text stringlengths 83 79.5k |
|---|
H: $H$ and $K$ are supposed to be closed in $X$, Why?
Let $X$ be a topological space and $A$ a closed subspace of $X$. Let $H$ and $K$ be closed in $A$. Now $H$ and $K$ are supposed to be closed in $X$, Why?
AI: The complement of $H$ in $A$ is open in $A$, so is the intersection of some open set $U$ in $X$ with $A$. T... |
H: proof reading and a little question and refine english about college-algebra
Welcome to edit my post to revise any mistakes, both English and Math, thanks.
Theorem 12
polyIC=integer polynomial
polyRC=rational polynomial
Suppose $f(x)=a_nx^n+a_{n-1}x^{n-1}+\text{...}+a_0$ is a polyIC, and $\frac{r}{s}$ is a rationa... |
H: Finding maximum volume
A box has corner (0,0,0) and all edges parallel to the axes. If the opposite corner (x,y,z) is on the plane $$ 3x+2y+z=1 $$, what position gives maximum volume? Show first that the problem maximizes
$$
xy-3x^2y-2xy^2
$$
Can somebody please explain how this should be solved?
AI: From the desc... |
H: If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
I want to prove or disprove this problem:
If there exist $\lim\limits_{x\rightarrow \infty} (f'(x)+f(x))=L<\infty$ then $\lim\limits_{x\rightarrow\infty} f(x) =L$.
When I assume problem below:
If there... |
H: An interesting problem using Pigeonhole principle
I saw this problem: Let $A \subset \{1,2,3,\cdots,2n\}$. Also, $|A|=n+1$.
Show that There exist $a,b \in A$ with $a \neq b$ and $a$ and $b$ is coprime.
I proved this one very easily by using pigeon hole principle on partition on $\{1,2\},\{3,4\},\dots,\{2n-1,2n\}$.
... |
H: Problem with a definition of $\succsim$-maximal element and $\succsim$-maximum
I found in a book the following definition of $\succsim$-maximal element and $\succsim$-maximum, but I really don't see the difference between what is written in the book (I know the difference between those two concepts). Am I wrong?
Le... |
H: What is the difference between multiplicative group of integers modulo n and a Galois Field
What is the difference between multiplicative group of integers modulo n and a Galois Field?
Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$?
Or is it the same as $\mathbb{Z}/n\mathbb{Z}$?
Sorry for the short and simpl... |
H: Stupid question about $1 - \frac{1}{2}-\frac{1}{4}+\frac{1}{3}- \frac{1}{6}-\frac{1}{8}+\frac{1}{5}\dots$
I have
$$1 - \frac{1}{2}-\frac{1}{4}+\frac{1}{3}- \frac{1}{6}-\frac{1}{8}+\frac{1}{5}\dots$$
Partial sum $S_{3n}$ of the above is:
$$(1 - \frac{1}{2}-\frac{1}{4})+(\frac{1}{3}- \frac{1}{6}-\frac{1}{8})+(\frac{1... |
H: A basic probability problem on conditional expectation
A coin having probability $p$ of coming up heads is successively flipped until two of the most recent flips are heads. Let $N$ denote the number of flips. How to find $E[N]$.
AI: Assume that $p\ne 0$. At any time during the tossing, we are in state $A$ if we ha... |
H: Proving a biconditional statement with an or
I want to prove a theorem in geometry of the form $p \iff q \vee r$. My plan is to prove:
$q \implies p$ as well as $r \implies p$
$p \text{ and } \lnot q \implies r$
Can I get someone to verify that I haven't sidetracked in my logic?
AI: Your planned approach is sound... |
H: Probability that two points are more distant than a third equidistant point
Say you have three points $x,y,z \in \mathbf{R}^n$ with standard Euclidean distance $d$ and $d(x,y) = d(y,z)$. Then what's the probability that $d(x,z) > d(x,y)$ for random $x,z$? For convenience, $y=\mathbf{0}$ and $d(x,y) = d(y,z) = 1$.
F... |
H: Finding the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$
I want to find the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$. In other words, I am looking for the largest $n$ such that $\frac{n^3-7}{n-7}$ is an integer. Can anyone provide me with a hint? Please do not post a full solution.
