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H: How to correctly represent a nested sum
Suppose I have a matrix:
$$
A = \begin{pmatrix}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9 \\
\end{pmatrix}
$$
For which i want to sum the elements.
My first inclination is to write:
$$ \sum_j \sum_i A_{i,j} $$
But I am unsure whether this is ... |
H: How many seven digit numbers are there that are divisible by eleven?
How many seven-digit numbers are there that are divisible by 11?
In other words, I want to find the number of seven digit numbers that are divisible by 11.
AI: Well, the last number divisible by 11 before 7 digits is 999999, so the answer is $$\l... |
H: Solve some unusual log/exponential equations
I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course):
$ 4^x = x + 10$
$x^x = 3$
$(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$
Are there specific algebraic steps that I can follow so that I solve them? Or... |
H: The definition of a sequence converging.
The sequence $\{a_n\}$ converges to t he number L if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$,
$$n > N \implies |a_n - L| < \epsilon$$
If no such number $L$ exists, we say that $\{a_n\}$ diverges. If $\{a_n\}$ converges to L... |
H: Bayes Rule Probability question??....
A survey of mathematics students at the college revealed that 44% consistently spent at least 1.5 hours on mathematics homework and 56% spent less. Of those who spent at least 1.5 hours on homework, 76% made an A or B in the course. Of those who spent less than 1.5 hours, 27% m... |
H: Probability question?.
Here is an estimate of the number of credit-card holders.
Age group % of population % of group having credit cards
______________________________________________________________________
Under 35 49.2 64.1
35-64 38.1 77.7
65 or over ... |
H: real analysis inquiry
LEt $(X, \mathcal{F})$ be measurable space equiped with measure $\mu$. Suppose $\{ F_n \}$ are in $\mathcal{F}$. Put $A = \bigcap_{n=1}^{\infty} ( \bigcup_{i \geq n} F_i ) $. Does it follow that
$$ \mu(A) \leq \mu\left( \bigcup_{i \geq n } F_i \right)$$
Is it required that the $F's$ be pairwi... |
H: Basic proof Galois Theory
i was looking a proof of the following:
Let $k\subset F, F\subset K$ be Galois field extensions,
then $Gal (K/F)$ is a normal subgroup of $Gal(K/k)$.
I understand the proof but it start using the fact
that $k\subset K$ is Galois and i can't see it.
Is it trivial?
Thanks in advance.
AI: ... |
H: Solving for $x$ in an equation involving rational powers
$128000-256x^{3/4}\left(\frac{256}{625x}\right)^{1/4}=0$
I need help doing the problem.
The answer is $625$.
I started off by isolating $128000,$ but now I'm stuck.
AI: As you said, we first add $-256x^{3/4}\left( \frac{256}{625x}\right)^{1/4}$ to both sides ... |
H: Solve $z^4 + 4 = 0$
I'm trying to solve it by using its polar form, but then I get
$$
\begin{align*}
z^4 &= (\rho e^{i\phi})^4 = \rho^4 e^{4i\phi}\\
&= -4 = -4 e^{0i}\\
\end{align*}
$$
From the definition of equality of complex numbers, $\rho^4 = -4$ and $4\phi = 0 + 2\pi k$ for some $k \in \mathbb{Z}$.
This w... |
H: How to tell if a map is a linear map?
Can someone run me through the process of showing whether a map is a linear mapping or not. For an example I have:
$T:\mathbb{R}^2 \to \mathbb{R}^2, T(x,y)=(x-y^2, 5x)$
I am aware that it must satisfy the conditions of:
1) $f(x+y)=f(x)+f(y)$
2) $f(kx)=kf(x)$
However I don't rea... |
H: Conditional probability questions??
At the Campus Coffee Nook, 55% of the customers order regular coffee and 45% order flavored coffee. Of those who order regular coffee, 55% drink it black and 45% use sugar or cream. Of those who drink flavored coffee, 15% drink it black and 85% use sugar or cream.
I know that P(... |
H: Finding Powers of Relations
I have been trying to work on this question and this up to were I was able to go, but I am stuck and I do not know if I am going the right way.
