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H: A bagel shop has plain muffins, cherry muffins, chocolate muffins, almond muffins, apple muffins, and broccoli muffins.
How many ways are there to choose:
1.) two dozen muffins with at least two of each kind?
2.) two dozen muffins with at least five chocolate muffins and at least three almond muffins?
3.) two dozen... |
H: characterization of uniform ellipticity
Let $B$ be a $n\times n$ matrix over $\mathbb{R}$ and define $A:=BB^*$. I read in a paper that the following two statements are equivalent:
(1) the matrix $A$ is uniformly elliptic; i.e. for all vectors $y\in\mathbb{R}^n$ and some constant $N>0$, it holds that $$y^*Ay \geq \f... |
H: Continuity of bounded variation functions
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a BV function if there exists $M<\infty$ for which $$\sum_{k=1}^N|f(x_k)-f(x_{k-1})|\leq M$$ for every sequence $x_0<x_1<\ldots<x_N$ and every $N$.
Any BV function has only jump discontinuities. That is, at any point $a$, t... |
H: Complex power series (or not quite so?)
I'm stuck with this problem. Any hints are appreciated.
It just says
$$
\mbox{"For what values of}\ z\ \mbox{is}\quad
\sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?}
$$
The thing is that it doesn't look like a power series, but I guess I should tr... |
H: Interval of convergence for $\sum_{n=1}^{\infty}9(-1)^nnx^n$
I need to find the interval and radius of convergence and I'm really confused with what I'm supposed to be doing. Here is the problem:
$\sum_{n=1}^{\infty}9(-1)^nnx^n$
I then used the ratio test to get $|9||x|\lt1$ and took a guess at my interval of conv... |
H: restriction of functions of several variables
Let $f: \Bbb R ^n \to \Bbb R $ will be differentiable function satisfying the condition
$$ \sum_{i=1}^{n} y_i \frac{ \partial f}{ \partial x_i } (y) \ge 0 $$ for every vector $ y=(y_1,y_2,...,y_n)$
Demonstrate that the function is bounded from below by $f(0)$
I don't... |
H: What is the congruence class of $x^3\mod x^3+x+1$?
I have a given Polynom congruence with a Polynom $x^3+x+1$ ... so the set of the congruence classes is $\{0, 1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$
But what would look this like?
$$x^3\mod x^3+x+1\equiv ?$$
I think the result must be one of the congruence classes of t... |
H: Trouble with factoring polonomial to the 3rd degree
I am having trouble factoring this problem:
$\displaystyle{-x^{3} + 6x^{2} - 11x + 6}$
I know the answer but i can't figure out how it is done with this.
I have tried by grouping and is doesn't seem to work. Can someone show me how to do this.
AI: $$\begin{align}
... |
H: The trace map in a finite field.
Let $p$ be a prime number, and consider the mapping called the trace
$$ Tr \quad : \quad \mathbb{F}_{p^n} \ \longrightarrow \ \mathbb{F}_{p^n} \quad : \quad x \ \longmapsto \ x + x^p + x^{p^2} + \cdots + x^{p^{n-1}}$$
My syllabus Abstract Algebra states the following:
Every element... |
H: Angle between vectors?
Here's the problem from my homework:
If the vector $\vec{a}+\vec{b}$ is perpendicular to the vector $7\vec{a}-5\vec{b}$, and if the vector $\vec{a}-4\vec{b}$ is perpendicular to the vector $7\vec{a}-2\vec{b}$, what is the angle between vectors $\vec{a}$ and $\vec{b}$?
So, if I use the fac... |
H: How to notice that $3^2 + (6t)^2 + (6t^2)^2$ is a binomial expansion.
The other day during a seminar, in a calculation, a fellow student encountered this expression:
$$\sqrt{3^2 + (6t)^2 + (6t^2)^2}$$
He, without much thinking, immediately wrote down:
$$(6t^2+3)$$
What bothers me, is that I didn't see that. Althoug... |
H: How to prove $(\frac{n+1}{n})^n
Prove that $(\frac{n+1}{n})^n<n\quad $ for any n=3,4,5... by using induction.
For n=3 is true. and lets assume $(\frac{n+1}{n})^n<n$ is true. we must show that $(\frac{n+2}{n+1})^{n+1}<n+1$ is true. How can I continue?
