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H: an exercise book for probability theory recommendation request
I'm looking for a good exercise book for probability theory, preferably at least partially with solutions to it. I want it to be detailed, not trivial, providing me solid fundamentals in the topic to be developed in the future. I'd wish it to be more of... |
H: A sensible/systematical way to deal with the following equation
Given that $y=x\varphi(z)+\psi(z)$ where $z$ is an implicit function of $x,y$, and $x\cdot\varphi'(z)+\psi'(z)\neq0$. Try to prove that
$$\frac{\partial^2z}{\partial x^2}\cdot\left(\frac{\partial z}{\partial y}\right)^2-2\cdot\frac{\partial z}{\partial... |
H: If $V$ is a neighbourhood of $a$ having $E= D \cap V$ , show that $a \in E'$
I'm having a little problem in understanding this question...
Let $f \colon D \rightarrow \mathbb{R}$ with $D \subset \mathbb{R}$ and $a \in D'$. If $V$ is a neighbourhood of $a$ and $E= D \cap V$, show that $a \in E'$.
The exercise a... |
H: errors, random and approximations
Hello good evening all!
If a reading is reported as R = 200.045 + 0.001 or 200.045 - 0.001 Ohm. Does +0.001 or -0.001 Ohm represents a systematic or random error?
Thanking you.
AI: It represents random error. Random error is a bit of 'spreading out', but systematic error means the... |
H: combinatorics counting sets with a one element having values in another set
counting the number of ordered subsets in a n-set is easy, it's $ 2 ^n -1 $ (assuming we don"t want the empty subset, it's the sum of $ n \choose i $ , i>0
now imagine I have a set s=(a, b, x), where x can have 2 possible values x1, x2
the ... |
H: Is an open linear map closed (to some extent)?
Suppose we have a surjective bounded linear operator acting between Banach spaces. By the Open Mapping Theorem it maps open sets in the domain to open sets in the codomain. Must the image of a closed linear subspace of the domain be closed then?
AI: A construction very... |
H: Cauchy in Norm and Weakly converge Implies Norm convergent
Let $X$ be a normed space and $(x_n)$ is a Cauchy sequence in the norm sense. Also assume the $x_n \rightarrow x_0 $ weakly. Then $x_n \rightarrow x_0 $ in norm.
What I did:Take $ \varepsilon >0 $ since $x_n$ is cauchy there a $n_0$ such that $ |\!| x_n-x... |
H: Does the sequence $\frac{n!}{2^n}$ converge or diverge?
Does the following sequence $\{a_n\}$ converge or diverge?
$$a_n=\dfrac{n!}{2^n}$$
AI: Consider writing "out" the sequence:
$$\tag 1\frac{{n!}}{{{2^n}}} = \frac{n}{2}\frac{{n - 1}}{2}\frac{{n - 2}}{2} \cdots \frac{4}{2}\frac{3}{2}\frac{2}{2}\frac{1}{2}$$
Not... |
H: Finding a conformal map to the upper half plane
Let
$$
\Omega = \{z \in \mathbb{C} : |z| > 1, z \notin \mathbb{R}_{< -1}, z \notin \mathbb{R}_{\ge 2}\}.
$$
Find a conformal map which maps the region $\Omega$ to the upper half plane.
I would want to know what I am supposed to do first. I tried to shift by $1$ to the... |
H: Calculating Points in Circles
I am trying to calculate the points around a circle with a certain distance in between each other.
Below is a graphical representation of what I am attempting to do.
The diameter is 70 (radius 35), and I want to find all the points that are 5 apart from each other.
AI: Do you want the ... |
H: Find all ordered pair of integers $(x,y)$
Obtain all ordered pair of integers $(x,y)$ such that $$x(x + 1) = y(y + 1)(y + 2)(y + 3)$$
I'm getting 8,
(0, 0), (0, -1), (0, -2), (0, -3)
(-1, 0), (-1, -1), (-1, -2), (-1, -3)
Please confirm my answer.
