Math 205a stanford reddit. Students are required to attend classes.

Math 205a stanford reddit Office hours: Tue 5-7PM. Course Content: A tentative list of topics can be can be found at 205a-content: Lectures: Math 205A - Fall 2016 Homework #3 Solutions Problem 1: (i) Show that step functions and continuous functions are dense in Lp[0;1];1 p<1. Let xhave two ternary expansions 0:a 1 a 2:::and 0;a0a0 2:::, with a tail of 2s in the rst, and let N, N0and b n, b0 n be the respective quantities of Math 205A - Fall 2014 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. Can someone share their experience about cs205l ? I lack solid math foundation but I survived ML classes in the past (not in Stanford). Sep 24, 2024 · Andrea Montanari. In this case, there 9msuch that a n = a0 It is also recommended to have taken at least one of Math 122, Math 145, or Math 154. Your average stanford student (eg people who did well with American high school math and got a 5 in calc bc, but have little experience with proof-based math) will find it pretty hard. Billingsley, Probability and Measure. My understanding is that Math 56 was created for students interested in gaining a mathematical foundation in proofs, but who aren't inclined to take the more rigorous Math 61/62/63 CM/DM series. Thus S= [nS nis a The complete Stanford Bulletin - course listings for the Academic quarter Painleve), and stochastic calculus. Syllabus. As for reasons to pick mit over Stanford… it’s in a much more urban area than Stanford if you like that kind of thing. For example, the mononotone convergence theorem and Fatou’s theorem are false if we don’t assume that the Math 205B Homepage, Winter 2020-2021 Real Analysis Instructor: András Vasy Office: virtual E-mail: andras "at" math. Math 202A (an entry-level graduate course) at Berkeley ( https://math. you wouldn't need to know calculus, but you could learn it if you had too), but the ability to learn math and to think about math in an organized manner. Right, that's a good question. If xhas two ternary expansions 0:a 1 a 2:::and 0;a0a0 2:::, then one of them has a tail of 2s, say 0:a 1a 2:::. Instructor: Eugenia Malinnikova. edu Tentative office hours: half an hour after every lecture (i. Evans and R. I know both schools are top-ranked in these areas, and I'm having an extremely difficult time deciding. Lecture notes for Math 205A, Version 2014 Lenya Ryzhik September 27, 2016 Nothing found here is original except for a few mistakes and misprints here and there. edu. Dec 18, 2021 · Can someone share their experience about cs205l ? I lack solid math foundation but I survived ML classes in the past (not in Stanford). Let S be the the set of numbers in the interval [0;1] such that the decimal expansion for xcontains a 7. Office: 383M Phone: 723-2226 E-mail: andras "at" math. ) Note on the infimum of the emptyset. Generally speaking, math 104 and Strang's class is roughly equivalent. Office hours: Tue 1:30-3:00PM, Wed 3:00-4:30PM Math 205A - Fall 2016 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. If you did AB in high school and you didn't feel that solid, do 21. Prerequisities: STATS 310A or MATH 205A. Math 52 is the multivariable integration class, so that's where what you learned in Calc BC will come in handy. Nov 5, 2024 · Math 205A – Real Analysis Andrea Montanari , Stanford University, Autumn 2024 Topics include: Measure theory, Lebesgue measure, integration, modes of convergence, Lp spaces, Radon-Nykodym theorem; product measures; differentiation. Solutions. The 2022 Math Placement Diagnostic opens on August 1st, and I'm curious as to what topics the test covers. edu/~bcwhite/205a/ teaches measure theory and Lebesgue integration. Homework 1: Due on 10/3 : Mathematics Department Stanford University Math 205A Sample Final Examination Unless otherwise indicated, you can use results covered in lecture, provided they are clearly stated. Homework assignment 2 (due before 11:59pm on Thursday, Oct 11. If F= ;, then K= [0;1] because each open interval has positive measure. The readers should consult the original books for a better Nov 5, 2024 · Math 205A – Real Analysis Andrea Montanari , Stanford University, Autumn 2024 Topics include: Measure theory, Lebesgue measure, integration, modes of convergence, Lp spaces, Radon-Nykodym theorem; product measures; differentiation. Math 137 didn't have many problem sets, but I spent several hours a week trying to catch up and learn the material. Let f n = 1 [n;1). nnNOTE: Undergraduates require instructor permission to enroll. Stanford doesn’t have applied math, so aside from the intro 50 series and like 2 classes, everything else is pure math. Lectures: TTh 9:00-10:20, 300-300. Show that: (i) (t) 1 tp jjfjj p Lp(). Office hours: T 12:30-1:30 pm and F 3:00-4:00 pm. 205A is designed to give a standard introduction, at beginning graduate level, to measure theory and the theory of the Lebesgue integral. Course Information Jun 24, 2020 · Math 205 series: Graduate real analysis sequence. edu Tentative office hours: After lectures on zoom and by appointment Class location: TTh 1:30-2:50, Room 381T. stanford. Outer Measure: 9/23 Usually, these kinds of courses require "mathematical maturity", as in, not any math skill in particular (e. ) Note the set of numbers with a 7 in the nth decimal place is a nite union of intervals, namely: S n= 10n[1 1 k=0 k 10n 1 + 7 10n; k 10n 1 + 8 10n : In particular each S nis a Borel set. 3 Q. Any other non-elementary fact must be proven. In this case, there 9msuch that a n = a0 Math 205A Real Analysis Homework 5 Andrea Montanari Due on November 9, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. 3 units Sep 23, 2024 · Prerequisites: 171 and 205A or equivalent. Basic measure theory and the theory of Lebesgue integration. ) 1. Textbook and Course material Mathematics Department Math 205A List of Main Topics 1. All 205A Files Sep 10, 2024 · Math 205A : Exam Rules Exams (midterm and final) will consists in a certain number of problems. berkeley. I don't intend to study super hard for it or anything, but in previous years, has it covered everything up to Calc BC (including sequences and series)? Hello I am trying to get linear algebra ready for cs229. Especially if it’s been awhile since you’ve taken math, if you’re having a hard time with taking derivatives, seeing a solution to a system of equations (or seeing a redundant equation in a system), or doing procedural optimization problems, then it’ll be hard to keep up with math 51 concepts while you’re sweating this stuff. Cohen, Measure Theory. Show that Lq(R) 6ˆLr(R) and Lr(R) 6ˆLq(R). (There’s also a version for locally compact spaces. Suppose 1 p<1. (i) A step function is a function of the form P n 1 c i˜ A i, where A i’s are nitely many disjoint interpals. Thus contains all the Borel sets and (1) is proved in case (X) <1. ) (a). Math 205A CA: Zeb Brady notzeb@math: Room 380-380G, Office Hours: Mon, Wed, Fri 10:00am-noon. If you did BC in high school, do 51. 4 Q. 6 Q. Stanford University Mathematics Department Math 205A, Autumn 2013 : General: Math. Math 205A Real Analysis Practice Problems for Midterm Andrea Montanari Due on October 17, 2024 Solutions should be complete and concisely written. In this case, there 9msuch that a n = a0 Math 205A Real Analysis Homework 1 Andrea Montanari Due on October 3, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. (a) Using lower and upper Riemann sums for ∫0 to n x^α dx, prove that for any integer n ≥ 1, we have A subreddit for current students and alums to talk about Stanford stuff. Measure Theory 1 1. Lots of proof writing. Handwritten solutions are Ok provided you write legibly. 1 Q. But R 0 = 0 <1= lim n R f n. Fix >0. edu/courses/fall-2020-math-202a-001-lec ) does the same. Math 205A - Fall 2015 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. You have 3 hours but you are not required to solve all the problems! Just solve those that you can solve within the time limit. Math 205A Real Analysis Homework 7 Andrea Montanari Due on November 21, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. In this case, there 9msuch that a n = a0 MATH 205A HOMEWORK 1 SOLUTIONS 1. Math 51 will be significantly easier than anything in Strang's class. Office hours: Tuesday 3:00-4:30PM. Since each F is a ˙-algebra, it