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H: What is a set function that returns another set of points called?
I have a set of points $S = \{x_i\}_{i = 1}^m, x_i \in \mathbb{R}^n \forall i$. Now, I have a set function $f$ which operates as follows:
$$f(S) = GX^T$$
where $G \in \{0,1\}^{m\times m}$ and $X = [x_1 x_2 \ldots x_m]$.
Is there a specific name for s... |
H: Definite integral of a periodic funtion, offset by the period, equals the original definite integral
Suppose $f: \mathbb R \to \mathbb R$ is Riemann integrable on every finite interval and periodic with period $T>0$. Then for every interval $[a,b]$:
$$ \int_a^b f = \int_c^d f,$$ where $c = a+T$ and $d = b+T$.
... |
H: If derivative of $e^{ax} \cos{bx}$ with respect to $x$ is $re^{ax}\cos(bx + \tan^{-1} \frac {b} {a})$
Problem: If derivative of $e^{ax} \cos{bx}$ with respect to $x$ is $re^{ax}\cos(bx + \tan^{-1} \frac {b} {a})$.
Then find $r$ when $a>0,b>0$
Solution: Differentiating $e^{ax} \cos{bx}$ w.r.t $x$ we get
$ ae^{ax} ... |
H: A question on generators
Suppose $I$ is an ideal in a ring $R$ which is finitely generated. Suppose on the other hand that there is some (possibly other) set of generators $\{g_t\colon t\in T\}\subset I$ which also generates $I$ as an ideal. Can we find a finite subset $T_0\subset T$ such that $\{g_t\colon t\in T_0... |
H: Relationship between positive definite matrix eigenvalues and the principal axes lengths of ellipsoid
In many textbooks of pattern recognition I have seen the following statement:
If a matrix $A$ in the quadratic form $F(\text{x}) = \text{x}^{T}A\text{x}$ is positive definite, then it follows that the surfaces of c... |
H: Commuting Exponential Matrices
Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$.
Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal.
$A, B$ square matrices.
AI: Hint: (1) What ODEs do $x$ and $y$ fulfill? Compute $x'$ and $y'$, try to find th... |
H: ball on the moon part 3
This the final parts to my previous question ( On the surface of the moon )
The original question reads "On the surface of the moon, acceleration due to gravity is approximately 5.3 feet per second squared. Suppose a baseball is thrown upward from a height of 6 feet with an initial velocity ... |
H: True Or not: Compact iff every continuous function is bounded
Let $X$ be a topological space. My question is:
If $f:X\to \mathbb{R}$ is bounded for all such continuous $f$, then is $X$ compact. Is is really?
If $X$ is the subset of $\mathbb{R}^d$, then it is clear, beacause with Heine-Borel we get what we want (clo... |
H: Conjecture on eigenvalue property of a matrix subspace
Suppose we have a full rank positive definite Hermitian matrix $A\in \mathbb{C}^{n \times n}$ with eigenvalues $\lambda_1>\lambda_2> \dots >\lambda_n$. Consider a semi-orthogonal matrix $X \in \mathbb{C}^{n \times p}$ (i.e., $X^* X = I_p$) spanning a subspace o... |
H: The national lottery
When playing the lottery you have to pick 6 numbers out of 45 possibilities. Since the order of the numbers don't matter, the number of possible combinations for the jackpot (and hence the 6 correct numbers) is given by:
$
\dbinom{45}{6} = C^6_{45} = \frac{45!}{(45-6)!6!} = 8145060
$
Assuming t... |
H: Powers of a unitary matrix
I'm trying to find the minimum exponent $M\in \mathbb{N}$, such that for a certain unitary matrix $F\in \mathbb{C}^{N}$,
$$F^M = 1_{\mathbb{C}^{N}}.$$
I don't think it matters, but it's the DFT matrix.
Now, I noticed that if $F^M = 1$, then $F^{M/2} = ({F^{M/2}})^\dagger$. However, I can'... |
H: Show that ($\ell^1$, $\|\cdot\|_1$) is complete
Show that the vector space $\ell^1 : = \{(a_n) : \sum_n|a_n| < \infty\}$ with the norm $\|(a_n)\|_1 : = \sum_n|a_n|$ where $(a_n)$ are sequences in $\mathbb C$ is complete.
