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H: Epp's proof that if a graph $G$ has a vertex of degree $k$ and $G$ is ismorphic to $G'$, then $G'$ has a vertex of degree $k$
I'm very confused by the this last paragraph shown here:
What exactly does she mean by "there are no edges incident on $g(v)$ other than the ones that are images under $g$ of edges incident... |
H: How to know the possible order of a coset when an order of an element is given?
I have a group $G$ and an element $a\in{G}$ which has an order of 17 ( o(a)=17 ). Also I'm giving that $H$ is a normal subgroup of $G$, and I'm supposed to know the possible orders of the right coset $Ha$ from this given.
I'm trying to ... |
H: Let $G$ be a group, $H\le G$. Define $X=\cup_{g\in G}gHg^{-1}.$ Prove $X=G$ when $[G:H]<\infty$.
Let $G$ be a group and $H \leq G$. Define $$X = \bigcup_{g\in G} gHg^{-1}.$$ I want to prove that $X=G$ when $[G:H] < \infty$.
I had couple of observations:
$g_1 H g_1^{-1} = g_2 H g_2^{-1} \iff g_1 N_G(H) = g_2 N_G(H)... |
H: Improper Integral Property of a Positive Function
Let $g:[0,\infty)\rightarrow\mathbb{R}$ a positive function satisfying
$$\int_{0}^{\infty}{\frac{dr}{g(r)}}=\infty.$$
Can say that $\int_{0}^{\infty}{g(r)dr}<\infty$ ? Thanks in advance!
AI: Take $g(r) = r$. Since the harmonic series diverges, then by integral test,... |
H: Why $\int_{ \mathbb{R}^2 } \frac{dx\,dy }{(1+x^4+y^4)} $ converges?
Why $\int_{ \mathbb{R}^2 } \frac{dx\,dy }{(1+x^4+y^4)} $ converges?
Apparently this integral is quite similar to the integral
$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?
and it converges.
So this is quite remarkable... |
H: Proving exponent law for real numbers using the supremum definition only
I was working on a problem displaying the expansion of the definition of exponents, and naturally the final question was to prove the exponent laws when the exponents are real numbers.
For $a>1$, define $a^x=\sup\{a^r:r\in\mathbb Q, r\le x\}$... |
H: Composition of orthogonal projections, $P_1 P_2 = P_2 P_1 \rightarrow P_1 P_2$ is the orthogonal projection on $W_1 \cap W_2$
Let $V$ be an inner product space of finite dimension. $P_1, P_2$ the orthogonal projections on sub-spaces $W_1, W_2$.
Prove that if $P_1 P_2 = P_2 P_1$ then $P_1 P_2$ is the orthogonal pro... |
H: Evaluate the integral $\int\limits_{0}^{b}\frac{dx}{\sqrt{a^2+x^2}}$
Show that $$\int\limits_{0}^{b}\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{b}{a}$$
However, when I use Maple or WolframAlpha to calculate the left integration, both gave me $-\frac{\ln(a^2)}{2}+\ln(b+\sqrt{a^2+b^2})$, which seems not agree with the ... |
H: Prove that there is a constant $ M $ such that $ \int|fg|dm\leq M \| f\|_{L^{p}} $ for all $ f\in L^{p}(\mathbf{R}) $.
I could not understand the last part of the proof of the following theorem: Let $ p\geq 1 $ and $ g $ be a measurable function such that $ \int|fg|dm<\infty $ for every $ f\in L^{p}(\mathbf{R}) $. ... |
H: If $f$ is differentiable for any value except $x=0$, and $e^{f(x)} = x$, show that $f’(x) = 1/x$
If $f$ is differentiable for any value except $x=0$, and $e^{f(x)} = x$, show that $f’(x) = 1/x$.
AI: Knowing that $e^{f(x)}=x$, we can take the derivative of both sides with respect to $x$:
$$\begin{align*} \dfrac{d}{d... |
H: To Prove $ \bigcap_{i \in I} A_i \in \bigcap_{i \in I} P(A_i) $
$$ \bigcap_{i \in I} A_i \in \bigcap_{i \in I} P(A_i) $$ , $ I \neq \phi $
MY ATTEMPT
I use proof by Contradiction. Assume $ \bigcap_{i \in I} A_i \notin \bigcap_{i \in I} P(A_i) $$
Let $ x \in \bigcap_{i \in I} A_i $
i.e $ \{ x \} \notin ( P(A_1... |
H: Flux through the positive part of a sphere centered at $(0,0,1)$
Let $$B=\{(x,y,z):x^2+y^2+(z-1)^2<4, \ z\geq 0\}$$ and consider the vector field $$F:(x,y,z)\mapsto(x^3,y^3,z)$$
I want to compute the flux of $F$ through $\partial B.$
We have $$\text{Div}F=3x^2+3y^2+1$$ and so by the divergence theorem we could co... |
H: What is the probability the the second toss is heads?
