is a proper subset of (I:LT(g)f), then LT(g) \in (I:f), and now repeat the argument for g-LT(g).<br /><br />The following facts will help you to prove that (I:f) = (I:m) for some monomial m:<br /><br />ii) Note that since I is monomial, one has that (I:f) = intersection of (I:f_i) for all the terms f_i of f. [You should prove this]<br /><br />iii) Recall a (jazzed up version of a) homework problem: If P is prime and the product of the ideals I_1, I_2, ..., I_t (denoted I_1I_2I_3...I_t) is in P, then I_j is in P for some j. You may just cite this version without giving the (easy) proof using induction and the previous homework problem.<br /><br />To show that if I is squarefree, then one can take m to be squarefree, you should look at Theorem 4.4.11 in the book.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-48019703281096115902010-05-08T10:30:00.000-07:002010-05-08T10:34:05.662-07:00Office hours this weekend<span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;">Unfortunately, I won't be able to hold my regular office hours this week. To compensate, I'm holding office hours this weekend:</span></span><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;">Saturday 3:30 to 5:30</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;">I can also see people on Sunday morning if you send me an e-mail</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;">Best of luck with your final exam.</span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:arial;"><span class="Apple-style-span" style="font-size:small;">Saúl</span></span></div>Saúl Blancohttp://www.blogger.com/profile/16162084691903257574noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-7978144147441532252010-05-07T11:00:00.000-07:002010-05-07T11:06:48.651-07:00Final ExamThe final exam is available <a href="http://www.math.cornell.edu/~frankmoore/exam2Math4370.pdf">here</a>.<br /><br />Note that the wording on problem 2 is changed slightly. One must assume that f is homogeneous in part a, and that I is a homogeneous ideal in part b) [see problem 1 for a definition of homogeneous].Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-90004765949427265352010-05-02T21:51:00.000-07:002010-05-02T21:53:07.950-07:00Office hours on 05/05/2010<span class="Apple-style-span" style="font-size: small;">My office hours for Wednesday May 5 will be from 1 PM to 2 PM in Malott 112.</span><div><span class="Apple-style-span" style="font-size: small;"><br /></span></div><div><span class="Apple-style-span" style="font-size: small;">Saúl</span></div>Saúl Blancohttp://www.blogger.com/profile/16162084691903257574noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-21244032015322901142010-04-30T07:33:00.001-07:002010-04-30T07:34:11.508-07:00Updated example fileThe updated Macaulay2 example file from class is available <a href="http://www.math.cornell.edu/~frankmoore/math4370apps1.m2">here</a>.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-88108077947513208542010-04-30T06:34:00.000-07:002010-04-30T06:35:38.683-07:00Homework #6 correctionI had the numbers wrong in the additional problem that was assigned. The correct problem is now listed in the homework (I changed the 'b' vector from [-323,1035,654] to [-323,389,8]).Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-23698347205399489352010-04-26T06:32:00.001-07:002010-04-30T06:34:27.966-07:00Homework #6Sorry for the delay in assigning the homework. Posting the problem set completely slipped my mind until Sunday evening. The due date for this homework will be May 6th to account for the delayed assignment time.<br /><br />6.1: 4<br />6.2: 4<br />6.3: 5,7,10<br />7.1: 5,8<br /><br />1. Find an optimal integer solution to the following integer programming problem using the cost function c(s_1,s_2,s_3,s_4,s_5) = 10s_1+100s_2+100s_3+10s_4+s_5, where<br />2s_1+5s_2-3s_3+s_4-2s_5=-323<br />s_1+7s_2+2s_3+3s_4+s_5=389<br />4s_1-2s_2-s_3-5s_4+3s_5=8Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-84851636239249862792010-04-21T12:49:00.000-07:002010-04-21T13:10:51.131-07:00HW 5, what to hand inI have received questions about what needs to be hand in for the computer problems. Here is what I think you need FOR ALL THREE PROBLEMS: The input (hypothesis, conclusions, edges, etc), the output, and a few sentences explaining what the output means and why it gives you (or not) what you want. It would be nice if you mark these clearly, so it is easier for me to find. Also, it is perfectly find to hand in printouts of what the input/output (no need to write them down with your own handwriting).<div><br /></div><div>Good luck with your HW.</div><div><br /></div><div>Saúl </div>Saúl Blancohttp://www.blogger.com/profile/16162084691903257574noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-48718970078509102732010-04-19T14:33:00.001-07:002010-04-19T14:41:27.452-07:00Homework #5 correctionThere was a slight problem with question #2. Here is the corrected version:<br /><br />2. a) Using a computer algebra system, show that the graph with edges<br />{(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,7),(3,4),(3,6),(3,9),(5,6),(6,8),(6,9),(7,8),(8,9)}<br /><br />is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring. How many colorings are there?