0
votes
0answers
6 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
0
votes
1answer
12 views

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$. I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure ...
-2
votes
0answers
7 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences.

Let $\begin{align*} A_0 &= 6 <br/> A_1 &= 5 <br /> A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$ There is a unique ordered pair $(c,d)$ such that ...
0
votes
1answer
12 views

Is there a divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
2
votes
1answer
8 views

Limit of non-linear multi-variable function

I'm trying to prove the limit of the following function is $0$: $\lim_{(x,y) \to (1,-1)} {x^3} - {2xy^2} + 1$ I know that I'm trying to find a $\delta$ s.t $ 0 < \sqrt{(x^2 - 1) + (y^2 + 1)} < ...
0
votes
0answers
10 views

Isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ such that lines become circular arcs

I'm currently working on a hobbyist math project that require taking lines on an infinite plane, and projecting them onto a finite (euclidean) surface such that intersections are preserved. Does ...
1
vote
0answers
25 views

(un)Intentionally funny titles of mathematical works

There are many mathematical/scientific works with very concise and relevant titles, but only few can be called 'funny' in some people's perspectives. For example, Fourier Transformation for ...
-1
votes
1answer
5 views

A 57“ arm has a vertical pin at one end, if the pin angle changes by .020” how much does the other end of the arm move?

this is prob pretty simple but I can't remember how to do it so a simple formula would be appricated.
0
votes
1answer
9 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
0
votes
1answer
10 views

How to integrate $1/\sqrt{(1+x^2)^3}$?

Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem. Is anyone able to assist me with ...
0
votes
0answers
7 views

If $\mu$ is a semifinite measure, every set of infinite measure contains a subset of arbitrarily large, finite measure

I need help in this question. Thank you. If $\mu$ is a semifinite measure and $\mu(E)=\infty$, for $C>0$ there exists $F\subset E$ with $C<\mu(F)<\infty$.
0
votes
2answers
15 views

Why $|x-y|<1\implies|y|\leq |x|+1$?

I have the following passage in one of the proofs in my workbook: $$|x-y|<1\implies|y|\leq |x|+1$$ Why is this valid?
0
votes
1answer
22 views

Simple Trig Integration. Why is my answer wrong?

$$\int \dfrac{\cos x+\sin 2x}{\sin x}dx=\int \dfrac{\cos x+2\cos x\sin x}{\sin x}dx=\int \dfrac{\cos x\left(1+2\sin x\right)}{\sin x}dx$$ Substitute $u=\sin x$ and $du=\cos x\ dx$: ...
0
votes
2answers
9 views

How many additional crews should be brought in to minimize the cost of an oil spill?

An oil spill has fouled $200$ miles of Pacific shoreline. The oil company responsible has been given $14$ days to clean up, after which a fine will be $10000$ \$/day. The local cleanup crew cleans $5$ ...
0
votes
1answer
14 views

Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
1
vote
1answer
10 views

What does the spectrum of the Grothendieck ring of varieties look like?

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation ...
0
votes
0answers
7 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
0
votes
0answers
18 views

proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let n ≥ 0 and k ≥ 0 be integers. 1) How many bitstrings of length n + 1 have exactly k + 1 many 1s? 2) Let i be an integer with k ≤ i ≤ n. What is the number of bitstrings of length n + 1 that have ...
0
votes
1answer
13 views

Multiplying brackets in $n(n+1)/2+n+1$

Why does: $$n(n+1)/2+n+1 = (n^2+3n+2)/2 $$ and not $$ (n^2+2n+1)/2 $$ ? Additionally, why is: $$(n^2+3n+2)/2 = ((n+1)(n+1)+1)/2$$ rather than: $$((n+1)(n+1)+1n)/2$$
1
vote
3answers
20 views

Determine a matrix knowing its eigenvalues and eigenvectors

I read through similar questions, but I couldn't find an answer to this: How do you determine the symmetric matrix A if you know: $\lambda_1 = 1, \ eigenvector_1 = \pmatrix{1& 0&-1}^T;$ ...
-2
votes
2answers
18 views

prove of topology and metric spaces

Prove or disprove $f: A \to B$ a function from $A$ to $B$. $A_i$ subset of $A$ and $B_i$ subset of $B$. If $A_0 \subset A_1$ then $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
0
votes
0answers
15 views

Expected time until beating an initial try

Consider the following problem: Let $X,X_1,X_2,...$ be i.i.d. random variables. We execute the following experiment. One samples $X$. Then, one samples $X_1$,$X_2$ and so on until the first time the ...
-1
votes
2answers
13 views

Prove that if y>1, then ∀M∈R, there exists an N in the natural numbers s.t. n≥N implies (y^n)>M.

For y in the reals, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N$ in the natural numbers such that $n≥N \implies y^n>M.$ I'm not used to proving these kinds of questions so ...
0
votes
0answers
5 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
0
votes
0answers
26 views

What's “a little computation” here?

I tried to ask the other person in a comment but he's not replied so I'm asking this here: How did he arrive at this formula $$2\pi i f(w) = \int_{\partial D} \frac{f(z)}{z-w}\,dz + \iint_D ...
1
vote
4answers
36 views

Why is it ok to factor an equation with no limit so it has a limit?