One bag contains two coins. One is fair, the other is biased with Heads probability = $0.6$. One coin is randomly picked and it is tossed. It lands heads up. What is the probability that the same coin will land heads up if tossed again?
Now, the probability that... |
H: Get a random circle C inside a bigger circle, and C encompasses a specific point
So basically I want to draw a random circle of radius R inside a bigger one, but the drawn circle should encompass a specific point.
If my math is right, it comes down to solving the following and getting the valid ranges of x and y:
x... |
H: How to integral $z^{\prime\prime}+ 2 \eta z^\prime=0$?
From the given ODE, $$z^{\prime\prime} + 2 \eta z^\prime = 0 $$
The rest procedure is as below
$$\int \frac{z^{\prime\prime}}{z^\prime} d \eta = \int -2 \eta \, d \eta$$
$$ \ln z^\prime = - \eta ^2 + c_0 $$
$$ z^\prime = \frac{d z}{d \eta} = C_1 \exp (- \eta ^2... |
H: How to prove the following, for $n>4$?
Let $G$ be an undirected graph with $n>4$ nodes. Prove that there is a cycle in $G$ or in $G'$?
Note: $G'$ is the graph which includes all nodes in $G$ and includes edges iff those edges do not exist in $G$.
AI: If $G$ does not contain cycles then it is a forest, hence $|E|<|V... |
H: transformation of Bern random variable
I've encountered this question:
Let X ~ Bern(1/2) and let a and b be constants with a < b. Find a simple transformation of X that yields an r.v. that equals a with probability 1 - p and equals b with probability p.
I have been working on this for sometime without answer, is it... |
H: Norm of Position Operator in $L^2[0,1]$
I was wondering what is the norm of the position operator $Xf(x)=xf(x)$ in $L^2[0,1]$. I have two different results.
The first one is the simplest and the reasonable:
$$||X|| \overset{||f(x)||=1}{=} \sup||xf(x)||=\sup||x||=1, $$
since $x\in[0,1]$.
The second method is the ... |
H: Proving that the inverse in subgroup $H \leq G$ is the same as the inverse in $G$ : does it follow from uniqueness?
Let $H$ be a subgroup of $G$. I am trying to show that the inverse element of some $h \in H$ is the same as its inverse in $G$. I know how to prove it without uniqueness, but I am trying to understand... |
H: Knight on a $3\times 4$ board: Hamiltonian graphs
A chess knight sits on a $3\times 4$ board. Is it possible for the knight to jump into the $12$ squares without jumping twice in any of them and ending and starting in the same box? What if it starts and ends in the different boxes?
I have drawn the graph that repre... |
H: How to show $(a_1a_2\ldots a_n)^{\frac{1}{n}}\leq \frac{\sum_{i=1}^{n}a_i}{n}$
How to show $(a_1a_2\ldots a_n)^{\frac{1}{n}}\leq \frac{\sum_{i=1}^{n}a_i}{n}$ with $a_i$ positive.
Well, I tried by induction: with $n=2$ then $\sqrt{ab}\leq \frac{a+b}{2}$ is equivalent to say (elevate square in both side) $4ab\leq a^2... |
H: In which ratio does the point $P$ divide the segment $\overline{AN}$?
In an arbitrary triangle $\triangle ABC$, let $M\in\overline{AC}$ s. t. $|AM|:|MC|=2:1$ and let $N\in\overline{BC}$ s. t. $|BN|:|NC|=1:2$. Let $P$ be the intersection point of the segments $\overline{AN}$ and $\overline{BM}$. In which ratio does... |
H: Subspace $\operatorname{null}(T^2+bT+c)^j$ has even dimension when $b^2<4c$
Question: Let $V$ be a finite-dimensional real vector space and $T$ be a linear operator on $V$. Let $b,c\in \mathbb{R}$ such that $b^2<4c$. Prove that for every $j$, $\operatorname{null}(T^2+bT+c)^j$ has even dimension.
What I have done is... |
H: The interval $(a,b) \subseteq \mathbb{R}^{2}$ is bounded - metric spaces
I am trying to show that the interval $(a,b) \subseteq \mathbb{R}^{2}$ is a bounded set.
