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### Monotone convergence theorem for a generic $f(x,y)$ instead of $f_n(x)$

I was wondering if it is safe to say that the monotone convergence theorem $$\lim_{n\to\infty} \int_X f_n(x) = \int_X \lim_{n\to\infty} f_n(x)$$ (where $f_n(x)$ is a non-decreasing sequence, etc.) is ...
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### Finding integer solutions of K for equation floor(A/K) = B

How to find integer solutions of K for equation floor(A/K) = B, in terms of A and B where A and B are non-negative integers? What I tried: floor(A/K) = B then B <= A/K < B + 1 then BK <= A &...
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### $\forall\delta, \sigma \in F$ where $\delta \land \sigma$ are contrad., $\exists\theta$ so that $\delta\land\neg\theta$ & $\sigma\land\theta$ contrad.

I'm in my first logic class ever and I'm trying to wrap my head around this obscure question... Show that for all pairs $\delta, \sigma \in F$, where $\delta, \sigma$ contradict themselves there ...
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### Find the minimum value of the expression given that a,b,c are positive and real.

$${(a+3c)\over(a+2b+c)} + {4b\over(a+b+2c)} - {8c\over(a+b+3c)}$$ Here was my attempt: If c tends to infinity and a and b are small we get 1/3. Now , i took $b=0$ and we get $(a-c)^2\over(a+c)(a+3c)$ ...
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### $Tor(A,B) = 0$ if $A,B$ is torsion free

I don't understand the proof given in Hatcher p.265 of $Tor(A,B) = 0$ if $A,B$ is torsion free. The proof is the following : The line I don't get is "This means [...] can be reduced to $0$ by a ...
1answer
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### What is $DE.DA$ here?

In $\triangle ABC$, let $D$ be the midpoint of $BC$ and $E$ be a point on $AD$ so that $\angle BEC +\angle BAC = 180^\circ$. If $BC = x$, then find $DE.DA$. How to approach this? Please do not work ...
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Let $(X,\mathcal{A},m)$ be some probability space where $m=\frac{1}{p}\sum_{j=0}^{p-1}\delta_{f^jx}$ for some fixed $x\in X$ that is $p$-periodic with respect to the measure-preserving transformation $... 1answer 27 views ### Solving the system$ab=9-12i$,$ac=-16-12i$,$db=36$,$dc=-48i$for complex$a$,$b$,$c$,$d$I need to find four Unknown Variables,$a$,$b$,$c$,$d, and I have four equations: \begin{align} ab &= \phantom{-1}9-12i \tag1\\ ac &=-16-12i \tag2\\ db &=\phantom{-}36 \tag3\\ dc &... 0answers 19 views ### Does this group G have a subgroup isomorphic to G/Z(G)? Assume that 1° G is a group of order 180; 2° Z(G) has order 3; 3° G/Z(G) is isomorphic to A_{5} (fifth alternating group); 4° every nontrivial characteristic subgroup of G has order ... 1answer 19 views ### Prove by induction for n\geq1, n \in \mathbb{N} , 2^{2n+1}\equiv 9n^2-3n + 2(\mod54). Question taken from the book by Andre Weil, titled Introductory number theory, chapter 5, question #V.3. Prove by induction that, if n is a positive integer, then 2^{2n+1}\equiv 9n^2-3n + 2(\mod54)... 0answers 12 views ### What is the first and second order derivative of the following function? What is the first and second order derivative of the following function?\sum_{l=1}^{k}(\sum_{j\in B_l} Y_j^{\frac{1}{\Theta_l}})^{\Theta_l}$$0answers 10 views ### Finding analytic continuatiuon of a branch I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases. The ... 0answers 9 views ### Probable existence of an almost integer contained in a limit I found this almost integer in studying the limit :$$\lim_{x\to \infty}\Gamma\left(\sin^2\left(\frac{1}{x}\right)\right)\Gamma\left(\sin\left(\frac{1}{x}\right)\right)-x^3=-\inftyWell my goal was ... 0answers 6 views ### How to construct a bump function from a given diffeomorphism I encountered a problem from, An introduction to chaotic dynamical systems 2nd edition by Robert. L. Devaney(pdf version available online), which reads: Using a bump function, show that the ... 0answers 11 views ### Positive function not vanishing in a neighbourhood If a positive function in C[-1,1] does not vanish in any neighbourhood of -1, then it has to be strictly positive in some neighbourhood of 1. It seems obvious to me when I try draw pictures, ... 0answers 4 views ### Sign Error: solved differential equation y'' = -b\,y'-g I got solved a differential equation here, but it still seems to differ from the textbook solution by a wrong sign: again it's about y'' = -b\, y'-g. The solution process goes like: \begin{align*} &... 1answer 18 views ### Showing that F(x) = x + f(x) defines a homeomorphism when f : E \to E, and where E is a Banach space. Let E be a Banach space and f : E \to E a contraction. Show that the equation F(x)=x+f(x) defines a homeomorphism F:E \to E that is Bilipschitz. Since f is a contraction the following to ... 0answers 13 views ### \lim_{n \to \infty}\sup_{k \geqslant n} (\frac{1+a_{k+1}}{a_k})^k \geqslant e [duplicate] Prove:\lim_{n \to \infty}\sup_{k \geqslant n} (\frac{1+a_{k+1}}{a_k})^k \geqslant e$for any sequence$\{a_n\}$where$a_n > 0$. I made a likely proof myself, but it's so complicated that ... 0answers 9 views ### Survival probability of a random walk I was looking for the probability that a discrete random walk stays below a certain level for$n$steps.$x_0$denotes the initial position and the position at the$i$-th step is$x_i=x_{i-1} + \eta_i$... 0answers 6 views ### Supporting hyperplane for convex set in Hilbert space So, I have a closed and convex set$C$in a Hilbert space$H$. Say, we have a point$x_0\in\partial_{\rm rel}C$(relative boundary of$C$). Does there exists a vector$y\in H\backslash\{0\}$such ... 0answers 30 views ### Mathematical research institutes similar to Banff and Oberwolfach What other institutes such as these two exist for a visit by a scientist for an undisturbed period of short research? Ideally with a good landscape. Dagstuhl is another one I found. 1answer 32 views ### Tangents to circles: What do I do now? everyone! I am confused about what to do here and how to do it. May I have some help? So, I know that$\overline{FI}$and$\overline{IE}$are both radii of the small circle and$\overline{JK}$and$\...

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