AI: Hint: From $x-y=c$ and $y-z=c,$ we cannot conclude that $x-z=c.$ However, we can we conclude that $x-z=2c.$ (Hence, we can only conclude tha... |
H: Evaluating $\int_{x=0}^{x=\sqrt6}\int_{y=-x}^{y=x}\mathrm{d}y\,\mathrm{d}x$
How would you evaluate this integral?
$$
\int_0^{\sqrt6}\int_{-x}^x\mathrm{d}y\,\mathrm{d}x
$$
AI: Just integrate first with respect to $y$ and then with respect to $x$:
$$
\int_0^\sqrt{6}\int_{-x}^x dy\,dx=\int_0^\sqrt{6}\left[y\right]_{y=... |
H: Fourier series coefficients proof
Can somebody help me understanding the fouries series coefficients?
I know that if we have:
$$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$
And because of the $\sin(a+b)$ formula:
$$ f(t) = \sin(2\pi nt + Ph) = \sin(2\pi nt )\cos(Ph_n) + \cos(2\pi nt )\s... |
H: Mean and variance for a Binomial distribution.
I found a proof http://www.proofwiki.org/wiki/Expectation_of_Binomial_Distribution but the question here asks to consider differenciation (see hint) how do I do that?
$$E[X] = \sum_{x=0}^\infty x \left(
\begin{array}{c}
n \\
k
\end{array}
\right)
p^{x}(1-p)^{n-x}$$
... |
H: Integrating this complicated integral for statistics
I want to show that :
$$ \int_{-\infty}^{\infty} e^\frac{-u^2}{2} du = \sqrt{2\pi} $$
Is there an elementary way using the tools of Calculus II to do this type of integration? I have not studied numerical analysis yet.
AI: The usual trick is to let $I=\int_{-\i... |
H: Probability of index at which sequence stops decreasing
Let $X_1,X_2, \dots $ be a sequence of independent and identically
distributed continuous random variables. Let $N \ge 2$ be such that
$X_1 \ge X_2 \ge X_{N-1} < X_N$. That is, $N$ is the point at which
the sequence stops decreasing. Find $P\{N \ge n... |
H: Notation for an eigenspace?
The set of $\lambda$ satisfying
$$
(A - \lambda I_n)\vec{x} = 0
$$
is called the eigenspace of a matrix $A$ corresponding to $\lambda$.
Now I want to write eigenspace = the set of eigenvalues $\lambda$ satisfying ... as a set. So I want to use notation like $\lambda_A$ or $E_A$ or even b... |
H: How to find the limit of $\frac{\ln(n+1)}{\sqrt{n}}$ as $n\to\infty$?
I'm working on finding whether sequences converge or diverge. If it converges, I need to find where it converges to.
From my understanding, to find whether a sequence converges, I simply have to find the limit of the function.
I'm having trouble ... |
H: Probability of winning multiple games
I would like to know a formula for the probability of winning a certain number of games.
What is the probability of me winning 2 games if I have a 1/3 chance of winning each time and I play 5 times?
What is the probability of me winning 2 or more games if I have a 1/3 chance of... |
H: Why is $\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)$?
Background: We are looking at the wave equation on $\mathbb{R}^n$ via the Fourier transform. If $u(x,t)$ solves $\Delta u = u_{tt}$ in $\mathbb{R}^n$, with $u(x,t) = f(x)$ at $t=0$ and $u_t(x,t) = g(x)$ at $t=0$, then I ... |
H: Prove that \frac{2n}{3n+7} converges (Question about my method of proof)
Show that the sequence $<\frac{2n}{3n+7}>$ converges. Find its limit.
This is the work I have currently:
$\displaystyle \lim_{n \to \infty} \frac{2n}{3n+7} = \frac{2}{3}$
Let $\epsilon > 0$, and observe that:
\begin{align*}
\left|\frac{... |
H: Meaning of Quotient in this context
I was seeing the following problem a couple days ago:
Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected?
I understand $R\backslash F$ in the case where F is an equivalence relation but in this i... |
H: Different kinds of marbles
Ben Chusey walks into a marble store wanting to buy 12 marbles for his kid. The store has red, blue, green, yellow, and orange marbles.
How many different kinds of marbles can he select?
How many different kinds of marbles can he select which he has at least one of each kind?
I dont under... |
H: Laplace Transform of the product of two functions
What is the Laplace Transform of the product of two functions? Specifically, my function is $$\sin(5t) \cdot \cos(5t)$$, but I'd like to know a general principle if it's available. What's the easy way to compute this with a general formula without integrating?
AI:... |
H: question about partitions
Im trying to solve the following exercise. Im including a definition.
