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H: Is this recursion relation proof correct?
Recurrence relation:$$a_0 = 1$$
$$a_{n+1} = 2a_n$$
I'm trying to prove that for any n ∈ N, $a_n = 2^n$. I want to use induction.
What I have is, assume that $a_n = 2^n$ is true for $P(n)$.
Then $P(n+1)$ would be:
$$a_{n+1} = 2^{n+1}$$
$$a_{n+1}=2\cdot(2^n)$$
Because $a_n =... |
H: A function that brings back the prime number just before it?
Is there a function that brings the prime number just before it?
I.e P(18)=17 P(6)=5 P(28)=23;
I know how weird that sounds.
AI: Standard notations are $p_n$ for the $n^{th}$ prime and $\pi(n)$ for the number of primes less than or equal to $n$. Combinin... |
H: The Product Rule of Square Roots with Negative Numbers
In the statement $\forall a, b \geq0, \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, why is it necessary to restrict $a$ and $b$ to being $\geq 0$? It seems that one should be able to say, for example, $(-3)^{1/2} \cdot (-3)^{1/2} = (-3 \cdot -3)^{1/2} = 9^{1/2} = 3$, s... |
H: Sum of one, two, and three squares
If a square $n^2$ can be written as the sum of two nonzero squares as well as the sum of three nonzero squares, then can we conclude that it can be written as the sum of any number of nonzero squares up to $n^2 - 14$ nonzero squares?
Example: $13^2 = 12^2 + 5^2 = 12^2 + 4^2 + 3^2$... |
H: Proving $u_n(x)\le v_n(x), \sum\limits_{n=1}^{\infty}v_n(x)\to f$ uniformlly $\Longrightarrow\sum\limits_{n=1}^{\infty}u_n(x)\to g$ uniformly
Prove if $u_n(x)\le v_n(x)$ and $\sum\limits_{n=1}^{\infty}v_n(x)$ converges uniformly then also $\sum\limits_{n=1}^{\infty}u_n(x)$ converge uniformly
I thought solving it ... |
H: Quotient with Non-Normal Subgroup
This has been brought up here but I'd like to bring up a few more questions.
Taking the quotient of group $G$ with subgroup $H$ is well-defined iff $H$ is normal in $G$. Well, what happens when $H$ is not normal? The left- and right cosets of $H$ in $G$ don't coincide. I have this ... |
H: Confused with proof that all Cauchy sequences of real numbers converge.
First the textbook proves that all Cauchy sequences are bounded, and so have a convergent subsequence, $\{a_{n_{k}}\}$ that converges to a limit, say $L$. Now we use this to prove that all Cauchy sequences are convergent.
So an $N_1$ exists suc... |
H: Express $[\cos(x) + \sqrt3 \sin(x)]$ in the form $[r\cos(x-a)]$
Express $[\cos(x) + \sqrt3\sin(x)] $ in the form $[r\cos(x-a)]$, where $r>0$ and $ 0\leq360$, hence solve the equation $[\cos(x) + \sqrt3\sin(x)= \sqrt2]$
This is as far as i have completed. I don't know whether the question is wrong or i just cant get... |
H: Do isomorphic structures always satisfy the same second-order sentences?
I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false.
This is probably a completely obvious question, but is it true that whenever two mathematical structures are is... |
H: Find k such that f is density function
I have the following function: $f_X(x, \theta) = \left\{
\begin{array}{lr}
k/x^3 & : x \leq \theta \\
0 & : x > \theta
\end{array}
\right.$ and $\theta >0$.
I should find $k$ such that $f$ is a density function.
What i know: $\displaystyle\int_{-\inf... |
H: Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges
Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series
$$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$
converges.
AI: According to MathWorld, this is called MacLa... |
H: $\neg P \implies \neg T$ and $P \implies \neg T$. Where do I go next?
I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and I should be able to use that fact as such in ... |
H: what are first and second order logics?
