text stringlengths 83 79.5k |
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H: Assume $a_n \rightarrow a$ for $n \rightarrow \infty$. Use the sequence $x_n = a_n - a$ to show that $b_n \rightarrow a$ for $n \rightarrow \infty$.
Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ with $b_n = (a_1 + ... + a_n) / n$
I've proved that $b_n \rightarrow 0$ for $n \rightarrow \infty$ when $\lim... |
H: Open and Closed Sets Examples
Ok so well Im struggling to find examples for the first two parts and for the last, well I don't think it is open but can't again find an example. Thanks.
AI: Since the second question is not answered in the link I gave, I will put its answer here. If $X$ is closed and bounded then it... |
H: Continuity in Metric Spaces
$8.\,\,\,$Let $d$ and $d'$ denote the usual and discrete metrics respectively on $\Bbb R$. Show that all functions $f$ from $\Bbb R$ with metric $d'$ to $\Bbb R$ with metric $d$ are continuous. What are the continuous function from $\Bbb R$ with metric $d$ to $\Bbb R$ with metric $d'$?
... |
H: Question about proving something about subgroup
$G$ is a group, and $\left\{H_i|i\in I\right\}$ is family os subgroups of $G$ (I is not a matrix or something similar...)
I need to prove that $$\bigcap_{i\in I}H_i$$ is a subgroup of $G$.
Any ides?
Thank you!
AI: This is a very standard problem.
http://groupprops.sub... |
H: Average score of a batsman's innings
A batsman's runs just before the last match of the season, adds up to $750.$ In his last $2$ innings, he scores only $6$ runs, and his average drops by $2.$ Find his final average of the season.
$\sum x_{n-1}=750, x_{n-1}+x_{n}=6, \frac{\sum x_n}{n}=\frac{\sum x_{n-2}}{n-2}-2... |
H: if $M$ is a monoid then $a \in M$ is invertible iff $\exists b \in M: \text{ } aba=a \text{ and } ab^2a=e$
Apparently this should be an easy question, but I couldn't solve it on my own. I used the search option on MSE and I don't think a similar question has been asked before.
Suppose that $(M,\star)$ is a monoid a... |
H: Does the $\gcd(2n-1,2n+1)=1?$
I am posting this to ask if my proof is correct as I haven't taken number theory in a year and I feel a bit rusty. If it isn't correct, please tell me where I went wrong so I can fix it.
I want to prove that the $\gcd(2n-1,2n+1)=1$ for all $n$.
Using the Euclidean Algorithm, we have ... |
H: What happens if one multiplies two elements belonging to two different groups?
What happens if one multiplies two elements belonging to two different groups? Where does the result lie? Let's say that $a \in \mathbb{Z/pZ}$ and $b \in \mathbb{Z/p^2Z}$, then where does $a \cdot b$ belong?
AI: Motto: One cannot add app... |
H: If $\sum_{n=1}^\infty f_n(x)$ and $\sum_{n=1}^\infty g_n(x)$ converges uniformly, then $\sum_{n=1}^\infty [ f_n(x)+g_n(x)]$ converges uniformly
Prove that if $\sum_{n=1}^\infty f_n(x)$ and $\sum_{n=1}^\infty g_n(x)$ converges uniformly on $x\in X$,
then $\sum_{n=1}^\infty [ f_n(x)+g_n(x)]$ converges uniformly on $x... |
H: Matrix Eigenvector in Opposite Direction to WolframAlpha?
I've been asked to find the Eigenvalues and Eigenvectors for the following matrix:
$$
A=
\begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 1 \\
\end{bmatrix}
$$
Which I have calculated the Eigenvalues to be:
$$
\la... |
H: $\det(B\cdot A\cdot B^T)\neq0$ if and only if $\ker(B^T)=\{\bar{0}\}$
If we have:
$A$, $n\times n$ matrix non singular.
$B$, $m\times n$ matrix.
How would we prove that $\det(B\cdot A\cdot B^T)\neq0$ if and only if $\ker(B^T)=\{\bar{0}\}$.