AI: Hint: It is easily proved that the product of four consecutive... |
H: Explanation of Proof of Zorn's lemma in Halmos's Book
I have been reading Halmos's book on naive set theory on my own and have got stuck in the Chapter on Zorn's lemma. The 2nd and 3rd paragraphs are not very clear to me. Here is the text:
Zorn's lemma. If $X$ is a partially ordered set such that every chain in $X... |
H: Common factors of the ideals $(x - \zeta_p^k)$, $x \in \mathbb Z$, in $\mathbb Z[\zeta_p]$
I'm trying to understand a proof of the following Lemma (regarding Catalan's conjecture):
Lemma:
Let $x\in \mathbb{Z}$, $2<q\not=p>2$ prime, $G:=\text{Gal}(\mathbb{Q}(\zeta_p):\mathbb{Q})$, $x\equiv 1\pmod{p}$ and $\lvert ... |
H: Is it possible to prove that the Grz axiom is valid in a modal frame iff the frame is reflexive and transitive?
We need to prove (or disprove?) that $ \square (\square (A \rightarrow \square A) \rightarrow A) \rightarrow A $ is valid in the Kripke modal frame $ F = <S, R> $ iff R is transitive and reflexive.
I thin... |
H: Finding $f'(0)$ when $f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt$
I need to show that $f'(0)=0$ for
$$
f(x)=\int\limits_0^x\sin\left(\frac{1}{t}\right)dt
$$
But fundamental theorem of calculus is unapplicable here. What should I do?
AI: Here is one approach:
Use integration by parts to show that $f(x)=x^2\... |
H: Which of these statements about $Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$ is correct?
Pick the correct option regarding $Q$.
$$Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$$
Pick one option:
$Q>1\qquad$ 2. $Q\leq \frac{1}{3}\qquad$ 3. $\frac{1}{3}<Q\leq \frac{2}{3}\qquad$ 4. $\frac{2}{3}<Q\le... |
H: About the stablizer of an element, $G_\alpha$
After reading some notes about permutation groups, I have tackled with this really simple question I hope it is not a ridiculous question. :)
From the first chapters of any book about these kinds of groups and knowing how a group $G$ acts on a set $\Omega$, we are face... |
H: Binomial sum - generating functions
Find a closed form for $a_n:=\sum_{k=0}^{n}\binom{n}{k}(n-k)^n(-1)^k$ using generating functions.
AI: We have
$$\begin{eqnarray*}
\sum_{k=0}^{n}(-1)^k \binom{n}{k}(n-k)^n
&=& \sum_{k=0}^{n}(-1)^k \binom{n}{n-k}(n-k)^n \\
&=& \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^n \\
&=& \le... |
H: How to define rational composition of functions, appropriately?
Let $f: X \rightarrow X$ and $f^{1/n}:X \rightarrow X$ such that $f$ is equal to $n$ times composition of $f^{1/n}$, i.e. $$f^{1/n} \circ f^{1/n}\circ \cdots \circ f^{1/n} = f $$
then $f^{1/n}$ is an $n$-th root of $f$.
Such a function may well not e... |
H: Is it true that the Laplace-Beltrami operator on the sphere has compact resolvents?
We consider the Riemannian structure on the sphere $\mathbb{S}^n$ seen as a submanifold of $\mathbb{R}^{n+1}$ and the Laplace-Beltrami operator defined on $C^\infty(\mathbb{S}^n)$ by the equation
$$\Delta f= -\operatorname{div}\oper... |
H: Are Complex Substitutions Legal in Integration?
This question has been irritating me for awhile so I thought I'd ask here.
Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals:
$$\int_{0}^{\infty} \sin x^2\, dx=\operatorname {Im}\left( \int_0^\infty\c... |
H: Does the series converge or diverge? $\sum_{n=1}^\infty\frac{4^n+n}{n!}$
Does the following series converge or diverge?
$$\sum_{n=1}^\infty\frac{4^n+n}{n!}$$
AI: $$\sum_{n=1}^{\infty} \dfrac{4^n + n}{n!} = \sum_{n=1}^{\infty} \dfrac{4^n}{n!} + \sum_{n=1}^{\infty} \dfrac{n}{n!} = \exp(4)-1 + \sum_{n=1}^{\infty} \df... |
H: Perfect square modulo $n = pq$
I've been stuck on this problem for a while. Any insights to the problem would be great!
We start with $n = pq$, where $p, q$ are distinct odd primes. In addition, $\gcd(a,n) =1$. If $x^2 \equiv a \pmod n$ has any solutions, then it has four solutions (where the book does not specify ... |
H: Simple equation solving/transformation
Im really stuck here, and feel quite dumb, because it looks so simple. However, I'm sure I'm overlooking something here.
