n points on circle how many different square and Curved square you can build My solution is: $C(n,4)$ to the curved squre and $C(n,4)*3$ to the squre which makes total sense, for me. and I know it's not enough

enter image description here I have solved the above using the below method. x= 12 + 1/(2+(1/2+x)) After solving for x, i got the answer as 11.7515 and -1.41824 So what is the real value of x, it should be one value for the expression. Why am I getting two values? Please assist

Find unitary matrix $U\in\mathbb{C}^{2\times2}$ so that $D=UAU^*$ is diagonal where $$A=\begin{bmatrix} 3 & -4\\4 &3 \end{bmatrix} \in \mathbb{R}^{2\times 2}$$ I know that for unitary matrix it holds that $U^* U = U U^* = I$ and if we know that $D$ is diagonal: \begin{align} D&= U^* A U \;\;\;|...

Let $ \{ a_{n} \}_{n=1}^{\infty} $ be a Cauchy sequence of rationals, and let $x$ be a real number. Show that if $a_{n} \leq x $ for all $n \geq 1 $, then $\lim_{n \rightarrow \infty} a_{n} \leq x $.

So, I am working on some analysis homework and created a lemma to help me prove something. The problem is I don't know if it's true or false, and I don't want to waste a whole bunch of time attempting to prove a false lemma. Is the following lemma true? Proofs are not required, but greatly apprec...

I read on internet that " Fractal is special example of Cantor set". My question, Is Cantor set uncountable? Hence, Fractal is also an example of uncountable set? Thanks

In a triangle ABC from vertex C median of a line segment AB, angle $\angle|BCA|$ bisector and line segment perpendicular to a line segment AB divides angle $\angle|BCA|$ to four equal parts. Task is to compute angles in a triangle ABC. Thanks for any help.

How can I prove that a degree sequence satisfy Chvatal’s criterion? I know that i must prove that sequence A is Hamiltonian if and only if A' is hamiltonian but i am lost on where i should start. for example does this satisfy Chvatal’s criterion? (3, 3, 4, 4, 4, 6, 7, 7, 8, 8)

how to Prove that e^-i|x| is not a characteristic function

Consider numerically approximating the integral $$I = \int_{a}^{b}f(x)dx$$ using the open Newton-Cotes with $n = 2$,i.e., 3 points $$I_{2}^{(o)} = \frac{4}{3}h_2[2f(x_0) - f(x_1) + 2f(x_2)]$$ 4b.) Suppose global step halving is used to define a coarse grid with $m$ intervals of size $H_c$ and a...

2 (2 pts) A window in Prof. Xu' ss office is by the function f where f is given by f (s) = 5 sin ((π/32)(s^2 + 16)). Every evening, before she leaves her office, Prof. Xu draws the curtain over the window. Every morning she draws the curtain open, starting at time t = 0. If she draws the cu...

I need to prove the for every 0 < k <3 there are at least 2 real roots to the fucntion: $f\left(x\right)=x^5-8x^2+k$

What is the derivative of a vector with respect to a matrix? Specifically, $\frac{d(Ax)}{dA} = ? $, where $ A \in R^{m \times n}$ and $x \in R^n$.

How to prove that every irrational number with eventually periodic continued fraction expansion is a root of a polynomial of degree 2?

Please let me know the values of Q1 and Q2 and what does this say? (Image attached)

Define the following, $$F_2(x) := \frac{1}{2}+\frac{(2x)}{1!} B_2\Big(\tfrac{1}{2}\Big)+\frac{(2x)^2}{2!}B_3\Big(\tfrac{1}{2}\Big)+\frac{(2x)^3}{3!}B_4\Big(\tfrac{1}{2}\Big)+\dots \infty$$ $$\color{brown}{F_3(x)} := \frac{1}{3}+\frac{(3x)}{1!} B_2\Big(\tfrac{1}{3}\Big)+\frac{(3x)^2}{2!}B_3\Big(...

I came across this symbol while reading about number theory what does it mean? Thanks for any help.

Let $ {a_k}$ be an unbounded,strictly increasing sequence of positive real numbers and $x_k=\dfrac{a_{k+1}-a_k}{a_{k+1}}$. Prove that $\sum_{k=m}^n x_k>1-\dfrac{a_m}{a_n}\forall n\geq m $ and $\sum_{k=1}^\infty x_k$ diverges to $+\infty.$ What I thought: $\sum_{k=m}^n x_k=\sum_{k=m}^n(1-\dfrac...

