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H: non-intuitive results in random sampling
I play a card game (Magic The Gathering) that involves creating a deck of cards that can loosely be split into two categories: land and spells. You are permitted to play only one land per turn, and the land are used to cast your spells. Due to the one-per-turn limitation, it... |
H: $P\in \operatorname{Syl}_p(S_n)$ implies that $P\in \operatorname{Syl}_p(A_n)$ and $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$
$\newcommand{\Syl}{\operatorname{Syl}}$
This is an exercise (with hint about the second part) in my own language book in Group theory, however, maybe it is a lemma or theorem in an standard book... |
H: Are bimodules over a commutative ring always modules?
Let $R$ be a commutative ring. It is true that every module over $R$ is an $(R,R)$-bimodule. Is the converse true? In other words is it possible that there is an $R$-module where left multiplication and right multiplication do not co-incide?
I thought perhaps a... |
H: $L_p$ space convergence revised
Here's what I really wanted to ask
Suppose that $f_n \rightarrow f$ a.e.
If we have $\int{\left|\lvert f_n\rvert^p - \lvert f\rvert^p\right|} \rightarrow 0$, is it true that $\|f_n\|_p \rightarrow \|f\|_p$?
What I want to do is let $g_n = \lvert f_n\rvert^p, g=\lvert f\rvert^p$ and t... |
H: Orders of moves in the Rubik's cube
Playing with my Rubik's cube, I was thinking of facts about it that are immediate to mathematicians but novel to others. Here's one:
Given a Rubik's cube in the solved state, any sequence of moves, if repeated long enough, will eventually return the cube back to the solved state
... |
H: Is independence preserved under the Girsanov transformation?
Let $(B_{t},\mathcal{F}_{t})$ be a standard 1-d Brownian motion on some $(\Omega, \mathcal{F}, \mathbb{P}) \ $, and let's assume $\mathcal{F}_{t}$ is the augmentation of $\mathcal{F}_{t}^{W}$.
Let $T>0$ and define $Z_{T}=\exp\{W_{T}-\frac{1}{2}T\} \ $, th... |
H: Brute force method of solving the cube: How many moves would it take?
Given that Rubik's cube has finitely many positions, one possible "brute force" method to solve it would be to determine once a sequence of moves which eventually reaches every possible position of the cube, and then whenever you want to solve a ... |
H: Compact subspace
Is the subspace $$C^k([0,T]) \subset C^{k-n}([0,T])$$ compact? I think the answer is no. But since $C^k$ is compactly embedded in $C^{k-n}$, it seems like it should be yes in some way. Can I do anything here?
($C^k$ is the space of $k$ times continuously differentiable functions)
AI: What is true, ... |
H: Hermitian/positive definite matrices and their analogues in complex numbers
I've heard a couple of times some people say that in a way, Hermitian matrices are to matrices as real numbers are to complex numbers. I know two examples where this is sort of true:
Complex conjugate vs conjugate transpose We can realize c... |
H: Why Does Finitely Generated Mean A Different Thing For Algebras?
I've always wondered why finitely generated modules are of form
$$M=Ra_1+\dots+Ra_n$$
while finitely generated algebras have form
$$R=k[a_1,\dots, a_n]$$
and finite algebras have form
$$R=ka_1+\dots +ka_n$$
It seems to me that this is an flagrant a... |
H: Exponent and group quotient
We know tha if $1\not=H\trianglelefteq G$ which consists of all the elements
of $G$ which have finite order, then in $G/H$, no elements(except element
neutral) has finite order.
My question is: If $1\not=H\trianglelefteq G$ ($H\not=G$) has exponent
finite, then what can say on $G/H$?
AI:... |
H: Period of $x^n \mod y$?
Let $x$, $y$, $n$ be positive integers, where $n$ and $y$ shall be constants and $x$ a variable. Then it is trivial that the period of
$x \bmod y$
in $x$ is $y$, since the function simply drops to zero and starts over when $x$ reaches $y$. Now, for me at least, it is much less obvious, wha... |
H: A regular expression for the words that don't contain the sequence $ab$ over $\{a,b,c\}$
The following is an exercise in a book I am reading:
Let $\Sigma=\{a,b,c\}$, define $L$ to be the language of all words
over $\Sigma$ that do not contain $ab$ as a sub-word.
