1
vote
1answer
50 views

Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have ...
0
votes
1answer
16 views

What is the involvement of the Gaussian primes in a positive integer being expressed as the sum of two squares?

I am going through a fairly comprehensive sketch of a proof of the following theorem: A positive integer $n \in \mathbb{Z}$ may be expressed as a sum of two squares $n = x^2 + y^2$ for $x,y \in ...
1
vote
2answers
27 views

Countable set can be listed in a sequence

Let $S$ be a countable set, i.e. exists bijection between $\mathbb{N}$ and $S$. Why elements of $S$ can be listed in a sequence? EDIT: I guess that bijection is not crucial. It's sufficient that ...
1
vote
0answers
5 views

Please verify the solution about Brownian motion process.

Problem Let $Y(t)$ denote the amount of time by which the racer is ahead when $100t$ percent of the race has been completed. $\{Y(t), 0 \leqslant t \leqslant 1\}$ is modeled as a Brownian motion ...
0
votes
5answers
46 views

First Year Calculus Problems

I'm doing a few past exam papers for my calculus test. I came across a few problems which I thought would be worth asking about. 1.) Consider the function $f(x) = \begin{cases} x+1, & \text{if ...
0
votes
0answers
8 views

An extemal combinatorial design question. “Weak” steiner stystems.

A Steiner system $S(t,k,\nu)$ is a collection $X$ of $\nu$ points and a collection of subsets of $X$ of size $k$ (frequently called blocks) such that each $t$ element subset of $X$ occurs in exactly ...
2
votes
1answer
34 views

Prove this diophantine equation $2^a-3^b=5~,a,b\in N^{+} $ has no postive integers solution

show that the diophantine equation $$2^a-3^b=5~~~~,a>5,b>3,a,b\in N^{+} $$ has no postive integers solution maybe is old problem,But I try somedays,can't solve it by now
1
vote
2answers
29 views

Parameters leading to an elementary integral

For which values of $a,b$ the following integral is an elementary function, and which elementary function? $$\int \frac{x^2+ax+b}{(x-1)^2}\,e^x\, dx$$ I tried to solve this integral but it ...
0
votes
2answers
33 views

Can you “solve” any equation with 2 variables?

Here on Stack Exchange Mathematics are many questions about "How can you solve this equation?". That made me wonder if you can solve any equation with two variables. What I mean with "solve" is not ...
0
votes
0answers
18 views

Finding the roots of this multivariable polynomial?

My polynomial is this ten term monster $P(x,y,z) = 6561 x^3+486 x^2+12 x+6561 y^3+1944 y^2+192 y+6561 z^3+6318 z^2+2028 z+223$ It's simplest form is ${1 \over 81} \left( (81x+2)^3 + (81y+8)^3 ...
0
votes
1answer
22 views

Inverse trigonometry problem2

Let $\sin^{-1}x + \sin^{-1}y=\sin^{-1}(x \sqrt{1-y^{2}}+ y \sqrt{1-x^{2}})$,then what is the area represented by the locus of point (x, y)? I'm totally blank about this question so please explain ...
4
votes
1answer
22 views

Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$

Matrix of linear operator $\mathcal A$:$\mathbb R^4$ $\rightarrow$ $\mathbb R^4$ is $$A= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1\\ 1 ...
1
vote
2answers
24 views

Proving that things are or aren't rational numbers

My question comes in three parts: Suppose $x,y\in \Bbb Q$. Prove that $2x-5y\in \Bbb Q$ Prove that $3^{1/2}\not\in \Bbb Q$ Suppose $x\in \Bbb Q$. Prove that $x^2+3^{1/2}\not\in \Bbb Q$ In the ...
0
votes
3answers
22 views

How to setup this triple integral

Let $D$ be the region in $\mathbf R^3$ that lies above the surface $z=x^2+y^2-1$and below the surface$z=\sqrt{1-x^2-y^2}$. Sketch the region of $D$ together with its projection on the $XY$-plane ...
3
votes
3answers
29 views

How to solve trig equations and get all the solutions using graphs, $\cos(2x-\pi/3)=\cos(x)$

The question is to solve $$\cos\left(2x-\frac{\pi}{3}\right)=\cos(x)$$ I originally approached this using the addition formulae but the mark scheme showed a way by first replacing $x$ on the right ...
4
votes
2answers
237 views

A condition that implies two numbers to be coprime?