Thanks in advance.
AI: Consider a Cauchy sequence $\{x^n\}$in $l_1$. Where $x^n = (x_1^n, x_2... |
H: Is $\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$ a natural number for $n$ even?
Probably this is a easy question, but I was unable to solve it. Let $n$ be a even natural number. Is true that the following number is natural for all $n$? $$\frac{n!}{\left(n/2\right)!\left(n/2\right)!}$$
I can see, for example, that... |
H: Trigonometry question: Find in simplest surd form: $\cos 195^{\circ}$
Find in simplest surd form: $\cos 195^{\circ}$.
Ive recently been doing the trigonometry topic form textbook and have oftenly come across these questions. Can someone please justify how you do this question? Ive tried many times but no luck.
AI: ... |
H: Understanding Power Series Multiplication Step
Working on Spivak's Calculus problems, I searched online, trying to understand the solution provided for Problem 4a of Chapter 2. I found the question I needed:
Spivak's Calculus - Exercise 4.a of 2nd chapter.
However, the answer provided there started with the followi... |
H: Group Theory Normal Subgroups
Let $G$ be a group of order $8$ with $x\in G$ such that $o(x)=4$.
Prove that $x^2\in Z(G)$, where $$Z(G)=\{ x \in G \mid xg=gx\text{ for all }g\in G\}.$$
AI: HINT: Any subgroup of index $2$ is normal in $G$. |
H: Is an automorphism of the field of real numbers the identity map?
Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map?
If yes, how can we prove it?
Remark An automorphism of $\mathbb{R}$ may not be continuous.
AI: Hint: Let $\phi$ be a field automorphism of $\mathbb R$. Then prove:
$\phi... |
H: A function in a Real Vector Space V (where V is the set of all complex-valued functions f on the real line ), which is NOT real-valued.
Question:
Let $V$ be the set of all complex-valued functions $f$ on the real line such that
(for all $t \in \Bbb R) \ \ f(-t) = \overline {f(t)} $.
The bar denotes complex conjugat... |
H: Rational number to the power of irrational number = irrational number. True?
I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question.
Now, his inital solution was like this: let's tak... |
H: Predicting the number of decimal digits needed to express a rational number
The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$ Meanwhile, it takes 49 digits to express the number $1/221$, since a string of 49 digits repeats: $$\... |
H: Is a differentiable probability density function bounded?
Let $g$ be a lebesgue probability density function, differentiable, such that $\sup_x|g'(x)| < \infty.$
Is $g$ bounded?
AI: Let $M := \sup_x \lvert g'(x)\rvert$. Let $n \geqslant 2$. Suppose there exists an $x_n$ with $g(x_n) = n$. Let $x_{n-1}$ be a closest... |
H: A not smooth distribution
Is exist the not smooth distribution which satisfying:
$\left ( D_{t}^{2}-D_{x}^{2} \right )u(x,t)=0$
I can't find at least one not smooth distribution like this...
Thanks for the help!
AI: A classic! This example upset a lot of people c. 1800 and even before. It it the one-dimensional wav... |
H: Statistics I - Bayes's Theorem
A desk has three drawers. The first contains two gold coins, the second has two silver coins and the third has one gold and one silver coin. A drawer is selected at random and a coin is drawn at random from the drawer. Suppose that the coin selected was silver. Use Bayes's Theorem to ... |
H: Parametric Equations rotation of axes
For old coordinates $(x,y)$ the new coordinates $(u,v)$ are related like this:
$x = u\cos(\theta) - v\sin(\theta)$
$y = u\sin(\theta) + v\cos(\theta)$
So would it be correct to say that to rotate the axis for a parametric equation defined by $x = f(u)$ and $y = g(u)$ I need to ... |
H: On linear transformations
Do there exist continuous linear maps $T\colon V \to V$ such that $T^{-1}\colon V\to V$ exists but is not continuous? Clearly if $V$ has finite dimension the answer is no.