<br />b) Show that if you add edge (1,3) to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-59931225009395676102010-04-14T11:06:00.001-07:002010-04-14T11:20:53.209-07:00Homework 5 due dateI said in class, yet forgot to post on the blog, that homework #5 is due on April 22nd. By the way, I think I will assign one more homework on the 22nd that will be due May 4th. The final will also be assigned on May 4th, and will be due on the day the final is scheduled to be for our class, which is Thursday May 13th.<br /><br />Also, <a href="http://www.math.cornell.edu/~frankmoore/math4370apps1.m2">here</a> is the Macaulay2 file that illustrates the automated geometric proving and colorability of graphs applications from class.<br /><br />I have also placed the book <a href="http://www.ams.org/bookstore?fn=20&arg1=gsmseries&ikey=GSM-3">An Introduction to Grobner Bases</a> by Adams and Loustaunau on reserve in the math library. The material on colorability of graphs, as well as the applications to linear programming are covered in chapter 2 of that book.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-67061124124877156882010-04-09T07:41:00.000-07:002010-04-09T08:41:18.887-07:00Homework #5Problems from the text:<br /><br />4.5: 3,8,11,12<br />4.6: 9<br />6.4: 3,8<br /><br />Additional problems:<br /><br />1. Using the Grobner basis technique from class (and of course, a computer algebra system), prove that Euler's nine point circle theorem is true. You may use either the fraction field method, or the saturation method to do this problem, but since the conclusions do not follow strictly from the hypotheses, you must use one or the other. A nice picture and discussion of the theorem is available <a href="http://www.calvin.edu/~venema/eeg/eeg-7.pdf">here.</a> <br /><br />The statement is the following. Let ABC be a triangle. Then the midpoints of the sides, the feet of the altitudes, and the midpoints of the line segments joining the orthocenter (which is the intersection of the altitudes) and the vertices of the triangle, all lie on a single circle. The circle theorem of Appolonius is Euler's theorem in the case of a right triangle.<br /><br />When I worked this problem out, I had 3 'u' variables, and 20 'x' variables, and therefore 20 'hypotheses'. There should be six 'conclusions' that you have to check. One way to proceed would be to consider the circle that goes through the midpoints of the sides, and show that the other 6 points are on this circle.<br /><br />2. a) Using a computer algebra system, show that the graph with edges {(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,7),(3,6),(3,9),(5,6),(6,8),(7,8),(8,9)} is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring.<br />b) Show that if you add a new edge {(1,3)} to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.<br /><br />3) Use the method of linear programming we learned in class to solve the following system of equations in the nonnegative integers:<br />3s_1+2s_2+s_3+2s_4=10<br />4s_1+3s_2+s_3=11<br />2s_1+4s_2+2s_3+s_4=10Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-25838053320992375692010-04-02T09:25:00.000-07:002010-04-02T09:29:02.477-07:00Exam Solutions<a href="http://www.math.cornell.edu/~frankmoore/exam1Math4370Solns.pdf">Here</a> are solutions to the exam.<br /><br />FrankFrankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-57331242051876523022010-03-30T17:31:00.000-07:002010-04-02T09:29:30.472-07:00Homework #43.3: 11a, 13, 14<br />4.1: 1, 4<br />4.3: 11, 12<br />4.4: 2,9<br /><br />This homework is due on April 8th.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-29517518360239741282010-03-12T16:05:00.000-08:002010-03-12T16:12:05.263-08:00Exam and Macaulay2 file<a href="http://www.math.cornell.edu/~frankmoore/lagrange.m2">Here</a> is the Macaulay2 file alluded to in the <a href="http://www.math.cornell.edu/~frankmoore/exam1Math4370.pdf">exam</a>.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-35398886589489788022010-03-08T20:06:00.000-08:002010-04-02T09:29:53.128-07:00Homework #3 CommentPlease take a look at the comment on the homework #3 post.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-44362704241647736252010-03-02T11:12:00.000-08:002010-03-02T11:19:19.460-08:00Macaulay2 scriptThe Macaulay2 script solving the Lagrange multiplier problem can be found <a href="http://www.math.cornell.edu/~frankmoore/lagrange.m2">here</a>.<br /><br />One can perform this computation in a <a href="http://www.sagenb.com">Sage Notebook</a> screen with macaulay2 enabled as I mentioned in an earlier blog post, or you can try installing Macaulay2 and emacs on your personal machine by following the installation instructions on the <a href="http://www.macaulay2.com">Macaulay2 Home Page</a>.<br /><br />If you have any problems setting it up, then I can try and help you via email.