AI: If $(\frac{n+1}{n})^n<n$ is true, then
$(\frac{n+1}{n})^n*\fr... |
H: Exponential of a complex line
Is there an "elementary" way to prove that if $D$ is a one-dimensional vector space in $\mathbb{C}$ (considered here as a real vector space), then $\exp(D) \neq \mathbb{C}^{\ast}$ ?
AI: What counts as elementary?
The restriction of $\exp$ to any subset whose diameter is smaller than $2... |
H: Cantor construction is continuous
I define a function $f:\mathbb{R}\to\mathbb{R}$ as follows:
$f(x)=0$ for $x\le 0$. $f(x)=1$ for $x\ge1$.
$f(x)=\dfrac12$ for $x\in\left[\dfrac13,\dfrac23\right]$.
$f(x)=\dfrac14$ for $x\in\left[\dfrac19,\dfrac29\right]$, $f(x)=\dfrac34$ for $x\in\left[\dfrac79,\dfrac89\right]$.
and... |
H: Inner product on tangent space and metric tensor
In our class we talked about integrating on submanifolds and as a short side remark our teacher told us that by knowing the metric tensor, it is possible to define an inner product on a tangent space and then he said: $\alpha(t):=\phi(tb)$, where $t$ is a real number... |
H: Bi-implication theorem proving
While proving a theorem, i came across a situation like as follows
(P has a property) $\leftrightarrow $ $(x=y)$
(P has a property) $\leftrightarrow $ $(y=z)$
Now can i infer the following fact from the above two facts ?
(P has a property) $\leftrightarrow $ $(x=z)$
$\leftrightarrow ... |
H: Why can't you solve this probability problem in this way?
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-... |
H: Generating Function for Binary String Question
The following is an assignment question I have been trying to work out for quite some time.
Let $C(x,y)=\sum_{n,k \geq 0} c_{n,k} x^{n} y^{k}$, where $c_{n,k}$ is the number of binary strings of length $n$ with $k$ blocks. Prove that
\begin{equation}
C(x,y)=\frac{1-... |
H: Proving $\lim_{n \to\infty} \frac{1}{n^p}=0$ for $p > 0$?
I'm trying to prove 3.20a) from baby Rudin. We are dealing with sequences of real numbers.
Theorem.
$$\lim_{n \to {\infty}} \frac{1}{n^p} = 0; \hspace{30 pt}\mbox {$p > 0$}$$
Proof. Let $\epsilon > 0$. Because of the Archimedan property of real numbers, the... |
H: Convergence functions
Let X be a nonempty set. I define a convergence function on X to be a partial function from the set of all sequences in X, to X, that satisfies the five additional conditions:
Every constant sequence is assigned that constant.
If a sequence converges, so does any sequence obtained from alter... |
H: Limit of convolution of measures is Cantor function
For positive integer $k$, let $\mu_k=\dfrac{1}{2}\left(\delta(x)+\delta\left(x-\dfrac{2}{3^k}\right)\right)$. Show that $$\lim_{k\rightarrow\infty}(\mu_1\ast\mu_2\ast\cdots\ast\mu_k)((-\infty,x))=C(x),$$ where $C$ is the Cantor function.
The definition of measur... |
H: Question on Rudin sequences?
In baby Rudin, Rudin shows that $$\lim_{n \to \infty}\sqrt[n]{p} = 1.$$
In the proof of limit he tries to prove that the limit is $1$. So he takes $x_n = \sqrt[n]{p} - 1$.
I have never noticed this before, but I could have tried to prove the limit for any $n \neq 1$ (and obviously it wo... |
H: exponential equation with a sum of exponents
I'm trying to solve the following exponential equation:
$e^{2x} - e^{x+3} - e^{x + 1} + e^4 = 0$
According to the the text I am using the answer should be $x = 1,3$ but I can't derive the appropriate quadratic $x^2 -4x + 3$ from the above equation using any of the method... |
H: Linear congruence proof, show congruence has exactly two incongruent solutions
Let p be an odd prime and k a positive integer. Show that the congruence $x^{2}$ $\equiv 1 \ mod p^{k}$ has exactly two incongruence solutions, namely, $x \equiv \pm 1\mod p^{k}$.
I'm not sure what to do after this:
$x^{2}$ $\equiv 1 \ m... |
H: Korean Math Olympiad 2000 (floor function, quadratic mod)
Let $p$ be a prime such that $p ≡ 1\ (\mathrm{mod}\ 4)$. Evaluate $\displaystyle\sum\limits_{k=1}^{p-1}\left({\left\lfloor\frac{2k^2}{p}\right\rfloor}-2{\left\lfloor\frac{k^2}{p}\right\rfloor}\right)$.
