Category: Uncategorized

  • Mazur Manifold – Class Lecture

    Let G = < A | R > be a presentation of a group. Tietze Transformations (finite versions) are ways that a group presentation can be altered without changing the group. \(T_I \) – add a relator that is a consequence of the other relators. example: \(aba^{-1}b^{-1} , aba \) implies \(a^2b \) \(T_{II} \) – add […]

  • Genus Two Symmetry

    Thurston 1.2.6 (a) Julian Rosen created a photo and answer:  Imagine three lines of longitude on a unit sphere (embedded in ℝ3), at angles 0, 2π/3, and 4π/3. Let G be the union of those three lines of longitude. Then G has three-fold rotational symmetry and the set of points in ℝ3 that are distance exactly […]

  • একটি সূত্রের জন্ম

    (ফরাসি গণিতজ্ঞ, সেড্রিক ভিলানি, ‘থিয়োরেম ভিভান্ত’ গ্রন্থ রচনা করেন ২০০৮ সালে। বইটি খানিকটা ডায়রি লেখার ঢঙে রচিত। মূল ফরাসি থেকে বাংলায় অংশ বিশেষ অনুবাদ করছি স্রেফ খেলার ছলে। কোনো বাণিজ্যিক উদ্দেশ্য নেই।)।  লিয়ঁ, ২৩শে মার্চ, ২০০৮ রবিবার দুপুর ১টা। প্রায় জনহীন গবেষণাগারে দু’জন ব্যস্ত গণিতজ্ঞ ছাড়া বোধকরি আর কেউ নেই। লিয়ঁর এই গবেষণাকেন্দ্রে আমি গত […]

  • The Alexander Trick

    Here is the original paper: J. W. Alexander, On the deformation of an n-cell (A 2-page paper that influenced a remarkable amount of later work).

  • Combinatorics Problem List for AIME

    This is a collection of combinatorics and probability problems that have appeared in AIME. Two dice appear to be standard dice with their faces numbered from \(1\) to \(6\), but each die is weighted so that the probability of rolling the number \(k\) is directly proportional to \(k\). The probability of rolling a \(7\) with […]

  • Chapter 11 (Rings) – Artin – Section 1 (Definition of a Ring)

    1. Note that a complex number \(\alpha \) is algebraic if it is a root of a(nonzero) polynomial with integer coefficients. a) Sine we need \(7 + \sqrt [3]{2} \) to be root of an integer polynomial, there fore \[1]x – (7 + \sqrt [3]{2}) = (x-7 – 2^{\frac{1}{3}}) \) is one of the factors. […]

  • Shrinking the diameter of Vitali Set

    Claim: The diameter of Vitali Set V on [0, 1] can be shrinked as much as we please Proof:  Let R be an equivalence relation defined on [0,1] such that x is related to y if x – y is rational. Let E be an equivalence class corresponding to the equivalence relation R. In order to […]

  • A SAGE experiment on Nested Approximate Subgroups

    (this is a continuing write-up mostly for personal records) Spoiler Before you read on, here is what I have found curious so far (this section is constantly changing): A little computation reveals that though group action conjugation preserves most elements of a subgroups in majority of cases, there exists some subgroups of \(S_4 \) which are ill […]