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-55255897368592499632010-02-23T06:38:00.000-08:002010-04-02T09:30:09.643-07:00Homework #3Problems from the book:<br /><br />2.6: 1, 4, 12<br />2.7: 2, 7<br />2.8: 8<br /><br />1) Let I be a monomial ideal and > any term order. Show that any generating set for I is also a Grobner basis.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com1tag:blogger.com,1999:blog-731550782065269753.post-32996455131314330042010-02-22T19:26:00.000-08:002010-04-02T09:30:28.036-07:00HW Comments<span class="Apple-style-span" style="font-family: Helvetica; font-size: medium; "><div style="font-size: 18px; "><span class="Apple-style-span" style="font-family: Helvetica; font-size: medium; "><div>The fist HW set has been graded. In general people did pretty well. Here are a couple of comments to consider for future assignments. </div><div><br /></div><div>1) Please be neat. Your paper should not have scribbles or crossed out words. It should be correctly stapled, and the writing should be legible.</div><div><br /></div><div>2) Write in full sentences. Look at the proofs in your textbook, they are all written in full sentences, and not as a collection of equations.</div><div><br /></div><div>If there are any complaints about the grading, don't hesitate to come to office hours to talk about it. My office hours are Monday and Wednesday from 1:30 to 2:30 PM in Malott 112. </div><div><br /></div><div>Thank you,</div><div><br /></div><div>Saul</div></span></div></span>Saúl Blancohttp://www.blogger.com/profile/16162084691903257574noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-85224740660575805102010-02-22T17:53:00.000-08:002010-04-02T09:30:49.837-07:00Timetable up until Exam 1Now that we are nearing the first exam time (already!) I thought I would let you know the timetable that I had in mind for the next assignment, as well as the first exam:<br /><br />HW #2 due Feb 25th<br />HW #3 assigned Feb 25th<br />HW #3 due Mar 9th<br />Exam #1 assigned Mar 9th<br />Exam #1 due Mar 18th<br /><br />FrankFrankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-16848038469570330632010-02-22T10:00:00.000-08:002010-02-22T13:38:35.277-08:00HW ConfusionIn my original homework post, I said that only 15a of section 1.5 was required for the homework. Later, I posted how one could go about doing problem 15b online via Sage, making mention that it was in fact an assigned homework problem. Just to clarify, problem 15b will *not* be required of homework #2.<br /><br />Sorry for any confusion.<br /><br />FrankFrankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-75729212918768631592010-02-22T07:49:00.000-08:002010-02-22T08:00:57.556-08:00HW ExtensionIt has come to my attention that the monomial ideals problem that I assigned is much easier with one more lemma under your belt. So, what I propose is that I extend the homework deadline to Thursday of this week, and add the lemma as an exercise that makes the rest of your homework much easier. It will also be useful at various points in the course. <br /><br />Lemma:<br />Let I be an ideal of k[x_1,...,x_n]. Suppose that I has the property that for any f in I, every term of f also is in I. Show that I is a monomial ideal.<br /><br />One can view this as the converse to Lemma 3 in section 2.4 of IVA; it is assigned as problem 1 in the exercises in that section. In some texts, this is the <span style="font-style:italic;">definition</span> of a monomial ideal.<br /><br />So, to recap, homework is now due on Thursday Feb 25th, and proving the preceding lemma is added to the list of assigned problems. This lemma should be extremely useful in the problem on sums, intersections, and radicals of monomial ideals, especially the radical problem. One can fumble through the sums and intersections problems without it, but trying the radical problem without it requires significantly more mathematical stamina.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-75004495668068876192010-02-19T11:43:00.000-08:002010-02-19T12:25:03.032-08:00Computer Algebra SystemThis is just a quick tutorial on how to use Sage Notebook to answer the 'computational' problem on this homework, namely problem 15.b in section 1.5. One does not need to use a computer to do this problem (it is not *that* hard to find compute the GCD), but for those of you that want to use a computer, here is how it can be done.<br /><br />Go to www.sagenb.com and create a log in. You will be presented with the (currently empty) list of worksheets you have created. Create a new worksheet by clicking on the "New Worksheet" button along the top row. We will work through an example; note that this is *not* the example that you are asked to compute in the problem.<br /><br />Near the top, you will see a drop down box that says 'sage'. Select 'macaulay2' from the dropdown box. After you type in each cell, be sure to click 'evaluate' at the bottom of the cell to see what the result of running each command will be.<br /><br />In the first cell, type in the following:<br /><br />R = QQ[x]<br />f = x^9 - 29*x^8 + 371*x^7 - 2747*x^6 + 12968*x^5 - 40460*x^4 + 83392*x^3 - 109440*x^2 + 82944*x - 27648<br /><br />This tells sage that we are working in the ring QQ[x], and we are going to try to find the squarefree part of f. In the next block, put<br /><br />fprime = diff(x,f)<br /><br />This is just the derivative of f with respect to x. In the next box, put<br /><br />gcd(f,fprime)<br /><br />And then finally, in the next box put.