follows that Ac 2F for all , and hence Z 2ˇ 0 F(t) 0(t)dt= Z 2ˇ 0 f(t) 0(t)dt Z 2ˇ 0 Z t 0 g(s) 0(t)dsdt Z 2ˇ 0 f(0) 0(t)dt Z 2ˇ 0 f(t) 0(t)dt Z 2ˇ 0 Z 2ˇ s g(s) 0(t)dtds | {z } Fubini f(0)( (2ˇ CS graphics is maybe the only part of CS which can make serious use of differential equations (which doesn't mean you need to take Math 53 right away), and knowledge of material as in Math 143 (not a very proof-oriented class) on the geometry of surfaces in space is useful too. Suppose that (X;A; ) is a nite measure space and that fis a -integrable function. The MCAT (Medical College Admission Test) is offered by the AAMC and is a required exam for admission to medical schools in the USA and Canada. Letters are preferred Prerequisites: 171 and 205A or equivalent. If your background in such matters is shaky, then you should take math 171 (and perhaps also math 161) instead of math 205a; you can then take the undergraduate measure theory course (math 172) in the winter, and/or math 205a next fall. Make examples to show that in the \big" theorems about integration we’ve proved (monontone convergence, dominated convergence, fatou, egorov), the hypotheses really are necessary. That is, A2F for all . I don't think a bad grade in Math 51 is going to sink your consulting prospects. Office: 383 Y. The lecture on May 19 will be shorter, starting around 2pm, TBC. Sep 26, 2024 · Andrea Montanari. Let xhave two ternary expansions 0:a 1 a 2:::and 0;a0a0 2:::, with a tail of 2s in the rst, and let N, N0and b n, b0 n be the respective quantities of Math 205a Homework (Fall 2018) Prepare your hw with LateX if possible. Math 205A Real Analysis Homework 1 - Solution Andrea Montanari Due on October 3, 2024 Problem 1 (a) Let G= T F , with each F a ˙-algebra. General outer measures, and Caratheodory’s criterion for measurability. You have 80 minutes but you are not required to solve all the problems! Just solve those that you can solve within the time limit. NOTE: Undergraduates require instructor permission to enroll. In this case, there 9msuch that a n = a0 MATH 205A HOMEWORK 4 SOLUTIONS (FALL 2018) 0. 205A Home; 205A General Info. CS 109 and Stats 116 both teach probability but the approach and teaching vary drastically for both. Show that every compact set is the support of some Borel measure. Prerequisites: 106B or X; MATH 51; or equivalents. Location: Sequoia 105 Math 205A - Fall 2014 Homework #3 Solutions Problem 1: (i) Show that step functions and continuous functions are dense in Lp[0;1];1 p<1. I am trying to decide between cs205L and Math104 next quarter. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in Dec 6, 2024 · Notes from Stanford . The math major is beautiful because it's so light — the general plan for most is 10 units AP Calc BC credit, 15 units intro series (either 51, 52, 53 or 61, 62, 63), which most math majors tend to do their frosh year, at least in what I've seen, 8 other math courses at 4 units each, and then 15 units of electives, which are pretty broad Stanford Mathematics Department Homepage of Math 205A, 2013 . I am debating if I should take cs205l or cs224n this winter. Thus if we let zDfA2 WXnA2Ag then Azis a ˙-algebra containing all the closed sets, and hence Azcontains all the Borel sets. I haven't taken math 53 but I assume it's similar to math 52 in terms of difficulty, which I took alongside physics 63 and cs106b (and a super chill think class that had on average maybe 1 hour of work a week). /r/MCAT is a place for MCAT practice, questions, discussion, advice, social networking, news, study tips and more. 5 Q Sep 16, 2024 · Math 205A : References. Let xhave two ternary expansions 0:a 1 a 2:::and 0;a0a0 2:::, with a tail of 2s in the rst, and let N, N0and b n, b0 n be the respective quantities of Math 205A - Fall 2016 Homework #2 Solutions Problem 1: Show that there is a support of every Borel measure. Homeworks will be posted every Thursday starting on 9/28. debray@math. There will be one homework due every week, on Thursday, one in-class midterm and one in-class final Wᴇʟᴄᴏᴍᴇ ᴛᴏ ʀ/SGExᴀᴍs – the largest community on reddit discussing education and student life in Singapore! SGExams is