Find a regular expression for $L$.
I am unable ... |
H: Proof of first isomorphism theorem of group
(Proof of (first) Isomorphism Theorem) Let $f : G \rightarrow H$ be a surjective group homomorphism.
Let $K = \operatorname{ker} f$. Then the map $f' : G/K \rightarrow H$ by $f'(gK) = f(g)$
is well-defined and is an isomorphism. Proof: If $g'K = gK$, then $g' = gk$ w... |
H: Summing digits of powers of 2 to get 1 2 4 8 7 5 pattern
If you add the decimal digits of multiples of 9,possibly repeatedly, you get 9. For instance, with 18 we have $$18\mapsto 1+8=9$$
and with 909 we have $$ 909\mapsto 9+0+9=18\mapsto 1+8=9.$$ This is proven.
If you do the same with powers of two, you develop a ... |
H: Integral $\int_0^\infty x\exp(-\frac{x^2}{2}\ )\mathrm{d}x$
I was trying to solve
$$\int_0^\infty x \exp \left( { - \frac{{{x^2}}}{2}} \right)\;{\text{d}}x$$
I was trying to use $$(f(g(x)))'=f'(g(x))\cdot g'(x)$$
(So $[e^{f(x)}]'=[e^{f(x)}]\cdot f'(x)$ ?), but I got the wrong answer. (I substituted $0$ and $\inf... |
H: How do you go from $x\in\cup\{\mathcal{P}(A)|A\in\mathcal{F}\}$ to $\exists A\in\mathcal{F}(x\in \mathcal{P}(A))$?
These are statement 4 of example 2.3.6 and its solution from section
2.3 in Daniel J. Velleman's "How to Prove It - A Structured Approach"
(great book), where the author asks the reader to analyze the ... |
H: An exercise from a linear algebra book (hoffman and kunze)
I am not sure how to solve this exercise:
Find all solutions of $$\begin{align*}2x_1-3x_2-7x_3+5x_4+2x_5&=-2\\
x_1-2x_2-4x_3+3x_4+x_5&=-2\\
2x_1\qquad-4x_3+2x_4+x_5&=3\\
x_1-5x_2-7x_3+6x_4+2x_5&=-7\;.
\end{align*}$$
my row-reduced echelon matrix is:
$$\pm... |
H: Find $w$ such that $w^8=15-15i$
Find the complex number, lying in the second quadrant, and having the smallest possible real part, which satisfies the equation
$$w^8=15-15i$$
AI: $$z=15-15i\Longrightarrow |z|=15\,\sqrt 2\,\exp({7\pi i}/{4}+2k\pi i),\,k\in\Bbb Z$$
$$\Longrightarrow w^8=z\Longrightarrow w=z^{1/8}=15... |
H: collinear three points in a linear equation $ax+by+z$
So, there is a plane equation $ax+by+z = 0$. Suppose there are three points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$ that is on the given plane.
Then, $\begin{vmatrix} x & y & z & 1\\x_1 & y_1 & z_1 & 1\\x_2 & y_2 & z_2 & 1\\x_3 & y_3 & z_3 & 1 \end{vm... |
H: Evaluate $\sum_{i=1}^n \frac{i}{n^2}$
My book asks to simplify this problem:
$$\sum_{i=1}^n\frac{i}{n^2}$$
This equals:
$$\sum_{i=1}^n\frac{1}{n^2}i$$
Now, shouldn't
$$\sum_{i=1}^n\frac{1}{n^2}=\frac{1}{n}\text{ ?}$$
Because you are summing $1/n^2$, $n$ times, so that $n / n^2 = 1/n$.
So it should be
$$\frac{1}... |
H: Function is pointwise limit of integrals
This is a question from some old masters exams.
