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Mathematics

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Induced morphism between image and ker (category theory)

Context question: “Differential on a object of an abelian category” Let $C$ be an abelian category and $x$ an object of $C$. Consider a morphism $f: x\to x$ such that $f\circ f=0.$ I can see two ways to define a morphism $\text{im}\, f\to \ker f:$ Consider the diagram where the morphisms $i$ and $j$ are […]

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Why is addition countable?

can someone explain to me why addition is countable when it is a finite sequence of terms separated by + signs: i.e.(x1 + x2 + ….+xk) or (13, 2 + 3 + 3, 10 + 11 + 12) because i know to be countable it needs to have a one-to-one correspondence with Natural numbers. But […]

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Is there a series representation for $\frac{1}{\log(x)}$?

for $x>0$ it is known that: $$\log(x)=2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}$$ Is there a series representation for $\frac{1}{\log(x)}$ in the following form? $$\frac{1}{\log(x)}=a_1(x)+a_2(x)+a_3(x)+\dots=\sum_{k=1}^\infty a_k(x)$$ Or, is there a way to transform the expression $$ \frac{1}{2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}}$$ into the aforementioned form?

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Vector space, direct sums involving linear maps and polynomials

Let be $V$ a vector space over $K$ and lets $a\in K\setminus\{0\}$ and $T:V \to V$ a linear map s. t. $a^2T-3aT^2+T^3=0$. Show that $V = Ker(T) \oplus Im(T)$. My professor told me an observation: This result is more general, that is, if I have a polynomial $P$ such with $P(0) = 0$ and $P'(0) […]

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Should we divide by 0 in limited question

I got a huge argument between me and me doctor. ends me kicked from the classroom. It was a limit question and we was have e to the power 1 divided be x and the x was get closer to zero and the doctor end the question by dividing by zero and saying infinite and […]

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Need help proving continuity for $\frac{2x}{3x-1}$ at $x=1$ using Epsilon-Delta

The function is given as: $\frac{2x}{3x-1}$ I must prove continuity at $x=1$. I understand the definitions using the Epsilon Delta proof but my issue comes when I arrive at $|f(x)-f(1)|=|\frac{1-x}{3x-1}|=\frac{|x-1|}{|3x-1|}$< $ \epsilon$. I know I must relate that result with $|x-1| < \delta $ but I am stumped on how to bound the denominator of […]

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Solution to the Massive Laplacian Operator

Let $\phi(x,y)$ be the solution of the PDE: $$(-\Delta + m)\phi(x,y) = \delta(x-y)$$ where $m > 0$ and $\Delta$ is the usual $d$-dimensional Laplace operator. I’ve read that the solution of this equation can be written as: $$\phi(x,y) = \frac{1}{(2\pi)^{d}}\int e^{-ik(x-y)}(k^{2}+m^{2})^{-1}dk $$ but I’m having trouble proving it. The obvious way is to use Fourier […]

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what is this question asking for? basic metric concepts

I am studying basic concepts of metrics. and I ve been practicing axioms of metrics to prove some functions can be metrics or not. then this question showed up and I do not understand what am I being asked for. how is related to axioms of metric? am i supposed to find a function? $X=\{a,b,c\}$ […]

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Conditional probability mass function of the sum of independent geometric random variables.

$X, Y \sim Geom\left(p\right)$, and $X, Y$ are independent. Find $p_{X | X+Y}\left(i | n\right)$. My approach: pmf of geometric distribution: $p_X(i) = {p(1-p)^{i-1}}$ Then, by the law of total probability: $p_{X | X+Y}\left(i | n\right) = \frac{p_{X, X+Y}\left(i,n\right)}{p_{X+Y}\left(n\right)} $ [Is using the law of total probability here correct as opposed to using Bayes theorem?] […]