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H: Eigenvalues of a particular matrix
Given two vectors $\boldsymbol\alpha=\left(\alpha_1,...,\alpha_N\right)$ and $\boldsymbol\beta=\left(\beta_1,...,\beta_N\right)$, is there an easy way to compute the eigenvalues of a matrix $M_{k,q}$ whose entries are expressed as
$$
M_{k,q}=\frac{\alpha_k}{\beta_q},\quad k=1,...,... |
H: Arithmetic problem
Prove that $(a^2:a^2+16)=1 \lor 4 \lor 16$
I know that $(a:b)=(a:b+ka)$ with $k\in\Bbb Z$
So:
$(a^2:a^2+16)=(a^2:a^2+16-a^2)=(a^2:16)=1\lor4\lor16$
This is enough to prove this?
AI: Maybe you need to explain a little more why $(a^2:16)=1\lor4\lor16$.
For example why $(a^2:16)=2$ is impossible?
Hi... |
H: What is the equation for a line tangent to a circle from a point outside the circle?
I need to know the equation for a line tangent to a circle and through a point outside the circle. I have found a number of solutions which involve specific numbers for the circles equation and the point outside but I need a specif... |
H: What is the order of $e^{\large \frac{4\pi i}{5}}$ in the circle group $U_{20}$?
We talk about the Circle group.
What is the order of $e^{\large \frac{4\pi i}{5}}$?
The power is $\frac{4\pi i}{5}$ if it's not clear...
Thank you!
AI: If this question stems from your earlier question, where you were talking about $\o... |
H: Inferenecs from the given: x = nonhomogenous + homogenous solution solve Ax = b. (GStrang P161, Ex 3.4B.4)
Given: All solutions to $\mathbf{Ax = b}$ have the form $\mathbf{x} = (1, 1, 0)^T + c(1, 0, 1)^T$.
$\Large{\color{red}{1. [}}$ Then $A$ must have $n = 3$ columns. $\Large{\color{red}{]}}$ With $(1, 0, 1)$ in... |
H: Urn probability function
Suppose I have an urn with an infinite number of balls which can be either red or white. I do not know what the proportion of each colour is, but I do know it's a fixed proportion. After drawing $N$ balls, I have observed $r$ red ones and $w$ white ones.
I believe the probability that I wil... |
H: $p$-polynomial of $n$'th degree, $q(x)=p[x,x_1,x_2,...,x_k]$, prove that q has the same leading coefficient.
So I have a polynomial $p$ of $n$'th degree and q given by $q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for $x$ it gives back the leading coefficient in interpolation of $p$ on points $x,x_1,...,x_k$. Prove tha... |
H: "Rising sun" function
Let $f:[0,1] \to \mathbb R$ be bounded and
$$
f_\odot :[0,1] \to \mathbb R: x \mapsto \sup \{f(y) : y \in [x,1] \}
$$ This is well defined since $f$ is bouned.
Claim: If $f$ is continuous then $f_\odot$, too.
I begun as follows: Let $x_0 \in [0,1]$ and $\epsilon > 0$. First: Find $y_0 \in ... |
H: How to evaluate the limit $\lim_{x\to0}\frac{\sqrt{x+1}-\sqrt{2x+1}}{\sqrt{3x+4}-\sqrt{2x+4}}$
First I tried direct substitution, which resulted in the indeterminate form. Then because of the square roots, I tried rationalizing (both numerator and denominator), but still get the indeterminate form. I can't use L'Ho... |
H: What is the difference between logarithmic decay vs exponential decay?
I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.
AI: The "Square is a rectangle" relationship is an example where the squ... |
H: Reducing a fraction, divisibility and indeterminate symbol
Quick question about validity, just to make sure.
When I have a fraction in a form:
$$\frac{3a + 3b}{a+b}$$
and I extract the common factor 3 out to get:
$$\frac{3(a+b)}{a+b} \;=\; 3\frac{a+b}{a+b}$$
is it valid now to reduce the fraction to 1 to end up wit... |
H: Calculating percentage to compensate for percent discount.
Missing something very basic here and cannot pin point it.
We need to charge a client \$100 for a product. Let's say our payment processor charges us 10% on every transaction. We make this transparent to the client and charge them accordingly:
$\$x = \$100 ... |
H: $\sum\frac{(k!)^2}{2k!}$ serie converge or diverge
Does the following series diverge or converge.
$\sum\frac{(k!)^2}{2k!}$
I decide to apply the powerful ratio test.