Let $\phi_{n}$ be a sequence of continuous, real functions on $\mathbb{R}$ such that
$\phi_{n}(x) = 0$ for all $|x|\geq 1/n$ and $\phi_{n}(x)\geq 0$ for all $x\in\mathbb{R}$, and
$$\int\limits_{-1}^{1}{\phi_{n}(x)dx}=1$$
For e... |
H: Given a list of axioms, can one prove that a statement is unprovable without exhibiting a model?
Suppose I am given the following axioms for a vector space $V$ over the field $F$:
For addition of vectors
Closure
Associativity
Existence of an additive identity
Existence of additive inverse
Commutativity
For multip... |
H: Partitioning $\mathbb{N}$ into distinct AP’s
Can $\mathbb{N}$, the set of natural numbers, be partitioned into a finite number of subsets that are in arithmetic progression with distinct steps ?
AI: No. See https://mathoverflow.net/questions/25313/finitely-many-arithmetic-progressions. |
H: Tensor-commutative abelian groups
Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$.
The first question is somewhat vague:
Question 1. Can one characterize the tensor-commutative abelian groups?
It is easy to see that s... |
H: $g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$. For what value of $p$, $g(x+p)=g(x)$.
$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$.For what value of $p$, $g(x+p)=g(x)$.
$g(x+2)+g(x)=g(x+1)$
AI: This is really a question about sequences hence assume that $x_{n+2}-... |
H: Group Theory - Proving $(a*b)^{-1} = (a^{-1}) * (b^{-1}) $
I'm trying to prove the below equation, where $ a,b \in G $ and $(G, *) $ is a group.
$(a*b)^{-1} = (a^{-1}) * (b^{-1}) $
I'm not really sure how to do it though. I tried doing something like
$(a*b)^{-1} * (a*b) = e = a * a^{-1} * b^{-1} * b $
$(a*b)^{-1} ... |
H: Evaluate $\lim_{s\to0}\frac{s\sqrt{e^s}}{e^s-1}$ without using L'Hospital rule.
Evaluate $\lim_{s\to0}\frac{s\sqrt{e^s}}{e^s-1}$ without using L'Hospital rule.
I've done this: Since $\sqrt{e^s}\to1$as $s\to1$, so $\lim_{s\to0}\frac{s\sqrt{e^s}}{e^s-1}=\lim_{s\to0}\frac{s}{e^s-1}=1$
Are my steps correct? Is step 1... |
H: How does the supremum of the cardinalities of a set of ordinals relate to the supremum of the ordinals?
Suppose we have a set of ordinals such that their supremum is a cardinal (not in the original set).
$\sup\{\alpha\}=\kappa$
I'm interested in the supremum of the cardinalities of those ordinals: $\sup\{|\alpha|\}... |
H: How to calculate blood viscosity formula using $cm^3$ (lt) and $cm^2$ (area) units
I'm reading a blood viscosity related theory and I have an example formula which I don't understand. It turns lt (fluid liters) or $cm^3$ into $cm^2$ and comes out with a number. I don't understand how the conversion is made from are... |
H: Modular Multiplicative Inverse & Modular Exponentiation Equation
I was solving a problem containing that equation.
$$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$
Given:
$1 \le a \le 2,000,000,000$
$0 \le n \le 2,000,000,000$
$2 \le m \le 2,000,000,000$
$a$ and $m$ are coprime
My Approach:
To Solve the SECOND part
... |
H: Which symbol should be used for an empty set?
Currently, a discussion started on the German Wikipedia article for Empty Set (the German discussion), whether $\emptyset$ or $\varnothing$ should be used or is more common as a symbol for an empty set.
Do you know any sources for this question? Do you know a source for... |
H: What is linearity of Expectations?
In reading about the average case analysis of randomized quick sort I came across linearity of expectations of indicator random variable I know indicator random variable and expectation. What does linearity of Expectation mean ?