AI: By the open mapping theorem, that can't happen when $V$ is a complete metrizable TVS (over $\mathbb{R}$ or $\mathb... |
H: Solution of pendulum linked to Weierstrass $\wp$-function
I've been working through a question about the equation of motion of a pendulum.
I have to now solve the equation of the form:
$$u'^2=u^3+au+b,$$ where $a=(\frac{g^2}{l^2}-\frac{c^2}{3})$ and $b=(\frac{g^2c}{3l^2}-\frac{2c^3}{27})$, using separation of va... |
H: Ideas about a transformation matrix
I have a problem in 3D where I have three vectors: $\boldsymbol{\omega}$, $\boldsymbol{m}$ and $\boldsymbol{T}$.
The first $\boldsymbol{\omega}$ is the variable I solve for, $\boldsymbol{m}$ is fixed and the last is related to the others by
$$ \boldsymbol{T} \sim (\boldsymbol{... |
H: How to find the sum of the sequence $\frac{1}{1+1^2+1^4} +\frac{2}{1+2^2+2^4} +\frac{3}{1+3^2+3^4}+.....$
Problem :
How to find the sum of the sequence $\frac{1}{1+1^2+1^4} +\frac{2}{1+2^2+2^4} +\frac{3}{1+3^2+3^4}+.....$
I am unable to find out how to proceed in this problem.. this is a problem of arithmetic prog... |
H: Sums and products involving Fibonacci
In summary, if $\phi$ is the golden ratio, I want to show:
\begin{align}
\sum_{n=1}^\infty \frac1{F_n} &= 4-\phi \\
\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{F_nF_{n+1}} &= \phi-1 \\
\prod_{n=2}^\infty \bigg( 1 + \frac{(-1)^n}{F_n^2} \bigg) &= \phi.
\end{align}
AI: The first equatio... |
H: Combination and Permutation
In how many ways can the letters of the word VANESSA be arranged so that no two
vowels are together?
The answer is $900$ but I get:
$2!/2!\cdot\binom{3}{2}\cdot 2!/2! = 3$.
AI: Let's lay down the consonants first. There are $4!/2!=12$ ways to arrange the letters $V,N,S,S$.
Let's take $SN... |
H: $\nabla \varphi \overset{?}{=} \nabla \cdot \varphi \bar{\bar{I}}$ where $\varphi$ is scalar, $\bar{\bar{I}}$ is identity tensor
I am trying to determine if these two are equivalent. I have a function written with both terms, and this is the only discrepancy. The gradient increases the rank of the scalar to a vecto... |
H: Create function F() from Points
I would like to recreate a function only by knowing points on the graph.
So I would have the points
A(x/y)
B(x/y)
C(x/y)
and would like to create its f()
Is this possible?
I heard this should be possible with a Taylor Series but to do a Taylor Series wouldn't I need a f() and its de... |
H: Complex Exponent of Complex Numbers
How does one find the algebraic solution of a Complex number raised to the power of another Complex number?
Here is the work I have done so far, if there are any mistakes please inform me.
A real number with a Complex Exponent:
$$A \in \Bbb R, \space z \in \Bbb C,\space z = x + i... |
H: proof of $ A - \left (B \cap C \right)= \left (A - B \right) \cup \left (A - C \right)$
I am trying to prove $ A - \left (B \cap C \right)= \left (A - B \right) \cup \left (A - C \right)$
I came up with this proof:
Let $ S \{ x | x \in A - \left(B \cap C \right)\}$ Let $ Q \{y | y \in \left (A - B \right) \cup \l... |
H: Difference between basis and subbasis in a topology?
I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?
AI: Bases and subbases "generate" a topology in different ways. Every open set is a union of basis elements.... |
H: Proof of Hilbert's Nullstellensatz, weak form.
The statement of Hilbert's Nullstellensatz, weak form, as given here is "Let $f_1,f_2,\dots,f_n$ be polynomials in $K[x_1,x_2,\dots,x_n]$, where $K$ is an algebraically closed field. Then $1=\sum{g_t f_t}$ for suitable $g_t\in K[x_1,x_2,\dots,x_n]$ if and ony if the al... |
H: Does an analytic function maps a simple connected region into a simple connected region?