AI: This is false. Take
$$
B=\pmatrix{1&0}\quad B^T=\pmatrix{1\\0}\quad A=\p... |
H: Prove an isomorphism via abstract nonsense
Suppose we are working in an abelian category and we have a commutative diagram with exact rows
$$
\newcommand{\ra}[1]{\kern-1em\xrightarrow{#1}\kern-1em}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
%
\begin{array}{llllllllll... |
H: Let $\gamma $ be an ordinal. Prove there is an ordering preserving $f:\gamma \to \mathbb{R}$ iff $\gamma < \omega_1.$
Let $<$ be the usual ordering on $\mathbb{R}.$ If $\gamma$ is an ordinal, then $f:\gamma \to \mathbb{R}$ is ordering preserving if $\forall \alpha \in\gamma \forall \beta \in \gamma [\alpha \in \bet... |
H: Show that every monotonic increasing and bounded sequence is Cauchy.
The title is kind of misleading because the task actually to show
Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy
without knowing that:
Every bounded non-empty set of real numbers has a least upper
bound. (Supre... |
H: Equivalence classes
I'm posting this question and answers to see if I am on the right track here, just want to be sure I understand or don't understand.
Bellow I will list some equivalence relations over the set $ S= \{1,2,3,4\} $ the assignment is to find the equivalence classes to $ [1] $
$\{<1,1>,<2,2>,<3,3>,<4,... |
H: Compute Galois group
Compute the Galois group $Gal\left(\mathbb{Q}\left(\sqrt{2}\right)/\mathbb{Q}\right)$ and Galois group of a normal closure of $\mathbb{Q}\left(\sqrt[5]{7}\right)/\mathbb{Q}$
AI: As you already noticed, $\mathbb{Q}(\sqrt{2}) | \mathbb{Q}$ is a quadratic extension and hence normal. The Galois gro... |
H: probability to get the black ball when taking 4 elements from a set of balls
I have a question about probability that seems to be more difficult than I thought:
We assume we have a set of $x$ balls: all of them are white except one that is black ($x-1$ white ball and $1$ white ball)
We take $4$ balls from the set ... |
H: What is the use of Delta symbol in set theory?
What is the use of $ \Delta $ in set theory?
AI: The $\Delta$ in set theory is the symmetric difference of two sets.
$A$ $\Delta$ $B$ $=$ $(B-A) \cup (A-B)$ |
H: Prove by contradiction that a real number that is less than every positive real number cannot be posisitve
This is an question from the book "A concise introduction to Pure Mathematics". I understand that it looks like a homework question but it's the first chapter and there are no answers for even questions. As I ... |
H: Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.
Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$.
How to prove inequality
$$
ab^2+bc^2+ca^2\le 4.\tag{*}
$$
In other words, if $a,b,c$ are non-negative real numbers, then how to prove inequality
$$
27(ab^2+bc^2+ca^2)\le 4(a+b+c)^3.\tag {**}
$$
$\color{g... |
H: Sitting arrangement around octagonal table
8 people A,B,C,D...H are sitting around an octagonal table.A does not want to sit beside D or opposite to him.B and C wants to sit together. In how many ways can this be done?
The answer says (8*4*4*2!*4!)/8. But how?
AI: First we seat $A$, who has $8$ choices. Then there... |
H: Why can't I use trigonometric functions here?
I tried solving this answer by using trig functions to get the answer (E). First I calculated angle FBC which was 30 and then i used sin 30 to get the length of FC. then I got the length of line BF (Using Pythagoras) and I used the formula of the area of a triangle to ... |
H: Autocorrelation functions of 2 correlated stationairy processes
I have some trouble solving the following problem:
Given are the stationairy processes $X_t$ and $Y_t$:
$X_t = Z_t*\sqrt{7+0.5X_{t-1}^2}$
$Y_t = 2+(2/3)*Y_{t-1}+X_t$
Where $Z_t$ is distributed IID $N(0,1)$
Now I simply need to find the ACF's (autocorre... |
H: For relations to be reflexive, symmetric and transitive is the property true for just the single subset $A$ or $A\times A$?
I was going over my notes on what it means for relations to be reflexive, symmetric and transitive and I'm unclear on one thing: is it for every $x$ in a set $A$ or set $A\times A$? So my unde... |
H: Partition of set
Show that the number of integer partitions of 2n into three parts, such
that the sum of any two parts is greater than the third, is equal to the
number of integer partitions of n with exactly three parts.
AI: HINT: For each partition $\langle p_1,p_2,p_3\rangle$ of $n$ consider the partition $\lang... |
H: Find $n$ , given $\sum_{i=1}^ni$
I would like to find the value of $n$ given $\sum_{i=1}^ni$. For example: If I have the number $5050$, how do I find that $n$ is $100$ here? Request your help.
