0
votes
0answers
17 views

Points of contacts of tangents of the curve $y=\sin x$

Prove that the points of contacts of tangents of the curve $y=\sin x$ drawn from origin lie on the curve $1/x^2 -1/y^2 = -1$
1
vote
1answer
29 views

Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$.

Let $I = [0,1]$ and $χ_I : \Bbb R → \Bbb R$ be the characteristic function on $I$. Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$. ...
1
vote
0answers
21 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
0
votes
0answers
19 views

An elementary residue problem

Given $A,B\in\Bbb N$ with $|A-B|=o(\min(A,B))$ how to find $C,\psi\in\Bbb N$ such that $|\psi-A^6|=o(A^6)$, $|\psi-B^6|=o(B^6)$, $|C-A^5|=o(A^5)$, $|C-B^5|=o(B^5)$ holds with ...
6
votes
1answer
66 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
2
votes
4answers
129 views

Squaring both sides when units are different?

Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true? $((3) \text{inches}) = ((0.5) \text{yards})$ $((9) \text{inches}) = ((1.5) ...
1
vote
0answers
23 views

Proving a compact Lie group admits a biinvariant metric

At the end of a lesson in Differential Geometry, my teacher said: Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica ...
2
votes
2answers
27 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
0
votes
1answer
17 views

System of functions

How can I solve this system? $$ \left\{\begin{matrix} 8x^3-6x+6xy^2=0\\ 4y^3-4y+6xy^2=0 \end{matrix}\right. $$ I do $$ \left\{\begin{matrix} 8x^3-6x+6xy^2=0\\ 4y^3-4y+6xy^2=0 \end{matrix}\right. ...
2
votes
0answers
13 views

About a perfect FN-group

the consept perfect group is new for me. I was reading the following: If a group $G$ is a FN-group and $G'=G$ then $G$ is finite. I tried to prove this and here is my attempt: let $G$ be a ...
1
vote
1answer
40 views

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$ and $\frac{\alpha}{\beta}$,$\frac{\beta}{\alpha}$ are the roots of $x^n+1+(x+1)^n=0$, then $n$ is Odd Even ...
1
vote
2answers
20 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
0
votes
1answer
14 views

How to solve a boundary value problem of a Laplace equation?

Suppose $x,y$ are in the range $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 1$, I can use separation of variables to get $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ...
0
votes
0answers
19 views

How to find the Cartesian equation of a plane in this example (in details)? [on hold]

I'm solving an A Level paper, and came across this question. Basically, they have given plane $p$ has the equation $(\mathbf r-3\mathbf i)\cdot(2\mathbf i-3\mathbf j+6\mathbf k)=0$. Now, I can see ...
4
votes
2answers
23 views

Pair of friends and a pair of “enemies” in each group of three students

The problem: There is a class. In each group of three students in the class there is a pair of friends and a pair of "enemies". Find the maximum number of students in the class. I tried to play with ...
0
votes
1answer
35 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
-1
votes
0answers
18 views

Is $ \lfloor {\log(n)} \rfloor!$ or $ \lfloor {\log(\log(n))} \rfloor!$ polynomially bounded? [on hold]

Which of these is is polynomially bounded: $ \lfloor {\log(n)} \rfloor!$ $ \lfloor {\log(\log(n))} \rfloor!$ I think both are but I can't prove it.
0
votes
0answers
6 views

What it means for a generalized function to be periodic or radially symmetric??

Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric. A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in ...
0
votes
0answers
17 views

If and Else of time in Matlab [on hold]

I just want to ask a matlab code (if-else statement) of time. Example is if 7AM-5PM output is 1 .. the rest would have an output of 0. Thanks
-2
votes
1answer
19 views

Infimum and supremum of two variable function [on hold]

How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$? Thanks
0
votes
0answers
16 views

Field of definition of an Ideal

I am trying to prove the following statement from Introduction to commutative algebra and algebraic geometry by Ernst Kunz p,16 Q9 Let $I$ be an ideal of the polynomial ring $K[X_1,X_2,...,X_n]$ over ...
3
votes
1answer
33 views

In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
0
votes
1answer
25 views

Conditional Proof in Trigonometry

If $\sin\theta + \sin\alpha=m$ and $\cos\theta + \cos\alpha=n$, prove that: $$\frac{\sec(\theta+\alpha)}{2}=\frac{\sqrt{m^2+n^2}}{2}$$ My attempt\ given: $$\sin\theta+\sin\alpha=m$$ $$2 ...
-2
votes
0answers
36 views

mathematician who changed the world [on hold]

I’m interested to know about mathematician and mathematical problems that changed the world. For example enigma machine + code braking, if you know amazing stories like this please comment them below
0
votes
0answers
20 views

probability,calculus

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
1
vote
1answer
32 views

Finding the First Derivative ( 1 question)

Using the Definition of a limit: [ Of form $\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$] Find $f'(x)$ when $x=9$ for $f(x)=\frac{2}{\sqrt{x}}$ I tried simplifying it but got jumbled when trying to multiply ...
0
votes
2answers
23 views

