# All Questions

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### Sum of a cosine series

I have a short question: It turns out that the following holds: $\sum_{i=1}^{k-1} \cos(\frac{2\pi i}{k}) = - 1$. Why is that? Thank you!
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### Distributional derivative of a characteristic function

I need some help with this exercise about distributional derivatives: If we have $N=2$, and a function $g=\chi_{C}$, where $\chi$ is the characteristic function, and $C$ is the unitary cube ...
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### how do you rewrite a recursive formula to find its roots

Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac12 + \sqrt{x_n}$. Rewrite the recursive formula in the form $$x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c$$ ...
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### Using Euler's method and Taylor polynomial to solve differential equation

Consider the initial value problem $dy/dx=x+y^2$ with $y(0)=1$ a) Use Euler's Method with step-length $h=0.1$ to find an approximation to $y(0.3)$. HINT 1: :Numerical methods. HINT 2: ...
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### What is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree 2, 3,..., n (i.e. a tile of an arbitrary degree) would be? Asked another way- what is it that ...
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### $(p^m-1) \mid (p^n-1) \Leftrightarrow m \mid n$

Prove that $(p^m-1) \mid (p^n-1) \Leftrightarrow m \mid n$. The $\Leftarrow$ part is ok, the $\Rightarrow$ part should be easy but I'm stuck with it! Thanks in advance.
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We were told, in recitation class, about a test for sequences convergence (not series) Which goes as follows: if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}=L$ then $\lim_{n \rightarrow \infty} ... 1answer 126 views ### Prove that subsequences of a sequence with infinite range contained in a compact set X converges to a point in X. I have a question about a step regarding the following proof from Rudin theorem 3.6a), Proof: Let E be the range of the sequence${p_n}$since E is a infinite subset of a compact set it has a limit ... 2answers 47 views ### Maximize expected return making a practice exam I had to make the following problem which I couldn't solve unfortunately... Problem In the springtime, a student has$N$days to find a summer job for one month. Each day, ... 1answer 47 views ### Possible number of names from a certain alphabet I am trying to solve the following problem, but I am a bit stuck. The question is as follows. The language of a certain island has only the letters A, B, C, D, E. Every place name must start and ... 1answer 86 views ### convergence on$L^p$space Let$ \displaystyle{ f \in L^p (\mathbb R^n), 1\leq p <\infty }$and let$ \upsilon \in \mathbb R^n$. For$h>0$define$\displaystyle{ f_h(x) = \frac{1}{h} \int_0^h f(x+s \upsilon) ds }$. ... 0answers 123 views ### Local trivializations for orthonormal frame bundle Let$(E,\pi, M)$be a real vector bundle of Rank$N$. Then one can define its frame bundle$GL(E)$as follows:$GL(E)_x:=\{\text{ordered bases of }E_x\}$(for$x\in M$).$GL(E):=\bigcup_{x\in M} ...
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Suppose $( \Omega_1, \mathcal{F}_1), ( \Omega_2, \mathcal{F}_2)$ are measurable spaces then My question is with regard to the definition of the collection $\mathcal{G}$. I don't really ...
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### Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$

Given two matrices $A$ and $B$. How would one prove that the Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$?
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### Least Squares & Normal Equations

I encountered the problem below, and I know how to do the least squares for a system of equations with no solutions (inconsistent system) where the number of equations (rows) is greater than the ...
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### Optimal strategy in a match picking game

I faced this exercise making a practice exam and couldn't find the answer: Question: 2 players play the following game with matches. Each player can take in turn some matches from a bunch of ...
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### Evaluate $\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$

I have been working on this sum for a while. The question asks to evaluate the double integral. $$\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$$ where $p$ is equal to $x^2$. I know that I have to ...
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### $\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$

I'd like to calculate $\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$. What I've thought of is: $\hat n= \hat R$ and using: ...
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### Minimal difference between classical and intuitionistic sequent calculus

Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an ...
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### Prove that $|G| = |Z(G)| + \sum_{i' \in I'}|G:C_G(x_{i'})|$

Let $G$ be a finite group. Define $\rho: G \to Sym(G)$ by $\rho(g) = c_g,$ where $c_g:G \to G (y \mapsto gyg^{-1}).$ Denote $(\rho(g))(x) =g(x)$ For $x\in G,$ define $G_x: = \{g\in G| g(x) = x\}$ ...
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### Sequence in span with disjoints supports has (?) block subsequence.

Assume $b_k \in <e_i, \text{with coefficients} \ a_i^k \geq 0>$ is a sequence and $a_i^k$ have disjoint supports(support is the set where $a_i^k \neq 0$). Is there a way to prove or disprove ...
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### Solving construction problems?

I recently encountered 'construction problems' in geometry. These were quite new to me and I didn't know the requirements they expected and prerequisites to solve them. I'll explain with an example. ...
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### Complex numbers problem

I have to solve where n is equal to n=80996.
### Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$
Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.
### Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.
Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of ...