0
votes
1answer
42 views

Proving Existence of Field Extension

This seems so simple that it doesn't warrant a proof, but how would one prove this if asked to? If $K$ is a field and $f(x)\in K[x]$ is monic, then prove that there exists some field $L\supseteq K$ ...
2
votes
1answer
74 views

Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
3
votes
1answer
258 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
1
vote
2answers
104 views

Find the smallest integer x s.t. x congruent to 1 (mod 1,2,3,4,5,6,7,8,9,10)

Don't really understand this question. If this is asking to find an x for each mod then the answer would be just be x+m...If this is asking to find an x that satisfies all mods, then this cant be ...
0
votes
1answer
46 views

What is the equivalence class of a relation's element?

I'm studying about equivalence relations. My book has the following definition for an equivalence class: If $R=(G,A,A)$ is a relation of equivalence over the set $A$, the equivalence class of ...
0
votes
1answer
117 views

Is it possible to solve this equation?

Here is an equation that I found is quite impossible to solve without graphing or approximating the answer. $$\sqrt{x} = 1+\ln(5+x)$$ I tried squaring both sides and factoring the $ln$ out, but it was ...
1
vote
1answer
57 views

Is this function continuous at $(0, 0)$?

Suppose a function $f(x, y)$ is defined as follows like this: $f(x, y)=\frac{xy^3}{x^3+y^4}$ when $(x, y)\neq (0, 0)$ and $f(x, y)=(0, 0)$ when $(x, y)=(0, 0)$. Is this function continuous at $(0, ...
0
votes
1answer
666 views

Find an ordered basis $B$ for $M_{n\times n}(\mathbb{R})$ such that $[T]B$ is a diagonal matrix for $n > 2$

I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference. "Let T be the linear operator on $M_{n\times n}(\mathbb{R})$ ...
0
votes
0answers
2k views

The Conjugate Roots Theorem for Irrational Roots

The Conjugate Roots Theorem for Irrational Roots states that for a polynomial $f(x)$ with integer coefficients, if a root of the equation $f(x) = 0$ is expressed as $a+\alpha$, where $a\in\mathbb{Q}$ ...
0
votes
1answer
42 views

Help showing that this ideal is principal.

This is not for homework, and I would really like a hint please. The question asks If $P = \{ 2a + (1 + \sqrt{-5})b : a, b \in \mathbb{Z}[\sqrt{-5}] \}$ is an ideal in $\mathbb{Z}[\sqrt{-5}]$, ...
0
votes
2answers
46 views

Why this is false?!

$\Gamma \models (\alpha \vee \gamma)$ iff $(\Gamma \models \alpha$ or $\Gamma \models \gamma)$ [$\Longrightarrow$] If $\Gamma \models (\alpha \vee \gamma)$ then ($\Gamma \models \alpha $ or $\Gamma ...
0
votes
1answer
51 views

Proof of the limit of a sequence of functions.

The questions is this. Let $f_n(x)=\frac{1}{n}sin(nx).$ Each $f_n$ is a differentiable function. Show that (a) $\lim f_n(x)=0,\forall x \in R$ (b) but $\lim f'_n(x)$ need not exist [at $x=\pi$ for ...
3
votes
2answers
196 views

rational solutions of Pell's equation

1) $D$ is a positive integer, find all rational solutions of Pell's equation $$x^2-Dy^2=1$$ 2) What about $D\in\Bbb Q$ ?
1
vote
0answers
51 views

differentiable sets of an arbitrary function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an arbitrary function. I have proved that the set $E$ of points where $f$ is continuous is a Borel set. For $k\in \mathbb{N}$ and $p,q,r\in \mathbb{Q}$, ...
9
votes
2answers
175 views