AI: Let $\xi_1,\xi_2:\Omega\to\mathbb R$ be two r... |
H: Prove that: $\frac{a}{a+1}+\frac{b}{(a+b+1)^2}+\frac{1}{a+b+1}\le1 $
If $a\geq0$, $b\geq 0$ then the following inequality holds:
$$\frac{a}{a+1}+\frac{b}{(a+b+1)^2}+\frac{1}{a+b+1}\le1 $$
There are at least three things to try here:
a). Use AM-GM for the denominator of the second fraction, $(a+b)\le \frac{(a+b+1)^2... |
H: Is $f$ is computable by a finite automaton, the dual of $f$ is thus computable also?
et $A$ be a finite alphabet. Let $A^*$ denote the language of all words in $A$, and $\epsilon$ the empty word. Let $\rho : A^* \to A^*$ denote the "reverting" map, that transforms $a_1a_2\ldots a_n$ into $a_na_{n-1} \ldots a_1$. I... |
H: Polynomial of degree $-\infty$?
I'm reading E.J Barbeau Polynomials. I'm in a page where he asks a polynomial of degree $-\infty$. Then I thought about $77x^{-\infty}+1$, but when I went for the answers, the answer to this question was zero.
Then I thought about making $n^{-\infty}$ on Mathematica and it outputed $... |
H: Dual of a 2-category
Let $C$ be a $2$-category. There are two ways of dualizing $C$: The first one is well-known and also generalizes to arbitrary $(\infty,1)$-categories: We dualize "at each stage". The second one only dualizes the $2$-morphisms: We keep the underlying $1$-category, however we define $2$-morphisms... |
H: Savings when driving the inside edge of a curve.
I was wondering how many kilometers/meters I would save if I always drove on the inside edge of the curves between my home and workplace, compared to driving in the middle of the lane.
If the lanes are approximately $3 m$ wide and I estimate that over $100 km$ I dri... |
H: Sequence that maps to arbitrary positive real number
How do you construct a sequence of functions $f_n(x)$ such that
$$s = \limsup_{n\rightarrow \infty} \sqrt[n]{f_n(x)}$$
for all $s > 0$?
I know it's possible to this with a different sequence
$$s = \limsup_{n\rightarrow \infty} (1 + \frac{x}{n})^n$$
where $x = \lo... |
H: Is there a sign for "x's negation equals y"?
For example, let M and N be two real numbers. M is smaller than N. Now I negate the inequality, such that now M is greater or equal to N.
$M < N ≟ M \geq N$
Is there a sign to replace '≟' in the expression above?
AI: It looks like you're talking about equivalence of a pr... |
H: Equations - Solving for x
I have this problem:
$$9x^3 - 18x^2 - 4x + 8 = 0$$
However, I'm not sure how to find the values of $x$. I moved the 8 over and factor out an $x$, but the trinomial it created can't be factored. Could someone enlighten me?
AI: Factor the function.
$9x^3-18x^2-4x+8=9x^2(x-2)-4(x-2)=(x-2)(9x^... |
H: Proving that a bounded, continuous function has a supremum
Theorem: Let $f$ be a continuous real-valued function on a closed rectangle $R$ in $\Bbb R^2$. Then,
(a) $\quad f$ is bounded on $R$
(b) $\quad $There exist points $c$ and $d$ in $R$ so that $f(c) = \sup_{p\in R} f(p), \quad f(d) = \inf_{p\in R} f(p)$.
(c)... |
H: Multiplying Reciprocal Exponents
This is the problem I have:
$$y^{\frac{3}{2}} = 5y$$
What I tried so far was raising $y^{\frac{3}{2}}$ to the $\frac{3}{2}$, making it equal $1$, but I had trouble raising $5y$ to that power.
AI: You have the right idea, you are trying to get rid of the fractional power. However, th... |
H: Triangular grid with 4 edges per vertex
I am trying to create a triangular grid/mesh for a rectangular domain in $\mathbb{R}^2$ with the property that each vertex is shared by (at most) four edges. Is this possible to accomplish?
AI: Only if you have few enough triangles. You can split the domain with a diagonal i... |
H: What is the rationale behind the evaluation of the Expectation operator?