A... |
H: Motion with acceleration and deceleration
Stumbled on this while working on a program code to display motion animation. The scenario is, the program will have a function to move a picture from its current position to move right. The program took only 2 variables, which is the travel distance and time to cover the d... |
H: Product topology with finer/coarser comparison
There is a question in Munkres' Topology which has me a little confused:
Let X have topologies $\mathfrak{T}$, $\mathfrak{T'}$, and Y have a topologies $\mathfrak{U}$,$\mathfrak{U'}$. Show that if $\mathfrak{T} \subset \mathfrak{T'}$ and $\mathfrak{U} \subset \mathfrak... |
H: Train and horseman velocity
This is a more or less easy exercise but there is one point I do not understand.
We have a train starting at $(0|0) $with velocity $v_t=20 m/s$ straight in the $y$-direction and a horseman at $H=(100|100)$ with velocity $v_h=15 m/s$
There are two questions now:
1) In which direction does... |
H: Find all functions $f$ so that $d(f(x))=x$ for every natural $x$.
Help me find all functions $f(x)$, $f:\mathbb N \to \mathbb N$, so that $d(f(x))=x$ for every natural number $x$ where $d(x)$ is number of divisors of $x$.
My works until now: Clearly $f(1)=1$. We also have $d(n)\leq2\sqrt{n}$ so $d(f(x))=x\leq2\sqrt... |
H: Conjecture on combinate of positive integers in terms of primes
Along a heuristic calculation, I am struggeling with a possible proof for my following conjecture:
Every positive integer $n\in \Bbb N$ can be written as a unique combination of $a,b \in \Bbb N$, $m\in \Bbb N_0$ and $p \in \Bbb P$ (a prime), such that:... |
H: Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
I'm quite sure I'm missing something obvious, but I can't seem to work out the following problem (web search indicates that it has a solution, but I didn't manage to locate one -- hence the formulation):
Prove that there exists a surjection $... |
H: What is the number of possible seating arrangements?
I have an exercise to do: How many possibilities for placing 12 apostles if it is important who is sitting next to who but it is not important from which side?
Answer possibilities:
a)
b)
c)
d)
It is possible that more than one option is correct and also ... |
H: Inner product on a von Neumann algebra
Let $M$ be a $\sigma$-finite von Neumann algebra (one which admits a faithful normal state) acting on a Hilbert space $H$. Denote its faithful normal state by $\omega$.
We can define an inner product on $M$ by
$$\left< x, y \right> := \omega (y^*x).$$
Let $(\pi, K, \xi)$ be a... |
H: Isn't every subset of a compact space compact?
Let $\mathfrak{C}$ be an open cover of a topological compact space $X$, and let $\mathfrak{B}\subseteq \mathfrak{C}$ be its finite subcover. Then every subset of $X$ also has the same cover and subcover! Shouldn't every such subset also be compact then?
Motivation: I r... |
H: Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols
Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols, such that $\sigma$ does not fix any symbols?
$1. \;18$
$2.\; 30$
$3.\;15$
$4.\; 28$
could any one just give me hints?
AI: Think about the ord... |
H: $f=g\; ; \;\bar{\mu}$-a.e. vs $\mu$-a.e.
Let $(X,\mathcal{M},\mu )$ be a measure space and let $(X,\overline{\mathcal{M}},\overline{\mu})$ be its completion. Let $f,g$ be in the set of the union of $\mathcal{M}$ and $\overline{\mathcal{M}}$-measurable functions.
It seems to me that $f=g$ a.e. is the same for both $... |
H: Find function $f$ having it's gradient
I need to find function $f$ having it's gradient.
$$
\operatorname{grad} f = (x \cos y − 2)j + i \sin y
$$
I think $i$ and $j$ are some sort of vectors, but I'm not sure..
how to find function f?
I know, that a gradient is a vector of the fastest growth, but I don't see any ve... |
H: Probability question
Suppose , there are 3 red balls and 2 black balls in a box .
I have to find probability of choosing 2 black balls .
Then what is wrong with this solution .
for 1st ball being black 2/5 , and for second ball being black it is 1/4 , after 1st ball is black .
Then total probability is 1/10 .