My attempt:
Let $\mathcal{Q}$ be a partition of $\Omega$ such that $\mathcal{Q}$ refines both $\mathcal{P}_i $ for $i=1,2$. So by definition, if we take arbitrary $A \in \mathcal{Q}$, then can write
$$ A = \bigcup_{i=1... |
H: condition of changeability of function
Let $(f_n)$ be a sequence in $\Bbb R \to \Bbb R$ that converges to a continuous function $f(x)$. Is it true that $\lim_{x \to a} f(x) = \lim_{n \to \infty} f_n (a)$?
AI: Since $f$ is continuous you have $\lim_{x \to a}f(x)=f(a)$ by definition. And also the definition of $f_n \... |
H: How do you validate that two math expressions are equal?
Let's say you have a few expressions like the following:
$$\begin{array}((x+17)^2 \\
x^2 + 34x + 289 \end{array} \\
288 + \frac{x^2}{2} + \frac{x^2}{2} + 34x + 1 \\
[...]
$$
You get the idea: there's an infinite number of ways of representing the same express... |
H: How find this matrix value of this $\det(A_{ij})$
Find this value
$$\det(A_{n\times n})=\begin{vmatrix}
0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\
a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\
a_{3}+a_{1}&a_{3}+a_{2}&0&\cdots&a_{3}+a_{n}\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
a_{n}+a_{1}&a_{n}+a_{2}&a_{n}+a_{3... |
H: Compactness and the suspension of a topological space
I would like to prove the following statement: A topological space $X$ is compact if and only if its suspension $SX$ is compact.
The proof in one direction is pretty easy: If $X$ is compact, then $SX$ is compact since products of compact sets are compact, and qu... |
H: Let $R$ be the relation on $\mathbb Z^+ \times \mathbb Z^+$ such that $(a, b)R(c, d)$ if $gcd(a, b) = gcd(c, d)$?
I need to find out:
Prove that $R$ is an equivalence relation.
(I am not clear on definition of an equivalence relation)
What is the equivalence class of $(1,2)$?
Give an interpretation of the equivale... |
H: Compute this line integral $\int_{C}(x+y){\mathbf{i}}+(x-y){\mathbf{j}}\ d\alpha $
Calculate the line integral of the function $$f(x,y)=(x+y){\mathbf{i}}+(x-y){\mathbf{j}}$$ around the ellipse $$b^2x^2+a^2y^2=a^2b^2$$ counter to clockwise.
My approach is the following: The equation can be written in the form $$\f... |
H: How many ordered triples of integers which are between 0 and 10 inclusive do we actually have if $a * (b+c) = a * b +c$
How many ordered triples of integers $(a,b,c)$ which are between 0 and 10 inclusive do we have if:
$a * (b+c) = a * b +c$
AI: $a*b+a*c=a*(b+c)=a*b+c$ if and only if $a*c=c$ by cancellation. We se... |
H: Relationship between Nilpotent Matrix and Matrix with all zero diagonal factors.
solving Linear Algebra HW, I suddenly became curious about the relationship between Nilpotent Matrix and matrix with all zero diagonal factors such that $A_{11} = A_{22} = \cdots = A_{nn} = 0$
Does Nilpotent Matrix implies the matrix w... |
H: Bounded variation function is countinuous except at countably many points
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 $N$.
I want to show that a BV function $f$ is co... |
H: Question about $(A - \lambda I_A)\vec{x} = 0$.
Finding a solution to $C\vec{x} = (A - \lambda I_A)\vec{x} = 0$ is the equivalent of considering the determinant of $C$ when it is zero. This means the matrix is linearly dependent and has infinite sol'ns.
My question is, the determinant is $zero$ when the matrix has n... |
H: About a domain of random variable $S_n=X_1+X_2+...+X_n$
I have a question about a random variable $S_n=X_1+X_2+...+X_n$ in the probability theory.
Assume that $X_k$ is a random variable on $\Omega$ for each $k$ and that each $X_k$ has the same distributions.
In probability theory, we study a random variable $S_n=X_... |
H: Why the lower limit of this integral is 1?
I solve this differential equation using Mathematica. But I don't understand the solution.
Why the lower limit of this integral is 1?
