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H: Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$.
Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$ for some $q\in \mathbb{Z}$.
Any ideas on how to start? Do I use a proof by contraposition? Also what's the definition of a perfect square?
AI: Hint: A perfect ... |
H: Summation of reciprocal of Product of Factorials.
How can this summation be evaluated:
$${∑ {1\over {a_1!a_2!....a_m!}}}$$
Where $$a_1+a_2+.....+a_m=n$$
Also $a_i !=n $ and $m<n$.
AI: $\sum\frac{n!}{\prod_ia_i!}=\sum\frac{n!}{\prod_ia_i!}\prod_ix_i^{a_i}$ with $x_i=1$;
hence $\sum\frac{n!}{\prod_ia... |
H: Show that 2S = S for all infinite sets
I am a little ashamed to ask such a simple question here, but how can I prove that for any infinite set, 2S (two copies of the same set) has the same cardinality as S? I can do this for the naturals and reals but do not know how to extend this to higher cardinalities.
AI: In o... |
H: Is the sum of factorials of first $n$ natural numbers ever a perfect cube?
If $S_n = 1! + 2! + 3! + \dots + n!$, is there any term in $S_n$ which is a perfect cube or out of $S_1$, $S_2$, $S_3$, $\dots S_n$ is there any term which is a perfect cube, where $n$ is any natural number.
AI: All factorials above $8!$ hav... |
H: Determine if the given integral is convergent
$$\int_0^{\pi/2}{\log x\over x^a}\,\mathrm dx,\quad a<1$$
I tried solving using the $\mu-test$. so if I consider $\mu=1$ then
$\lim\limits_{x\rightarrow 0} {x\log x\over x^a}$ Solving further, I get a limit $0$, so the integral must be divergent. But according to the bo... |
H: Minimisation of a distance sum
I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\}
\subseteq L$ in such a way value $S$ of the those K numbers is minimum.
$$
S = \sum_{0< i < j <= k} \left| x_i-x_j\right|
$$
Suppose N=4 and list is {10,20,30,100} and K = 3.
Then we can cho... |
H: Integer values of $a$ for which the expression $(x+a)(2013+x)+1$ is a Perfect Square
Calculation of all Integer values of $a$ for which the expression $(x+a)(2013+x)+1$ is a Perfect Square.
$\underline{\bf{My\;Try}}:$ Given $(x+a)\cdot (x+2013)+1 = y^2$
So $x^2+ax+2013x+2013a = y^2\Rightarrow x^2+(a+2013)x+(2013a-y... |
H: Inequality, what is wrong with $\log(-1) = - \log(-1)$?
Can anyone tell me what is wrong with the following line of argument:
$$
\log(-1) = \log(-1) - \log(1) = - \bigg( \log(1) - \log(-1) \bigg) = - \log \Big( \frac{1}{-1} \Big) = - \log(-1)
$$
Considering the complex logarithm the left-hand-side evaluates to $ i... |
H: Projection and direct sum
I want to show that for every projection $A^2=A$ we have that there exists a subspace $U_1 \subset ker(A)$ and $U_2$ such that $A|_{U_2} = id$ such that $V = U_1 \oplus U_2$. Does anybody here have a hint how to show this?
AI: Hint: (1) For $x \in V$ we have $x = (x-Ax) + Ax$.
(2) What ca... |
H: Number of ways $n$ distinct objects can be put into two identical boxes
The number of ways in $n$ distinct objects can be put into two identical boxes, so that neither box remains empty.
My Try:: If the question is the numbers of ways in $n$ distinct objects can be put into two Distinct boxes so that no box remains... |
H: An 'obvious' property of algebraic integers?
I am looking at the book A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer. I like the section on page 1 'the ring of integers' as it gives a motivation for choosing which elements we would like to regard as integers and how we get the definition in te... |
H: Show that ON-sequence is a base
I have a Hilbert space $H$ and a base $(e_n)_{n=1}^\infty$ and a ON-sequence $(f_n)_{n=1}^\infty$.
Given
$$
\sum_{n=1}^\infty ||e_n - f_n||^2 < 1
$$
show that $(f_n)_{n=1}^\infty$ is a base.
My work:
It is straight forward to rewrite the sum
$$
\sum_{n=1}^\infty (1 - \langle e_n, f_n... |
H: Does this sequence of inverse-binomial numbers have a name?
