0
votes
1answer
8 views

Convert from one format to other

Sorry for my lack of knowledge but I need help with, it seems, a basic math algebra. I want to know how they got this: $(1 - t)P1 + tP2$ from this: $P1 + t(P2 - P1)$ I did the math and they are ...
1
vote
0answers
10 views

Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
0
votes
0answers
29 views

To show an entire function is zero

$f$ is entire, satisfying $$|f(z)|\le \frac1{|\text{Re}(z)|}.$$ Show that $f$ is identically zero. It suffices to show boundedness. Since $f$ is bounded on the real line, then if $f$ is ...
-1
votes
0answers
14 views

If a conjugacy class intersect with its centralizer what can be said about its elements?

Suppose that $G$ is a finite group and let $x\in G$. If $y\in x^{G}\cap C_{G}(x)$, what can be said about the relationship of $x$ and $y$, or anything about $x$?
1
vote
1answer
21 views

Definition of length of a sequence

In definition $3.11$, the author defined the following rank: If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of ...
0
votes
0answers
6 views

Clarity on Boolean Algebra and Rings

I'm trying to wrap my head around Abstract Algebra, Boolean rings, and it's a little difficult. So I understand the ring (I believe it's a ring) <ℤ ,x, +, -, 0, 1 > is normal integer arithmetic ...
0
votes
3answers
34 views

How do you find the domain of this function?

$$f(x) = x^2 - 3x$$ My Working : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
0
votes
0answers
7 views

Application of the EGZ theorem

Given $r$ numbers $a_1,a_2,...,a_r$ and $n=qP$ where $P$ is the product of these $r$ numbers. $q$ is a natural number such that $q \geq 2$. Also given is a matrix $A$ of the following form: $$A=\...
1
vote
1answer
27 views

Find the shortest distance between tangent to the ellipse $x ^2\sin^2 \alpha+y^2=a^2$ and $z= x \tan \alpha$

Find the shortest distance between tangent to the ellipse $x ^2\sin^2 \alpha+y^2=a^2$ and $z= x \tan \alpha$ Ellipse is in $xy$- plane and line is in $xz-$ plane. When I write the equation of ...
0
votes
2answers
41 views

value of $\alpha$ in rational expression.

For all real values of $x\;,$ Given that $$\frac{4x^2+1}{64x^2-96x\sin \alpha+5}<\frac{1}{32}\;,$$ Then $\alpha$ lie in the interval $\bf{My\; Try::}$ We can write it as $$\frac{32(4x^2+1)-(64x^2-...
0
votes
2answers
14 views

Extending the single parity check code

I recently posted a question if it was possible to extend a code with odd minimum distance in some other way than the single parity check if you want to increment the minimum distance. This question ...
3
votes
2answers
415 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
0
votes
1answer
60 views

Find the limit $\lim_{x\to 0} x^{-3}\int_0^{x^2}\sin{(\sqrt t)}dt$

I use the fundemental theorem of calculus $$ \displaystyle\lim_{x\to 0}\frac{\displaystyle\int_0^{x^2}\sin{(\sqrt t)}dt}{x^3}=\frac{F_{(x^2)}-F_{(0)}}{x^3}="\frac{0}{0}" $$ Than apply L'hopital rule ...
0
votes
2answers
24 views

Which probability is greater, given minimal info

which probability is greater, given that $X$ and $Y$ are independant, positive random variables? There is also the option that it's impossible to know as we don't have enough information. I'd ...
0
votes
1answer
20 views

Expressing Tornheim sums in terms of Riemann's Zeta

If $$T(a,b,c)=\sum_{r\geq1}\sum_{s\geq1} \frac{1}{r^as^b(r+s)^c}$$ How to prove that : $$T(3,1,2)=-\frac13 \zeta(6)+\frac{\zeta^2(3)}{2}$$ I tried some algebraic manipulations but did not work. Can ...
-1
votes
1answer
19 views

If G and H are two gaphs then what does $G \Delta H$ indicate in graph theory?

I came across this notation in a book titled " Combinatorial Optimization Theory and Algorithms" by Bernhard Korte and Jens Vygen.
-4
votes
2answers
27 views

Is the following statement provable?

The statement is: $f$ is a real fucntion on $\mathbb R$. Then if $f'(x)=f(x)$ and $f(0)=1$, then $f(x)\neq 0.$
2
votes
2answers
58 views
11
votes
7answers
1k views

What does it mean when dx is put on the start in an integral?

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
6
votes
1answer
64 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$$ I believe that we must use the Dirichlet integral $$\int_0^{\infty} {\frac{\sin{x}}{x}}\ ...
5
votes
1answer
40 views

Preserving equality between different mathematical objects.