AI: Use the arithmetic sum formula of
$\sum_{i=1}^n i = n(n+1)/2 $
Solve for $n$ |
H: Shear in Summation Convention
I have the linear map for fixed $\vec{a}, \vec{b}, \lambda\in\Bbb{R}$ where a and b are orthogonal unit vectors:
$$ S: \vec{x}\to \vec{x'} = \vec{x}+\lambda(\vec{b}.\vec{x})\vec{a} $$
I am looking to turn the map into a matrix $S_{ij}$ in terms of the components of $a$ and $b$ such tha... |
H: In the symmetric group $S_{10}$, every element of order $14$ is odd permutation.
Show that: In the symmetric group $S_{10}$, every element of order $14$ are odd permutations.
AI: The order of a permutation is equal to the least common multiple of the lengths of its cycles. Since we can have no cycle of length $14$ ... |
H: Selection in Circular Table
Number of ways to select k people from a round table of n people so that no two selected people sit next to each other.
I tried the followings.
k = 1, Number of ways = n
k = 2, Number of ways = n * (n-3)
k = 3, for the first person, we have n choices
for the second person:
Case ... |
H: Is an infinitely small percentage of infinity infinite?
I'm not a mathematician, but this question intrigues me: Is an infinitely small percentage or part of infinity infinite? Do the two infinities "cancel out", leaving you with a real number? It seems like they would, but what number would be left? I've done some... |
H: Factoring a polynomial over finite field $\,F_3$ that has a root
A question I am struggling with.
We are asked to factor $\,f(x)=x^2+x+1$ over the field $F_3 =\{0,1,2\}$
So, I checked for a root, and I saw that $f(1) = 1^2+1+1 =0$ (because $3=0$ in $F_3$)
that means I can write f(x) as $(x-1)g(x)$ but how do i find... |
H: Counting in Arrow's theorem
I seem to be really confused with the counting system in Arrow's theorem. Can I have a simple explanation how they determine the outcome? I can't determine the outcome using rules from my notes. It says the roles are 1) If all vote the same that would be the outcome. 2) the ranking of A ... |
H: Semisimple ring problem
Prove that:
$R$ is a semisimple ring $\Longleftrightarrow$ Every right $R$-module is injective (projective)
My try: $R$ is semisimple ring $\Longleftrightarrow$ Every right $R$-module is semisimple $\Longleftrightarrow$ Every submodule is direct summand
Please explain that why since every s... |
H: Help finding the general solution of a (partial?) differential equation.
I've been asked to find the general solution of the differential equation:
$$
y^{'} - y^3 = y^3e^x\qquad\text{, satisfying}\quad y(0)=1
$$
To solve it I did the following:
$$
y^{'} - y^3 = y^3e^x \Rightarrow y^{'} = y^3 + y^3e^x\qquad... |
H: Why $\frac{d}{dy} \int_{-\infty}^{\frac{y-b}{a}}f(x)dx=\frac{1}{a}f \left ( \frac{y-b}{a} \right)$?
Could somebody explain why:
$$\frac{d}{dy} \int_{-\infty}^{\frac{y-b}{a}}f(x)dx=\frac{1}{a}f \left ( \frac{y-b}{a} \right)$$
I don't get where this $\frac{1}{a}$ comes from. I assume that in general the outcome is ba... |
H: Constructing a circle through 2 points
We have a triangle ABC with a circumscribed circle. Somewhere between BC we place a point D. There is a circle which goes through D and whose tangent at AB is A. This circle also intersects the circumscribed circle of ABC at a point E. Construct it.
So we're just going to c... |
H: Markov Chain Initial Distribution
Suppose $\{X_0,X_1,X_2,\dots\}$ is a discrete-time Markov chain taking values in a finite set $\{1,\dots,N\}$ with initial distribution $p_i(0) = P(X_0 = i)$ for $i\in\{1,\dots,N\}$ and transition probability matrix P where the $ij$-th entry $p_{ij} = P(X_k = j \mid X_{k-1} = i)$ f... |
H: What is the inverse limit of $...\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}$ (multiplying by all positive integers)?