I'm just starting out in calculus, so please bear with me if this is not a sensible question. In the book I'm reading, the author gives the example of the problem of finding the limit of ...
0
votes
0answers
19 views

Is there a formalism for a universal mathematical representation of algorithms?

I don't know if my question is correct so excuse me if I'm not 100% clear about what I would want to know. Is there a formalism which can capture all possible algorithms (mathematically speaking) ? ...
3
votes
1answer
45 views

How important is the own talent for research of your PhD supervisor?

Currently I am in the process of finding a PhD. Some potential supervisors are more didactical than others, some are nicer and warmer than others, and some are more famous mathematicians than others. ...
0
votes
0answers
9 views

Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for the minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
0
votes
0answers
15 views

Angle quadrisection in a triangle

In triangle ABC, AB=84, BC=112, and AC=98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is ...
0
votes
3answers
12 views

Finding a point on a curve where the tangent is parallel to another line noted

Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$ I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks
1
vote
3answers
23 views

What is a conventional name for a set of values having no properties except that values are distinct?

I know essentially nothing about math but I'm interested in very low-level concepts. I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe ...
0
votes
1answer
14 views

Inverse of a product in a group can be wriiten as the product of the inverses of each element in reverse order

Let $(G,\circ)$ be a group and let $g_1,...,g_n\in G, n\in\aleph$. Prove that $(g_1\circ ...\circ g_n)^{-1}=g_n^{-1}\circ ...\circ g_1^{-1}$ I tried this by induction but was unsure how to take out ...
0
votes
2answers
10 views

Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
0
votes
0answers
8 views

How can I know the stiffness

How can I formulate this system? I would like to know the stiffness of this system. $$\begin{cases} x'_1=-0.01x_1+10^6x_2x_3 \\ x'_2= 0.01x_1-10^6x_2x_3-5\cdot10^{7}x_2^2 \\ ...
-5
votes
1answer
19 views

Use De Morgan's Laws to simplify the following sets [on hold]

Simplify the following sets: $$ℝ\setminus \bigcup\limits_{n=1}^∞ (-1/n,1/n)\tag1$$ $$\bigcup\limits_{n=1}^∞ (ℝ\setminus[1/n,2+1/n])\tag2$$
0
votes
0answers
6 views

How to calculate recurrence $F(n) = F(n/u) + \Theta(n^k)$ where $u,k \in \mathbb{N}$

$\Theta$ is used as in Bachmann-Landau notation (often called as Big-O notation convention). How does one in general the recurrence relation of the following from: $$F(n) = F(n/u) + \Theta(n^k) ...
0
votes
0answers
18 views

Find basis of subspaces

I don't know how to create basis of V1 and V2. If I want to prove M1^2=M1, do I need to find matrix representation of M1 first? Thanks!!!!!!
1
vote
0answers
15 views

How to find the orthogonal of a vector space

Let $V$ be a vector space over a field $F$ equipped with a symmetric bilinear form $B$. Let $W$ be a vector subspace of $V$. I know that we define the orthogonal complement $W^\bot$ to be ...
1
vote
1answer
8 views

How to get from the statement $(AB'+C'A'+C'B')$ to equivalent statement $(AB'+C'A')$?

I've been working a Boolean algebra problem for probably 2 hours at this point, and while I arrive at a much simplified equivalent expression, there's a simpler one yet. Basically, I start out with a ...
0
votes
0answers
9 views

Finding the probability density function of $U=Y_1+Y_2$

Let $(Y_1,Y_2)$ denote a random sample of $n=2$ from the uniform distribution function on (0,1). (1) Find the probability density function $U=Y_1+Y_2$ (2) Find $E(U)$ I am unsure of how to bound ...
3
votes
2answers
42 views

Is every arrangement reachable by shuffling this way?

Suppose we have a vertical stack of $n$ distinguishable coins, each of which is either heads-up or tails-up. Let a shuffle be the following procedure: divide the stack at will into a top- and ...
2
votes
2answers
23 views

Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can ...
3
votes
1answer
53 views

The Purpose of Master Thesis

I am posting this question in the aftermath of the earlier posting in this link. Here are what I would like to know more about master and PhD thesis: (1) I understand that schools have different ...
0
votes
1answer
29 views

$\mathrm{lcm}(b,c)$ from $\mathrm{lcm}(a,b)$ and $\mathrm{lcm}(a,c)$

Given that lcm$(a,b)=60$ and lcm$(a,c)=270$, find lcm$(b,c)$ I believe you're supposed to use the rule lcm$(a,b)=p_1^{\text{max}(r_1,s_1)}\cdots p_m^{\text{max}(r_m,s_m)}$ Here's my work so far: ...
0
votes
1answer
10 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
2
votes
1answer
9 views

Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
2
votes
1answer
13 views

Difference between necessary and necessary but not sufficient?

This is from Discrete Mathematics and its applications I read up on necessary and sufficient from here What is the difference between necessary and sufficient conditions? If p->q (p implies ...
0
votes
0answers
5 views

Groups of order 36 - another step in lemma 5.4.

This is a follow up to my question last night Groups of order 36 where I was confused about the first step of Lemma 5.4 of http://matwbn.icm.edu.pl/ksiazki/fm/fm92/fm9211.pdf. I am now confused about ...
0
votes
1answer
10 views

Strong Topology and Strong Operator Topology on Hilbert Space

Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong ...

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