0
votes
0answers
2 views

Proof of Simson line theorem using barycentric coordinates.

Is there anybody who knows how to prove Simson line theorem using barycentric coordinates only?
0
votes
0answers
3 views

Find expected step, on which half-normal RV exceeds a scalar value?

I have defined a following problem. Given is a non-negative integer variable (steps) $s\in[0,1,...)$, and a scalar random variable as a function of $s$ $R(s)$. Random variable is half-normally ...
1
vote
0answers
8 views

simultaneous diagonalize

$A = \begin{pmatrix} 18 & -9 \\ -9 & 9 \end{pmatrix}$ $B = \begin{pmatrix} 3 & 2 \\ 2 & -2 \end{pmatrix}$ Find a real invertible matrix such as $P^tAP = I_2$ and $P^tBP$ is diagonal ...
0
votes
0answers
4 views

Finding the packing radii and the covering radii

Lets say I have $C=\{00000, 01101, 10110, 11011\}$ I know the covering radii is given by $e=\frac{1}{2} (d-1)$ So in this example $e=1$. Regarding the covering radius, this is the definition I have. ...
0
votes
0answers
6 views

Expressing a solution to a differential equation in a more compact form

We know that $Ae^{ikx}$ is a solution for $k \geq 0$ and $Be^{ikx}$ is a solution for $k \leq 0$. So does the $N$ here depend on whether or not $k \geq 0$ or $\leq 0$? Also I do not understand how ...
0
votes
0answers
3 views

Proving statement for a tree-graph theory

So i need help with this: Let T be a tree. And degree of every vertice is an odd number. So i need to prove that there is an odd number of paths in that tree. So i basically need to prove that there ...
0
votes
0answers
5 views

Compactly generated complete lattice is Heyting Algebra

Let $L$ be a lattice. $a\in L$ is said to be compact if whenever $a\leq \bigvee A$ for some $A\subseteq L$, then there is some finite subset $B\subseteq A$ such that $a\leq \bigvee B$. A lattice is ...
0
votes
1answer
17 views

is the followng function $f$ surjective?

$f$ is a function from plane $V$ to $V$ mapping $x$ axis to plane $V$ defined by if $P(x,0)$ then $f(P) = (x,x^2)$ ? I am not so sure for my following answer : to show that $f$ is surjective, for ...
0
votes
0answers
5 views

prove that a non constant linear functional from a normed linear space X is discontinuous if and only if Z(f) = {x in X | f(x) = 0} is dense in X.

I could prove that if Z(f) is dense in X then the functional must me discontinuous but I am not able to prove the other way round
1
vote
1answer
28 views

when $\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k$ exist?

When does $$\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k\ \ ?$$ My first way : Since $$\sum_{k=0}^n\binom{n}{k}x^k=(1+x)^n$$ the limit exist when $x\in ]-2,0]$ et it's limits is $0$ when $x\in ...
0
votes
1answer
17 views

Lebesgue Measure of $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$

Let $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$ and $m_2$ Lebesgue Measure of $\mathbb{R}^2$. I want to determine $m_2(A)$. So. I know that Lebesgue Measure of interval is b-a. And ...
0
votes
0answers
6 views

Torus Cylinder intersection

Two surfaces in 3D .. a Torus and a Cylinder with parametrizations respectively as: $$ \{ b + a \cos(u) ) \cos(v), (b + a \cos(u) \sin(v), a \sin(u )\}, $$ $$ \{a + b \cos(p), b \sin(p) ...
0
votes
2answers
22 views

Finding square of cube?

A cube is built using $64$ cubic blocks of side one unit. After it is built, one cubic block is removed from every corner of the cube. The resulting surface area of the body (in square units) after ...
0
votes
0answers
6 views

Calculating the residue of $\frac{1}{f(z)}$ at the simple pole $z=z_0$.