AI: Hint: Prove that
$$\lfloor2x\rfloor-2\lfloor x\rflo... |
H: Addition in finite fields
For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of degree 3, and creating the splitting field for that polynomial.
The polynomial I chose for t... |
H: Showing that $E[X|X
I would like to show that:
$\hspace{2mm} E[X|X<x] \hspace{2mm} \leq \hspace{2mm} E[X] \hspace{2mm} $ for any $x$
X is a continuous R.V. and admits a pdf. I'm guessing this isn't too hard but I can't come up with a rigorous proof. Thanks so much.
AI: Hint: $$E[X] = E[X|X<x]P(X<x) + E[X|X\geq x]... |
H: I don't understand this PDE solution involving Fourier coefficients and orthonormal eigenfunctions.
$u_{xx} + u_{yy} = 0$ with $x \in (0,\pi)$ and $y \in (0, \pi)$
Initial Conditions:
$$
u(x,0) = x^2
$$
$$
u(x,\pi) = 0
$$
Boundary conditions:
$$
u_{x}(0,y) = 0 = u_{x}(\pi, y)
$$
I performed separation of variables,... |
H: Finding vector and parametric equations provided only one point.
Normally to answer these questions I have a point and one or two vectors. However, for this one I only have a point. How can I concoct these equations provided there is limited information?
Find vector and parametric equations of the plane in $R^3$ t... |
H: Composition relations and powers
Let $R$ be the relation on $\Bbb Z$ such that $xRy$ iff $x-y=c$
a.) Define $R^2$
b.) Define $R^i$ for abitrary $i\ge1$.
Well the problem I'm having with this is trying to figure out how get the power of a relation and how does $xR^2y$ for example. Well I know for starters that i... |
H: Prove: The infinite series $x_n$ converges if and only if the infinite series $2^nx_{2^n}$ converges.
I am in desperate need for hints to get me in the right direction for this proof.
Let $(x_n)_{n\in \mathbb {N}}$ be a monotone decreasing null sequence. Prove that:
$$\sum_{n=1}^\infty x_n \text{ converges } \ \if... |
H: Equivalence of definition for weak convergence
The Wikipedia page on convergence of measures says:
In the case $S=\mathbb{R}$ with its usual topology, if $F_n, F$ denote the cumulative distribution functions of the measures $P_n$, $P$ respectively, then $P_n$ converges weakly to $P$ if and only if $\lim_{n\rightar... |
H: Hunting Birds probability
A hunter locates 20 geese, 25 ducks 40 eagles, 10 ostriches, and 5 flamingos. He randomly selects 6 birds to target. What is the probability at least one of each species is targeted?
My reasoning $20 \choose 1$ $25 \choose 1$ $40 \choose 1$ $10 \choose 1$ $5 \choose 1$ $5\choose 1$ for the... |
H: The Inverse Laplace Transform
What's the inverse Laplace transform of $\frac{s}{(s-5)^4}$? I'm thinking of adding zero to the top and dividing out to get rid of the top s.
AI: Hint:
Do the partial fraction expansion as:
$$\dfrac{s}{(s-5)^4} = \dfrac{5}{(s -5)^4} + \dfrac{1}{(s -5 )^3}$$
Now, use a Table of Laplace... |
H: Calculus: finding integral
I need to compute the following two integrals: $$\int_0^{\infty}y^be^{-y/2} dy$$
$$\int_0^{\infty}\frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy$$ can I do this? Please show me the process.
How to use definition of gamma function to solve these?
AI: For the first integral, substitute $u=y/2$ with $2... |
H: k critical graph cannot have k + 1 vertices
$k$-chromatic graph is called $k$-critical if removal of any vertex from graph makes it $k - 1$ vertex colorable.
Now i have to prove that if $G$ is a $k$ critical graph then it cannot have $k+1$ vertices.
