# All Questions

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### Is there a theorem that can help us count easily the number of subgroups of any given finite group?

Is there a theorem that can help us count easily the number of subgroups of any given finite group?
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Consider real diagonal (known) matrices $A$, $B$, and $C$, and a comforming matrix $\Pi$. $C + \Pi B + \Pi A \Pi = 0$ I have been trying to solve this system using elementary algebra. I have two ...
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### Examples of measurable and non measurable functions

I'm a new in measure theory and I want to understand measurable functions. As I expect measurable function is the function that maps one set to another where preimage of measurable subset is ...
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### how to show that g(t) is continuous?

how to show that g(t) is continuous? $g(t) = piecewise(t \neq 0, 0, t=0, 1)$ $\lim_{t \rightarrow 0} g(t) =1$? So it should be continuous at $t=1$? Well it seems this question doesn't meet you ...
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### characterization of compactness

When $X$ is an arbitrary topological space, I need to know which of the followings are true or false $1$. If $X$ is compact, then every sequence in $X$ has a convergent subsequence. $2$. If every ...
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### Showing that the equivalence class of partial orders order isomorphic to $(X, \leq)$ is not a set

I'm trying to do the following (original image): EXERCISE 25(R):(a) Argue on the grounds of the architect's view of set theory (which was outlined in the Introduction) that $\mathbf{PO}$ cannot be ...
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### A Borel-Cantelli lemma related question

I am learning about the Borel-Cantelli lemmas, which I think allow us to conclude that things converge "almost surely" to other things. I am having trouble with a homework question: Show: If ...
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### Weak derivative of $\operatorname{sgn}(x_1)$

Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak ...
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### Spectral radius and positive definite of matrices

Denote $\rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
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### Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
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### Why does this cyclic group have 8 elements?

The cyclic group of $\mathbb{C}- \{ 0\}$ of complex numbers under multiplication generated by $(1+i)/\sqrt{2}$ I just wrote that this is $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$ making a polar angle ...
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### Sur- in- bijections and cardinality.

I think about surjection, injection and bijections from $A$ to $B$ as $\ge$, $\le$, and $=$ respectively in terms of cardinality. Is this correct? And extrapolating from that, are these theorems ...
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### Finding a basis containing v

Let $w = \{ (a,b,c,d)^T \mid 2a+3b+c+2d=0 \}$ in the vector space $\mathbb{Z}_5^4$. $(2,2,1,2)$ is in $w$. ($w(2,2,1,2) = 15$, $15\,\mbox{mod}\,5=0$) How do I find a basis for $w$ that contains ...
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### Having difficulty understanding cosets. [duplicate]

Possible Duplicate: What is a quotient ring and cosets? I am taking an algebra class and having trouble understanding the elements (cosets) and ideals of quotient rings. Could someone ...
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### N piles of hidden cards of known marginal probability distribution, then a card is revealed in one of the piles.

I am currently trying to use probability theory to help solve a programming problem involving Monte Carlo Tree Search with Information Sets and have hit a roadblock. The problem can be described as ...
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### connectedness of two particular set of matrices

I need to know whether the $1)$ The set of all symmetric positive definite matrices are connected or not? Well I guess, This set is convex set, Let $M$ be a symmetric positive definite so ...
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### Solve equation for alpha

$E_1 = Ch_1^\alpha$ How can I solve for the exponent alpha? Is it just $\frac{\frac{E_1}{C}}{ln(\alpha)}$?
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### Maximising the correlation coefficient

How do I show that the correlation coefficient $\rho(Y,f(X))$, where $f$ is any measurable function, is maximized in absolute value when $f(X)$ is linear in $E[Y|X]$. I know that for 2 r.v.s X and Y ...
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### Min,max distances between point and spherical surface

I have a spherical surface defined by four points on an ellipsoid centered at (0,0,0). That is, the four points define a bounding box projected onto the ellipsoid. I have another point, P at some ...
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### The meaning of the Jacobi Symbol and its efficient evaluation

Are the following jacobi symbol evaluations correct?: $(\frac{35}{53}) = -1$ $(\frac{68}{233}) = -1$ $(\frac{126}{509}) = 1$ $(\frac{672}{1297}) = 1$ $(\frac{1235}{3499}) = -1$ Also what is the ...
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### Showing a metric space is bounded.

This is from a review packet: Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$ i) Show that $(\mathbb{R},d)$ is a bounded metric space. ii) Show that $A=[a,\infty)$ ...
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### extension of analytic to meromorphic function

could any one help me to show that any $f$ analytic in the annulus $1\le |z|\le 2$ and $|f|$ is a constant on $|z|=1$ and on $|z|=2$ (not necessarily the same constant) admits a meromorphic extension ...
If we have two identical sets $A_1 = A_2$, and we were asked to get the maximum sum of multiplying one distinct element from $A_1$ by another distinct element of $A_2$, for all elements in $A_1$. ...
The 3-dimensional representation of the group S3 can be constructed by introducing a vector $(a,b,c)$ and permute its component by matrix multiplication. For example, the representation for the ...