I need to show that the residue of $\frac{1}{f(z)}$ at it's simple pole $z=z_0$ is $\frac{1}{f'(z_0)}$. I have tried using the residue theorem together with Cauchy's integral theorem but not really ...
1
vote
1answer
12 views

Is there an hamiltonian path on a $4 \times 4$ chessboard

If you have a $4 \times 4$ chessboard: Is it possible to make a Hamiltonian graph such that each step is like a move of the knight?
0
votes
0answers
4 views

The w*-extension of a bounded linear functional

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ ...
0
votes
0answers
13 views

probability combination

McGyver is faced with the problem of opening a safe with 10 buttons numbered from 0 to 9. The safe can be opened by pressing three buttons, not necessarily distinct, in correct order. Realizing that ...
0
votes
1answer
22 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
0
votes
0answers
7 views

Curvature of a curve and curvature form

Is there any relation between the curvature of a curve defined in a classical way as the reciprocal of the radius of the osculating circle in a point and the curvature form on Riemannian manifolds?
1
vote
0answers
11 views

Show that $\Sigma_n^0 \not= \Pi_n^0$ holds for every $n \in \mathbb{N}$ (arithmetical hierarchy)

I want to know, how I can prove $\Sigma_n^0 \not= \Pi_n^0$ in the arithmetic hierarchy. For $\Sigma_1^0$ it is easy, since the diagonal halteproblem would be one example. But how to prove it for all ...
0
votes
0answers
19 views

In order to show or refute: Given a real function $f$ and $a, b \in R$ then $a\leq b \Rightarrow f(a)\leq f(b)$, what should I regard?

Is it enough to show that $f$ is increasing or decreasing in any interval $I$ that contains both numbers $a$ and $b$?
0
votes
0answers
12 views

approximation of a trigonometric sum

I have a trigonometric sum as below $$\sum_{r=0}^{N-1}\frac{\sin^2(\frac{\pi(Ne-e-r+n))}{N})}{\sin^2(\frac{\pi(r-n+e))}{N})}$$ and I want to show analytically that for small $e$ ($e<0.2$) and large ...
0
votes
0answers
7 views

How to handle $y=0$ values when using logarithmic $y$ scale?

Imagine as example the following 2d data: x = (1, 2, 3, 4, 5); y = (4, 3, 2, 1, 0); I know that $\log(0)=-\infty$ but somehow I would like to "make visible" also ...
1
vote
0answers
8 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
1
vote
0answers
36 views

How many solutions are $ a+b+c+d = 30 ,(a\leq b\leq c\leq d) $?

I would appreciate if somebody could help me with the following problem: Q: How many solutions are there to the equation $$ a+b+c+d = 30 ,(a\leq b\leq c\leq d) $$ where $a,b,c,d\in ...
0
votes
1answer
9 views

Well-ordering and proper subsets

I'm trying to show: $\forall m, n \in \omega (n < m \iff n \subset m)$ I have shown the forward direction, but I'm confused for the reverse. I have stated $n \subset m \iff \forall z(z \in n ...
0
votes
0answers
25 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
-1
votes
0answers
27 views

Area of an isosceles triangle

Let us consider an isosceles triangle $\triangle AOC$ if $AO=OC$ And $OB$ and $AG$ are altitude then, $$\psi=OC\sqrt{\frac{GC\cdot OB}{2}}$$ Where $\psi$ indicates area of triangle And now i have ...
1
vote
0answers
15 views

How to get an index of element in vector?

Suppose we have a vector: $$a=(a_1,...,a_n)$$ What is the simplest way to define (using mathematical notation) a function which returns an index of given element? Example: $$a=(10,20,30,30)$$ ...
0
votes
0answers
5 views

Why can I plug the roots of a partial derivative of a linear optimization objective E into E without changing it?

As an example, to fit a line to 2D data $\boldsymbol x_i$ with the parameters $\theta = (a\;\;b\;\;c)^T$ with the normal equation $\langle \boldsymbol x, ...
-1
votes
0answers
13 views

Finding distribution of $Y$ and $P(Y>1)$

$X$ is a random variable which has normal distribution with mean $4$, variance $9$ and $Y=3X-8$. What is the distribution of the random variable $Y$? How should I find $P(Y>1)$?
0
votes
0answers
21 views

show that the Power series solve the differential equation

$$f(t)=\sum\limits_{k=0}^{\infty}\frac{ (-1)^k\,(t/2)^{2k}}{(k!)^2}$$ show $t\, f''(t)+f'(t)+t\, f(t)=0$ By differentiating it, I got $$f'(t)= \sum\limits_{k=0}^{\infty}\frac ...
0
votes
0answers
14 views

Solve recurrence equation $T(n) = 9T\left(\frac{n}{18}\right) + 3T\left(\frac{n}{9}\right) + T\left(\frac{n}{3}\right) + n^\frac{3}{2}$