I can see that the property is true as a triangle is 3 critical. A... |
H: How prove this $\int_{a}^{b}xf(x)dx\le\int_{a}^{b}xg(x)dx$
let $f(x),g(x)$ is continuous on $[a,b]$,and such
$$\int_{a}^{x}f(t)dt\ge\int_{a}^{x}g(t)dt,x\in[a,b)$$
and
$$\int_{a}^{b}f(t)dt=\int_{a}^{b}g(t)dt$$
show that:
$$\int_{a}^{b}xf(x)dx\le\int_{a}^{b}xg(x)dx$$
my try: we only prove this
$$\Longleftrightarrow ... |
H: separable space and open covers
If a topological space $X$ is separable, then every open cover of $X$ must be countable?
since $X$ is separable , then there exists a countable dense subset $S$. This implies, in every open cover any set must intersect with $S$.
AI: No, definitely not. The Mrówka space $\Psi$ is a s... |
H: Initial Value Problem with Laplace Transform
How do you solve the following with Laplace Transform?
$$
{\rm y}''\left(t\right) - 10\,{\rm y}'\left(t\right) + 25\,{\rm y}\left(t\right)
=
24\,t\,{\rm e}^{-2t}\,;
\qquad\qquad
{\rm y}\left(0\right) = -2\,,\quad {\rm y}'\left(0\right) = -10
$$
AI: Hints:
$\mathcal{L} ... |
H: Finding the formula of a sequence.
I have a sequence $\{0,1,1,2,2,3,3,4,4,5,5,...\}$ and I am supposed to find a formula for the n-th term. I do not see a pattern between the terms except that each positive integer is repeated twice. I cannot see a relation between each term to determine whether it is arithmetic or... |
H: How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola?
How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola, knowing the canonical form is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=\pm1$ where $a$ and $b$ are constants? Thanks !
AI: Taking the '+' sign, $$y^2-x^2=x+y+1\implies\left(y-\frac12\right)^2-\left(x+\frac12\... |
H: Selecting Cards
A special deck of 51 cards constists of 25 pairs and 1 wild card. The deck is distrubuted evenly between 3 players (17 cards each). What is the probability that your hand has only two pairs that is the probability you have 13 single cards.
My reasoning was I have $51 \choose 17$ hands. There are $25... |
H: Public Key Cryptography
Assuming that a message has been sent via the RSA scheme with $p=37$, $q=73$, and $e=5$, what is the decoding of the received message "34?"
So far, I have $x^5 \pmod{37\times73} \equiv 34$. How do I invert a mod?
AI: We need the totient function of the modulus, hence we get:
$$\varphi(N) = ... |
H: How to prove a set is nowhere dense?
Im trying to solve a problem from chapter 8, Real Analysis, Carothers, 1ed, talking about compactness of metric space, :
I've finished the first problem actually.
For the second problem. I've proved the range of $f$ is closed with claim as following:
$f$ is continuous and $[0,1... |
H: Way to find the value of $ a_{2011}$
Let us consider the sequence $ a_1 , a_2 , a_3 ,\ldots $ where $ \frac{1}{a_{k+1}} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}$ for $k >1$ and $a_1 = 2^{2009}$. How can I find the value of $ a_{2011}$?
AI: Observe that $ \frac{1}{a_{k+1}} = \frac{1}{a_k} + \frac{1}{... |
H: How many ways the set D can be constructed?
The following relations hold for four non-empty sets $A,B,C,D$:
$ A \cup B \cup C \cup D = A \cap B $
$ B \cup C \cup D = B \cap C $
$ C \cup D = C $
If $A = \{1,2,3,4\} $ then in how many ways can the set $D$ be constructed ?
AI: (3) tells you that $D\subseteq C$; wh... |
H: Limit $\lim (\frac{n!}{n^n})^{\frac{1}{n}}$
I need to calculate $$\lim_{n\rightarrow \infty} \left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$$
My try:
When $n!$ is large we have $n!\approx(\frac{n}{e})^n\sqrt {2\pi n}$ (Stirling approximation)
$$\lim_{n\rightarrow \infty} \left(\frac{n!}{n^n}\right)^{\frac{1}{n}}=\lim_{n... |
H: problem with recurrence relation for series solution for ODE
I have
$$y''-xy'-y=0$$
and I'm trying to find the series solution around the ordinary point $x_0=1$. My last post I muscled through to the solution when the ordinary point was $x_0=0$, but this is proving to be tougher. Now I have obtained through power... |
H: Drawing a simple graph with a certain number of vertices
I am supposed to see if it is possible to draw a graph with 8 vertices given the degree sequence: 1,1,2,3,5,5,6,7 First I tried the handshaking lemma and it holds.
