# All Questions

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### What are some definite integration techniques?

Are there any definite integration techniques which I could learn (calc AB student)? I mean techniques which don't require you to find the anti derivative. Thanks!
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### what is $\lim_{x\to6}\frac{5}{(x-6)^2}$

$$\lim_{x\to6}\frac{5}{(x-6)^2}$$ Is it undefined or infinity and why ?
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### Distribution for Response Times

I have samples from a response time population for a web transaction. I want to be able to use them to describe a distribution for the population but don't know a proper one to use. I have shied away ...
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### $V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$

If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
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### Differentiable Operator Continuous?

Consider the space $C^{\infty}[a,b]$ with norm $||f|| = max_{[0,1]} |f(x)|$, with $f\in C^{\infty}[a,b]$. Is the differentiation operator $\frac{d}{dx}$ continuous on $C^{\infty}[a,b]$? I'm very ...
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### RSA encryption theory - modulo theory

I'm a bit mathematically challenged and have been working on the RSA cipher (good start). I can find the public and private keys and know how to work do modulo operations on a calculator. The problem ...
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### Prime and semiprime ideals of $A=T_3(D)$, the ring of $3\times 3$ upper triangular matrices over $D$

Let $D$ be a division ring. Could anyone tell me which are the prime and semiprime ideals of $A=T_{3}(D)$, where $A=T_{3}(D)$ the ring of $3\times 3$ upper triangular matrices with coefficients in ...
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### Normalizing the Second Moment of $n$ Discs

Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($||P_i - P_j||\geq d, i\neq j$). Graham and Sloane use the second moment as a measure of compactness for ...
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### Does the splitting principle define chern classes for vector bundles if they are known for line bundles?

Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$. Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is ...
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### Construction binary tree

First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$. We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$ ...
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### Lagrange polynomial interpolation on matlab

i am trying to write a code that will input data sets and produce coefficients of the lagrange polynomial. I have the code below but when i try and run it it tells me that xin is undefined. Can ...
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### property of an increasing or decreasing function

For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether $$af(x)\lesseqgtr f(ax)$$ for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
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### Calculate the Riemann Stieltjes integral

This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class. Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ ...
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### Anonymous graphs and graph embeddedness

What are anonymous graphs, what is graph embeddedness, and how do they relate to each other? Very confused - I could not find short answer. Thanks.
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### How to solve $at + b = 0 \pmod {(a-t)}$?

Is there any way (except trying for $t=0,1,2,\ldots,a-1)$ to solve the following equation for $t$ when $a$ and $b$ are known? $$at +b = 0 \pmod{(a-t)} \text{ with } a,b,t \in N$$
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### Does this polynomial factorise further?

I just did a national exam and this question was in it, I am convinced this does not work: Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorise this cubic fully. My attempt 1 | ...
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### How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
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### Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?

I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
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### Is a series (summation) of continuous functions automatically continuous?

I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a ...
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### Is there a flaw in Hao-cong Wu's Riemann hypothesis paper? [closed]

Hao-cong Wu published a paper in the EUROPEAN JOURNAL OF MATHEMATICAL SCIENCES, titled: Showing How to Imply Proving The Riemann Hypothesis Is there a flaw in it?
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### Why is the Derangement Probability so Close to $\frac{1}{e}$?

A derangement is a permutation of $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and ...
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### Lipschitz constant for $F(t,y,z)=(z,f(y,z)\sin t)$

Let $f\in C^1(\mathbb R^2,\mathbb R)$. Prove that all solutions for $x''=f(x,x')\sin t$ such that $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ have period $2\pi$. I'm in the process of solving the above ...
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### Can't establish a lower bound on a supremum

I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by $$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$ for all $k\geq 1,1\leq j\leq k$. This serves as an example of ...
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### About continuous functions and aritmethic progression

I've try solve this question, but I haven't sucess... The problem is the following: A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
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### Bolza example like Question

I have to find $u$ minimizing $\int_0^1 F$ with $F(x,u(x),u'(x)) = (1-(u'(x))^2)^2+(u(x))^2$ with $u(0) = 0$ and $u(1) = 1$. I'm relatively new to CoV and got told i should try ...
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### A simple 2 grade equations system

If we have: $$x^2 + xy + y^2 = 25$$ $$x^2 + xz + z^2 = 49$$ $$y^2 + yz + z^2 = 64$$ How do we calculate $$x + y + z$$
### How to prove that $A\cap B\subseteq C$ and $A^c\cap B\subseteq C$ imply that $B\subseteq C$?
How do you solve this problem?? Suppose that $A\cap B\subseteq C$, and $A^c\cap B\subseteq C$. Prove that $B$ is a subset of $C$. I don't know where even to begin Can anyone help? Thank you