The only knowledge I have on logic is due to a book I read a couple of years ago called Introduction to logic: and to the methodology of deductive sciences by Alfred Tarski. And in it he talks about free variables and sentences and quantifiers. But he never says what order l... |
H: A question about bounds, least and minimal elements, and partial vs strict ordered sets
It's not very clear to me if the concepts of bounds, least elements and minimal elements (also, greatest elements and maximal elements, etc. ) apply only to partial orders or if the definition applies to general ordered sets.
I... |
H: Calculating $1819^{13} \pmod{2537}$ using Fermat's little theorem
Can anyone make me understand how to calculate $1819^{13} \pmod{2537}$ using Fermat's little theorem? Here $p=2537$ and $p-1=2537-1=2536$.
I am unable to understand how to express $1819^{13}$ in terms of $1819^{2536}$.
AI: If you check description of... |
H: Can we conclude from $V=\ker(T) \oplus\operatorname{im}(T)$ the invariance of both subspaces?
Can we conclude for an endomorphism $V \in \operatorname{End}(V)$ where V is a finite dimensional vector space from $V=\ker(T) \oplus \operatorname{im}(T)$ that nullspace and image are invariant subspaces?
I somehow "feel"... |
H: For what kind of a subset its sums equal $\mathbb{R}^4$
For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$.
Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$.
For what values $a,b$, $B$ equals $\mathbb{R}^4$?
In general, what conditions can w... |
H: Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$.
Again, I am new to volume of bodies and I am struggling with it.
Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$.
Now from a previous question, I know that I can do it by $\iint_{D} {z_2(x,y) - z_1(x,y)dxdy}$.
In this case,... |
H: Prove automorphism is trivial
I would like to prove the following:
Let $L\subset L'$, where $L'$ is a quadratic extension of $L$, and $\rho\in\text{Aut}(L'/L)$, the automorphism group of $L'$ which fixes $L$. Also, let $\mathfrak{p}$ be a prime ideal of $L$ which ramifies in $L'$. Then $\rho$ acts trivially on the... |
H: Factor Equations
Please check my answer in factoring this equations:
Question 1. Factor $(x+1)^4+(x+3)^4-272$.
Solution: $$\begin{eqnarray}&=&(x+1)^4+(x+3)^4-272\\&=&(x+1)^4+(x+3)^4-272+16-16\\
&=&(x+1)^4+(x+3)^4-256-16\\
&=&\left[(x+1)^4-16\right]+\left[(x+3)^4-256\right]\\
&=&\left[(x+1)^2+4\right]\left[(x+1)^2-4... |
H: What do countable transitive models of ZFC look like?
According to Cantor's Attic (link):
Not all transitive models of ZFC have the $V_\kappa$ form, for if there is any transitive model of ZFC, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.
Questio... |
H: Complex Differential Equation: $f'(z)=bf(z) \iff f(z)=ae^{bz}$
Let $f\colon G\to\mathbb{C}$ be holomorphic on the domain $G\subseteq\mathbb{C}$ and $b\in\mathbb{C}$. Show that the two following statements are equivalent:
1) $f(z)=ae^{bz}$ on $G$ with a constant $a\in\mathbb{C}$ 2) $f'(z)=bf(z)$ on $G$
1) $\... |
H: Which of these sets is a subspace of F?
Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$:
$F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$,
$F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$,
$F_3 := \{ x \in F:\ \text{$x$ is a zero sequence}\}$,
$F_4 := \{ x \in F:\ \text{$x_n ... |
H: Concept Of Double Integration
Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.
AI: It's basically this: When you do single variable integration, you are fitting rectangles under a curve and letting the width of the rectangles get smaller and s... |
H: Non-trivial Topology
I can't understand the differences between a non-trivial topology and a trivial one.
Whuat's the meaning of "non-trivial" topology?
Is there a link with connection's properties?
For example, could we say that a moebius strip has a "non-trivial" topology while an ordinary strip has a trivial one... |
H: Does every countable subset of the set of all countable limit ordinals have the least upper bound in it?
