2
votes
0answers
59 views

Complex structure of HyperKaehler manifold

Let $X$ be a hyperKaehler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
7
votes
2answers
2k views

Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
1
vote
0answers
61 views

A convex polynomial inequality.

Let $\phi(x)$ be a convex polynomial of degree $m$ at least two. Note that for $x,q \in \mathbb{R}$ $$\phi(x) + \phi(q) - 2\phi(\frac{x+q}{2}) = ...
4
votes
0answers
79 views

Split short exact sequences and the associated graded algebra

Let $G$ be a group and $K$ be a field. Let $K(G)$ be the group ring and $I$ be its augmentation ideal. You are given that there exists a section of the projection $\pi_s:I^s\to I^s/I^{s+1}$ for all ...
1
vote
3answers
84 views

When is the set statement: (A\B)⊕(A ∩ B) = A true?

"When is the set statement: (A\B)⊕(A ∩ B) = A true? Is it sometimes true, never true, or always true? If sometimes, state the specific cases where it is. A & B are arbitrarily selected ...
2
votes
1answer
56 views

Question on tensor calculation in Reimannian geometry

Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We ...
2
votes
2answers
173 views

What is a necessary and sufficient condition of that $A$ has $n$ linearly independent eigenvectors?

If $A$ is an $n$-by-$n$ matrix with complex entries, (i.e., $A\in M_n(\mathbb{C})$,) $A$ must have $n$ eigenvalues, counting algebraic multiples. But it is not always true that $A$ has $n$ linearly ...
3
votes
1answer
1k views

linear dependence and coplanar vectors

I am confused about the coplanarity of vectors, and the relation of coplanarity to linear dependence. If I have real vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$, with $\mathbf{w}$ a linear ...
1
vote
2answers
102 views

Using the limit point property in metric spaces to prove existence of maximum distance.

Let $M$ be a metric space and suppose that $K \subset M$ is a non-empty compact set. So if $p$ is any element of $M$, then there is a point $q$ that belongs to $K$, such that $d(p,x)\leq d(p,q)$ for ...
0
votes
2answers
84 views

some question on polynomial function

Let $f,g$ be polynomials. There are two questions : (1) If $f(n)\geq 0$ for all $n\gg 0$ and $n$ integer, then the leading coefficient of $f$ is a positive. (2) If $f(n)=g(n)$ for all $n\gg 0$ and ...
5
votes
2answers
179 views

Does $\mathbb{Z}_5(\sqrt[3]{3})$ make sense? Or, can we always extend a field by a root of a reducible polynomial?

I'm preparing assignment questions for a course in ring/field theory. We'll shortly be looking at extension fields, and the students are meant to understand what notation such as $\mathbb{R}(i)$ ...
3
votes
3answers
777 views

Give algebraic and geometric descriptions of the $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$

Give algebraic and geometric descriptions of $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ where $a_1 = (1, -1, -2), a_2 = (3, -3, -1), a_3 = (2, -2, -4), a_4 = (2, -2, 1)$ So far, I have: $$ ...
0
votes
5answers
187 views

When is the set statement: (A⊕B) = (A ∪ B) true?

"When is the set statement: (A⊕B) = (A ∪ B) a true statement? Is it true sometimes, never, or always? If it is sometimes, state the cases where it is." How would you go about finding the answer ...
4
votes
1answer
212 views

I calculated $\lim_{x\to 0} {\sin x\over x}$ as 0.01745 instead of 1

When using a graphing utility to generate a table, I got 0.01745 as the limit rather than 1. Where did I go wrong?
0
votes
1answer
50 views

Can standard Linear Programming algorithms return all valid solutions without losing their efficiency?

I have a (generalized) Linear Programming problem to solve. I anticipate exactly two equally valid optimizations of my objective function. I would be happy if I could receive both these points; it ...
0
votes
2answers
455 views

Need Homework Help: A small corportion borrowed $500,000, some at 9%, 10% and 12%. Use a system of equations--how much was borrowed at each rate if…

A small software corporation borrowed 500,000 cash to expand its software line. The corporation borrowed some of the money at 9%, some at 10%, and some at 12%. Use a system of equations to determine ...
4
votes
1answer
721 views

Chebyshev Polynomials

I am trying to prove a something regarding Chebyshev polynomials. Given the polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined by $$\begin{cases} T_0(x) = 1\\ T_1(x) = x \\T_n(x) = ...
3
votes
1answer
204 views

Asymptotic analysis Big O Big Omega

When we have $F(n) = \Omega(H(n))$ and $G(n)=\mathcal{O}(H(n))$. Can we prove that $G(n)/F(n) = \mathcal{O}(1)$? I tired to use the definitions of $\mathcal{O}$ and $\Omega$ but all I ended up with ...
3
votes
3answers
482 views

When is the set statement: (A ∪ B) ⊆ (A ∩ B) true?