We know that the expectation operator is defined for a random variable $x$, as such:
$$
\mathbb{E} \left\{x\right\} = \int_{-\infty}^{\infty} x \: p_x(x) \; \mathrm{d}x
$$
Where $p_x{x}$ is the PDF of the random variable $x$.
If there is an ar... |
H: Derivative of function defined by integral
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and let $a$ be a nonzero real number. Show that the function
$F(x)=\frac{1}{2a}\int_{-a}^{a}{f(x+t)dt}$ is differentiable and has continuous derivative.
My thoughts were that
$\frac{d}{dx}(\frac{1}{2a}\int_{-a}^{a}{f(x... |
H: Function to represent curve between two 2D points
I need a (simplest) function that interpolates values in range from predefined point $A$ to $B$ with rules:
it must be smooth curve
direction near $B$ must be the same as predefined $D$ vector
I have a variant to build circle arc but is too complex. Maybe some so... |
H: A list from each element of another?
Sorry to edit this so much at this late stage, but the question and answers are confused so much by my incorrect use of terminology and the such, I feel that I should clear this up.
Where $a$ and $b$ are lists.
$$a = \{\mathbf{true, false, true, false, false}\}$$
$$b = f(a) = \... |
H: What is the essential difference between the terminologies "associative" and "semi-group".
Can only a map
$$*:S\times S\rightarrow S$$
be associative?
If I look at
$$(a* b)* c=a* (b* c),$$
then it seems I have to rule out the more general case
$$*:A\times B\rightarrow C.$$
But $A=B=C$ is really the only... |
H: Sum independent of the rearrangements of its terms?
Is it true, that in every concept of an infinite sum in a Banach space, encountered in an introductory functional analysis course, convergence is independent of the rearrangement of its terms ?
AI: $\mathbb R$ is a Banach space, and its concept of infinite sums is... |
H: Evaluating $\int e^{ax} x^b (1-x)^c \mathrm{dx}$
Edit: clarify question
The integrand looks kind of like a gamma density function, and kind of like a beta density function, so maybe it has a somewhat nice solution?
$$\int e^{ax} x^b (1-x)^c \mathrm{dx}$$
Wolfram alpha does not want to do it.
AI: You can expand out ... |
H: Name a ring of 2 by 2 matrices where $a^3 = a$ and a belonging to this ring?
I need an example of a ring consisting of 2 by 2 matrices where $a^3=a$ with $a$ belonging to this ring. If someone can list the elements I would be satisfied.
What I'm trying to get at it is conceptualize why a ring $R$ is always commmuat... |
H: Derivative of vector field
I have just started learning differential geometry and I am trying to answer the following exercise (it's not for homework):
Let $M$ be an $n$-dimensional manifold with chart $(U, \varphi_U)$ and let $(x^1, \dots, x^n)$ be the corresponding local coordinates. Define a vector field on $U$... |
H: calculator errors and rational root theorem
In this question we were asked for roots of $9x^3 - 18x^2 - 4x + 8 = 0$ and I remarked on the rational root theorem. In fact, the roots are $2, \frac 23, \frac {-2}3$. When checking them it would be nice to use an exact form of the fractions, but in fact many will use a... |
H: Lifting a homomorphism
In a set of lecture notes, I recently came across the remark that any continuous homomorphism from $\mathbb{R} \to S^1$ can be lifted to continuous homomorphism from $\mathbb{R} \to \mathbb{R}$. How do I conclude that the lift is a homomorphism too? I tried many ways to solve this but failed.... |
H: Can this type of series retain the same value?
Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation only in those blocks, like this (for brevity illustrated only for the fi... |
H: Comparison of two infinity
Possible Duplicate:
Different kinds of infinities?
Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be infinity. If it doesn't have any particular value, how can we say that this infinity is small and other one is greater. Ca... |
H: An explicit calculation of Galois group
This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$.
Here is a brief of what I can think of.
