# All Questions

3 views

### confused about meaning of a expectation of a function

https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff#Derivation well,in the "Derivation" part of the wiki link. i don't figure out why $E(f)=f$, does it imply that the function $f$ is ...
2 views

### Universal Property of free objects

I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I ...
2 views

### cofinite topology

If we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of ...
6 views

### Number of root of a non-negative polynomial

Let $g\in\mathbb{R}[X,Y]$ of degree $d$ and assume that $g=0$ is a smooth curve. Further, assume that $f\in\mathbb{R}[X,Y]$ is non-negative on $g=0$. Assuming that the number of roots of $f$ on ...
5 views

### Algebraic topology theorem 57.1

i read right now the theorem 57.1 on munkres book: if h:s^1 to s^1 is continuous and antipode-preserving, then h is not nulhomotopic. at the end of the proof there is written: h* is injective . so ...
3 views

### Covariance matrix for $p_1\delta_{e_1}+…+p_d\delta_{e_d}$

I would like to know the covariance matrix for $p_1\delta_{e_1}+...+p_d\delta_{e_d}$, with $\delta$ as the dirac measure and $e_i$ as the $i$-th unit vector for $i=1,...,d$.
8 views

### Finding limit of series of functions

Let $s>1$. I want to find $$\lim_{x\to 0}\sum_{n\in\mathbb{Z}} |\sin(2\pi nx)-2\sin(4\pi n x)|^s.$$ I think it equals 0 but I don't know how to prove it. I thought applying dominant convergence ...
11 views

### Showing a finite field is a local ring

I am asked to find an integer $n$ such that $\mathbb{Z}_n$ is a local ring and then to prove my answer. Here is my attempt: We know that $Z_p$ is a finite field for prime $p$. Since $\mathbb{Z}_p$ ...
8 views

### the number of epimorphisms

Find the number of epimorphisms of free groups$$F_2$$ of rank 2 on groups $$\mathbb Z_{p^2} \oplus\mathbb Z_{p^2}$$ and the number of normal subgroups $$F_2$$, which quotient groups are ...
13 views

### pdf is defined as $f_X(x)=P(X=x)$ but isn't $P(X=x)=0$

When we define a probability distribution function, we say: $f_X(x)=P(X=x)$ and thats equal to some function such as a gaussian But isn't $P(X=x)=0$ for a continuous random variable $X$. Is it ...
8 views

### Jumps in a flow to the outermost area enclosure of a surface

While studying some geometrical properties of some flows of surfaces, I encountered this problem: I consider some surfaces $E_t$ flowing to infinity. I also define $E'_t$ to be the outermost minimal ...
13 views

### Computation of an integral depending on the Legendre polynomials

Let $P_l$ be a Legendre polynomial ($l$ is an integer). I want to know why the quantity $$v_l(k):=(-i)^l\int_{-1}^{+1}\mathrm{e}^{ikx}\,P_l(x)\;\mathrm{d}x$$ is real?
9 views

### Elementary 3D geometry

This is surely trivial, but my old brain can't remember how to do it. Assume a plane. A second plane intersects, forming line $AB$. The angle of intersection is $\theta$. A third plane intersects, ...
20 views

### Show that $X$ is Hausdorff.

Suppose that $X$ is a space with the property that for any point $p \in X$ there is a map $f: X \rightarrow \mathbb{R}$ such that $f^{-1}(1) = \{p\}$. Show that $X$ is Hausdorff. ...
11 views

### Direct sum of nonzero groups

Please help with the decision Find the number of expansions in a direct sum of nonzero groups $$\mathbb Z_{p^2} \oplus\mathbb Z_{p^2}$$ p - simple.
48 views

### Find $\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}.$

Find $$\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}.$$ Source. I don't know how to start. Hints are also appreciated.
15 views

### Prove that $\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2 .$

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2$$ where ...
20 views

### Convergence of series

Is this series convergent? $$S_{N}=\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}}$$ where $c_{n}^{N}$ is coefficient of $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
23 views

### Calculating the determinant as a product without making any calculations

My problem is on the specific determinant. All i can do is prove the factor (n+1) and i think that we have to work only on one column and the do the exact same thing to the others.
28 views

### Cannot be simultaneously rational

Let $a,b \in \mathbb{N}^{*}$. Prove that $\sqrt{13a^2+b^2}$ and $\sqrt{a^2+13b^2}$ cannot be simultaneously rational. If $(a,b)=(k,k\cdot6)$, then $\sqrt{13a^2+b^2}$ is rational, but I do not know if ...
14 views

### set theoretic equivalence in quotient ring

If I am given a ring $R$ and a 2-sided ideal $K\subseteq R$, I know that I have a well-defined quotient ring $R/K$. My question is the following: We know that if we have $a,b\in R$, then ...
27 views

22 views

### Non measurable set

In first answer of this question The construction of a Vitali set, how did we came to conclusion that $A$ is not measurable ? How that series doesn't converge $\Rightarrow A$ is not measurable ?
8 views

### Two problems from Avner Friedman's PDE book.

The problems are as follow: Prove that if $Lu=0$ for any $u\in C^m(\Omega)$, then $L\equiv 0$ - that is all the coefficients of $L$ vanish identically. Prove that the assertion of the previous ...
15 views

If a is a quadratic residue, and ab is a quadratic residue, how can I show that b is also a quadratic residue? Would appreciate a hint. So far I thought about the problem a little and I have: $a^2$ ...
10 views

16 views

### The probability that the output of the experiment is Y is ___?

Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop. Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, ...
A function $f : N^+ → N^+$, defined on the set of positive integers $N^+$, satisfies the following properties: $f(n) = f(n/2)$ if $n$ is even $f(n) = f(n+5)$ if $n$ is odd Let $R = \{i|∃ j : f(j) = ... 0answers 31 views ### requirement of multiple choice question for linear algebra I want to prepare myself for a multiple choice examination. Is it ok if someone introduce a good and complete multiple choice question books for Linear Algebra and Calculus to me? Thanks, 4answers 53 views ### Finding coefficient of polynomial? The coefficient of$x^{12}$in$(x^3 + x^4 + x^5 + x^6 + …)^3$is_______? Somewhere it explain as: The expression can be re-written as:$(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$... 1answer 12 views ### MATLAB rand(m,n) I want reconfirm what I believe is the answer. So this is the question: "Make a vector in MATLAB with n = 10 random elements, and then ﬁnd the percentage of those elements that are less than 1/2." ... 1answer 16 views ### Understanding the definition of the direct sum of subspaces of a vector space I have a question regarding the definition of direct sum of a vector space in relation to subspaces. Definition: A vector space$V$is called the direct sum of$W_1$and$W_2$if$W_1$and$W_2$... 1answer 24 views ### I don't understand how Kirchhoff's Theorem can be true Kirchhoff's Matrix-Tree theorem states that the number of spanning trees of a graph G is equal to any cofactor of its Laplacian matrix. Wouldn't this imply that all cofactors of a Laplacian matrix ... 2answers 35 views ### Limit - Applicability of L'Hopital's Rule I am required to find$\lim\limits_{x \to 0^+} \frac{xe^x}{e^x-1}$. My attempt:$\lim\limits_{x \to 0^+} \frac{xe^x}{e^x-1}$=$\lim\limits_{x \to 0^+} e^x\cdot\lim\limits_{x \to 0^+} ...
A point $x$ in a topological dynamical system $(X,f)$ is called (positively) recurrent if $x \in \omega(x)$, where $\omega(x)$ denotes the $\omega$-limit points of $x$. $R$ denotes the set of all ...