I was inspired by considering the following:
$$\left(\sum_{i=1}^n i\right)^2=\sum_{i=1}^n i^3$$
Are there exact formulas for the sums of the powers of the integers? For example, we have:
$$\sum_{i=1}^n i={n(n+1)\over 2}$$
$$\sum_{i=1}^n i^2={n(n+1)(2n+1)\... |
H: How prove this $a+b\le 1+\sqrt{2}$
let $0<c\le b\le 1\le a$, and such $a^2+b^2+c^2=3$, show that
$a+b\le 1+\sqrt{2}$
My try: let $ c^2=3-(a^2+b^2)\le b$
AI: As Sun stated, given a valid tuple of $(a,b,c)$, replace it with $(A, b, 0)$ where $ A^2 = a^2 + c^2$. Observe that $a + b \leq A + b$, hence it remains to sho... |
H: What to do when $x$ in $\Gamma(x)$ is a negative integer?
I have the following likelihood calculation:
\begin{align}\mathcal{L}(s|\alpha) = \sum_{i=1}^O\Biggl\{\ln\frac{
\Gamma(\alpha_0 )}{
\Gamma( B )}-
\sum_{k=1}^K
\ln \Gamma(\hat{\mathcal{S}}_k^i+1)
+ \ln
\biggl[
\sum_{k... |
H: sketch graph by given data
suppose that we are given following data,clearly in this case it is talking about linear form right?i meant $y=k*x+b$ form, in this case we can simply choose any two point for example $(1990,11)$ and $(1992,26)$,calculate slope and finally find
$b$,it is required right or we should ... |
H: Needing help picturing the group idea 2
Needing help picturing the group idea.
A further extension relating to the question.
Give an example where $XY$ is not a subgroup of G, where $X$ and $Y$ are subgroups of $G$. Show that $XY$ is a subgroup if $XY=YX$. Hence show in particular that $XY$ is a subgroup if either ... |
H: Order of subgroups and number of elements of order $3$ in a group of order $9$
Let $G$ be a group of order $9$.
1) State the possible orders of subgroups and elements in $G$.
2) Find the number of elements of $G$ of order $3$ in the cases where (a) $G$ is
non-cyclic, (b) $G$ is cyclic.
I have a problem with t... |
H: Limits problem: Factoring a cube root of x?
Disclaimer: I am an adult learning Calculus. This is not a student posting his homework assignment. I think this is a great forum!
$$\lim_{x\to8}{\frac{\sqrt[3] x-2}{x-8}}$$
How do I factor the top to cancel the $x-8$ denominator?
AI: Actually you need to factor the den... |
H: Prove $\int_0^1|f(t)-g(t)|dt \le (\int_0^1|f(t)-g(t)|^2dt)^{1/2} \le \sup_{t\in[0,1]}|f(t)-g(t)|$
Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$.
Let these be 3 metrics on $C$.
$p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$
$d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$
$t(f,g)=\int_0^1|f(t)-g(t)|dt$
Pr... |
H: Limits problem with trig? Factoring $\cos (A+B)$?
Disclaimer: I am an adult learning Calculus. This is not a student posting his homework assignment. I think this is a great forum!
$$\lim_{h\to0} \frac{\cos(\frac{\pi}{3}+h)-\frac{1}{2}}{h}$$
Do I use the angle addition formula to do this?
I did that, and have no id... |
H: algebra, equivalence relation regarding associates
If f(x) ~ g(x) if and only if f and g are associates,
prove this is an equivalence relation
have tried to prove this both ways, struggling
AI: Well you need to show 3 things :
Reflexivity : Take $u=1$
Symmetry : If $u$ works in one direction, then $u^{-1}$ works i... |
H: Two form of derivative $ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$
Why I can write formula derivative $$ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ in this form: $$ f'(x)=\frac{f(x+h)-f(h)}{h}+O(h)$$
I know, that it's easy but unfortunately I forgot.
AI: The formula $f'(x) = \lim_{h \to 0} \frac... |
H: Is this an Equivalence Relation and why?
if $I$ is a set of positive integers and relation $\def\R{\mathrel R}\R$ is defined over the set $I$ by $x\R y$ iff
$x^y = y^x$.
Is this an Equivalence Relation and why?
AI: Reflexivity and symmetry are trivial, so let's test transitivity:
Assume $x^y=y^x$ and $y^z=z^y$. We... |
H: How to solve $q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}$
I have come across this equation recently. All the variables are positive and real too.
$$q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}.$$
Under what conditions can this be solved for $q$?
