All Questions

0
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0answers
3 views

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 ...
0
votes
2answers
9 views

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, ...
0
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0answers
8 views

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 ...
0
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0answers
12 views

$f(x)/x \to l$ 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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0answers
13 views

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 ...
-4
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0answers
21 views
0
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0answers
7 views

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 ...
0
votes
1answer
17 views

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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0answers
10 views

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 ...
0
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0answers
11 views

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. ...
-2
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0answers
17 views

Compactness of $S\subseteq \mathbb{C}^2$

Let $M$ is a positive operator on a complex Hilbert space $F$. My goal is to show that the following set $$S_M=\{x\in F\,;\;\langle Mx,x \rangle=1\},$$ is not always compact. If $M=\begin{pmatrix}...
-1
votes
1answer
20 views

Arithmetic Progression (question find the sum of n terms)

Find the sum of the n terms: $$\left( 4-\frac 1n\right)+\left( 4-\frac 2n\right)+\cdots +\left( 4-\frac nn\right)$$ Note: the answer is meant to be $$\frac {7n-1}2$$
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0answers
4 views

Eigenvectors of half-derivative operator

The eigenvectors of the derivative operator $D$ are $\exp(kx)$, as talked about in this post and computed with Wolfram Alpha here. The eigenvectors of $D^2 = D \circ D$ with eigenvalue -1 are linear ...
0
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0answers
16 views

How to prove that $(A\cdot (B\times C))D=(A\cdot D)(B\times C)+(B\cdot D)(C\times A)+(C\cdot D)(A\times B)$

I got: $A_i\epsilon_{i,j,k}B_jC_k=(A\cdot D)(B\times C)+(B\cdot D)(C\times A)+(C\cdot D)(A\times B)$ $(A\cdot D)(B\times C)+(B\cdot D)(C\times A)+(C\cdot D)(A\times B)=A_iD_i\epsilon_{i,j,k}B_jC_K+...
0
votes
1answer
11 views

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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0answers
11 views

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?
1
vote
2answers
12 views

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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0answers
11 views

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 ...
0
votes
1answer
15 views

simple question on gamma functions monotonicity

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 $(...
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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 ...
0
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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 $\...
-4
votes
0answers
25 views

Some limits to solve [on hold]

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 ...
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0answers
8 views

Symmetric step function

Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $J=3$ $\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$ $\mu_j\in \mathbb{R}$ $...
1
vote
4answers
35 views

Let $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$.

Prove or disprove: let $G$ be a group and $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$. I saw the following example which tries to disprove the theorem: $G=\mathbb{Z}_{10}$ and $a=2,b=8$....
0
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0answers
15 views

Compare two normal probabilities

Assume $X_1,X_2,X_3$ are IID $N(0,1)$ random variables. I want to show that $$P(X_1>0,X_1+X_2+X_3<0)>P(X_1+X_2>0,X_1+X_2+X_3<0)$$ I know I can show this by calcultating these two ...
1
vote
1answer
19 views

What matrix has only negative or zero real part for all the eigenvalues?

Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
0
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0answers
11 views

Finite double sum $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$; any advanced summation technique?

Let $M,N,c$ be positive integer. It was astonishing when trying to solve $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$ to obtain this rather complex looking result \begin{align*} ...
0
votes
1answer
18 views

How to prove that $(A\cdot (B\times C))D=(A\cdot D)(B\times C)+(B\cdot D)(C\times A)+(C\cdot D)(A\times B)$?

I got: $A_i \epsilon_{ijk}B_j C_k$ I'm not know how to prove this identity.
1
vote
3answers
25 views

Is it possible to know the result of a match between three teams with only knowing their goals for and goals against?

I've found this problem in my book and left clueless on how to make the right interpretation to come up with a solution. The situation involves a hockey match. Typically when you have a goals for and ...
0
votes
1answer
12 views

Infinity matrix norm is maximum row sum norm

I want to prove that the infinity matrix norm is maximum row sum norm. I've shown that for $\|x\|_{\infty}=1$ $$||Ax||_{\infty} = \max_{i}\left|\sum^n_{j=1}a_{ij}x_j \right| \leq \max_{i}\sum^{n}_{j=...
1
vote
0answers
14 views

In a commutative, Noetherian ring, $d(A/J) = d(A/{J^{m}})$

Let $R$ be a commutative, Noetherian ring. Let $d$ be a dimension function: for each finitely generated $R$-module $M$, we assign a natural number, or zero, such that for every $N \leq M$ a submodule ...
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0answers
13 views

Unnecessary premises in proposition about base change (Görtz-Wedhorn)

Here is proposition 4.20 from Görtz and Wedhorn's Algebraic Geometry I. It seems to me that the right square in (4.5.1) is completely unnecessary: We can always choose $S = X$ making the right square ...
0
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1answer
19 views

Prove that in any non-empty chain $L$ in $\left\langle A, \le \right\rangle $ exist the smallest element

In the set $A=\left\{ \left\langle k,l\right\rangle \in \mathbb Z \times \mathbb Z: k<l\right\} $ we introduce relation of a partial order $\le$ for which $k' \le k$ and $l \le l'$.Prove that in ...
2
votes
1answer
18 views

Proof explanation : If $K$ is a convex open set of $R^d$ and $v_0 \notin K$ , then $K$ and $v_0$ can be separated by a hyperplane .