I'm sorry if the question is that kind of trivial, I just feel uncertain about these ordinals all the time. Is the answer to the following question "yes":
Denote by A the set of all countable limit ordinals. Doe... |
H: Fundamental theorem of Morse theory for $\Omega(S^n )$
Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can I attack these cells? For example $\Omega(S^2) \simeq e^0 \c... |
H: A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact
Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$
Does it follow that $C$ is pre-compact? In particular I am trying to p... |
H: Help for solving this sequence
I couldn't solve the following sequence and couldn't even see any pattern:
AI: Check the OEIS. In particular, we have: $prime(n) -2n$ as seen here. |
H: Factor Equation
Help me with this,
Question: factor $x^3y-x^3z+y^3z-xy^3+xz^3-yz^3$.
Solution:
$$\begin{eqnarray}&=&x^3y-x^3z+y^3z-xy^3+xz^3-yz^3\\
&=&x\left(z^3-y^3\right)+y\left(x^3-z^3\right)+z\left(y^3-x^3\right)\\
&=&x\left[(z-y)\left(z^2+zy+y^2\right)\right]+y\left[(x-z)\left(x^2+xz+z^2\right)\right]+z\left[(... |
H: The Analog of the Cube in The Fourth Dimension
I was just wondering how a "cube" would look in 4-D. I know that in 1-D it is a line, in 2-D it is a square, in 3-D it is a cube.
Is it possible to envision it? If it is, how would the axes be defined? (i.e: 3-D as the x,y, and z axes)
P.S.: Not sure what tag this woul... |
H: Closed set in $l^1$ space
Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote $\delta^{(n)}:=(\delta_j^{(n)})_{j=0}^\infty$ and $E := \{ \delta^{(n)} : n \in \mathbb N\}$.
I want to... |
H: Real valued analytic function defined on a connected set is constant
Let $G$ be a connected set and $f : G \rightarrow \mathbb{C}$ a real valued analytic function. Prove that $f$ is constant.
My idea to prove the result is to prove a subset $A \neq \varnothing$ of the connected set $G$ is both open and closed. S... |
H: parallel and normal projections
I have a vector $v$ given by $(v_x, v_y, v_z)$ which makes an angle $\theta$ with the $x$-axis. The projection of $v$ onto $x$ is given by the dot product
$$v\cdot x = \cos\theta\sqrt{v_x^2+v_y^2+v_z^2}$$
Say I want to find the projection of $v$ onto the $yz$-plane ($v$ has an angle ... |
H: Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$
Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?
That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?
AI: It is not hard to see that $x^2+x+1$ does not have a root in $\Bbb{Z... |
H: proof of equivalence of continuity and continuity in terms of limits of sequences
I am currently working on the axiom of choice and was looking for easy applications. A common example is the proof of the equivalence of continuity with continuity in terms of limits of sequences. Usually the proof uses the axiom of c... |
H: Is a subset of $\mathbb{R}^{n}$ that is homeomorphic to $\mathbb{R}^{n}$ necessarily open?
Let $A$ be a subset of $\mathbb{R}^{n}$, such that $A$ is homeomorphic to $\mathbb{R}^{n}$.
Is $A$ open in $\mathbb{R}^{n}$?
AI: Yes. This is a special case of the theorem of invariance of domain. |
H: differentiability: a question involving interchange of limits
Let $f:[a,b]\to\mathbb{R}$ continuous and $C^1(\,]a,b])$.
Suppose
$$ f'(x)\xrightarrow[x\to a+]{}{}l $$
1) If $l\in\mathbb{R}$ I manage to prove that $\exists\,f'(a)=l$ (I used uniform continuity of $f'$), hence $f\in C^1([a,b])$.
2) Now if $l=\infty$ I ... |
H: Approximating sum by Gaussian integral - how big is the error?