By $(a,b) \subseteq \mathbb{R}^{2}$ I am meaning $(a,b) \times \{0\} = \{(x,y) \in \mathbb{R}^{2}: a<x<b, y=0\}$
Bounded:
If I can show that $(a,b) \times... |
H: Two definitions of singular cohomology
I am reading Hatcher's Algebraic Topology and Milnor's Characteristic Classes. In these two books, the definition of singular cohomology is little bit different, as follows: Fix a topological space $X$ and a (commutative) ring $R$ (with $1$). First, in Hatcher, we form a chain... |
H: Behmann's proof of Infinitude of primes.
I am having difficulty in understanding the proof of Behmann of Infinitude of primes. Can someone please explain the last part 'The proof is concluded by noticing....' which is in page $178$?
Any help would be appreciated. Thanks in advance.
AI: Suppose there are only $m$ pr... |
H: Find Recursive Definition from given formula
I've read some ways about how to derive a formula from a recursive definition, but what about this one?
I started solving this formula
$$ a_n = 2^n + 5^n n , n \in \mathbb{N} $$
gives you the recursive definition of what and how you go about figuring that out? Any tips f... |
H: Show that length of sine is equal to length of cosine on the same interval.
Let $$f(x)=\sin(x)\\\ g(x)=\cos(x)$$
Let $L_1$ be $$\int_0^{2\pi}\sqrt{1+\cos^2(x)}\space dx$$
And $L_2$
$$\int_0^{2\pi}\sqrt{1+\sin^2(x)}\space dx$$
I.e. L is a length of sine/cosine during it's period interval.
Numerical approach shows th... |
H: Jensen inequality in measure theory : why doesn't the convex function need to be nonnegative?
This section of the Wikipedia article on Jensen's inequality states that if $g$ is an integrable function on a measure space with mass $1$ and $\varphi$ is a convex function, then
$$\varphi \left( \int g \right) \leq \int ... |
H: Finding a function to fit a curve.
I would like to fit an equation to the curve shown below. A selection of the data points (x,y) are given too.
I have tried to fit equations $y = a \, e^{(-b \, x)}$ and $y = a x^{-b}$ using software minitab 19 without getting a good fit.
So if an equation can be found or suggeste... |
H: Relation of Adjacency matrices of 2 graphs
Let G be the subgraph
of G¯(prime) such that there is an edge between vi ∈ V and vj ∈ V in G if and only if there is at least one edge between vi and vj in G¯(prime).
Any tips or hint to solve about finding A from A¯(prime) ?
AI: $A_{ij}$ is equal to 1 if $\bar A_{ij}$ is... |
H: If $\nabla\cdot u=0$ and $w=\operatorname{curl}u$, then $\int w=0$
Let $\Lambda\subseteq\mathbb R^2$ be open, $u\in C^1(\Lambda,\mathbb R^2)$ with $\nabla\cdot u=0$ and $$w:=\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}.$$
How can we show that $$\int_\Lambda w=0?\tag1$$
Since $\nabla\cdot u=... |
H: How many ways can we make a 3-senator community where no 2 of the members are from the same state?
Part (a): There are $2$ senators from each of the $50$ states.
We wish to make a $3$-senator committee in which no two of the members are from the same state. In how many ways can we do it?
Part (b): Suppose for this ... |
H: How is $nq-1 = \sum_{m=0}^n (m-1){n \choose m} p^{n-m}q^m$
$$nq-1 = \sum_{m=0}^n (m-1){n \choose m} p^{n-m}q^m$$
I found this formula in my facility planning book. The math part of it I don't understand.
I tried taking it apart like this $\sum_{m=0}^n (m){n \choose m}p^{n-m}q^m-\sum_{m=0}^n {n \choose m}*p^{n-m}q^m... |
H: bounds for conditional expectation and variance
let X and Y be random variables with density:
$$f_{x,y}(x,y)=\frac{1}{\pi}\mathbf{1}_{\{x^2+y^2\le1\}}$$
Find $$E[X|Y] \& Var(X|Y)$$
So what I did was:
$$E[X|Y]=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{x}{2\sqrt{1-y^2}}dx=0$$
$$Var(X|Y)=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y... |
H: Proof that the perpendicular bisectors of the sides of a general triangle meet at a point
I have been studying Lang's basic math and I am stuck on the problem below:
This is my representation:
I am not sure on how I should proceed in this proof, I think this page is related to the problem, if so should I use a si... |
H: An application of the dominated convergence theorem to approximate a function.
In "Measure theory and probability theory (pag. 58)" by Krishna and Soumendra we can found the following:
Let $\mu$ be a Lebesgue-Stieltjes measure, and $f \in L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ with $0<p<\infty$.
Define $B_... |
H: Maximum likelihood estimator for bombing planes.
The bombing planes are intersecting two lines of anti-aircraft
defense. Each plane, regardless of the others, has probability
$\theta$, true resemblance, and can be knocked down by the first line
of defense and second line of defense. Probability $\theta$ is not
kno... |
H: If $f: X \to [0,\infty]$ is measurable, $\lim_{n \to \infty} \int_X f^n d \mu$ exists.