So since drawing that graph is tedious I decided to remove the vertex with degree 7 since I kn... |
H: Initial Value Problem with Repeated Eigenvalues
Given the matrix
$$
A=\left(\begin{array}{ccc}
1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
$$
For $X'= AX.\quad$
$X\left(0\right)=\left(\begin{array}{r}1 \\ 0 \\ -2\end{array}\right)\,.\quad$
What is the solution ?.
AI: We find the eigenvalues and eigenvecto... |
H: Restriction of $f$ to $B(x_0,\delta)$ attains a maximum
Let $f: U\to \mathbb{R}$ $U\in \mathbb{R}$ be twice differentiable at $x_0\in U$. If $f'(x_0)=0$ and $f''(x_0)<0$ then the restriction of $f$ to $B(x_0,\delta)$ attains a maximum at $x_0.$
I know that by defintion $$\lim_{x\to x_0} =\frac{f'(x)-f'(x_0)}{x-x_0}... |
H: Exponentials in complex numbers
If $\displaystyle z-\frac1z=i$, then find $\displaystyle z^{2014}+\frac{1}{z^{2014}}$.
The answer should be in terms of $1, -1,\;i\;or\;-i$. I am not able to understand how to simplify the given expression so that its $2014^{th}$ power can be found out easily.
AI: $(1)$
We have $\di... |
H: How to solve this gamma integral
Let we have the (p.d.f) of x which is:
$$f(x)=\frac {\Gamma{(n-1)/2)}}{\Gamma{(1/2)} \Gamma{(n-2)/2)}}x(1-x^2)^{(n/2)-2}$$
then to find the $E(x) = $$\int_{-\infty}^{\infty} x *(f(x) dx$ ; (Expectation of Mean)
Thus, we construct that
$$\int_{-\infty}^{\infty} (x) \frac{\Gamma{(n-... |
H: Find solutions to equation (ring/field theory, residue class)
I'm trying to solve this problem:
A residue class ring mod $n$ is a field if n is prime. Let $\mathbb{Z}_p$ be a residue ring, p prime. Let $a \in \mathbb{Z}_p$.
What are all solutions $x \in \mathbb{Z}_p$ to the following equation?
$$\widetilde{x^2} = ... |
H: Verify that the unit closed ball in $c_0$ has not extremal points
I want to understand this proof of the fact that $c$ and $c_0$ aren't isometrically isomorphic, but I have very little experience in working with extremal points.
So how can I verify that the unit ball in $c_0$ didn't have one of these points? I trie... |
H: Number of solutions in a given interval
How many solutions does the equation $\cos^2x-\cos x-x=0$ have in the interval $\displaystyle \left[0,\frac\pi2\right]$?
Clearly, $x=0$ is a solution. Are there any other? I couldn't proceed after differentiating it to find the extremes.
AI: You don't need to take the derivat... |
H: How prove this matrix equation $AXB=C$ has a solution
Qustion:
if Matrix equation $AY = C$ and $ZB = C$ has solution,
show that:
the equation of $AXB= C$ has solution
This problem is from this PDF(page 3) problem 2 :http://wenku.baidu.com/view/d6625d1cff00bed5b9f31dba.html
My idea:we have
$$\left(A\bigotime... |
H: Rank-one update of singular matrix
My problem is the following:
Given a column vector $\mathbf{e''}\in\mathbb{R}^{3}$ with $||\mathbf{e''}||=1$ and a $3\times 3$ matrix $B$ such that $rank\{B\}=3$, I would like to find a vector $\mathbf{v}\in\mathbb{R}^3$ garantying that $rank\{A(\mathbf{v})\}=3$, where $A(\mathbf{... |
H: What is meaning of strict weak ordering in layman's term?
I gone through many pages using Google, but not understand exact meaning of Strict-weak Ordering term.
I have this requirement while sorting strings.
AI: This notion, which sounds somewhat like an oxymoron, is not very commonly used in mathematics, but it i... |
H: A doubt in finding the number of real roots of a given polynomial using derivative
I have learnt so far that a polynomial of degree $n$ has $n$ roots. To find out the number of real roots it has, we have to take its derivative, equate it to $0$ and then find the roots. Those roots are the extremes in the graph of t... |
H: What is the value of $ a_{2009}$
I have the following sequence :
$a_0 = 3 $
$ a_n = 2 + a_0 a_1 a_2\text{ ...}a_{n-1} $
How can I find the value of $a_{2009} $ ?