For the first two, the splitting field is $\mathbb{Q}(\sqrt{2},i)$. Since $X^2+1$ and $X^2-2$ are... |
H: Number of Zeros of a Section vs Integral First Chern Class
I've often read that the first chern class can be seen as "the number of zeroes a section must have".
How precise can this statement be made?
I'm only interested in Line bundles. I actually know how a generic section looks like on my manifold. From this I... |
H: How to find the eigen vector for the given matrix?
Sorry, I made error here. Please check Edit!!
Given matrix is
$$
A = \begin{bmatrix}
2 & 5\\
4 & -2
\end{bmatrix}$$
How to find the eigen-vector? I'm stuck here
$$\begin{bmatrix}
2 - 4i & 5\\
4 & -2 -4i
\end{bmatrix} \times \begin{bmatrix}
x\\
y
\end{bmatrix}... |
H: Find the domain of $f(x)=\ln(3x+2)$
Find the domain of $f(x)=\ln(3x+2)$
I can find domain, but is it the same for a $\log$ function? And also, do I have to rid the equation of the $\ln$ before I can find the domain? I'm really confused as to how to start out this problem. Please help.
Edit: I got the answer and n... |
H: Direct Sum and Sum
We know if $R$ is semi simple ring and $M$ is $R$-module then for any submoldule $N$ of $M$ there exists submodule $N'$ such that $N \oplus N'=M$.
Now I have trouble with sum and its supposed answer.
let $R$ be ring (Not semi simple) and $M$ is $R$-module and $N$ is submodule of $M$. Does exist s... |
H: Calculate a whitening matrix without using inverses?
Consider a random column vector $\mathbf{x}$, of dimension $m$. That is, it is a random vector, composed of $m$ random variables. The PDF of the random vector $\mathbf{x}$ is thus the joint-PDF of its $m$ random variable components.
Let us assume that one wishes ... |
H: Graph $f(x)=e^x$
Graph $f(x)=e^x$
I have no idea how to graph this. I looked on wolframalpha and it is just a curve. But how would I come up with this curve without the use of other resources (i.e. on a test).
AI: You'd like to capture some of the important behavior of the graph, yes? For example, end-behavior (w... |
H: What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$?
Let $V$ denote an $n$-dimensional real vector space, and $\wedge^n$ denote the $n$-fold exterior product. What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$?
In the book Introduction to Manifolds, by Loring W. Tu, 3rd edition, Chapter: Orie... |
H: Graph $f(x)=\ln x+2$
Graph $f(x)=\ln x+2$
And find all intercepts and asymptotes.
I know exactly how the graph looks and I have a sketch in front of me. Now, for the intercepts, $x$-int=$-1$ and $y$-int$=-\infty+2$ which I'm guessing is just $-\infty$ because adding $2$ wouldn't make much of a difference on tha... |
H: Galilean transformations
How do you prove that every galilean transformation of the space $\mathbb R \times \mathbb R^3$ can be written in a unique way as the composition of a rotation, a translation and uniform motion? Thanks!
