# All Questions

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### Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
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### How to solve problems on alligation and mixture when three types are given?

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...
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### How to prove every term of this sequence is not a natural number

Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question: Let's take a look at this sequence: (1) $[a_1,a_2,a_3,a_4,...,a_x]$ where ...
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### What is the “Cumulative Distribution of the magnitude of the N-dimensional standard gaussian”

I am confused by this line from a paper: "Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative ...
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I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., Z_\infty(G)=\bigcap\limits_{S\in ... 1answer 26 views ### Trouble Solving a system of 3 equations I'm having trouble solving a system of 3 equations. The set of equations in question is shown below C_a=\frac{R_a}{\frac{R_a}{r_a}+\frac{R_b}{r_b}+\frac{R_c}{r_c}}, \quad ... 1answer 21 views ### Proper way to solve function notations? I'm just starting to use function notation and I'm wondering if I'm solving correctly. If f(x) = 4x - 11, determine a. f (1/4) f(x) = 4x - 11 f(1/4) = 4 (1/4) - 11 f(1/4) = 1 - 11  ... 0answers 7 views ### Choose \rho such that \rho-norm minimizes the matrix condition number I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes. In my proposed answer I didn't ... 0answers 38 views ### A proposed method for further abstracting prime numbers [on hold] I previously posted this but I framed it as a question and only inserted my results as an edit several days after the original post was created. Ulam's Spiral is a wonderful discovery. Obviously it ... 1answer 15 views ### How to solve  \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0 How to find a solution to the following equation \begin{align*} \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0 \end{align*} where c and A are some constants such that c\ge 1 ... 0answers 15 views ### integrate very long expression using orthogonality in maple I have very long expression and i must integrate it. i try to apply "orthogonality" on my equations to eliminate "X" and "Y" variables. Image Shows examples of Orthogonal Functions After execute the ... 0answers 14 views ### Taking derivate wrt a vector I'm trying to read through Wiki's description of the Levenberg-Marquardt algorithm. I've taken linear algebra, but I've always been fuzzy about taking derivatives with respect to a vector and just ... 0answers 10 views ### Notation for polynomials and equating coefficients I am reading a paper where they define P_k(s_1,s_2|t) as a polynomial of degree k in s_1 and s_2 given t. What does it mean "given t"? (I was thinking that each term looks like ... 2answers 19 views ### Proof that any finite subset of a lattice has a supremum. Let C\subseteq D where C = \{c_{1}, c_{2},\ldots, c_{n}\} and D is a lattice. I believe that this supremum is ((c_1 \vee c_2)\vee c_3)\vee \ldots \vee c_n) (sorry about the notation). However ... 3answers 21 views ### Exponential Growth and Decay Question: A Bacteria Culture Grows with Constant Relative Growth Rate. A Bacteria Culture Grows with Constant Relative Growth Rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. a) What is the relative growth rate? Express your answer as a ... 3answers 61 views ### Rule for squaring arbitrary powers? This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how: (q^{2^{n+1}})^2 = q^{2^{n+2}} $$I thought it would ... 2answers 32 views ### Right answer, wrong explanation, probability of grids? Two unit squares are selected at random without replacement from an n\times n grid of unit squares. Find the least positive integer n such that the probability that the two selected squares are ... 0answers 17 views ### Which directed graphs correspond to “algebraic” diagrams? Any diagram for which the question of commutativity make sence is a directed graph, but not any directed graph make the question meaningful. \require{AMScd} \begin{CD} A @>>> B @. A ... 3answers 77 views ### Finding all real numbers x such that x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88 Question: Find all real numbers x such that x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88. The notation \lceil x \rceil means: The least integer which is not less than x. My ... 