# All Questions

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### Need help in algebra question

I hope u can help me out For number 8 and 9 find each value if f(x) = -3x^3 + 2x^2 8) f(-1) 9) f(1/2)
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### Is a subobject classifier logically equivalent to set-inclusion?

Can one subsume the notion of set-inclusion $\subseteq$ and $\subset$ with the notion of a subobject classifier, expressed via an injective morphism $\hookrightarrow$? Specifically: Are the ...
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### Bounding random variable

Let $y_1,...,y_k$ be random variables that get only 1 and 0 such that $P(y_i=1) \ge 2^{-1} +q$ for all $i$ and let $Y$ be the sum of all these variables. I want to bound $P(Y \le k/2)$ using chernoff ...
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### why absolute value function is not differentiable at X=0?

They say that the right and left limit does not approach a definite value hence it does not satisfy the definition of derivative.But what does it mean verbally in terms of rate of change?
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### Finite models for Systems of Incidence and Parallelism Axioms?

I'm reading Shafarevich's Basic Notions of Algebras, my question is at the end of these prints. I don't understand what's the connection of that linear equations (in the end of the ...
1answer
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### Why is the equation $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2}$ in the residue theorem accurate?

I don't understand the reasoning here: $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2}$
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### Extension field of finish degree

Let $E$ and $F$ two finish extension of a field $K$ of degree $[E:K]=m$ and $[F:K]=n$ such that $\gcd(m,n)=1$. Show that if $\alpha\in F$ has degree $r$ on $K$ therefore $\alpha$ has degree $r$ on ...
1answer
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### Finding the maximum and minimum of sets

1) Does $[0, \sqrt{2}] \cap \mathbb{Q}$ have a minimum? Maximum? Minimum is $0$ since $0$ is also a rational number. $\sqrt{2}$ cannot be the maximum because it is not a rational number. How can I ...
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### Equality of explicit recursively enumerable sets independent of ZF

Are there two recursively enumerable sets defined by explicit enumerators whose equality is independent of ZF? If so, can such sets each have finitely many elements? I don't know how exactly to define ...
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### Write an equation in standard form for the line that is parallel to the graph of -8x= 5 - 4ynd has y-intercept -0.5

I hope somebody can help in my homework Question: Write an equation in standard form for the line that is parallel to the graph of $-8x= 5 - 4y$ and has y-intercept $-0.5$
1answer
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### Let $p$ be a prime number. How I can simplify this expression

My question is: Let $p$ be a prime number. How I can simplify this expression: $$z=2∑_{j=0}^{2^{p-3}}C_{2^{p-2}}^{2j}2^{2^{p-2}-2j} 3^{j}$$ where $C_{2^{p-2}}^{2j}=((2^{p-2}!)/((2j)!(2^{p-2}-2j)!))$
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### How to find a conserved quantity in this differential equation.

Consider the system $\ddot x = x^3 -x$, what is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...
1answer
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### Problem with the answer of a basic ratio problem

Here is the solved example. The answer given is 175, but my calculation comes out to be 375. I thought that it was just a typo, but it has been written twice, so I'm not sure. What am I doing wrong?
1answer
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### A question on nilpotent linear operator on finite dimensional vector space with dimension same as degree of nilpotency

Let $T$ be a nilpotent linear operator of index $n>1$ ($T^n$ is the null operator but $T^{n-1}$ is not ) on a vector space of dimension $n$ ; then how do we prove that there is no linear operator ...
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### How to show that 1/4 is in the standard Cantor set?

I have studied the method that shows 1/4 belongs to the standard Cantor set, proving that it has only {0,2} as digits in its ternary expansion. But how can I do this in some other way?
1answer
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### What is the length of the longest decreasing sequence in integer matrix?

Given a finite $m \times n$ matrix $M$ with all distinct integers, we travel it following two simple rules: The travel can start from any cell, say, $M[i,j]$. At each cell $M[i,j]$, it computes the ...
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### If $x \in \{1,2,3, \cdots, 9\}$ and $f_n(x) =xxxx\cdots x($ n digits) then find the value of $f_n^2(3)+f_n(2)$

If $x \in \{1,2,3, \cdots, 9\}$ and $f_n(x) =xxxx\cdots x($ n digits) then find the value of $f_n^2(3)+f_n(2)$ If x =1, then $f_n(1) = ?$ Please suggest how to expand such series. Thanks
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### Beautiful bent triangles

Xº yº Zº are angles of tight rope triangle ABC. A killer bully bent its sides one degree each. So he got a concave hexagon with (x-1), 181, y-1, 181, z-1, 181 as its angles (all degrees). Proving that ...
3answers
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### Probability of $X = Y$ for a bivariate pdf

Can somebody explain to me why the probability of $X = Y$ is $0$ for a continuous bivariate pdf and not the integral of $f(x, x)$ over all possible $x$?
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### Commutativity of special type of permutations

Let $p_1,p_2$ be permutations on the set $S=\{1,2,...,n \}$ and define $U_1:=\{k\in S:p_1(k)\ne k\}$ , similarly we can define $U_2$ . Now I can prove that if $U_1 \cap U_2=\phi$ , then ...
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### How to show that the integral converge or diverge?

I have the following integral and wish to see whether it converge or diverge. where the integration contours $-l_1$ to $l_1$ belong to real numbers. I have tried by comparison test like the ...
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### Basis of a set of vectors

Assume that we have a set of vectors as $S=\{v|v=(\sum_{i=r+1}^nx_ic^i_1,\dots,\sum_{i=r+1}^nx_ic^i_r,x_{r+1},\dots,x_n)^T,\forall x \in \mathbb{R}^{n-r}\}$. Here the first $r$ elements of the vector ...
1answer
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### How is a coordinate system called where values increase to the bottom instead to the top?