I did
$\frac{(k+1)!^2}{2(k+1)!}$
which then becomes
$\frac{(k+1)!(k+1)!}{(2k+2)!}$*$\frac{2k!}{(k!)(k!)}$
I get the
$k\rightarrow\infty$
$\frac{(k+1... |
H: almost everywhere convergence
Let $E\subseteq\mathbb{R}^l$ be s.t. $E$ is Lebesgue measurable and $m(E)>0$. Let for all $k\in\mathbb{N}$, $f_k:E\rightarrow \mathbb{R}$ be measurable functions. If for all $\epsilon>0$ we can find $F\subseteq E$ s.t. $F$ is closed and $m(F)\leq \epsilon$ AND $f_k\rightarrow f$ unifor... |
H: $\sum\frac{k^k}{3^{k^2}}$ series be convergent or divergent
Would the following series be convergent or divergent.
$\sum\frac{k^k}{3^{k^2}}$
I applied the root test to the following series and got.
$(\frac{k^k}{3^{k^2}})^{1/k}$
I got $\lim$ $k\rightarrow\infty$
$\frac{k}{3^k}$
using Le Hospitals rule I got
$\frac{1... |
H: Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?
I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm wrong). Assume I have a square integrable fun... |
H: Power Series Expansion Asymptotics
From my text:
Given $\cos^n(x),$ set $x=\frac{\omega}{\sqrt{n}}$, then a local
expansion yields:
$\displaystyle\cos^n(x)=e^{n\log\cos(x)}=e^{(-\frac{\omega^2}{2}+O(n^{-1}
\omega^4))}$
the approximation being valid as long as $\omega = O(n^{1/4})$
I do get the approximation as... |
H: Is $ n^2-14n+24 $ a prime number?
How many are those positive integers n such that Is $ n^2-14n+24 $ is prime ? I have tried to solve this problem by putting different values of natural number . Is it a right way ?
AI: $$n^2−14n+24=(n-2)(n-12)=p \text{ (p prime number)}$$
so
$$(n-2)(n-12) = p*1\text{ or }1*p$$
so
... |
H: discrete mathematics , sequences, characteristic equation
Today in my discrete mathematics class we started combinatorics and also solved some some recurrence relation sequences using the characteristic equation. So my question is can you guys point me to some exercises about recurrence relation sequences using the... |
H: $\sum\sin(\frac{k\pi}{4})$ absolute convergent, conditional convergent,divergent?
Would the following be abs convergent, conditional convergent or divergent.
$\sum\sin(\frac{k\pi}{4})$
I know sin(x) is between $-1<x<1$
y=sin(x) is oscillating would it be $(-1)^n$
AI: Sorry about the old answer, that was just garba... |
H: Prove « If P(A) is a subset of P(B) => A is a subset of B »
I need to prove «If P(A) is a subset of P(B) => A is a subset of B», generally, I understand the main way I should prove it, but the problem is in the formal, pedantic language I have to use to prove such statement.
General proof is:
1. Let suppose «a» i... |
H: Compute $ I(n) = \int^{+\infty}_{-\infty} \frac{e^{n x}}{1+e^x}\, dx$
I need compute this definite integral for values 0 < n < 1. I am not sure how to begin at all. I was able to compute the indefinite integral for the special case of n = 1/2, but I am unable to use the same strategies of substitution.
$I(n) = \int... |
H: $x^{311} \equiv 662 \mod{713}$
We have to find $x$ such that $x^{311} \equiv 662 \mod{713}$
The example given in my notes has a few typos and the professor is unavailable.
AI: $$713=23 \cdot 31 $$
By Chinese Remainder Theorem, you need to solve
$$x^{311} \equiv 662 \pmod{23} \\
x^{311} \equiv 662 \pmod{31}$$
or equ... |
H: Probability: Random Variables
Let's $T_1$ be a random variable with pdf:
$$f(t) = \frac{6+2t}{7}$$
and $T_2 \sim Exp(\frac{1}{3})$
Knowing that $T_1$ and $T_2$ are independent calculate
$$P(T_1 + T_2 > 1) $$
During my classes we produced the following result:
$$\int_0^1 \int_{1-t}^{\infty} \frac{6+2t}{7} \frac{1}{... |
H: Preservation of moments under convergence in distribution
Let $X_n$ be a sequence of random variables with first moment uniformly bounded.