also more than a subreddit - we're a registered nonprofit that organises initiatives supporting students' academics, career guidance, mental health and holistic development, such as webinars and mentorship programmes. (i) A step function is a function of the form P n 1 c i˜ A i, where A i’s are nitely many disjoint intervals. The Math 19-20-21 courses will also provide you with a level of understanding that is deeper than in high school calculus. It is easy to show that if f xhas two expansions, a n and n then there exists Nsuch that a k = k for all k N, N+1 = 0, k = 2 for all k>N+ 1, a N+1 = 1 and a k Nothing you learned in Calc BC is used in Math 51, iirc. Here are some examples to keep in mind. Let f: R !R be a Lebesgue measurable function such that R jfjp d <1. You're gonna want some type of multivariate calculus-based probability from any of CS 109, Stats 116, or Math 151. Culturally we’re a very applications/startup oriented school which could be good or bad. CS may be a bit different because some SWE jobs demand a CS degree, but I know plenty of math majors working in Software/Data Science. It really depends on the other classes that you're taking and how willing you are to grind through psets. Math 205A - Fall 2016 Homework #3 Solutions Problem 1: (i) Show that step functions and continuous functions are dense in Lp[0;1];1 p<1. By Egorov’s theorem, there exists N and a set B with jB j such that for all n N , jf n(x)j in [0;1]nB , with to be chosen later. The students in the 60 series are typically math majors or those with heavy theoretical interests and/or mathematical background. Homework 1: Due on 10/3 : For <C=(2R 1 j 2 = : 1 1 function 1. edu Kevin Rizk. edu/math21oae (where 21 is your course number, you need to change it to your own course number 53, 61cm, 104, 171, 210a, etc) to upload it and detail the specific accommodations you will need in this course. Members Online The three huge lies Stanford told our incoming freshman class, 20+ years ago. Math 205C Homepage, Spring 2019-2020 Real Analysis Instructor: András Vasy. Math 210 series: Graduate algebra sequence, goes up to Lie theory. Clearly The only classes providing a formal treatment to measure theory and Lebesgue integration are Math 172 and 205A. Measure Zero, Lebesgue’s Theorem on the Riemann integral 2. Gariepy "Measure Theory and Fine Properties of Functions", Math 205A - Fall 2016 Homework #2 Solutions Problem 1: Show that there is a support of every Borel measure. Math 172 is also recommended. These lecture notes are based on material from the following books: H. Contents Part 1. Royden "Real Analysis", L. Since F a ˙-algebra, we have that 2F , and as this applies for all , it follows that 2G. I live-TEXed them using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to a. For example, Stanford's Math 205A (an entry-level graduate course) http://web. Some probability classes (math 136, 228 and 230A, for example) provide a brief treatment of sigma algebras for the purpose of formalizing probability spaces, but Lebesgue integration is not the emphasis. [Not to hand in. Homework assignment 1 (due before 11:59pm on Thursday, Oct 4. Math 205A Real Analysis Homework 8 Andrea Montanari Due on December 5, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. Homework 1: Due on 10/3 : Jun 24, 2020 · Math 205 series: Graduate real analysis sequence. I don't know much about consulting except what I hear from my friends. These notes are simply a record of what I cover in class, to spare the students the necessity of taking the lecture notes. In math 121 and 122, I had to stay up very late to finish one or two problem sets. Lebesgue outer measure in Rn; Caratheodory’s criterion for measurability. Math 205A - Fall 2016 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. (Thus we write 1 instead of :999:::, etc. sigma-algebras in abstract space, Borel sets in topological space. edu Viet The Vu. >0 0 :). Thus strict inequality is possible in Fatou’s theorem. Otherwise, we claim that K= O c, where O= ([F(p;q)) . Office: 381 A. (Not to turn in). utexas. Clearly MATH 205A NOTES ARUN DEBRAY AUGUST 21, 2015 These notes were taken in