# All Questions

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### Riemann sum and partitions

If f is riemann integrable and if $(P_n)$ is any sequence of tagged partitions of [a,b] such that $\lVert P_n \rVert$ -> 0, prove that $\int_a^b f = lim_n S (f;P_n)$. I am confused as to how to ...
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### Diagonal decomposition, square root and eigenvector / eigenvalue of a matrix

I have encountered a problem of finding eigenvector and eigen value of a matrix of type $$A = \dfrac{1}{2} \begin{pmatrix} 4&1&-2\\ -4&1&6\\ 2&0&-2 \end{pmatrix}$$ Also I ...
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### Still stuck on simplifying terms while doing linear combinations

So I'm currently trying to wrap my head around finding gcd through the Euclidean Algorithm in order to write the integers as a linear combination. For example, a problem is to express the ...
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### Show $ex \leq e^x$ for all $x \in \mathbb{R}$

So far all I have is this: Let $f$ be a function where $f(x)=ex-e^x\leq 0$ $f'(x)=e-e^x \leq 0$, so $f$ is decreasing. I'm stuck here. Can someone help me with the next steps?
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### determine whether a combination number is odd or even

Let $k$ be a given positive integer (fixed). I want to determine whether $$2n-k\choose n$$ is even or odd, for each positive integer $n$. Is there any general result? My attempt: Case (1). ...
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### Why are structures with no relations called algebras?

Elementary algebra has at least one relation: the equality (or identity) relation, signalized by the symbol "=" The equality relation is quintessential to linear algebra and algebraic equations, such ...
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### What did I do wrong with this combinatorics question?

I was given the following problem. "A teacher wants to choose a captain and vice-captain among 12 volleyball players. In how many ways can she do so?" I tried to solve it by multiplying 12 by 11 ...
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### How is $K(x)$ pronounced, where $K$ is a field?

Let $K$ be a field. The ring of polynomials $K[x]$ is pronounced "$K$ adjoin $x$", right? How is the field of rational functions $K(x)$ pronounced? (Sorry if this is a silly question. I am ...
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### Describing polar coordinates in a window

I'm having trouble with the following problem and have no idea what to do. I tried drawing a horizontal and vertical line down the middle of the window but got nowhere. A window is in the shape of a ...
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### Probability of choosing one of two desired boxes out of five on the second try.

I've got a problem that I'm not understanding how a given probability is being found. If you have five boxes, three of which are empty and two of which have items you want, if you choose boxes in a ...
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### Geometric distribution converges to exponential distribution

For $n\in \mathbb{N}$ let $X_n$ be geometric with parameter $p_n \in (0,1)$, that means $\mathbf{P}[X_n = k] = p_n(1-p_n)^k$, $k\in\mathbb{N}_0$. How must the sequence $(p_n)_{n\in\mathbb{N}}$ ...
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### Differentiability & continuity proof?

Prove that if the absolute value of f is differentiable at a and f is continuous then f is differentiable at a. I've started off by writing down the formal definition of differentiability and the ...
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### What is a function which differs when differentiated with respect to x and then with y to function differentiated with respect to y and then x?

What is a function which differs when differentiated with respect to x and then with y to function differentiated with respect to y and then x? Even if it differs at one particular point.
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### Functions and range

$$a \colon \mathbb{R}\setminus\{0\} \to \mathbb{R} \;\text{ defined by }\; a(x)= 6/x \\ b \colon \mathbb{Z} \to \mathbb{R} \;\text{ defined by }\; b(x) = 3x + 1$$ a) State the range of ...
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### Let X_1,..,X_n i.i.d negative binomial. Find the best unbiased estimator for P(x<=3)

I am not sure where I should even start with this problems. I know that the sum of negative binomial random variables is itself a negative binomial random variable. I am sure that I can show that the ...
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### What is $p+4 \pmod {13}$

$p=\sum_{m=1}^{12} (m \cdot m!)$, and the question is just the title. I tried applying Wilson's theorem i.e $(p-1)!+1\equiv0 \pmod p$ but did not get much help. Thanks for helping
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### $a,b,c,p$ are rational number and $p$ is not a perfect cube

Given that $a,b,c,p$ are rational number and $p$ is not a perfect cube, if $a+bp^{1\over 3}+cp^{2\over 3}=0$ then we have to show $a=b=c=0$ I concluded that $a^3+b^3p+c^3p^2=3abcp$ but how can I go ...
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### tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
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### If $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes?

Show that if $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes. This might be a silly question but I really don't see it? The thing I ended up was: let $n=a^3 + b^3 + c^3$, ...
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### Iterative method for a positive definite matrix

Given A-> a positive definite matrix, if W(x)= 1/2- is a functional form and x-> x + a_i * v_i is the i-th step iteration( v_i is the ith eigen vector of A) 1. how do I find the step size a_i 2. Show ...
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### Boundary Value Problem and solutions

The linear ordinary differential equation $y'' + y = 0$ has the family of solutions $y = A \sin(x) + B \cos(x)$ Determine whether $y(0)=2, y'(\pi/2)=3$ is a unique solution. If not, does it have no ...
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### Relationship between ball speed and distance traveled [on hold]

What is the relationship between the speed of a thrown baseball, and the distance it travels? In other words, if a baseball is thrown 300 feet, what does its speed need to be? What about 325 feet? ...
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### Integrable but not differentiable function

Suppose $f(x)$ is continuous on $[a, b]$ except at a point $c$ in $(a, b)$ at which $f(x)$ has a jump discontinuity. For $x$ in $(a,b)$, set $F(x)=\int_a^xf(t) \, dt$. Show that $F(x)$ is continuous ...
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### Linearising equations about a base state.

Consider a shallow-water system with mean depth H, where the base state consists of the flow (u,v)=($u_{0},$0), with a sloped water surface $\eta_{0}$(x,y) = - $\gamma y$, where u$_{0}$ and $\gamma$ ...
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### Series and sequences- Help

Let $a_n=n^x(n^(1/n^2)-1)$ for n in natural numbers and assume that lim(n goes to infinity)ln(n)/n^r = 0 for any r>0. Let ln(x)=integral(from t=1 to x)dt/t for x>0. Prove the inequality h/1+h < ...