If this sequence of random variables converges to $X$ in distribution,
then we have the inequality $ \lim E (|X_n| ) \geq E (|X|). $
I am aware of examples where the inequality ... |
H: Radius and Interval of Convergence of $\sum (-1)^n \frac{ (x+2)^n }{n2^n}$
I have found the radius of convergence $R=2$ and the interval of convergence $I =[-2,2)$ for the following infinite series:
$\sum_{n=1}^\infty (-1)^n \frac{ (x+2)^n }{n2^n}$
Approach:
let
$a_n = (-1)^n \frac{ (x+2)^n }{n2^n}$
Take the ratio ... |
H: Where is the tensor product of two unit vectors projection onto?
I know that $\bar{e} \otimes \bar{e}$ is a projection onto $\bar{e}$.
Then, I start to think where is then $\bar{e}_{i} \otimes \bar{e}_{j}$ projection onto.
Where is the expression $\bar{e}_{i} \otimes \bar{e}_{j}$ projection onto?
It is likely a t... |
H: Showing S is an equivalence relation in X when we know R is a reflexive and transitive relation in X.
I have this question and can't quite grasp it..I'll write down what it says then go through what I've tried.
Let $R$ be a reflexive and transitive relation in $X$. Let $S$ be a relation in $X$ such that $(x,y) \in ... |
H: Contrapositive clarification
Let's say I have this statement:
∀ real numbers x, if −x is not irrational, then x is not irrational.
Which one of the following statements is equivalent to this? [because −(−x) = x],
1.∀ real numbers x, if x is rational, then −x is rational.
2.∀ real numbers x, if -x is rational, then... |
H: Values of $\frac{e^{-x}}{1+x}$
For $$f(x)={\frac{e^{-x}}{1+x}}$$ where $ x\not=-1$ decide which values $f(x)$ could take.
Should I take the limit as $x$ → -1 in each direction? And then as x → ± $\infty$? If so, I'm not sure of how that would be done.
AI: You want to find the range of the function $f$:
$$
f(x) = \... |
H: Lagrange Multipliers where no solutions(s) satisfy the constraints
$f(x) = x-y$ subject to constraint $x^2-y^2=2$
Using the method of Lagrange Multipliers, we get:
$(1, -1) = \lambda(2x, -2y)$ which gives $x=y$ but this does not satisfy $x^2-y^2=2$
Is this because the Lagrange method is not applicable here? Or i... |
H: What is the difference between `convergence radius` and `convergencee interval`?
I have a power series $ \sum^\infty_0 = a_nx^n $ , and I have to find the convergence interval and convergence radius.
The convergence radius is $\lim_{n \to \infty} \frac{1}{\sqrt[n]a_n} $, but what is the convergence interval?
AI: Th... |
H: Relation between blowing up at a point and at a variety
maybe this is an idiot question, but I have to ask it. Usually, in the classical background, one defines the blow up $Bl_Y(X)$ at a variety $Y$ in as the closure of the graph of the function $f: X/Z(f_1, f_2, ...f_m) \longrightarrow \mathbb{P}^{m-1}$ where $f$... |
H: If E is a Hilbert space and $T \in B(E)$ is compact, show $T(E)$ does not contain a closed infinite dimensional subspace
It's the problem from "Essential Results of Functional Analysis," R.J. Zimmer, Chapter 3, problem 3.1.
I try to prove this problem and I am confused with the condition "closed infinite dimensiona... |
H: If $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, 1-1, and on-to, then f maps Borel Sets to Borel Sets
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous, on-to, and 1-1. Prove that if $A$ is a Borel set, then $f(A)$ is a Borel set.
AI: Let's not worry about whether $f$ is a homeomorphism, it ends up not p... |
H: Use single variable calculus methods to find the area of the region
Use single variable calculus methods to find the area of the region in the first quadrant bounded by the curves $y^2=3x$, $y^2=4x$, $x^2=3y$, $x^2=4y$
Could someone please show me how to go about this?
AI: First, express all the functions as functi... |
H: Will somebody look over my real-analysis/calculus solution of $\lim\limits_{x\to 0}\frac{1}{x} \int_0^{x} \sqrt{9+t^2}\mathrm{d}t$
Evaluate $\lim_{x\to 0}\frac{1}{x} \int_0^{x} \sqrt{9+t^2}\mathrm{d}t$.
Immediately, taking,
$$\begin{align}
&\lim_{x\to 0}\frac{1}{x} \int_0^{x} \sqrt{9+t^2}\mathrm{d}t \\
=&\frac{\lim... |
H: Is this ODE solvable?
this equation popped up when I was trying to apply math:
$$Af^2+B\left(\frac{df}{dx}\right)^2=C$$
Where $A,B,C$ are positive constants. Also, $~f(0)=D$, another positive constant. What are the solutions? Is this solvable? If not, are there any texts you recommend to read?