Suppose $f$ is analytic, say, in $\mathbb{C}$, and suppose $\Omega$ is a bounded simple connected open domain whose boundary we denote as $\Gamma$, then is $f(\Omega)$ also a simple connected domain whose boundary is $f(\Gamma... |
H: real and imaginary part in $\sin z$ where z is complex
I wanted to know, how can I determine the real and imaginary part in
$\sin z$ where $z \in \Bbb{C}$?
Well, this is a part of a series of questions comprising the same in
$\log z$ and $\tan^{-1} z$
I was able to solve this but no idea on how to solve for $\... |
H: About Regulated and Bounded Variations in Banach Spaces
In the following definitions, we assumed that $(X,\left\|\cdot\right\|)$ is a Banach space.
Definition 1. $f:[a,b]\to X$ is of bounded variation on $[a,b]$ if
$$\operatorname{Var}(f;[a,b])=\operatorname{sup}(D)\sum \left\|f(v)-f(u)\right\|<+\infty$$
where the... |
H: Zeroes of a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ for $n\geq 2$
Why can the zeros of a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $n\geq 2$ have no isolated zeros (or poles if we write it as meromorphic)?
Someone says the $n$-times Cauchy Integral formula is enough, but how does it work?
AI: This ... |
H: Expansion of $\sin x$
I wanted to know, how can I derive:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots$$
AI: Use Mclaren series : $f(x) = f(0)+f^{'}(0)\frac{x}{1!}+f^{''}(0)\frac{x^2}{2!}+f^{'''}(0)\frac{x^3}{3!}+\cdots$
for $f(x)=\sin x$ |
H: Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct?
Problem
The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:
$a_1=\sqrt2$,
$a_{n+1}=\sqrt{2+a_n}$.
Evaluate the limit $\lim_{n\to\infty} a_n$.
Solution
Show that the sequenc... |
H: Properties of Triangle - Trigo Problem : In $\triangle $ABC prove that $a\cos(C+\theta) +c\cos(A-\theta) = b\cos\theta$
Problem :
In $\triangle $ABC prove that $a\cos(C+\theta) +\cos(A-\theta) = b\cos\theta$
My approach :
Using $\cos(A+B) =\cos A\cos B -\sin A\sin B and \cos(A-B) = \cos A\cos B +\sin A\sin B$, we... |
H: What is $\mathfrak a$?
I'm currently reading Mendelson's Introduction to topology and have came across this theorem:
Theorem 3.8:
Let a neighborhood in a topological space be defined by Definition 2.2 and an open set in a neighborhood space be defined by Definition 3.5. Then the neighborhoods of a topological spac... |
H: How to resolve this probability question?
today I was answering a exam and I get a problem which I have no idea how to resolve it. Here is the announcement
$500$ people attend a nightclub. Those who are members of the club
pay 14 dlls, and those who are not members paid 20 dlls
All ($100\%$) of those who are ... |
H: Prove that in an obtuse triangle $\angle HAO = \angle B - \angle C$
Consider the following triangle with orthocentre $H$ and circumcentre $O$. Prove that $\angle HAO = \angle B - \angle C$.
I am familiar with the proof for this when $ABC$ is acute, I wanted to prove it when it is obtuse.
$\angle HAO = \angle HAC... |
H: $PGL(n + 1)$ acts on sets of $n + 2$ points in $\mathbb{P}^n$ transitively: proof without determinants?
It is "well known" that if $p_1, \dots, p_{n + 1}$ are points in $\mathbb{P}^{n -1}$ (over $\mathbb{C}$, say) in general position, and $q_1, \dots, q_{n + 1}$ are another set of such points, then there is a uniqu... |
H: Solving Summation Expressions
I would like to know how do you solve summation expressions in an easy way (from my understanding).
