0
votes
0answers
4 views

Given the matrix, find a matrix such that

Given $T(\begin{bmatrix}1\\-2\end{bmatrix}) = \begin{bmatrix}3\\10\end{bmatrix}$ $T(\begin{bmatrix}-2\\-1\end{bmatrix}) = \begin{bmatrix}-1\\-5\end{bmatrix}$ Find a matrix such that: $T(...
0
votes
0answers
6 views

$\dfrac{x^2+y^2}{x+y}$ is a divisor of $1978$

Two nonzero integers $x,y$ (not necessarily positive) are such that $x+y$ is a divisor of $x^2+y^2$, and the quotient $\dfrac{x^2+y^2}{x+y}$ is a divisor of $1978$. Prove that $x = y$. Let $A = \...
-1
votes
1answer
3 views

how to know if a set of arbitrary vectors are a basis?

so if we're given that {x,y,z,w} is a basis of R^4, how do we show that {x+w, y+w, z+w, w} is also a basis of R^4? i know that for a set to be a basis, it has to be linearly independent and it also ...
0
votes
0answers
17 views

Question about congruence

Find the smallest positive n that satisfies the system of congruences $$n \equiv 3 \pmod 4$$ $$n \equiv 4 \pmod5$$ $$n \equiv 5 \pmod 7$$ Approach: Not very useful $4|3-n$, $5|4-n$, $7|5-n$ $3-...
0
votes
0answers
12 views

Idempotents in commutative ring of characteristic 2 form a subring

Question: In a commutative ring of characteristic $2$, want to show that the idempotents form a subring. Subring Test is probably the way to go. It is easy to verify the identity element in ...
0
votes
0answers
7 views

Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
0
votes
1answer
14 views

Relation between these two series

Assume a constant $\alpha$ and $N$ positive integers $\{n_1,...,n_N\}$. What is the relation between $\frac{N\alpha}{\sum_i{n_{i}}}$ and $\sum_i{\frac{\alpha}{n_{i}}}$ when $N\rightarrow\infty$? $N$ ...
0
votes
0answers
5 views

terminology help: do we say a hyperplane “embedded” in higher dimensional space?

Let $L$ be a line in $\mathbb R^2$. I have a function $f$ defined form $\mathbb R^1$ to $\mathbb R^1$ and I want to use this function $f$ on $L$ and define the set $$ S:=\{(x_1,x_2)\in\mathbb R^2,\,\, ...
0
votes
0answers
8 views

Street problem validated with SMV. How to model this problem?

I'm new with SMV and I am trying to solve the following problem using SMV. (http://www.kenmcmil.com/smv.html) Problem: Environment: A cross with three times signal (the first for pedestrians, the ...
1
vote
1answer
18 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
3
votes
2answers
38 views

Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math?

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. For example, I have no idea what eigenvalues, ...
0
votes
0answers
11 views

Proving Gerretsen's Inequality

Today in class we were shown Gerretsen's inequality: $$16Rr-5r^2\leq s^2 \leq 4R^2+4Rr+3r^2$$ Where $R$, $r$, and $s$ are the circumradius, in radius, and semiperiter of a triangle. After some ...
0
votes
1answer
11 views

What is the difference between exponential growth and decay?

A colleague came across this terminology question. What are the definitions of exponential growth and exponential decay? In particular: 1) Is $f(x)=-e^{x}$ exponential growth, decay, or neither? 2) ...
0
votes
0answers
18 views

Number of graphs having a specific structure

Let $\mathcal{N} = \{1,2,\ldots,N\}$ and $\mathcal{N}^i = \mathcal{N}\setminus \{i\} $. For each $i \in \mathcal{N}$ and for each $S \subset \mathcal{N}^i$, we have a vertex $C_i^S$. For example, if $...
0
votes
1answer
11 views

Isomorphism between $\mathbb{T}$ and $\mathbb{R}^n/\Gamma$

In the Analysis on Manifolds via the Laplacian page $51$, they define the torus as $\mathbb{T} = \mathbb{R}^n/\Gamma$. This quotient is unintuitive that it defines the torus. Is there exists a natural ...
1
vote
3answers
24 views

Arranging numbers around a square

In how many ways numbers 1 to 12 can be arranged on a sides of squares (5 places on each sides i.e 20 places total) leaving 8 places empty? I am getting answer as 12c5(selecting 5 numbers)*7c5(...
-2
votes
0answers
18 views

Is this true or false? Tangent Space isomorphism [on hold]

Is this a consequence of the third isomorphism theorem? Let $L \subset N \subset G$, then $T(G/N) \simeq T(G/N) / T(N/L) $?
-3
votes
0answers
18 views

Why we use Convolution Integration?

Can anyone explain me why we do convolution integration instead of numerical integration. I am not clear about this.
0
votes
2answers
20 views

If $f: A \rightarrow B$ is surjective, and $A, B$ are nonempty sets, and $X \subseteq A$, does $f(A) - f(X) = f(A - X)$?

I'm working on a proof, and the proof will be complete if this is true... but I can't find a theorem in my book that explains whether or not this is true.
-1
votes
0answers
17 views

Existence of submersions between manifolds

I have a ton of problems, where I need to prove (or disprove) the existence of submersions between given manifolds. I will give you some examples, and hopefully I can learn the techniques to solve all ...
0
votes
0answers
20 views

Non-injective embedding

Embedding is defined to be a one-to-one structure preserving mapping. The question is of the one-to-one condition is critical. Like if low-dimensional maps such as PCA, could be considered an ...
4
votes
2answers
29 views

How can I prove or disprove that there exists a function such that…

Suppose we have a function $f$ of $bx-ay$ where $a$ and $b$ are two real constants, if we have for example $e^{bx-ay}$ then obviously it is a function of $bx-ay$. Can we find a function $f$ such ...
1
vote
0answers
12 views

Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
1
vote
0answers
18 views

Combinatorial formula for Legendre Polynomials

Using the recursion formula for the solution of the Legendre equation: $$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x) = 0$$ With solution $P_n(x)$ such that $P_n(1) = 1$, show that $$P_n(x) = \sum_{k = 0}^{n}\...
0
votes
1answer
27 views

Is sup of max, same as max of sup?