I run:
$$\text{DSolve}\left[y'(x)+y(x)=Q(x),y(x),x\right]$$
the solution:
$$\left\{\left\{y(x)\to e^{-x} \int_1^x e^{K[1]} Q(K[1]) \, dK[1... |
H: $1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$
The sum $1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$ is equal,
a) ${1\over 1006}+{1\over 1007}+{1\over 1008}+\dots+{1\over 2013}$
b) ${1\over 1007}+{1\over 1008}+{1\over 1009}... |
H: $a,b\in\mathbb{R}\ni a
$a,b\in\mathbb{R}\ni a,b>0, a<b<{1\over a}$ and $$x=(a+{1\over a})-(b+{1\over b})$$
Then $a) x>0$
b) $x<0$
$c) x=0$
d) no such conclusion can be drawn about $x$
just confirm me that answer is $d$?
AI: $a<\frac{1}{a}$ so $a<-1$ or $0<a<1$. In the first case $y=t+\frac 1t$ is strictly increas... |
H: Recurrence relation - repeated substitution
I am having some trouble with solving a recurrence relation with repeated substitutions.
$$a_n = 3\cdot2^{n-1}-a_{n-1}$$
I show some work:
$$a_n = 3\cdot2^{n-1} -(3\cdot2^{n-2}-a_{n-2})=3\cdot2^{n-1}-3\cdot2^{n-2}+a_{n-2}$$
Then I guess the pattern looks like this:
$$a_n ... |
H: Matrix representaiton of linear operator by inner product
Let $V$ be a finite-dimensional inner product space, and let $\beta = \{\alpha_{1}, \cdots \alpha_{n}\}$ be an orthonormal basis for $V$. Let $T$ be a linear operator on $V$ and $A$ the matrix of $T$ in the ordered basis $\beta$. Prove
$$
A_{ij} = (T\alpha_... |
H: Left-continuous function defined on measure
Let $\mu$ be a measure on $\mathbb{R}$ that assigns a finite non-negative number to each compact set. Show that there exists a left-continuous function $f$ such that for $-\infty<a<b<\infty$, we have $$\mu((a,b))=\lim_{\epsilon\rightarrow 0^+}(f(b-\epsilon)-f(a+\epsilon)... |
H: Constant coefficients and intuition behind multiplication by $x^m$
Short (well...) version of the question
I am having a bit of a problem understanding one thing about the solution to non-homgeneous second-degree linear differential equations using the constant coefficients method.
Given a DE of the form:
$$
a_2y''... |
H: Is there a simple method to finding orthonormal basis given a partially complete set
I have a question
Find the indicated projection matrix for the given subspace, and find
the projection of the indicated vector $<2,-1,3>$ on
$sp(<2,1,1>,<-\frac{8}{6},\frac{11}{6},\frac{5}{6}>)$ $ R^3$
I feel like there shou... |
H: Each open cover of a sequentially compact metric space has Lebesgue number
I want to query, whether I'm right. (I'm sorry if don't use the correct words in my translation, please feel free to correct, and give me hints.)
I have a metric space $(X,d)$ which is sequentially compact (that means every sequence has a co... |
H: Prove projection is self adjoint if and only if kernel and image are orthogonal complements
Let $V$ be an IPS and suppose $\pi : V \to V$ is a projection so that $V = U \oplus W$ (ie $ V = U + W$ and $U \cap W = \left\{0\right\}$) $ \ $ where $U = \ker(\pi)$ and $W = \operatorname{im}(\pi)$, and if $v = u + w \ $ ... |
H: we need to find $m+n$.
I just dont understand this question, could any one tell me how to solve this one?
A pen costs $13$ dollar and a notebook costs $35$ dollar, let $m$ be the maximum number of items that can be bought for $1000$ dollars and $n$ be the minimum number of items that can be bought for the same am... |
H: Question about topological spaces
Let's mark the standard topological space on $\mathbb{R}$ with $\tau$.
We'll define new topology on $\mathbb{R}$, $\tau_l$ with the following base:
$B_l = \{[a,b)|a,b \in \mathbb{R},a<b\}$
I have to prove that a function $f:(\mathbb{R}_l,\tau_l)\to(\mathbb{R},\tau) $ is continuos i... |
H: change of unit normally distributed random variable
Assume that $X_{1}$,$X_{2}$,$X_{3}$ are independent continues random variables with $\mathcal{N}(30,12)$, what is the normal distribution of $X_{average}$ (average of $X_{1}$,$X_{2}$,$X_{3}$)
solution given by teacher:
$X_{average} = 1/3(X_{1}+X_{2}+X_{3})$ the... |
H: Why can't I simply use algebra to solve this inequality?