<br /><br />h = f // gcd(f,fprime)<br /><br />To verify our work, check that both elements factor to give the right thing, so in two boxes, put<br /><br />factor f<br /><br />and<br /><br />factor h<br /><br />This is exactly what the problem told us would happen :)Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-70422818513671902662010-02-05T08:08:00.000-08:002010-02-11T06:19:41.703-08:00Homework #2Here is the next homework assignment:<br /><br />By the way, below I use (f_1,...,f_s) to denote the ideal generated by the {f_i} instead of the angle brackets, since HTML doesn't like it when I try to use angle brackets.<br /><br />1.5: 12, 13 (use linearity of the formal derivative, and prove this for monomials first!), 14, 15a<br />2.2: 2<br />2.3: 5<br /><br />1. Let I and J be ideals of k[x_1,...,x_n]. Show that I intersect J, and I+J are ideals, where I+J = {f + g | f is in I and g is in J}.<br /><br />2. Let I be an ideal of k[x_1,...x_n]. We say I is a <span style="font-style:italic;">monomial ideal</span> if I can be generated by monomials m_1,...,m_s in the variables x_1,...,x_n. For example, (x^2) is a monomial ideal of k[x], while (x+x^2) is not.<br /><br />a) Show that intersections and sums of monomial ideals are monomial.<br />b) Let I = (m_1,...,m_s). Compute a generating set for the radical of I, and show that it is also a monomial ideal.<br />c) Show that a monomial ideal I is radical if and only if I has a monomial generating set where each monomial is <span style="font-style:italic;">squarefree</span>, i.e., the power on each variable in the monomial is 1. (i.e., xyz^2 is not squarefree, but xyz is).<br /><br />The due date is February 23rd.<br /><br />FrankFrankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-8681488835856605552010-02-05T07:22:00.000-08:002010-02-05T08:16:48.170-08:00Ideals and VarietiesIn this past week, we have introduced a fair amount of new objects, so lets recap what we have done, as well as give an indication as to where we are headed.<br /><br />We defined an affine variety V to be the set of points in affine n-space k^n satisfying a system of polynomial equations f_1 = ... = f_k = 0. There are many times in applications where one wants to understand properties of such a set. Varieties can describe many things, such as allowable positions of robotic arms, solutions to Sudoko puzzles, and others.<br /><br />The description of an affine variety as a zero locus of a collection of polynomials is called an implicit description. When using such a description, it is easy to determine if a point in k^n is on the variety, but hard to find all the points. A description of (most of) the points on the variety via rational functions is called a parameterization of the variety. Parameterizations are nice because they make it easy to find points on the variety, but hard to know whether a given point is in the variety.<br /><br />We did several examples of both polynomial and rational parameterizations of some different varieties, namely the circle, ellipse, parabola, and the tangent surface to the twisted cubic. One can always go from a rational parameterization of an affine variety to an implicit description using elimination theory, but not all varieties have a single rational parameterization of 'most' of the points. We'll see this later on in the class.<br /><br />In linear algebra, one introduces the idea of a subspace in order to study solutions of linear systems. In order to study varieties, one introduces the notion of an ideal. An ideal is a subset of the polynomial ring that is closed under sums and "polynomial scalings", that is, if r is a polynomial, and f is in the ideal, then rf is also in the ideal. For us, we will be interested in the smallest ideal containing a set of polynomials f_1,...,f_k. This ideal can be thought of as the set of 'polynomial consequences' of the equations f_1 = 0, ..., f_k = 0. We'll spend a fair amount of time studying ideals, and algorithms that pertain to them, in order to study varieties.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0tag:blogger.com,1999:blog-731550782065269753.post-43334289855181547062010-02-02T06:47:00.000-08:002010-02-02T06:55:33.700-08:00Homework #1The first homework will be the following problems from Cox, Little, and O'Shea:<br /><br />1.1: #5<br />1.2: #6,7,8<br />1.3: #6,8<br />1.4: #6<br /><br />Also:<br /><br />Find a Lagrange multiplier problem in your old calculus book (you still have one don't you ;) that has some messy equations that one must solve to get an answer, and solve it (how messy is up to your own discretion). We'll learn a much easier way of solving these systems later, and I will use some of these examples in class later on.<br /><br />The due date for this assignment is February 11th.Frankhttp://www.blogger.com/profile/12982629384914135540noreply@blogger.com0