BTW you might want t... |
H: Does the Gauss' Trick Really Belong to Gauss?
We all have heard the story of the young Guass, summing 1 to 100 by writing the sum backward below the original one.
In this article, just two books are referred for the trick. I looked at both of them but the story was just mentioned briefly without any firsthand refer... |
H: Closed ball is not compact
Show that the closed ball in $C([0,1])$ of center $0$ and radius $1$ is not compact.
I thought it will be compact since every closed and bounded set in $\mathbb{R}$ is compact?
Why is it not compact and how can I prove it?
AI: Let $f_n$ be zero except on $[\frac{1}{2}(\frac{1}{n+1}+\frac{... |
H: Notation in group theory?
I have three questions on notation.
What does the squiggly line mean in the following:
$Aut(G) \cong Aut(G) \wr \mathbb{Z}_2$
What does $\rtimes$ mean in the following:
$ \varphi :(Aut(G)\times Aut(G))\rtimes\mathbb{Z}_2\rightarrow Aut(G)$
What does $\succeq$ mean in the following:
For so... |
H: Reducibility of Equivalent Polynomials
This feels like a trivial question but somehow I couldn't come up with an immediate solution. Given polynomial $P$ in a multivariate polynomial ring over some base field $F$, if $P$ is irreducible over $F$, does it follow that any equivalent polynomial is also irreducible over... |
H: Combinatorics: complete set of solution to the congruence
$$111x \equiv 112 \bmod 113$$
I've tried all the theorems. The only thing I found is that there is only one solution. Other than trying all 1-113 possible values that x takes, is there any efficient way to do it?
AI: Use negative numbers. We have $111\equiv ... |
H: How do you prove translation invariance of Fourier transform?
Let $f$ be a rapidly decreasing function in the sense that it lies in the Schwartz space $\mathcal{S}(\Bbb{R})$.
Then $\widehat{f(x+h)} = \hat{f}(\omega) e^{i 2 \pi h \omega}$, where $\hat{f}(\omega)$ is the Fourier transform of $f(x)$.
How do I prove t... |
H: Showing that $|\phi(\mathcal{N})| = \kappa$ s.t. $\mathcal{M} \equiv \mathcal{N}$ with $|\mathcal{N}| = \kappa$
Problem: Suppose $\mathcal{M}$ is an $L$-structure and $\phi \in L_n$ ($n > 0$) is such that $\phi(\mathcal{M})$ is infinite.
Then show that for every cardinal $\kappa$ with $\kappa \ge |L|$ there is an $... |
H: Bound of power series coefficients of a growth-order-one entire function
An entire function $f(z)$ satisfies
$$|f(z)| \leq A_\varepsilon e^{2\pi(M+\varepsilon)|z|}$$
for every positive $\varepsilon$. I want to show that
$$\limsup_{n \to \infty}\ [f^{(n)}(0)]^{1/n} \leq 2\pi M.$$
Alternatively, we can state the resu... |
H: $E[X]=1.8$, where $X$ is the total number of successes of 3 trials. What is the largest/smallest $P\{X=3\}$ can be?
Since each trial has the same probability of success, $p$, can you not uniquely solve for $p$? I.e:
Let, $X_{i} = 1$ if the $i^{th}$ trial is a success ($0$ otherwise). Then, $X=\sum_{i=1}^{3}X_{i}$,... |
H: Limits of integration for a joint PDF
I have $f_{X,Y}(x,y) = \lambda^2e^{-\lambda y}$ for 0 < x < y.
If I want to show that this is a joint PDF, I need to do a double integral and show that it is equal to 1. Do I set my integration limits up as 0,y and x,$\infty$ for x and y respectively?
AI: There are two ways. On... |
H: Confusion in Burnside's proof that any $2$-generated group of exponent $3$ is finite?
I'm reading a proof of Burnside's theorem that groups of exponent $3$ are finite, but have some problems.
Let $G=\langle x,y:z^3=1\rangle$ be a group generated by $x$ an $y$ with exponent $3$. Let $a=yxy^{-1}$ and $b=y^{-1}xy$. T... |
H: How to parametrize the curve of intersection of two surfaces in $\Bbb R^3$?
I have to parametrize the curve of intersection of two surfaces. The surfaces are:
$$y^2 = z \text{ and } x + y = 4$$
Could someone please show me how to do this step by step?