Let $(X, M, \mu)$ be a finite measure space. Let $f: X \to [0,\infty]$ be a measurable function. Prove:
a) $\lim_{n \to \infty} \int_X f^n d \mu$ always exists on $[0,\infty]$.
b) The latter limit is finite iff $\mu\{x \in X : f(... |
H: If $G$ is a directed and finite graph whose underlying graph is a clique, then does $G$ have a root?
Since I have failed to prove the following I think it's mostly false.
Let $G$ be a directed and finite graph such that its underlying graph is a Clique.
Then, $G$ has a root.
Can someone give a contrary example?
... |
H: RMO 1990 question
Prove that the inradius of a right angled triangle having integer sides is also integral
I tried it and got something like
$r=\frac {(a.b)}{(a+b+c)}$
How to proceed after this.
AI: First, note that $$r=\frac{a+b-c}{2}$$
Also, the Pythogorian theorem says $$a^2+b^2=c^2$$
$\textbf{Case 1: }$ If both... |
H: Typo on page 92 of Spivak's Calculus 4th edition?
On page 92 of Spivak's calculus 4th edition in the middle of the page it states:
"We just have to be 1,000,000 times as careful, choosing $|x- a|<\epsilon/3,000,0000$ in order to ensure that $|f(x) - a|<\epsilon$."
Is it not $|f(x) - f(a)|<\epsilon$ or $|f(x) - l|<\... |
H: For $a,b$ in abelian group $G$ of orders $m,n$ where $\gcd(m,n) = 1$, $|ab| = mn$.
I am trying to prove this following result.
Let $G$ be an abelian group. Let $a,b$ be elements with orders $m,n$, respectively, where $m$ and $n$ are relatively prime. Prove that $|ab| = mn$.
Here is my attempt. There is one final ... |
H: Computing the signed curvature of a surface in an arbitrary direction
If I have a surface defined as the graph of the function $z = f(x,y)$, is there a closed-form expression for the signed curvature of this surface in an arbitrary direction? That is, if $x(t) = x_0 + t\Delta x$ and $y(t) = y_0 + t\Delta y$, how ca... |
H: Show that: $\left[\underset{n\to \infty }{\text{lim}}\int_1^{\infty } \frac{\sin (x)}{x^{\{n+1\}}} \, dx\right] = 0 $
Show that: $$\left[\underset{n\to \infty }{\text{lim}}\int_1^{\infty }
\frac{\sin (x)}{x^{\{n+1\}}} \, dx\right] = 0 $$
My attemp:
I using Taylor series:
$$
\sin(x)= x-\frac{x^3}{6}+\frac{x^5}... |
H: Metric spaces with two conditions
If $X$ is an non-empty set and $d: X \times X \rightarrow \mathbb {R}$ has the following properties
$d(x,y)=0$ if and only if $x=y$
$d(x,y) \leq d(x,z)+\color{red}{d(z,y)}$
Prove that d defines a metric on X.
I need to prove that
$d(x,y) \geq 0$
$d(x,y)=d(y,x)$
I know this result.
... |
H: If $f$ is continuous and satisfies $|f(x) - f(y)| \ge \log(1+|x-y|)$, how do I show that $f$ is bijective?
Let $f:\mathbb R\to \mathbb R$ be a continuous function satisfying $|f(x)-f(y)|\ge \log(1+|x-y|)$ for all $x$ and $y$ in $\mathbb R$. Prove that $f$ is bijective.
$f(x)=f(y)$ at once implies $x=y$ and hence $f... |
H: proves and disproves about inner product spaces
$l_p=\{[{a_n}]_{n=1}^ {\infty}|\sum_{n=1}^{\infty}|a_n|^p < \infty \}$
with the norm $||a_n||_p = (\sum_{n=1}^{\infty}|a_n|^p)^\frac{1}{p} $
prove or disprove:
$L_2\subset L_1$
I know its true for functions but is it also for sequnces?
If $\lim\limits_{n \to \inf... |
H: How Was Tarski's Undefinability Theorem Used?
How is Tarski's undefinability used in the following stack-exchange answer(s)? To recall, the result is that for language $L$ for which diagonal lemma (i.e. $ZFC, PA)$ applies there is no formula $T$ with one free variable such that for all wffs $\phi$
$$L\vdash \phi\le... |
H: Triangular numbers divisible by $3$
I can't understand any of sentences from the images below. Since I don't understand almost every possible lines, I'm very troubled for what I should even ask. But I'll try to.