According to a modified answer of this question, the direct limit of the sequence
$$
\mathbb{Z}\xrightarrow{1}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{4}...
$$
in the cat... |
H: Question about describing a sub-group of $S_{\mathbb{R}}$
We had the function $f(x)=x+1$.
What is $\langle f\rangle$ of $S_{\mathbb{R}}$?
How can I describe $\langle f\rangle$?
Thank you!
AI: $f^n(x)=x+n$, $f^{-1}(x)=x-1$, so $\langle f\rangle =\{x+n:n\in\Bbb Z\}$. |
H: Compute that Galois group $Gal\left(F/\mathbb{Q}\right)$, with $F$ is the splitting field of the polynomial $X^{4}-2X^{3}-8X-3$
Compute the Galois group $Gal\left(F/\mathbb{Q}\right)$, with $F$ is the splitting field of the polynomial $X^{4}-2X^{3}-8X-3$
AI: $X^{4}-2X^{3}-8X-3=(X-3)(X^3+X^2+3X+1)$.
Then $Gal (F/\m... |
H: Can sets contain objects of different types?
Working on some basic proof work. The conjecture is
There exists a set $\mathrm X$ for which $\mathbb R \subseteq \mathrm X$ and $\emptyset \in \mathrm X$.
My reasoning was that this is false because the members of $\mathbb R$ are numbers, and $\emptyset$ is a set, henc... |
H: When can we interchange Fourier transform and countable sum?
When does $\mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x))$ where $\mathcal{F}$ the Fourier transform operator.
AI: Basically, it is Fubini's theorem. Equality holds at fixed $\lambda>0$ if and only if you ... |
H: A lot of terms to calculate lim
I'm trying to prepare for exam and I came across a limit to calculate.
$$
\lim_{n->\infty} \frac{2^n + \left(1+\dfrac{1}{n^2}\right)^{n^3} + \dfrac {4^n}{n^4}}{\dfrac {4^n}{n^4} + n^3\cdot 3^n}
$$
When I'm trying to extract $4^n$ I end up with nothing.
And I managed to tell that $(... |
H: Inverse of polynomial over $\mathbb F_3$ finite field, quotient space
A question about quotient spaces, something I do not fully understand yet, and can use some help.
$A = \mathbb F_3[x]$, $P = x^3-x+2\in A$
1) Show that $P$ is irreducible (I did it, it has no roots in $\mathbb F_3$).
2) Find the inverse of $x^2+P... |
H: How do I show this is a basis?
Suppose $\beta$ is a basis for $\mathbb R$ over $\mathbb Q$ and let $a \in \mathbb R$, $a \neq 1$.
Show that $a \beta= ${$ay\ |\ y \in \beta$} is a basis for $\mathbb R$ over $\mathbb Q$ for all $a \neq 0$.
Okay, isn't the dimension of $\mathbb R$ over $\mathbb Q$ uncountably infini... |
H: If a set $S$ is infinite, then it can be put in 1-1 correspondence with proper subset.
This is a problem from Curtis' Abstract Linear Algebra. We have the following definition of infinite set:
A set $T$ is infinite if it contains a subset $U\subseteq T$ which can be put into a one-to-one correspondence with $\math... |
H: What does it mean to solve a math problem analytically?
I'm reading a Calculus book for my own edification and at the beginning the pre-calculus introduction has the problem,
$3x+y=7$
They talk about solving the problem graphically, analytically, and numerically. The subject is the basic graph, Rene Descartes, etc... |
H: Another condition for bipartite graphs
Let $G$ be a graph. Then prove $G$ is bipartite if and only if for all subgraphs $H$ of $G$ with no isolated vertices. $\alpha(H)=\beta'(H)$.
Here $\alpha(H)$ is the size of the largest independent set. $\beta'$ is the number of edges in a minimal edge covering of $H$.
So fa... |
H: Conformal Maps and Homeomorphisms
Is every conformal map from an open subset $U\subseteq\mathbb{C}$ to an open subset $V$ a homeomorphism? Here is why I think it is. A conformal map is holomorphic (hence continuous and open) and bjiective. Seems really easy but just want to make sure.
AI: No. For instance a conf... |
H: Prove in any set of 51 positive integers,there are 11 integers $d_1
Show that in any set of 51 positive integers, there are 11 integers
$d_{1} < d_{2}\ < --- < d_{11}$ with the property that the sum
$5^{d_1} + 5^{d_2} + + 5^{d_{11}}$is an integer-multiple of 11:
Using FERMAT's Little Theorem, each one of the t... |
H: Why the ellipse circumference shows minor axis as 10 times?