Thanks.
AI: Let $y=t$. Then $x=4-t$ and $z=t^2$ and so $\vec ... |
H: Existence of certain set
Problem: Let $0<a<1$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Show that:
(i) There exists a closed set $A\subseteq[0,1]$, which does not contain any non-empty open sets, such that $\lambda(A)=a$.
(ii) There exists an open, dense set $B\subseteq[0,1]$ such that $\lambda(B)=a$.... |
H: Let m be an odd integer. Since gcd(2,m) = 1,2 is invertible modulo m. What is the inverse of 2 modulo m?
Let m be an odd integer. Since gcd(2,m) = 1,2 is invertible modulo m. What is the inverse of 2 modulo m? Justify your answer.
I know that m being odd has a crucial part in this solution and what does the notatio... |
H: Stabilizer subgroups - proof verification
I have a problem that I would like help on. I'm preparing for an exam, and I have provided my work below.
Problem statement: Let $G$ act on $X$, and suppose $x,y\in X$ are in the same orbit for this action. How are the stabilizer subgroups $G_x$ and $G_y$ related?
My attemp... |
H: Differential Equations/IVP: $\frac{dy}{dt} = 4 - y^3$ and $y(-1)=2$
Transform the given initial value problem into an equivalent problem with the initial point at the origin.
$$\cfrac {dy}{dt} = 4 - y^3 \\ y(-1)=2$$
I have no idea about how to solve it.
Could you please help?
AI: Your I.V.P. looks like $y'=f\circ y... |
H: Non-negative, real matrix $\Rightarrow$ non-negative, real eigenvalues?
Does a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such?
Ideas: Perhaps we can multiply a prospective eigenvector so its biggest entries are positive, and then show that it i... |
H: How do you generalize the Laplace transform to more variables?
Just what the title says. How can I take the Laplace transform of $f(x,y,z)$ ?
AI: For multivariable version of Laplacian transformation, assume $t=(t_1,t_2,t_3)$, $X=(x,y,z)$, then
$\mathcal L\{f(x,y,z)\}=\int e^{t\cdot X}f(X)$. |
H: Any two paths in $X = \mathbb{R}^n$ having same initial and end point are homotopic
Suppose $X = \mathbb{R}^n$. Let $\gamma, \alpha : [0,1] \to X $ be to paths such that $\gamma(0) = \alpha(0) = x_0 , \; \; \gamma(1) = \alpha(1) = x_1$. We want to show $\gamma$ and $\alpha$ are homotopic.
My try: Take $F(s,t) = f_... |
H: How to find a transformation matrix, given coordinates of two triangles in $R^2$
I am an undergraduate student, and today I was given two triangles, $T_1$ (green) and $T_2$ (blue) in $R^2$:
I was then asked to find the transformation matrix transforming $T_1$ to $T_2$.
What I understand from this is, that I need ... |
H: Is it possible that all subseries converge to irrationals?
Does there exists a positive decreasing sequence $\{a_i\}$ with $\sum_{i\in\mathbb{N}} a_i$ convergent, such that $\forall I\subset\mathbb{N},\sum_{i\in I}a_i$ is an irrational number?
Such an example would give rise to a closed perfect set containing no ra... |
H: Show that if $R_n$ is prime then $n$ must be prime.
this is an exam practice question:
For each positive $n$ define $R_n = \frac{1}{9}(10^n-1) $ (so that in
the usual base 10 notation, $R_n = 111,\ldots,1$ where there are n
digits).
Show that if $R_n$ is prime then $n$ must be prime.
Here is what I have so fa... |
H: How to prove even subsets equal to odd subsets?
There is question that I don't know how to prove.
we have set $A=\{1,2,3,\ldots,n\},\; O=\{B\mid B⊆A,\text{ odd }B\},\; E=\{B\mid B⊆A,\text{ even }B\}$ it ask to prove that subsets even equal to subsets odd by proving that $f:O\to E$ is an injective and surjective fun... |
H: Differential Equation: $y'=ty+1$ , $y(0)=0$
For
$y'=ty+1$, $y(0)=0$
determine $w(n)$ for an arbitrary value of $n$. (Picard's iteration method).
I found a solution, but not sure if it true or not.
$$w(n) = \sum_{n = 1}^{\infty} \frac{t^{2n - 1}}{3 \cdot 5 \cdot 7 \cdots (2n - 1)}$$
Need some help.