Theorem : If $K$ is a convex open set of $R^d$ and $v_0 \notin K$ , then $K$ and $v_0$ can be separated by a hyperplane . In the sense that there exist a non-zero linear functional $l$ and a real ...
1
vote
1answer
38 views

Since $\lim_{n\to\infty}\pi(n) = \frac{n}{ln(n)}$, can't this be used to prove Legendre’s conjecture?

Legendre's conjecture states that for all $n$, there is a prime number between $n^2$ and $(n+1)^2$. It has been proven that $\lim_{n\to\infty}\pi(n) = \frac{n}{\ln(n)}$. It can be proven that $\...
0
votes
0answers
13 views

Discrete Event Arrivals

If a concert starts at $8$ AM, assume that all passengers arrive between $7$:$00$ and $8$AM. If the show is sold out with $1000$ seats and one had to guess the inter-arrival times of patrons, what ...
2
votes
1answer
25 views

Can $\min_{a,b} E[(V -(aU+b))^2]$ be found without differentiation?

Let $(U,V)$ be joint random variables (assume zero means for simplicity). It is well known that \begin{align} \min_{a,b} E[(V -(aU+b))^2] \end{align} is minimized by $a= \frac{E[VU]}{E[U^2]}$ and $b=...
3
votes
1answer
13 views

Showing weakly continuous operators are continuous without using weak topology

Let $X$ and $Y$ be Banach spaces, and let $T:X\rightarrow Y$ be a linear map such that $f\circ T$ is continuous for all $f\in Y'$. Show that $T$ is continuous. Now I think this problem is trivial ...
0
votes
0answers
18 views

Relation between characters corresponding to hook partitions

Let $(n-k,1^k)$ be a hook partition. I want to know if there is a relation involving any two, or maybe all three, of the quantities: $$\chi_{(n-k,1^k)},\quad \chi_{(n-k-1,1^k)},\quad \chi_{(n-k,1^{k-...
2
votes
1answer
24 views

Solving PDE using characteristic method

I am trying to solve the partial differential equation $x\ u_ x - u\ u_y = y$ with the initial condition $u(1,y) = y$ , using the mathod of characteristics. My problem is with y and z , I mean $$\...
7
votes
1answer
62 views

Why does $\int_a^b f(x)h'(x) \, \mathrm{d}x=0$ imply that $f$ is constant?

As an assignment, I have to prove the following: If $f(x)$ is a piecewise continuous function and $\int_a^b f(x)h'(x) \, \mathrm{d}x=0$ for all piecewise continuously differentiable $h(x)$ that ...
1
vote
1answer
12 views

Understanding the factorization of the restriction of a function

I was reading Steve Awodey's book Category Theory and one concept I really do not understand. On the page $94,$ we have In $\textbf{Sets},$ take a function $f: A \to B$ and a subset $V \subset B.$ ...
0
votes
3answers
28 views

What is the area of $ABCD$ parallelogram where $E$ is mid-point of BC and the area of $BEC$ is 126?

$ABCD$ is a parallelogram. Point $E$ divides $BC$ into two equal lengths. If the area of $BEF$ is 126, what is the area of $ABCD$? Source: Bangladesh Math Olympiad 2017 Junior Category. I can not ...
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0answers
15 views

Why is the limit of $\int {\frac{1}{z-z_0-x} - \frac{1}{z-z_0+x}}dz$ on a vertical segment as $x$ approaches $0$ not $0$?

A solution to the problem below is given by parametrization, converting the integral from $dz$ to $dt$. The result would be $ -2\pi i$, which would be correct. My problem is why is it not $0$, which ...
3
votes
0answers
19 views

Machine learning book with robust linear algebra approach

I am looking for machine learning book - neural network, deep learning etc etc - that use linear algebra in a robust manner. I found satisfactory the old book of Simon Haykin : Neural Networks : A ...
0
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0answers
22 views

Odds of winning a video game combat

We have a combat between 2 characters in a video game. Character 1 has starting hit points = 'c1_hit_points' Character 2 has starting hit points = 'c2_hit_points' Every turn character 1 deals a ...
1
vote
0answers
7 views

Koopman-von Neumann Decomposition Properties

Sorry if the title of this question is vague--I'm open to suggestions. For this question, we're working in a probability space $(X, \mathcal{M}, \mu)$. In a proof of "ergodic Roth's theorem" given ...
1
vote
2answers
25 views

“Cycles Premiers” In EGA

I am reading through EGA IV, 3.1, but I can't figure out what "cycles premiers" translates to. I don't know any french, really. I think it means "prime cycles", but an English search for that doesn't ...
0
votes
0answers
9 views

Programming Equivalent PBL (Project Based Learning) in Math

I have been doing programming for about 4 years now and I have learned alot by working on projects such as: 2D Games, Websites, Web Templates etc... and I worked on these projects with my own will, ...
0
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0answers
44 views

Show that $\mathbb E[X_{n}]\xrightarrow{n \to \infty} \infty$ while $X_{n} \xrightarrow{n \to \infty} 0$ a.s.

Say I have a biased coin that shows heads with probability $p \in ]1/3,1/2[$ and I initially have capital of $100 $EUR. Every time heads is shown, my capital is doubled, in the other case I pay half ...

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