I was looking to find the supremum of this set of real numbers $$\Big\{ \vert \sum^{\infty}_{n=1} \frac{a}{n^{2}+a^{2}} \vert: a \in \mathbb{R} \Big\}$$ I was able to show (I hope this is right) that for any $a \in \mathbb{R}$, that $\vert \sum^{\infty}_{n=1} \frac{a}{n^{2}+a^{2}} \vert< \vert...

Is there a language of affine spaces theory ? Is it first order ?

If in a group $G$ there is an element $a$ of order $360$, what is the order of $a^{220}$ ? How to proceed ?

I have the following simple optimization problem $$ \text{max} \ x^2 + 3xy + y^2 \text{ subject to } x + y = 100$$ and the exercise tells me to check whether I have found the actual solution by checking its convexity. The way I would solve would be to break the function into $(x+y)^2 + xy$ and...

I need to eliminate $t$ from, $$ x = t + \cos \frac{t}{k} $$ and $$y = -\sin \frac{t}{k}$$ to get an equation in terms of $x$ and $y$ only. Essentially, I am trying to convert it from parametric form to Cartesian form. Please help, I am stuck. :(

Can someone please help me prove this relation in ABC conjecture The ABC conjecture says that for every real number e > 0, there exist only finitely many triples (a; b; c) of positive integers, with gcd(a; b; c) = 1, that satisfy a + b = c and c > (rad(abc))^(1+e)

How can I show that every finite subset of $\{0,1\}^*$ is recursive ?

Can anyone explain how they came up with the product of E and H ? I don't understand why the exponent of E cross H are multiplied by 2. Thanks enter image description here

I often encounter both spellings Lagrangian or Lagrangean. I just want to know which one is correct or commonly used in mathematical journals.

Let G be a group of order 3n for some $n\in\mathbb Z$. Suppose all the elements of G of order 3 are conjugate .Then which are true? G must be cyclic. G canot be abelian G must be abelian and not cyclic G must be abelian and may or may not be cyclic. Since all the elements of order 3 are conj...

How to evaluate $\int_c \dfrac{dx+dy}{|x|+|y|}$ , where $C$ is the square with vertices $(1,0);(0,1);(-1,0);(0,-1)$ traversed once in a counter-clockwise direction ?

Let each edge of $K_n$ be given an orientation and let $Q$ be the incidence matrix. Determine $Q^+$:($Q^+$ is more penrose inverse of $Q$)

Let $F={P\over Q} $ be a fraction with $deg(P)< deg(Q)$ and such that $P$ and $Q$ are polynomials with real coefficients. Suppose $Q$ has the form $Q=(X-a)(X-z)(X-\bar z)$ with $a\in \mathbb R$ and $z$ is a non real root of $Q$ and $\bar z$ is the conjugate of $z$. The partial fraction decomposi...

I need to prove this equation to be true not sure how to solve. I know I have to first use one and then plug in k+1 but what am I plugging it into a or n and then how do I solve? Thanks

So basically I want to integrate over two complex variables, so my integration will look something like this $\int uv\cdot e^{-uv}dudv$ where u and v are complex coordinates, in this case two dimensional (u=x+iy) and v=(x-iy). Im wondering how to solve this integral. I have seen a number of dif...

Prove that Fermat's Last Theorem is true for all large n, i.e., for all large integers n>=3, the equation x^n + y^n = z^n has no solution in positive integers x, y and z.

I need help with this prove it is part of my homework for school so I was stuck on this question and don't know how to do this so here is the question: enter image description here

I have to give an example of absolutely continuous function. Can somebody help? I know that if $f$ is constans on an interval, then $f$ is absolutely continuous.

Find a general formula for the largest number less than $10^n$ that has exactly $8$ divisors. I was wondering if there was a way to find a general formula for this or if it is possible to create one.

Suppose that M is a TM that crashes on some inputs. How would I modify M so that the new machine accepts the same language but does not crash on any input ?

I received some question for my exam preparation, but I'm not sure about the answer. So I need some reasonable explanation of the following tasks: I. Every day Sonja arrives at the railway station between 6 and 7 pm, but her exact time of arrival is random within this one hour interval. The trai...

Limits: $$\lim\limits_{x\to1}{\frac{\sqrt{2^x+7}-\sqrt{2^{x+1}+5}}{x^3-1}}$$ I multiplied by the conjugate but nothing happened

Let $(X,d)$ me a metric space and pick a real interval $(a,b)$. Why is $d^{-1}((a,b))$ open in the product topology on $X\times X$ induced by the metric topology on $X$?

Prove that $\lim_{x\to0^+}[1+[x]]^{\frac{2}{x}}=1$,where $[x]$ represents the floor function of $x$ $\lim_{x\to0^+}[1+[x]]^{\frac{2}{x}}=\lim_{x\to0^+}[1]^{\frac{2}{x}}$ Because $\lim_{x\to0^+}[x]=0$ But i am stuck.Please help me.Thanks.