Stanford’s Math 205a class in Fall 2014, taught by Lenya Ryzhik. Prove that Sis a borel set. Math 205A Real Analysis Homework 2 - Solution Andrea Montanari Due on October 10, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. g. 7 Q. Lecture notes for Math 205A Lenya Ryzhik December 4, 2008 Essentially nothing found here is original except for a few mistakes and misprints here and there. I had already taken an intro linear algebra and ODE course before Math 51 and I still hated it/felt like the concepts were taught in a confusing way. ) Solutions. The material in this class can be found in any of these textbooks: P. May 27, 2021 · For example, Stanford's Math 205A (an entry-level graduate course) http://web. ] If you don’t already know it, learn the Baire Category Theorem (and its proof) for complete metric spaces. To check (2) in case (X) <1we can just apply (1) to XnA: thus for each iD1;2;:::there is an open set U i Math 51 is known to suck. Math 205a - Fall 2011 Problem Set 1 Problem 1 - 1 Show that f is well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. Let 1 q<r<1. MATH 205A HOMEWORK 5 (FALL 2018) 3 Thus for this g, kfk p= Z fgd = lim n Z f ngd liminf n kf nk pkgk q (by Holder) = liminf n kf nk p (since kgk q= 1. Clearly Math is obviously good but not full of imo people like mit. Also the better place to dabble in the humanities/social sciences. The #1 social media platform for MCAT advice. MATH 205 A Real Analysis Autumn 2019. till 4pm), except Wednesday when until 4:30pm, and on Monday (starting week 2) an extra hour after half an hour break: 4:30-5:30pm. Office hours: T 3:30-5:30 pm, W 4:30-5:30 or by appointment. MATH 205A HOMEWORK 4 (FALL 2018) 0. (Not to turn in. Points assigned to each problem are Math 205A - Fall 2016 Homework #3 Solutions Problem 1: (i) Show that step functions and continuous functions are dense in Lp[0;1];1 p<1. MATH 205A HOMEWORK 1 0. Continuous mathematics background necessary for research in robotics, vision, and graphics. 5 Q. Math 205A - Fall 2014 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. e. From what I understand, you actually don't need to do the Math 51-53 series. If necessary, continue solutions on backs of pages Note: work sheets are provided for your convenience, but will not be graded 3 Hours Q. Let F= f(p;q) : p;q2Q; ((p;q)) = 0g. Students are required to attend classes. But really what you should do is email the instructors of Fall Math 19 (Schaeffer) and Fall Math 20 (Lucianovic), both of whom have extensive experience teaching such courses here. The only classes providing a formal treatment to measure theory and Lebesgue integration are Math 172 and 205A. Clearly Math 205A Real Analysis Homework 3 Andrea Montanari Due on October 17, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. Totally doable. Math 205A - Fall 2014 Homework #4 Solutions Problem 1: Let f2Lpand (t) = mfx: jf(x)j>tgthe distribution function of jfj. ) 6. Math 205A : Homeworks. Math 205A - Fall 2015 Homework #3 Solutions Problem 1: (i) Show that step functions and continuous functions are dense in Lp[0;1];1 p<1. Location: Sequoia 133. He later graduated from the Econ major with honors and got into MBB. D. Homework 1: Due on 10/3 :. Course Assistant: Joey Zou. Prerequisite: Math 171. These notes are si 104 has been revamped to be much more like an actual math class, with more proofs and much less mind numbing calculations (which is why math 51 kinda blows lol). 2 Q. They are due the following Wednesday, submitted via Gradescope. If 205 is a great course then I’ll take it and take 231n in spring as 224 and 231 both deals with DL. Then for n N Math 205A Real Analysis Homework 2 Andrea Montanari Due on October 10, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. STANFORD MATHEMATICS . It is also recommended to have taken at least one of Math 122, Math 145, or Math 154. Is this true for L1? (ii) Show that if fis integrable in E, then lim t!0 R E jf(x+ t) f(x)jdx= 0. Please, mark clearly the beginning and end of each problem. Then f n 0, and f n(x) !0 for all x. Another thing you could look into is taking a different class in the fall quarter and taking Math 51 in the winter or something. Stanford vs. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc. email: montanari@stanford. For quant finance jobs, math is probably significantly preferred to Econ, since trading needs more proficiency in stats and theoretical math than most Econ majors would have. Math 205A Real Analysis Homework 4 Andrea Montanari Due on October 31, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. Welcome to r/UCSD! This is a forum where the students, faculty, staff, alumni, and other individuals associated with the University of California San Diego can discuss, share, advise, and collaborate among themselves! Math 205A - Fall 2015 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. 4. Math 205 series: Graduate real analysis sequence. Real Analysis (MATH 205A) 1 year ago Let α > 0 be a (fixed) real number. You will have 80 mins for the midterm and 3 hours for the final. Math 205A Lecture Supplement 3 are in A. Mathematics Department Stanford University Math 205A 2013|Lecture Supplement #3 Di erentiability Theory for Functions and Measures As a preliminary to the discussion of di erentiation of functions and measures, we need the following important covering lemma, which we state and prove in Rnbut which clearly has a natural extension to metric spaces: Math 205A Real Analysis Homework 6 Andrea Montanari Due on November 14, 2024 • Return your assignments via Gradescope • Solutions should be complete and concisely written. It varied widely: in math 120 I could usually work on the problem sets over two or three afternoons a week and be done. Again, 21 and 51 are very different classes, so I think their recommendation is more to ensure that you've covered everything you need to cover in calculus before moving on, but it isn't because you need to have done 21 level math to do 51. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in MATH 205A HOMEWORK 5 (FALL 2018) 0. MATH 205A: Real Analysis I, Autumn 2022 ; MATH 205B: Real Analysis II, Winter 2023 ; MATH 210A: Modern Algebra I, Autumn 2022 ; MATH 210B Math 205A Real Analysis Practice Final Andrea Montanari Due on — Solutions should be complete and concisely written. You can reference results/statements in either of the textbooks. Let us use the nite decimal expansion when there is one. Harvard for math/applied math I'm in the position of choosing btw these two schools with the intention of doing either math or applied math as an undergrad. 8 Oct 18, 2024 · If you already have an Academic Accommodation Letter, please use the googleform tinyurl address https:goto. You can try taking some other CS math electives like Math 108 or Math 110, which are proof based. I enjoy it, but definitely not for the faint of heart. Math 205A - Fall 2015 Homework #2 Solutions Problem 1: Show that there is a support of every Borel measure. Possible topics: linear algebra; the conjugate gradient method; ordinary and partial differential equations; vector and tensor calculus. 1. Mathematics Department Stanford University Math 205A Sample Final Examination Solutions Unless otherwise indicated, you can use results covered in lecture, provided they are clearly stated. I have taken Math51, but honestly don't feel like I retained enough from it. Math 151 for math majors, it requires analysis (like actually requires 115) and teaches probability from the analysis-perspective. Nov 9, 2022 · The only classes providing a formal treatment to measure theory and Lebesgue integration are Math 172 and 205A. In this case, there 9msuch that a n = a0 Math 205A - Fall 2016 Homework # 1 Solutions Problem 1: Show that fis well-de ned, monotone, continuous and that it is constant on every interval contained in the complement of the Cantor set. 3. I had a friend who failed Math 51 and had to re-take it. Then for n N Oct 9, 2024 · Math 205A : Grading. Suppose now that A2G. Prove that for every >0, there is a step function gsuch that kf gk p < : My high school math classes were significantly more intellectually rigorous, and I really doubt you would learn much being a grad student taking a class like this. 5. vlkfpkrt soacv lkeg razp wkm buef lbciz haf pkdn xhsp