If I ... |
H: Seeking formal explanation of definite integral over infinitesimal interval
Why does :
$$\frac{\displaystyle\int_t^{t+h}f(s) \, ds}{h} = f(t)\text{ as }h \rightarrow 0\text{ ?}$$
Intuitively this makes sense, because the value of the integral is infinitesimally close to $f(t)$, you have $h$ of them, and you divide ... |
H: Default way of defining measure on subset of measure space
Let $(X,\mathcal{M},\mu)$ be a measure space and let $E\in \mathcal{M}$. I'm interested in knowing what the default way to define a measure space $(E,\mathcal{M}_E,\mu_E)$ is?
My guess would be the $\sigma$-algebra $\mathcal{M}_E = \{F \cap E : F\in \mathca... |
H: Using integration by parts to evaluate an integrals
Can't understand how to solve this math: use integration by parts to evaluate this integrals:
$$\int x\sin(2x + 1) \,dx$$
can any one solve this so i can understand how to do this! Thanks :)
AI: $$\int x \sin (2x+1)\;dx$$
parts by integration:$\int u(x)\cdot v(x) ... |
H: Does a linear quotient map have sections
Suppose $V$ is a vector space with vector subspace $N$. Then there is
a natural projection
$$
\pi_N: V \to V/N
$$
from the vector space $V$ to the quotient space $VN$ of $V$ modulo $N$.
Does $\pi_N$ have sections?
Means are there 'section maps' $i: V/N \to V$ with $\pi_N\... |
H: How to write this gradient as a vector?
How to write this gradient as a vector using brackets []?
Is this:
$$
\operatorname{grad}f=(x\cos y−2)j+i\sin y
$$
equal to this:
$$
\operatorname{grad}f=[\sin y,x\cos y−2]\text{ ?}
$$
Thanks
Regards.
AI: you have to understand of this manner :
i : indicates the first coordin... |
H: Question: What does "nonzero" polynomial mean?
What does "nonzero" polynomial mean?
Thank you
AI: Since there is no answer, here is what I stated in the comments:
Usually, a nonzero polynomial $f$ is a polynomial of where not every coefficient is zero, i.e.
$$f(X)=\sum\limits_{k=0}^n a_kX^k\quad(n\ge0)$$
and one of... |
H: Matrix Inverses
So in class we have been discussing matrix inverses and the quickest way that I know of is to get a matrix A, and put it side by side with the identity matrix, like $[A|I_{n}]$ and apply the Gauss-Jordan algorithm until it is of the form $[I_{n}|A^{-1}$], where $A^{-1}$ will show up assuming A is in... |
H: Cumulative Probability Distribution Example Question
Flip a fair coin three times. let T be the random variable that denotes the number of tails that occur given that at least one head occurred. Calculate the Probability and Cumulative Distribution Functions.
I was having a little trouble with this question so ... |
H: Permuting and finding combinations of bit strings
I am working on typical computer science math and one theme in many text books is to find different permutations/combinations of binary strings:
1011011101 is an example op a binary number of length 10. How many
binary numbers of length 10 end up with 111 and con... |
H: What's the formula for $\sum_{n=0}^{\infty}\left ( an+b \right )x^{n}$=?
Use the two formulas
$\sum_{n=0}^{\infty}=\frac{1}{1-x}$ and $\sum_{n=0}^{\infty}\left ( n+1 \right )x^{n}=\frac{1}{(1-x)^{2}}$ to find a formula for this $\sum_{n=0}^{\infty}\left ( an+b \right )x^{n}$ for all pair of constants $a$ and $b$ (b... |
H: A question about chain rule for partial derivatives
I know this isn't supposed to be difficult, but I'm not sure how to use the chain rule.
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable and define $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $u(x,y):=e^{2x}f(ye^{-x})$. I want to show that $u$ satisfies... |
H: Is $\lim_{r\to 0}\frac{|A\cap B(x,r)|}{|B(x,r)|}\neq 1$?
Let $A\subset\mathbb{R}^N$ be a Lebesgue measurable set and denote by $|\cdot|$ the Lebesgue measure. Fix some $x\in A$ and $a\in\mathbb{R}^N$. Consider the set $$S_a=\{v\in S^{N-1}:\ \langle a,v\rangle\geq\frac{1}{2}\|a\|\}$$
Suppose that for each $v\in S_a$... |
H: Trying to prove a theorem about convergence of a sequence.
