Let $k$ be an algebraically closed field and $m\leq n$. Suppose $\pi:\mathbb{A}^n\to \mathbb{A}^m$ is map which sends $(a_1,\ldots,a_n)\to (a_1,\ldots,a_m)$. If $V$ is an affine algebraic set, then is it true that $\dim(V)\geq \dim(\overline{\pi(V)}) ## Subset of Measurable set is measurable (with conditions) February 9 1 I know that in general a subset of a measurable subset need not be measurable (the general example being Vitali sets). However I've been working on the following and hit a wall:Let U and V be Lebesgue measurable subsets of [0,1]. Assume A is a subset ## Expected value problem: flip$6$fair coins before we obtain$3$heads and$3$tails February 9 1 How many times on average (expected value) must we flip$6$fair coins before we obtain$3$heads and$3$tails? I know I need$∑ xp(x)$. I just don't know how to apply it. Call three heads and three tails when tossing six coins a success. Then the p ## Existence of a subset$S\subset\mathbb R$s.t.$\forall a February 9

I am trying to either find an example of such a set, or prove that no such set exists. I know of examples of dense sets with measure $1/2$ on specific intervals, such as $[0,1]$, but I haven't been able to find any set that satisfies this more genera

## Distribution of ceiling function and absolute value of random variable February 9    1

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is exponentially distributed, so it would be sufficient to

## Translate the following into predicate calculus. State assumed universe of discourse February 9

This is my first assignment on these, so I would greatly appreciate your help.Translate the following into predicate calculus. For each answer, also state the assumed universe of discourse. a) "Anyone who was an ancient Roman citizen and tried to kil

## Prove this simple graph is not planar. February 9

I need to show this graph is not planar. I've attempted to find $K_5$ and $K_{3,3}$ as a subgraphs but haven't been successful yet. It's possible but unlikely this graph is planar but I have ...

## Is there no difference in symbols between the floor and the ceiling of x February 9

Source: Discrete Mathematics with Applications, Susanna S. EppThe symbol of floor of x is [x] and so is the symbol [x] of ceiling of x. Is it correct that there's no difference in symbols be ...

## Evaluate $\int ( \frac{1}{\sin x}\sin x\sin x \log(\sin x) ) dx$ February 9    1

This one looks easy but I still could'nt figure it out.$$\int ( \frac{1}{\sin x} - \sin x - \sin x \log(\sin x) ) dx$$I tried substituting $\log(\sin x)=z$ but that's not working.Any suggestions?Hint: Integrate by parts with the logarithm term2

## Distribution and expecation value of ceiling function of poisson February 9

There is poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is lower than 1).How can i find distribution of $Z$.? And what is expection val

How to prove that $\dfrac{1}{(1-x)^3}$ is the generating function for the triangular numbers? The $n^{\text{th}}$ triangular number is defined as $T_n = \displaystyle{n+1 \choose 2}$. I used calculus ("cheating") and found out that $$\displaysty ## Eigenvector of a matrix of ones associated with \lambda =0 February 9 An n\times n matrix consistent of all ones, will have two eigenvalues: 0 and n. The eigenvector associated with n will be (1,1,...,1), but are there then infinite solutions for the eigenvector associated with 0? Because, in order for the ## group action same thing as homomorphism February 9 1 A linear group action of a group G on a vector space V is the same thing as a homomorphism from G to the general linear group GL(V).attempt: Suppose a linear group action of a group G on a vector space V is given. For each g \in G we obta ## Find the radius of convergence and interval of convergence of the series February 9 1 Find the radius of convergence and interval of convergence of the series:\sum_{n=1}^{\infty}n^n x^{n^4}I'm really lost as to how to approach this problem. The other power-series problems were all very straight-forward compared to this one. I attemp ## How to find integral of the form e^xf(x) February 9 I always face trouble with these type of integrals.I need to find$$\int{e^x \frac{x(\cos x -\sin x)-\sin x}{x^2}}dx$$My problem would be solved if can express f(x) like g(x)+g'(x) but identifying g(x) by trial and error method is sometimes ted ## Square Free congruence modulo n February 9 I am trying to show that if a^n\equiv a\pmod n for all integers a that n is square free. I have an idea to start with the contradiction that suppose n=p^2m for some prime p, then n does not divide a^{p^2m}-a for some integer a. Any hint ## how to write as geometric series \dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)} February 9 How would I write \dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)} as a sum of geometric series? ## Compute (df)_a in chart \varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U) February 9 1 Suppose that for a submanifold H of \mathbb{R}^3 we have two charts$$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)\varphi_2:U=\{(x,y,z)\in\mathbb{R}^3:y\neq0\}\rightarrow\varphi_2(U).$$s.t. \varphi_1(x,y,z)=(y,z) and ## Compute in the chosen charts of M and S^1 the expression of DF_{(5,0,-4)} February 9 1 Let me show my work before presenting the problem itself.Let M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}.We can easily see that M is a submanifold of \mathbb{R}^3 of dimension 1.We can also see that we can construct a global atlas for  ## Let p=(5,0,-4) and v \in T_{(5,0,-4)}M. Compute (F^{*}\omega)_p(v). February 9 Let me show my work before presenting the problem itself.Let M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}.We can easily see that M is a submanifold of \mathbb{R}^3 of dimension 1.We can also see that we can construct a global atlas for You Might Also Like • If \mathfrak{g} is a finite-dimensional Lie algebra, then it is very known that the Universal enveloping a ... • Let \Omega be an open set in \mathbb{R}^n and now consider the weak formulation of Poisson's equation$$ ...
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