2
votes
3answers
97 views

$f\cdot g=0 \implies f=0 $ or $g=0$.

I know this is kind of an obvious thing to say: Let $f,g \in \Bbb K[x]$, then $$f\cdot g=0 \implies f=0 \text{ or } g=0$$ But to my surprise I couldn't prove it. What's a simple way to do this?
0
votes
0answers
17 views

Within what angle does she need to throw her stone at to hit her opponents?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
3
votes
1answer
15 views

Question on complete spaces, longer, more specific question.

Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, ...
0
votes
0answers
20 views

Singularities of an integral

We have the integral : $$I(t)=-i\int_0^\infty \frac{\log\left[\frac{\sin(t\log\sqrt{1+ix})}{\log(1+ix)} \right ]-\log\left[\frac{\sin(t\log\sqrt{1-ix})}{\log(1-ix)} \right ]}{e^{2\pi x}-1} \, dx$$ I ...
1
vote
0answers
11 views

What is the difference between a lower bound and an upper bound in an Interval Graph $G(I)$

As I know that the maximal size of an independent set $IS$ of an interval Graph $G$ is a lower bound. Now what is exactly the upper bound, and when they might be equivalent to each other. are there ...
0
votes
0answers
3 views

How to interpret Realized Volatility and TSRV using R

I am looking at some high frequency data and I would like to know how to interpret and compare Realized volatility (RV) and Two Scale Realized Volatility (TSRV). References below. Given X is the log ...
-2
votes
1answer
16 views

How to get the equation for a one-to-one function given the coordinates. [on hold]

The one-to-one function is defined as: f(x)={(-8,4)(-6,1)(3,-6)(8,-2)}
-2
votes
1answer
28 views

Tricky word problem

Two crews were assigned the job of setting a total of $960$ conduit hangers. By the time the job was completed, one crew had set on $7/8$ as many conduit hangers as the other crew. How many conduit ...
0
votes
1answer
17 views

Directional derivative for differentiable function

In the directional derivative formula $$\frac{\partial f}{\partial v} = \nabla f \cdot v$$ why must $v$ be a unit vector?
0
votes
1answer
15 views

Closed form of a set

Let $V$ be a vector space with $W_1, W_2$ subspaces of $V$. Also, we have the set $\{e_1, e_2, e_3\}$ of linearly independent vectors such that: $$W_1 = \langle e_1, e_2, e_3 \rangle$$ and $$ W_2 = ...
0
votes
2answers
23 views

Permutation of Indistinguishable Objects

How many number of two digit numbers can be formed using $\{4,5,6,6\}$ without repetition? I know that $\{45,46,54,56,65,64,66\}$ are the possible answers, but I am wondering if there is any formula ...
0
votes
2answers
52 views

Multinomial Coefficients Definition in expansion of $(1+x+x^2+\cdots+x^l)^n$

The literature defines multinomial coefficients (or extended bnomial coefficients) as $$ \binom{n}{r_1,r_2,\cdots,r_l} = \frac{n!}{r_1!r_2!\cdots r_l!}$$ where $$ r_1+r_2+\cdots+r_l = n$$ Which is ...
0
votes
1answer
16 views

Proof Matrix L with respect to Basis B and C

Good one guys! I'm doing the conceptual exercises of my Linear Algebra book, and I ran up to the following exercise: I tried to use the following theorem: That came from: But it got messy ...
1
vote
0answers
12 views

Can we view the connected component of the Picard scheme $\text{Pic}_0(X)$ as a “kernel” of the first Chern class?

So on a curve, $\text{Pic}_0(X)$ is just the Jacobian variety, and just correspond to degree $0$ divisors. One way to extend the notion of divisors corresponding to a vector bundle is taking the first ...
1
vote
2answers
24 views

Prove that the normed spaces $(C[0,1], \| \cdot\|_2)$ and $(C[a,b], \|\cdot \|_2)$ where $\| \cdot\|_2$ is the Euclidean norm are isometric.