The following is the problem that I am working on.
Prove that if $\{s_n\}$ converges, then $\{|s_n|\}$ converges.
The following is the idea of the proof I'm trying to make, so it's a little loose. However, I'm not 100% sure if I'm doing it right.
proof:
I... |
H: Multiplicity of the simple $R$-module $M$ in the semisimple ring $R$
I'm confused about the conclusion of Wedderburn's structure theorem for semisimple rings.
Let's consider the special case where $R=M^n$ as modules for some simple module $M$.
Wedderburn's theorem says that $R=M_n(D)$ for $D=\text{End}_R(M)^{o}$.
T... |
H: Probability of k out of m microbes splitting.
A microbe either splits into two perfect copies of itself or else disintegrates. If the probability of splitting is p, what is the probability that there will be m microbes in the nth generation. Furthermore, what is the probability that out of these m microbes, k will ... |
H: Finding minimal polynomial
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, such that $A$ is not of the form
$A=c I_n, c \in \mathbb{R}$ and $(A-2I_n)^3 (A-3I_n)^4=0$. Find the minimal polynomial of $m_A(x)$of $A$.
I know that $m_A(x) | (x-2)^3(x-3)^4$, but I am stuck here. Any help appreciated.
AI: Here ... |
H: Bijective conformal map from half disc to upper half plane
I'm trying to find a bijective conformal map from the half disc $\{z: |z| < 1, \Re(z)>0\}$ to the upper half plane $\{z: \Re(z) > 0\}$. Any help is appreciated. Thanks!
AI: Use a Möbius transform to bring one vertex of the half disk to $\infty$ and the ot... |
H: How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$?
How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$?
I found the inequality while reading a TCS paper, where this inequality was taken as a fact while proving ... |
H: Equivalence between these definitions of ordinal numbers
Von Neumann defines an ordinal $\alpha$ as a transitive set whose elements are well-ordered with respect to the membership relation $\in$.
Meanwhile, in Naive set Theory, Halmos defines an ordinal $\alpha$ as a well-ordered set such that $\forall \zeta \in \... |
H: A Complete k-partite Graph
A complete $k$-partite graph is a graph with disjoint sets of nodes where there is no edges between the nodes in same set, and there is an edge between any node and other nodes in the other subsets.
The subsets are $X_1,\dots,X_p$, and the number of nodes is $N$. What is the maximum numbe... |
H: Derivate of finite sum?
I'm having trouble finding the derivative of the following finite sum:
$$\frac{\mathrm{d}}{\mathrm{d}n} \sum\limits_{k=1}^{n-1} \ln\frac{n-k}{n}$$
I get the following:
$$\frac{1}{n-1}+\frac{1}{n-2}+\ldots+\frac{-1}{n}+\frac{-1}{n}$$
I know the closed form is:
$$\frac{\mathrm{d}}{\mathrm{d}n}... |
H: Combinatorics: Is it true to say $s \in F$ where $s \in S$ and $F$ is a set of subsets of $S$
From this problem: (from Measure Theory by Paul R. Halmos, Springer-Verlag, 1950)
Let $S$ be a set. Suppose that $s$ is an element of $S$, $T$ is a subset of $S$, and $F$ is a set of subsets of $S$. How many statements o... |
H: What will happen if I try to print an impossible solid into a 3D printer?
What would be the result of a 3D modeled impossible solid, like the Penrose Triangle, printed out of a 3D printer?
AI: Gershon Elber of the Computer Science Department at Technion in Isreal works on creating similar objects by way of introduc... |
H: Binomial Expansion Word Problem (Creating a Equation)
I was working on my math textbook (Nelson Functions 11) and came across the following word problem. This question is shown in the "Pascal's Triangle and Binomial Expansions" section of the book.
"Using the diagram at the left (shown below), determine the number... |
H: How do you solve this equation: $10 = 2^x + x$?
Is it possible to solve this equation?
\begin{align}
a &= b^x + x \\
a-x &= b^x \\
\log_b(a-x) &= x
\end{align}
If $a$ and $b$ are known, how do you find $x$?
AI: As noted in the comments, since the function is monotonically increasing, there is only one real solutio... |
H: Relationship Between Tangent Function and Derivative
Is there a relationship between the trigonometric function tan(x) and the derivative of y with respect to x? Are they just named similarly by coincidence?