AI: Let $q = e^x$, so that $\ln q = x.$ Also, put $A = \ln{n},$ $B =... |
H: How to factor and find zeros of $2x^6+5x^4-x^2$
In the equation $y=2x^6+5x^4-x^2$, how can I factor it in order to find the zeros or x-intercepts? This is what I've gotten to, but I can't see what to do next: $x^2(x^2(2x^2+5)-1)$
AI: $$y=2x^6+5x^4-x^2 = x^2(2x^4 + 5x^2 - 1)$$
For the factor $\;2x^4 + 5x^2 -1,\;$ pu... |
H: Morphisms of complexes chain
I have a small question:
Why is the following true?
"If we have a continuous mapping between two topological spaces $f:X\rightarrow Y$, we can associate a morphism of chain complexes $f_*\colon C_\bullet (X)\rightarrow C_\bullet(Y)$ "
thank you.
AI: I suppose you are considering singu... |
H: Simplifying pre-calc
I'm doing my homework and I'm stuck on this problem:
$$ \frac{\frac 1 {\sqrt{a}} + \frac {\sqrt{a}} 2}{\frac 1 {\sqrt{a}} - \frac {\sqrt{a}} 2} $$
I tried playing around with the powers, but failed pretty hard.
AI: Expand by $2\sqrt{a}$ and your'e done. |
H: Automorphism of $Q_8$
Is there anyone could help me to prove that $Aut(Q_8)=S_4$?
Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas?
Thank you!
AI: In $Q_8$ the central elements $1,-1$ are clearly fixed by all automorphisms. The remaining thr pairs $(i,-i)$, $(... |
H: cell-by-cell constraints within a positive-definite matrix?
One simple constraint in a positive definite matrix relates each off-diagonal cell to the corresponding on-diagonal cells:
$$ |m_{ij}| \lt \sqrt{m_{ii}m_{jj}} $$
While this may be a necessary condition, I guess it's not sufficient?
If we take a positive d... |
H: Homology of wedge sum is the direct sum of homologies
I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a neighborhood of $A$ that deformation retracts onto $A$).
What I tried:
Si... |
H: Hausdorff dimension of support of harmonic measure in complex plane
I know that harmonic measure $\omega$ in complex plane $\mathbb{C}$ is absolutely continuous with Hausdorff measure $\mathcal{H_{h_k}}$ $(\omega << \mathcal{H_{h_k}})$, where
$$
h_k(t) = t e^{k\sqrt{\log\frac{1}{t}\log\log\log \frac{1}{t}}}
$$
wit... |
H: Bound on surface gradient in terms of gradient
Let $S \subset \mathbb{R}^n$ be a hypersurface and define the surface gradient of a function $u:S \to \mathbb{R}$ by
$$\nabla_S u = \nabla u - (\nabla u \cdot N)N$$
where $N$ is the normal vector.
Is it possible to obtain a bound of the form
$$|\nabla u |_{L^2(S)} \leq... |
H: A question about Pearson correlation coefficient
Suppose that we have two vectors
$x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n)$ is the following correct about their Pearson correlation coefficient?
$\operatorname{corr}(x,y)=\operatorname{corr}(x+a,y+b)$ where $a$ and $b$ are non-zero real numbers.
AI: Yes. This is becau... |
H: Problem similar to the birthday problem
a biased coin is tossed $n$ times (each toss is independent) with probability $h$ for heads. I need the smallest $n$ that lets the probability of at least one head to be $0.9$.
I found p (no heads)=n(1-h) then p (at least one head)=1-n(1-h) then I found n terms of h. I'm not ... |
H: Ideals in $F[X]$ are of the form $(f(x))$ where $f$ can be chosen to be monic. How?
I am reading a statement whereby it says that
In $F[X]$, where $F$ is a field, any ideal is of the form $(f(x))$ where $f$ can be chosen to be monic.
I don't get this part of the statement '$f$ can be chosen to be monic'. If we hav... |
H: Finding the limit $\lim_{x\to-\infty} (2x)/(2x-1)^2$.
Studying for a midterm:
Let $f(x)=\frac{2x}{(2x-1)^2}$
Then $\lim_{x\to-\infty} f(x)$ is:
Now keep in mind I'm shaky on how to do infinity limits.
I have $f(x)=\frac{2x}{(2x-1)^2}$
Remove x by dividing by the highest common denominator:
$=\frac{2+\frac1x}{(2-\fr... |
H: Computing the limit.