The probability that a seed will germinate is 0.47. Suppose 156 seeds are planted. Use the Central Limit Theorem to determine the probability that at most 72 seeds germinate

Suppose we have two samples with known correlation (should be relatively high). Say both samples have $n$ data points. What if now we still know the correlation factor but one sample only consistent of the first 5 data point. Could one still construct the remaining data points solely using the ...

can you help me please with limit solving without using L'hopital rule $\lim_{x \to \pi/2 }\frac{\cos3x}{1+\cos2x}$. Thanks a lot!

Oh, well, the title actually describes what kind of question will this question be, but let us do some warm-up before stating the question as clearly as possible. Suppose first that everything we do we do on some set of the form $[a,b]$. Suppose secondly that we have some sequence $f_k$ ; $k \i...

For general nonlinear system, can a union of equilibrium point (sink) and Heteroclinic orbit be a omega-limit set. Thanks

I posted quite an interesting question yesterday about a maze algorithm, but nobody responded yet. Would you mind taking a look? Runtime of maze algorithm

Let $k$ be a natural number and for each $k$ let $r_k$ be the minimum number $n$ so that if we colour the edges of $K_n$ with $k$ colours then we can find a monochromatic triangle. I have so far showed that $r_k − 1 ≤ k(r_{k-1} − 1) + 1$ and now I have been asked to do the following: Use induc...

Given an $n\times n$ non-negative with dominant eigenvalue $\mu$. Let $m>\mu$. Consider the infinite summation $$\sum_{k=0}^\infty \frac{M^k}{m^k}$$ Is the following equality correct: $$\min_{i\le j\le n} \sum_{i=1}^n\sum_{k=0}^\infty\frac{M^k}{m^k} = \sum_{k=0}^\infty \frac{1}{m^k} \min_{1\le...

I want to know the possible images of the group homomorphism that maps D10 --> G (where G is some arbitrary group) Things I know: G/ker(f) is iso to Im(f) normal subgroup to D10 are {e}, $<sigma>$, D10 I do not know how to connect all this information, so please help!

In the image attached, enter image description here Can you help find two non-congruent right angled triangle that have rational side lengths and area equal to 5

On $\mathbb P^n_k\times_k \mathbb P^m_k$, is it true that $T_{\mathbb P^n_k\times_k \mathbb P^m_k}\otimes \mathcal O_{\mathbb P^n_k\times_k \mathbb P^m_k}(d,e)\simeq p_1^*T_{\mathbb P^n_k}(d)\oplus p_2^*T_{\mathbb P^m_k}(e)$ ? ($n,m\geq 1$ and $p_i$ are the two projection).

I'm having troubles because of the -u double prime

I'm a first-time Calc I student currently struggling in class. Yesterday we started on Substitution and Integration with integrals. One problem our professor put on the board was: $\int \frac{\left(x^2+2.1x\right)}{\left(x^3+3x+12\right)^6}dx$ And he refused to solve it, said nobody in the room...

If you have 2 sets, one being called A and another B, which B is a subset of A. You need to determine the formula in general for A to intersect B who contains only one element. My idea, was to take a and say that the cardinality of A = i, and the cardinality of B = j. Then devising a general ...

Find the tangent equation hows slop is $1$ and is tangent to $f(x)=18x^5-17$ I have got two tangent function: $y_a=x-17.25$ and $y_b=x-16.73$ can it be?

Given a sequence $(a_n)_{n \in \mathbb{N}} \subset \mathbb{R}$ and a set $M = \{a_n : n \in \mathbb{N}\}$, is $|M| < \infty$, then $(a_n)_{n \in \mathbb{N}} $ has a limit point. I believe that $(a_n)_{n \in \mathbb{N}}$ possesses no limit points. Here is why: From the definition of a limit poin...

For city we have simplified its weather forecasting as such. If it rains then the probability for rain the next day is $0.2$. If its sunny then the probability for sunny day the next day is $0.7$. Vector $$x_{k}=\begin{bmatrix}\text{probability for sunny weather at day } k \\ \text{probability f...

Please is there a way to input this fourier series into wolfram alpha ? I was given: f(x)= x for (-pi, pi). So I need to calculate A0, An, Bn coefficients. Thanks for help. Martin

How do I show absolute convergence for the series $$\sum_{n=0}^{\infty} \frac{n}{\sqrt{2n^5 +1}}$$ I have already showed by Comparison test that it is convergent. I am after the way of showing $\sum |a_n|$ is convergent. I tried ratio and root test but it gives me a limit of 1 so I need to do a...

