# All Questions

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### Matrix factorization overview

Does this hold? All $\lambda_i$ are different ⇒ eigenvectors are linearly independent. I am, for the sake of matrix factorization such as svd, $QR$ and $UDU^H$ interested in the relationship ...
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### how to compute the last 2 digits of 3^3^3^3 to n times?

input n, output the last 2 digits of the result. n=1 03 3=3 n=2 27 3^3=27 n=3 87 3^27=7625597484987 n=4 ?? 3^7625597484987=?? Sorry guys, the formula given ...
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### Proof cofactor-matrix cofac(AB) = cofac(B)*cofac(A)

Let $A \in K^{nxn}$ and $Cofac(A)$ be the cofactormatrix to A. I have to show (1) $cofac(AB) = cofac(B)*cofac(A)$. In fact I have: $^t(cofac A) = cofac (^t A) = adj(A).$ Then I have (I have ...
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### Intersection of two circle orthogonally.

The Circle $x^2+y^2+2g_{1}x-a^2 = 0$ and $x^2+y^2+2g_{2}x-a^2=0$ cut each other orthogonally. If $p_{1}$ and $p_{2}$ are perpendiculars from $(0,a)$ and $(0,-a)$ on common tangent of these circle. ...
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### Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1$$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x)$$ We also have, ...
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### Minkowski operations on normed vector spaces. Closedness of set.

This is a seemingly easy question, but I am having trouble grasping the solution: Given a closed $B\subset X$ prove that given $\beta \in K$, $$\beta B:=\{\beta b: b\in B\},$$ is closed. $X$ is our ...
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When you $\tt{diff}$erence discrete observations of a function $f$ you lose one observation each time you apply $\tt{diff}$. When you $\partial$ifferentiate a $\mathcal{C}^n \ni$ function $f: ... 5answers 45 views ### Proving that$\dfrac{\sec\theta\cdot\sin\theta}{\tan\theta+\cot\theta}=\sin^2\theta$The question is: Prove that: $$\dfrac{\sec\theta\cdot\sin\theta}{\tan\theta+\cot\theta}=\sin^2\theta$$ My proof is shown below. If anyone has an alternate proof please, please post it. Thanks! 2answers 20 views ### Diagonalization of a strange transformation Let be$V$a vector space on$\mathbb C$and$\dim V=4$and let be$f \in \operatorname{End}(V)$such that$\operatorname{Im}(f^2+a \cdot \operatorname{id}) \subset \ker(f+id)$where$a=\det f$, ... 1answer 160 views ### where is the mistake in this “paradox”? In the middle of page 34 in Bruns and Herzog, Cohen-Macaulay Rings, the authors present the following situation: Let$k$be a field and let$R=k[X,Y]$be a graded ring with grading induced by ... 3answers 41 views ### Square root of negative integer Can I write:$-\sqrt{(2)}$=$\sqrt{(-2)}$and vice versa? Or, say, we have,$(-\sqrt{(x - 4)}$Can this be changed into$(\sqrt{(4 - x)}$by taking the minus sign inside the square root? How? 2answers 330 views ### Probability of circle given by randomly chosen diameter falling inside a square Two dots are thrown into a square with side length 1 cm. The line ending in these two dots is the diameter of a circle. What is the probability that the circle lies in the square? 0answers 13 views ### Maximizing the number of groups The problem is as follows, There is a restaurant which has N number of chairs each chair has a unique number written on it so the array of chairs is like [1,2,....N-1,N] , there are G number of groups ... 2answers 23 views ### Equivalent logical quantifier statements? I was doing an exercise that said convert the statement "Jane saw a police officer, and Rodger saw one too" into the logical equivalent using quantifiers. My answer was: $$\exists x(P(x)\implies ... 0answers 17 views ### How to prove that two curves are not path homotopic I have a unit circle around origin.And another unit circle around (2,0). Consider the domain R^2 / \{(0,0)\}. I am able to clearly see that both are not homotopic but i am unable to prove it ... 0answers 26 views ### How did Fourier find the formula for the fourier series coefficients? The modern proof use the dot product but did he use that ? 5answers 132 views ### how to prove e^{A \oplus B} =$$e^A$ $\otimes$ $e^B$ where A and B are matrices? (Kronecker operations)

how to prove $e^{A+B} = $$e^A$$e^B$ where A and B are matrices? The operations '+' and '*' are defined such that AI + IB = A+B, where I is the identity matrix. I suppose these are called Matrix ...
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### Why the following is a seminorm rather than a norm

I really don't understand why the following is a seminorm rather than a norm? $$p_k(u)=\sum_{|α|\le k}\sup_{x∈R^n}(1+|x|^2)^{k/2}|D^α u(x)|,$$ for all $u \in C^\infty$. I do understand if ...
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### Characterization of Volumes of Lattice Cubes

Let $C$ be a lattice cube in $\mathbb{R}^n$. Characterize all possible volumes for $C$. I broke this proof into three cases, the last of which I am having trouble with in one direction. We will let ...
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### Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
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### give an example of a test statistic that is not a pivotal quantity

give an example of a test statistic that is not a pivotal quantity. what is the relationship between test statistics and pivotal quantity ? i know that pivotal quantities are fundamental to the ...
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### true or false: $\infty$ = 1? [on hold]