Essentially I'm looking for a bijection $f: C[0,1]\to C[a,b]$ such that$$\|f(x) \|_2=\| x\|_2$$ I don't know how to go about finding this function, but I do know that it is possible. $$\| x \|_2 = ...
0
votes
0answers
27 views

conditional expectation

I got somewhat confused about this condition expectation here. Can anyone help me please?$$$$ Let $v_1,v_2,v_3$ be 3 continuous random variables from an i.i.d distribution, does the equality below ...
1
vote
1answer
35 views

show that the function $\{x_n\}\mapsto \sum_{n=1}^\infty 2^{-n}x_n$ is continuous

This problem comes from an old Preliminary exam: Consider the space $[0,1]\times [0,1]\times \cdots$ (the countably infinite product of $[0,1]$ with the product topology) An element of $X$ may be ...
0
votes
1answer
16 views

Geometric meaning of directional derivative

Suppose $f(x,y)$ is a differentiable function and $v = (a, b)$ is a vector. If $(x_0,y_0) \in D_f$ and $\frac{\partial f}{\partial v}(x_0,y_0) = 0$. What is the meaning of this? Along the direction ...
1
vote
2answers
19 views

Poisson Distribution Word Problem

The image you are looking is a solution to a problem that has been cropped out. I'm certain the solution is incorrect since it does not include P(X=2). Just to be on the overly safe side, I decided ...
0
votes
0answers
11 views

Implicit function theorem conclusion notation?

I am working through implicit function theorem for the first time, and I have the following understanding. Given a system of $n$ equations, \begin{equation} f_i(x_1,\dots ,x_m,y_1,\dots , y_n)=0,\ \ \ ...
2
votes
1answer
33 views

Frazzle game question

In $7^{th}$ grade, in order to learn divisibility, memory, and focus, my math teacher had my pre-algebra class play a game called Frazzle. To play the game Frazzle, each person went around the room ...
0
votes
1answer
22 views

Tensor Products of bimodules over commutative rings

Suppose that $ R $ and $ S $ are commutative rings with identity, $ R \subset S $, $ 1 _{R} = 1_{S} $, $ M $ is a $ (S,R)$-bimodule, $ N $ is a $ (R, S)$-bimodule, $ T = M ...
-3
votes
3answers
34 views

my problem is about probability [on hold]

8 people have to go to hospital during a particular week.what is probability that on atleast one days,atleast two people will go to hospital?
1
vote
1answer
15 views

Partial derivative of polinomial root

I have a characteristic equation of the form $P(x,y,z) = 0$. $P$ is a polynomial in $x$ with degree of 3 and is a first order polynomial in $z$. I computed the value of $x=F(z)$, such that ...
1
vote
1answer
11 views

Understanding Hyperbolic Householder Transformations

Note 1: This post is a continuation of a previous post on Householder transformations. I'm using this post to document my understanding. Please provide your valuable comments. Note 2: All vectors in ...
0
votes
0answers
42 views

Better approximation for prime numbers [on hold]

The prime number theorem gives this formula for approximating the number of primes up to $n$: $\frac{n}{\ln(n)}$. By looking at this image from Wikipedia, I noticed that the function giving ratio ...
0
votes
1answer
35 views

Show that a group with 21 elements contains at max 3 subgroups with 7 elements

I don't know if lagrange's theorem apply here anyway. I just know that this group can have a subgroup with 3 elements, but I don't know about any theorem that talks about number of elements of the ...
0
votes
1answer
36 views

Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
-4
votes
0answers
21 views

Adjoint Proof Question [on hold]

How do I go about proving $adj(A^n)=(adj(A))^n$
0
votes
1answer
20 views

How do I solve for A in the matrix equation $A - B(A./C) = D$?

I've got $A - B(A./C) = D$, and I want to solve for $A$.* $A$ is an unknown 2x1 vector, $B$ is a 2x2 matrix, $C$ is a 2x1 vector, and $D$ is a 2x1 vector. *The notation $A./C$ means each element of ...
0
votes
0answers
35 views

Is this $\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{cosk\theta}{k} $ alternating series for all values of $\theta$?