AI: Rewrite $a_{n+1} = a_n(a_n-2) + 2$ as $a_{n+1} - 1 = (a_n -1)^2$
To clarify further.
If $b_n = a_n -1$, then we have $b_{n+1} = (b_n)^2$, and so $b_{... |
H: Represent boolean OR opperator in non-boolean math notation
I'm trying to represent the boolean opperation OR in a regular formula, I am familiar with the boolean algebra notation, I came up with this (A+B)/(A+B)
this works for all binary values except if both A=0 and B=0
is there a simple alternative that also wor... |
H: Nspire CAS spitting out a wrong answer?
Consider the integral:
$$\int \frac{8x+11}{(2x+3)(x+1)}$$
My Nspire CAS tells me that the answer to this is
$$\ln\left\lvert(x+1)^3 \cdot (2x+3)\right\lvert$$
This is not the correct answer according to my calculations and Wolfram Alpha
Any ideas what's going on?
AI: HINT:
U... |
H: Using the intermediate value theorem for derivatives to infer that a function is strictly monotonic
My textbook Elementary Classical Analysis claims that by Darboux's theorem (the intermediate value theorem for derivatives), if a function $f:\mathbb R\to\mathbb R$ has a nonzero derivative on $\mathbb R$, then is $f... |
H: I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process.
Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a
Gaussian process. Find its mean and covariance functions.
thanks .
AI: We have to prove that if $(t_1,\dots,t_d)$ are non-negative numbers and $(a_1,\dots,a_d)$ real nu... |
H: Integration of parabola
I have this homework question I am working on:
The base of a sand pile covers the region in the xy-plane that is bounded by the parabola $x^2 +y = 6$ and the line $y = x$: The height of the sand above the point $(x;y)$ is $x^2$: Express the volume of sand as (i) a double integral, (ii) a tri... |
H: Calculation of a square root of a big number
How can I calculate the following number:
$$ \sqrt{444 \cdots (2n \text{ digits}) + 111 \cdots (n+1 \text{ digits}) - 666 \cdots (n \text{ digits})}.$$
My trying :
I have tried to calculate these data by observing the pattern of $ 7^2 , 67 ^2 ,667^2 , 6667^2 , \do... |
H: Computing a probability of finding defects by random sampling
This is a problem from my semester end exams (which have got over). I suspect that the problem below is vague or open for misinterpretation. I would really like to know the actual answer to the problem if they are correctly asked, if not, I would like ho... |
H: Definition of logarithm function derived from its useful properties
My understanding of the point of logarithms is that they turn multiplication into addition, and exponentiation into multiplication.
i.e.
$$ \ln cx = \ln c + \ln x $$
$$ \ln x^c = c \ln x $$
Let's call the above two statements about logarithms their... |
H: Every projective f. g. module is f. p.
I want to show that if $P$ is a finitely generated (f.g.) projective module then $P$ is finitely presented (f.p.).
AI: As $P$ is f. g. we have an exact sequence $0\rightarrow Q\rightarrow A^n\rightarrow P\rightarrow 0$, $Q$ denoting the kernel of the map $A^n\rightarrow P$. A... |
H: How find this invertible matrix $C=\left[\begin{smallmatrix} A&B\\ B^T&0 \end{smallmatrix}\right]$
let matrix $A_{n\times n}$,and $\det(A)>0$, and the matrix $B_{n\times m}$,and such $rank(B)=m$,and let
$$C=\begin{bmatrix}
A&B\\
B^T&0
\end{bmatrix}$$
Find this Invertible matrix
$C^{-1}$
my try: I found this matrix... |
H: Question about tensor product of modules and ideals
Trying to prove some properties of tensor product with a given module, I came up with questions some of them I can't prove. Maybe it is also because Im not very used to work with tensor products and I think they can be tricky.
So let $A$ be a ring and let $M$ be a... |
H: Prove that f(x) can have any value between A and B
I need to prove the following statement:
Let f be a bounded and continuous function in the interval (a,b). if $\lim_{x\to a+}f(x)=A$ and $\lim_{x\to b-}f(x)=B$ and $A\neq B$ then f can get any value between and A and B in the interval (a,b).