Consider the inequality:
$\frac{(x+3)(x-5)}{x(x+2)}\geq 0$
Why can't I simply multiply both sides by $x(x+2)$ and get $(x+3)(x-5)\geq 0$ ?
Which would yield: $x^2-2x-15\geq 0$ and I could then use the quadratic formula to derive the answer..?
This seems algeb... |
H: Householder matrix confusion
I read that: If $(I-2ww^T)x=y$ and $x \neq 0$ ($w^Tw=1$) then
$$w= \dfrac{(x-y)}{\|x-y\|_2}.$$
I tested this for $x=[9,2,6]^T$ and $y=[-11,0,0]^T$ and it worked.
But for some reason for $x=[1,2,3]^T$ and $y=[1,0,0]^T$ it doesn't work... the result was $[1,-2,-3]^T$ instead of $[1,0,0]^T... |
H: Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.
Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.
I've proven that $X^4 + X + 1$ is irreducible, so $L$ is a field. I also know ... |
H: Problems on Sylow Theorems
Let $G$ be a finite group, let $p\in\mathbb{N}$ be a prime and let
$$(ab)^p=a^pb^p,~~ \forall a,b\in G$$
Prove that $G$ has a unique Sylow $p-$subgroup.
AI: By the Sylow theorems, $a \in G$ is an element of some sylow $p$-subgroup of $G$ if and only if $a^{p^n} = 1$ for some $n>0$. Cons... |
H: Prove $|e^{i\theta} -1| \leq |\theta|$
Could you help me to prove
$$
|e^{i\theta} -1| \leq |\theta|
$$
I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?
AI: By the fundamental theorem of calculus
$$e^{i\theta}-1=\int_0^\theta ie^{it}\mathrm{d}t$$
Hence..... |
H: Is this set infinite?
If we say that $B = L_1 \cup L_2$, $L_1 \cap L_2 = \emptyset$,also $B, L_1,L_2$ are infinite and we are given that $A \subset L_1$ and $B \backslash A$ is infinite, does that say that $L_1 \backslash A$ is infinite?
AI: Not necessarily. Take $A=L_1$. |
H: Borel sum of $ 1!+2!+3!+.... $
I know that the Borel sum of $ \sum_{n=0}^{\infty}(-1)^{n}n! $ is $ \int_{0}^{\infty} dx \frac{e^{-x}}{1+x} $
but what happens with the sum $ \sum_{n=0}^{\infty}n! $
the Borel sum should be $ \int_{0}^{\infty} dx \frac{e^{-x}}{1-x} $ which has a pole at $ x=1 $ using Shothotsky's fo... |
H: About the continuity of a function in the closed graph theorem proof
I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. Thanks
$X ,Y$ are $F$-spaces, $f: X \rightarrow... |
H: $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle
I am completely lost in this one
$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
AI: From equating first and third equations, you get $y=x+1$. Use this in another two of the equations to get value... |
H: If X is a random variable how do we show that $E(|X|)=0 \iff P(X=0)=1$
If X is a random variable how do we show that $E(|X|)=0 \: \iff \: P(X=0)=1$
I see that $-|X|\le X \le |X|$ and so $|E(X)| \le 0$ and thus $E(X)=0$ but how do I show that this implies $P(X=0)=1$.
AI: Well, $|X|\geq 0$. So if $|X|$ is positive wi... |
H: Trigonometric Limit: $\lim_{x\to 0}\left(\frac{1}{x^2}-\frac{1}{\tan^2x}\right)$
I cannot figure out how to solve this trigonometric limit:
$$\lim_{x\to 0} \left(\frac{1}{x^2}-\frac{1}{\tan^2x} \right)$$
I tried to obtain $\frac{x^2}{\tan^2x}$, $\frac{\cos^2x}{\sin^2x}$ and simplify, and so on. The problem is that ... |
H: Weird sequence of numbers
Supply the missing number in the following sequence:
10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ___, 100, 121, 10.000.
I've spent like 1,5 hours on this weird sequence, but still haven't worked out complete algorithm of n-th element of this sequence. Any ideas?
AI: The missing number is... |
H: Is this function injective / surjective?
A question regarding set theory.
Let $g\colon P(\mathbb R)\to P(\mathbb R)$, $g(X)=(X \cap \mathbb N^c)\cup(\mathbb N \cap X^c)$
that is, the symmetric difference between $X$ and the natural numbers.
We are asked to show if this function is injective, surjective, or both.
I ... |
H: Limit of strange function composition
Known facts.