AI: First write ... |
H: Set of orthonormal vectors can be identified with set of matrices
Let $V_k(\mathbb{R}^n)$ be the set of all orthonormal $k$-tuples of vectors $v_1,\ldots,v_k\in\mathbb{R}^n$. Let $M_{k,n}$ be the set of all $k\times n$ matrices. Let $W=\{A\in M_{k,n}\mid AA^t=I_k\}$, where $I_k$ is the identity $k\times k$ matrix. ... |
H: Show that $(0,1)$ is open in $\mathbb{R}$ and that $[0,1]$ is not open in $\mathbb{R}$
My Question: I want to show that the interval $(0,1)$ is open in $\mathbb{R}$ and that $[0,1]$ is not open in $\mathbb{R}$. I proved the latter, but I felt that it was clumsy and could be refined. If you could also show that $(0... |
H: A question about reduciblility
Why a polynomial $f(x)＝2x^2+4$ is reducible over $\Bbb C$? Isn't 2 a unit on $\Bbb C$? Hop someone can explain it clearly. Thanks.
AI: Our polynomial factors for example as
$$2x^2+4=(2x-2\sqrt{2}i)(x+\sqrt{2}i).$$
Remark: It is true that $2x^2+4=2(x^2+2)$ does not prove reducibility, ... |
H: Computing derivative of function between matrices
Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map $\phi(A)=AA^t$. Show that $D\phi(A)$ can be identified with the map... |
H: Limit of Fuctions
Let
$f(x)=
\left \{
\begin{array}{cc}
x & x\in \mathbb{Q}\\
0 & \,\,\,\,\,\,x\in \mathbb{R}\setminus\Bbb{Q}
&
\end{array}
\right . $
Determine all $a \in \mathbb{R}$ for which $\lim_{x \rightarrow a} f(x)$ exists.
I see that the answer is $a=0$, but I don't know how to prove it.
AI: Consider ... |
H: Show that $\frac{a+b}{2} \ge \sqrt{ab}$ for $0 \lt a \le b$
I have to prove that
$$\frac{a+b}{2} \ge \sqrt{ab} \quad \text{for} \quad 0 \lt a \le b$$
The main issue I am having is determining when the proof is complete (mind you, this is my first time). So I did the following steps:
$$\begin{align}
\frac{a+b}{2} &... |
H: sum of independent random variables where $N$ is a random variable
I want to show $E[S_N]=E[N]E[X_j]$ where:
$X_1,X_2,\ldots$ is a sequence of independent random variables, and $N$ is a random variable independent of the sequence.
$S_n=\sum_{i=1}^n X_i$, $S_N=X_1+X_2+\ldots+X_N$,
So far, I have
$$E[S_N]=E\left[\su... |
H: modular arithmetic with exponents
I'm looking at the solution manual of a book and it lists a solution for
$$[19^3\mod {23}]^2 \pmod {31} \equiv [(-4)^3\mod {23}]^2 \pmod {31} \equiv [-64\mod {23}]^2 \pmod {31}\equiv 5^2 \pmod{31} = 25$$
How does it get from $[19^3\mod {23}]^2 \pmod {31}$ to $[(-4)^3\mod {23}]^2 \p... |
H: smooth bijective functon's derivative $\ne 0$?
$f\colon (\alpha,\beta)\to (a,b)$ is a smooth bijective functon, that is, the derivative $f^{(n)}(x)$ exists for all $n\ge 1$.
1) If the inverse function $f^{-1}\colon (a,b)\to (\alpha,\beta)$ is also smooth, does it follow that the derivative $f^\prime(x)\ne0$ for all... |
H: Placing K knights in an nxn board such that no two attack each other
This is a problem from spoj
A and B are playing a very interesting variant of the ancient Indian game 'shatranj(also known as chess)' on a 'maidaan'(chessboard) n×n in size.
They take turns to put game pieces called 'ghoda'(knight) so that no two... |
H: Finding the number of solutions to this equation in range 50 to 100
$x^{2}\; -\; \mbox{floor}\left( x \right)\cdot x\; =\; 83.26$
I was able to find it by graphing and counting the number of intersections within the range, but there has to be an easier way to solve this.
AI: On Ross Millikan's suggestion, I'm posti... |
H: $f:X\rightarrow Y$, $g:Z\rightarrow Y$ then $h:X\times Z\rightarrow Y$ continuous
$\textbf{Lemma}$: If $f:X\rightarrow Y$ and $g:Z\rightarrow Y$ are continuous functions with $X,Y,Z\subseteq \mathbb{R}$ then $h:X\times Z\rightarrow Y$ via $h(x,z)=f(x)g(z)$ is continuous.
