All Questions

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Find the quadratic equation whose roots are $2+i$ and $3-i$. $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
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$9999\ldots9\cdot9999\ldots9$ will always contain exactly one $8$?

Will a number consisting of only the digit $9$, multiplied with another number consisting of only the digit $9$, always result in a number that contains exactly one $8$ digit, and how can one know ...
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Use the Stokes' Theorem to find the work of the vector field $\overrightarrow{F}$

I have the following exercise: "Use the Stokes' Theorem to find the work of the vector field $\overrightarrow{F}=x^2 \hat{i}+2x \hat{j}+z^2\hat{k}$ along the anti-counterclockwise oriented area of ...
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Lower Exponent P Central Series

The lower exponent p central series for a p-group G, defined by G=P1(G)>P2(G)...>Pc(G)=1, where Pi(G)=[Pi-1(G), G]Pi-1(G)^p. If Gi=G/Pi(G), and A: Gi+1------->Gi, Then KerA=Pi(G)/Pi+1(G). Now my ...
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Fidelity of measurement using conditional probabilities

Let me begin by saying that I'm not entirely sure if this is the correct forum, or if Cross Validated would be more suitable. The problem I'm about to describe is statistical in nature, but I believe ...
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If $0<r<1$ show that $r^n$ goes to $0$ as $n$ goes to 1.

If $0 < r < 1$ show that $r^n$ goes to $0$ as $n \to 1$. If $|r| < 1$ then $r^2 < r$ similarly $r^4 < r^3 < r^2 < r$ so $r^n$ as $n \to +\infty$ will be equal to $1$ how can ...
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Find the remainder of a division

Which is the remainder of the division $985^{423}:98$? That's what I have tried so far: Let $a=985,n=98$. Then $(a,n)=1$ and $\varphi(n)=42$. So, we have that $985^{42}\equiv 1 \pmod{98}$. Hence, ...
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Limits of the Fourier Integral Transform of a top hat function?

Would the Fourier transform of the following function: be the integration of -(X+1)e^(-ewjt) between the limits of -1 and +1 or -1.5 and -0.5? Would there a shift of limits?
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If $n$ and $m$ are integers and $d$ is their $GCF$, prove all factors of $n$ and$m$ are factors of $d$.

Here's what I know: The basic outline of the proof. I know how to do the proof, but one part doesn't click with me. That is, how do we know that the GCF's prime factorization, and all of its ...
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prove or disprove $F_{\sigma}$ characteristic function is Baire-1?

prove or disprove $F_{\sigma}$ set's characteristic function is Baire-1? I know this reslut:A characteristic function A is Baire-1 if and only if $A\in \Delta_{2}^{0}$ can see this PDF ...
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Tangent Plane Question.

Find the tangent plane to the surface $$4x^2 + y^2 - z^2 = 4$$ at the point $(1,-2, 2)$. Sketch the level curves for $z = k$ and for $y = k$ where $k = -1, 0, 1, 2$. Hence sketch the surface. Find ...
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Moment generating function with binomial coefficients

I am trying to calculate a moment generating function, and I have obtained the following result: ...
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How to visualize four dimensional tic-tac-toe?

I have played three dimensional tic-tac-toe with three players before, and we had no problem visualizing it. We drew three layers on a sheet of paper and just remembered all the different ways you ...
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Random sample taken, what is probability?

It was determined that 22% of all stock investors are retired people. In addition, 38% of all U.S. adults invest in mutual funds. Suppose a random sample of 20 stock investors is taken. a. What is ...
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Prove Theorem 1. Now let A,B,C,D,E be the following five sets

Theorem 1. If H is a subgroup of the symmetric group S_5 with order 60, then H = A_5. The group S_5 is included in S_6 in the usual way: if σ is a permutation of {1; 2; 3; 4; 5} we extend it to a ...
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Does the equation converges uniformly on open disk D(0, 1/3)

Does the function converges uniformly on open disk D(0, 1/3)? Σ (n/(n+1))z^n, start value n=1 to infinity. I do understand that the series converges but what are the steps I need to do to know that it ...
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What structure does this set of mapping have?

Suppose two sets $B = \mathbb R$, and $A$. $F$ is a set of mappings from$A$ to $B$, such that $\forall f_1, f_2 \in F$, there exists a bijection $g: B \to B$ , such that $f_2 = g(f_1)$, ...
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$\lim(x_n) = \infty$ iff $\lim\inf(x_n) = \infty$

Given a sequence $(x_n)$ in $\mathbb{R}$, show that $\lim(x_n) = \infty$ if and only if $\lim\inf(x_n) = \infty$. Is this still true if we replace $\lim\inf$ with $\lim\sup$?
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Numerical estimation of $\pi$ with the Buffon's needle

Is there a way of estimating $\pi$ with the Buffon's method without assuming $\pi$ known? To be more precise: in a Monte Carlo simulation of the experiment invented by Buffon I would (ideally) ...
Suppose we have a ruled surface $\pi : X \to C$. Let $\sigma : C \to X$ be a section of $\pi$, $D := \sigma(C) \in \operatorname{Div}(X)$, and $\mathscr{E} := \pi_\ast \mathcal{O}_X(D)$, so that ...