$f$ and $g$ are continious and exist in the neighbourhood of $x=0$.
$\lim_{x\to 0}a(x)=\lim_{x\to 0}b(x)=0$.
$\lim_{x\to 0} \frac{f(x)}{g(x)}$ exists.
$\lim_{x\to 0} \frac{a(x)}{b(x)} = 1$
Problem.
I want to investigate and possibly also prove the statement
$$\l... |
H: Compute $H^1(X,\Bbb{Z}_U)$
Let $X = \mathbb{A}^1_k$ with $k$ infinite and $U = X - \{P,Q\}$ and $\mathbb{Z}_U= i_{!}(\mathbb{Z}|_U)$, $\Bbb{Z}$ the constant sheaf. I want to say that $H^1(X,\mathbb{Z}_U) \neq 0$. If it is zero we see exact sequence
$$0 \to H^0(X,\mathbb{Z}_U) \to H^0(X,\mathbb{Z}) \to H^0(X,\mathbb... |
H: Example of functions that grow faster than the exponential functions and/or factorial functions?
What is example of functions that grow faster than the exponential functions and/or factorial functions?
AI: The busy beaver function can be shown to grow faster than any computable function. |
H: Are convex function from a convex, bounded and closed set in $\mathbb{R}^n$ continuous?
If I have a convex function $f:A\to \mathbb{R}$, where $A$ is a convex, bounded and closed set in $\mathbb{R}^n$, for example $A:=\{x\in\mathbb{R}^n:\|x\|\le 1\}$ the unit ball. Does this imply that $f$ is continuous? I've searc... |
H: Prove using mathematical induction that $2^{3n}-1$ is divisible by $7$
So, i wanna prove $2^{3n}-1$ is divisible by $7$, so i made this:
$2^{3n}-1 = 7\cdot k$ -> for some $k$ value
$2^{3n+1} = 1+2\cdot1 - 2\cdot1 $
$2^{3n+1} - 1-2\cdot1 + 2\cdot1 $
$2^{3n}\cdot2 - 1-2\cdot1 + 2\cdot1$
$2(2^{3n}-1) -1 +2$
$2\cdot7k+... |
H: Changing lower limits integral.
$\displaystyle\int_2^\infty\dfrac1{(x-1)^3}\,\mathrm dx\quad$ Let $u=x-1 \\ \mathrm du=\mathrm dx$
$\displaystyle=\int_1^\infty\dfrac{\mathrm du}{u^3}=\lim_{R\to\infty}\int_1^R\dfrac{\mathrm du}{u^3} \\\displaystyle
=\lim_{R\to\infty}\dfrac{-1}{2u^2}\Bigg|_1^R=\lim_{R\to\infty}\left... |
H: A question on an Erdős proof
I am currently going through a proof by Erdős and I am having difficulty understanding one of his arguments. He first gives the following Lemma:
If ${n}\choose{k}$ is divisible by a prime power $p^a$, then $p^a \leq n$.
Now he says this:
Let $\pi(k)$ denote the number of primes less tha... |
H: Show that in a ring with $u^2 = 0$, $1 + u$ is a unit
I eventually worked this out through trial and error and found that $(1 + u)(1 - u) = 1$
But is there a more calculated way of determining this? I just kept trying different values until I found one that satisfied the equation, it seems a very inefficient way of... |
H: Bounded variation functions have jump-type discontinuities
I read on the Wikipedia page that bounded variation (BV) functions have only jump-type discontinuities. Why is that? Suppose at some $a\in\mathbb{R}$, the limit $\lim_{x\rightarrow a^+}f(x)$ doesn't exist (or is infinite). Why would such a function $f$ not ... |
H: Solve equation with logarithm
Let $f(x)$ be some distribution function.
Let $a\in \mathbb{R}$ and $b>0$.
Find $a$ and $b$, such that
$$
\ln f(x)=a+bf^{1/2}(x),
$$
in addition, it is known that if $f_*(a,b)$ is a solution for the above equation, then $\int f_*(x)dx=1$.
Thank you.
AI: Suppose $f(x)$ is the distributi... |
H: GCD and LCM of three numbers
Given two positive integers G and L, could you tell me how many solutions of (x, y, z) there are, satisfying that gcd(x, y, z) = G and lcm(x, y, z) = L? gcd(x, y, z) means the greatest common divisor of x, y and z, while lcm(x, y, z) means the least common multiple of x, y and z. Also, ... |
H: Find the dimensions of a cylinder of given volume V if its surface area is a minimum.