I need to solve the following equation $$T(n) = 9T\left(\frac{n}{18}\right) + 3T\left(\frac{n}{9}\right) + T\left(\frac{n}{3}\right) + n^\frac{3}{2}$$ The depth of the recursion tree is $log_3n$. I ...
0
votes
1answer
16 views

Compact space with a discrete subspace

I'm looking for an example (or a proof of nonexistence) of a compact space with discrete and uncountable subspace.Thank you for all your answers.
0
votes
0answers
15 views

Gauss-Jordan method - why does it work? [on hold]

I was wondering why the Gauss-Jordan method works and how to come up with that method.
0
votes
4answers
67 views

Factoring $x^4 - 16$

I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$. Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$?
1
vote
4answers
39 views

To prove $\sum_{n=0}^\infty \binom{r}{x}\binom{N-r}{n-x}=\binom{N}{n}.$

To prove $$\sum_{x=0}^n \binom{r}{x}\cdot \binom{N-r}{n-x}=\binom{N}{n}.$$ I tried comparing the coefficients of $(1+x)^{(n+k)} = (1+x)^n(1+x)^k$ but couldn't reach the answer.
0
votes
0answers
12 views

In a sweepstakes giveaway scenario, how does having 2 chances to win the same prize affect the overall odds?

In a sweepstakes giveaway scenario where total entries are expected to result in final odds of 1:93,150.685 for/against a single entrant (after adjusting for multiple entries) and can be won by either ...
2
votes
0answers
26 views

what exactly is arc length element $ds$ or area element $dA$

I am reading a book on complex analysis and it has something like: The spherical arc length element on the Riemann sphere ($S^2$) works out to be $ds=\frac{|dz|}{1+|z|^2},$ and the spherical area ...
0
votes
1answer
11 views

Volume of the body around z- axis

Let $F:=\left \{ (x,0,z) : x \geq 0, x^3 \leq z\leq 8\right \}$ and K is body, which is made by rotation of F around z - axis. How can I calculate volume of K?
0
votes
1answer
16 views

Matrix representation of Lie Algebra $B_2$

I'm writing some practical examples where to calculate the Killing form, the Cartan Matrix, Dynkin diagrams etc. Does anybody have on or two nice matrix representations of the $B_2$ Algebra? It would ...
0
votes
0answers
27 views

Fourier Series in Functional analysis

Would you please solve this question? I really have problem with this kind of questions.
0
votes
0answers
6 views

Small roots of multivariate polynomials

I'm looking for a program/algorithm (or even "theory"?) which checks if a given multivariate polynomial, say $P(x_1, \dots, x_n)$ has a root in some given region, say a closed $\varepsilon$-Ball ...
0
votes
0answers
20 views

Equation to the circle.

How to show that the equation to the circle of which the points $(x_1,y_1)$ and $(x_2,y_2)$ are the ends of a cord of a segment containing an angle $\theta$ is, $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2) ± ...
0
votes
0answers
22 views

Existence of a continuous surjection

Does there exist a continuous surjection from R (set of real numbers) to [0,2*pi]? (the set is closed). It seems apparent to me that this should be affirmative, but I haven't been able to provide a ...
0
votes
1answer
23 views

how can we define closed set or open set for a set of matrices?

Suppose we consider the set of all matrices in $M_{2}$(R) such that neither eigenvalue is real .Is the set open or closed?
0
votes
3answers
26 views

No solutions to diophantine equation

I am trying to deduce that $x^2-5y^2=0$ having shown that $x^2 \equiv 5y^2 (mod 7)$ has no integer solutions (not 0). How do I go about this?
2
votes
1answer
13 views

Topology on $\mathcal{L}(E,F)$ : infinite dimensional case

We take two infinite dimensional normed space $E,F$, we perform the vector space of continuous linears maps $\mathcal{L}(E,F)$. We use the the sup norm for a linar map. Let $f: U \to F$ where $U$ is ...
0
votes
0answers
12 views

Wave equation with constraind force

I got to solve the wave equation with the damping force $f(x,t)=L(x^2-x)$. It means the equation need to be solved is $\frac{\partial ^2}{\partial x^2} \psi - \frac{1}{c^2} \frac{\partial ^2}{\partial ...
16
votes
1answer
70 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...

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