AI: Do you mean $\frac {dy}{dx}$ is reminiscent of the triangle definition of tangent $\tan \theta=\frac{\t... |
H: Proof of correctness of Putzers algorithm
I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into somewhat readable english.
What I don't understand is the line:
$$\sum^{n-1}_{k... |
H: Retract and Homotopy extension property
See picture below the following picture.
According to Hatcher, homotopy extension property implies that for a pair $(X,A)$ where $A$ is a subspace of $X$,
$X\times I$ should retract to $X\times\{0\}\cup A\times I$ .
My question is whether the retract given in the picture is... |
H: How to show $\frac{a_1+\cdots+a_n}{n}\le S_h\sqrt[n]{a_1\cdots a_n}$
For positive numbers $a_i$ with $0<m\le a_i\le M$, how to show
$$\frac{a_1+\cdots+a_n}{n}\le S_h\sqrt[n]{a_1\cdots a_n},$$ where $h=M/m$ and $$S_h=\frac{(h-1)h^{\frac{1}{h-1}}}{e\log h}$$
AI: Let
$$
f(b_1,\dots,b_n)=\frac{b_1^n+\dots+b_n^n}{nb_1... |
H: What is the difference between a proposition and an observation?
As far as I know, a proposition is a statement which might be used to prove a theorem but is also of independent interest.
How would you differentiate it from an observation? Would you say it is a proposition with an easy proof?
AI: I agree with your ... |
H: The limit $\lim\limits_{n\to\infty} (\sqrt{n^2-n}-n)$. Algebraic and intuitive thoughts.
I am working on the following problem.
Find the limit of $$\lim_{n \to \infty} (\sqrt{n^2-n}-n)$$
Intuitively, I want to say it's $0$ because as $n \to \infty$, $\sqrt{n^2-n}$ behaves like $n$ and subtracting $n$ makes it $0$... |
H: Finding an unbiased estimator for the negative binomial distribution
Consider a negative binomial random variable Y
as the number of failures that occur before the r
th success in a sequence of independent and identical success/failure trials. The pmf of $Y$
is $$nb(y;r,\theta)=\begin{cases}
{y+r-1 \choose y}\... |
H: Is polynomial $1+x+x^2+\cdots+x^{p-1}$ irreducible?
Let $p$ a prime number, is the polynomial
$$1+x+x^2+\cdots+x^{p-1}$$
irreducible in $\mathbb{Z}[x]$ ? Thanks in advance.
AI: Mr. Eisenstein certainly thinks so!
The polynomial can be rewritten as $\frac{x^p-1}{x-1}$. Setting $x=(y+1)$ has no effect on (ir)reduci... |
H: Is the square root of $4$ only $+2$?
Why is $4^{1/2}=+2$?
It should also be $-2$ since both squared just give two only. Also why do we always represent root of $x$ on the right side of the number line?
AI: It is by convention: with real numbers, we agree to take the positive square root. This allows us to define $... |
H: Finding the upper and lower limit of the following sequence.
$\{s_n\}$ is defined by $$s_1 = 0; s_{2m}=\frac{s_{2m-1}}{2}; s_{2m+1}= {1\over 2} + s_{2m}$$
The following is what I tried to do.
The sequence is $$\{0,0,\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{3}{8},\frac{7}{8},\frac{7}{16},\cdots \}$$
So the even t... |
H: The security guard problem
There was a security guard in a bank. In front of him were 100 lockers in rows of 10. He thought of something as he saw all the lockers were closed. He started opening all the lockers whose lock numbers were multiples of 1, then closed all lockers whose number was a multiple of 2,and did ... |
H: What am I doing wrong with solving $2\tanh^2x-\text{sech}~x=1$?
$2\tanh^2(x)-\text{sech}(x)=1$
$\tanh^2(x)=1-\text{sech}^2(x)$
$2(1-\text{sech}^2(x))-\text{sech}(x)=1$
$2\text{sech}^2(x)+\text{sech}(x)-1=0$
$\text{sech}(x)=\frac{1}{2} $ Not possible. And $\text{sech}(x)=-1$ Also not possible
What am I doing wron... |
H: Sets and bijection
I am asked to do determine if the following set is countable and if so, perform a bijection with the Natural set of numbers.
