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  1. M

    Challenge Math Challenge - November 2021

    I seem to be on a slightly different page from others, or just further behind, so let me explain my viewpoint. I apologize if this is more algebro - geometric than desired, and/or too elementary to be useful. I was trying to relate problems 4a and 4b by letting a matrix act on the riemann...
  2. M

    Challenge Math Challenge - November 2021

    martinbn, I understood you meant they were isomorphic, but I think you still have to prove it. One way to prove it is to give a bijection between the riemann sphere and the projective line under which the two actions correspond. Perhaps you were taking all that for granted. (As a naive person...
  3. M

    Challenge Math Challenge - November 2021

    Nice! I was thinking they should be equivalent, but was trying to do it via 4b acting on the real plane, the complement of infinity. Your idea seems to use the realization of the riemann sphere as the projective complex line. Maybe you could give a little more detail. \\SPOILER: #6: as for...
  4. M

    Challenge Math Challenge - November 2021

    SPOILER: solution for 2: Let the two disjoint lines be L and M, and p the point not on either of them. Then there is a unique plane A spanned by L and p, and a unique plane B spanned by p and M. Then A and B are distinct planes, since one of them contains L, the other contains M, and L and M...
  5. M

    Challenge Math Challenge - November 2021

    SPOILER: solution of #10: By definition, V(Y^2-X^2) denotes the set of points in (X,Y) space whose coordinates satisfy the polynomial Y^2-X^2. Hence the real points consist of the points on the two lines X=Y and X+Y = 0, in the real plane. By definition Spec(R) is the set of prime ideals in...
  6. M

    Challenge Math Challenge - November 2021

    SPOILER: solution for 9: By definition of the tensor product, or by use of its basic property, a linear map out of VtensW is equivalent to a bilinear map out of VxW, which then is equivalent both to a linear map from V to W*, and to a linear map from W to V*. In particular, given f as in the...
  7. M

    Challenge Math Challenge - November 2021

    SPOILER, solutions of 8c, 8d: 8c: To be reducible it suffices to have a root, say X=a, since then by the root factor theorem, (X-a) is a factor. In each of the last three fields the polynomial has a root: in the reals, by the intermediate value theorem every odd degree polynomial takes both...
  8. M

    Challenge Math Challenge - November 2021

    Nice work! Linear equations can always be used to find inverses as you do. Another trick is to find the "minimal polynomial" of an element, since if X^3 + aX^2 + bX + c = 0, then X^3 + aX^2 + bX = -c, and so X(X^2 + aX + b) = -c, and since we know the inverse of -c in the field of...
  9. M

    Challenge Math Challenge - November 2021

    Suggestion for #8b: First prove that F7 is a field, then try to use the same proof to show F7[T]/(T^3-2) is a field. Here is an example of the (easier) problem of finding an inverse, this time of T: since T^3 = 2, then 4T^3 = (4T^2)T = 8 ≈ 1 (mod 7), so T^-1 = 4T^2. For a reprise of basic...
  10. M

    Challenge Math Challenge - November 2021

    Question on #6: Pardon me for ignorance, but I wonder just what is wanted in problem 6, since I thought 6a is sometimes taken as a definition of an orientable differentiable manifold. Of course it is possible to define the orientability of any continuous manifold, in terms of continuous...
  11. M

    Challenge Math Challenge - November 2021

    Hint for #9: To show two vector spaces are isomorphic, of course we first try to find a linear map from one to the other, and then hopefully it is bijective. We don't quite have that here, but we have something close. I.e. remember the defining property of the tensor product, Hom(VtensW, X)...
  12. M

    Challenge Math Challenge - November 2021

    In reference to #2, probably none of us in the US had a course in elementary geometry, even though we may have seen topological vector spaces, advanced calculus, field extension theory, tensor products, and free groups! So this is a review of simple geometric facts about projective 3 space...
  13. M

    Calculus Just got Courant's calculus text -- Need some tips to work through it

    yes, spivak is essentially a modern version of courant. i.e. spivak himself perhaps took calculus from courant (i.e. Courant's book) and then wrote his own take on the material, with some proofs lifted from courant, but with many additions and elaborations. Basically you just need to start...
  14. M

    Solving polynomial equations

    I believe Cardano's method used to be taught in high school, but before my birth. Consequently, the clearest explanation I have found is in the "elementary algebra" book by Euler, from which I adapted this presentation for bright young children, like your granddaughter, in a summer camp. All...
  15. M