What I did:
Let $ g(x) ... |
H: Exact sequence in a category with zero morphisms
Let $C$ be a category with zero morphisms (equivalently, $\mathsf{Set}_*$-enriched), for example it could be a linear category. Then we can talk about kernels and cokernels of morphisms in $C$. I wonder if the following definition is already established and appears s... |
H: Range of function
Given the function, $y=f(x)=\frac3{2-x^2}$, find its domain and range.
The domain is of course = $R - \{-\sqrt2,\sqrt2\}$.
However, the range I got was wrong(rather incomplete).
Rewriting the function for x in terms of y, I got $x=\pm \sqrt{\frac{2y-3}{y}} $
$\frac{2y-3}{y} \ge 0 \implies y\ge \f... |
H: Limit problem. Probability picking an item once out of M picks where p(pick) = 1/M+k.
What is the probability of picking an item one or more times if $p(pick) = \frac{1}{M} + \epsilon$ and we pick $M$ times?
$$p(pick~one~or~more) = 1-p(not~pick)^M = 1 - (1-(\frac{1}{M}+\epsilon))^M$$
I am interested in what happens... |
H: How to find the expresion such that its derivative must meet a certain condition
Suppose $a$ and $b$ are expressions in terms of the variable $x$.
We know:
$\begin{align}
a &= a \cdot\frac{b}{b} \\
&= \frac{ab}{b} \\
\end{align}
$
Is there a systematic way to find $b$ such that:
$\frac{db}{dx} = ab$
For example:
If... |
H: The arithmetic mean of $X$ when arithmetic mean of $X^2 = 29$.
Sorry if my question is a beginner because of my mathematical knowledge is low.
arithmetic mean is :
$$
\overline x=\dfrac{x_1+x_2+\cdots+x_n}n
$$
What method can solve it?
$$
\overline{X^2}=29\quad\Rightarrow\,\text{Not }\left(\overline X\right)^2 \... |
H: A question about infinitary proofs and First Order Peano Arithmetic
In certain proof systems, infinite proofs are allowed; a common example is a version of Induction:
Given $\Sigma \vdash \phi(S^n 0)$ for all $n \in \Bbb N$, infer $\Sigma \vdash \forall x \phi(x)$.
What is the relation with the system described i... |
H: Lower triangular matrices
Is the inverse of an invertible and lower triangular matrix still both lower triangular and invertible?
AI: Yes. The lower triangular matrix $L$ is invertible if and only if its determinant is nonzero, and since its determinant is the product of its diagonal entries, if and only if its di... |
H: $\frac{\mathrm d}{\mathrm dt} \exp\{\lambda((q+pe^t)-1)\}$
$$\frac{\mathrm d}{\mathrm dt} \exp\{\lambda((q+pe^t)-1)\}$$
How do I do this? Do I use the chain rule?
$$= \exp\{\lambda((q+pe^t)-1)\} \frac{\mathrm d}{\mathrm dt} (\lambda((q+pe^t)-1)) \\
= \exp\{\lambda((q+pe^t)-1)\} \lambda pe^t$$
But the solution appe... |
H: Intuitive Understanding of Projective Modules
I was wondering if anyone could give me any sort of intuitive explanation of what a projective module is or a useful way to think about them. I know the definition(s) in terms of lifting, split exact sequence, direct summand but I have no intuitive understanding of what... |
H: Conditional Expectation Probability
Walking in the street for 10 minutes, the number of people you cross has a
Poisson distribution with mean λ. Suppose that each has a cold with probability p. During
those 10 minutes, what is the expected number of walkers you cross who have a cold?
Attempt:
Let X~Poi(λ) where x=... |
H: If a function is smooth is 1 over the function also smooth
If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
AI: If $f$ is differentiable and non-zero at some point $a$, then $1/f$ is differentiable at $a$, and $(1/f)'=-f'/f^2$. This is a "base case" of... |
H: The normal approximation of Poisson distribution
(I've read the related questions here but found no satisfying answer, as I would prefer a rigorous proof for this because this is a homework problem)
Prove: If $X_\alpha$ follows the Poisson distribution $\pi(\alpha)$, then
$$\lim_{\alpha\rightarrow\infty}P\{\frac{X_... |
H: How to find this limit:
The question is to find this limit without using H.R (Hopital Rule):
$$\lim_{x\to 1}\frac{x^4 -1}{x^3 -1}$$
So this will be $\frac{0}{0}$
Which is indetermined but using H.R we find it is
$\frac{4}{3}$
But I need another method to get the answer so how could I get it?