"Where A is an arbitrary set and B is an arbitrary set, when is the statement: (A ∪ B) ⊆ (A ∩ B) true? Is it true all the time, sometimes, or is it never true? If it is sometimes true, explain ...
2
votes
1answer
251 views

Find the order of a group G from its presentation

Suppose G is a group defined by the presentation $G=\langle u, v\mid uv^2=v^3u, ~u^2v=vu^3\rangle$, is $G$ finite or infinite? If it is finite, what is its order? In general, I want to know whether ...
0
votes
1answer
88 views

Numerical Analysis Interpolation

I have a question that I would just like a little bit of clarification about. Find a and b, 0 < a 1, 0 < b 1 such that max x is element of [−1,1] |(x + b)(x + a)(x − a)(x − b)| = max x is ...
4
votes
1answer
80 views

If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.

THEOREM: If $ \text{ran } F \subseteq \text{dom } G $ then $\text{dom }(G \circ F)= \text{dom }F$ PROOF: if $ F\subseteq A \times B$ and $ G\subseteq B\times C$ then by definition ...
0
votes
3answers
95 views

Calculating Operating Costs to give a fixed profit margin

We are importing sales data from a company we just purchased. We want to calculate the logistics value so that the profit margin is 13%. We have the following rules in our system: ...
1
vote
1answer
723 views

Geometry problem calculate arc, inner triangle.

I learned how to solve this in geometry in middle school, but I can't remember. This isn't homework. I want to calculate the distance between points A˜B and the ...
3
votes
2answers
140 views

Show that $c^{\varphi(m)/2} \equiv 1 \pmod{m}$ if $m$ has two odd prime divisors

The following problem is one of the exercises in Topics in the Theory of Numbers (Erdős et al.) Show that if the positive integer $m$ has at least two distinct odd prime divisors, and $c$ is ...
3
votes
2answers
93 views

Understanding Multiplication in a Linear Algebraic Group

I am reading Springer's Linear Algebraic Groups and have a question about how ordinary group multiplication ( in the ordinary group theory sense) translates to that in term so linear algebraic groups. ...
4
votes
2answers
771 views

Does $\sqrt{x}$ have a limit for $x \to 0$?

I am taking a calculus course, and in one of the exercises in the book, I am asked to find the limits for both sides of $\sqrt{x}$ where $x \to 0$. Graph for sqrt(x) from WolframAlpha: This is how ...
1
vote
2answers
232 views

Function with a Modular Inverse

For a combinatorics problem I have a function, $h(x)$ that is always divisible by five, but it is calculated in pieces, e.g. $h(1) = 43 + 7$. The final function that I need is $f(x) = (h(x) / 5) ...
0
votes
2answers
565 views

Markov Chain Reach One State Before Another

I'm stumped on a problem. Here's my transition matrix: $$P = \begin{bmatrix} \frac{3}{4}&\frac{1}{4}&0&0&0&0&0 \\\\ ...
1
vote
1answer
383 views

Bisect an angle

I have a simple question, and I've looked over the internet and can't find a numerical way of doing it. I only found how to do this using a ruler and a compass, which I can't use, since I'm doing a ...
8
votes
1answer
333 views

Sequence of convex functions

If a sequence of convex functions $\{ f_n \}$ on [0,1] converges pointwise to a continuous function f, then is the convergence uniform? The questions is almost identical to this one, except that the ...
2
votes
1answer
612 views

Oscillation of a continuous $f$

The original theorem is a as follows: THEOREM Let $f$ be continuous on $[a,b]$. Then for any $\epsilon>0$ there exists a finite partition of $[a,b]$ such that the oscillation of $f$ on each ...
2
votes
1answer
261 views

How to solve this linear homogeneous diff equation when A is not constant

I know how to solve the differential equation $\dot{x} = A x$ when $A$ is a constant $n\times n$ matrix. However, I cannot solve the problem when $A$ also depends on $t$. To be more specific, $A ...
3
votes
3answers
141 views