I received some question for my exam preparation, but I'm not sure about the answer. So I need some reasonable explanation of the following tasks: I. Every day Sonja arrives at the railway station between 6 and 7 pm, but her exact time of arrival is random within this one hour interval. The trai...

I need an exemple of the conformal aplication of $n$-dimensional ball $B^n \subset R^n$ in ball $n-1$ - dimensional $B^{n-1} = \{ (x_1,...,x_n) \in R^n ; x_1^2 + ... + x_{n-1}^2 = 1 and x_n = 0 \}$. Can some help me pleaser?

Let L be the language consisting of all strings in (a+b)* that have an even number of letters and do not have aaba as a substring. Into how many L-equivalence classes is (a+b)* divided?

I have the following series, Series I have tried comparing with the limit comparison test to a p series. But the limit is hard to evaluate. Any suggestions to prove convergence? Thanks

For example, $\int_a^b (c-x)^2 g(x)\,dx$, where $g(x)$ is left unspecified. I have a bunch of terms like this and I want to see if it all adds up to $0$.

This problem arised while I'm reading [this]( physics.stackex change.com/questions/221972/ combination-of-simple- harmonic-motions ) question on physics stackexchange. But it seems this platform is good for asking my doubt. In the above mentioned question, author asked about the periodicity of fu...

What is dyadic Green's function? Is it related to coupled ODEs? Please refer to some good texts.

We have the following relation: $R_1=\{(x,y) \in R^2:-1 \le x \le 1,-3 \le y \le 2 \}$ Could anyone tell me how to make the graph for the above relation?

Let $\Delta^n$ be the standard $n$-simplex. Denote with $H_0$ the (simplicial) $0$-homology. In my book it is written that $H_0(\Delta^n, \partial \Delta^n)=\mathbb{Z}$. But $\Delta^n$ and $\partial \Delta^n$ have the same number of vertices, so $H_0(\Delta^n, \partial \Delta^n)=0$. Is it just ...

Does anybody can explain me in plain english what's the real point with the Gelfand–Naimark Theorem. I know it's crucial, but I think I'm missing how much it's crucial.

Im reading a book about and discoverd a inequality, i have Problems with. $$(\sum_{2^l\leq|m|<2^{l+1}}|\hat{f}(m)|)^2\leq(\sum_{2^l\leq|m|<2^{l+1}}1^2)\sum_{2^l\leq|m|<2^{l+1}}|\hat{f}(m)|^2$$ Surly it is true that: $$(\sum_{2^l\leq|m|<2^{l+1}}|\hat{f}(m)|)^2\leq\underbrace{(\sum_{2^l\leq|m|<

Don't know how to prove if converges or not. which converging test do i need to use? $\sum_{n=1}^{\infty} \frac{(1}{nln^2(n)}

Solve the system of n equations: $$ \begin{cases} 2x_1^3+4=x_1^2(x_2+3)\\ 2x_2^3+4=x_2^2(x_3+3)\\ .........\\ 2x_{n-1}^3+4=x_{n-1}^2(x_n+3)\\ 2x_n^3+4=x_n^2(x_1+3)\\ \end{cases} $$

Let $E_0$ and $E_1$ be two ellipses contained in the projective plane $\textbf{P}$. Each ellipse bound a disk on one side and a Möbius band on the other. We assume that the disk bounded by $E_1$ contains $E_0$. We normalize by a projective transformation so that $E_0$ is the unit circle and $E...

If $f\in L^{+}$ and $\int f < \infty$, for every $\epsilon > 0$ there exists $E\in M$ such that $\mu(E) < \infty$ and $\int_{E} > \left(\int f\right) - \epsilon$ Proof: Let $\epsilon > 0$, and $f\in L^{+}$ where $f$ is simple, we can define $$\int f d\mu = \sup\{\int \phi d\mu: 0\leq \phi \leq f...

I have 9z=−2x+7y and I need to convert it to cylindrical coordinates in the form of r=f(θ,z)=____ I started by substituting x=rcos(theta) and y=rsin(theta) in but got confused as to what to do next.

This is my original problem: Find the number of elements of order 7 in $S_7$. Find the number of Sylow 7-subgroups of $S_7$. Since 7 is prime, we know that this must be a 7-cycle in $S_7$ (since no $P_1 * P_2 = 7$). My next thought is to calculate the total number of combinations a number ...

I believe that the above is equal to $\infty$ but I don't have any formal reason why.

How do I find the groups D10/N where N is the normal subgroups to D10? I know that the definition is aN for all a $\in$ D10. But I am unsure what the group, for example, D10/ D10 looks like? Any help would be much appreciated.

