0
votes
0answers
7 views

If $Y$ is a quasi-affine variety, then dim$Y$ = dim$\overline{Y}$

Reading through the proof of proposition 1.10 in Hartshorne's Algebraic Geometry I found some f in to be unnecessary. Is the following proof correct or can you point out my flawed logic. Let $Z_0 ...
0
votes
0answers
6 views

Find limiting distribution

The question is like this: $X_i$ are i.i.d with $P(X_i\leq x)=e^{-x}$. $S_n=X_1+\cdots+X_n$. Find the limiting distribution of $\sum_{i=1}^nP(X_iS_n>1)$. It seems that the problem is related to ...
0
votes
1answer
21 views

what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$?

if we have $a+b+c=1$ and $ab+bc+ac=\frac{1}{3}$ then what is the value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$ and $$\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}$$. from the hypothesis we have ...
0
votes
1answer
15 views

Grade 8 simple algebra equation help

I find this question hard, please help It is given $x+(1/x)=3$ and $x^2+(1/x^2)=7$ Please find the value of $x^3+(1/x^3)$ Please show the steps
0
votes
0answers
8 views

How to find Difference equation of this block diagram. PLEASE HELP!

Can anyone please help me with this question. Like for part b), it says i need to convert all the z terms to negative powers. How do i do that and how do i find the transfer function?
0
votes
0answers
5 views

The set of ordinals $< \alpha$ of a given cofinality $\kappa< \text{cf}(\alpha)$ is stationary

I don't understand the last line of the following proof : I understand that we climbed up to $g(\kappa)$ in $\kappa$ steps (i.e. cf$\big(g(\kappa)\big) \leq \kappa$) but can we be sure that we did ...
-6
votes
0answers
16 views

probability of getting lucky in exam?

In an examination, you are given a choice to pick up a chit, which has a question, there are ten of those chits(randomly arranged), only half you have prepared(you know all the question but you're ...
0
votes
0answers
26 views

All solutions to functional equation $f(x+1)-f(x)=1$

I was thinking of the possibility of finding all solutions other than $f(x)=x$ for the functional equation: $f(x+1)-f(x)=1$ If there are other solutions, what will be the restrictions for the ...
1
vote
2answers
18 views

Taylor series in two variables?

how can I calculate the taylor series for a two-dimensional function? Example: \begin{equation*} f(x,y) = Log(1+x+y). \end{equation*} I have $f_x = (1+x+y)^{-1} = f_y$. $f_{xx} = -(1+x+y)^{-2} = ...
0
votes
1answer
23 views

Does $O(\log^2(x))$ imply $O(x)$

Does $O(\log^2(x))$ imply $O(x)$ I have to prove the following: $$\sum\limits_{\substack{n\in\mathbb N\\n\le x}}\Lambda(n)\log(n)=\psi(x)\log(x)+O(x)$$ Applying partial sum I get; ...
0
votes
0answers
9 views

How to get this very simplified demographic forecast?

I'm working on the simulation of a population growth. The variables and hypothesizes are the following: Lifetime: X years (X constant for everybody, yeah !) Initial population: Y people (with always ...
0
votes
0answers
9 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are 3 teams observing the movement of the birds. Team 1 observes that on their first flight the birds move a short distance across a branch ...
-1
votes
0answers
14 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
0
votes
0answers
10 views

limsup facts - which imply which?

According to this answer on my previous question, $\limsup X_n = \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq k} \bigcup_{m \geq n} ...
0
votes
2answers
27 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
0
votes
2answers
16 views

Proper use of indicator function

Given a set $X$ and a subset $A \subseteq X$ the indicator function $\boldsymbol{1}_{A} : X \rightarrow \{0,1\}$ of $A$ is defined as $$\boldsymbol{1}_{A}(x) = \begin{cases} 1 & \text{if } x \in A ...
3
votes
3answers
30 views

How many possible words of this type can be formed?

We are making $10$ letter words using the letters $A,C,G,T$. How many possible words are there of the form $A...AC...CG...GT...T$ This is where all of the $A's$ go before the all of the $C's$ and ...
0
votes
0answers
10 views

Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...
1
vote
0answers
12 views

Example 2.3.4 in Hartshorne algebraic geometry

I am reading Hartshorne algebraic geometry. Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed ...
1
vote
1answer
27 views

Weird contradiction between equations

A guy that I tutor came to me with the following question: The time it takes for body $A$ to pass 160 km is 5 hours longer than the time it takes for body B to pass 90 km. The speed of body A is ...
0
votes
0answers
48 views

verify that the set $\{0,1,2,3\}$ is not a group under multiplication modulo $4$

Given the set $\{0,1,2,3\}$: -Associativity holds for this set -Closure holds for this set (constructing the Cayley table, all entries in the tables are in this set). -there is an identity element ...
1
vote
0answers
8 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
0
votes
1answer
15 views

Speed at which hands of clock approaching one another.

A clock's hour hand's length is $1$, and its minute hand's length is $r$. First I had to find the distance between the tips of the hands at 4:00. I did this using the law of cosines. This gives me ...
0
votes
0answers
10 views

How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
-1
votes
1answer
42 views

Eyebrow calculation

Given a width of 71 and a height of 35, what are the following dimensions: left side, right side, radius, and base?
-9
votes
0answers
30 views

Limit of a particular series [on hold]

Please help me find the limit of the following series: $$1+1+3/4+1/4+5/16+3/16+7/64+5/64+\cdots$$
0
votes
1answer
14 views

Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation?

