All Questions

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What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
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Prove that the arithmetic-geometric mean inequality holds for any list of numbers whose length is a power of 2

I am self-studying and currently reading How to Prove it by Velleman. I tried to prove the above by induction (I proved that this holds true for $n=2$), but I think my proof is wrong. I only started ...
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Maple: Symmetries of strange modified heat-equation

I thought about the following PDE: $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(x=0,t).$$ How can i determine the symmetries of this PDE with Maple? And how ...
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Intersection of minimal affine halfspaces

We have a subset S which is the sum of two sets, A and B. (S= A+B) A is the set of a vector: $x\begin {pmatrix} 0 & 0 & 1\end{pmatrix} \in \mathbb{R}^3$ where $x \geq 0$. B is the set of a ...
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Why does Maple include x in the solution of this definite integral?

I have the following function defined in Maple: $$f(x) := (2 - a + ax^2) \sqrt{1 + 4a^2x^2}$$ And I want to calculate the definite integral of this from -1 to 1: $$\int_{-1}^{1}{f(x)dx}$$ I do ...
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Throwing dice and finding limits

We throw an honest dice $n$ times. Let $S_n$ be the number of throws with even number of dots on the dice. 1.) Calculate the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n)$$ 2.) Express the value ...
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“A reflection across one line in the plane is, geometrically, just like a reflection across any other line.”How?

How can this statement be represented geometrically?-"A reflection across one line in the plane is, geometrically, just like a reflection across any other line." (i tried it by drawing some ...
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Principale value, how can we consider it?

How can we consider principal value where as, it's something that "doesn't exist". For example, $$\int_{-\infty }^\infty \frac{1}{x}\mathrm d x$$ doesn't exist, but the principal value is nulle. What ...
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Why do we need axiom of choice for this?

The answerers on this question say that we need AoC (or some variant thereof) to prove that every infinite set has a countably infinite subset. In my view, choice is not needed but the answerers are ...
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$p \vee q \leq p+q$ for $p,q$ projection

I am wondering if $p \vee q \leq p+q$ for $p,q$ projection acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ordering ...
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A proposition about a stochastically continuous process with independent increments.

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with filtration $F=\left\{ \mathcal{F_t} \right\}_{t\geqslant0}$ , and $X=\left\{ X_t \right\}_{t\geqslant0}$ is an ...
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Proving that the following is a convex subset

It is given that $I(S)=\{z|\exists x,y \in \mathbb{R}$ such that $(x,y,z)^t \in S$}$\subseteq \mathbb{R}$. How do I show that I(S) is a convex subset? We know that $S \subseteq \mathbb{R}^3$ is ...
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image of composition of upper triangular integral matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $im(AB)$ in terms of $im(A)$, $im(B)$, unions, intersections, determinants, etc?
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Combinatorial commutative algebra

Let G is simple graph and Δ(G) is clique complex, IΔ(G) has a 2-linear resolution if and only if, for any subset W ⊂ [n], one has H˜i(Δ(G)W ; K) = 0 unless i=0
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How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
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Calculating the normalization constant in spherical harmonics?

Anyone know a presentation of the calculation of the normalization constant in spherical harmonics. Specifically, how has $$\sqrt{\frac{2l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}}$$ been found in Y_l^m(\...
Consider we need to form differential equation of the following: $Ax^2 + By^2 = 1$ where A & B are constants. Now one approach I know of is two differentiate this equation 2 times (since 2 ...