I have given this equation:
$x=c_0y + c_1z$ where $x,y,z$ are some variables and $c_0,c_1$ are constants.
My goal is to get an equivalent equation in the fo... |
H: Given k % y, how can I adjust the dividend (k) to preserve the modulo when the divisor (y) is incremented by one?
In a programming algorithm, I'm using the result of k % y. I need to understand how to adjust k when the value of y is incremented by one to preserve the same modulo result.
In other words, solve for x... |
H: Tricky derivative with chain rule.
Let $z = 1/x$ and $y = f(z)$, find $\dfrac{d^2y}{dx^2}$
So the answer was $$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} z}\frac{\mathrm{d}z }{\mathrm{d}x }$$
Where $$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} z}\frac{\mathrm{d}z }{\ma... |
H: Finding a function and a sequence of recursively defined sets that gives a counterexample.
Let X be a set and $f:X\rightarrow X$. Define the sequence $(A_n)$ recursively by $A_1=X$ and $A_n=f(A_{n-1})$ for $n>1$. Let $A=\bigcap_{n\in N}A_n$. My problem is asking me to show $f(A)⊊A$. I have already shown $f(A)\subse... |
H: Find $\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$
$$\lim\limits_{x \to \infty} \frac{\sqrt{x^2 + 4}}{x+4}$$
I have tried multiplying by $\frac{1}{\sqrt{x^2+4}}$ and it's reciprocal, but I cannot seem to find the solution. L'Hospital's doesn't seem to work either, as I keep getting rational square roots.
... |
H: Smallest number in a set
A is the set of seven consequtive two digit numbers, none of these being multiple of 10. Reversing the numbers in set A forms numbers in set B. The difference between the sum of elements in set A and those in set B is 63. The smallest number in set A can be :
I tried to write some sets an... |
H: Is $M$ the midpoint of this line?
$${{\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet}\atop O \;\quad\quad M\quad\quad\; P}$$
Given that $OM = x + 8$, $MP = 2x - 6$, $OP = 44$, is $M$ the midpoint of $OP$?
AI: HINT $$OM + MP = OP$$
Move your mouse over the g... |
H: Please recommend books on calculus, linear algebra, statistics for someone trying to learn Probability Theory and Machine Learning?
I am tackling some topics in Probability Theory and Machine Learning and while I have plenty of resources dedicated to those disciplines I am lacking in a good basic math foundation.
D... |
H: Powers of $2 \times 2$ matrices expressed in linear form
I recently reopened an old high school math textbook and came upon the matrices unit. Some of the questions were those rewrite-in-linear-form problems: given, say, $M^2 = 2M - I$, express in linear form ($aM+bI$) the matrices $M^3$ and $M^4$. The method of so... |
H: how to find maximal linearly independent subsets
Given a set of vectors, we can compute the number of independent vectors by calculating the rank of the set, but my question is how to find a maximal linearly independent subset. Thanks!
AI: One method is:
Place the vectors as columns of a matrix. Call this matrix $... |
H: paradox of square matrix
I remember that every embedding is injective, and every projection is surjective. For square matrices, they're both embedding and projection, which means they're bijective, so they should be invertible. But obviously, not every square matrix is invertible. I don't which part is wrong in my ... |
H: Self-Linking Number on 3-Manifolds
We can assign a framing to a knot $K$ (in some nice enough space $M$) in order to calculate the self-linking number $lk(K,K)$. But of course it is not necessarily canonical, as added twists in your vector field can remove/add crossings.
Two things are stated in Witten's QFT paper... |
H: Proving $\sup\left\{ r\in\mathbb{Q}:r^{2}<3\right\}=\sqrt{3}$
Let $E=\left\{ r\in\mathbb{Q}:r^{2}<3\right\}$. Prove that $\sup E=\sqrt{3}$.
Since $E$ is bounded from above by $\sqrt{3}$ and is nonempty, $\alpha:=\sup E$ must exist by the Least Upper Bound Principle. Now I am stuck. Suppose $\alpha<\sqrt{3}$. ... |
H: Silly question about Fourier Transform
What is the Fourier Transform of :
$$\sum_{n=1}^N A_ne^{\large-a_nt} u(t)~?$$
This is a time domain function, how can I find its Fourier Transform (continuous not discrete) ?