AI: You should deco... |
H: Triple Integral with bounds in first octant
I am really confused on how to get my integrating function because I don't know, even after graphing, how the tetrahedron intersects the x-y-z axis.
I am supposed to find the triple integral for the volume of the tetrahedron cut from the
first octant by the plane $6x + 3... |
H: How to find the limit of an algebraic function
The question is to find this limit:
$$\lim_{x\to\infty}\frac{2x^\frac{5}{3}- x ^\frac{1}{3}+7}{x^\frac{8}{5} +3x + \sqrt{x}}$$
I need any hint to help since I tried so much and couldn't solve it.
AI: As $x$ tends to infinity, the dominant term of polynomial is the on... |
H: Unable to understand combination of quantifiers and set notation
I know what universal and existential quantifiers are but following is confusing,may be its comibination of set notation and quantifers. What does the following statement means?
∀xP(x)
AI: For all $x$, the statement $P(x)$ is true.
It helps to separat... |
H: How to solve this integral: $\int_{-1}^{1} x^k (1-x^2)^{(n/2)-2} \, dx$
How to solve this integral step by step:
$$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx=??? $$
In my text book, it shows the result like below:
$$\int_{-1}^{1} (x^k) (1-x^2)^{(n/2)-2}dx= \frac{(x^{(1+k)} ~_2F_1((1+k)/2, ~2-n/2, ~(3+k)/2, ~x^2))}... |
H: Conditional Probability/ Bayes' Theorem puzzle
I always believed that problems on conditional probability could be solved with common logic without using Bayes' theorem (because I cannot understand Bayes' theorem intuitively and I didn't bother because I knew another way).
But this problem gives me varying answer... |
H: What's the difference between isomorphism and homeomorphism?
I think that they are similar (or same), but I am not sure.
Can anyone explain the difference between isomorphism and homeomorphism?
AI: Homomorphism - an algebraical term for a function preserving some algebraic operations. For a group homomorphism $\ph... |
H: Cauchy integral formula for derivatives intuition
A nice way to remember a formula is to connect it to something you already know. For example, to remember the Cauchy integral formula, I remember that $f(z_0)=\frac{1}{2\pi i}\int_\Gamma\frac{f(z)}{z-z_0}\,dz$ because if you write out the Laurent series for $\frac{f... |
H: Please help me with this Integral
Calculate the integral (complex):
$$\oint_{D(0,1)}\overline ze^z \mathrm dz$$
While $D(0,1)$ is the unit circle.
AI: Collecting the hints from the comments in an answer:
On the unit circle, we have $\overline{z} = \dfrac{1}{z}$. Hence the integral can also be written
$$\int_{\lve... |
H: Show that $|\sin\frac{1}{n^2}|<\frac{1}{n^2}$, $n=0, 1, 2, \dots$
As part of showing that
$$
\sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right|
$$
converges, I ended up with trying to show that
$$
\left|\sin\left(\frac{1}{n^2}\right)\right|<\frac{1}{n^2}, \quad n=1, 2, 3,\dots
$$
since I know that the su... |
H: Prove that there is no homomorphism from $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ ont0 $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$
Prove that there is no homomorphism from $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ ont0 $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$.
My idea for the proof : Let $\phi$ be such homomorphism. Since Ker ... |
H: if the matrix such $B-A,A$ is Positive-semidefinite,then $\sqrt{B}-\sqrt{A}$ is Positive-semidefinite
Question:
let the matrix $A,B$ such $B-A,A$ is Positive-semidefinite
show that:
$\sqrt{B}-\sqrt{A}$ is Positive-semidefinite
maybe The general is true?
question 2:
(2)$\sqrt[k]{B}-\sqrt[k]{A}$ is Positive-semide... |
H: Average value of double integral
I am trying to work on this homework problem but I am lost. The only thing I know to do here is divide the double integral by 8 or multiply by 1/8 since area = l*w and 4*2=8.
But now, please help me understand how I can convert the C to x and y and get the bounds for integration. I ... |
H: The question related to a regular surface.
Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$.
I know these following definitions:
(1) a smooth surface $S$ is regular if for any ... |
H: Throwing balls onto cliffs
A man throws a ball from the ground towards the top of a cliff. Suppose that as soon as the ball reaches its maximum height, the ball lands on top of the cliff.
Let time t=0 be the moment the ball is released from the man's hand. For simplicity's sake, let the ball reach a maximum height... |
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