I have the following infinite sum:
$$S=\sum_{n=1}^{\infty}e^{-an^2}$$
Where $a$ is a positive constant. Is there a simple way to estimate the error when approximating $S$ by:
$$S \approx \int_0^ \infty e^{-ax^2}dx .$$
Does this depend at all on the valu... |
H: Asymptotic formula for complex gamma function at $+\infty+i \times y$
I am currently looking for the behaviour of the complex gamma function at real infinity:
$\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$
and more particularly for asymptotic formulas for the following functions:
$f_1\left(y\right)=\text{Re}\l... |
H: Can't see how this function is differentiable Spivak's Calculus on Manifolds Exercise 2-4
The problem is as follows:
Let $g$ be a continuous real-valued function on the unit circle $\{x \in \mathbb{R}^2 : \lvert x \rvert = 1\}$ such that $g(0,1) = g(1,0) = 0$ and $g(-x) = -g(x)$. Define $f: \mathbb{R}^2 \to \mathbb... |
H: Cubic with turning point near zero
I want a bunch of cubics which have a turning point near the $x$-axis, both above and below the $x$-axis.
That way, the graph might not easily show whether there is a zero there, and Newton's method might give the answer.
I want a bunch so that each student gets a different one. ... |
H: How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$?
Let $V=K^{n+1}$ be a vector space of dimension $n+1$ and $\mathbb{P}V$ the projective space associated to $V$. How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$? Thank you very much. I thin... |
H: How to find closed form by induction
How can I find the closed form of
a) 1+3+5+...+(2n+1)
b) 1^2 + 2^2 + ... + n^2
using induction?
I'm new to this site, and I've thought about using the series 1 + 2 + 3 +...+ n = n(n+1)/2 to help me out But isn't that technically using prior knowledge and hence invalid? Am I on... |
H: Prove that $\frac{100!}{50!\cdot2^{50}} \in \Bbb{Z}$
I'm trying to prove that :
$$\frac{100!}{50!\cdot2^{50}}$$
is an integer .
For the moment I did the following :
$$\frac{100!}{50!\cdot2^{50}} = \frac{51 \cdot 52 \cdots 99 \cdot 100}{2^{50}}$$
But it still doesn't quite work out .
Hints anyone ?
Thanks
AI: $$... |
H: diagonalize quadratic form
I have this quadratic form
$Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$
And they ask me:
for which values of $x,y$ and $z$ is $Q=0$?
and I have to diagonalize also the quadratic form.
I calculated the eigenvalues: $k_{1}=0=k_{2}, k_{3}=14$,
and the eigenvector $v_{1}=(-2,1,0), v_{2}=(1,2,3),... |
H: approximating $\frac{S^2}{\sigma^2}$
Let $Y_1,\ldots,Y_n$ be independent random variables from a normal distribution with expected value $\mu$ and variance $\sigma^2$ and let $S^2 = \dfrac{1}{n-1} \sum^n_{i=1} (Y_i-\bar{Y})^2$ be the sample variance. use the Central limit theorem to show that the distribution of $... |
H: Revolution of a solid - mandatory disk method
I know I am doing something wrong. Anyways
$x = 2$
$x = 3$
$y = 16 - x^4$
$y = 0$
about the y axis
So about the y axis means I need everything in terms of y. Easy enough, that is just one term.
$$y = 16 - x^4$$
$$x = (y - 16)^\frac{1}{4}$$
Then I intgrate with respect t... |
H: Determining whether these two groups are isomorphic
Consider the following group of matrices with multiplication:
$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \ B = \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}, \\ C = \begin{pmatrix} -1 & -1 \\ 0 & 1 \end{... |
H: Taking the derivative of an Integral
I would like to express the derivative of an integral in as elegant a form as possible. However, I am struggling at the moment. I would like to find the derivative $f'(y)$ of the function
$f(y) = \int_{h(y)}^{g(y)}u(x,y)\,\mathrm{d}x$
in terms of only the functions $g$, $h$ and ... |
H: Going from the Poisson distribution to the Gaussian.