Studying for a midterm.
Compute the following limit:
$$\lim_{x\to 4} \frac{x+4}{x^2+3x-4}$$
Factor the denominator:
$$\lim_{x\to 4} \frac{x+4}{(x+4)(x-1)}$$
The $(x+4)s$ cancel out:
$$\lim_{x\to 4} \frac1{(x-1)}$$
$$\lim_{x\to 4} \frac1{(4-1)}$$
$$\lim_{x\to 4}=\frac13$$
Just wondering if someb... |
H: If $f$ is a function of moderate decrease then $\delta \int f(\delta x) dx = \int f(x) dx$
A function of moderate decrease is a map from $\mathbb{R}$ into $\mathbb{C}$ such that there exists $A \in \mathbb{R}$ such that $\forall x\in \mathbb{R}, \ |f(x)| \lt \frac{A}{1 + |x|^{1+\epsilon}}$. Let $\epsilon$ be fixed... |
H: Prove the identity
$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8} = \frac{\sin x}{ 8\sin \frac{x}{8}}$$
Conjecture a generalization of this result and prove its correctness by induction.
Ps: I have tried using identities, but I keep running on a loop. I wanted to use identities first to have an id... |
H: Good upper bound for $(1-x)^r$
The Bernoulli's inequality gives a lower bound on numbers of the form $(1-x)^r$:
$$(1-x)^r\geq 1-rx$$ for integer $r\geq 0$ and real number $0<x<1$.
Is there a corresponding upper bound for $(1-x)^r$? In particular, when $r$ gets large, $(1-x)^r$ becomes very small. I suspect there sh... |
H: How can a subset be disjointed?
I have the proof: Suppose A,B, and C are sets. Prove that C⊆A∆B iff C⊆A∪B and A∩B∩C=∅.
If I suppose that C ⊆A∆B, how can the three sets be disjointed?
AI: Here's the direction that you seem to be having trouble with.
Let $x\in C$ and suppose that $C\subset A\triangle B$, then $$x\in ... |
H: Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$
How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$?
Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the maximum is at $x= e^{1+e}$.
$W$ is the Lambert-W function.
AI: Using the fact... |
H: Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix?
By definition, we have
$$
\|V\|_p
:= \sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}
\qquad \text{and} \qquad
\|A\|_p
:= \sup_{x\not=0}\frac{||Ax||_p}{||x||_p}
$$
and if $A$ is finite, we change sup to max.
However I don't really get how we ... |
H: Doubt in proof of special case of implicit function theorem
I've been studying the implicit and inverse functions theorems and I've started with one special case of the implicit function theorem. The book I'm reading states the theorem as follows:
Let $f : U\to\mathbb{R}$ be a function of class $C^k$ with $k\geq... |
H: A question about an epsilon-delta proof
Currently, I am stuck on a question:
Let $ g : [ 0 , \infty ) \mapsto \mathbb{R} $ be defined by $g(x)= \left\{
\begin{array}{ll}
x^2 & \mbox{if } 0 \leq x \leq 1 \\
3x & \mbox{if } x > 1
\end{array}
\right.$
Prove: For $\epsilon =1$, $\forall \delta >0$, $\exists x ... |
H: Differentiation of $x!$, where $x\in \mathbb{N}+\{0\}$
Calculation of $\displaystyle \frac{d}{dx}(x!) = $, where $x\in \mathbb{N}+\{0\}$
My Try:: We Know that $x! = (x)\cdot (x-1)\cdot(x-2)...........(3)\cdot(2)\cdot(1)$
Now Taking $\bf{\ln}$ on both side
$\bf{\ln(x!)} = \ln (x)+\ln(x-1)+\ln(x-2)+................+\... |
H: Given $f(x)$ its inverse function, domain and range
$f(x) = \frac{{2x + 3}}{{x - 1}},\left[ {x \in {R},x > 1} \right]$
I've got the inverse function to be:
${f^{ - 1}}(x) = \frac{{x + 3}}{{x - 2}}$
How would I go about working out the range and domain of this function? The range is possibly $f^{-1}(x)>1$ if im not ... |
H: What is the name of this delta operator
In Euler-Lagrange Equation: $${\delta \over \delta y}F \equiv {\partial F \over \partial y}- {d \over dx} ({\partial F \over \partial {y'}})$$
What is the name of operator $\delta$ here?