AI: Tips:
The Fourier transform is linear; $$\mathcal{F}\left\{\sum_l a_lf_l(t)\right\}=\sum_l a_l\ma... |
H: How to solve the maximum subset sum problem?
I've just read this article about solving the below minimum subset sum using dynamic programming technique.
Given a list of $N$ coins, their values $(V_1, V_2, ... , V_N)$, and the
total sum S. Find the minimum number of coins the sum of which is $S$
(we can use as ... |
H: Flirtatious Primes
Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many?
AI: These are tabulated at the Online Encyclopedia of Integer Sequences. It appears to be known that there are infinitel... |
H: For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$
Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that strictly contains $F$. Why is $x$ algebraic over $K$?
Thanks a lot!
AI: The thing is you need to understand what are the subfields in between $F$ and $F(x)$... |
H: $f(x)=x^n+5x^{n-1}+3$ can't be expressed as the product of two polynomials
Let $f(x)=x^n+5x^{n-1}+3$ where $n\geq1$ is an integer. Prove that $f(x)$ can't be expressed as the product of two polynomials each of which has all its coefficients integers and degree $\geq1$.
If the condition that each polynomial must h... |
H: deciding whether $125$ is a primitive root modulo $529$
I'd like your help with the following: I need to show that $5$ is a primitive root modulo $23^m$ for all natural $m$ and to decide if $125$ is a primitive root modulo $529$.
For the first part I need to show that $5$ is a primitive root for $23$ ( I showed tha... |
H: Do I need to check solution while solving an equation?
I am facing a strange problem in solving the following equation for x and y
$$\frac{4x^2+(x^2+y^2-1)^2}{(x^2+(y-1)^2)^2}=1$$
$$4x^2+(x^2+y^2-1)^2=(x^2+(y-1)^2)^2$$
On solving the terms in brackets, we get
$$4y^3-8y^2+4y+4x^2y=0$$
$$4y(y^2-2y+1+x^2)=0$$
$$4y[(y-... |
H: Quadratic equation : Two solutions or one solution?
I have an equation to solve for y:
$$\frac{y^2}{y}=1$$
Normally, I would cancel out one $y$ and get $y=1$ as a single solution.
But If I think of it as quadratic equation
$$y^2=y$$
$$y^2-y=0$$
$$y(y-1)=0$$
$$y=0 \space \text{or} \space y=1$$
to have two solutions.... |
H: What does this mean: $\mathbb{Z_{q}^{n}}$?
I can't understand the notation $\mathbb{Z}_{q}^{n} \times \mathbb{T}$ as defined below. As far as I know $\mathbb{Z_{q}}$ comprises all integers modulo $q$. But with $n$ as a power symbol I can't understand it. Also: $\mathbb{R/Z}$, what does it denote?
"... $ \mathbb{T... |
H: Is the product of square singular and non singular matrices always singular?
Given $A,B\in R^{n\times n}$ such that A is singular, and B is non-singular. Is $(AB)$ always singular? If so, how do I prove it?
AI: The easiest way to see this is by looking at the determinant, since $\det(AB) = \det(A)\det(B)$ and a m... |
H: Sum of the series $\frac{1}{n!}$
I wish to find sum of a finite and infinite series$$\sum \frac{1}{n!}$$
I am aware, that this is a standard series and thus has a straight forward (well-known) answer BUT I am not recollecting it.
A hint Please.
AI: Do you remember that $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}\ ?... |
H: Why is $f(x) \equiv 0 \;\bmod \; p$ for all $p \in \mathbb{P}$?
I try to find a reason/proof for the following statement: Let be $f(x)=x^2+x$ an integer polynomial. Why is $$x^2+x \equiv 0 \pmod p$$ for all $p \in \mathbb{P}$?
I made a list for the first primes and obviously it's true, but I can't find a proof for ... |
H: All subgroup are Normal
All subgroups of a abelian group are normal. But the converse is not true. If every subgroup of a group is normal, then what more can we say about the group?
AI: If $G$ is a finite non-abelian group where all subgroups are normal, then $$G \cong Q_8 \times A \times B$$ where $A$ is an elemen... |
H: Fundamental Theorem of Calculus question
While revising, I came across this question:
Let $h(x)=\int_0^{x^2}e^{x+t}dt$.