Calculating mean after removing smallest items

I'm having a tough time wrapping my head around this question. Lets say that I am doing an experiment where I roll 10 dice. Each time I roll the dice, I record the average value and repeat the ...
4
votes
2answers
151 views

Playing with plane curves

Let $\phi: \mathbb{R}^1 \longrightarrow \mathbb{R}^2$ be the map given by $t \mapsto (t^2,t^3)$. I'm trying to show that any polynomial $f \in \mathbb{R}[X,Y]$ vanishing on the image $C = ...
2
votes
2answers
81 views

Maximum and minimum ratio of matrix calculation

Suppose you have a matrix : $$A = \begin{pmatrix} 13 & -4 & 2 \\ -4 & 13 & -2 \\ 2 & -2 & 10 \end{pmatrix}.$$ I want to find the maximum and minimum values of the ratio ...
0
votes
1answer
359 views

Whats the probability that a certain amount of batteries life longer than 3.25 hours?

The mean and standard deviation of the lifetime of a battery in a portable computer are 3.5 and 1.0 hours respectively. So, what is the probability that the mean lifetime of 25 batteries exceeds 3.25 ...
1
vote
1answer
177 views

How to prove that a function f(n) exists/belongs to bigTheta?

So as per the title, I'm trying to prove that a function $f(n) = n^2 + 8n$ exists in $\Theta (n^2)$. What I'm having trouble with is the logic/concept behind doing so. By definition, it would mean ...
0
votes
1answer
673 views

finding the significant digits for relative error

How exactly do you go about finding the number of significant digits? From what I've found I am suppose to find t where relative error (Re) $ \le$ 5*10^-(t) But I don't understand how you find t. ...
1
vote
1answer
151 views

Bijection between the set of classes of positive definite quadratic forms and the set of classes of quadratic numbers in the upper half plane

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$. Let $\alpha = \left( \begin{array}{ccc} ...
0
votes
1answer
82 views

Contracting an angle (using straightedge and compass)

In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable. Shouldn't this be $\frac{\pi}{n}$ instead of ...
1
vote
2answers
805 views
0
votes
1answer
57 views

Paracompactness and Filters

If one use ultrafilters to describe a compact space, one gets Tychonoff Theorem as a trivial result. So, im just asking if there is such a "useful" equivalence, but concerning paracompact spaces ( ...
5
votes
1answer
423 views

Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$

Following my previous question the book then asks Use this to determine when the field extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$ is biquadratic (where $a,b\in\mathbb{Q}$) ...
0
votes
1answer
65 views

what is the fourier series of $x(at)$ when $a = 0, a>0$ or $a<0$? How do I go about solving this?

Given Fourier series co-efficient of $x(t)$ is $X$; how do I go about solving fourier series of $x(at)$ when $a = 0$ ? I can easily get to the step where I insert $x(0)$ into the Fourier coefficient ...
2
votes
1answer
246 views

Showing that the Poisson process is characterized by five properties

Klenke defines the Poisson process as a family of non-negative integer valued random variables with independent increments, whose increments are distributed according to the Poisson distribution ...
4
votes
2answers
114 views

“$f$ is differentiable at $x_0$” implies…

Just making sure I understood: $$\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$$ At a first glance I didn't understand why the above is true. It's because (in the case above) we can say that ...
5
votes
4answers
174 views

Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $

Prove that $$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $$ I'm trying induction, this is what I have so far: Basecase $(n=6): 64<720<729$ is true. Inductive Case: Assume ...
3
votes
2answers
104 views

Inequality. $(n+1)\left(a^{n+1}+b^{n+1}\right) \geq (a+b)\left(a^n+a^{n-1}b+\ldots+b^{n}\right). $

Let $a$ and $b$ be positive numbers, and $n \in \mathbb{N}$. Prove that (using Rearrangement Inequality) $$(n+1)\left(a^{n+1}+b^{n+1}\right) \geq (a+b)\left(a^n+a^{n-1}b+\ldots+b^{n}\right). ...
5
votes
2answers
205 views

Easy highschool number theory equation

I have some problems with my homework. We have two integers $a$ and $b$ which satisfy the equation $a^b - b^a = 1008$. Show that $a$ and $b$ have the same remainder when divided by $1008$.

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