In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value of the Poisson distr. increases, the devation from the mean beh... |
H: Trigonometric equations with more than one function
This is a general question about how to solve trigonometric equations which involve different functions. I have been multiplying and dividing the functions but have not been able to attain an expression with just one function. I'm encountering questions such as:
$... |
H: Succeed with one multi-choice test out of five if answers are guessed
I'm taking a class in probability, and I have some issues understanding some concepts. Thing is, I am trying to calculate the probability that you can succeed with at least one multi-choice test out of five, if all the answers are guessed. In eac... |
H: Show that two tori that identified $\left(x,y\right)\mapsto\left(y,x\right) $ is homeomorphic to $\mathbb{S}^{3} $
The boundary tori $\mathbb{S}^{1}\times\mathbb{S}^{1}$ of two copies of the solid torus $\mathbb{S}^{1}\times D^{2}$ are identified be a map $\left(x,y\right)\mapsto\left(y,x\right)$. Show that the res... |
H: Arithmetic operations in ternary number system
In a ternary number system, how are the $4$ arithmetic operations defined?
AI: Strictly speaking, one should call it the ternary numeral system. What is different from base-10 is not the numbers, but the numerals.
Addition, subtraction, multiplication, and division ar... |
H: Why is the discrete metric said to be so important
Can anyone enlighten me as to why the discrete metric is considered to be important in mathematics? The only real use I can see of it is that it shows the existence of a metric on any non-empty set.
I wonder if there is something I'm missing: maybe it used as a tec... |
H: Will this Trigonometric give the following answer?
If $n$ is an integer, can $$\cos[(2n-1)\pi/2]-\cos[(2n-1)\pi/4]$$
be equal to $\;\;\cos[(2n-1)\pi/4],\;\;?$
I have tried the formula for $\cos A-\cos B$ but that would give result in $sine$
AI: Hint: $$\cos \left((2n-1)\frac{\pi}{2}\right)=\cos \left(\pi n-\frac{\p... |
H: Why $\sum_{k=1}^n \frac{1}{2k+1}$ is not an integer?
Let $S=\sum_{k=1}^n \frac{1}{2k+1}$, how can we prove with elementary math reasoning that $S$ is not an integer?
Can somebody help?
AI: Hint: Recall the (elementary) proof that $\sum_{i=1}^n \frac{1}{i}$ is not an integer (for $n>1$):
Let $2^k$ be the largest po... |
H: Will the feasible region always be convex in linear programming?
In linear programming we find a feasible region , is this region always convex? . if a concave region is found where objective is minimization , I think then a solution exists .
Advance thanks.
someone deleted the answer of my previous post , although... |
H: Method of moments and maximum likelihood
I have the following function: $f_X(x, \theta) = \left\{
\begin{array}{lr}
\theta/3 & : x = -1 \\
\theta/3 & : x = 0 \\
1-2\theta/3 &: x= 1
\end{array}
\right.$
What is the method of moments of $\theta$?
Here's my attempt:
1º Method of momen... |
H: Proving by induction: $2^n > n^3 $ for any natural number $n > 9$
I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$
Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical induction.
I tried the problem for a long time, but got ... |
H: Finding the coordinates of the point on the graph of $f(x) = (x+1)(x+2)$
Im trying to find the coordinates of the point on the graph of $f(x) = (x+1)(x+2)$ at which the tangent is parallel to the line with the equation $3x - y - 1 = 0$
How do i find the coordinates?
What I've tried is:
Since it is parallel, i know ... |
H: Algorithm for finding the longest path in a undirected weighted tree (positive weights)
I'm trying to find from a undirected weighted tree of only positive weights the longest path (diameter of a tree, I'm told?) I know the most common algorithm is one where you pick a random node $x$, use DFS from that node to fin... |
H: Selecting a committee of $5$ out of $20$ with conditions
A company has $20$ employees, $12$ males and $8$ females. How many ways are there to form a committee of $5$ employees that contain at least one male and at least one female?