    I Roots of polynomials in C

    You know it just dawned on me that it seems in a way odd that I think of using the intermediate value theorem to prove a sqrt of 2 exists, or some sort of limiting process. I.e. I am assuming given the real numbers, which to me is the natural outgrowth of the points on the Euclidean line...
  16. M

    Geometric Construction (bisecting an angle with a compass and straightedge)

    That's what it was. I did not design a presentation intended to suit all 8 year olds. I was given a class of about 30 bright, some brilliant, 8,9,10 year olds and asked to give a presentation appropriate to them. I would guess however that once that presentation exists it might suit some in a...
  17. M

    I Roots of polynomials in C

    @lavinia: Thank you. I think I understand your meaning now of solutions obtained by manipulating the coefficients, to mean that the solutions can be expressed using algebraic operations and nth roots, starting from the coefficients and rational numbers, i.e. that the equation is what we now...
  18. M

    Geometric Construction (bisecting an angle with a compass and straightedge)

    @mpresic3: in case of possible interest, here is my presentation on the topic of impossible constructions, to my class of bright 8,9,10 year olds. Remarks on impossible constructions: I want to say something about why the only regular p-gons that can be constructed with prime p, are for the...
  19. M

    I Roots of polynomials in C

    good points. to be more precise, i wondered how does one prove it without analysis, i.e. limits. dedekind cuts and cauchy sequences seem like analysis, and existence of "diagonals" do not seem to be very algebraic either. so my curiosity was about how one proves existence of roots of...
  20. M

    I Roots of polynomials in C

    Just a question, re post #27. I don't see why the quadratic formula is purely algebraic. I.e. how does one prove that a square root exists, even for every positive real number, without analysis? e.g. how does one prove sqrt(2) exists without taking a least upper bound?
  21. M

    Admissions Admission GPA

    By the way, I apologize for apparently leading the other thread astray. I intended my post, #1 above, to be relevant to the OP for determining whether Va Tech was a reasonable choice for him, in the sense of helping him gauge whether he would get in. Since the posted data on admissions to Va...
  22. M

    B What kind of definition is this?

    remark: the cantor set is infinite, bounded and closed. thus it contains some of its own accumulation points, at which the value of a continuous function is determined by its values on the other points.
  23. M

    Admissions Admission GPA

    heres where i read that 4.04 gpa, it seems to be claimed as an average, not a minimum: https://www.prepscholar.com/sat/s/colleges/Virginia-Tech-SAT-scores-GPA
  24. M

    Admissions Admission GPA

    Moderator's note: thread split off from https://www.physicsforums.com/threads/university-search-for-a-physics-major.1008062/ To an old guy like me these admissions standards look strange. It said online that to get into virginia tech one must have at least a 4.04 GPA. In my day no one had...
  25. M

    Schools University Search for a Physics Major

    That is a good point. You probably know I am a retired college professor formerly involved in acceptance of graduate students and faculty hiring. I never turned down an overqualified grad applicant (never saw one in fact), but was occasionally slow to offer jobs to people so highly qualified...
  26. M

    Schools University Search for a Physics Major

    I think this list in #26 looks very useful, as an answer to your specific question, but you should realize it still consists of very competitive schools. Brown has always been one of the very most difficult schools to gain entrance to in my experience. I.e. many students have long preferred to...
  27. M

    Courses Should I retake Real Analysis I?

    As a professor many students considered "bad" myself, I read this with interest. The only bad quality you mentioned about your professor, other than his choice of notation, was that his standards were higher than you were used to. Of course he may have been poor at explaining as well. But low...
  28. M

    Prove that a locally constant function is constant on a connected X

    If f is not constant and takes value c somewhere, it seems that the set A where f equals c and the set B where it does not, are disjoint, non empty, and both open. Done.
  29. M

    I Proving a function f is continuous given A U B = X

    By one definition of continuity, it suffices show that the inverse image C of an open set of Y under f, is open in X. By hypothesis its restriction, i.e. intersection, with both A and B are open. So we want to show that if X = AunionB, and both A and B are open, and if CmeetA is open in A, and...
  30. M

    Intro Math Maths needed to advance my physics understanding

    Indeed that is true. My philosophy is however, that since math in general and calculus in particular is hard for students, not to give them easy methods when they exist, is in fact a sin against honest teaching. In this example, although using calculus as you say is a straight forward...
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