I am computer science student analyzing for loops and finding it's time complexity.
e.g
Code:
for i=1 to n
x++
end for
Summation:
n
∑ 1
i=1
Solving:
= ∑ [n-1+1] (to... |
H: Every infinite subset of a countable set is countable.
Here is the proof I tried to weave while trying to prove this theorem:
Theorem. Every infinite subset of a countable set is countable.
Proof. Let $A$ be a countable set and $E\subset A$ be infinite. Then $A\thicksim\mathbb{N}$. This implies that there is a seq... |
H: What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?
Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$
My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point
Here $A,B$ and $C$ are the point of a $... |
H: Evaluate $\int_0^R \frac{r^2}{(1+r^2)^2}dr.$
I am trying to evaluate the following integral: $$\int_0^R \frac{r^2}{(1+r^2)^2}dr.$$ I might substitute $u=r^2$, but I don't find $du$ anywhere. Obviously the integral should be bounded on $R\in [0,\infty)$.
Any ideas?
AI: Substitute $r=\tan{t}$; $dr = \sec^2{t} \, dt$.... |
H: Is it really true that $A^2 = -A \Leftrightarrow (I + A)^2 = A$?
$A$ is a generic square matrix and $I$ is the identity matrix.
I failed to prove that, but I managed to disprove it:
\begin{align*}
A &= [-1] \\
A^2 &= ([-1])^2 = [1] = -A \\
(I + A)^2 &= ([1] + [-1])^2 = [0] \neq A
\end{align*}
So... am I missi... |
H: Exterior Product $d\Phi_1\wedge d\Phi_2$ and spherical coordinates
One short question:
If $\Phi\colon\mathbb{R}^3\to\mathbb{R}^3$, defined by
$$
\begin{pmatrix}r\\\vartheta\\\phi\end{pmatrix}\mapsto\begin{pmatrix}r\sin \vartheta\cos \phi\\r\sin \vartheta \sin\phi\\r\cos\vartheta\end{pmatrix},
$$
what are ... |
H: Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$.
Let $f(x,y)\in C^3(\mathbf{R}^2)$ and let $u=x+y$ and $v=y$.
Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$.
I'm supposed to use the chain rule, how do I go about?
Thanks!
Alexander
AI: If $f$ is of the form $f... |
H: Prove that $ n < 2^{n}$ for all natural numbers $n$.
Prove that
$
n < 2^{n}
$
for all natural numbers $n$.
I tried this with induction:
Inequality clearly holds when $n=1$.
Supposing that when $n=k$, $k<2^{k}$.
Considering $k+1 <2^{k}+1$, but where do I go from here?
Any other methods maybe?
AI: Proof by inductio... |
H: Geometric description of points in the complex plane
What does the following inequality look like if sketched in the complex plane:
$$\operatorname{Re}(az+b)>0$$
The $a$ and $b$ are both complex numbers (function parameters). I understand that $\operatorname{Re}(z)>0$ will yield all of the complex numbers to the ri... |
H: The Schwarz Reflection Principle for a circle
I'm working on the following exercise (not homework) from Ahlfors' text:
" If $f(z)$ is analytic in $|z| \leq 1$ and satisfies $|f| = 1$ on $|z| = 1$, show
that $f(z)$ is rational."
I already know about the reflection principle for the case of a half plane, so I tried u... |
H: Gaps between primes
I recently watched a video about the recent breakthrough involving the gaps between primes. I have an idea that I'm sure is wrong, but I don't know why.
If you take the product of all prime numbers up to a certain number and call it x, won't x-1 and x+1 always be primes?
And since they always... |
H: Does $2^{\mathfrak m}=2^{\mathfrak n}$ imply $\mathfrak m=\mathfrak n$?
Suppose $\mathfrak m$ and $\mathfrak n$ are infinite cardinals. Does $2^{\mathfrak m}=2^{\mathfrak n}$ imply $\mathfrak m=\mathfrak n$?
AI: This is independent of ZFC. It is implied by GCH for example, but there exist models where (say) $2^{\al... |
H: Get the position by only knowing a landmark and the relative position to it
I have a little problem finding a solution for this problem:
My position is unknown. I can see a landmark (angle, range) and so I know the position $LM_{local}$ in my local coordinate frame.