Find $h'(1)$.
I tried using the substitution $t=u^2$, then $dt=2u du$. The integral becomes $h(x)=\int_0^x{e^{x+u^2}}\cdot 2udu$.
Then by the Fundamental Theorem of Calculus, $h
'(x)=e^{x+x^2}\cd... |
H: What does the negative in this anwer depict.
For the following question:
Peter lives 12 miles west of school and Bill lives north of school.Peter finds the direct distance from his house to Bill is 6 miles shorter than the distance by way of shcool.How many miles north of school does Bil live?
For this after co... |
H: An introductory textbook on functional analysis and operator theory
I would like to ask for some recommendation of introductory texts on functional analysis. I am not a professional mathematician and I am totally new to the subject. However, I found out that some knowledge of functional analysis and operator theory... |
H: Is there any difference between the definition of a commutative ring and field?
Is a commutative ring a field? A set equipped with addition and multiplication which is abelian over those two operations and it holds distributivity of multiplication over addition?
AI: A key difference between an ordinary commutative ... |
H: Finding an orthogonal basis from a column space
I'm having issues with understanding one of the exercises I'm making.
I have to find an orthogonal basis for the column space of $A$, where:
$$A = \begin{bmatrix}
0 & 2 & 3 & -4 & 1\\
0 & 0 & 2 & 3 & 4 \\
2 & 2 & -5 & 2 & 4\\
2 & 0 & -6 & 9 & 7
\end{bmatrix}.$$
The fi... |
H: How do I combine two matrix equations into one?
I have a discretely sampled 2D function:
S =
1 2 3 4
1 2 3 4
1 2 3 4
I want to find finite difference matrices, DX and DY such that:
$\ S_x=DX*S,S_y=DY*S $
where subscript denotes partial derivation with respect to.
... |
H: Algebra of Functions
There seems to be an interesting algebra of functions. Does it already exist in literature?
Given functions $f_1 : X_1 \to Y$ and $f_2 : X_2 \to Y$, if $f_1(x) = f_2(x)$ for all $x \in X_1 \cap X_2$, then their sum is defined as $(f_1 + f_2) : X_1 \cup X_2 \to Y$, where
$$
(f_1 + f_2)(x) =
\beg... |
H: Help understanding a $3n+1$ problem programming project
I was going through this website where I found 3n + 1 problem.
I was not able to understand what the output should be.
I understood if the input is 1 and 10
then the numbers between 1 to 10 need to be in consideration, and the output will be 20.
But I am wonde... |
H: Section of a circle
I need some help with some of my homework, I can't figure it out.
I have a radius on a circle and a height from the circle to the chord.
I found this formula
$$
h=r \left(1-\cos \frac{v}{2} \right)
$$
And isolated it to
$$
v = \arccos \left( \frac{h/r -1}{2}\right)
$$
Not sure if that's correct.... |
H: Best statistical method for determining likely subset length
Let's say I have a string set such as "AAAABBBBAAAABAAAA" and I want to have some quantitative measure of most likely subset length. In the above example (4 sets of length 4, one set of length 1), it is "humanly" evident the next set will be 4 in length, ... |
H: What are the subsemigroups of $(\mathbb N,+)?$
While trying to solve a somewhat bigger problem, I realized that I don't know what the subsemigroups of one of the most important semigroups, $(\mathbb N,+)$, are. (I assume $0\not\in\mathbb N$.) I've tried to characterize them but I haven't managed to do it fully. Wha... |
H: Defining a Limit Point of A Set
Limit Point is defined as:
Wolfram MathWorld: A number $x$ such that for all $\epsilon \gt 0$, there exists a member of the set $y$ different from $x$ such that $|y-x| \lt \epsilon$.
Proof Wiki: Some sources define a point $ x \in S$ to be a limit point of $A$ iff every open neighbo... |
H: Cauchy sequence of functions and the limit inferior
I'm trying to understand a step in a proof. I don't get a special trick that is used several times in the book I am reading, so this does not get out of my head. I try to explain the prerequisites and what I don't understand:
Let $Y$ be a Banach space and let $S$ ... |
H: first 1 in a bitmask using log2
I am trying to get the last 1 in a bitmask. More mathematically speaking, I have a number k, that can be written in its binary form as a sequence of 1 and 0. I want the "weight" or "index" of the last 1 in that number (from the right).
