0
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0answers
4 views

linear transformations of same form

Let m and n be positive integers and F be a field . Let f1 , . . . , fn be linear functionals on F^n . For any element a in F^n ,define : T(a) = ( f1(a) , . . . , fn(a) ) If T is a linear ...
1
vote
0answers
6 views

A question about the definition of complete dg Lie algebras in a paper of Lazarev and Markl

In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23): Definition: A complete differential graded Lie algebra is an inverse limit of ...
0
votes
0answers
12 views

Find the sequence defined by the recurrence equation $x_{n+1} = 4x_n − x_{n−1}, (n ≥ 1)$

Find the sequence defined by the recurrence equation $x_{n+1} = 4x_n − x_{n−1}, (n ≥ 1)$ with $x_0 = 1$ and $x_1=2$. Find an odd prime factor of $x_{2015}$. I've found the characteristic equation to ...
0
votes
0answers
9 views

Functional Equation: $f(x^2-y^2)=xf(x)-yf(y)$

Let $\mathbb{R}$ be the set of Real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2-y^2)=xf(x)-yf(y)$$ for all pairs of real numbers $x$ and $y$. This is a problem ...
1
vote
0answers
9 views

For $0<x<\frac{\pi}{4}$,prove that $\frac{\cos x}{\sin^2x(\cos x-\sin x)}>8$

For $0<x<\frac{\pi}{4}$,prove that $\frac{\cos x}{\sin^2x(\cos x-\sin x)}>8$. I have no idea how to solve this problem.Somewhat i tried. We need to prove that $\cos x>8\sin^2x(\cos ...
0
votes
0answers
4 views

What is a metric to determine if a set of points were sampled from a curve?

Suppose we have a curve with its equation given as a spline. We also have a set of ordered (x,y) coordinates. Is there a metric that would indicate whether the points were sampled from the spline? The ...
0
votes
0answers
7 views

Group of polynomial $x^4+2$ in $\mathbb Q[x]$

Describe the Galois group of the polynomial $x^4+2 \in \mathbb Q[x]$. I've been able to see how to do this for $x^4-2$ and $x^4+1$ but am unsure how to do this for the polynomial above. Based on the ...
0
votes
0answers
4 views

Simple Equation Required for Break even Amount

Simple Equation required using all the variables listed in the image to calculate the value of "X".X2 Formula Paint
0
votes
0answers
5 views

how to calculate the limit as $\lim_{s \to 0 }$ of this large equation?

I am having a hard time calculating this limit: $$\lim_{s \to 0 } \frac{-R_{4}}{R_{3}}\frac{sC_{2}\frac{R_{3}}{R_{3}+R_{4}}\frac{R_{5}+R_{6}}{R_{6}} } {\frac{s^2C_{1}C_{2}R_{1}R_{2}R_{5}}{R_{6}} ...
0
votes
0answers
8 views

Casting an expectation as an integral

I probably picked the most ambiguous title possible for the question I am about to ask. Sorry for that. I have two random variables, $X$ and $Y$. I am about to define conditional densities and I am ...
0
votes
1answer
11 views

Using simple matrix algebra to solve for a specific matrix (Beginner question)

The matrix $AB = C$ where $A$, $B$ and $C$ are all $2 \times 2$ non-singular matrices. How would I go about to solve for the Matrix $A$ and express it in terms of $B$ and $C$? There are two methods ...
1
vote
0answers
11 views

Show that all the solutions of the given differential equation are bounded

Let $ f:[0,\infty)\rightarrow \Bbb R$ be a bounded and continuous function. Show that every solution of the differential equation $$y''+2y'+5y=f(t,)\quad t\ge 0$$ is bounded on $[0,\infty)$. By using ...
0
votes
0answers
9 views

Understanding steps to obtain derivative of $|x_n|^{\frac{3}{7}}$

I was trying to solve the following derivative $$|x_n|^{\frac{3}{7}}$$ as follows $$(|x_n|^{\frac{3}{7}})' \\= \frac{3}{7}(|x_n|^{\frac{3}{7} - 1}) \cdot (|x_n|)' \\= ...
1
vote
0answers
4 views

Need of cusps (with respect to Modular Forms)

In every introduction about Modular Forms (on $SL_2(\mathbb{Z})$ and congruence subgroups) one reads the term 'cusps'. A Modular Form should be holomorphic in the cusps. Can anybody explain to me, ...
0
votes
0answers
8 views

How to evaluate these two euler sums?