I recognized this landmark, and so I also know ... |
H: Finding the angles of the start and endpoint of an arc
I have a line $100$ $mm$ long and I want to draw an arc from endpoint to endpoint with a height of 3mm.
I use this formula to find the radius of the arc $$\frac{W . W}{ 8 H} + \frac{H} {2}= 418.1667.$$
But the CAD software I'm using, FreeCAD, uses radius and... |
H: Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?
$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis.
My intuition behind this is that $\frac{d}{dx}e^{ix}=i\cdot e^{ix... |
H: Find the upper and lower limits of the sequence
Finding the results (4.1) (4.2) (4.3) was really easy.
But it's hard to understand what lines in the red box function or work in this solution.
What I understood about lines in the red box is that
if there are infinitely many odds and evens, then it is not Cauchy, and... |
H: The preimage of $\triangle$ is a compact zero-dimensional manifold.
$f: X \to Y$, $g: Z \to Y$ and $Z$ are appropriate for intersection theory ($X,Y,Z$ are boundaryless oriented manifolds, $X,Z$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim Y$), $f$ is transversal to $Z$.
(1) If $\trian... |
H: Probability of drawing a pair from a poker hand, unordered with replacement?
I am wondering what is the probability with which you can draw a pair in a 5-card hand from a standard 52-card deck, if order does not matter in the context of cards in the hand, and if the cards can be replaced with every draw?
AI: To fin... |
H: Combinatorics - placing numbered balls (distinguishable) into distinguishable bins
Say we have 10 balls, numbered $1$ through $10$, and $30$ distinguishable bins. How many ways are there to distribute the balls among the bins?
I think the answer is just $30^{10}$ for this. Is that correct? If not, what am I missing... |
H: Problem about intermediate fields in the extension
Let $E,K$ be intermediate fields in the extension $L/F$
(a) If $[EK:F]$ is finite, then
$$[EK:F] \leq [E:F][K:F] $$
(b) If $E$ and $K$ are algebraic over $F$, then so is $EK$
For (a), I try two ways:
(1) I try to prove that $[EK:K]\leq [E:F]$ and $[EK:E]\leq [K:F]... |
H: Number of non-negative integer solutions for linear equations with constants
How do we find the number of non-negative integer solutions for linear equation of the form:
$$a \cdot x + b \cdot y = c$$
Where $a, b, c$ are constants and $x,y$ are the variables ?
AI: Not a complete answer, but a relatively simple one ... |
H: Examine for absolute and conditional convergence
Examine the following series for absolute/conditional convergence: $$S_n = \sum_{n=1}^{\infty}{n^2\sin(n\pi/2)\sin(\pi/n^3)}$$
Attempt: I see that $$\sin(k\pi/2)= 0 ,\quad k=2n$$ and $$\sin(k\pi/2)=(-1)^k, \quad k=2n-1$$
In other words, the series terms are zero for ... |
H: Why do I keep getting a vertcat error in MATLAB?
For some reason the following code gives me the following error:
Error using CI2 (line 94)
Error using vertcat
Dimensions of matrices being concatenated are not consistent.
The code is:
clear all
% Odds Ratio for no clinically significant change, according to origin... |
H: How many 32-bit strings have fewer 1s than 0s?
Now I know the answer is the summation of $32$ choose $0$ all the way to $32$ choose $15$. My class came up with this formula. I thought I understood how we got it but now I don't. If anyone would explain this method of shortening that calculation that would be great.
... |
H: Is this a Combination problem?
In how many ways can the letters of the word ADRIENNE be arranged so that no two vowels are together? Simplify your answer.
My answer is the following:
$\dfrac{8!}{2!2!} - \dfrac{4!}{2!} \cdot \large ^5C_4 \cdot \dfrac{4!}{2!} = 9,360$ . But my instructor's answer is $720$. Why?
AI: ... |
H: How would you interpret this question focusing on problem solving?
The first step of problem solving is to understand what the problem is asking, that is where I am stuck.