As an example, for the number 0b 10 0100 (36 in... |
H: How to solve $x'(t)=\frac{x+t}{2t-x}$?
I wish to solve $x'(t)=\frac{x+t}{2t-x}$ with the initial condition
$x(1)=0$. I noted that $x'(t)=f(\frac{x}{t})$ where $f(y)=\frac{y+1}{2-y}$
so I denoted $y(t)=\frac{x(t)}{t}$ and got that $y'(t)=\frac{f(y)-y}{t}$
so I can write something like $\frac{dy}{dt}=\frac{f(y)-y}{t}... |
H: Asymptotic behavior of the expression: $(1-\frac{\ln n}{n})^n$ when $n\rightarrow\infty$
The well known results states that:
$\lim_{n\rightarrow \infty}(1-\frac{c}{n})^n=(1/e)^c$ for any constant $c$.
I need the following limit: $\lim_{n\rightarrow \infty}(1-\frac{\ln n}{n})^n$.
Can I prove it in the following way?... |
H: lego geometry: Segments bent to a given angle make an n sided polygon
I swear this isn't homework. I'm actually ordering lego pieces from Pick-a-Brick. They have pipe segments that bend at 180 degrees (straight), 157.5, 135, 112.5, and 90 degrees. I need to know the number of sides of a polygon with those internal ... |
H: Period of the sum/product of two functions
Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link for this.
Also I know that period of $\sin(x)$ is $2\pi$, but what about... |
H: Measure of "boundary" of a nowhere dense set
Suppose $E$ is a nowhere dense set. For simplicity, assume it is in $R$. Is it true that the Lebesgue measure of $\overline{E}-E$ is zero? I.e., $m(\overline{E}-E)=0$.
The statement is not true in general. If $E$ is allowed to be open then take the complement of a fat c... |
H: Fireworks under inverse-cube gravity
What is the path of a projectile under an inverse-cube gravity law?
Imagine that the law of gravity was changed overnight from
$F(r) = G m_1 m_2 / r^2$
to
$F(r) = G' m_1 m_2 / r^3$.
To be specific, suppose
$G' = G r_E$ where $r_E$ is the radius of the Earth,
so that the forc... |
H: How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$
How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$
AI: Note that $\displaystyle\frac{(\log n)^p}{n}=p^p\cdot\left(\frac{\log k}k\right)^p$, where $k=n^{1/p}\to\infty$ when $n\to\infty$. |
H: Normal subgroup of prime index
Generalizing the case $p=2$ we would like to know if the statement below is true.
Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
AI: This is a standard exercise, and the answer is that the statement is true, but the... |
H: Decimal expression of reals
Let $x>0$ be real. Then $A_1=\{n\in \mathbb{N}\mid x<n\}$ is nonempty since $\mathbb{R}$ is dedekind complete. Since $\mathbb{N}$ is well ordered, $A_1$ has a least element $k$. Thus $k-1$ is the largest element of $A_2=\{n\in \mathbb{Z}\mid n\leq x\}$.
Thus for every $0<x\in \mathbb{R}$... |
H: expected value for this question
A manufacturer buys an item for 1600 dollar and sells it for 2000 dollar. The probabilities for
a demand of 0, 1, 2, 3, 4, “5 or more” items are 0.05, 0.15, 0.30, 0.25, 0.15, 0.10
respectively. How many items he must stock to maximize his expected profit?
AI: There is often no magic... |
H: How do I know which method of revolution to use for finding volume in Calculus?
Is there any easy way to decide which method I should use to find the volume of a revolution in Calculus? I'm currently in the middle of my second attempt at Calculus II, and I am getting tripped up once again by this concept, which see... |
H: Solving a complex integral
I need help solving an integral from John Conway book.
Lets $\alpha$ complex number different from 1 find integral $$\int\frac{dx}{1-2\alpha\cos{x}+{\alpha}^2}$$ from 0 to $2\pi$ in unit circle $$(z-\alpha)^{-1}(z-\frac{1}{\alpha})^{-1}$$
AI: Substitute: $z = e^{i x}$ then:
$$\cos x = \f... |
H: What is this digital operation called?