Consider the natural number $n=2^r\cdot s+1$, where $s$ is odd and suppose $a\in \mathbb N$ with $1<a<n$ is such that: $\exists j\in \mathbb N: j<r \wedge a^{2^j\cdot s}\equiv -1 \pmod n$. ...
0
votes
0answers
10 views

Prove if $G$ and $H$ are graphs on the same vertex set, then $dg(G∪H)≤dg(G)+dg(H)$

Prove if $G$ and $H$ are graphs on the same vertex set, then $dg(G∪H)≤dg(G)+dg(H)$ $dg(G)$ is the the minimum k such that $G$ is k-degenerate. I know it can be proved with respect to graph coloring, ...
4
votes
2answers
68 views

Periodic functions problem

What is the period of $(2007)^{\sin x}$? Please explain how to proceed and what's the technique to generally solve these kind of problems.
1
vote
1answer
12 views

Help to prove an expression about sums of binomials coefficients using Complex Power Series theorem.

I'm solving some exercises from Kreszig's Advanced Math book and I got stuck in one: (10th ed, chapter 15.3, problem 18): Using $(1+z)^p*(1+z)^q=(1+z)^{p+q}$, obtain the basic relation: ...
1
vote
3answers
38 views

Parametrization of the sphere and the torus.

Is there a way to find easily the parametrization of the sphere and the tore ? I see on wikipedia that for the sphere it's $(x,y,z)=(\sin \theta\cos \varphi,\sin\theta\sin\varphi,\cos\varphi)$ with ...
0
votes
1answer
29 views

Can you prove the following formula for hypergeometric functions?

I wanna prove the following identity for big values of $N\gg 1$ $$ {}_3F_1\left(-N+1,1,1;2;-\frac{1}{N}\right)\to\frac{1}{2}\bigg({}_2F_1\left(1,1;2;1-\frac{1}{N}\right)+\log 2+\gamma\bigg) $$ where ...
2
votes
1answer
42 views

Show that $a+b+c=1$ implies $\exists x, y \in \{a-ab, b-bc, c-ca\}$ so that $x \leq \frac{1}{4}$ and $y \geq \frac{2}{9}$

Let $a, b, c$ be three positive real numbers such that $a + b + c = 1$. Prove that among the three numbers $a − ab, b − bc, c − ca$ there is one which is at most $1/4$ and there is one which is at ...
1
vote
1answer
28 views

Why aren't CDFs left-continuous?

Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous: Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be ...
1
vote
1answer
24 views

coordinate geometry in polar coordinate

Let $G= \{(x, f(x)) \mid x \text{ lies between } 0 \text{ and } 1 \}$ Let $(1,0)$ belong to $G$. It is given that tangent vector to $G$ at any point is perpendicular to radius vector at that point. ...
0
votes
0answers
20 views

find maximum of $| f(z) |$ in n $[0,2\pi] \times [0, 2\pi]$ for $f(z)=1+\sin^2(z). $

Find the maximum of $|f(z)|$ in $[0,2\pi] \times [0, 2\pi]$ for $$f(z)=1+\sin^2(z). $$ It is obvious to use maximum principle to do that. But I find the expression of the $|f(z)|$ is very ...
0
votes
0answers
7 views

Mixed convection - Matlab

So I am trying to solve the following mixed convection problem in Matlab: for $x=0: u=v=0, T=Th$ (heated wall); for $x=L: u=v=0, \frac{\partial T}{\partial x}=0$; for $y=0: u=v=0, \frac{partial ...
0
votes
0answers
21 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
3
votes
1answer
33 views

A subset of the set of numbers of seven non zero digits

Someone visits me knowing that I'm indisposed for now. Courteously, brings me a problem without imagining it is on my “bête noir”, Combinatorics. I post it with its solution, 151200, I think am ...
-2
votes
1answer
35 views

How to eliminate the $\delta$ function? [on hold]

I have the following equation: $$f(\omega)\delta(\omega+\omega^{\prime})=a(\omega)b(\omega^{\prime})\delta(\omega+\omega^{\prime})$$ Then which of the following is true? ...
0
votes
1answer
43 views

Inverse trigonometry problem

What is the value of $\cos(\tan^{-1}(\tan 2))$ ? Am I thinking correct? $\tan 2$ is negative so $\tan^{-1}$ and $-\tan 2$ cancel each other....giving $\cos(-2)$ ... which finally gives the answer as ...
-1
votes
0answers
40 views

Combinatorics :: In how many ways can ten boys and four girls sit in a row?