One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the le... |
H: The tangent space of $\mathrm{Aut}(T_eG)$
Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence.
“$\operatorname{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its tangent space at the identity is naturally identified with $\operato... |
H: What is the function to the following graph?
What is the function to the following graphs?
I am just looking for a rough estimate. It doesn't need to match the exact graph.
AI: It depends on what you think is happening to the right. Clearly it goes to infinity for $x=0$ so there is a denominator of $x^n$. If they... |
H: Searching a function expressing $\sin x$ versus $\cos x$
I'm searching a specific function, I searched everywhere and didn't find anything. Here is my problem with $X=\cos x$ I'm trying to find $f(X)= \sin x$, so that this function shall use only $ \cos x$ as variable.
Can someone help me?
If you think that ain't a... |
H: Diophantine Equation: $xy+ax+by+c=0$
How to find integer solutions $x,y$ of $xy+ax+by+c=0$ for given $a,b,c \in \mathbb{Z}$?
Is there somewhere a treatise on this kind of equations?
AI: Remark: We will assume that the equation is $xy+ax+by+c=0$. If it really is $xy+(a+b)x+c=0$, then it can be done in the same way a... |
H: Isomorphism of polynomial rings in several variables
I have been struggling with the following problem:
How can one prove that if there is an isomorphism between several variable polynomial rings over a field $K$, $ \varphi : K[X_1, \dots, X_n] \to K[X_1, \dots, X_m]$ such that $\varphi$ restricted to $K$ is $id$... |
H: Weak flat condition?
Let $R$ be a unit ring (not necessarily commutative). Then it is clear that for a right $R$-module $M$ we have:
$M$ is flat $R$-module $\Rightarrow$ for any left $R$-module $E$ with $E\otimes_{R}M=0$ and any submodule $E'\leq E$, we have $E'\otimes_{R}M=0$.
Can anyone give me a counterexample ... |
H: Why is it necessary $n\geq 2$ in this problem about boundary and isolated point?
Let $A\subset\mathbb{R}^n$ be an open set, where $n\geq2$. Prove that given $a\in\mathbb{R}^n-A$, are equivalent:
(i) the set $A\cup\{a\}$ is open;
(ii) $a$ is an isolated point of the boundary of $A$;
(iii) There exists $r>0$ such tha... |
H: Combinations: A Generalization on a Classic Problem
I'm asked the following question:
How many integers from $1$ to $10,000$, inclusive, are multiples of $5$ or $7$ or both?
I've got the answer using this:
unsigned a_5 = 0;
unsigned a_7 = 0;
unsigned a_both = 0;
unsigned a_35 = 0;
for( unsigned i = 1 ; i <= 10... |
H: Showing that if an equation has a unique solution for one variable, then it has unique solutions for all.
I have a problem and a proposed solution. Please tell me if I'm correct.
Problem: Let $A$ be a square matrix. Show that if the system $AX=B$ has a unique solution for some particular column vector B, then it h... |
H: Planar Graph with Maximum Number of Edges and 3-Colouring in Eulerian
Show that a planar graph with $n$ vertices and $3n-6$ edges with $\chi=3$ is Eulerian.
$\chi=3$ means there is a optimal vertex colouring with three colours. Eulerian means that the graph admits an Eulerian cycle (a cycle which contains each ed... |
H: $Ax=b\Leftrightarrow b\in\left(\ker A^*\right)^\perp$
Let $A:\mathbb{R}^m\to\mathbb{R}^n$ be a linear map and $A^*:\mathbb{R}^n\to\mathbb{R}^m$ be the adjoint of $A$ (that's $\langle Ax,y\rangle=\langle x,A^*y\rangle$ for all $x\in\mathbb{R}^m,y\in\mathbb{R}^n$). Given $b\in\mathbb{R}^n$, are equivalent:
(i) there ... |
H: The largest number to break a conjecture
There are several conjectures in Mathematics that seem to be true but have not been proved. Of course, as computing power increased, folks have expanded their search for counterexamples ever and ever upwards.