I have an assignment for next week and I'm stuck with these two questions : a) Let G be a simple graph on 8 vertices with exactly 25 edges. Can G be Eulerian? How about with 24 edges? What I did : a) A graph on 8 vertices contains at most C(8,2)=28 edges. So G is a complete graph -3 edges. The...

So I fully understand that a=b is a partial order relation as it is the identity relation. a+3<=b however is where I am being stumped.

I have a problem with the definition of Hilbert Space and Banach Space. What is the difference between a Hilbert Space and a Banach Space?

Prove that any finitely generated subgroup of $(Q,+)$ is cyclic.

f(x) is continuously differentiable here. Using separation of variables, I think I might have shown that the equality form of the statement is true, but I'm a bit wary of trying separation of variables for a differential inequality. There was no solution provided for the problem.

Why does $b^{log_bx} = x$? Can someone break this down by showing me the steps as to why this is true?

There is a general mantra in math which says that what is independent of bases shall be defined independent of bases. Well, it is well known that the elementary divisors of a linear map $M\xrightarrow{\ \ f\ \ }N$ of finitely generated free modules over a principal ideal domain $R$ are independen...

I can find a continuous map that has no fixed points for the case $n=1$ but fail to see how this generalizes.

Given an alphabet $\Sigma = \{0, 1 \}$, to represent the set of infinite (bit) strings, it is usually used the notation $\{0,1\}^{\omega}$. To represent the fact that I take an element of this set, it is of course used the notation $str \in \{0, 1 \}^{\omega}$. What I need is to make explicit t...

When considering the SIR model, are there any equilibrium points other than when the whole population is recovered?

Consider a sequence of real-valued random variables, $\{X_n\}_{n}$. Suppose that $X_n \rightarrow_p X$. Consider (*) $\lim_{n \rightarrow \infty} P(X_n>0)$. What I have seen doing in several books is the following: since $X_n\rightarrow_p X$ then (**) $\lim_{n \rightarrow \infty} P(X_n>0...

The sequence converges to zero at a rate that seems to be slightly faster than $1/n$. What are the best known results on the convergence rate of this sequence?

What is e^(it) or e^(-it) equal to? In the example he finds the eigenvalues r=i and r=-i and for the general solution he writes: y=c1cost+c2sint.

Here is the following problem: Let $g_0$ be the Euclidean metric on $\mathbb C=\mathbb R^2$. Let $M=\{z \in \mathbb C| \ |z|<1 \}$ and equip it with the Riemannian metric $g=\frac{1}{(1-|z|^2)^2}g_0.$ Let $N=\{z \in \mathbb C| \ \text{Im} \ z>0 \}$ and equip it with the Riemannian metric $...

I'm stuck with this question. 1. Let G be a simple graph on 21 vertices with at least 200 edges. Show that G is Hamiltonian. I tried to use Dirac's theorem to prove it but it is inconclusive because I get δ(G) = 10 < 21/2. And I don't think we have seen Ore's theorem in class. Is there another...

Consider a finite set of real numbers. Let its variance be $V$. Let the highest number be $h$ and the lowest number be $l$. Let $x$ be an arbitrary number with $l < x < h$. Now create a new set by removing one element equal to $h$ and replacing it with $x$. Call the variance of this new set $V...

Let $a$ and $b$ belong to a group $G$. If $o(a)=12$, $o(b)=22$ and $<a> \cap <b> \neq \{e\}.$ Prove that $a^6=b^{11}.$

How could we prove this? thanks!

X and Y are independent and identically distributed exponential random variables with parameter 10. Compute the joint density of (U,V) where U=X+Y and V=exp(X).

Please I have this two conditions: $$(H_1)~~\lim_{|t|\rightarrow0}\frac{f(t)}{g_1(t)|t|}=0,\,\,\,\, (H_2)~~\lim_{|t|\rightarrow\infty}\frac{f(t)}{g_2(t)|t|}=0$$ $$(H_3)~~\lim_{|t|\rightarrow0}\frac{B(t)}{G_1(t)}=0,\,\,\,\, (H_4)~~\lim_{|t|\rightarrow\infty}\frac{B(t)}{G_2(t)}=0$$ such that $$G...

How would I use Brzozowski's derivatives method to construct a minimal DFA recognizing the language defined by the rational expression: L = (ab + b)* ba

In a box we have 5 dices, 3 of them are proper, one has two sixes, second has six sixes. Given the events A - after first throw we have a six, B- after next two throws we have six and something else. Find probability: $P(A \cap B | proper dice)$. My attempt is: $$ (1/6)\times(1/6)\times(5/6)=(5/6...