The following is the question :
Find the dimensions of a cylinder of given volume V if its surface area is a minimum.
The cylinder has a closed top and bottom.
2 formula :
(1) $V=r^2\pi h$
(2) $A=2r\pi h+2r^2\pi$ -> $A=2r\pi \... |
H: Differentiating under the integral sign chain rule
Can someone explain to me why
$$
\frac{\partial}{\partial x}\int_{0}^{x\nu}u^{c - 1}{\rm e}^{-u/2}\,{\rm d}u
=
\left(\nu x\right)^{c - 1}{\rm e}^{-\nu x/2}\,\nu\quad {\large ?}
$$
I know it has to do with the application of Chain rule to the integral, but I'm not u... |
H: What is a coordinate shifting?
I need to find the limit:
$$\lim_{(x,y,z)\to(1,3-1)}\frac{(x-1)(y-3)+(z+1)^2}{(x-1)^2+2(y-3)^2+3(z+1)^2}.$$
A hint written below says: Perform a coordinate shifting to (0,0,0).
What does coordinate shifting means, and how should I use it in this case?
AI: The suggestion is to shift yo... |
H: Show that every operator norm is consistent
Is the following a correct way to show that operator norms are consistent?
$$
\|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} \frac{\|Bx\|_\alpha}{\|x\|_\alpha}\le \max_{y \ne 0} \frac{\|Ay\|_\alpha}{\|y\|_\a... |
H: How to calculate anti-log using calculator?
I have a calculator that does not have antilog function. All it has is log to base 10 and natural log functions.
I was wondering if it is possible to calculate antilog using the log to base 10 function. Can this be done ? I am only concerned about log to base 10 and ant... |
H: Why can any type be realized?
I couldn't find this question asked previously, which means it's probably an especially daft question.
Given an $\mathcal{L}$-structure $\mathcal{M}$, my textbook defines an $n$-type over $A\subseteq M$ to be a set $p$ of sentences all in the same $n$ free variables such that $p\cup Th... |
H: Dual Vector Spaces with Orthonormal Basis
I'm really stuck on the following quesiton.
Let $U$ and $V$ be finite dimensional vector spaces over the complex numbers with bases $e_1,..,e_n$ of $U$ and $f_1,...,f_m$ of $V$. They also have dual spaces $U^*$ and $V^*$ with bases $e^i$ and $f^i$ respectively.
Then assume ... |
H: Entropy of a distribution over strings
Suppose for some parameter $d$, we choose a string from the Hamming cube ($\{0,1\}^d$) by setting each bit to be $0$ with probability $p$ and $1$ with probability $1-p$. What is the entropy of this distribution on the Hamming cube? Clearly, if $p=\frac{1}{2}$, then the entropy... |
H: Formality and mathematics
Why is it important to be formal in mathematics? Is formality beneficial for students? Or is it just to scare students away from mathematics?
AI: I will not be so ambitious as to write a full answer to this question. That would require a table of contents listing lots of long chapters.
If... |
H: prove that the following sequences are bounded from above
let $a_n = \frac{1}{3^1+1} + \frac{1}{3^2+1} + ... + \frac{1}{3^n+1}$
$b_n = \frac12 + \frac1{2 * 4} + ... + \frac1{2*4*...(2n)}$
its easy to show the 2 sequences are monotone rising, but how do i prove that they are bounded from above?
i couldent find ano... |
H: $\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$
Is it true that
$\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$
I know that for an adapted process $\Delta(t), t\geq 0$, the integral
$\int_0^t \Delta(u)dW(u)$ is a martingale in $t$, and therefore had expectation $0$, if
$\mathbb{E} \int_0^T \Delta^2(t) dt... |
H: name of matrix of inner products $\langle f_i, f_j\rangle$
Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with
$$
M_{i,j} := \langle\phi_i, \phi_j\rangle
$$
have any particular name?
AI: That matrix is called the Gram-matrix of the vectors $\{ \phi_i \}$. |
H: How to calculate the tangent angle with the axis of an ellipse
I hava an ellipse. I know its' equation.