I've recently come across an operation that has interesting ties to the digital root operation. Instead of finding the sum of all digits, one would find the difference between each digit, repeating this until a single number is obtained. One would then find the digital root of... |
H: Partial fraction with same denominator
Is the following fraction (actually a Laplace transform) a kind of partial fraction?
$$\frac{4s+3}{{s^2}+3}$$
Can this be solved this way?
$$\frac{A}{s}+\frac{B}{s+{\frac{3}{s}}}$$
If not can you please tell me how to find inverse transform?
AI: If you want to keep everything ... |
H: Direct sum of submodules
I'm trying to prove that the following statements are equivalent for a commutative ring $R$
A: $_RR=_RN \oplus_RM$ for some submodules $_RN$, $_RM\subseteq_RR$
B: There exists an element $e=e^2\in R$ such that $N=Re$ and $M=R(1-e)$
I have no idea how to show $A \implies B$, but $B\implies A... |
H: Properties of complex function
Let be $f$ a complex function in zone $\Omega$ can someone help me how to prove that if $f$ and $f^2$ are harmonic in $\Omega$ then $f$ and $f^2$ are holomorphic in $\Omega$
AI: $f$ is harmonic if $f_{z\bar z}=0$ (you should check this if you haven't seen this before). Another way to ... |
H: Proofs with steps of division and setting things not equal to zero
I am self studying from A Book of Abstract Algebra by Charles C. Pinter. In chapter 3 problem set B problem 4 it asks to show whether $(a,b)\star(c,d)=(ac-bd,ad+bc)$ on the set $ \{(x,y) \in \mathbb R^2 | (x,y) \not= (0,0)\} $ is a group. Currently ... |
H: Exact sequence of sheaves in Beauville's "Complex Algebraic Surfaces"
On the first pages of Beauville's "Complex Algebraic Surfaces", he has a surface $S$ (smooth, projective) and two curves $C$ and $C'$ in $S$. He defines $\mathcal{O}_S(C)$ as the invertible sheaf associated to $C$. I'm assuming that if $C$ is giv... |
H: Find max $x\cdot y$ where $x\in S= \{(x_1,\ldots, x_n)\in \mathbb{R}^n$
Let $S= \{(x_1,\ldots, x_n)\in \mathbb{R}^n$; $|x_1|^p+\ldots+|x_n|^p=1\}$, where $p>1$ is real(and fixed), consider a fixed $y\in\mathbb{R}^n$ and $T:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $T(x) = x\cdot y$, where $x\cdot y = x_1y_1+\ldo... |
H: doubt on $C^{\infty}$ vector field
Warner page-37:
Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve $$\gamma_m:(a(m),b(m))\rightarrow M$$ such that
a) $0\in (a(m),b(m))$ and $\gamma_m(0)=m$
b)$\g... |
H: How many times do you need to double previous result to get at least $10^{82}$?
This is pretty straightforward, but I'd like to study, how find out, how many times do you need to double previous result of calculation to get some sum, for example: $10^{82}$
$1\times 2 = 2$
$2\times 2 = 4$
$4\times 2 = 8$
$8\times 2... |
H: Linear combination of cosines: so close to perodic.
I'm interested in linear combinations of cosines:
$$f(x) = \alpha_1 \cos(2\pi \theta_1 x) + \alpha_2 \cos(2\pi \theta_2 x) +
\cdots + \alpha_k \cos(2\pi \theta_k x)\enspace,$$
where $\alpha_i \in \mathbb{Z}$ and the $\theta_i$'s are linearly dependent
irrational n... |
H: Angle between vectors when their dot product and norm of cross product are equal.
I had a question in the final exam that asked what the angle between vectors a and b is if:
$$\vec a \cdot \vec b=|\vec a\times\vec b| $$
Any hints please.
AI: It is not hard to show that $$\|u\times v\| = \|u\|\|v\| \sin(\theta),... |
H: Is the arithmetic most mathematicans use a modelled within first or a second order logic?
I often read that arithmetic in first order logic has problems and you really want to do it in second order logic.
However, aren't the Zermelo–Fraenkel axioms written down in the language of first order logic?
AI: Note that Z... |
H: Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$?
Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$ where $R$ is a ring and $I,J$ are ideals?
