All questions

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How to find a finite set of generators for $I \subset k[x_1, ..., x_n]$

Suppose that we have a set of polynomials $f_1, ..., f_m \in C = k[x_1, ..., x_n]$ where $k$ is an algebraically closed field, and let $V$ denote the set on which they all simultaneously vanish. Define $I(V)$ to be the ideal of polynomials in $C$ which vanish on all of $V$. In general, $I(V) = \text{rad}(f_1, ..., f_m)$ by Hilbert's nullstellens...
2022-07-25 17:47:06
0

Axiom of choice and compactness.

I was addressing an inquiry lately that managed compactness as a whole topological rooms, and also just how compactness falls short to be equal with consecutive compactness unlike in metric spaces. The only counter - instances that took place in my mind called for hefty use axiom of choice: well - getting and also Tychonoff is theory. Can a pe...
2022-07-25 17:47:06
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An injection $F$ from the set of real numbers to itself with $F(x)-F(y)\not\in\mathbb Q$ for $x\neq y.$

Please aid me locate a shot, $F$, from the set of actual numbers right into itself such that $F(x) - F(y)$ is an illogical number for any kind of 2 distinctive actual numbers $x$ and also $y$. Thanks.
2022-07-25 17:47:06
0

Applications of parity formula on connected planar graph

I've been offered the adhering to trouble as homework: A graph is drawn in the plane and has 78 faces, all of them triangles. Prove the outer face is not a 19-gon. We are additionally offered a tip to make use of the formula:|V|-|E|npls|F|= 2 (for planar, attached chart, where|F|= the variety of faces). What I've attempted until now: I unders...
2022-07-25 17:47:06
1

Surface integral (stokes?)

I intend to address the adhering to trouble, I intend to locate $$ \iint_S x \, \mathrm{d}S $$ where S is the component of the allegorical cyndrical tube that exists within the cyndrical tube $z = x^2/2$, and also in the first octant of the cyndrical tube $x^2 + y^2 = 1$ I was clearly thinking of switching over to round works with, yet I have ...
2022-07-25 17:47:02
0

Question On the Proof of The boundedness Theorem

let $f:[a,b]\rightarrow\mathbb{R}$, f continuous on $[a,b]$. I shall prove that $\exists A,B\in\mathbb{R}, \forall x\in[a,b], A\le f(x)\le B$. Proof: Let's define $g(x)=|f(x)|$, we need to prove now that $\exists A\in\mathbb{R}, \forall x\in[a,b], g(x)\le A$. let's suppose that this claim is false, therefore we get: $\forall A, \exists x\in[a,b]...
2022-07-25 17:47:02
0

Difference in limits because of greatest-integer function

A Problem: \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} causes the remedy: $1$ But the very same function confined in a best integer function causes a $0$ \begin{equation}\lim_{x\to 0} \left\lfloor{\frac{\sin x }{x}}\right\rfloor\end{equation} Why? My ideas: ¢ [The value of the first function often tends to 1 as a result of t...
2022-07-25 17:47:02
0

Simple proof of integration in polar coordinates?

In every instance I saw of integration in polar coordinates the Jacobian component is made use of, not that I have a trouble with the Jacobian, yet I asked yourself if there is a less complex means to show this which will certainly additionally offer me some even more instinct concerning the Jacobian. If I attempt to merely write the differenti...
2022-07-25 17:47:02
0

Compact complex surfaces with $h^{1,0} < h^{0,1}$

I am seeking an instance of a portable facility surface area with $h^{1,0} &lt; h^{0,1}$. The bound that $h^{1,0} \leq h^{0,1}$ is recognized. In the Kähler instance, $h^{p,q}=h^{q,p}$, so the instance can not be (as an example) a projective selection or an intricate torus. Does any person recognize of such an instance? Many thanks.
2022-07-25 17:47:02
2

Fréchet differentiability from Gâteaux differentiability

Let $X$ be a Banach space and $\Omega \subset X$ be open. The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle g,th \rangle]=0$$ How can I prove the following: If $f$ has a continuous Gâteaux derivative on $\Omega$, then $f \in C^1(\Omega,\mathbb R)$.
2022-07-25 17:46:59
0

If the closures of two subsets of a topological space are equal, are the closures of the images of the two subsets under a continuous map also equal?