I'm taking an 'Intro to Higher Mathematics'-type course right now, were we learn about basic set theory, number theory, algebra, etc. and I had the following thought: Say you're trying to solve a ...
-1
votes
0answers
37 views

Trigonometry inequality 1

Let $n, r, s$ be the non negative integers such that $(\frac{1}{2})^2+(n+\frac{1}{2})^2=(r+\frac{1}{2})^2+(s+\frac{1}{2})^2$. Then prove that $\cos (\theta/2)-\cos(r+1/2)\theta\geq \cos(s+1/2)\theta- \...
0
votes
0answers
8 views

reference request on Wong Sequence

I am trying understand properties of matrix pencils and pair of matrices etc and I need to learn Wong Sequence , so could anyone tell me name of books, authentic papers, article to start with? Thanks
0
votes
1answer
50 views

Why bother showing $S^{1}$ covers itself?

I've just been introduced to covering spaces, and one of the examples I've been shown is that $p: S^{1} \to S^{1}$, $p(z)=z^{n}$ is a covering map for every $n$. My question is: why would you care? ...
0
votes
2answers
15 views

Convergence to supremum using Squeeze Theorem and Limit Theorem?

Suppose that $B$ is a nonempty set of $ \mathbb{R} $ that is bounded above. Let b = sup($B$). Prove there is a sequence $b_n \in B$ that converges to b. So I know we can use the Squeeze Theorem, and ...
0
votes
3answers
47 views

Non prime number test

Suppose $a\geq 3, n\geq 3$ are integers. I claim that if $gcd(a, n)=1$, then $a^n+n^a$ is not a prime. I am trying find a counter example but till now I did not reach there.
0
votes
1answer
21 views

Expected number of permutations required to sort a list of numbers

Given a list of N numbers if we are performing random permutations each time and checking whether the list is sorted, what will be the expected number of permutations required to sort that list?
0
votes
1answer
11 views

cylindrical shells method

I am really struggling with this cylindrical shells problem. i really do struggle with the rotating around the y axis. Find the volume of the solid that results by revolving the region enclosed by ...
5
votes
4answers
202 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
4
votes
0answers
24 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 $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$
5
votes
1answer
142 views

Number of zeros in Fibonacci sequences mod $p$

We know that Fibonacci sequences are periodic in mod $m$. For example, for $p\equiv \pm 1 \pmod 5$ and $\pm 2 \pmod 5$ the periods for Fibonacci sequences modulo $p$ divide $p-1$ and $2p+2$ ...
2
votes
2answers
30 views

Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
0
votes
0answers
19 views

Basic Limit Theorem for Markov Chain (Knowing the odds)

In the book "Knowing the Odds", Basic Limit Theorem for Markov Chain is stated as follows. Theorem 7.41 (Basic Limit Theorem). Suppose j is a recurrent aperiodic state in an irreducible Markov chain....
2
votes
1answer
134 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
3
votes
1answer
41 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
1
vote
0answers
13 views

Finding valuations/uniformizers for the branches of the blow up of a singular curve

I understand that for a nonsingular curve $C(x,y)$, the uniformizer at a point $(a,b)$ is either $x-a$ or $y-b$, since the partial derivatives with respect to $x$ and $y$ are not both 0. However, if ...
-7
votes
1answer
42 views

Is $x \mapsto x^3 - 2x$ an injection from $\mathbb{Q}$ to $\mathbb{Q}$ or not? [on hold]

As the title suggests, is $x \mapsto x^3 - 2x$ an injection from $\mathbb{Q}$ to $\mathbb{Q}$ or not?
-1
votes
0answers
21 views

sufficient reason for a function to be bijective

I know of course that an application $\Phi: A \rightarrow B$ is bijective if it is injective and if it surjective. I also know that for all bijective function, there exists an inverse. My question ...
1
vote
1answer
15 views

Primitivable and Riemann integrable discontinious function

Are there any known examples of functions that are both Riemann integrable and primitivable (they admit antiderivatives), but not continious?
6
votes
1answer
518 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 i'...
4
votes
2answers
122 views
+50

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ||x||\...
0
votes
1answer
38 views

Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
2
votes
1answer
332 views

A differential equation (nonlinear First-Order)

how to solve this equation: $(Px-y)(Py+x)=h^2P$ that $P=\frac{dy}{dx}$ and $h$ is a constant.
21
votes
10answers
1k views

Why is the complex plane shaped like it is?

So it's always taken for granted that the real number line is perpindicular 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 ...
0
votes
0answers
8 views

Customer problem in poisson process (two products)

Customers arrive at a shop according to a Poisson process at rate $\lambda$ (/minute), where they choose to buy either product $A$ (with probability p) or product $B$ (with probability $1-p$), ...
0
votes
0answers
22 views

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$?

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$? I am trying to understand what are in these two structures. My thought is that, if we look at the derivative of $X^p - t$, we ...
3
votes
0answers
37 views

Is quotient of open invariant subset open?

I am reading GIT book by Mumford. He needs special cases of the following conjecture several times. Conjecture Let $G$ be a reductive algebraic group acting on an irreducible affine scheme $X=Spec ...
0
votes
0answers
8 views

Solving $I^* = \arg\min_{I'} \left( \|\phi_\ell(I) - \phi_\ell(I')\|_2^2 + R(I') \right)$ with gradient descent

I am trying to create the results from this a paper that is trying to understand the types of features a convolutional neural network is learning to recognize. I don't think understanding ...

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