The set is: all bit strings not containing the bit $0$
I have determined that this is a countable set. Performing the bijection is the part I am unsure of.
My first though... |
H: Showing reflection in a plane containing origin is represented by symmetric orthogonal matrix
I'm sure this question is pretty straightforward but I've been scratching my head at this for a small while
Assuming that any isometry $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ may be written in the form $T({\bf{v}}) = ... |
H: Is this the best solution (or a correct one) to this recurrence relation.
The $r_k$'s are terms of a sequence of positive integers satisfying $\sup r_k=\infty$. I am looking for a solution $F_k$ to
$$F_{k}\ge 2r_{k+1} F_{k+1}-F_{k+1}\ (k\ge1);\ F_1=1 \ (*)$$
(I want $F_{k}$ to be the smallest possible solution to... |
H: Trying to understand implication
I'm currently slogging through propositional calculus and making my brain do impressions of a pretzel, but I'm slowly getting it though I'd like to see if that's actually true for the problem below. Is my answer correct for the below proposition?:
(p $\rightarrow$ p) $\lor$ (q $\rig... |
H: On the definition of graded Betti numbers
Let's use as reference the slides 19-31.
Let $S=k[x_1,\dots,x_n]$ and $M$ a finitely generated graded $S$-module. Then by Hilbert's Syzygy Theorem, $M$ has a minimal, graded, free resolution of length at most $n$, i.e.,
$$0 \rightarrow F_m \rightarrow F_{m-1} \rightarrow \c... |
H: Does introducing penalties for getting true/false questions incorrect result in higher skill penetration (less luck/variance)?
A student is asked to answer 50 true/false questions and he would get 35 right and 15 incorrect if he had to put his best guesses for each question down.
Now, for each question he has a cer... |
H: How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$?
How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?
This should be easy, but I'm not seeing how. A hint would be appreciated.
AI: For positive $x, y$, we have:
$$\sqrt{x + y} \leq \sqrt{x} + \sqrt{y} \iff \left(\sqrt{x + y}\right)^2 \leq ... |
H: Big-Oh Notation
I'm given to the following relationship:
$$C(x) = C(\lfloor(\frac x2)\rfloor) + x, C(1)=2$$
I do not understand how my teacher says to calculate big O. Any help to start?
AI: HINT: Put $x=2^n$ and solve to find the series. |
H: Give example of a distribution.
Give examples of distribution
(1) such that $X$ and $1-X$ have the same distribution.
(2) such that $X$ and $\dfrac1X$ have the same distribution.
For the first one I think $X$ is $\text{Uniform}(0,1)$. Since $1-X$ is also $\text{Uniform}(0,1)$. I don't know the second distribution... |
H: Real sample data source
I'm about to start working on a final project for a college level statistics course. The problem is that I have to analyse real data and so far I haven't got many options. Do you know about any website or perhaps books where I can find data used in previous statistical surveys or something?
... |
H: If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map
This is from a past qualifying exam.
Here is the question:
If $\alpha= (1+\sqrt{-19})/2$ then any ring homomorphism $f : \mathbb{Z}[\alpha] \rightarrow \mathbb{Z}_3$ is the zero map.
Here is ... |
H: How can I construct a matrix?
Construct a matrix whose one eigenvector is $(1,-1,1,-1)$.
Here only one eigenvector is given and I need to construct a matrix with this. I don't know how to proceed. Please help.
AI: Short answer -- Every nonzero vector is an eigenvector of the identity matrix.
Long answer -- For a ... |
H: What is the name for the region enclosed by an $n$ dimensional object?
In a $1$ dimensional object, the name for the region enclosed by it is the length of the object. In a $2$ dimensional object, the name for the region is the area of the object. In a $3$ dimensional object, the name is the volume of the object. W... |
H: Clarification on 2 'E's for expected value in a conditional probability
The text I am reading defines the Expected Prediction Error as the squared difference between the actual Y value and the predicted Y value (f(X) in the text). Then it conditions on X. The trouble I'm having is understanding the notation of the ... |
H: How to show that there is $x_{0} \in X$ such that $f_{n}(x_{0}) \notin \mathbb Q$ for every $n$. $X$ is a Banach space
Let $X$ be a Banach space. $\{f_{n}\}$ is a sequence of nonzero bounded linear
functionals on $X$. Show that there is $x_{0} \in X$ such that $f_n(x_0) \notin \mathbb Q$ for
every $n$.