Ellipse of having minor axis 0.692200628 and major axis 1.444667861 has circumference 6.9229....... which seems quite close to be minor axis 0.6922006.... multiplied by 10 but deviation occurs at 6.922''9''... after ''22''? Why?
AI: What you have here is a... |
H: Proving that $1/x$ and $1/x^2$ limit does not exist
1) If I am to prove that limit of $ \frac1x$ doesn't exist at $x\to0$ is it sufficient and rigorous enough to show that the left hand and the right hand limits are not equal(EDIT: are not equal NUMERICALLY) ? Or should I approach it by contradiction to be "RIGOROU... |
H: Local max and local min
If we have a function $f(x)$ with a derivative which changes sign infinitely many times when $x$ approaches zero and $f'(x)=0$ when $x=0$, why can't $x=0$ be a local max or local min?
For example, $f(x)=x^4sin(\frac{1}{x})$ when $x\neq 0$ and $f(x)=0$ when $x=0$.
For this function, we know ... |
H: Determine the cube roots of -8 in polar form
Exam time tomorrow and I am not entirely sure if I am doing this right.
I first write -8 as a complex number
$z^3 = -8 = -8 \times 0i$
Calculate the modulus of z
$|z| = \sqrt{-8^2} = 8$
Get the arg of z
$tan^{-1} = \frac{0}{-8} = 0 = \pi$
Write the number in polar form
$... |
H: Prove that if $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies $f(-1)=f(1)$, then $f(B)=f(B-1)$.
Suppose $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies $f(-1)=f(1)$. Prove that $\exists B \in [0,1]$ such that $f(B)=f(B-1)$.
I tried by looking at a new function $g(x)=f(B)-f(B-1), x\in [0,1]$. In order fo... |
H: How many positive 4-digit integers are there?
How many positive 4-digit integers are there?
Ans= 9*10*10*10=9000
I don't get it, what about the number after 9000.
9000-9999 are still positive 4-digit integers right?
AI: The first 999 numbers are only three or fewer digits long; 9999 is the last four digit number.... |
H: How to solve this linear algebra problem using mathematical induction
please consider this question:
Let $S,T$, be two linear transformations such that $ST-TS=I$. Prove that $ST^n-T^nS=nT^{n-1}$ for all $n\ge 1$.
thanks
AI: Consider your base case ($n=1$): $ST^1-T^1S=1T^{1-1}$. Well we know that $ST-TS = I$ and t... |
H: Problems with Integration
$$ \lim_{x \rightarrow \infty} x\left(\frac{1}{x^2}+\frac{1}{(x+1)^2}+...+\frac{1}{(2x-1)^2}\right)$$
My answer is 7/24, but the correct answer provided by the book is 1/2.
Could anyone help me?
Thanks for any insights.
AI: $$\lim_{n\to\infty} n\left(\frac1{(n+0)^2}+\frac1{(n+1)^2}+\frac1{... |
H: How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$
$$\cos^2x=\frac{1+\cos2x}{2}$$
Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at face value and memorize?
AI: Ho... |
H: Find the area of a circle that is inscribed in a circular sector with a radius $R$ and an angle $2x$.
The circle within the sector touches the radii R and the arc.
So what is the area of the inscribed circle?
The answer is actually $$S = \pi R^2\frac{\sin^2x}{(1+\sin x)^2}$$
How can I derive this?
AI: Call $\;O\;$... |
H: how to show $P_{\hat X}=P_{X}$.where $P_{X}$ is distribution.
Let $X$ be a random variable on the probability space $(\Omega,\mathcal B,P)$, with distribution $P_{X}$. Consider the random variable $\hat X$ on the probability space $(\mathbb R,\mathcal B_{\mathbb R},P_{X})$,defined by $\hat X(x)=x$ . Then $P_{\hat X... |
H: Domain of function is not correctly evaluated by WolframAlpha?
The domain of function $$\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{\sqrt{1+x}}}$$ should be $-\infty < x < -2$ and $-2 < x < -1$ and $-1 < x < \infty$.