This is what I got: $12\times19\times18\times17\times16-12\times11\times10\times9\t... |
H: Prove by induction: $\forall n\in\mathbb{Z}_{\geq1}:3\ |\ (6n^2-12n+3)$
I'm not sure how to start this induction problem.
I was told that we start doing induction by using a base case $n=1$. Then we set $n=k$ to prove $n=k+1$. But how do I prove that $6(k+1)^2-12(k+1)+3$ is divisible by $3$ if $6k^2-12k+3$ is?
I... |
H: Combinatorics question about Taking Days Off
A cashier wants to work five days a week, but he wants to have at least one of the Saturday and Sunday off. In how many ways can he choose the days he will work?
So, in this case, what should I count first? How do I start? I know how to solve this if the cashier doesn... |
H: Volume of a solid of revolution: $y = x^3$, $y = x^{1/3}$, $x \geq 0$ rotated about $y$-axis
I am trying to find the volume:
Rotate about $y$.
$$y = x^3,\quad y = x^{1/3},\quad x \geq 0$$
Simple enough.
$x = y^3 \implies x = y^{\frac{1}{3}}$
$$\pi \cdot \int_0^1 y^{(1/3)^2} - y^{3^2}dy$$
$$\pi\cdot \left(\frac... |
H: discretize a function using $z$-transform
I would like to discretize the following continuous function using $z$-transform:
$$G(s)=\frac{s+1}{s^2+s+1}$$
The process I am using is to take the inverse Laplace transform of $\frac{G(s)}{s}$ and then take the z-transform of it. Finally I multiply it the result for $1-z^... |
H: Need help with non-recursive definition
So I'm trying to find a non-recursive definition for $b_n$. I'm given $$b_0=1$$ $$b_{n+1}=2b_n-1$$
Does this mean I'm trying to find a number for $b_n$ that fits that algorithm?
Update:
Proof by induction. Let P(n) be that for any $n$, $b_n=1$.
As our base case, we prove P... |
H: mathematical maturity
So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel like I lack the intuition necessary to be able to come up with concepts on my own. I feel like ... |
H: Find the derivative of $y = f(x^2 - 2x + 7)$ where $f'(10) = 2$
Determine the derivative if $y = f(x^2 - 2x + 7)$ and $f'(10) = 2$
Ok so honestly, I dont know how to solve this, or even know where to start. All i know is that we are given a point $(10, 2)$. But what is $f(x^2 - 2x +7)$ supposed to mean?
AI: Suppose... |
H: If three cards are selected at random without replacement. What is the probability that all three are Kings?
This a two part question:
$1$: If three cards are selected at random without replacement. What is the probability that all three are Kings? In a deck of $52$ cards.
$2$: Can you please explain to me in lay m... |
H: What is the difference between exponential symbol $a^x$ and $e^x$ in mathematics symbols?
I want to know the difference between the exponential symbol $a^x$ and $e^x$ in mathematics symbols and please give me some examples for both of them.
I asked this question because of the derivative rules table below contain b... |
H: Bounds on a mapping from unit disc to left half plane
I've recently started studying for qualifying exams and have been having trouble with the following question: Let $f$ be a nonconstant analytic function on $\mathbb{D}$ such that $f (\mathbb{D}) \subset \{z \in \mathbb{C}: Re(z)<0\}$ with $f(0)=-1$. Prove that f... |
H: Splitting the electricity bill
This is a similar question, but not quite - click
Problem:
3 people stay at a flat and they need to divide the electricity bill fairly. They took one meter reading when they moved in and 1 meter reading $k$ weeks later. For those $k$ weeks, person $A$ has been in the flat for $x$ wee... |
H: $\pi_0$ in the long exact sequence of a fibration and quaternionic projective space
I am doing a past paper for an introductory course in algebraic topology. The question is
Calculate the homology of the quaternionic projective space. What can you say about its homotopy groups?
