9
votes
2answers
607 views

A slightly problematic integral $\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$

Question. To find the integral of- $$\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. Any help?
-1
votes
1answer
28 views

what is the ordered triple of postive Integers, ABC = 2104000 [on hold]

What is the ordered triple of positive integers (a,b,c) satisfy abc = 2104000 Sorry but I do not know anything about this problem.
1
vote
1answer
13 views

Are cyclotomic polynomials irreducible modulo a prime?

I am actually interested on cyclotomic polynomials of the form $x^n+1$ (i.e. $n$ is a power of 2, for large $n$). Are these polynomials irreducible modulo a prime $q$? I already saw that $X^2+1$ is ...
0
votes
2answers
16 views

How many five-digit numbers are there that have number 4 as at least one digit?

How many five-digit numbers are there that have number 4 as at least one digit? How to do this? I don't know how to start.
41
votes
12answers
5k views

Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someome ...
1
vote
0answers
18 views

Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
0
votes
0answers
10 views

Regarding pedigrees

Here's a errrhmm.... a modelling problem. Applied math. Or something. We're given pair of sets $X,Y$ (for notational simplicity, assumed disjoint). Let $T_{X,Y}$ denote the following disjoint union:...
0
votes
0answers
20 views

Covering map of classifying space

We know that for any $m\in\mathbb{N}$ the map $p_m:S^1\to S^1$ is an $m$-sheeted covering of $S^1$. Suppose that $BG$ is the classifying space of an arbitrary group $G$. Does there exist such a map $...
1
vote
1answer
27 views

Geometric interpretation of the coefficients of the quadratic equation.

The quadratic equation has three general forms: $ax^2+bx+c$ $a(x-r_1)(x-r_2)$ $a(x-h)^2+k$ $r_1$ and $r_2$ are the zeroes of the quadratic. $h$ is the horizontal position of the ...
1
vote
2answers
42 views

Shouldn't $t^n : \mathbb{A}^1 \rightarrow \mathbb{A}^1$ ramifies at $0$?

Yo, this is probably the stupidest question ever that I've asked here. Let $$\varphi: \mathbb{A}^1 \rightarrow \mathbb{A}^1$$ be the map of schemes (over a field $k$) such that $\varphi (x) = x^n$. ...
0
votes
0answers
14 views

Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
0
votes
0answers
7 views

Aproximate values of holomorph function by identity principle

Let's say I have two sequences of complex numbers $\{a_n\}_{n \in \mathbb{N}}$ and $\{b_n\}_{n \in \mathbb{N}}$ such that $a_n \to a$, and $b_n \to b$, and I construct a complex function $f$ by saying ...
1
vote
2answers
253 views

Stopping times, Filtration, Martingales,

I am new here and I have a question. Definition: Let $ \tau$ be a stopping time, then $\mathcal F_{\tau}=\left\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \mathcal F_{n}\...
0
votes
0answers
7 views

Nonlinear first-order differential equation with periodic bounded solution

Let $f:\mathbb R\times \mathbb R\longrightarrow\mathbb R$ be a $C^1$ function, periodic in the first variable, and such that $f(t,1)\leq > 0\leq f(t,0)$ for all $t$. Consider the ...
0
votes
0answers
3 views

linear recursive bi-sequence closed form

Let $(u_{n,m})_{n,m}$ a bi-sequence of real numbers, and we assume that the sequence is recursive : $$ u_{n+1,m}=\sum_{k=0}^r a_k u_{n-k,m}\qquad \forall m\geq 0\\ u_{n,m+1}=\sum_{l=0}^p b_l u_{n,m-l}\...
-2
votes
2answers
49 views

is cos(iz) injective?

I had an exam, and there was a question - is cosh(z) injective? I presented it as cos(iz) (or cos(-iz), doesn't matter because it's an even function), but it didn't ...
2
votes
0answers
25 views

A problem about the supremum of countable stopping times

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with a countable filtration $F=\left\{ \mathcal{F}_0,\mathcal{F}_1,\cdots,\mathcal{F}_n,\cdots \right\}$. $\left\{ T_n \...
0
votes
0answers
3 views

From Bezier Curve basis to B Spline basis functions

Bezier basis functions can be determined using recursion: $B_{i,p} = (1-t)B_{i,p-1}+tB_{i-1,p}$ So for a quadratic bezier basis, we get: $1-2t+t^2$ $2t-2t^2$ $t^2$ So for a quadratic bezier ...
2
votes
0answers
34 views

How to find all the integral solutions of $x^2-by^2=z^k$ where $k$ is an odd integer >2 and $b>0$?

Consider the bivariate quadratic polynomial of the form: $$ x^2-by^2=z^k$$ where $k$ is an odd integer>2 and $b>0$. It's well-known that Euler's method: $$x^2-by^2=(p^2-bq^2)^k $$ provides a class ...
2
votes
0answers
17 views

Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book: Аssume that $f$ satisfies $\nabla f(x) \ge 0$ for all $x$, but is not nondecreasing, i.e., there exist $x,y$ with $x < y$ and $f(y) < f(x)$. By ...
1
vote
2answers
15 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
0
votes
1answer
8 views

How to find the linear application given the function on the basis vectors.

Say I am given a linear application $f$ from $R^2$ to $R^2$, and I am told it maps $e_1$ to $(1,3)$ and $e_2$ to $(-2,7)$ In this case I know how to find how the linear application acts on a generic ...
0
votes
0answers
8 views

Approximating $L_n[1/3, 1.92]$ for GNFS

Approximating the RHS of $T(n) = L_n[1/3, 1.92]$ Perhaps related to this earlier question on the cost of running the GNFS, I am looking for an approximation for solving equations of this form, when $...
2
votes
1answer
17 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
4
votes
1answer
30 views

What is the name for a function that behaves symmetrically when its arguments are scaled?