AI: It's called the functional or variational derivative with respect to $y$. |
H: Is my alternative proof that the intersection of nested compact sets is nonempty valid?
Let $\{A_n\}_{n=1}^\infty$ be a collection of nested closed sets in a compact space $X$. Since $A_n$ is closed, it is compact, and consequently limit point compact. Let $\varepsilon > 0$ and define a sequence $(x_n)$ where $$ x_... |
H: Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.
Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.
Clearly it's infinite dimensional, because if yo... |
H: If $f(x)$ is positive and decreasing, can $xf(x)$ have more than one maxima?
Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that $xf(x)$ has more than one maxima on $[0,1]$?
... |
H: A question from Eisenbud, Commutative Algebra
On page 35, the proof of corollary 1.8:
If k is an algebraically closed field and A is a k-algebra, then A = A(X) for some algebraic set X iff A is reduced and finitely generated as a k-algebra.
In the proof, it says: "... Conversely, if A is a finitely generated k-alge... |
H: How to show some sets below to a $\sigma$-field $\mathcal{F}$
Let $A$ and $B$ belong to some $\sigma$-field $\mathcal{F}$.
How would I show that $A\cap B,$ $ A\setminus B $ and $A \Delta B:=(A\setminus B)\cup(B\setminus A)$ below to $\mathcal{F}$ as well and
how would I find $P(A \Delta B)$ in terms on $P(A)$, $P... |
H: $E[X^4]$ for binomial random variable
For a binomial random variable $X$ with parameters $n,p$, the expectations $E[X]$ and $E[X^2]$ are given be $np$ and $n(n-1)p^2+np$, respectively.
What about $E[X^4]$? Is there a table where I can look it up? Calculating it using the definition of expectation looks like a lot o... |
H: What is the measure of $x \in [0.1]$ whose binary representations have percentage of ones that converge within a given range?
For binary representation of $x \in [0,1]$, i.e. $x = \sum_n a_n 2^{-n}$, (where all $a_n$ are binary, and using all trailing ones is chosen instead of rounding up), let $X(b,c) \subset [0,1... |
H: How do I find where a curve hits the $xy$ plane with vectors?
I need to find $r'(t)$ and $||r'(t)||$ of $r(t)=<t,t,t^2>$, and tell where the curve hits the $xy$ plane (if it does). Also, I need to say something about how the curve looks like (do I just plot some points and figure out how it looks like?). I understa... |
H: Additive norm $||a+b||=||a||+||b||$
I've read somewhere that there exist spaces where $||a+b||=||a||+||b||$ is true iff $a = \lambda b, \ \ \lambda>0$.
Could you tell me what spaces have that property and what spaces don't?
$||\lambda b + b|| = |\lambda +1| \cdot||b||$ by homogeneity and I don't know what conditio... |
H: Tennis balls counting problem
On a Friday morning, the pro shop of a tennis club has 14 identical cans of tennis balls. If they are all sold by Sunday night and we are interested only in how many were sold in each day, in how many different ways could the tennis balls have been sold on Friday, Saturday and Sunday?
... |
H: Lifting elements of $SO(3)$ to $SU(2)$.
Let $A$ an element of ortogonal group $SO(3)$ such that the orders of $A$ is $>2$. We have that $SU(2)$ is a $2$-fold cover of $SO(3)$:
$$ \mathbb{Z}_2 \to SU(2) \to SO(3) .$$
So how can I build a lift of $A$ to an element $\tilde{A}$ is $SU(2)$?
AI: Let $\mathbb{H}$ be the q... |
H: Solution of PDE with directional derivative
There is a partial differential equation containing directional derivative in the left-hand side:
$$
\vec{s} \cdot \nabla f = a f + b \\
$$
where $f, a, b$ are functions of $(x,y,z)$, and $\vec{s}$ is a unit direction vector. How to solve this type of equation? Standard ... |
H: Develop second-order method for approximating f'(x)
I am stuck on the following question:
Develop a second-order method for approximating $f'(x)$ that uses the data $f(x-h), f(x)$, and $f(x+3h)$ only.
Do you have any hints or tips? Thanks in advance.