I figured that we have a CW decompo... |
H: I know what I need to do but dont know how to apply: the question related to The first order approximation theorem
$\mathbf{Question:}$
Prove that
$\displaystyle \lim_{(x,y)\to (0,0)} \dfrac{\sin(2x+2y)-2x-2y}{\sqrt{x^{2}+y^{2}}}=0$
$\mathbf{My\ ideas:}$
I will use the First Order Approximation Theorem.
But how c... |
H: Why is matrix multiplication defined a certain way?
Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined this way?
Why can't a (1x3) by (3x1) yield a (3x3), or a (3x1) by (... |
H: Evaluate $\cos 18^\circ$ without using the calculator
I only know $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ as standard angles but how can I prove that
$$\cos 18^\circ=\frac{1}{4}\sqrt{10+2\sqrt{5}}$$
AI: Consider the isosceles triangle pictured below:
By considerin... |
H: Changing from x to y $x = y(4-y)$
$$x = y(4-y)$$
I am guessing I need some pretty advanced math to solve this for y. I am trying to use the shell method and I have to use opposite terms of the rotation axis so I am rotating around y so I need x variables.
I have a whole sheet of paper trying to solve this, is there... |
H: Why is the Jacobi symbol $(D/m) = (D/n)$ for certain $m,n,D$?
$m \equiv n$ mod $D$, $m,n >0$ and odd, and $D \equiv 0,1$ mod $4$, then $(D/m) = (D/n)$
I'm am sure that one can show this using quadratic reciprocity and the supplements. Any ideas?
AI: If $D\equiv 1\pmod 4$, then
$$ \left(\frac Dn\right)=(-1)^{\frac{... |
H: Is there a simple way to state continuity for $I$-adic topology?
Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a homomorphism $f:R\to S$ is continuous? I am guessing... |
H: How to calculate $\cos(6^\circ)$?
Do you know any method to calculate $\cos(6^\circ)$ ?
I tried lots of trigonometric equations, but not found any suitable one for this problem.
AI: I'm going to use the value of $\cos 18°=\frac{1}{4}\sqrt{10+2\sqrt{5}}$ obtained in this question.
$\sin^2 18°=1-\left(\frac{1}{4}\sq... |
H: Related rates, where do I start?
A revolving searchlight, which is $100$ m from the nearest point on a straight highway, casts a horizontal beam along a highway. The beam leaves the spotlight at an angle of $\frac{π}{16}$ rad and revolves at a rate of $\frac{π}{6}$ rad/s. Let $w$ be the width of the beam as it swee... |
H: Is determinant uniformly continuous?
The determinant map $\det$ sending an $n\times n$ real matrix to its determiant is continuous since it's a polynomial in the coefficients. Is it also uniformly continuous?
AI: The determinant function on the set of $n\times n$ matrices is a non-zero polynomial which is homogeneo... |
H: What is the integral of $\int e^x\,\sin x\,\,dx$?
I'm trying to solve the integral of $\left(\int e^x\,\sin x\,\,dx\right)$ (My solution):
$\int e^x\sin\left(x\right)\,\,dx=$
$\int \sin\left(x\right) \,e^x\,\,dx=$
$\left(\sin(x)\,\int e^x\right)-\left(\int\sin^{'}(x)\,\left(\int e^x\right)\right)$
$\left(\sin(x)\,e... |
H: Proof of the continuous function having tangent plane has directional derivatives
Suppose that the continuous function $f: \Bbb R^2 \to \Bbb R$ has a tangent plane at the point $(x_0, y_0, f(x_0, y_0))$
Prove that the function $f$ has directional derivatives in all directions at rhe point $(x_0, y_0)$
I guess th... |
H: For All Unique Combinations of 60 A's and 20 B's Number of Combinations that have BB
Here is my question. I have 60 A's and 20 B's and need to find out the number of unique combinations of those where B shows up consecutively at least once.
For example (6 A's and 2 B's):
AAAAAABB = 1
AAAAABAB = 0
AAAABBAA = 1
AI: L... |
H: Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$
The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar $\lambda$.
