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What is the interpretation of the expectation notation in the GAN formulation?

I'm confused about the expectation notation in the context of GAN loss functions. The GAN loss for the discriminator is binary cross-entropy. ie: is this real or not. real = $D(x)$ (ie: give ...
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Is $0$ the only vector in the kernel of every bounded linear functional?

Let $X$ is a normed vector space, and let $x_0\in X$ have the property that for every bounded linear functional $f:X\rightarrow K$, $f(x_0)=0$. Then does $x_0=0$? I think the answer is clearly yes, ...
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Procedure for producing a total order on infinite set

Suppose we want to prove that every infinite set, $X$, can be totally ordered. A probably faulty procedure I thought of was the following: by the Axiom of Choice we know that there exists a surjection ...
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$f(x)/x \to l$ and $f''(x) = O(1/x)$

$f \in C^2(\mathbb{R}, \mathbb{R})$ such that $f(x)/x \to l \in \mathbb{R}$ as $x \to + \infty$, and such that $f''(x) = O(1/x)$ at $+\infty$. Find : $$\lim_{x \to +\infty} f'(x)$$ Some thoughts : ...
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Computing the product $(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n$

I want to compute the product $$(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n,$$ for a natural number $n$. For $n$ equal to 0 or 1, the computation is very simple obviously, but for such a low number as 2 ...
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conditional probability for a simple Bayes network

I have a very simple, and maybe trivial question about this Bayes net. Given a Bayes net, A -> B <- C and the probability distribution p(B|A,C). How do we compute P(B = b| A = a, C)? I am so ...
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Confused about the use of arctan and the presence of 180.

In my high school math class I'm currently working on solving trigonometric equations. Solve for all possible values of x: 5 = 30cos(2x + 1) - 5 My teacher (on some examples posted online) ...
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Light rays in a prism

While studying about path of light rays while passing through a prism, I noticed that: Although the prism is a 3d object, only a cross section of the prism is considered enough to talk about light ...
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M lies on AB Justify.

Draw a 🔺 ABC with Angle A =90’ construct BP on AC and BQ on AC extended with angles PBC=QBC. Draw angle bisector of angle APB &BQC extend both the angle bisectors to meet at a point M. ...
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Reflexive graph, meaning of the reflection

Here on the page 1 there is a definition of reflexive graph. I need an intuition how it works the morphism $e:X_0\to X_1.$ What is it and to what edge in $X_1$ it sends a vertex from $X_0$?
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How to deal with the case that $f(z_0)=0$ for some $z_0\in C_R$?

In the content labeled by red box, if $f(z_0)=0$ for some $z_0\in C_R$, where $C_R$ denotes the circle of radius $R$ centered at origin, how to deal with it?
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Suppose we have 2 sets of billiard/pool balls, what is the probability of drawing 4 unique balls?

The Cue ball is not included in either set, so we just have 1-15 in each set (30 total balls).
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Two definitions of a connection

For me a connection $\nabla$ on a vectorbundle $E$ over a smooth manifold $M$ is a $\mathbb{R}$-bilinear map $\Gamma(TM)\times\Gamma(E)\rightarrow\Gamma(E)$ which is tensorial in the first slot and ...
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Consider $x> 1$ and $i=1,2,3...$. I do not know much about gamma functions. Is $\Gamma(x) < \Gamma(x+i), \forall i$? I know there is some property that gamma function is always increasing in $(... 0answers 19 views Is it possible to cross a tangled rope? So, I thought of this question a while ago and haven't found a clear answer for it yet. Here's the question: Suppose you have a rope that is tied to two ends of a room. You can tangle the rope ... 0answers 7 views Number of possible extensions of field into algebraically closed field I have understood completely the reasoning above Proposition 2.7. But I cannot understand how to prove the proposition itself. Let me ask my questions more precisely? 1) In the reasoning above$\...
How to solve these limits ? $$\lim_{n\rightarrow ∞}\int_{0}^1 \frac{f(x)}{x+n} dx$$ $$\lim_{n\rightarrow ∞}\int_{0}^1 x^n f(x) dx$$ $$\lim_{n\rightarrow ∞}\int_{0}^1 \frac{f(x)}{1+nx} dx$$ ( f is ...