Suppose $A$ and also $B$ are parts of a topological room $X$ such that $\newcommand{cl}{\operatorname{cl}}\cl(A) = \cl(B)$. Allow $f\colon X\to Y$ be a continual map of topological rooms. Does that mean that $\cl(f(A)) = \cl(f(B))$?
2022-07-25 17:46:59
0

Formal proof that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.

Cam any person give me the evidence of: that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is attached.
2022-07-25 17:46:59
0

Show that the set $\{ x \in [ a,b ] : f(x) = g(x)\}$ is closed in $\Bbb R$.

Let $f : [a,b] \to\Bbb R$ and also $g : [a,b] \to\Bbb R$ be 2 continual functions on $[a,b]$. Show that the set $\{ x \in [ a,b ] : f(x) = g(x)\}$ is enclosed $\Bbb R$.
2022-07-25 17:46:58
0

Proof that a continuous function $f : [a,b] \to {\mathbb Q}$ is a constant function.

Possible Duplicate: ¢ Let $f : [a,b] \to \mathbb Q$ be a continual function. Confirm that $f$ is a constant function.
2022-07-25 17:46:58
0

Prove $n+1$ items in $n$ buckets implies some bucket has $2$ items.

How would certainly you create and also officially confirm (from a marginal set of axioms) the adhering to declaration holds true? For all favorable integers $n$, if $n+1$ things are positioned right into $n$ pails, than among the pails have to have $2$ or even more things.
2022-07-25 17:46:55
0

cohomology group of $SO(n)$

I am computing the Alexander-Spanier cohomology $H^i(SO(n),\mathbb{Z})$. I embedded $SO(n)$ into $R^{n^2}$. Since the embedding $i$ is a monomorphism, the induced group homomorphism $i^*$ is an epimorphism. Since $R^{n^2}$ is homotopic to a point, $H^i(R^{n^2})=0 , \forall i \in \mathbb{Z}^+$. That gives us $H^i(SO(n))=0, \forall i \in \mathbb{...
2022-07-25 17:46:55
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Solving linear inequalities over rings

The concrete trouble: for any kind of offered $N\ge 1$ I have a system of $2^N-1$ straight inequalities over $\mathbb{Z}_6^N$ which resembles this: for every single nonempty $S\subseteq[N]$ there is some $b_S\in\mathbb{Z}_6$ and also the inequality $\sum_{i\in S}x_i\ne b_S$. I intend to locate a remedy to all the inequalities simultaneously, cer...
2022-07-25 17:46:55
0

Finding Lie Subgroups

I've been asked to locate all correct lie subgroups of $SU(2)$. I appear to remember assuming that $U(1)$ is the only nontrivial linked lie subgroup, yet I can not fairly bear in mind just how I thought of this (do not you despise that sensation when you did something, you go on to another thing to maintain energy, and afterwards neglect what y...
2022-07-25 17:46:55
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Finite Field Extensions and the Sum of the Elements in Proper Subextensions

Let $F$ be a limited area, and also allow $u,v$ be algebraic over $F$. Take into consideration the areas $F(u,v),F(u)$ and also $F(v)$. Must it hold true that $F(u,v) = F(u)$ or $F(u,v) = F(u+v)$?
2022-07-25 17:46:55
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Finding all possible paths from one corner to the other on a grid, without backtracking

Me once more "new to mathematics guy". Please inform me if the material of my inquiries are not an excellent suitable for the website. I'm currently onto and also it feels like there is some mathematical course searching for strategy I need to make use of. Checking out I've located chart and also tree traversal, djikstras fastest co...
2022-07-25 17:46:51