(1) (a) In how many ways can ten boys and four girls sit in a row? $$= 14!$$ (b) In how many ways can they sit in a row if the boys are to sit together and the girls are to sit together? ...
0
votes
4answers
34 views

Chinese remainder theorem/Fermat's little theorem problem

Prove that $p^4 \equiv 1 \pmod {240}$ for any prime $p>5$. I'm not sure how to go about this at all - I started with some computation to check it works for $\{7,11,13\}$ and they all end in $1$ - ...
0
votes
0answers
12 views

On characterization of MRE estimators

I have some touble with understand the second equality in the proof of theorem 6; Using the lemma we can just plug in $\delta_{0}-v$ and minimize over that w.r.t $v$, but howcome we have the ...
0
votes
1answer
12 views

What is the effect of applying toeplitz matrix on both sides of + semi-def matrix

I have a positive semi-def matrix $\mathbf{A}$ that has an eigen value decomposition $\mathbf{A}=\mathbf{V\Lambda V^T}$. I have a real toeplitz matrix $\mathbf{Q}$. Can I say anything definite about ...
1
vote
0answers
26 views

Varieties are isomorphism

Let $f:Y\to X$ be a birational morphism, Y is projective. Let $H$ be a very general ample divisor on $Y$. If $f^{*}f_{*}H=H$, is it true that $Y$ is isomorphic to $X$?
2
votes
1answer
50 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
5
votes
2answers
98 views

My conjecture $\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$ [duplicate]

$$\int_{0}^{1}{x^n-1 \over \ln(x)}dx=\ln(n+1)$$ Let deal with case $n=1$ $$I=\int_{0}^{1}{x-1 \over \ln(x)}dx=\ln(2)$$ $u=\ln(x)$ $\rightarrow du=\frac{1}{x}dx$ $x \rightarrow 1 ,u=0$ $x ...
1
vote
2answers
30 views

Irreducibility Over $Q[n]$

Prove that the $q(x)=X^2+3$ is irreducible over $Q[\sqrt{2}]$. Is MY proof Correct? Proof. Since it is a polynomial of second degree it factors iff it has 2 polynomials of degree 1.Since we are ...
0
votes
2answers
70 views

Are these independent probability events? Why or why not?

Suppose company {X} represents clients in buying inventory during automated (online) auctions. The client chooses the bid value beforehand, and {X} repeatedly makes that bid in the automated auctions ...
7
votes
3answers
253 views

Combinatorial formulas and interpretations

I found that $$ \sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a ...
1
vote
1answer
24 views

Find $v$, $a_1$ or $a_n$ in an arithmetic series

Following Data are given in an arithmic series: S$_{25}$=1275 (and so n = 25) When trying to find other elements of the formula's i have learned so far: $a_n = a_1 + (n-1)v$ $s_n = n \cdot ((a_1 ...
0
votes
0answers
8 views

Equality of Quotients of Probabilities from Combinatorics

Sorry the title is so vague - I don't know how else to ask this. Essentially, what is a real-life example showing why: $\frac{_aC_k}{_nC_k}=\frac{_aP_k}{_nP_k}, where \ a<n$ is true?
2
votes
0answers
18 views

The integral is the area under the curve. Is there a similar notion for stochastic integrals?

As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ...
2
votes
1answer
19 views

Showing Intermediate Value property and closed preimage implies continuity

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying the Intermediate Value property. Assume that for any $y \in \mathbb{R},$ the preimage $f^{-1}(\{y\})$ is closed. Prove $f$ is continuous. ...
-1
votes
2answers
22 views

The position of a point mass that moves in a straight line .. / Determine the units

The position of a point mass that moves in a straight line is given by $x(t) = At^2 + Bt + C$, where $t$ is time. Determine the units of $A$, $B$ and $C$. The answer to the question is [A] = ...
1
vote
2answers
52 views

If $|a|= \infty$, then $<a>$ is infinite subgroup

I'm trying to understand the proof of: Let $G$ be group and $a \in G$. If $|a|= \infty$, then $<a>$ is infinite subgroup. The proof I have here goes like: Let $|a|=\infty$. Then $a^i ...
0
votes
1answer
32 views

Integral $\int_{1}^{\infty}\frac{-4}{(2 \cos{x} - 2) x^3}\ \mathrm dx.$

I'm not sure how to proceed with the following integral: $$I=\int_{1}^{\infty}\frac{-4}{(2 \cos{x} - 2) x^3}\ \mathrm dx.$$ Mathematica could not find a closed form solution for it and I really have ...
0
votes
2answers
28 views

Polyhedron with 12 pentagons and 1 hexagon

In this answer http://mathoverflow.net/a/19823/5239, it is indicated that it is impossible to make a polyhedron (with 3 faces meeting at each vertex) out of 12 pentagons and 1 hexagon. There is ...
0
votes
0answers
24 views

Gaussian process for machine learnig

Here is my question in the equation 2.11 A is N by N matrix, so there is not feasible if N is large the textbook say in the euqation 2.12, we only need to invert size n by n. But I think K is 1 by ...
4
votes
2answers
183 views

What is an “algèbre augmentée sur un corps?” (EGA I)

In EGA I, Chapter 0, (1.1.10), Grothendieck is giving examples of terminal objects in different categories. He says "dans la catégorie des algèbres augmentées sur un corps $K$ (où les morphismes sont ...

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