Providing a counterexample to a conjecture with a very large numbe... |
H: Substring Occurrence in a String of Size $5$
Question:
How many strings can be formed by ordering the letters $ABCDE$ so that each string contains the substring $DB$ or the substring $BE$ or both?
Attempt:
There are four possible ways to order both the $(DB)$-substring and the $(BE)$-substring amongst the other... |
H: Martians and Jovians
In how many ways can five distinct Martians and eight distinct Jovians wait in line if no two Martians stand together?
AI: Make a lineup of $8$ letters $J$ like this:
$$J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad J \qquad$$
There are $7$ gaps between $J$'s that we could slip ... |
H: Eigenvalues of a second derivative
I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the eigenvalues of the second derivative operator should be negative definite. I was curious... |
H: Induction Proof for a series expansion of a function
I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final step to work.
In most induction proofs we start with a base cas... |
H: $\frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}$, right?
I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the article. Yo... |
H: The evaluation of the infinite product $\prod_{k=2}^{\infty} \frac{k^{2}-1}{k^{2}+1}$
How does one show that$$ \prod_{k=2}^{\infty}\frac{k^{2}-1}{k^{2}+1} =\frac{\pi}{\sinh \pi} ?$$
My attempt: $$ \begin{align} \prod_{k=2}^{\infty}\frac{k^{2}-1}{k^{2}+1} &= \lim_{n \to \infty} \prod_{k=2}^{n}\frac{(k-1)(k+1)}{(k-... |
H: Showing that two topologies on the unit circle are the same
Consider the unit circle, described two ways. The first is as a quotient space, as in What does it mean to "identify" points of a topological space?. (I'm using the first definition of its topology from http://en.wikipedia.org/wiki/Quotient_space) Let's ca... |
H: $X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?
Requesting a hint or solution.
X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.
AI: $\newcommand{\cl}{\operatorname{cl}}$You need to show that if $G_n$ is a dense open subset of... |
H: Exact form of pdf of maximum of normal random variables
$$
z = max(x+b,y)
$$
where
x ~ N(m1,s1) and y~N(m2,s2), b is a contant
What's the pdf of z?
Or exact form of E(z)? (E is expectation operator)
To the best of my guessing from the literature it is related with Weibull (https://en.wikipedia.org/wiki/Weibull_di... |
H: Men and Women: Committee Selection
There is a club consisting of six distinct men and seven distinct women. How many ways can we select a committee of three men and four women?
AI: There are "$6$-choose-$3$" $=\binom{6}{3}$ ways to select the men, and "$7$-choose-$4$" =$\binom{7}{4}$ ways to select the women. That'... |
H: Is this infinite series a Fourier series?
I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this.
$$e^{3i\pi/2}+2e^{3i\pi/2}+3e^{3i\pi/2}+4e^{3i\pi/2}+5e^{3i\pi/2}+\cdots$$
Any ideas? If changes to this series are necessary to make it a Fourie... |
H: What does it mean to "compute" a generic formula without values?
Very elementary math question here in regards to my discrete math class. I've got a problem here that says...
"Compute the following":
$$\sum_{j=1}^n \frac{1}{j(j+1)}$$
What on earth does it mean to "compute" this? I mean, without any values, does t... |
H: Why in a space in which compact sets have empty interior, the closure of open sets is not totally bounded?
I'm trying to understand the proof of the theorem:
The Baire space $\mathcal{N}$ is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty interior.
It i... |
H: Generalization of Jensen's inequality for integrals?
Jensen's inequality for sums says that for $f$ convex, $$f\left(\sum_1^n \alpha_i x_i\right)\leq \sum_1^n \alpha_i f(x_i), \,\,\,\,\text{for } \sum_1^n \alpha_i = 1.$$
I have read that a generalization of it is that $$f\left(\frac{\sum_1^n \alpha_i x_i}{\sum_1^n... |
H: The union of a sequence of countable sets is countable.
While working on the theorem below, I constructed the following proof:
Theorem. If $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ is a sequence of countable sets, then
$$
\bigcup_{n\in\mathbb N}E_{n}
$$
is countable.
Proof. Let $S=\left\langle E_{n}\righ... |
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