How find this limit $\lim_{n\to\infty}a_{n}$

let $f(x)=x\ln{x} (x>0)$, and $f_{1}(x)=f(x)$,and such $f_{2}(x)=f(f_{1}(x)),f_{3}(x)=f(f_{2}(x)),\cdots,f_{n+1}(x)=f(f_{n}(x))$ Assume that the sequnce $\{a_{n}\}$ such $f_{n}(a_{n})=1$ ...
1
vote
1answer
34 views

real values for a function

We know that $f(x)=ax^2+bx+c=0$ has two real solutions when $b^2-4ac \geq 0$ My question is, if we have a function \begin{equation} f(x)=\frac {D\ln((-0.5\sec x-1)/k)}{\ln a}-\frac ...
1
vote
0answers
478 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
4
votes
1answer
879 views

Combinations: Poker hands, full houses

Reading through my Probability book brushing up on some stuff: What is the probability that a poker hand is a full house? (Full houses are three cards of one denomination and two cards of another ...
4
votes
1answer
315 views

Proving properties of Isomorphic groups

I just wanted to practice my proofs and my understanding of Isomorphic so I decided to prove the following if I am wrong or need a better argument for anything please feel free to let me know so I can ...
2
votes
1answer
66 views

on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

if $A$is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$. Also on ...
0
votes
2answers
617 views

Positive and negative integer that is congruent to 0 (mod 5) and incongruent to 0 (mod 6)

I'm kind of confused by this because I thought 0 mod 5 = 0, and 0 mod 6 = 0 as well. So what's an integer that is congruent to one but not the other?
0
votes
1answer
75 views

Uniform Convergence of Taylor Polynomial

Let F: [a,b]-> R be infinitely differentiable on [a,b]. Let $|F^{(n)}(x)| \leq C$, where $F^{(n)}(x)$ is the $n^\text{th}$ derivative of $F$, and $C$ is a nonnegative constant. Let $P_n(x)$ be the ...
4
votes
1answer
149 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
0
votes
0answers
170 views

Finding the first, last, minimal and maximal elements in these relations.

I'm now learning about relations of order. This is what I gather: First element: Precedes all elements Minimal elements: Have no predecessors. The first element is always a minimal. Last element: ...
1
vote
0answers
65 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
1
vote
0answers
183 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
2
votes
1answer
141 views

pigeonhole principle divisibility proof

Let n be some positive odd number, prove that there exists some positive integer k such that n|(2k-1), prove in terms of the pigeonhole principle
0
votes
0answers
29 views

Finding dual of incredibly complex LP; any trick?

This is homework, so only hints please. This is about a LP relaxation of the minimum cost perfect matching problem, with another constraint that shrinks the solution space in a way that a lot of ...
1
vote
0answers
39 views

Can any reduced residue system be generated by two polynomials?

Let $S_m=\left\{r_1,\ldots ,r_{\phi (m)}\right\}$ be a reduced residue system modulo $m$. The question is: For every $m$ does there exist two cuadratic polynomials $f,g\in \mathbb{Z}_m[x]$ such that ...
0
votes
0answers
116 views

Graph question. how to determine for each vertex v…

1.there is path to every other vertex 2. there is path to only at most 20 vertices my line of attack function(G): for every vertex v in G ...
-1
votes
1answer
475 views

Functions and trig question, finding minimum value

Functions $f(x)$ and $g(x)$ are shown below: $$f(x) = 3x^2 + 12x + 16,$$ $$g(x) = 2 \sin(2x - \pi) + 4$$ Using complete sentences, explain how to find the minimum value for each ...
1
vote
1answer
232 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
0
votes
2answers
358 views

Dirchlet Riemann Integrable in certain interval

Considering the Dirichlet function f : f (x) = { 1 if x is rational 0 if x is irrational } I want to know if this function can be Riemann Integrable in ...
0
votes
1answer
97 views

Images of Lines

I'm studying for this exam and one of the questions I am stuck on is: Find the image of the line $$3x-y+1 = 0$$ under the transformation $$z \mapsto \frac{2}{z+1}$$ So I know I have to convert the ...
1
vote
2answers
561 views

Probability of selecting four letters from ENCYCLOPAEDIA

From a maths textbook: "Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one letter E will occur in the selection of 4 letters." Clarification - as I ...
5
votes
2answers
344 views

What are some physical, geometric, or otherwise useful interpretations of divergent sums?