If ultimate reality is real and infinite, then being the only $1$, $∞0\infty = 1$. There are two types of $1$, though. The $1$ ultimate reality (or infinite $1$) and all the parts of the ultimate ...
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### Solving set of equations involving summation/integration and squares

I can only get implicit results for these equations. I looked everywhere but I wanted to check whether you can see something I can't. I find implicit results for all of the variables.I need to find ...
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### Joint & Marginal pdf

Let $R$ be the region in the plane bounded by the line $y = x$ and the parabola $y = x^2$. A random point $(X, Y )$ is picked from $R$. (a) Find the joint pdf of $X$ and $Y$. (b) Calculate the ...
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### The Closest matrix of the symmetric matrix

Consider the space $M_{2\times 2}(\Bbb R)$ under the standard inner product between matrices. Prove or disprove the following statements: (a) If $C$ is the symmetric matrix closest to $A$, and $D$ is ...
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### Prove the curvature of a level set equals divergence of the normalized gradient

Suppose we have a function $\phi : \mathbb{R}^2 \to \mathbb{R}$, and a curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by a level set of $\phi$, ie. the codomain of $\gamma$ is ...
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### A basic question on distribution function and stieljes integral

Is it true that $$\int_{\Bbb R} G(x)dF(x) + \int_{\Bbb R} F(y)dG(y) = 1$$ where $F$ and $G$ are distribution functions of $X$ and $Y$ such that both have no common jump points i.e. $P(X=Y) = 0$ ? ...
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### Is Tietze's Extension Theorem overkill here?

Suppose $X$ is a normal space with a closed discrete subspace $A\subseteq X$. I want to extend a continuous function $f:A\to[0,1]$ to all of $X$. Tietze's Extension Theorem would certainly do the job. ...
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### Limit with logarithm

I am stuck with this limit and I can't get rid of the deminator $x^2-x_0^2$, I figured that I am going to solve this limit for two situation, 1) $x_0^2+x_0^4=1$ and 2)$x_0^2+x_0^4\not=1$, but that's ...
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### Abelian group C*-algebras

Let G is a locally compact Abelian group $C^*$-algebra, then $C^*(G)$ is an Abelian $C^*$-algebra, so C*(G) is isomorpohism with the C$_0$(X) for some locally compact Hausdorff space X, here X is the ...
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### Establishing the n-th order weak derivative.

Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where $AC$ stands for the space of absolutely continuous functions. Now, let $f,g\in{L_{loc}^{1}(a,b)}$ and ...
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### Set theory: Metamath Proof of the Pigeon-Hole Principle, Error?

I have recently came discovered Metamath. Supposedly the language is one that a computer may proof-check. I then began to look at concepts that I am familiar with, and decided to look up the pigeon ...
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### Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup?

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup? Isn't $C_0$ the set of continuous functions that vanish at infinity? Thanks and regards!
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### Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
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### Why does $\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2}$?

My question is: Why does $$\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2},$$ where $a_n$ is some complex number, $n$ an integer going ...
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### Finite rings and subrings isomorphic to $\mathbb{Z}_n$

My book has proven this: Every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z_n}$. The $\mathbb{Z_n}$ case arises if the parent ring has characteristic $r>0$ I ...
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### convex combination of known solutions

I have a system $Ax=b$, where $A\in\mathbb{R}^{m\times n}$ is a large rectangular matrix with $m<n$, $b\in\mathbb{R}^m$ is known, and $x$ a decision vector. Suppose that I have known vectors ...
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### convergence prove of $a(n) = \frac{n}{4^n}$

i need to prove that the following sequence converges: $$a(n) = \frac{n}{4^n}$$ in the assignment there is also a hint: prove that $2^n \gt n$ holds true for every $n \ge 0$ i can prove that ...
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### Tangent plane passes through origin

This is from a section in my course book on elementary differential geometry: Since the tangent plane $T_p S$ of a surface $S$ at a point $p \in S$ passes through the origin of $\mathbb{R}^3$, it ...
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### Difference between separable and linear? Differentials

My understanding was that a separable equation was one in which the x values and y values of the right side equation could be split up algebraically. I tried this once before and got the wrong ...
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### How can I solve Dynamic Problem?

I tried to solve this Dynamic Problem , But I do not know how to begin solving it. A minivan starts from rest on the road whose constant radius of curvature is 40 m and whose bank angle is 10°. The ...
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### Newbie Question - Difference Distribution

This may seem a simple stupid question but its driving me crazy for 4 hours. I have two values $-10.5$ and $+15.0$ These values sum $4.5$. For whatever reason that doesn't matter the end result ...
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### Why is the exponential of sets the function set?

I have asked a question about exponentials yesterday, but there is something that is not clear to me so I thought I should ask a different question. By looking at the case of sets, I am not sure ...
### Proving a continuous function $f:X\to Y$ is uniform continuous if $X$ is compact.
I'm reading the proof of "if there's a continuous function $f:X\to Y$ where $X$ is a compact metric space and $Y$ is a metric space, then $f$ is uniformly continuous on $X$." The proof proceeds thus: ...