Firstly, how is it possible for the triangular number with 3k+1 as the last number to be followed by the... |
H: $f$ is absolutely continuous implies that $f$ is continuous
I am going to show that if f is absolutely continuous it implies that f is continuous, the test seems somewhat trivial to me of the definition, however I would like to know if I am doing it well.
If $f$ is absolutely continuos on [a,b], let $\varepsilon >0... |
H: Does a first integral of $\dot x = f(x)$ satisfy $\nabla H \cdot f = 0$?
Let $\dot x = f(x)$ be an ODE, where $f: U\to \mathbb{R}^n$, and $U\subset \mathbb{R}^n$ is open. Then I know that a first integral is a function $H: U\to \mathbb{R}^n$ so that $H(\varphi(t,x_0))=const$ for every solution $\varphi(t,x_0)$ of t... |
H: Show that the card drawn at time $(S+1)$ is equally likely to be a club or a diamond.
Let $k \geq 2$ be an integer. At time $n=0,$ we shuffle a deck of $2k$ cards of which $k$ are clubs and $k$ are diamonds. We draw a card each turn without replacing. Denote by $C_{n}$ the number of clubs drawn by the turn $n$ and ... |
H: Determine whether $F$ is continuous as a map from $\big({\cal C}[0,1],d_1\big)$ to $\big({\cal C}[0,1],d_1\big)$.
Define $F:{\cal C}[0,1]\to{\cal C}[0,1]$ by $F(f)(x)=\int_0^x{f(t)\over\sqrt t}\,dt$.
I have a feeling that the map is continuous. I have shown that the map is Lipschitz continuous from $(\mathcal C[0, ... |
H: Bounded Variation Functions
My question is;
$f(x)=\left\{\begin{array}{cc}x^{2} \sin \left(\frac{1}{x^{2}}\right), & 0<x \leq 1 \\ 0, & x=0\end{array}\right.$
"Show that this function is Bounded Variation on [0,1] or not. "
i know , It's not a bounded variation function.
Because i read a theorem from a book;
where ... |
H: Implication of probabilistic ordering from convergence in probability
For a sequence $X_1, X_2, \ldots X_n$ with $E X_n = a$ I proved that $X_n \to_p a$, i.e.
$$
\lim_{n \to \infty}P(|X_n - a|>\varepsilon)=0
$$
Does this type of convergence imply a probabilistic $ordering$ on the sequence, i.e.
$$
P(|X_1-a|>\varep... |
H: Let $Y = X − [X]$, where $X\sim U(0,\theta)$. Show that $Y \sim U (0, 1)$
Let $X \sim U(0,\theta)$, where $\theta$ is a positive integer. Let $Y
= X − [X]$, where $[x]$ is the largest integer $≤ x$. Show that $Y \sim U (0, 1)$
Clearly, the support of $Y$ is $S_Y = [0,1]$.
In order to show this, I want to be able... |
H: [Proof Verification]: $\overline{A\times B}=\overline{A}\times\overline{B}$
Suppose $X$ and $Y$ are topological spaces, and $A\subset X$ and $B\subset Y$ are subspaces. I'm trying to prove $\overline{A\times B}=\overline{A}\times\overline{B}$. For the backward inclusion, choose $(x_1,x_2)\in \overline{A}\times \ove... |
H: The product of an arbitrary family of locally convex spaces is locally convex.
Let $\{E_\alpha\ : \ \alpha\in I\}$ be a family of a locally convex sets, where $I$ is an index family. I want to prove that
$$E:= \prod_{\alpha\in I}E_\alpha$$
is locally convex.
I know that, by definition, for each $\alpha\in I$, $E_... |
H: Integral Criteria for Functions to be Zero Almost Everywhere
While reading the proof of Lemma 2 in the following link, I realized they only proved the case of a nonnegative function $f$, but that's not an hypothesis of the lemma. So, what happens if $f$ takes negative values? Does the lemma remain true? Is the proo... |
H: $U \sim U(0,1)$ and let $X$ be the root of $3t^2 −2t^3 −U = 0.$Show that $X$ has p.d.f. $f(x)=6x(1−x)$, if $0\leq x\leq 1$
Let $U \sim U(0,1)$ and let $X$ be the root of the equation $3t^2
−2t^3 −U = 0.$ Show that $X$ has p.d.f. $f(x)=6x(1−x)$, if $0\leq
x\leq 1,\;=0$,otherwise.
I really don't know how to begin... |
H: Sum of divisors function inequality
Prove that if $n<m$ and $n$ divides $m$, then $\frac{\sigma(n)}{n} < \frac{\sigma (m)}{m}$, where $\sigma(x)$ denotes the sum of all the divisors of $x$.
