# All Questions

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### Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem y'=y^2, y(0) = l, exist for all real x. (b) Find a solution of the initial value problem in (a). On what interval does it exist? ...
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### Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
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### Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
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### The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N$

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
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### Cohomology space

Let M be a compact Riemannian manifold without boundary. a) If M is a sphere, prove that the cohomology space of order 1 is trivial $H ^{ 1} (M, R) = 0$ b) If $\omega = \delta\theta$ is the ...
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### The Sum of the first five terms of an arithmetic sequence is 65/2…

The sum of the first five terms of an arithmetic sequence is 65/2. Also, five times the seventh terms is the same as six times the second term. Find the first term and the common difference of the ...
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### numerical analize ,regula falsi

A nice method to find an approximate solution is to successively cut intervals in half, as follows: let's first rewrite this as $$f(x) = 3x + \sin x - e^x = 0$$ Now pick two values, $a$ and $b$, such ...
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### Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx$$
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### Altering a model to account for another variable

A question in my textbook asked me to find a general equation for depth fallen by an object of mass $75kg$ thrown from a bridge whilst tied to an elastic rope. Below the bridge there is a stream of ...
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### Security of such cryptosystem design?

Is one able to reveal $m$ when $$С = (m + r)^e \bmod N$$ $C$ is known $r$ is known $e$ is known $N$ is known and not prime
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### Prove that $\sin\theta_1.\sin\theta_2.\sin\theta_3=\frac{r^2_1}{16R^2}$

If $2\theta_1,2\theta_2,2\theta_3$ are the angles subtended by the circle escribed to the side $a$(opposite to vertex $A$) of a triangle at the centers of the inscribed triangle and the other two ...
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### Statics Find the range value of P

Answer is $29.3 N ≤ P ≤ 109.3 N$ I tried solving it for quite some time already which I don't understand why I don't get the values. Can someone help me? $Fv$ Vertical force $Fh$ Horizontal force ...
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### Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function f from A to B", ...
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### $A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
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### Kiselev's Book I Plainimetry Question 242 - Question in the Description

Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points ...
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### $\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_N(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
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### Definition of prime model extension over a set

Standard definition of prime model over a set is that: $M\vDash T$ is said to be a prime model extension of a set $A$ if $A\subset M$ and any partial elementary map $A\rightarrow N$ ($N\vDash$) extend ...
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### $G$ is $p$-supersolvable group . $Q \in Syl_{2}(G^{\prime})$. Show $Q \unlhd G$.

Let $G$ is a finite $p$-supersolvable group for odd prime numbers . Suppose $Q \in Syl_{2}(G^{\prime})$. Now i'll show $Q \unlhd G$. Since $G$ is $p$-supersolvable, then $G^{\prime}$ ...
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### Strongly Equivalent metrics

How to show any two metrics to be strongly equivalent? Please suggest me the proper way to show this. Also i want to know how to find the constants in the respective definition.
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### Find a smooth path along which a given function on the plane is not differentiable at the origin

From Bamberg & Sternberg’s A Course In Mathematics For Students of Physics, Exercise 6.1d: Let $F(x,y) = \frac{x^3y}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $F(0,0)=0$. Invent a smooth curve ...
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### Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$\begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
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### Geometric Progression sums and sums of squares

Sum of the first $4$ terms in GP is $30$ and the sum of their squares is $340$. Find the numbers. How do I solve this?
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### Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $\mathbb A^n$ a closed subset that is not equal ...
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### Showing a subset of $S_n$ is a subgroup

Let $P$ be the set of all the elements of $S_n$ which can be written as $\sigma\mu\sigma^{-1}\mu^{-1}$ for $\sigma, \mu \in S_n$. Show this is a subgroup. This doesnt seem to be as simple as ...
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### how Find the probability that the committee will consist of the following all dentists

A committee of four people is to be formed from six doctors and eight dentists. Find the probability that the committee will consist of only dentists
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### Ring homomorphism from field

If we have homomorphism from field K to ring R, does that mean that we have ring homomorphism but K is a field? I have trouble understanding this. Thank You very much for your help.
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### If G is finite p-group then $d(G)=d(\frac{G}{\Omega_1(Z(G))})$

Let $G$ be a finite p-group such that $G$ has no non-inner automorphism of order p leaving Φ(G) elementwise fixed If $\Omega_1(Z(G))\le G'\le \Phi(G)$ how we can get $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
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### What is the geometric meaning of the inner product of two functions?

When it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clear ...
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### A billinear transformation

Let $$w(z)=(az+b)/(cz+d)$$.Then $w(z)$ maps a straight line of $z-$plane to the circle $|w|=1$ in $w-$plane if $1.|b|=|d|$ $2. |a|=|c|$ $3. |a|=|d|$ $4. |b|=|c|$ My work: I started by considering ...
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### Chance of wining Uno 7 Times in a row

I am Intrigued to determine the odds of winning such a game with 4 players.. and in 7 times consecutively Would be great to have a clear answer on this Thks Phil
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### Minimum value function

It's just a very simple question, is there a function defined and that tells you the minumum and maximum value of a list of variables, like: min(4, 3) = 3 min(2, 19) = 2 max(1, 10, 3) = 10 Is that the ...
This function is Lebesgue-integrable:\chi(x)= \left\{ \begin{array}{ll} 1 & \text{if}~x~\text{is rational}\\ 0 & \text{if}~x~\text{is irrational}. \end{array} ...