When calculation it with WolframAlpha it gives empty domain, however when plotting this function it's c... |
H: What is the square root of 1 cm
If $1$cm = $.01$m then shouldn't the square root of $1$cm = the square root of $.01$m
but the square root of $1$ = $1$ while the square root of $0.01$ = $0.1$
So my dilemma, is the square root of $1$ centimeter = $0.01$ meters or $0.1$ meters?
AI: I think you need to think of the squ... |
H: Is the collection of hyperplane separating vectors Borel-measurable?
Let $C\subseteq\mathbb R^d$ be non-empty, convex, such that $0\notin C$. Let $$H=\{\alpha\in\mathbb R^d\mid\alpha\cdot c\geq 0 \text{ for all } c\in C\text{ and }\alpha\cdot c_0>0 \text{ for some } c_0\in C\}.$$ By the separation theorem in $\math... |
H: limit of nth root of a difference
I need some help finding the limit of the following sequence (as n goes to infinity):
$$a_n=\sqrt[n]{7^n-3^n}$$
I can limit it from above by $\sqrt[n]{7^n}$ but i can't see a way to limit it from below by anything that converges to 7.
Any help would be greatly appreciated :)
AI: Fo... |
H: Weighted coin probability
i'm having difficulty answering the following probability question...
Suppose two players, A and B take turns rolling a die. The first one who obtains a 3 wins. If A goes first what is the probability that A wins? What is the probability that B wins?
I can't really come up with a way of so... |
H: Let's throw with a regular dice (probability)
Let's throw with a regular dice twice (independently).
Let $X_{1}$ be the number of dots thrown on the first try and let $X_{2}$ be on the second.
We know, that $X=X_{1} + X_{2}$.
Calculate the expected value $\mathbb{E}[X_{1} \mid X = k ]$.
I'm stuck with this prob... |
H: How to transform and simplify the limit : $\;\lim_{x\to 0} \frac{\ln(1+5x)}{x}$?
The result of the limit $$\;\lim_{x\to 0} \dfrac{\ln(1+5x)}{x}$$ should be $5$. How do I get it?
AI: When evaluating a limit you want to let $x$ approach zero. For $f(x) = \frac{ln(1+5x)}{x}$ you get
$\lim_{x\rightarrow 0}f(x) = \frac{... |
H: Continuous bijection between open simply connected subsets of $\mathbb{C}$
Suppose $U,V \subseteq \mathbb{C}$ are open sets. I did a proof saying if $U$ and $V$ were conformally equivalent then $U$ simply connected implies $V$ is as well. I did this by showing the conformal map between the two was a homeomorphism... |
H: Find the number of 3 x 3 matrices with elements in $F_p$ such that determinant is non zero?
My question is: How do I find the number of $3 \times 3$ matrices $A$ with elements in $F_p$ such that the determinant is non-zero?
I don't really know how to go at it. I have a feeling that maybe Gaussian coefficients does ... |
H: Absolute value properties: inequality $|x|-|y| \le |x-y|$
I'm attempting to prove that $|x|-|y| \le |x-y|$.
I've come up with the following proof.
The proof relies on these results obtained from previous exercises:
$-|x| \le x \le |x|$
${|x-y|=|y-x|}$
Case 1. $x \le 0$, $y \le 0$. Here, we have $|x|-|y| = -x ... |
H: Derivatives of component maps
Given functions $f_1:\mathbb{R}^{a_1}\rightarrow\mathbb{R}^{b_1}$ and $f_2:\mathbb{R}^{a_2}\rightarrow\mathbb{R}^{b_2}$, and the function $f:\mathbb{R}^{a_1+a_2}\rightarrow\mathbb{R}^{b_1+b_2}$ is defined by $$f(x,y)=(f_1(x),f_2(y))$$ for $x\in\mathbb{R}^{a_1},y\in\mathbb{R}^{a_2}$.
Ta... |
H: An example of a total order (that is NOT a well-order) of the Natural numbers
I need an example of a total ordering of the Natural numbers, that is not a well-ordering. So the classic "less than or equal to" doesn't work in this case since it is well-ordered.
I've been wracking my brain for hours but I can't just c... |
H: How to find the limit of $\frac{1−\cos 5x}{x^2}$ as $x\to 0$?
What is the right approach to calculate the Limit of $(1-\cos(5x))/x^2$ as $x \rightarrow 0$?
From Wolfram Alpha, I found that:
$$\lim_{x \to 0} \frac{1- \cos 5x}{x^2} = \frac{25}{2}.$$
How do I get that answer?