I know that $\sigma (x)$ is multiplicative but not completely multiplicative. So, if $m=nk$ and $\gcd(n,k)=1$, then the inequa... |
H: Conceptual reason why height of unit tetrahedron is the same as the distance between opposite faces of an octahedron?
One of my favorite mathematics visualizations shows why attaching a tetrahedron to a triangular face of a square pyramid results in a polyhedron with five faces instead of the seven faces one might ... |
H: Proving $\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1}\sum_{j=n}^{n+i}\frac{\binom{n+i}j}{2^{n+i}}=0$
I found the following question online: How can I prove that
$$\lim_{n\to\infty}\frac1n\sum_{i=0}^{n-1}\sum_{j=n}^{n+i}\frac{\binom{n+i}j}{2^{n+i}}=0$$
?
One notices that the inner sum is equal to the probability $\maths... |
H: Checking the derivative of function of two variables
Lef $f:\mathbb R^2\rightarrow\mathbb R $ be defined by
$$f(x,y)=\sin\left(\frac{y^2}{x}\right)\sqrt{x^2+y^2}$$
if $x\neq0$ and $f(x,y)=0$ if $x=0$. Then show that $f$ is not differentiable at $(0,0)$.
It is clear that $f$ is continuous at $(0,0)$. Now by the defi... |
H: Curve family whose velocities are normalized exponentials
For a fixed $\gamma > 0$, consider the family of curves parametrized by $\Theta=(\vec a,\vec b)$:
$$\vec v_\Theta(t) = \frac{\vec a + e^{\gamma t}\vec b}{||\vec a + e^{\gamma t}\vec b||_2}$$
I'm interested in their integrals
$$\vec x_\Theta(t) = \int_0^t \ve... |
H: Solving $(D^2-1)y=e^x(1+x)^2$
I did like this:
$$\text{Let,} y=e^{mx} \text{ be a trial solution of } (D^2-1)y=0$$
$$\therefore \text{The auxiliary equation is } m^2-1=0$$
$$\therefore m=\pm1\\
\text{C.F.} = c_1e^x+c_2e^{-x}$$
$$\begin{align}
\text{P.I.}& =\frac{1}{D^2-1}e^x(1+x)^2\\
& =e^x\frac{1}{(D+1)^2-1}(1+2x+... |
H: Monotone Convergence theorem Application
$$
\lim _{n \to \infty} \int_{-\infty}^{\infty} \frac{e^{-x^{2} / n}}{1+x^{2}} d x=?
$$
My opinion is using Monotone Convergence Theorem here.
For every $x \in \mathbb{R}$ the sequence $\left\{e^{-x^{2} / n}\right\}$ monotonically increases and converges to
$e^0=1$. Thus b... |
H: Prove $(\mathbb{Z} \times \mathbb{Z})/ \langle (2,3)\rangle$ is isomorphic to $\mathbb{Z}$.
I'm trying to prove the following but im stumped:
Prove that $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 3)\rangle \cong \mathbb{Z}$.
My attempts so far have been to try and find a single generator of this group. Since its... |
H: Let $p$ be an odd prime, and $f(x):=x^{p^2-p} -1+(x+1)^p.(x+2)$
Solve $f(x)\equiv0\pmod p.$
Find the number of incongruent solutions of Solve $f(x)\equiv0\pmod{p^2}.$
I think i can raise solutions to the second one from the first one by Hensel's Lemma but i am not sure how to do the first part. Any help?
AI: Let's ... |
H: Applications of Integration: PDF
The probability of showing the first symptoms at various times during the quarantine
period is described by the probability density function:
f(t) = (t-5)(11-t) (1/36)
Find the probability that the symptoms will appear within 7 days of contact.
F(t) = (1/36)(8.t^2 -t^3/3 -55t)
P(x<... |
H: Unable to Solve a quiz question asked in mathematics exam ( Quantitative Aptitude)
I am self studying for an exam and I am unable to solve this quiz question.
Adding it's image ->
I tried by finding numbers in the sentences but couldn't find and I think that's a wrong approach.
Can anyone please tell how to solve ... |
H: area of triangle using sides (ratio?)
Given a triangle ABC, points D, E and F are placed on sides BC, AC and AB, respectively, such that BD : DC = 1 : 1, CE : EA = 1 : 3 and AF : FB = 1 : 4. A line parallel to AB is drawn from D to G on side AC. Lines DG and EF meet at X. If the area of triangle ABC is 120, what is... |
H: Unable to think about a question based on Directions in Aptitude
I am self studying quantitative aptitude and I am unable to solve this particular question.