Is there a way to show the above that uses... |
H: Demonstrating that $f(x) = x^2 + 1$ is bijective and calculating $f \circ f^{-1}(x)$
I did this exercise. I am not sure if my surjective proof is right - is it good? In fact, I'm not even sure what exactly am I trying to prove (I know that surjective means that each element in the codomain should have a preimage in... |
H: Map to symmetric matrices is surjective.
Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that $AA^t=I_k$. Let $f:M_{k,n}\rightarrow S_k$ be the map $f(B)=BA^t+AB^t$. Prove that $f... |
H: In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
Prove that in a triangle $ABC$, $\angle A = \angle 2B$, if and only if:
$$a^2 = b(b+c)$$
where $a, b, c$ are the sides opposite to $A, B, C$ respectively.
I attacked the problem using the Law of Sines, and tried to prove that if $\angle A$ was indeed equal to... |
H: Properties of $A + B$ related to $A$ and $B$. (GS2010)
Let $A$ and $B$ be subsets or $\mathbb{R}$. Define $A + B = \{ a + b : a \in A, b \in B\}$. Then which of the followings are true and which are false? Why? Please give a proof for truth and a counterexample or false. Will the situation be different if we consid... |
H: If $f$ is injective, demonstrate that $f \circ g = f \circ h \implies g = h$
I'm trying to do this exercise:
With functions:
$$f : A \rightarrow B$$
$$g : C \rightarrow A$$
$$h : C \rightarrow A$$
Demonstrate that if $f$ is injective, then $f \circ g = f \circ h \implies g = h$
So we have two premises:
$f$ is in... |
H: How should I express one $\log$ in terms of others?
Can someone please help me with this logarithmic question? I know it’s easy, but I need to refresh my memory on how to do it.
If $X=\log2$ and $Y=\log3$, express $\log0.6$ in terms of $X$ and $Y$ (assume all logs have a base of $10$).
AI: All you need to know is ... |
H: reducing enemies - geometry puzzle
You are given an irreflexive symmetric (but not necessarily transitive) "enemies" relation on a set of people. In other words, if person A is an enemy of a person B, then B is also an enemy of A. How can you divide up the people into two houses in such a way that every person ha... |
H: $\mathcal{C}_1 \subseteq \mathcal{C}_2 \implies \sigma( \mathcal{C}_1) \subseteq \sigma( \mathcal{C}_2) $
$\mathcal{C}_1$, $\mathcal{C}_2$ are collections of subsets of $X$,then Im having hard time seeing why the following is true. Can someone explain them to me?
$\mathcal{C}_1 \subseteq \mathcal{C}_2 \implies \si... |
H: Question on series
Suppose $ a_i $ be a sequence of positive real numbers such that $ \sum_{i=1}^{\infty}a_i < \infty $. Is it true that $ \sum_{i=1}^{\infty} \log(i) \cdot a_i < \infty $?
Thanks
AI: No, a counterexample is $a_n = \frac{1}{n \log^2 n}$. |
H: The Limitations of Vieta's Formula
I was attempting to find the roots of $f(x)=2x^3+10x^2+5x−12$ and since the the OP had already found one of the roots, I tried to recall a relation to help me find the other two easily. The first one that popped to mind were Vieta's formulas for Cubic Polynomials:
$$x_1+x_2+x_3=-\... |
H: Evaluation of $\int\frac{1}{(x^3+1)^2}\mathrm dx \cdots$
$\displaystyle \int\frac{1}{(x^3+1)^2} \mathrm dx$
$\bf{My\; Try}::$ Using Integration by parts
Let $\displaystyle I = \int\frac{1}{(x^3+1)}\cdot 1\; dx = \frac{1}{(x^3+1)}\cdot x + \int\frac{3x^2\cdot x}{(x^3+1)^2}dx$
$\displaystyle I = \frac{1}{(x^3+1)}\cdo... |
H: Product of randomly drawn numbers
Here are two code line to run in R:
prod(rnorm(100, mean=1, sd=0)) # (1)
prod(rnorm(100, mean=1, sd=0.2)) # (2)
$prod(..)$ returns the product of a sequence. The sequence it given by $rnorm(n, mean, sd)$. This function $rnorm(...)$ return n values randomly drawn from a normal dis... |
H: where to start reading theory of logics?
