## Real Analysis, Folland problem 2.14. Integration of Nonnegative functions November 28

This comes from Real Analysis, by Folland. We will use proposition 2.15 in part of the proof which is:Proposition 2.15 - If $\{f_n\}$ is a finite or infinite sequence in $L^{+}$ and $f = \sum_{n}f_n$, then $\int f = \sum_{n}\int f_n$Problem 2.14 - If

## Real Analysis, Folland Corollary 2.5 November 28

This Corollary follows from proposition 2.4:Let $(X,M)$ and $(Y_\alpha,N_\alpha)$($\alpha\in A$) be measurable spaces, $Y = \prod_{\alpha\in A}Y_\alpha, N = \bigotimes_{\alpha\in A}N_\alpha$, and $\pi_{\alpha}: Y \rightarrow Y_\alpha$ the coordinate

## Partial derivatives calculus November 28

If x^x.y^y.z^z=c then prove that del2z/delx.dely=(-xlogex)^-1. How to do this? I was first taking logarithm and then differentiating but did not work.

## Filling an Obtuse Triangle with Equilateral Triangles or a Pre-Defined Shape November 28

I am creating an obtuse triangle of undetermined proportions and I need to find how to fill it with equilateral triangles or a pre-defined shape that can fill it.Any math I've done has been, ...

## Help solving this GCSE question on angles/parallel lines/isoscelesh November 28

Problem:$ABC$ is an isosceles triangle with angle $\angle ABC=52^\circ$.$XY$ is parallel to $BC$.Work out the size of angle $\angle BAC$.Hi I need help solving this GCSE question under the t ...

## Why is the matrix multiplication defined as it is November 28

Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = \sum\limits_{k=1}^mA_i,_kB_i,_j$. I was wondering for what good reason did mathematicians define it like

## How to find points where a complex function has derivation November 28

I'm trying to find points where the function $F(z)= x ^ 3+ i y ^ 3$ has derivation.I applied the CR conditions and I get the result that in two lines $y = x , y = - x$ the CR equations are correct,can I guess now that in these points the function ma

I have to prove that $$\sum_{i=0}^n {{n \choose i} \times 2^I} = 3^n$$ Such that ${n \choose i} = \frac{n!}{i!(n-i)!}$ and $n$ is some arbitrary int I proved we can expand 2^I in a way such that $$2^I = {i \choose 0} +{i \choose 1} + ... + {i \cho ## Ergodicity under measure-theoretic isomorphism November 28 Suppose we have two measurable dynamical systems (X_1,\mu_1,T_1) and (X_2,\mu_2,T_2), with \mu_i(X_i)=1,\ i=1,2. Suppose they are measure-theoretically isomorphic (so there is an almost-sure one-to-one map \theta:X_1\to X_2 such that \theta ## Prove the inequality (law of total variance) November 28 Anyone can guide me on this question?E[Var(Y|X)]<=Var(Y)Thank you, appreciate your help ! ## A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest November 28 I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery:Let  H  be a group and  K = <x>  be a cyclic group (finite or infinite) and we define two group homomorphisms  \phi_i ## Parallelogram midpoint proof November 28 If ABCD is a parallelogram and M is the midpoint of AC, prove that M is also the midpoint of BD.I think vectors must be used to solve this problem You Might Also Like • I know that the conjugate of a binomial is the negation of the second part. So the conjugate of (a + b) woul ... • Prove that f(x)=x can have at most one solution if f'(x)\ne1What I did : Use g(x) = f(x)-x, then g'(x ... • Let B be an Infinite dimensional Banach Space and T:B\to B be an continuous operator such that T(B)=B ... • The question I am trying to show under what conditions$$\vec{A}\times(\vec{B}\times\vec{C}) = (\vec{A}\times ...
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