This is a pr... |
H: Prove this vector identity using vector identities
Let $f$, $g$ and $h$ be any $C^{2}$ scalar functions. Using the standard identities of vector calculus, prove that;
$$ \nabla \cdot \left( f\nabla g \times \nabla h \right) = \nabla f \cdot \left(\nabla g \times \nabla h \right)$$
Here is my working out so far;
... |
H: how to show that $Y= \{f\in L^{2}[0,1] \mid f(x)\geq x \text{ a.e.}\}$ is weakly closed in $L^{2}$
Problem: Let $Y= \{f\in L^{2}[0,1] \mid f(x)\geq x \text{ a.e.}\}$. Show that
$Y$ is weakly closed in $L^{2}$.
My thought about solving this problem is that consider a sequence $\{f_{n}\}$ which converges to some... |
H: Question about a solution to a problem involving Taylor's theorem and local minimum
I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem:
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that
$f(x)$ has a local minimum at $x = 0$. Prove t... |
H: If $X$ is a random variable, then $X^{\pm}$ are random variables
For $x\in\mathbb{R}$, let $x^+:=\max\{0,x\}$ be its positive part, $x^-:=-\min\{0,x\}$ be its negative part. Prove that if $X$ is a random variable, then $X^{\pm}$ are random variables too.
I have attempted to prove this by using definition of a ran... |
H: Integral of $\int \frac{1+\sin(2x)}{\operatorname{tg}(2x)}dx$
I'm trying to find the $F(x)$ of this function but I don't find how to do it, I need some hints about the solution.
I know that $\sin(2x) = 2\sin(x)\cos(x)$ its help me? It's good way to set $2x$ as $t$?
$$\int \frac{1+\sin(2x)}{\operatorname{tg}(2x)}dx... |
H: Finding the number of symmetric, positive definite $10 \times 10$ matrices having...
I was looking at old exam papers and I was stuck with the following problem:
What is the number of symmetric, positive definite $10 \times 10$ matrices having trace equal to $10$ and determinant equal to $1$ ? The options are:
$0... |
H: Fourier Transform from Discrete Fourier Transform
If I have the basic Discrete Fourier Transform from a discrete function $x[n]$, like this:
$$\displaystyle X[k] = \sum\limits_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn} $$
How can I get to the expression for the Fourier Transform:
$$\displaystyle X(j\Omega) = \sum\limi... |
H: Solve the following linear first-order equation
Question:
Solve the following linear first-order equation.
$(1+e^x)y '+e^xy=0$
I resolved:
$a_0(x)\acute{y}+a_1(x)y=g(x) => \acute{y}+p(x)y=Q(x)$
$\acute{y}+\frac{{e}^{x}}{1+{e}^{x}}y=0 , Q(x)=0, P(x)=\frac{{e}^{x}}{1+{e}^{x}}$
Integral factor in building:
$\mu (x)=... |
H: A challenging logarithmic integral $\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx$
While playing around with Mathematica, I found that
$$\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx = \frac{1}{3}\log^3(2)-\frac{\pi^2}{12}\log(2)+\frac{\zeta(3)}{8}$$
Please help me prove this result.
AI: Use your favorite program to comput... |
H: On Polar Sets with respect to Continuous Seminorms
In the following, $X$ is a Hausdorff locally convex topological vector space and $X'$ is the topological dual of $X$. If $p$ is a continuous seminorm on $X$ then we shall designate by $U_p$ the "$p$-unit ball", i.e,
$$U_p=\{x\in X: p(x)\le 1\}.$$
The polar set of $... |
H: A basic probability doubt on derangment
Is there any implication that the probability that a random permutation is a derangment is $\frac{1}{e}$ when $n->\infty$ ?
AI: I assume that by derangement, you mean a permutation not fixing any element of the underlying set.
My probability professor in college offered the f... |
H: A domain with only a (non-zero) prime ideal
What is an example of a domain $A$ such that Spec$A=\{(0),\mathfrak p\}$? For instance one could find a principal ideal domain that is also a local ring but I can't imagine such a ring.