In some computer graphics libraries the coordinate system is almost like the "usual" cartesian coordinate system. The only difference is that the $y$ values increas to the bottom, not to the top. ...
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### A counting problem related to Parallelogram

There are n distinct points in the plane, given by their integer coordinates. Find the number of parallelograms whose vertices lie on these points. In other words, find the number of 4-element subsets ...
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### Does $f ≥ g$ always imply $f^{-1} ≤ g^{-1}$ for invertible $f, g$?

If $I = (I, ≤)$ and $J = (J,≤)$ are linearly ordered sets, does $f ≥ g$ awalys imply $f^{-1} ≤ g^{-1}$ for invertible $f, g \colon I → J$? This is easy to prove if $f$ and $g$ are assumed to be ...
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### Measurability properties of processes that arise as limits of sequences of measurable processes

I try to reduce my problem to a more general statement from which I want to know whether this is true in general. I have a sequence of continuous-time stochastic processes $X_t^{(n)}, t \geq 0$ with ...
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### Simplifying expressions when taking limit to infinity

When taking the limit as $x$ approaches infinity of the function $[(x-2)^2+4x-x^2]/5$. How far should I simplify? I can simplify the part in parentheses, but simplifying further will get me $4/5$ and ...
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### second order differential equations

Determine the longest interval in which the given initial value problem is certain to have a unique twice-di erentiable solution. Please do not solve the equation. (t-1)y''-3ty'+5y=sint y(-3) = 2 ...
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### Any book on timeline of progress of Math concepts and applications

I was wondering if there is any book that chronicles the progress of Math over the centuries and also mentions about how/when applications of various theories were discovered/invented. I have been ...
1answer
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### If $A$ is a convex set in $R^n$ with a limit point $x_0$, can we have a line segment in $A$ with $x_0$ its limit point…?

If $A$ is a convex set in $R^n$ with a limit point $x_0$ outside $A$ can we have a line segment in $A$ with $x_0$ its limit point...?
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### Diagonalisable linear operator on infinite-dimensional vector space: definition problem

How to define a linear operator on an infinite-dimensional vector space to be diagonalisable? I have tried to search for the definition on internet but it seems not very fruitful as all are about ...
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### Proof by induction with variable other than $n$

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
1answer
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### Are prime ideals always comaximal?

This is easy to see in the ring of integers. In fact, the ideals don't even have to be prime. It's enough to be coprime. Then their GCD is 1, so 1 can be written as a linear combination of the ...
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### $\int_0^\pi \tan(\theta + \text{i}a)\text{d}\theta$ =?

Solve the integration $$\int_0^\pi \tan(\theta + \text{i}a)\text{d}\theta$$ Where $a\in \mathbb R$ and $a \neq 0$ The method I took is to let $$z=e^{i\theta}$$ and since ...
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### A simple arithmetic problem

In a field grass grows at uniform rate ; If the field can feed 36 cows for 4 days , 21 cows for 9 days , how many cows can be feed for 18 days ? My solution : Let $x$ be the initial amount of ...
1answer
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### showing $\psi: R\to \mathbb C$ is ring isomorphism.

Below is an example from I.N. Herstein: Let $R=\Bigg\{\left( \begin{array}{ccc} a & b \\ -b & a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of ...
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### series problem feedback needed

this is a example from my lecture notes and while studying i cannot understand how the final answer is found. the particular place where i am confused i have circled. shouldn't it be -1/n(n+1) ?
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### Question on abstract algebra about invertiblity?

Let $(G,*)$ be a group. Could anyone give me an example $a,b\in G$ such that $$a*b=e ~ and ~ b*a\neq e$$ Where e is the identity element. I would appreciate any help. Thanks in advance!
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### Attempted proof of expression for $p_2(n)$, the nth composite of two primes

This is an exercise to find an expression for the general 2-prime $p_2(n)$ from the generalized PNT. The 2-primes are numbers comprised of two primes factors, possibly repeated. The notation ...
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### How can I evaluate this double integral by using geometry/symmetry?

I have to evalute this integral. $\displaystyle\iint\limits_{D}(2+x^2y^3 - y^2\sin x)\,dA$ $$D=\left \{ (x, y):\left | x \right |+\left | y \right | \leq 1\right \}$$ At first, I evaluated simply by ...
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### Two surfaces that could be birational.

Suppose that $X$ and $Y$ are two non isomorphic non-singular complex projective surfaces. We also know that there is a morphism of varieties $f:X\longrightarrow Y$ which is bijective. At this point ...
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### Magnitude of the value of a line at a point which does not lie on it

This may be trivial but I had never thought of it before. I know that the sign of the value represents on which side of a line a point lies but don't know what the magnitude of that value represents. ...
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### Is $(\frac{1}{ x}\frac{d}{d x})^k(\frac{x}{\sinh x})$ always a bounded function?

Consider $f_{k}(x)=(\frac{1}{ x}\frac{d}{d x})^k(\frac{x}{\sinh x})$, $x>0$, $k=0,1,2,\cdots.$ Then, Is $f_{k}(x)$ always a bounded function? The only thing one need to care is the behavior when ...
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### what is The general sequence of: 1, 7, 12, 17, … [on hold]

i have an math question. what is The general sequence of beloow sequence? 1, 7, 12, 17, ... thanks
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### A question about characteristic classes

I have a map $\phi:BO(1)^n\rightarrow BO(n)$ which is given by sending any $n$-tuple in $BO(1)^n$ to an $n$-plane through the origin. Thus, this induces a group action on the symmetric group $S_n$ ...
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### Beautiful logical minimum construction

On a circle after equal intervals 25 points are located. On every point is a policeman. All policemen are numbered (from 1 to 25) in some way. Now they have to move to some other points through this ...

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