In other words, is there a name for this property of a function $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$
0
votes
2answers
47 views
+50

There are 4 nickle coins and 4 half nickle coins. How many different options are there for the sum of 5 coins.

I have this exercise in combinatorics: In a drawer there are 4 nickle coins and 4 half nickle coins, bob takes out from the drawer 5 nickles, how many different options are there for the sum of ...
4
votes
0answers
54 views

I am looking for a mathematical equation to warp an image [on hold]

Theoretically, I know that to warp an image, each pixel $(x,y)$ in the source image is transformed to $(x', y')$ using a function f (i.e. $x'=f(x,y)$ & $y'=f(x,y)$ ). But what mathematical ...
1
vote
2answers
25 views

Comparison of two slopes to get the best value

I'm trying to get something in excel that may be solvable by coming up with the proper formula. I believe it's a comparison of two slopes. The theory comes an economies of scale: How can I spend the ...
2
votes
4answers
234 views
+50

Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like ...
1
vote
1answer
25 views

Can a circuit in graph theory use a edge twice in the circuit

I need to find out a circuit in a graph that uses the edge ab, what I want to know is can the circuit use the edge ab, more than once.
0
votes
1answer
25 views

Find the number of tangent lines to a curve

The number of tangent to curve $$y^2 - 2x^3 - 4y + 8 = 0$$ that passes through $(1,2)$ My work Assuming tangent touch the curve at$(x_1,y_1)$ $$\frac{dy}{dx}=\frac{3x^2}{(y - 2)} $$ $$\frac{2 - y_1}{...
2
votes
0answers
15 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
1
vote
0answers
37 views

Construction of Lebesgue measure in Rudin's RCA book

This theorem from Rudin's RCA book. Here's one moment from it's proof which seems to me very weird. Rudin states that equality $\lambda(E)=m(E)$ holds for all Borel sets. But I think that it's ...
4
votes
0answers
56 views

Is there a concept of |x|<0?

I'm just curious, did any of the famous mathematicians consider |x|<0? Would the zero have to be then not the additive identity and what else would it be then? (assuming you could build a field ...
1
vote
0answers
13 views

Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
9
votes
2answers
323 views

What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

What is the derivative of $$x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$$ My effort: Let $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}\implies g(x)=x!^{g(x)}$$ Taking natrual log on ...
6
votes
2answers
257 views

Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
2
votes
1answer
31 views

Series Converges Pointwise but not Uniformly

Consider $\sum_{n=1}^{\infty}\frac{x^2}{n(n+x^2)}$ on $[0,\infty)$. To show pointwise convergence, I used $\frac{x^2}{n(n+x^2)}\leq\frac{x^2}{n^2}$ on $[0,\infty)\implies\sum_{n=1}^{\infty}\frac{x^2}{...
1
vote
3answers
42 views

Solving $\frac{x + 2}{\sqrt{1-\frac{3}{x}}}=12$

This is my attempt: $$\frac{x + 2}{\sqrt{1-\frac{3}{x}}}=12$$ $$\implies\sqrt{1-\frac{3}{x}}=\frac{x+2}{12}$$ $$\implies 1-\frac{3}{x}=\frac{x^{2}+4x+4}{144}$$ $$\implies -\frac{3}{x}=\frac{x^{2}+4x-...
3
votes
1answer
86 views
+50

Milnor's definition of smooth manifold

In Milnor's book "Topology from a differential viewpoint" on page one he defines a smooth manifold to be a subset $M \subset \mathbb R^n$ which is locally diffeomorphic to some open subset of $\mathbb ...
0
votes
3answers
43 views

Number patterns types

Write an expression in terms of n for the nth term in the following sequence $9,16,25,36,49$ The difference is $+ 7 , + 9 , + 11 , + 13, + 15 , + 17 ,$ etc The difference is not constant so it's ...
0
votes
1answer
12 views

How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $ C^1 $. How is the class related to ...
0
votes
3answers
292 views

Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
-3
votes
1answer
36 views

Can you solve this geometric question on triangles?

In a triangle $ABC$, $D$ is a point on the side $BC$.Given: $AD=10$,$BD=DC=8$ and $BC*AD=6$.What is the length of $BC$? a.$5$ b.$10$ c.$15$ d.$20$ That was asked in a newspaper quiz.
1
vote
0answers
19 views

A few questions about relations over finite sets.

I am trying to describe some property of relations and to figure out, whther it was studied before. I know, that a set with functions is called an algebra. Is there a name for a set with relations (...
25
votes
1answer
2k views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
0
votes
1answer
26 views

Splitting a file into $m$ pieces of size $1/n$, such that any $n$ pieces allow you to recover the file?

Let's say we have a file (which we could define as a finite sequence of 0's and 1's (or any other two symbols)). For $m > n$, can you create $m$ pieces (which are themselves files), each $\frac 1n$...
1
vote
1answer
24 views

Computing the distribution function

I have troubles solving this task: Let $U_1,U_2,\dots$ be an i.i.d. sequence of random variables with uniform distribution on $[0,1]$. We set for every integer $n\geq 1$ $$M_n=\max\{1/\sqrt{U_1},\...
2
votes
0answers
13 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
1
vote
2answers
56 views

Why does the normal distribution describe data collected in real life so well? [on hold]

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?

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