AI: Let $p$ be a prime number and let $\mathbb{Z}_{(p)} = \{ x \in \mathbb{Q} : p \tex... |
H: Set partition of strings by suffix
I have this question:
Is this collection of subsets a partition on the set of bit strings of length 8:
The set of bit strings that end with 111, the set of bit strings that
end with 011, and the set of bit strings that end with 00.
My answer would be no, this is not a partition,... |
H: Prove: If $g(A)$ is not scalar ($g(A) \neq \lambda I$) $\rightarrow$ $g(A)$ has no real eigenvalues for a given matrix and minimum polynomial
Given $A \in M_{n x n} (\mathbb R)$ such that $m_A(x) = x^2 + 1$ (the minimum polynomial), and let $g \in \mathbb R[x]$. Prove: If $g(A)$ is not scalar ($g(A) \neq \lambda I... |
H: Can't figure out this transformation matrix
So basically I want to write a transformation matrix to take me out of one coordinate system and into another.
The transformation has to be as follows:
1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate sy... |
H: Validity of residue outside the domain
Using the identity theorem I can see that $f(z)=\dfrac{2}{3+z}$ and hence 1 is true and 4 is false. This far is easy. But for 2 and 3 I can see that $f$ is not defined at $z=3$ and $-3$ is not a isolated singular point of $f$ considering the domain $D.$ Is this a valid logic ... |
H: How to express $a_n$?
Let $\{a_n\}$ be a sequence such that $a_{n+2}=a_{n+1}+a_{n}$ with $a_1=1$ and $a_2=1$, i.e., it is $\{1,1,2,3,5,8,13, \dots\}$. I don't know what is $a_n$, i.e., how to express $a_n$?
Thanks for your help:)
AI: The charecteristic equation is
$$r^2=r+1$$
and it's roots are
$$r_1=\frac{1+\sqr... |
H: shadow cast by a circle
A point source emits light at a circular disc (thickness negligible), and a shadow is left on a wall (XY plane) behind and parallel to the disc. The Z component of distance between the point source and the disc is 'a' units. The Z component of distance between disc and wall is 'b' units. The... |
H: Calculating determinant of matrix
I have to calculate the determinant of the following matrix:
\begin{pmatrix}
a&b&c&d\\b&-a&d&-c\\c&-d&-a&b\\d&c&-b&-a
\end{pmatrix}
Using following hint:
Calculate determinant of matrix $AA^{T}$ and use the theorem that
$\det(AB) = \det A \cdot \det B$
I simply don't see how can I ... |
H: Prove that if $\gcd{f, P_A} = 1$ for some matrix $A$ and polynom $f$ then $f(A)$ is invertible
Let $f \in F[x]$ and $A \in M_{n x n} (\mathbb F)$. Prove: If $\gcd\{f,P_A\} = 1 \rightarrow f(A)$ is an invertible matrix.
This is what I did so far:
If the $\gcd\{f,P_A\} = 1$ then $f$ and $P_A$ have no common diviso... |
H: A question on an unbounded function
Does there exist a function $f$ such that it has a finite value for each point $x$ of $[0,1]$, however for any nbhd of $x$ $f$ is unbounded?
Thanks for your help.
AI: I think, such function can be given by
$$
f(x) = \begin{cases}
0,\text{ if }x\notin\Bbb Q
\\
n,\text{ if }x = ... |
H: Arithmetic with the natural log
We have:
$$ \ln(p^3 + 4) - \ln(4) = 2$$
What I did is:
$$ \ln (p^3 + 4) = \ln(4) + \ln(e^2)$$
$$p^3 + 4 = 4 + e^2$$
$$ p = e^{2/3}$$
Why is this incorrect?
AI: Correct your mistake using this equality
$$\log (a)+\log(b)=\log(ab)$$ |
H: If $a \mid c, b \mid c, \gcd (a,b)=1$ then $ab \mid c.$
If $a \mid c, b \mid c, \gcd (a,b)=1$ then $ab \mid c.$
I understand that given problem is true. however im struggling with writing to prove.
I let A=2 , B= 3 , C= 6
2 l 6= 3
3 I 6=2
3*2 l 6=1
I have shown my work to prove that the theorem is true however I c... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.