For a given 't' how can I calculate angle "alpha". (I also know the coordinate of the tangent point)
AI: The tangent is in the direction of
$$\left(\frac{\mathrm{d}x}{\mathrm{d}t},\frac{\mathrm{d}y}{\mathrm{d}t... |
H: Combinatorial proof with binomial coefficients
I need to prove this with combinatorial arguments. I don't know how to start.
$$
\sum_{j = r}^{n + r - k}{j - 1 \choose r - 1}{n - j \choose k - r}
=
{n \choose k}\,,
\qquad\qquad 1\ \leq\ r\ \leq\ k\ \leq\ n
$$
AI: On the right hand side, I need to choose $k$ integ... |
H: Derivative of a map involving the matrix inverse
I have $f: U\rightarrow \mathbb{R}$, $f(X):=\operatorname{tr}(X^{-1})$, $U$ contains all matrices $X$, which are positive definite and symmetric. I want to show that $f$ is differentiable on $U$.
To do so, I have to figure out
$f(X+tY)=\operatorname{tr}[(X+tY)^{-1}... |
H: How to find a sequence by its limit?
Is there any way to construct non-trivial sequence by its limit? Something like
$\begin{cases}
a_1=2 & \\
a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right)
\end{cases}$for $\sqrt2$. I'm especially interested in square roots, trigonometric functions and alike. By non-trivial i mea... |
H: Solve the recursion $a_n=\frac{1}{4}2^{n-1}-1+3a_{n-1}$
Solve the recursion $a_n=\frac{1}{4}2^{n-1}-1+3a_{n-1}$
I'd know how to solve it if it weren't for that -1. Because of it, I can't divide the particular equation with $2^{n-2}$ to solve it. What can be done here?
Oh, sorry. The starting conditions are $a_3=1$,... |
H: Continuity of measures on intersection of intervals
Suppose we have a measure $\mu$ on the real line, and $a\in\mathbb{R}$. Is it necessarily true that $$\mu((-\infty,a])=\lim_{\epsilon\rightarrow 0}\mu((-\infty,a+\epsilon))?$$
This looks related to the continuity of measures, but to apply continuity of measures on... |
H: Addition and subtraction with exponents
I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did:
http://puu.sh/5w3XQ.png
What I'm wondering is about the 2^4 - 2^3. I know this is is 2^3 because 2^3 is the value multiplied by 2 to ge... |
H: Normal Operator that is not Self-Adjoint
I'm reading Sheldon Axler's "Linear Algebra Done Right", and I have a question about one of the examples he gives on page 130. Let $T$ be a linear operator on $F^2$ whose matrix (with respect to the standard basis) is $$\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix}$$ I can se... |
H: integer $m$ has primitive root if and only if the only solutions of the congruence $x^{2} \equiv 1 \pmod m$ are $x \equiv \pm 1\pmod m$.
Show that the integer $m$ has primitive root if and only if the only solutions of the congruence $x^{2} \equiv 1 \pmod m$ are $x \equiv \pm 1\pmod m$.
I don't quite understand wha... |
H: Continuity of measure finite on compact sets
This is related to this question.
Suppose we have a measure $\mu$ on the real line, and $a\in\mathbb{R}$. Suppose also that $\mu$ is finite on every compact subset of $\mathbb{R}$. Is it necessarily true that $$\mu((-\infty,a])=\lim_{\epsilon\rightarrow 0}\mu((-\infty,a+... |
H: vector subspaces of $(\mathbb Z/2\mathbb Z)^3$
How many possible vector subspaces of $(\mathbb Z/2\mathbb Z)^3$ are there?
My idea was, to proove this as follow:
$$U_b := \left\{\left(\begin{matrix}\lambda_1\\\lambda_2\\\lambda_3\end{matrix}\right) \Bigg|\, a\lambda_1 +b\lambda_2+c\lambda_3+d=0\right\}$$
For $a, b,... |
H: How can I prove that this sequence is bounded?
I need to prove that this sequence is bounded:
$${b_n} = {1 \over 2} + {1 \over {2*4}} + ... + {1 \over {2*4*2n}}$$
Any help would be appreciated!
AI: $${2 \over {2*4*..*2n}} \leq {2 \over {(2n-2)*2n}}={1 \over {2n-2}}- {1 \over {2n}}$$
Thus
$$2b_n ={2 \over 2} + {2\o... |
H: Comprehension about definition of dense set
Dense set in Real Analysis, Carothers, 1ed was defined in his homework and captured below:
What's the meaning of empty interior? Examples will be appreciated^_^
Can I claim that (0,1) is dense in [0,1]?
AI: Empty interior simply means that there is no non-empty open set... |
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