How to evaluate $$\sum_{n=1}^{\infty}\frac{H_n}{n^52^n}~~,~~\sum_{n=1}^{\infty}\frac{H_n}{n^62^n}$$ In this post M gives a closed form for the first series,but it contains a zeta function,can it be ...
1
vote
0answers
9 views

Show the embedding

I want to show the embedding $W^{1,p}(0,1) \subset C^0 [0,1]$. So we pick a $u \in W^{1,p}(0,1)$ and want to show that $u \in C^0 [0,1]$. Let $x_n \to x$. We want to show that $u(x_n) \to u(x)$. ...
1
vote
0answers
6 views

product of divisors

Let $D$ be a divisor on a curve $C$ (smooth projective over a field). I have seen people considering a zero-cycle on the surface $C \times C$ which they denote by $D \times D$. If $D=\sum_{P \in S} ...
4
votes
0answers
19 views

Sum of three consecutive cubes equals a perfect square

I have found this problem in an old German textbook: Find all sets of three consecutive integers such that the sum of their cubes is a perfect square. We can write $$S = (x-1)^3 + x^3 + (x+1)^3 = ...
0
votes
0answers
7 views

how to prove t(x)-x$\in F_{p}$ when $p(x) \in K$,K is a field whose characteristic is p(p is prime in this case)

Suppose K is a field whose characteristic is p(p is prime in this case), and L is a finite Galois extension of K. Also, we have an endormorphism h such that $h(x)=x^p-x$ for any x $\in L$. ...
0
votes
0answers
5 views

How to find open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$?

For purposes of illustrating Local Class Field Theory, let us play with the $3$-adic numbers. I'd like to find some open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$. I know about the ...
0
votes
0answers
9 views

Why is $T/S$ isomorphic to $k^{\ast}$?

I had a quick question about quotients of varieties. I am still not very good at them. Let $T$ be a torus, $\alpha$ a nontrivial character of $T$, and $S = (\textrm{Ker } \alpha)^0$. Since $T$ is ...
0
votes
0answers
5 views

How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n = {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
0
votes
0answers
4 views

Prove that a 3-regular graph with at most 2 cut-edges has a perfect matching

A $k-regular$ graph is a graph in which every vertex's degree is $k$. $w(G)$ is the number of connected components of graph $G$. A cut-edge of graph $G$ is an edge like $e \in E(G)$ such that ...
2
votes
0answers
13 views

Show that $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R}), \int_{\mathbb{R}} f dx=0\}$ is dense in $L^p(\mathbb{R})$

Show that $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R}), \int_{\mathbb{R}} f dx=0\}$ is dense in $L^p(\mathbb{R})$. Is the statement true if $\mathbb{R}$ is replaced by $[0,1]$? Also what can we say ...
1
vote
0answers
30 views

Proving that $\lim_{x \to \infty} \frac{e^{-x}}{x^{-n}} = 0$ without switching fraction

How could I prove that $$\lim_{x \to \infty} \frac{e^{-x}}{x^{-n}} = 0$$ with L'Hopitals rule without switching $$\frac{e^{-x}}{x^{-n}}$$ to $$\frac{x^{n}}{e^{x}}$$ I tried differentiating and it ...
0
votes
0answers
6 views

Finding the area of the different portions of a rectangle that lie in a grid.

I am an undergraduate student working on a large research project and one part involves calculating the portions of a rectangle that lie in different parts of a cartesian grid. In the figure below, I ...
0
votes
3answers
16 views

Unseen Problem based on area of triangle

In $\triangle ABC$, $BD=2CD$ and $AE=ED$, prove that $6\triangle ACE=\triangle ABC$ If $A,X$ is joined such that $X$ is the mid point of $BC$ then: $\triangle ABX=\triangle AXC$ Also, $\triangle ...
0
votes
0answers
8 views

Finding a function of a random variable that maximizes some expression

The following problem is part of my studies, so I would prefer hints or suggestions for self-study. Let $v_1$ be a random variable taking values in $[a,b]$ for $a,b\in \mathbb R$ and assume that the ...
2
votes
1answer
16 views

Express the statement that $x$ has at least one element in the language of set theory

For a past paper for a module that I am revising for, we are asked to express the Axiom Schema of Separation for the property that "$x$ has at least one element". I understand that the Axiom Schema ...
0
votes
1answer
11 views

Relations and functionss

I am unsure how to do this, is it possible someone could give me a step by step guide so I can have a good understanding of it. f(x) and g(x) are defined over the real number set R as follows: $g(x) ...
0
votes
0answers
16 views

Expected value and sum of independent variables.

EDIT: I've found my mistake. Flipped around the values because in my head I had them tails up at the start.. Not sure what to do with the question now... On a table there are three coins in a row, ...
0
votes
1answer
8 views

Residue of $\dfrac{1}{z^2+4z+1}$. Laurent series.