Could somebody provide some intuition why ... |
H: Infinite-dimensional extensions of $\mathbb Q$
I need help to solve the following exercise:
Let $X$ be an indeterminate over $\mathbb Q$ (so a transcendental number) and consider the field extensions $\mathbb Q\subseteq \mathbb Q(X^3)\subseteq\mathbb Q(X^2)\subseteq\mathbb Q(X)$. Prove that $$\mathrm{Fix}(\mathrm{... |
H: number of different addends which sums to 41
I have following equation:
$$\sum_{i=1}^{21} m_i = 41$$
where $m_i$ are non-negative integers.
How many different solutions are there. Note, that $41 + 0 + ... + 0$ is a different solution than $0 + 41 + 0 + ... + 0.
My only idea was to solve this recursive: let $P_a(... |
H: Discrete Mathematics: $x\leq y+\epsilon \implies x\leq y$
Let $x$ and $y$ be real numbers. Prove that if $x\leq y + \epsilon$ for every positive real number $\epsilon$, then $x\leq y$.
I would like a hint as to how to prove this. Thank you. Pictorial proof would be nice too.
So this is how I word-smithed the ans... |
H: proving inequality $0 < x^4+2x^2-2x+1$ for $x>0$
How can I elegantly prove the inequality $0 < x^4+2x^2-2x+1$ for $x>0$. I have plotted this function in a Sage (an open source and free CAS) and I can see that there is a local min between $0$ and $1$ that lies above the x-axis.
Therefore,I could show that the functi... |
H: Using Calculus to find total and maximum revenue and profit
I'm grappling with understanding how to use calculus to find rates of profit, revenue, and cost. I have the following problem:
$x = \text{ quantity }$, $12 < x < 48$
Total Cost: $C(x) = \dfrac 92x^2 -17x + 2700$
Price per item: $p(x) = -\dfrac{x^2}3 +\df... |
H: Work to pump water out of a tank with radius $10$
Water density is 1000, tank is a half sphere with radius $10$.
9.8 for gravity, 1000 for density, 2pi to find the volume and all of the rest gives me work.
$$ 9.8(1000)(2\pi) \int r^2 dy$$
To find the radius I just use $(10 - y)$
This is wrong, but why?
AI: The mas... |
H: Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$
Let $f\in L(X,\mathcal{X},\mu)$. This makes $\int f^+\,d\mu<+\infty$. Now, there exists a monotone increasing sequence of simple measurable functions $\phi_n$ that converge to $f^+$. By the monotone convergence theorem, we also ha... |
H: Minimum value of $f(x) = x^3 + 9x^2 + 5$ on $[0,3]$
For the function $f(x) = x^3 + 9x^2 + 5$ on the interval $[0,3]$, determine the minimum value.
I don't know how to do this. I think we have to find the derivative and set that equal to $0$, but that wasn't giving me the right answer. Any ideas?
AI: I assume tha... |
H: Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
Let $a,b,c$ be positive real numbers. Prove that $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$.
My (strange) proof:
$$
\begin{align*}
a^3+b^3+c^3 &\geq a^2b+b^2c+c^2a\\
\sum\limits_{a,b,c} a^3 &\geq \sum\limits_{a,b,c} a^2b\\
\sum\limits_{a,b,c} a^2 &\geq \sum\limits_{a,b,c} ab\\
a... |
H: Using complete induction, prove that if $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$, then $a_n=2^n$
Could anyone please explain to me how to do this problem by using the principle of complete induction? Thanks. :)
Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Prove that $a_n=2^n$ for all n... |
H: Change of basis and identity
Let $\beta = \{b_1,\dots, b_n \}$ be a base for $V$. Explain why the $\beta$ coordinate vectors of $b_1,\dots, b_n$ are the columns $e_1, \dots, e_n$ of the $n$ by $n$ identity.
The solution simply says $b_1 = 1b_1 + 0b_2 + \dots0b_n$.
Here is what I don't understand. If I take $b_1 =... |
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