We have simple cycle graph $G$ of $9$ vertices. I colour $3$ vertices blue, $3$ red, $3$ green. We assume the same graph coloring $G'$ and $G''$ if $G'$ = $R \circ G''$ , $R$ - rotation. Let coloring is right if every two adjacent vertices have different colour. enter image description here ...

I'm pretty dim when it comes to basis and span, please take that into account :) Help me prove this theorem: Let $V$, $W$ be vector spaces, also $T:V \rightarrow W$ is a linear transformation. If $\beta = \{v_1, v_2, ..., v_n\}$ is a basis for $V$ then $R(T) = span(\{T(\beta)\}) = span (\{T(v...

I am looking for examples of finitely generated solvable group that is not polycyclic. In Wikipedia Baumslag-Solitar group $BS(1,2)$ is an example. But how to prove this fact?

I want to prove that $=e^{A+B}=e^{A}e^{B}$ using differential equations. I found a proof here: LINK Here is how the proof goes: Given a square matrix $M$, the function $X(t):=e^{tM}$ is the unique solution of the linear differential equation: $X'=MX$ and $X(0)=I$. Now set $X(t):=e^{tA...

Four digit vehicle numbers sum up to 18 ...due to a maximum probability from central value as: $$ (0+9)/2 *4 $$ Is it correct?

Let $B_1 \to X_1=\mathbb P^n$ be the blow-up along a linear subspace $\mathbb P^k$. Let $B_2 \to X_2$ be the blow up of a quadratic cone $X_2$ in its vertex. I need to calculate $K_{B_i}$. As any blow up is an isomophism ouside of exceptional locus, $K_{B_i}=\pi^*K_{X_i}+D_i$, where a divisor ...

Let $z$ be a random variable, which is defined as $ Z = d^{\alpha}$. where $d$ is random variable. I tried to plot the CDF for $Z$ with different values of $\alpha$, which is: So, average value(50th percentile) of $Z$ decreases with increase in $\alpha$ values. Is it correct? Because if we in...

I need to calculate $\lim\limits_{(x,y) \to(0,0)} \frac{x^4y^2}{ (x^4+y)^5}$ I get $[0/0]$. i think it doesn't have a limit but i don't know how to prove it. Thank you.

The theorem goes: Let $V$ and $W$ be vector spaces and $T:V \rightarrow W$ is a linear transformation. If $V$ is finite dimensional, then $nullity(T)+rank(T)=dim(V)$

I've got another question about normal distribution and its confidence intervals interpretation. Your explanation will respectively help me to better prepare for my examination. One of the social surveys, which selects a nationally representative sample of the Canadian adult population, asked re...

Hi I would like to understand Banach-Tarski paradox, but well... my knowledge with set theory is very very limited. I know what a union is, I will tell you how to partition a simple set, know couple of things about sets being disjoint and well, that will be it. ANd every idea of proof (take awa...

I'm intrigued by this finding on the extraordinary portal WolphramAlpha. What is the reason why the solution has not been simplified, eliminating the factor $\frac{\sqrt x \sqrt{x+2}}{\sqrt {x(x+2)}} $? If not removal is justified, I would love to know why (I not discard that there is a strong...

$F(x,y)= \left(\displaystyle\frac{1-y^2}{(1+xy)^2},\displaystyle\frac{1-x^2}{(1+xy)^2}\right)$. I've been having some troubles to find the potential of $F$. To find it the idea was finding $\int F dx = g(x,y) +h(y)$, and having $g$ find the function $h$. Unless my calculations have a mistak...

How much the most edges have graph on $n$ vertices with 2 components? How much the most edges have graph on $n$ vertices with 3 components? Thank you for any hints.

I am trying to do an algorithm to construct the follow matrix $\begin{bmatrix} 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 \end{bmatrix}$ This is what I did: First I construct the first line of the matrix: $M(1...

I'm not even sure where to start. The problem is: ====================================================== Consider the subset H = { [3k] | k ∈ Z } of Z12. (a) Determine the distinct elements of H and construct an addition table for H. (b) A relation R on Z12 is defined by [a] R [b] if [a-b] ∈...

I'd like to find $\nabla \cdot(\nabla f(x))$ where $r=\sqrt{x^2+y^+z^2}$. I know that $\nabla f(x)=\partial_xf\vec{e_x}+\partial_yf\vec{e_y}+\partial_zf\vec{e_z}$. I don't underastnt how use this for my case when it's know what $r$ is equaled.

Find a (parametric) infinite family of solutions for the equation $a^5+b^5+c^5+d^5+e^5+f^5=a+b+c+d+e+f$, where $a,b,c,d,e,f$ are integers. Numerical experimentation suggests that this should be possible. For example, there are known parametric solutions for the equation $a^5+b^5+c^5+d^5+e^5+f^...