AI: Expanding on Mhenni Benghorbal's link:
$f(x+h)
=f(x)+hf'(... |
H: Show that $\overline{(\textbf{y},\textbf{x})}_A = (\textbf{x},\textbf{y})_A$
I want to show that $\overline{(\textbf{y},\textbf{x})}_A = (\textbf{x},\textbf{y})_A$ where $(\textbf{x},\textbf{y})_A =\textbf{x}^T A \textbf{y}$, $\textbf{x},\textbf{y} \in \mathbb R^p$ and $A$ is a real symmetric $p \times p$ matrix . ... |
H: Find all the integral solutions to $2x+3y=200$
What's the best way of going about this?
$$2x+3y=200.$$
AI: For this case,
if $2x+3y=200$,
an obvious solution is
$x=100, y=0$.
From this base solution,
all other integer solutions are
$x=100-3n$,
$y=2n$
for integer $n$.
If the solutions are to be
non-negative,
then
$... |
H: Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$
I have a problem to solve this exercise, I hope someone help me.
Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$
AI: One looks for the complex numbers $z=x+\mathrm iy$, $z\ne0$, such that $z=\bar z^2$.
In particular, $|z|=1$. Plugging $z... |
H: Relations in Propositional Logic
It is my understanding that relations are best described with predicate logic. I have a homework question that asks me to convert English sentences into propositional logic. The following list of sentences are similar to the homework, but do not reflect the actual assignment.
a) Hum... |
H: Inversing a function
I'm having some problems calculating the inverse of this function:
$f(u,v)=(u+v,v-u^2)$, its domain is $D=\{(u,v)$ in $\Bbb R^2 : u>0\}$
Thanks in advance.
AI: If you consider the inverse function $f^{-1}(a,b)$, you have to solve
$$\left\{\begin{array}{l}
u+v = a\\
v-u^2 = b
\end{array}\right.$... |
H: Simple notation question
Let A = {2, 3, 4, 6, 7, 9} and define a relation R on A as follows:
For all x, y ∈ A, x R y ⇔ 3 | (x − y).
Then 2 R 2 because 2 − 2 = 0, and 3 | 0.
What does the 3 | 0 notation mean here?
AI: $\;3\mid 0\; $ = three divides zero, which means there exists an integer $\;k\;$ s.t.
$$0=3\cdo... |
H: Why is the Brownian motion a multivariate normal distribution?
I have seen in class that for some reasons I forgot, the Brownian Motion has a Multivariate normal distribution, but I am unable to prove it easily. Could someone tell me why it's true?
From what I understand, I have to take a finite linear combination ... |
H: How to solve the equation $x^2=a\bmod p^2$
What is the standard approach to solve $x^2=a\bmod p^2$ or more general $x^n = a\bmod p^n$ ?
AI: The usual method for solving polynomial equations modulo $p^n$ is to solve it mod $p$, then use some method to extend a solution from mod $p$ to mod $p^2$, then to mod $p^3$, a... |
H: Why does substitution work in integrals
Let's say I have this integral:
$$\int_0^\infty e^{-t} \, dt$$
And I make the substitution:
$$t = nu$$
Then why I can say that:
$$dt = n\,du$$
and then put this into my integral like this:
$$\int_0^\infty e^{-nu}n\,du$$
What's happening in the background that allow this to be... |
H: Simplifying a trigonometric function
Can anyone show me the steps to get from:
$$\dfrac{\cos (x)}{1+\sin(x)}+\frac{1+\sin(x)}{\cos(x)}$$
To:
$$2\sec(x)$$
AI: First do the obvious thing and combine the fractions over a common denominator:
$$\begin{align*}
\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}&=\frac{\cos^... |
H: Calculating individual wheel velocities from a desired angle in a differential wheeled robot
I am working on a simulation of a two-wheeled robot, and at present am driving it by setting each individual wheel's velocity. The robot is similar to an ePuck:
What I would like to do is set an initial (and constant) ove... |
H: How does my professor go from this exponential equation to a logarithmic one?
How does the "therefore" portion work? How does that exponential equation come to equal n(lgn + 1)?
AI: In the first line, $n=2^k$, so from the next to last we substitute in. Given this definition, $k=\lg n, k2^k=n \lg n$ |
H: Is it true that if $n$ is even then $\sum_{k=1}^{n}(n \bmod k)<\frac{8}{45}n^2$?
Let $f(n,k)$ be the least non-negative integer such that $n\equiv f(n,k) \bmod k.$
$f(10,k)(k=1,2,\cdots,10)=0, 0, 1, 2, 0, 4, 3, 2, 1, 0.$
Hence $$\sum_{k=1}^{10}f(10,k)=1+2+4+3+2+1=13.$$
Question: Is it true that if $n$ is even the... |
H: All values of $z$ s.t. $e^z= 1+\sqrt{3}i$
I'm trying to find all values of $z$ such that $e^z= 1+\sqrt{3}i$ and am getting stuck.