I have tried to do other form of alternating series I got this: $$\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{cos k\theta}{k} $$ Can I say that the above series is alternating series for all ...
0
votes
0answers
7 views

Underlying utility function behind a linear two-product demand curve

I am trying to find the underlying utility function behind a linear two-product demand model. For that, I use two methods considering the following utility function: \begin{equation} U(q_1,q_2) = ...
0
votes
0answers
18 views

verifying Green's Theorem between 2 Circles

I get the same answer as my textbook only with a negative sign , so I am wondering who is right ...: Verify Green's theorem in the plane for {line integral of} x^2ydx + (y^3- xy^2)dy , where C is the ...
0
votes
0answers
11 views

Is there a name for this type of online optimization problem?

I have a sequence of items $1\leq i \leq n$ that arrive to me one at a time. Each item has a weight $w_j\geq 0$. If I pick up one item, I will not be allowed to pick up any of the next $k$ items ...
0
votes
2answers
33 views

Learning if the possible roots of an equation are different without resolving it

Is there anyway to know if a given equation will have different roots (all of them different to each other). Say: $x^3 - 17x^2 + 5x - \pi = 0$ Is there any property or theorem to know this for ANY ...
1
vote
1answer
12 views

Positive convergent sequence. Existence of another positive convergent sequence with same limit and larger elements

I have a positive sequence which converges to zero, i.e. $a_k \geq 0 \;, \forall k \in \mathbb{N}$ and $\lim_{k\rightarrow \infty} a_k = 0$. Does there exist another sequence $b_k$ with the property ...
0
votes
0answers
51 views

Set Theory (Real Numbers)

I have seen in a book that a number whose square is nonnegative is called real number. How can we explain what a real number is?
1
vote
1answer
29 views

Left and Right Eigenvectors - when are they the same?

Under what circumstances is any left eigenvector of a matrix also a right eigenvector (and vice versa)? My guess is that this is true if the matrix is symmetric, but is this necessary, and is it ...
0
votes
1answer
49 views

How to simplify this integrand,

I am trying to compute arc length in three dimensions but am currently stuck with integrating $$\sqrt{1+ e^{-2t} + 4e^{-2t}}$$ Can I get some hints on how to simplify? I didn't combine the second ...
0
votes
2answers
54 views

Let $a$, $b$, and $c$ be positive real numbers.

Let $a$, $b$, and $c$ be positive real numbers. Prove that $$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}$$ Under what conditions does equality occur? That is, for what ...
0
votes
0answers
13 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
1
vote
2answers
39 views

Given the adjoint $\mathrm{adj}(A)$, how do you find $\det(A)$ and $A^{-1}$?

Given $\mathrm{adj}(A)$ where $A$ is an $n\times n$ matrix, how do you find the value of $\det(A)$ and $A^{-1}$?
0
votes
2answers
65 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
9
votes
0answers
59 views

Does Nakayama's Lemma imply Cayley-Hamilton?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
1
vote
1answer
15 views

Mechanics: Projectiles involving a ball shot out of a cannon, moving in the opposite direction of the shot

I'm already quite familiar with projectiles and how to go about with most of those questions, but this is the first time I've seen a question in which the point from which the object is thrown from ...
3
votes
5answers
69 views

What is a set of bijections?

I am taking a course on abstract algebra, and the lector defined $T$ to be a set, and defined $G$ to be the set of all bijections from $T$ to itself: $$ G=\{\text{all bijections }g\colon T\rightarrow ...
1
vote
3answers
31 views

Isomoprhic and equal symbol for abelian groups

Let $A$ and $B$ be two abelian groups and $A$ and $B$ are isomorphic. Moreover suppose that $A$ and $B$ are not the subgroups of the same group. Is it correct to write $A = B$? Thank you.
0
votes
0answers
9 views

Proving a Partial Derivative Equivalence Using Taylor Series Expansion?

I'm studying computer vision, and one of the problems in my book is to prove that $\partial f/ \partial x = f(x+1) - f(x)$ It's been a while since I've touched Taylor Series, and so I'm not sure of ...
1
vote
1answer
21 views

Motivic measure

Somebody can give me some good references for start to read Motivic-measure, Now I`m studing the Grothendieck Ring, and is necesary undertand something of motivic theory for my case, so I need a good ...
2
votes
1answer
38 views

Determinant of a adjoint

If $A$ is an $n\times n$ nonsingular matrix then $\det(\operatorname{adj}(A^{-1}))=(\det(A))^{1-n}$ I tired using the fact $AA^{-1}=I$ but I ran around in circle.

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