When we could use the following equation: $$\mathbb{E}X^2=\int_0^\infty 2t \mathbb{P}(X>t)$$ I mean how is it possible to change $X^2$ to $2t$?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the connections between these ways are not clarified mostly by teachers. Once I read the calculus book by ...

Would this even assist math in the way that $i$ did? Or is this just outright pointless and/or too exclusive to call for a definition?

Is the series $$\sum_{n=1}^\infty\left(1-\frac1{\sqrt n}\right)^n$$ convergent?

How would you solve this equation? 500n = 4000(1.016)^n I tried using some logarithms but I could not do it. The only unknown variable is n but I'm having a bit of trouble getting there. Thanks.

Two functions f:N→N and g:N→N meet the following conditions. (1) For A={f(n)|n∈N}, B={g(n)|n∈N}, A∩B=Φ (2) A∪B=N (3) g(n)=f(f(n))+1 for n∈N (4) f(n+1)>f(n), g(n+1)>g(n) Question: Find f(240) This is a math contest problem but I have no idea. I tried: f(1)=1 or g(1)=1 i) If f(1)=1: g(1)=f...

1) Two boxes, both contain $6$ white $\&$ $5$ black balls. Random ball $A'$ is drawn from the first box and place into second box. Then random ball $A''$ is drawn from the second box. What is the probability that $A$ is white? 2) Three boxes, the first contains $3$ white $\&$ $4$ black balls,...

could you please help me with this equation? 3sin(3x+π/4)=0 I know that the answer is obviously π/4 (or k*2π + π/4). but could you give me a step-by-step solution? Could you also tell me how this equation is solved? sinh(x+1)=1 Thank you very much.

I'm trying to prove it but so far I can only write the antisymmetric tensor using the basis for all $k$-tensors. Can you please help me?

I am currently trying to solve this problem but I'm quite unsure if my answer is correct. I've build the tree different types of measures, MIN, MAX and AVERAGE but I might have some of them wrong. Can anyone check and tell me for sure please. Thanks in advance. Image 1 contains the maths problem...

Suppose $(a_n)$ is a sequence in $\mathbb{R}$ such that $a_n > 0$ for all $n \in \mathbb{N}$ and that there exists $f: [0, \infty) \to \mathbb{R}$ such that $f(n) = a_n$ for all $n \in \mathbb{N}$. Assume that $f$ is a continuous function and decreasing. Also $f > 0$. I would like to prove the ...

How can I show: Let {$f_n$} be a sequence of functions that are uniformly continuous on $(0,1)$. Show if {$f_n$} converges uniformly on $(0,1)$ then $f(x)$ is uniformly continuous on $(0,1)$. Also, would someone mind explaining what it means to have a uniformly continuous sequence?

I have two matrices $A_{a\times n}$ and $B_{b\times n}$ (as it is apparent from their sizes, they have the same number of columns). Now I want to filter each column of $A$ by its corresponding column in $B$ in Matlab without using for-loop (in a vectorized manner). Indeed I want to filter the fir...

So according to my math textbook the integral of ln|x+1/x+2| from 0 to Infinity is equal to ln(2). I don't understand how the limit of ln|x+1/x+2| as x approaches infinity is zero. I cannot find any explanation in the textbook.

Where A and B are ideal triangles in H^2 (upper sheet of hyperboloid). How do I get started with this proof?

i've been looking for some applications or some kind of usage for the zeta-function an what a proof of the Riemann hypothesis would mean to areas such as number theory, but I haven't been successfull researching the latter. Could someone go through some important things relying on the proof of th...

I have the following series: Sum from one to infinity ln(2(n+1))- ln(2n) How can I test it to show it is divergent?

I was thinking that i would write it out like this (10/3)(8/2)(6/2)*5 factorial. but i realized that would just show the total possible ways the word can be rearranged. if i was to guess as to what to do i would say that (10/7). which translates to (10factorial)/7factorial *3factorial)

If this subgroup does exists, how would I go about finding it?

I am almost to complete my first course in group theory. I have read Dummit and Foote into chapter 5. I know that I can always find an abelian subgroup isomorphic to C_k(1) X C_k(2) X ... X C_k(j) in S_n where n = k(1) + k(2) + ... + k(n). Specifically, the permutation group G = <(1 2...k(1)),...

Let $G$ be a group of nilpotency class $n$. Every proper subgroup of $G$ would have nilpotency class at most $n$. Do we know of an example where the commutator subgroup of a nilpotent group of class $n$ also has class $n$?