If $I \subset J$ then it follows from the third isomorphism theorem.
In $R= \mathbb Z$ with $I = 3 \mathbb Z$ and $J = 5 \mathbb Z$ we have $I/J = \{ \bar{0}, \bar{3}, \bar{6}, \bar{9}, ... |
H: Highschool math: $w=\frac{x+jy}{x+jy-1}$ separate (x and y) and j terms.
I'm stuck on the first step again:
$$w=\frac{x+jy}{x+jy-1}$$
I need to separate out (x and y) and j
The only thing I can come up with at this stage is:
$$w=\frac{x}{x+jy-1}+\frac{jy}{x+jy-1}$$
but that doesn't help me at all.
Aaaaargh!!!
AI: W... |
H: Absoluteness and categories
From the wikipedia article on the Skolem paradox:
A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed this is ... |
H: What happens to this theorem if we change the dimension of the spaces?
We have the following theorem
Let $T:X\rightarrow Y$ be a compact operator between the Hilbert spaces $X,Y$. Then there exist (possibly finite) orthonormal bases $\{e_1,e_2,\ldots \}$ and $\{f_1,f_2,\ldots \}$ of $X,Y$ and (possibly finite) num... |
H: Differentiable structure on the real line
The usual differentiable structure on real line was obtained by taking ${F}$ to be the maximal collection containing the identity map, Let ${F_1}$ be the maximal collection containing $t\mapsto t^3$. I need to show $F_1\neq F$, but $(\mathbb{R},F)$ and $(\mathbb{R},F_1)$ ar... |
H: Algorithms and generalisation of functions
I admit I'm little bit poor in functions in mathematics. But I'm in real urge to get this riddle out.
How to express $$x(n)=x(n-1)+x(n-2)+1,$$ where $n>1$ and $x(0)=0$ and $x(1)=1$, in terms of the function $$y(n)=y(n-1)+n,$$ where $n>1$ and $y(0)=0$ and $y(1)=1$?
I foun... |
H: Directed colimit in a concrete category
I recently found myself at a spot that I never believed I'll get (or at least not that soon in my career). I ran into a problem which seems to be best answered via categories.
The situation is this, I have a directed system of structures and the maps are all the inclusion ma... |
H: How to show growth without bound in only certain cases and not in others?
I encountered the following problem. (We're working in a finite-dimensional real vector space, here.)
Suppose $$A=\frac{1}{2}\left(\begin{array}{cc}-2 & 4\\1 & 1\end{array}\right).$$ Find a vector, $y$, so that $\lVert A^nx\rVert\to\infty$ a... |
H: variance of multiple regression coefficients
If I consider universal kriging (or multiple spatial regression) in matrix form as:
${\bf{V = XA + R }}$
where $\bf{R}$ is the residual and $\bf{A}$ are the trend coefficients, then the estimate of ${\bf{\hat A}}$ is:
${\bf{\hat A}}=(\bf{X^{T}C^{-1}X)^{-1}X^{T}C^{-1}V}$... |
H: finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents
I'd really love your help with finding the rational number which the continued fraction $[1;1,2,1,1,2,\ldots]$ represents.
With the recursion for continued fraction $( p_0=a_0, q_0=1, p_{-1}=1, q_{-1}=o), q_s=a_sq_{s-1}+q_{s-... |
H: Is there a analysis conjecture proven to be unprovable or a proof is non-existence?
Is there a analysis conjecture proven to be unprovable or a proof is non-existence?
So, is it once a math history milestone
AI: There is a large number of statements in analysis that have been proved independent of the axioms of ZFC... |
H: Direct sum of Hilbert Space subspaces - Notation?
This is a really basic question sorry, I just need to make sure I have my understanding correct.
Given an infinite dimensional Hilbert space $\mathcal{H}$ and two subspaces, also Hilbert spaces, $\mathcal{H}_1$ and $\mathcal{H}_2$. If one can write:
$\mathcal{H}=\ma... |
H: Unital nonabelian banach algebra where the only closed ideals are $\{0\}$ and $A$
This is a problem in exercise one of Murphy's book
Find an example of a nonabelian unital Banach algebra $A$, where the only closed ideals are $\{0\}$ and $A$.
But does such an algebra exist at all?
My argument is the following:
Le... |
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