AI: We can use L’Hôpital’s rule, where $f(... |
H: How to tell if a function is one-to-one or onto
We just learnt this today in Discrete Math, and now I'm trying to review from the textbook. However unfortunately during this lecture I was completely lost with no idea what was going on.
I know that for one-to-one, every $x$ has an unique $y$ and for onto, for all $... |
H: Congruence Inconsistency
I have a question about congruency...
I understand that:
$$
12 \equiv 7 \bmod 5
$$
$$
\text {is equivalent to:}
$$
$$
5|12-7
$$
but this doesn't seem to hold for:
$$
2 \equiv 8 \bmod 6
$$
$$
\text {the conclusion i come to is:}
$$
$$
6|-6
$$
are these equivalent??
AI: Yes, $6\mid-6$. That... |
H: How to prove that $f(x,y)=x-y$ is one-to-one?
Given $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, f(x,y) = x-y$. How can I prove that the function is one to one?
I know that f(x,y)=x-y is a plane, and just by visualizing it I can see that it is one-to-one. But I'm confused as to how to prove this for a f... |
H: Definition of the limit of a function at a point
What is the formal definition of the limit of a function at a point in Real Analysis?
Is it this one?
For a function $f: D\subseteq \mathbb{R}\rightarrow \mathbb{R}$ with the domain $D$ containing an open interval around $a$, except possibly at $x=a$, say that $\lim... |
H: Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?
True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
AI: Consider the mapping
$$f : \mathbb N\to\mathbb N, x\mapsto\begin{cases}1 & \text{if }... |
H: Cauchy sequence from Fourier coefficients
Let $f\in L^2(\mathbb{R})$, and let $f_1,f_2,\ldots \in L^1(\mathbb{R}),L^2(\mathbb{R})$ be such that $\|f_n-f\|_2\rightarrow 0$ as $n\rightarrow\infty$.
Is it true that $\{\hat{f_n}(y)\}_{n=1}^\infty$ forms a Cauchy sequence, where $$\hat{f_n}(y)=\int_{-\infty}^\infty f_... |
H: Probability of group
I have seen a problem in Probability that I am stuck :
Imagine that we have $12$ students in class and among this students there are $3$ honor students. Say that a teacher wants to assign a group project and wants to balance the groups out by forming 3 groups of students with exactly one honor ... |
H: Prove that functions are one-to-one
Given $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x) = x^{3}$
Proof: Assume $f(m)=f(n)$ for some $m, n \in \mathbb{R}$. Then $m^{3}=n^{3}$, and $m=n$. $f$ is one-to-one.
Given $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x) = 2^{x}$
Proof: $f'(x) = \ln(2) \times 2^{x} \neq 0$, so by... |
H: What is the gradient of this function
Imagine you have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$
$f(z):=z^TAz$, where $A$ is a symmetric matrix. Now I was wondering what
$\nabla f(x_1,...,x_{n-1},\gamma(x_1,...,x_{n-1}))$, where $\gamma$ is some differentiable function $\gamma:\mathbb{R}^n \rightarrow \mat... |
H: How to formally show "if $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$"?
Prove: if $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$
It's quite obvious, but I'm not sure what the proper approach is to proving a set problem that involves subsets, as none of the set identities give... |
H: Expressing vectors in an octagon
I'm having trouble with this question in my course.
I am to consider a regular octagon with vertices A, B, C, D, E, F, G and H in counter clockwise order. The vectors $\overrightarrow{AC}$ and $\overrightarrow{AD}$ make up a base for the plane. Then I am supposed to express the vect... |
H: How to prove that the partial Euler product of primes less than or equal x is bounded from below by log(x)?
How does one prove $\prod_{p \leq x}(1 - \frac{1}{p})^{-1} \geq \log(x)$?
AI: As commented by Daniel Fischer,
I think you mean
$\prod_{p \leq x}\frac1{1 - \frac{1}{p}} \geq \log(x)
$.
Let $P(x)$ be the set o... |
H: Prove if $f: A \rightarrow B, g: B \rightarrow C$, and $g \circ f: A\overset{1-1}{\rightarrow}C$, then $f: A \overset{1-1}{\rightarrow} B$
Statement: If $f: A \rightarrow B, g: B \rightarrow C$, and $g o f: A\overset{1-1}{\rightarrow}C$, then $f: A \overset{1-1}{\rightarrow} B$
Here's my proof by contradiction.
