# All Questions

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### Ordinals as Trees

I'm trying to understand countable ordinals and their tree representation. I understand that $\omega$ is the first "non branching tree" of infinite height. I also understand that the exponent of ...
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### Prove sequence $S_n$ converges

If $S_1 = \sqrt{2}$, and $S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....), prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$ This is one the questions ...
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### The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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### Integer solution to multiple modular arithmetic equations

So i understand how to do this when it is just x, but now with multiples of x I am a little confused, and there's no example in my textbook of this. I just need a push in the right direction for how ...
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### Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
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### The Residues of an even function or an odd function on $U$ subset open symmetric

I have to proof that for $f$ even function holomorphic with singularities isolated then $$res_{z}f=-res_{-z}f$$ an simmetric for $f$ odd, $i.e.$ $$res_{z}f=res_{-z}f$$ My hint is proof that laurent ...
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### A real matrix whose eigenvalues have all negative real parts

While taking a look in some lecture notes of an ODE course, I found the following claim, which appeared in the text as an exercise: Let $A$ be a real $n\times n$ matrix whose eigenvalues have all ...
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Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v$ where $v \in T_a A$ $E_2 F: A ... 1answer 99 views ### Is the induced map between quotient vector spaces automatically an inclusion? If$T: V_1 \to V_2$is a linear map and$W \subset V_1$and$W \subset V_2$with$T(W) \subset W$, then the (induced) map$T: \frac{V_1}{W} \to \frac{V_2}{W}$given by $$T(x + W) = T(x) + T(W)$$ is ... 1answer 89 views ###$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$,$|c_{k}| \leq \frac{1}{k}\}$is compact Let$H$be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume$\{u_{k}\}_{k=1}^{\infty}$be an orthonormal set in ... 2answers 42 views ### How would this equation be converted to polar coordinates? How would this equation be converted to polar coordinates?$(x^2 + y^2)^2 = 2xy$First, I changed$(x^2 + y^2)^2$to$(r^2)^2 = r^4$Then, rewrote the equation to$r^4 = 2xy$What needs to take ... 2answers 50 views ### How do I find this partial derivative I have the following function u(x,y) defined as: $$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero, and$u(0,0)=0$I want to compute its partial derivative$u_{xy}$at ... 2answers 44 views ### From which equation of motion was this formula derived from in physics When solving problems involving projectile motion I use:$\sqrt{2 * \dfrac{\text{height above ground}}{9.8}}$Eg calculate the time it takes for a bomb to impact if it is travelling 4.9km above ... 2answers 336 views ### How to convert a big number back to decimals when you divide 1/15000 using a basic simple calculator? This question has always stumped me since using a simple basic calculator over 20 years ago. I'm using the basic calculator on windows or it can be any for that matter. I input the following into ... 2answers 118 views ### modular arithmetic congruence Simplify the following congruence: $$−169 \equiv \text{ ?} \mod 52$$ (By simplify, we mean find the smallest non-negative whole number which is congruent to$-169$modulo$52$.) Simplify the ... 1answer 47 views ### Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence. Consider$\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$with the metric$d((a_n$), ($b_n$)) =$[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let$A = \{(a_n) : |a_n| < 1/n ...
Let $M$ be an orientable $2$-manifold in $\mathbb{R}^n$, with area form $dA$. Let $\phi: U \to M$ be a local coordinate system for $M$, with $U \subseteq \mathbb{R}_{uv}^2$. It is stated that ...