Is there any practical application to discussing the 'sum' of sequences that are not convergent under Cesàro summation? Why would we want to assign a value to an otherwise divergent sequence and ...
1
vote
0answers
66 views

Mirror images of knots and Kauffman and HOMFLY polynomial

Let $K$ is a knot and let $\bar{K}$ be the mirror image of $K$. I want to confirm this relationships. Let $f_K(t)$ be the Kauffman polynomial of $K$. To get the mirror image we swap every right ...
3
votes
1answer
140 views

What is the relationship between the spectrum of a matrix and its image under a polynomial function?

Clearly if $\lambda$ is an eigenvalue of $A$ then $p(\lambda)$ is an eigenvalue of $p(A)$ where $p$ is a polynomial. And there are cases where $A$ may have eigenvalues other than these. Is there a ...
0
votes
2answers
74 views

How to solve $(2x+2, 6x) = x+1$

I'm looking at an old discrete mathematics test preparing for my test tomorrow and one question says solve for $x$ and gives the above. I'm thinking that this is the $\gcd(2x+2, 6x)$ but have never ...
1
vote
2answers
261 views

Proof of geometric congruence using linear algebra

We may assume some set $T$ of all triangles within the same plane. Let $R$ be defined on $T$ where $a\ R\ b$ if the triangles $a, b$ are congruent. We may assume congruence to be defined as follows ...
2
votes
0answers
106 views

A group theory exercise in Brown's group cohomology book

My question is taken from exercise IV.3.4(b) in Brown's group cohomology book. Let $E$ be a finitely generated group, and suppose that $C$, the center of $E$, has finite index in $E$, say $[E:C]=n$. ...
1
vote
2answers
154 views

Proving a limit exists

Supposing that a function $g(x)$ is differentiable on the interval $(0,1]$, and $g'(x)$ is bounded on $(0,1]$: show that the limit of $g(1/n)$, as $n$ goes to infinity, exists. My initial guess was ...
6
votes
0answers
81 views

The theorem on formal functions

Recall the Theorem on Formal Functions [Hartshorne, III.11.1] Let $f:X \to Y$ be a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$ and let $y\in Y$. Then the ...
1
vote
1answer
74 views

Find a quantity that is constant along the solutions of the ODE

I have the following problem. Find, for any $r > 0$, a quantity that is conserved along the trajectories of the system $\ddot{x} = r - e^x$ How do I exactly solve this question? I had a ...
2
votes
2answers
3k views

Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
1
vote
1answer
165 views

Expected value of a number of events

If I have a set of random events say ${\{A_{1}, A_{2}, ... A_{n}\}}$ and a number ${N=\sum_{1}^{n} I(A_{i})}$ where ${I(A_{i})}$ is the indicator function (basically ${N}$ is the number of events that ...
1
vote
2answers
53 views

Group theory: Let $H=\{0,\pm 3, \pm 6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.

I am having trouble understanding the following homework question, Let $H=\{0,\pm 3, \pm6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$. I know the answer is $H$, $1+H$, and $2+H$ but ...
0
votes
1answer
41 views

help with nand circuit

I tried to make a circuit from this expression but it's not working right. Here's the expression and circuit: Expression ...
1
vote
2answers
97 views

Determination of the prime ideals lying over $2$ in a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let ...
0
votes
2answers
201 views

Why is the sequence $ a_n = \left(1+\frac{1}{n}\right)^n $ Cauchy?

I was looking at the post: Cauchy Sequence that Does Not Converge And the top answer was this sequence: $ a_n = \left(1+\frac{1}{n}\right)^n$. I understand that this sequence converges to $e$, ...

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