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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $(phi_alpha)_alpha>0$ be a familiy of mollyfiers, $phi_alpha:mathbbR^n rightarrow mathbbR$ defined as: beginalign phi_1(x)=left{beginarrayrcl c cdot exp(frac-11-vert xvert^2) &,& vert x vert < 1\ 0 &,& textotherwise endarrayright. endalign with $c>0$ such that $int_R^n phi_1(x) dx=1$ and $phi_alpha(x)=alpha^-n phi_1(x/alpha)$ Consider the function $w(x)=(y_1-y_2) int_0^1 phi_epsilon(x-y_1+t(y_1-y_2)) ~dt$ with $y_1,y_2 in mathbbR^n $ and calculate its divergence $div~ w(x)= sum_i=1^n frac partial w_i(x)partial x_i.$ The result should be $div ~w(x) = phi_epsilon(x-y_2)-phi_epsilon(x-y_1)$. How do I calculate the divergence in this case? So far I tried to use the fundamental theorem of calculus, however the result was not the same. calculus divergence share | cite | improve this question asked Jul 18 at 8:32 akwa 30 5 add a comment  |Â...

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite There are $n$ triangles of positive area that have one vertex at $(0,0)$ and their other two vertices with coordinates in $0,1,2,3,4$. Find the value of $n$. I know that other than $(0,0)$ the vertices are $$(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4),(2,0),(2,1),(2,2),(2,3),(2,4),(3,0),(3,1),(3,2),(3,3),(3,4),(4,0),(4,1),(4,2),(4,3),(4,4)$$ but how do I select the two points which satisfy the given condition. combinatorics share | cite | improve this question edited Jul 18 at 9:58 Parcly Taxel 33.6k 13 65 88 asked Jul 18 at 8:58 learner_avid 682 4 13 closed as off-topic by Alex Francisco, Isaac Browne, Strants, Xander Henderson, José Carlos Santos Jul 19 at 22:54 This question appears to be off-topic. The users who voted to close gave this specific reason: " This question is missing context or other details : Please improve the question by providing...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite This seems to be a simple problem, but I can not come up with a counterexample. I would be happier if it were true! Set up: Consider $A_i_iinmathbbN$ a sequence of measurable subsets of $[0,1]$. Fix $epsilonin (0,1/2)$ and assume that the Lebesgue measure of each set is bounded below by $|A_i|>1-epsilon$ for every $iinmathbbN$. Question: Can we find an infinite set of indexes $i_n_ninmathbbN$, such that the Lebesgue measure of the intersection of all these sets is bounded below by $$ left|bigcap_ninmathbbN A_i_nright|>1-2epsilon? $$ real-analysis analysis measure-theory share | cite | improve this question edited Jul 18 at 9:42 Asaf Karagila ♦ 292k 31 403 733 asked Jul 18 at 9:23 Sloth-Meister 3 4 I'm unclear on your quantifiers. Are you asking if we can find such a sequence for each $A_i$, or if there is an $A_i$ such that we can find such ...