0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 $i=1,...,d$.
0
votes
0answers
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 ...
0
votes
0answers
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 $\mathbb{Z}_p$ ...
-1
votes
0answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
1
vote
1answer
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?
1
vote
0answers
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, ...
2
votes
1answer
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. ...
-1
votes
0answers
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.
1
vote
4answers
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.
0
votes
0answers
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 ...
0
votes
0answers
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. ...
0
votes
1answer
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.
1
vote
2answers
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 ...
1
vote
0answers
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 ...
1
vote
1answer
27 views

Identifying the limit in the function

We see in the third part part that limit of 2$\theta$ is between -$\frac{\pi}{2}$ and -$\pi$ which become $-\frac{\pi}{2}<\pi + 2\theta<0$. How do we know that $2\theta$ became $\pi + ...
0
votes
0answers
9 views

A review of an example given by C. A. Akemann

I am witting concerning the example II.6 given in the paper "The General Stone-Weierstrass Problem" by C. A. Akemann: Let me review this example. Let $H_i$ be the two dimensional Hilbert space ...
0
votes
1answer
16 views

Completeness and orthogonal projection

a. Which are the properties that define an orthogonal projection? Give a precise definition. b. What does completeness mean? Please state both the definition and an example (without proof) ...
0
votes
0answers
6 views

Arrangement of Convex Discs in the plane is independent of the choice of origin?

This is the Problem 3.1 in 'Combinatorial Geometry' by J. Pach, and P. Agarwal. Problem: Prove that if C is any arrangement of convex discs in the plane, then $\bar{d}$$(C,\mathbb{R}^2)$ and ...
1
vote
4answers
25 views

Statistics: Conditional Probability

$P(A│B)=\frac25$ ,$P(B)=\frac14$, $P(A)=\frac13$. Find $P(A\land B)$ $P(B|A)$ Here is what I did: Part 1. $$P(A\land B) = P(A) \cdot P(B)\\ = \frac13\cdot\frac14=\frac{1}{12}$$ Part 2. ...
0
votes
0answers
11 views

sequential characterisation of limits

There is a proposition left on a 2nd year vector calculus notes provided with no proof. I always having trouble writing these kind of proof and I hope someone could provide an answer for future ...
0
votes
1answer
9 views

Computing shortest path including specific edge

Consider the weighted undirected graph with $4$ vertices, where the weight of edge $\{i, j\}$ is given by the entry $W_{i, j}$ in the matrix $W$. $$W = \begin{bmatrix} 0&2&8&5\\ ...
2
votes
1answer
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 ?
0
votes
0answers
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 ...
1
vote
1answer
15 views

Product of Quadratic Residues

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: $a^2$ ...
0
votes
0answers
10 views

Updating array and counting permutations with some criteria

We have an array $A[n]$, and an integer $D$. $P$ is a permutation of numbers $[1 2 3 .... n]$ $P$ is a valid permutation if $$ A[P_1]+D>A[P_2] $$ $$ A[P_2]+D>A[P_3] $$ $$ .... $$ $$ ...
1
vote
3answers
14 views

Conditional Probability

1) A card is drawn from a standard deck of $52$ cards. The card is drawn from the pack and not replaced. A second card is then drawn from the deck. Determine the probability: $a)$ that the ...
1
vote
1answer
51 views

Is this plot of heart for real?

I came across this equation while surfing through the internet.Is this for real?If it is how is it done?
0
votes
1answer
20 views

Calculus: Sketching a graph that satisfies the following conditions

The following question is from our reviewer for our upcoming exams: $\ast$ 3. Sketch the graph of a function $f$ satisfying the following conditions: (i). $-5,-3,-2,-1,0$ and $2$ are the only ...
2
votes
1answer
10 views

Proof that distinct numerator polynomials are equal for all x when over the same denominator polynomial

I am just curious about this part of the proof. The question is this: Suppose that F, G, and Q are polynomials and $\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}$ for all x except $Q(x)$ = 0 Prove that ...
0
votes
1answer
23 views

differentiate by a differential

This is a difficult question to phrase so I will show it mathematically. Let $f(\theta) = \sin(\theta)$ Is it possible to do: $\frac{d f(\theta)}{d \dot{\theta}}$ I have tried to do $ t = ...
0
votes
0answers
14 views

How could an estimator be biased but consistent according to mathematical definition?

According to the definition, an estimator can be biased, if $E_{\theta}[\hat{\theta}]\ne\theta$, with $\theta$ as parameter for a distribution we want to get from samples. While the estimator can be ...
0
votes
0answers
12 views

Construction of the Area Function

I am following calculus by Tom M Apostol in which he has given the Axiomatic definition of the Area Function We assume there exists a class M of measurable sets in the plane and a set function a, ...
0
votes
0answers
3 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Suppose $\mathbf{A}$ is a Hermitian $n\times n$ matrix with eigenvalues $\lambda_i(\mathbf{A})$, $i=1,\ldots,n$. Suppose $\mathbf{B}$ is an $n \times n$ complex-valued matrix and $b\neq 0$ is a ...
2
votes
1answer
20 views

Change of variables, integration

In a finite element analysis, I am evaluating the following integral: $$\int_{0}^{h}\left ( 1-\frac{x}{h} \right )*\left ( x \right )dx$$ but I want to apply a transformation from x to integrate ...
0
votes
3answers
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, ...
1
vote
1answer
26 views

The maximum possible size of $R$ is_____?

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) = ...
0
votes
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,
3
votes
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: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$ ...
0
votes
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." ...
1
vote
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 $W_2$ ...
1
vote
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 ...
1
vote
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^+} ...
0
votes
0answers
5 views

Non-wandering point which is not in the closure of recurent points

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 ...

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