Prove that all three metrics induces the same topology on $X_1\times X_2$

Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way: ...
0
votes
1answer
17 views

Deriving formula for externally tangent circle to internally tangent circle

($x^2+(y+1)^2=R^2$ should say $x^2+(y-1)^2=R^2$) I am trying to derive a formula for the radius of the circle that is externally tangent to the internally tangent circles of the quarter-circle, and ...
1
vote
0answers
16 views

Relating to representation of real numbers. [on hold]

Can someone tell me which representation is better for representing real numbers: fixed point representation or floating point representation? If the answer is circumstance dependent, please specify ...
1
vote
0answers
6 views

Is integral curve a embedded 1 dimensional submanifold of the given manifold?

I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or ...
1
vote
0answers
20 views

Path to self learning Calculus

I am currently self-learning Calculus from an old and cheap edition of Calculus (by James Stewart) which contains both single and multivariable calculus.I am aware that Stewart's book is regarded as ...
0
votes
3answers
26 views

Is every bounded sequence of random variables in $L^1$ convergent? [on hold]

If $\{X_n\}_{n>0}$ is a bounded sequence of random variables is it true that $E(X_n)$ converges?
0
votes
0answers
9 views

Matlab code date and time controller

I am struggling a lot in making the code for my controller. I successfully made the code wherein the input is the time and day of what my computer has but what I need next will be the controller in ...
0
votes
0answers
25 views

How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : ...
0
votes
1answer
15 views

Trigonometry Proving

If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$ Attempts: Here $\sin\theta + \sin\alpha=x$ ...
0
votes
1answer
17 views

what is non trivial basis for cofinite topology on non empty set $X$ [on hold]

what is non trivial basis for cofinite topology on non empty set $X$??? when $X $ is infinite set.
0
votes
0answers
3 views

Is it possible to work out each one of these variables if it is the only unknown? $d=c_1c_2\ln(\cosh(t/c_2))$

I have an equation that defines $d$: $$d=c_1c_2\ln(\cosh(t/c_2))$$ It is very simple to work out $c_1$ if it is the only unknown: $$c_1={d\over c_2\ln(\cosh(t/c_2))}$$ Each variable is a real ...
2
votes
2answers
27 views

Find bases of the kernel and image

Find the rank and the nullity of the following linear map $T : U \to V$ , and find bases of the kernel and image of $T$. $U = \Bbb R^4 , V = \Bbb R^4$, $$T(α, β, γ, δ) = (α − γ, γ − δ, α − β, β − ...
0
votes
5answers
41 views

find the area of a kite with integration

A stunt kite has the shape in the diagram below: How can I find the area using calculus integration. Can anyone help me start this question, I am not looking for the full answer. I assume I only ...
4
votes
3answers
91 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
-2
votes
0answers
18 views

Does the Riemann Hypothesis consider mirror symmetry on its non-trivial zeros?

Setting the bottom corners of the square 1 on the center of two intersected circumferences and taking as center of symmetry the center of that intersection, it's possible to project the square 1 ...
0
votes
1answer
13 views

Surface Integral of $3z^2 d\sigma$

Let $S$ be the bounded surface of the cylinder $x^2+y^2=1$ cut by the planes $z=0$ and $z=1+x$ Then how to show that the value of the surface integral $∬3z^2 d \sigma $ over $S$ is equal to ...
1
vote
2answers
38 views

motivation for the direct limit [on hold]

I know just the very basics on Category Theory and that's why I'm going to ask a stupid question. I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books ...
0
votes
0answers
5 views

Ellipse set with one fixed focus, co-tangential at origin

Find equation of an ellipse tangential to x-axis at origin and whose one focus is fixed at P $ (-a,-b), $ another is variably placed at Q $ (a\, m, b \,m).$
0
votes
0answers
6 views

Modelling the ballot theorem as a martingale.

The page 19 in the link http://www.imada.sdu.dk/~jbj/DM839/FL15.pdf provides the explanation of what a ballot theorem is and how we can prove that it is a martingale. It takes a random variable $S_k$ ...
0
votes
1answer
17 views

If P(i) is true for all integers i with 2≤i≤k as inductive hypothesis, then why also p(t) is true by the inductive hypothesis?

"Let P(n) be the property n is divisible by a prime number. We prove that P(n) is true for all integers n with n> 1. Basis step. If n=2, then P(n) is true because 2 is a prime and every ...
0
votes
1answer
24 views

Is there any identity for this series?

While solving inequality and finite series problem I often come across this series- $$(n+1)(n+2)(n+3)...(n+n)$$. Is there a general solution to this form of a series? Thanks for any help!!
0
votes
1answer
46 views

Conjecture $\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}…$

$$I=\int_0^{1}\frac{{(\rm arcsin})^2({x^2})}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}{\pi^3}-\frac{\pi}{2}ln^2{2}-2{\pi}\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ This result seems to me digitally ...

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