I want to calculate : $$ \int_0^{2\pi} \dfrac{\mathrm{d} \theta}{2+\cos(\theta)} $$ I use $z=\mathrm{e}^{\mathrm{i} \theta}$ and residue theorem : $$\int_0^{2\pi} \dfrac{\mathrm{d} ...
0
votes
1answer
9 views

How to find parametric equation between two points in line integral?

[In this example how can we find parametric equations of x and y.] [1] [question]: http://i.stack.imgur.com/lTOnW.png [1] [Solution]: http://i.stack.imgur.com/l8ao7.jpg
0
votes
1answer
13 views

Why if $\sup f(x) = \inf g(y)$ then there is $z,w$ such that $f(z) - g(w) < \epsilon.$

Let $f : A \to \mathbb{R}$, $A\subset \mathbb{R}^n$, $g: B \subset \mathbb{R}^m \to \mathbb{R}$, $f(A), g(B)$ bounded. Why is true that if $\sup f(x) = \inf g(y)$ then there is $z \in A$ and $w \in B$ ...
0
votes
0answers
8 views

return coefficients matlab

I'm trying to return coefficients from a 3rd order estimated equation using Matlab R2016a ...
0
votes
0answers
10 views

“Opposite” point on ellipsis by axis (or vector)

I'm currently working on a little game and am stumped as to how I'd solve this math problem. What I'm trying to do is get the "rotation" needed for B, where B is always opposing A on the Y axis no ...
1
vote
3answers
17 views

Third Order Differential Equations

I am having trouble solving the third order differential equation $y'''+y'=0$ It was given to me in a quiz (which I got wrong) with boundary conditions $y(0) = 0$ $y'(0)=2$ $y(\pi)=6$ I know that ...
1
vote
1answer
45 views

Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$

I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!
0
votes
0answers
11 views

Power function exponential distribution

I am trying to find the power function for a test. I know that the power function is calculated by $\beta(0) = P_0(x \in R)$ where $R$ is the rejection region. What I know about this test is that $X ...
0
votes
0answers
20 views

How can I tell if a matrix transformation is injective/surjective?

Determine whether or not $\mathbf v_1=(-2,0,0,2)$ or $\mathbf v_2=(-2,2,2,0)$ is in the kernel of the linear transformation $T:\mathbb R^4\to\mathbb R^3$ given by $T(\mathbf x)=A\mathbf x$ where ...
0
votes
1answer
10 views

How to develop a formula for a function?

What are the general tips and techniques to define an explicit formula for a function when the mapping of that function is known. Say f: N to Z (N is natural numbers and Z is integers). In this ...
1
vote
0answers
12 views

How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
0
votes
0answers
10 views

Complex Function Identities

Newcomer to Complex Analysis, I can't see any reason why these identities wouldn't hold, if taking multi valued log and exp the whole time. Am I correct?
0
votes
0answers
24 views

Every morning the lecturer chooses pairs of students

There are 10 students in a class 7 males and 3 female. Every morning the lecturer chooses pairs of students in a random. X - numbets of teams, including a man and a woman (together) I thought ...
0
votes
0answers
9 views

A nice question related to method of characteristics

let $ \alpha$ be real number and $h=h(x)$ be a continuous function in R.Consider following initial value problem: $yu_x + xu_y=\alpha u, u(x, 0) =h(x) $.Then a) Find all points on ${(y=0)} $ where $ ...
0
votes
1answer
5 views

Finding basis for column space of matrix

To find a basis for the column space of a matrix one finds the RREF of the matrix. The columns in the RREF are not a basis for the column space, but the same columns in the original matrix are a ...
0
votes
0answers
6 views

Use past and future data to predict or estimate missing values

I have a huge data set for a single variable $z$ , say WEATHER, not necessarily complete one. That is it has many holes in it(missing data) $z \hspace{3mm} is \hspace{3mm} a \hspace{3mm} 6000\times1$ ...
0
votes
2answers
20 views

Solving a Three Variable Equation 3;

Today i am facing a problem which involves three variable. Question: $$\frac{1}{x}+\frac{2}{y}+\frac{2}{z}=4$$ $$\frac{2}{x}+\frac{1}{y}+\frac{2}{z}=3$$ $$ \frac{6}{x}+\frac{-4}{y}+\frac{1}{z}=0$$ I ...
1
vote
2answers
27 views

$2^{49}$ ways to choose a set of integers $\leq 50$ with odd sum

Problem: Show that the number of ways one can choose a set of distinct positive integers, each smaller than or equal to $50$, such that their sum is odd, is $2^{49}$. My attempt: Suppose set ...
0
votes
1answer
8 views

Minimum of a bivariate quadratic function

According to (hope my calculation below is correct) https://en.wikipedia.org/wiki/Quadratic_function a bivariate quadratic function is a second-degree polynomial of the form $$ ...

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