Question is -> Starting from Point A you fly one mile south, one mile east, then 1 mile north which brings you back to Point A. Point A is not North Pole. The... |
H: How do I show that $\lfloor{(\sqrt 2+1)^{2020}}\rfloor\equiv 1\left(\text{mod}~ 4\right)$?
I think I can use that $$(a\sqrt b+c)^{2n}+(a\sqrt b-c)^{2n}\in\mathbb{Z},$$ but I have no idea about the next step.
AI: Let $r,s$ be given by
\begin{align*}
r&=1+\sqrt{2}\\[4pt]
s&=1-\sqrt{2}\\[4pt]
\end{align*}
Noting that ... |
H: Separable metric spaces have countable bases.
Suppose $X$ is a separable metric space, and $E$ is a countable dense subset of $X$. I want to show that the set
$$B=\{B_r(x)\mid (r,x)\in\Bbb Q^+\times E\}$$
is a basis for $X$. We want to show $$X=\bigcup_{(r,x)}B_r (x),$$
The backward inclusion is easy. For the forwa... |
H: Without calculating, tell which is a perfect square: 1022121; 2042122; 3063126; 4083128
I am trying aptitude questions but was struck on this problem.
Which of the following numbers is a perfect square?
A) $1022121\quad$ B) $2042122\quad$ C) $3063126\quad$ D)$4083128$
(original problem image)
As a perfect square ... |
H: Continuous mapping not exist on $\mathbb{R}^2$
Suppose that $\Delta\subseteq\mathbb{R}^2$ is a closed triangle area, $\partial \Delta$ its boundary (namely the 3 sides of the triangle). I'd like to show that there doesn't exist a continuous mapping from $\Delta$ to $\partial \Delta$, which maps each side of $\parti... |
H: Proving a function$f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is onto.
Question: Let $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a continuously differentiable function satisfying $||f(x)-f(y)||\geq||x-y||$ for all $x,y\in \mathbb{R}^n$. Show that $f$ is onto.
1. I know that, a function $f:\mathbb{R}\longrightarr... |
H: Orthogonal complements really confuse me, I think its the notation?
For example what do I do here, I know wha to do for part a but then...?
Let$$W=\operatorname{Span}\left\{\left(\begin{array}{c}1\\1\\0\\0\end{array}\right),\,\left(\begin{array}{c}0\\1\\1\\-1\end{array}\right)\right\}$$
(a) Show $v=\left(\begin{ar... |
H: Proving that $0
I need to prove that $0<e-S_{n}<\frac{4}{(n+1)!}$ $\quad\forall n\in\mathbb{N}\quad$ for $\quad S_{n}=1+1+\frac{1}{2!}+\ldots+\frac{1}{n!}$
Previously, I proved that $2<e<4$, and that $S_n$ is taylor polynomial of $e$.
I thought on proving it using induction:
Base: $S_1=1<2<e<4$
Step - for $n+1$:
$e... |
H: Show that for any positive integer $m$ there are $m$ consecutive positive integers each of which has at least $10$ positive divisors.
In other words, show that for any positive integer $m$ there are $m$ consecutive positive integers $a+1, a+2, ..., a+m$ such that $τ(a + i) ≥ 10$ for each $i = 1, 2, ..., m)$
I don't... |
H: how to determine dimension of bases?
Hello guys let's say we have 3 vectors $u,v,w$ in $R^4$. Does bases dimension depends on dimension of subspace? For example in $R^4$ can we have bases which has $\dim=2$ or $\dim=3$? I was confused when I saw dim=3 bases in $R^4$ and then in another page I saw dim=4 bases in $... |
H: Proof Verification: Cartesian Product of Finitely Many Nonempty Sets is Nonempty (Without AoC)
Synopsis
Please verify my proof. I would also appreciate any tips on how I might improve my mathematical writing. Thank you.
Exercise
Assume that $H$ is a function with finite domain $I$ and that $H(i)$ is nonempty for ea... |
H: Backwards direction of Cauchy Criterion for Sequences of Functions
I am reviewing the proof of the Cauchy Criterion for sequences of functions and have a question regarding the backwards direction.
Statement: Let $A\subseteq \mathbb{R}$ and $(f_n)$ be a sequence of real-valued functions with domain $A$. Then $(f_n)... |
H: Can one show that there exists a matrix $M$ such that $BA=M^k$ with the condition obtained?
In a problem, I got
$
AB=N^k
$ where $\det(N)=1$.
Can one show that there exists a matrix $M$ such that $BA=M^k$, where all $A,B$, and $M$ are $3 \times 3$ matrices with integer entries.