I am a student who is working Lie Theory. I want to start read theory of logics. I just need some reference and I have few questions regarding this,
i) will studying theory of logics will improve my theorem proving ability in 'other branch' of mathematics?
ii) is there any r... |
H: If $G$ and $H$ are nonisomorphic group with same order then can we say that $\operatorname{Aut}(G)$ is not isomorphic to $\operatorname{Aut}(H)$?
We know that nonisomorphic groups may have isomorphic automorphism groups. As an example, you can think klein four group and $S_3$ since their automorphism group is isomo... |
H: Linear subspace of Banach space containing unit ball
Am I right that any linear subspace of Banach space which contain unit ball is whole space?
AI: Yes, take any vector in the space and divide it by (a little more than) its and you'd end up with a vector within the unit ball. Hence, every vector in a Banach space ... |
H: Inverse Percentage.
Sorry for asking this foolish question.
Here is the data i have.
I purchased the product as $5 and additional fee is 2%.
So Here is the total dollor
$total = 5 + (5*2/100) = 5.1 total dollor i have.
But now i want to revert back to original price
in this case i have the data as only
total amou... |
H: Stokes' theorem for an annulus
From Wiki, I'm looking at this definition:
"If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation... |
H: Integral of $dy/dx$ confusion
Why is $\displaystyle \int \dfrac{dy}{dx} dx = y + c$, but for example $\displaystyle \int \dfrac{dy}{dx} y dx = \dfrac{1}{2} y^2 + c$ instead of $\dfrac{1}{2} y^3 + c$?
AI: Using and abusing the mathematical notation as sometimes is done when dealing with differential equations, what ... |
H: Question on pointwise convergence of a function
Let $f_n:\mathbb R^+\to\mathbb R$ for $n \in \mathbb N$ be given by
$$f_n(x)=\begin{cases}n & x \in \left(0,\frac1n\right)\\
\frac1x & x \in \left[\frac1n,\infty\right) \end{cases}$$
I have to show that it would converge pointwise to $\frac1x$ for all $x \in \mathbb ... |
H: Angle between $\vec a$ and $\vec b $.
We got the same size vector $\vec a$ and $\vec b $.
We know that the vector $\vec a +2\vec b$ and $5\vec a-4\vec b$ are perpendicular?
$(\vec a +2\vec b) \perp (5 \vec a-4\vec b)$
What is angle between $\vec a$ and $\vec b $?
AI: $$
(\vec{a} + 2 \vec{b})\cdot(5\vec a-4\vec b) ... |
H: Set Theory Symbols
I get easily confused when it comes to the symbol based terminology in Set Theory. Could someone please elaborate on what the following expressions mean? It would really help me out.
$S_1$ = knowledge of a subject matter
$S_2$ = problem solving related to this subject matter
$S_3$ = ability ... |
H: Is it possible to generalize the Mean value theorem for integral not on compact set?
I wonder it is possible to extend the mean value theorem not on compactness.
In more detail, Let $f : A \rightarrow \mathbb{R}$ be continuous on $A \subset \mathbb{R}^n$. The mean value theorem for integral states that if $A$ is c... |
H: Ordinals: if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$
I'm trying to proof that if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$, where $\omega$ is the least infinite ordinal.
I started with transfinite induction on $\beta$ by proving it for $\beta = 0$ ... |
H: finding the lim of the following set
I'm given:
$$a_0,\ldots,a_k\ge0$$
for the following set:
$$\lim_{x \to \infty} \sqrt[n]{a_kn^k+a_{k-1}n^{k-1}+\cdots+a_1n+a_0}$$
I need to find the limit of the set. Any help appreciated.
AI: Let assume $a_k\ne 0$ so since
$$n^i=_\infty o(n^k)\quad \forall i<k$$
then
$$\sqrt[n]... |
H: Proving that $R_1 = R_2$ (linearly ordered sets)
Let $R_1$ and $R_2$ be linearly ordered sets in set $X$.
Prove that if $R_1R_2$ is linearly ordered set, then $R_1 = R_2$
I understand the defnitions of $R_1R_2$ and totally ordered sets, but when I start to prove, it seems that I still lack some knowledge. Definitio... |
H: Homework - How many non negative solutions?
$a+b+c+d+e = 30$
we know that $10\leq e$ and that $4\leq d \leq 7$
How many non negative solutions does this equation have?
AI: Let $a'=a$, $b\,'=b$, $c'=c$, $d\,'=d-4$, and $e'=e-10$; $30-(4+10)=16$, so you’re trying to count solutions in non-negative integers to
$$a'+b... |
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