Is it true that the number of arbitrary constants in the solution (if solutions exist) always equal to order of an ordinary differential equation? If yes, how to "prove" such a statement, if it can be ...
1
vote
1answer
19 views

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?

Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$? By the way, I do not know how the adverb "explicitly" is strictly ...
-1
votes
2answers
69 views

Why is this true? $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$

Why is this True? $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 \Rightarrow A+B+C = \pi$$ with this assumption that $$0\leq A,B,C<\frac{\pi}{2}$$
-1
votes
1answer
18 views

Show $\lim X_k < \infty$ is in tail sigma-algebra

Show $\lim X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
-1
votes
0answers
14 views

Show $\sum_k X_k < \infty$ is in tail sigma-algebra

Show $\sum_k X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
1
vote
0answers
10 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
-1
votes
1answer
13 views

Inversion of a matrix in a system of linear inequalities

I would like to know if someone knows sufficient conditions on $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^{n}$ such that for all $x\in\mathbb{R}^{n}$: $$Ax\leq b \Rightarrow x\leq A^{-1}b \text{ ...
0
votes
3answers
26 views

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $P(x) = x^4 + 9x^{2} + 15 $.

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $p(x) = x^4 + 9x^{2} + 15 $. Find $[\mathbb{Q}(\alpha\ ):\mathbb{Q}(\alpha^2 + 3)]$. My attempt: By Eisenstein's ...
-1
votes
0answers
9 views

Integration of a tensor product

I would like to compute the integral of a diadic tensor on the sphere, but I can't see how to do this : $$\int_{\Bbb S^2}\mathbf{\vec n} \otimes\mathbf{\vec n}\mathrm{d}^2\mathbf{\vec n} $$ Where ...
0
votes
1answer
13 views

Largest number for which a laurent series converges

For part $(a)$ I got summation from $\sum^{\infty}_{n=0}(-1)^n\frac{z^{2n}}{(2n+1)!}$ Is this correct? Could someone explain how to do part (b) because I have no idea where to start Thanks
0
votes
0answers
2 views

An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$

Consider a functional $$J = \int_{a}^{b} F(x, y, y^{'}),$$ where $F(x, y, y^{'}) = \frac{1 + y^{2}}{(y^{'})^2}$ for admissible function $y(x).$ Which of the following are extremals for $J$? $y(x) = ...
1
vote
0answers
18 views

Is it 3-D Catalan numbers?

I am studying Catalan numbers recently but I think that how about 3-D Catalan? So that I imagine following situation ; A man travel through the path-way parallel to $ x, y, z $ axis from O ...
0
votes
2answers
23 views

How to find kth smallest value of a linear equation

Here's a question that was asked in IOITC 2009 India. Even though it should have a solution related to algorithms, yet I post it here as it is pretty "number-theoretic". Indraneel loves posing ...
-4
votes
0answers
56 views

IMO 2015 Problem 3 [on hold]

Let $n$ and $k$ be positive integers. Prove that if $n$ is relatively prime with $30$, then there exist integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2-b^2+k}{n}$ is an ...
0
votes
1answer
14 views

Finding surface of revolution isometric to helicoid

I'm trying to find a function $f(x)$ such that the two surfaces given below are isometric: $$f_1(x,y) = (ax \cos(y), ax \sin(y), y)$$ $$f_2(x,y) = (f(x)\cos(y), f(x)\sin(y), x)$$ Now I understand ...
0
votes
0answers
15 views

An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\ a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\ b_k = \frac{1}{\sqrt{\pi}} ...
2
votes
1answer
35 views

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV?

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV? Not sure how to do this one. Obviously there are $\frac{10!}{4!}$ anagrams of KOLMOGOROV but I'm not sure how to account ...
0
votes
0answers
7 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
1
vote
0answers
27 views

Assume that the sum of absolute values of all entries of $A$ equals to $1$. What is the maximal possible value of $\det(A)$?

Let $A$ be an $n \times n$ matrix and assume that the sum of absolute values of all its entries equals to $1$. What is the maximal possible value of $\det(A)$? My attempt: We know that $|a_{i,j}| ...
0
votes
2answers
33 views

Proof with subspaces [on hold]

Prove: If $V$ and $W$ are three-dimensional subspaces of $\Bbb R^5$, then $V$ and $W$ must have a non-zero vector in common. (Hint: start with bases for the two sub-spaces, making six vectors in all) ...
2
votes
0answers
14 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,J_{0.5}(b\cos x)\,J_{0.5}(b\sin ...
0
votes
0answers
17 views

Newton method for $p$-adic fields

I want to understand where the last line comes from. I.e. why there is the $p^{2n-2ka}$ term. I tried to use the estimate formula for the reminder but it doesn't work for me...
0
votes
0answers
14 views

Does a $K_n$ with $n$ pendants have a name?

Consider the graph we get by taking the complete graph on $n$ vertices, and then attaching a pendant vertex to each of the $n$ vertices by an edge. Does such a graph have a name, i.e. do such graphs ...
0
votes
0answers
13 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...

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