Let $\sigma_1, \sigma_2 \dots \sigma_n$ be functions of $\omega \in \mathbb{R}_+$. Is $\sup_{\omega}(\max_{i=1:n} (\sigma_i))$ same as $\max_{i=1:n}( \sup_{\omega}(\sigma_i))$? Could you also please ...
0
votes
4answers
20 views

Expected value of a roll of a fair die given that the number rolled is at least 4

I am trying to understand the solution to a probability problem, and I am having trouble understanding where some of the numbers are coming from. The textbook gives this definition for conditional ...
0
votes
2answers
16 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate element of GF(256) meaning that assume I would like to know what is alpha ^(32) or alpha^200 in polynomial form? given the primitive polynomial is D^8+D^4+D^...
3
votes
2answers
27 views

How may I use a 3x3 matrix to simulate a larger square matrix?

I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ...
0
votes
0answers
9 views

How to know whether a contact form is only defined locally or globally?

As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$: $$ \omega = dz + \sum_{k=1}^n x_k dy_k$$ Similarly, the following is the standard contact form on $S^{2n+1}$: ...
0
votes
1answer
15 views

Product rule for derivatives: one is bounded and the other is differentiable

Let $S \subset \mathbb{R}$, $f: S \to \mathbb{R}$ be bounded and $g: S \to \mathbb{R}$ be differentiable at $c \in S$ such that $g^{\prime}(c) = g(c) = 0$. Show that $h = fg$ is differentiable at $c$ ...
2
votes
1answer
49 views

Any hints on how to prove that $\ln{1\over 2\sin\left({90\over \pi}\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}$?

How do you prove that $$\ln{1\over 2\sin\left(\frac 12\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}?\tag1$$ where $B_{2n}$ is bernoulli number Any hints?
2
votes
2answers
38 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
2
votes
0answers
24 views

Show that there aren't negative eigenvalues.

I've been trying to solve this Sturm-Liouville theory problem. Show that the problem: $$\left\{\begin{matrix} y''+(x+\lambda)y = 0\\ y(0)=0\\y(1)=0\end{matrix}\right.$$ doesn't have ...
0
votes
0answers
22 views

Angle between Two Lines in 3D Space

Since my two lines are orthogonal (one in y-z plane and one in y-x), I KNOW there's a (probably simple) formula for the calculation I need, but somehow I haven't been able to find it. Equations of the ...
0
votes
1answer
40 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
1
vote
0answers
12 views

Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
2
votes
3answers
26 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...
0
votes
2answers
26 views

Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$

Let $x$ and $y$ be integers not congruent to $0$ modulo $p$ where $p$ is a prime. Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$. I thought about proving this ...
0
votes
0answers
16 views

Intuition/derivation for Cauchy's repeated integral formula?

https://en.m.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration I'm referring to this formula due to Cauchy. The wiki page has a proof, but what I'm looking for is a more direct derivation or ...
2
votes
2answers
46 views

Prove that the integral $\int_0^1 f(t) P(t - x)dt$ is a polynomial in $x$

So suppose $f$ is a generally complex, continuous function on $[0,1]$ and $P$ is a polynomial defined on the real numbers. Evidently $$ \int_0^1 f(t)P(t - x)\,dt $$ is a polynomial in $x$ but that ...
-10
votes
0answers
27 views

Quadratic term answer fast please [on hold]

let k be a real number such that k≠0.if a and b are non zero complex numbers satisfying a+b=-2k and a²+b²=4k²-2k,then a quadratic equation having (a+b/a)and (a+b/b)as its roots is equal to
2
votes
0answers
43 views

Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$

Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}...
-1
votes
0answers
13 views

Convolution form and $n$-th partial term of $\sum_{k=1}^{+\infty} kp^k$

With $p\in(0,1)$, let $\displaystyle x(k)=\sum_{k=1}^{+\infty}k$ and $\displaystyle \phi(k)=\sum_{k=1}^{+\infty}p^k$ So that $\displaystyle (x*\phi)(k)=\sum_{k=1}^{n}x(k)\phi(n-k)=\sum_{k=1}^{n} kp^{...
1
vote
2answers
34 views

Prove the n-th Fibonacci number is less than $2^n$ for all n greater than zero using strong induction

I need to prove the n-th Fibonacci number is less than $2^n$ for all $n \geq 0$ using strong induction. I have been exposed to the idea that strong induction differs from weak induction in that the ...
1
vote
1answer
22 views

If A is positive definite (but not necessarily symmetric) can you decompose it?

If A is a $2 \times 2$ matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that $A=B^TB$?
-8
votes
0answers
28 views

Quadratic term answer fast [on hold]

let sin x and sign y be the roots of the quadratic equation a sin square theta + b sin theta + C is equal to zero,such that sin x + 2 Sin y is equal to 1 then the value of a square + 2bsquare + 3ab+ ...
0
votes
0answers
19 views

Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
2
votes
0answers
21 views

Proof verification: Show that the Frobenius map is surjective.

I would like to prove the following but I would like someone to check my proof. For an algebraically closed field $K$ with characteristic $p$, the Frobenius map $F(x) = x^p$ is surjective What I ...

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