I'm solving the 1D wave equation \begin{equation} \frac{\partial^2 \eta}{\partial t ^2} - \frac{\partial^2 \eta}{\partial x ^2} = 0 \end{equation} with boundary conditions \begin{equation} \frac{\partial \eta}{\partial x} = 0 \qquad \qquad \text{on} \qquad \qquad x = 0, L \end{equation} using Fi...

First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for the last part: e[n+1]=Ce2ne[n+1]=Cen2 but how can we conclude that the relations is telling us the ac...

I have already found that (0,0,1) is a local maximum (and the only critical point), but I cannot figure out how to show that there is no global max.

Let $A$ be set of $n = 9k$ elements. Let $(A_1,A_2,A_3)$ be partition of $A$: $A = \cup_i A_i$, $|A_i|=\frac{n}{3}=3k$, $|A_i\cap A_j| = 0$, $i\neq j$ We assume two partitions same if set collections for partitions are same. For example $(A_1,A_2,A_3)=(A_2,A_1,A_3)$. Let $B$ be subset of $...

I do not figure out how to solve $$ L= \frac{1}{2}\int_1^4 \sqrt{(16t^2+t+4)}\;dt $$ The key is probably in simplifying the polynomial equation, but I don't find a way that is really simplifying the integral. Can someone help? Thank you.

Given a parallelogram with congruent diagonals, you are asked to prove that the parallelogram is a rectangle. Would saying: A parallelogram must be a rectangle if the diagonals are congruent. be a valid proof for this question?

I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One theorem in which they rely on it is as follows: Let $n$ be a positive integer and define $g(n) ...

I apologize if this question isn't appropriate for this site but I am looking for advice I think other math students might better be able to give me. I am an undergraduate math major about enter into my second year. What I have found is that I love the theoretical discussions of mathematics an...

Let build a set A by this rule: for $i$ from $0$ to $n - 1$ with probability of $\frac{1}{2}$ we add this number to set $A$. Then we generate set $B = A + A := \{a_1 + a_2: a \in A, a_2 \in A\}$ And the question is to find upper boun on probability $P(k \notin B)$ as precise as possible. I h...

Prove that $(Q,+)$ is not isomorphic to $(H,+) \neq (Q,+)$, a subgroup of $(Q,+)$. $Q$ is the rationals. I thought about taking the contradiction direction. If we do that then we have $f:Q\to H$ such that $f$ is an isomorphism. That means that that it is surjective and injective, but $(H,+)...

I seem to be stuck with this one right here: 2^x + 2^x+1 = 3^x+2 + 3^x+3 Mind to help me with it?

Consider the subset H = { [3k] | k ∈ Z } of Z12. (a) Determine the distinct elements of H and construct an addition table for H. (b) A relation R on Z12 is defined by [a] R [b] if [a−b] ∈ H. Show that R is an equivalence relation and determine the distinct equivalence classes. No idea where t...

) Dan travels from his home to the store and back. On the way to the store, he travels at a speed of 50 km/hr for the initial half distance and 100 km/hr for the latter half. On the way back, he travels 50 km/hr for the first half time and 100 km/hr for the latter half. The total distance from hi...

Prove that for n in the set of natural numbers, n is greater thean or equal to 2: For all a belonging to the set of natural numbers, For all b belonging to the set of natural numbers, a is modular congruent(subscript n) to b -->a^2 (subcript n)b^2. Sorry for writing in words, i am new to the si...

Let $$ S = \{ a_1,a_2, \cdots , a_k \}$$ be comprised of divisors of $n \in \mathbb{N}, n>1$ and $n$ not prime. Suppose we select $p$ elements from the set $S$ (possibly more than once for each divisor), and the $p$ chosen elements have a total sum of $q$. Prove that it is always possible to sel...

I have the following series Sum from 1 to infinity ne^(-n^2) How would one go about showing convergence? Which test can be used?

got stuck proving that $(A \times B)^c = (A^c\times U)\cup(U\times B^c)$. would appreciate your help, this is what I got so far:

I am having difficulties forming the inverse of this f(x) = 3*2^(3x+1)*5^(3x-1) What I have done so far: 3*2^3y*2^1*5^3x*5^-1 3*2*1/5*(5*2)^3y 6/5*10^3y .ln ln6/5*ln10*3y That is all.

Suppose $\{f_n\}_{n=1}^{\infty}\subset L^{+}$, $\lim_{n\rightarrow \infty} = f$ pointwise, and $\int f d\mu = \lim_{n\rightarrow \infty}\int f_n d\mu < \infty$. Then $\int_{E}f d\mu = \lim_{n\rightarrow \infty}\int_{E}f_n d\mu$ for $E\in M$ Proof: Let $\{f_n\}\subset L^{+}$ and $f_n\rightarrow f...