1answer 19 views ### A question on use of square integrable functions I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function f(x) satisfies the condition ... 1answer 37 views ### Differentiability at a point x with f differentiable in \mathbb R\backslash\{x\} I have a real function that satisfies: f:\mathbb R\rightarrow\mathbb R is differentiable at x for x\neq x_0. There is a full-measure set T such that for any sequence t_n\in T with ... 4answers 33 views ### How to find perpendicular vectors in 3D Find all values of a such that the vector q = \langle 2, a, –2\rangle is perpendicular to the vector p = \langle –3, a, 5 \rangle. 0answers 11 views ### Is the constant group scheme for \mathbb{Z} affine? Is the constant group scheme for \mathbb{Z} affine? It is said no in Gille's notes "INTRODUCTION TO REDUCTIVE GROUP SCHEMES OVER RINGS" 3.1, but I don't see why! 2answers 57 views ### Is the following a characterization of \Bbb Q\cap\cal C, where \cal C is the Cantor set? Let A be an ordered set, with the following properties: A is countable A has a least and greatest element Between any two points with successors are points without successors; between any two ... 0answers 17 views ### The set T=\{l\in\mathbb{N}: ml=nl \ \text{implies} \ m=n \} is inductive. I'm trying to prove the following statement: ml=nl implies m=n for every m,n,l\in \mathbb{N}. So I defined the set T=\{l\in\mathbb{N}: ml=nl \ \text{implies} \ m=n \} and if I prove that ... 3answers 45 views ### Find the maximum value of the fraction Let a and b be positive integers satisfying \frac{ab+1}{a+b}<\frac{3}{2}. The maximum possible value of \frac{a^3b^3+1}{a^3+b^3} is \frac{p}{q}, where p and q are relatively prime ... 1answer 18 views ### Linear algebra: proving transformation matrix between orthogonal basis is unitary The vector space V is equipped with a Hermitian scalar product and an orthonormal basis \{e_1,\ldots,e_n\}. A second orthonormal basis \{e_1',\ldots,e_n'\} is related to the first one by ... 1answer 12 views ### Property of an almost additive sequence of functions We say that a sequene of functions \Phi=(\phi_n)_n is almost additive if there exists a constant C > 0 such that for every n,m \in \mathbb{N} and x\in \Lambda we have \begin{equation*} -C + ... 0answers 11 views ### How to multiply the elements within a vector using matrix operations (e.g., dot product)? Suppose a vector \vec{v}^T=(v_1, v_2, \ldots, v_n)^T. To sum the elements within the vector, I can use the dot product with a column vector of ones, \sum_i v_i = \vec{v}^T \cdot \vec{1}. My ... 1answer 10 views ### Properties of unimodal functions A probability density function f is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ... 0answers 16 views ### how to understand a matrix with order O(n^{-1}) I am reading a paper in which an assumption is that a matrix (for example A_{n\times n}) is O(n^{-1}). I have difficulty to understand that assumption. Does that mean the norm of the matrix is ... 0answers 22 views ### limits of computation [on hold] Design a Turing machine to accept strings over \{0, 1\} that are palindromes, that is, w = wR. (Hint: Modify the machine that accepts wwR (even-length palindromes) so that it also ... 0answers 45 views ### Any hint on : Every A_{n} elemnt is n-cycles product. [on hold] [Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of A_{n} is ... 1answer 23 views ### C*-Algebra: Cyclic Elements Given a locally compact Hausdorff space \Omega. Consider the C*-algebra: ... 2answers 32 views ### Generators in group Z^*_{p} show that g=2 is a generator of group Z^*_{19} Can anyone explain me how i can show in this example and generally that an element is a generator in a group? 1answer 17 views ### Show that there exists a non-negative integer r s.t. ker(T^r) = ker(T^{r+1}). Question: Let V be an n-dimensional complex vector space, let T: V \to V be a linear transformation. Show that there exists a non-negative integer r s.t. ker(T^r) = ker(T^{r+1}). My ... 1answer 79 views ### Does square difference prove that 1 = 2? I was mathematically shown 1 = 2 by a function that states the following$$x^2-x^2 = x^2-x^2 x(x-x)=(x-x)(x+x)$$dividing by (x-x) we get...$$x=x+x x=2x1=2 I can see ...
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I'm trying to solve that equation: $x^2-3x-5\equiv0\pmod{343}$ I've completed the square as follows: \$x^2-3x-5 \equiv x^2+340x-5\equiv(x+170)^2-170^2-5\pmod{343}\\ (x+170)^2 \equiv 93\pmod{343}\\ ...

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