I know
$$e^z=e^{x+iy}$$
from this I've done
$$e^{z}=e^x\cos(y)+e^x\sin(y)i=1+\sqrt{3}i$$
giving that
$$e^x\cos(y)=1$$
$$e^x\sin(y)i = \sqrt{3}i$$
but can't find a value for $y$ in the... |
H: Quick Algebric Trig question
The question is:
I need to solve the equation for all values of x between 0 < x < 360
Now I got it to here and got a bit stuck:
= 0
AI: Quadratic equation: $$cos(x) = \frac{-1 \pm \sqrt{17}}{2}$$
Can you continue? |
H: Proof regarding GCD
I'm trying to prove that if $a,b$ are two primes between themselves then $a+b$ and $a^2+ab+b^2$ are also prime between themselves.
That is, we have to prove that $\text{gcd}(a,b)=1\Rightarrow \text{gcd}(a+b,a^2+ab+b^2)=1$ .
Should I try Bezout ? Any hint on what should I proceed on doing ... |
H: Simplify $\sum_{k=0}^n \frac{1}{k!(n-k!)}.$
Is there a way to simplify the expression
$$\sum_{k=0}^n \frac{1}{k!(n-k)!}?$$
This came up when I was trying to determine $\mathbb{P}(X+Y =r)$ given a joint mass probability $$m_{X,Y}(j,k) = \frac{c(j+k)a^{j+k}}{j!k!},$$ where $j$ and $k$ are non-negative integers and $a... |
H: Prove $T(x)=0$ (Linear Algebra)
Prove that for any five linear transformations mapping $\mathbb R^2$ into $\mathbb R^2$, there exists some non-zero scalars $r_1, r_2, r_3, r_4, r_5$ such that $T=r_1T_1+r_2T_2+r_3T_3+r_4T_4+r_5T_5$ such that $T(x)=0$ for all $x$ in $\mathbb R^2$.
What I did
$$T : \mathbb R^2 \right... |
H: Linear algebra - solving equaltion
System -1
-x-2y = -2
x+2y=2
System 2
x+3y=6
-x-3y=6
What is the solution to the system 1 and 2 separately?
It has no solution
Unique solution
Infinitely many solution
I am seriously confused like if in sys 1 , i add both the eq then 0 = 0, does this mean it can have infinite sol... |
H: Does "gerrymandering" matter?
In the United States, it is often a raw point discussing the issue of redistricting, or so unfavorably called, "Gerrymandering."
Background: In the United States House of Representatives, each State is allotted a number of representatives based on the State's population. Within the s... |
H: How would I know if I'm good in logic?
I've always been interested in logic, but unfortunately my school contains no logicians. What are some good logic puzzles/books and how would I know if logic is right for me?
Also, what can I do with logic? Is it strictly academic? Can I do math with it?
AI: There are several ... |
H: Tangent plane to the surface $\cos(x)\sin(y)e^z = 0$?
The surface is as in the title, $$\cos(x) \sin(y) e^z = 0$$
I'm looking for the tangent plane at the point $(\frac{\pi}{2},1,0)$
I know the equation of a tangent plane for $z = f(x,y)$ is
$$z-z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)$$
But in the surface ... |
H: Probability: balanced die is rolled repeatedly until the same number appears
Suppose that a balanced die is rolled repeatedly until the same number appears on the two successive rolls, and let $X$ denote the number of rolls that are required. Determine the value of $Pr(X=x)$, for $x=2,3,\dots$.
I guess the answer... |
H: How would you "count" $\omega^\omega$
$\omega^\omega$ can be seen as the limit of $\omega^n$ which are all countable sets, and is thus countable. For the latter sets, there is an "easy" way list the elements out, but how would you do it for $\omega^\omega$? That is, what would be a bijection from the positive integ... |
H: Metric on a Class of Functions Mapping to a Metric Space
Let $X$ be a nonempty set and $(Y,\rho)$ a nonempty metric space. Let $Y^X$ denote the set of mappings from $X$ to $Y$. Define $\pi:Y^X\times Y^X\to[0,\infty)$ as follows: for any $f,g\in Y^X$, let $$\pi(f,g)\equiv\min\left\{1,\sup_{x\in X}\rho(f(x),g(x))\rig... |
H: Find the vector, not with determinants, but by using properties of cross products
$(i + j)\times(i − j)$
I know how to use the right hand rule for the cross product, but how do you find the exact vector without using determinants?