Pro... |
H: normal as approximation to binomial
Among 784 checks, 479 had amounts with leading digits of 5, but checks issued in the normal course of honest transactions were expected to have 7.9% of the checks with amounts having leading digits of 5. Is there strong evidence to indicate that the check amounts are significantl... |
H: Delta function proof in QM
I'm actually working with some QM problems at the moment but I've hit a wall with a delta potential involved.
The problem asks me to verify that
$$
\frac{d \phi_{x=0^{+}}}{dx} -\frac{d \phi_{x=0^{-}}}{dx} = -\frac{2mV_{0}}{ \hbar} \phi_{x=0}
$$
which I believe is a continuity proof.
I k... |
H: Dimension of linear subspace of inear subspace.
Task: There is a vector space $X$ in $\mathbb R^5$. $Dim(X)=3$
Let $V=\{A \in \mathbb R^{5,4}: imA \subset X\}$
Show that V is linear subspace in $\mathbb R^{5,4}$ and find $Dim(V)$.
My work so far:
$A\in \mathbb R^{5,4}$
$imA=\{y\in \mathbb R^5: \ \exists x \in \m... |
H: probability and combinatorics mixed question
A bus follows its route through nine stations, and contains six passengers. What is the probability that no two passengers will get off at the same station?
no detailed solution is required here but an idea of the general line of thought could be nice...
AI: The problem... |
H: ETCS set theory: Are empty sets isomorphic?
Just a quick question about ETCS: Are any two empty sets isomorphic? Here, a set $X$ is empty if there exists no $x \in X$, i.e. no functions $x: 1 \to X$.
The reason I'm asking is that I need this to show that empty sets are initial sets.
Thank you!
AI: In the case of th... |
H: Dealing a 5 card hand with exactly 1 pair
Here's the question:
A 5-card hand is selected from a standard deck of playing cares. (A standard deck has 13 cards from each of
4 suits{clubs, diamonds, hearts, and spades. The 13 cards have face value 2 through 10, jack, queen, king,
or ace. Each face value is \kind"... |
H: Existence of homeomorphism between $[0,1]$ and parabola
Does a homeomorphism
$$f:([0,1],d_E)\to\left(\left\{\left(x,x^2\right):x\in[0,2]\right\},d_E\right)$$
exist?
I suppose that it exists. Of course, the second set is a parabola in $\mathbb{R}^2$. But I don't know how to find a function which will be a bijection.... |
H: The term "maximal solution" for PDE
A solution $x(t)$ of the ODE is called maximal if it is defined on an
open interval and cannot be extended to any larger open interval.
from "Ordinary Differential Equation". Alexander Grigorian. University of Bielefeld. Lecture Notes, April - July 2008.
How the term "maximal... |
H: Summation formula for $x^2+x$
Since I learned easier ways of calculating summations I've been curious as to how I could find formulas for as many equations as possible. I came across the equation $x^2+x$, I've spent quite some time on this problem and could not find a solution. If someone has maybe already done thi... |
H: Why is this relation reflexive?
$S$ is this set of all graduates from a university. $xRy$ means that student $x$ first attended the university at the same year student $y$ did.
The answer key says $R$ is reflexive but isn't it possible that only one student entered the university in a year? Does this matter for bei... |
H: In predicate logic, can existential variables be used interchangeably?
When doing a derivation in predicate logic, am I allowed to use two different existential variables interchangeably?
For instance, is $\forall xPx$ (or $∃xPx$) equal to $\forall yPy(∃yPy)$? If I cannot directly mix the two, am I allowed to do ... |
H: Open subsets of perfect Polish spaces
Is it true that every non-empty open subset of a perfect Polish space
is uncountable?
It is true that the space itself is uncountable but I was not able to show
that every non-empty open subset is uncountable.
AI: $G_\delta$-sets in a completely metrizable space are completely... |
H: Question about proof of Morse Lemma
I am working on a problem for my differential geometry course. We are proving the following special case of the Morse lemma:
Let $U \subseteq \mathbb{R}^n$ be open and containing the origin $\mathbf{0} \in \mathbb{R}^n$ (denote coordinates on $U$ by $x = (x_1, \dots, x_n)$). Supp... |
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