AI: I'm not entirely sure what the qu... |
H: Find Solution set of $(\log_4x)^2+4\sqrt{(\log_4x)^2-\log_2x-4}=\log_2x+16$
My attempt :
$$\dfrac{1}{4}(\log_2x)^2+4\sqrt{\dfrac{1}{4}(\log_4x)^2-\log_2x-4}=\log_2x+16$$
$$(\log_2x)^2+16\sqrt{\dfrac{1}{4}(\log_2x)^2-\log_2x-4}=\log_2x+16$$
Let $a=\log_2x$
$$a^2+8\sqrt{a^2-4a-16}=a+16$$
Because this is fourth degree... |
H: Probability of getting 3 blue skittles in even-sized allotments from the same bag
Let's say Skittles come in 3 colors: red, green, and blue. If I pour a bag of 90 Skittles into 10 bowls of 9 Skittles each, the probability of getting >= 3 blue Skittles in any bowl is 1/3. Is the probability of getting >= 3 blue Skit... |
H: Conditional Probability within regards to Discarding.
An urn contains four (4) red chips and six (6) white chips. Two (2) chips are drawn out and
discarded and a third chip is drawn. What is the probability that the third chip is red?
Would Hyper geometric Distribution be the best possible method for solving this?
... |
H: If $\sum_{k=0}^2 c_kf^{(k)}(x) \ge 0$ show that $f(x)$ is a non-negative polynomial
If for some polynomial $f(x)$ with real coefficients there exist $c_1,c_2 \in \mathbb{R}$, satisfying $c_1^2\ge 4c_2$, such that for all real values of $x$ the following inequality holds:
$$\sum_{k=0}^2 c_kf^{(k)}(x) \ge 0, \ \ (c_... |
H: Differentiability of $x^2\times\sin(1/x)$
Using first principle, when we try to check the differentiability of $x^2\sin(1/x)$ at $x= 0$,we get 0.
But if we differentiate the function first, and then try to find differentiability at x=0,we we find it's not differentiable.
I have encountered similar questions on stac... |
H: Do not understand this basic conditional probability
This example came from Tsitsiklis' Intro to Probability 2nd edition.
I don't understand why is $P(A|B)=2/5$ when $m=3$ or $m=4$. Surely it should be $4/5$?
I get that $P(A|B)=1/5$ when $m=2$, because when we restrict our domain to the highlighted area (i.e. $B$),... |
H: Probability question: choosing between two options
I am having trouble understanding the following exercise in probability and statistics:
A person has to choose between two jobs. In Job1, they have a profit of 12k with a 75% probability, while they have a damage of 3k with probability 25%. In Job2, they have a pro... |
H: Left adjoint of exponential functor
Does the contravariant exponential functor have a left adjoint ? And if so what is it ?
To elaborate, I know that in many categories, like $\textbf {Set}$ for example, the covariant exponential functor $\left(\_ \right)^A$ has a left adjoint (which is the product functor $\left(\... |
H: Does the Fundamental Theorem of Calculus tell us that integration is the 'opposite' of differentiation?
I have often read that the Fundamental Theorem of Calculus (FTC) tells us that integration is the opposite of differentiation. I have always found this summary confusing, so I will lay out what I think people mea... |
H: A non-unit which is neither irreducible, nor a product of irreducibles.
I am having trouble with this exercise
Find a non-unit $d \in D = \{f \in \mathbb{Q}[x] \mid f(0) \in \mathbb{Z}\}$ where $d$ is neither irreducible, nor a product of irreducibles.
I find it contradictory that an element can be non-irreducibl... |
H: Logical implication and conjunction in transitive relation definition
In the properties of relations, the transitive relation is defined as follows:
If I read it informally, it says, "If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$
What surprised me was when the author of the paper said that this definition... |
H: Why is this assumption needed in Cauchy's theorem?
I am studying complex analysis and Cauchy's theorem states:
Suppose that a function $f$ is analytic in a simply connected domain $D$ and that $f'$ is continuous in $D$. Then for every simple closed contour $C$ in $D$, $\oint_C f(z)dz = 0$
Next after this theorem ... |
H: Naive Bayes Classifier - With Lagrange Variable- Derivation
I am running through this link to understand better the derivation for MLE for Naive Bayes:
https://mattshomepage.com/articles/2016/Jun/26/multinomial_nb/
In particular, i am confused as to this part:
$L=\sum_{i=1}^{N}\sum_{j=1}^Pf_{ij}\log(\theta_j)+\lamb... |
H: Calculating the volume with triple integral
Calculate the volume of $V$ limited by planes:
$$x+y=1, x+y=2, x=0, y=0, z=0, z=1$$
The projection on $xy$ is
I tried finding the volume without integrals first:
$$V=Sh=\frac{1}{2}(2\cdot2-1\cdot1)\cdot (1-0)=\frac{3}{2}$$
And I got a completely different answer when cal... |
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