AI: The cross product has to be orthogonal to both $i+j$ and $i-j$. Let $v$ be this c... |
H: Why is this function not locally Lipschitz?
I was reading an exercise, and supposedly this function: $$\chi \colon \Bbb R\times\Bbb R\to\Bbb R, \quad \chi (t,x)=3x^{2/3}$$ is not locally Lipschitz (in the second variable). In the notes this isn't proved, so I assumed that it was easy to see, but I've been trying to... |
H: Maximum distance between two unit norm vectors
I have 2 random vectors. I want to limit the euclidean distance between those two vectors to a certain number (say 2) by normalizing them. I think that if I normalize them such that they have a unit (L2) norm then any two vectors arbitrarily selected of any dimensiona... |
H: How to check that probability adds up to $1$
When I asked this, I got one comment as "A good sanity check is to see if the probabilities over all $x$ add up to $1$."
The right answer to the question I linked is $\left(\frac{5}{6}\right)^{x-2} \cdot \frac{1}{6}$ and I want to know how it adds up as $x$ goes from $2$... |
H: Is $\complement(A\setminus B)=(\complement A) \setminus (\complement B)$ true or false?
The problem I have is to calculate this term $(\complement A) \setminus (\complement B)$ when I (forexample) let $A=\left \{ a,b,c,d \right \}$ and $B=\left \{ b,c,e,g\right \}$. How do I calculate it? I've tried, but never got ... |
H: Prove connectivity of graph with vertices of degree $\geq \lfloor \frac n2 \rfloor$
Claim: A graph with vertices of degree at least $\lfloor \frac n2 \rfloor$ where $n = $ number of vertices and $n \geq 3$ is connected.
I tried to prove this by contradiction, but I didn't know what to make of the $\lfloor \frac n2 ... |
H: Compact subset of a Banach space of infinite dimension
Let $X$ be a Banach space of infinite dimension. And let $K\subset X$ be a compact subset of $X$. Can we conclude something about the interior of $K$? Is it true that it's empty?
I don't know how to attack this problem. I have not even examples of compacts on t... |
H: The diameter of the open interval ,$(a,b)$
Suppose that $a,b \in R$ and $a<b$.
Now the $diam(a,b)$ = $b-a$
I am slightly confused at this point, because, by definition,
The diameter of a subset $A$ of a metric space $X$ is the $sup${$d(a,b)$|$a,b\in A$},
But in the above case $a$,$b$ do not belong to $(a,b)$ ,then ... |
H: Reducing a product-of-sums expression
f = ($x_1$ + $x_3$ + $x_4$) * ($x_1$ + $\overline x_2$ + $x_3$) * ($x_1$ + $\overline x_2$ + $\overline x_3$ + $x_4$)
I've been working on this problem for a while but I cannot for the life of me figure out how to simplify the function without distributing everything. The follo... |
H: Generalized Heron's formula for n-dimensional "n-angle" instead of "triangle"
Is there a generalized version of Heron's formula for calculating the equivalent of a "volume" of an n-dimensional "n-angle" based on the length of it's sides? I've seen the equivalent formula for a tetrahedron, but I'd like to keep exten... |
H: How to solve $2^x = 36$
I need to solve $\log$ of $36$ in base $2$
The logarithm result $= x$.
$$
\log_ 2 36 = x.
$$
How do I determine value of $x$ in
$$
2^x=36
$$
I don't know how do it, since there's perfect square of this number.
AI: With a calculator, you can simply calculate:
$$x= \log_2 36 = \log 36 / \log 2... |
H: How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$
Consider the following expression:
(a - b) mod N
Which of the following is equivalent to the above expression?
1) ((a mod N) + (-b mod N)) mod N
2) ((a mod N) - (b mod N)) mod N
Also, how is (-b mod N) calculated, i.e., how is the mod of a negative number c... |
H: Pick 9 balls from piles of different balls
How many ways are there to pick nine balls from large piles of (identical) red, white, and blue balls plus one pink ball, one lavender ball, and one tan ball? What is correct answer?
Is it ${11\choose9} + {{10\choose8} * 3}$?
AI: ${{3}\choose{0}}{{11}\choose{9}}+{{3}\choos... |
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