# All Questions

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### Volume of water [on hold]

Please Calculate volume of water in a sphere container with radius r that is filled with water up to the height h.
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### Modular calculus and square

I want to prove that $4m^2+1$ and $4m^2+5m+4$ are coprimes and also $4m^2+1$ and $4k^2+1$ when $k\neq{m}$ and $4m^2+5m+4$ and $4k^2+5k+4$ when $k\neq{m}$. Firstly : Let $d|4m^2+1$ and $d|4m^2+5m+4$ ...
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### Force non-consecutive colours for pie chart

Background Calculating colours for pie chart wedges. Consider: \begin{align} d(n)&=\frac{\theta}{t}\times n\\ \end{align} Where: $\theta$ is the degrees in a circle (360) $t$ is the ...
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### $S^1$ with length metric is not isometric to any subset of Euclidean plane (metric given by restriction)

Let $S^1$ denote point whose radius is 1 from the center. Metric is given by distance between two point is the shortest distance, that is the length metric. Prove that $S^1$ with this metric is not ...
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### Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
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### True or False. Convergent subsequence

Is the statement true or false? If $(x_n)$ has a convergent subsequence,then $(x_n)$ is bounded. The statement is False. However, can someone please show me an example of a sequence with ...
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### I'm new to proving inequalities. How does one prove this?

If a, b, and c are non-negative real numbers and $a + b + c = 2$, prove that $2 \ge a^2 b^2 + b^2 c^2 + c^2 a^2$
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### Generating function for number of integer solutions, no computer

Use a generating function to solve the number of integer solutions for $$x_1+x_2+x_3=17$$ Where $2\leq x_1 \leq 5, 3\leq x_2 \leq 6, 4\leq x_3 \leq 7$ Now all this takes is doing: ...
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### How to get the ratio from a function of N?

The exercise gave us a chart which showed the running time as a $N$ increases: \begin{array}{c|c} N & \text{seconds}\\\hline 256 & 0.000\\ 512 & 0.000\\ 1024 ...
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### Max no. of piece in k cut

Suppose I have large piece of rectangular sheet. Cutting is allowed only vertically and horizontally. My approach is if no. of cut is even then max. no of piece is (n/2)*(n/2) if no of cut is odd ...
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### Unique extrema of sum of monotonically increasing and decreasing functions on an interval

If I have two functions, f and g, defined on the interval [0, 1] with both f and g non negative (i.e. f(x), g(x) >= 0) f(x) is monotonically increasing, while g(x) is monotonically decreasing. and ...
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### The number of Balanced Boolean functions

Suppose we have n-variable Boolean function (BF) and we know that the weight of a Balanced BF is $2^{n-1}$. The total number of BFs are $2^{2^n}$, Affine BFs are $2^{n+1}$ and Linear BFs are $2^n$. In ...
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### How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
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I need to solve $$\int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$$ I tried to use symmetric properties of the trigonometric functions as is commonly used to compute $$... 1answer 17 views ### Vector Analysis (Parametized curve) The question is find a familiar parameterized curve that has the property r(t) \times\dfrac{dr}{dt}=0. The only curve that I can see that works is the line through the origin. I was just wondering ... 1answer 35 views ### Pulsating waves of zeta function Below is an animation of the partial sums of \operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t}, for 1\leq N\leq100 ... 1answer 28 views ### Probability: student passing an exam by randomly guessing (no calculator) Assuming you can't use a calculator, how do you estimate the answer to the following problem? Suppose an exam has 40 questions, all multiple choice. Each question has 5 choices and you need 20 ... 1answer 20 views ### Probability of Fair Die Hi everyone I have a question about probability: Fair die thrown two times, final score is calculated as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and ... 1answer 275 views ### Rank of matrix as a difference of ranks If X is an n \times p matrix of rank r and C = AX for some q \times n matrix A with rank(A) = q, how do I show that rank (X(I-C^{-}C))= rank(X)- rank(C)? I can show that rank ... 5answers 47 views ### Is \{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\} open or closed? I am trying to figure out if the set \{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\} is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ... 2answers 26 views ### every irreducible polynomial has a root in some field extension We know the following fact from field theory. Let F be a field and p(X) an irreducible polynomial in F[X]. Then we can find a field extension L of F such that p(X) has a root in L. ... 0answers 81 views +50 ### What is the smallest possible angle of this polygon? A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ... 0answers 10 views ### How to visualize probability distributions in terms of sets - joint and marginal? Let there be two sets, \mathcal{X},\mathcal{Y}, both finite, and they represent the set of values that the discrete random variables, X,Y can take. \mathcal{P}_{Y|X} be all possible ... 3answers 37 views ### Logic problem?? [on hold] When Paul's age in years and Evan's age in years are written down one after the other, they form a four digit number. Each of them are over 10 years old. The resulting number is a perfect square. In ... 1answer 27 views ### If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate] I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ... 0answers 43 views ### numerical method (Implicit) for nonlinear pde \newcommand{\lbar}{\underline{\lambda}} I need a numerical method (implicit , backward difference or forward difference) for estimate A in this nonlinear PDE:$$ A_t + \mu(\lambda -\lbar ) ...
I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...