AI: As mentioned, this problem appears in V.I Arnold's "Mathematical Methods of Classica... |
H: A natural question about convergence
Let $u,u_k \in C^{0}(K)$ where $K \subset \mathbb{R}^{n}$ is a compact set. Assume that $u_k \rightarrow u$ uniformly. Is these hypotheses sufficient to guarantee that
\begin{equation}
\mbox{med}(\{u_k>0\}) \rightarrow \mbox{med}(\{u>0\})
\end{equation}
or
\begin{equation}
\mbox... |
H: $(p,\frac{n}{p^\alpha})=1$ then $p\nmid\binom{n}{p^\alpha}$
Let $n=p^\alpha m$ wherein $(p,m)=1$. Then we have $$p\nmid\binom{n}{p^\alpha}$$ What I have done is just playing with $\binom{n}{p^\alpha}$ like:$$\binom{n}{p^\alpha}=\frac{n!}{(n-p^\alpha)!p^\alpha!}=m\binom{p^\alpha m-1}{p^\alpha-1}$$
Honestly, the rest... |
H: Statistical Significance Dice Probability
I think I just need to be pushed to the right formula or algorithm... Imagine you've got a "real" dice which is not an optimal one. And you want to know the confidence intervals for each result. So you rolled the dice a couple of times and get the following absolute probabi... |
H: Obtaining a monotone subsequence from a sequence of functions
It is true that given any sequence of numbers, there is a monotone subsequence within. I thought obtaining a monotone subsequence out of a sequence of functions might be too much to ask. But given a sequence of $L^1$ functions, could we say that it admit... |
H: An $(R,S)$-bimodule is a left $R \otimes_k S^{\text{op}}$-module
Let $k$ be a commutative ring, and let $R,S$ be $k$-algebras. To me "$R$ is a $k$-algebra" means that $R$ is a $k$-module such that $a(rs)=(ar)s=r(as)$ for all $a\in k$ and $r,s \in R$. Let $M$ be a $(R,S)$-bimodule. I am trying to show that $M$ is a ... |
H: Expansion of $x^4\over1+x^2$ into a power series
I calculated the generating function $A(x)$ of the recurrence $a_n = a_{n-2} - 2a_{n-3}$, $(n \ge 0, a_0 = a_1 = 0, a_2 = 2)$ and I have no clue on how to expand it into a power series in order to read the coefficients.
$$A(x) = {-2x^4 \over 1+x^2}.$$
AI: Your genera... |
H: Proving that $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
Prove $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
So, LS=
$$\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}$$
$$\left(\dfrac{1}{\sin\theta}\cdot \dfrac{\tan\theta}... |
H: Lebesgue measure: $\mu(f-g(D))=0\Rightarrow\mu(f(D))=\mu(g(D))$
Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be Lebesgue measurable functions, and let $D$ be a subset of $\mathbb{R}$. If $f(D)$ and $g(D)$ are measurable subsets of $\mathbb{R}$, and $(f-g)(D):=\{f(x)-g(x):x\in D\}$ is a measurable subset of $\mathbb{... |
H: Euler's theorem on a polynomial
The following theorem is said to be due to Euler.
However, it seems not so well known.
How do you prove this?
Theorem
Let $K$ be a field.
Let $n \geq 1$ be an integer.
Let $\alpha_1,\dots,\alpha_n$ be distinct elements of $K$.
Let $f(X) = (X - \alpha_1)\cdots(X - \alpha_n)$.
Then
$\s... |
H: What do finite, infinite, countable, not countable, countably infinite mean?
Possible Duplicate:
Different kinds of infinities?
Those terms keep confusing me. Could somebody give a clear and unambiguous definition? How do we prove a set is countable, finite, ...? I know that there are more real numbers than ther... |
H: Geometric interpretation of linear transformation
I have a linear transformation, given by the following matrix
$$
\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}
\mapsto
\begin{pmatrix}
2 & 2\\
-1 & -1\\
\end{pmatrix}
\begin{pmatrix}
x_1\\
x_2
\end{pmatrix}
$$
How can I determine what this corresponds to geometrically, wh... |
H: Localizations of the Boolean Ring P(X)
Given a set $X$, we can construct the Boolean ring whose elements are the power set of $X$. The multiplication therein is intersection, and the addition is symmetric difference. I am interested in finding information on the study of such rings, for instance, information about... |
H: Strategies for arranging coins in a circle
Suppose you have $n$ identical circular coins and you would like to arrange them on the table so that their edges touch and their centers lie on a circle.
Mathematically, there is no trouble. "Just" put the center of each coin at $re^{2ik\pi/n}$ for $k$ in $\{0, 1\ldots, n... |
H: Span of an empty set is the zero vector
I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone".
Is Nering defining the span of the empty set to be the set containing the zero vector o... |
H: Strong Law of Larger Numbers with Coin flips
I am reviewing some probability for a class I am about to take, and I came up with this example to try to help my intuition.
Suppose I take the the space of all sequences of the two symbols $H,T$. Let $X_i$ be the function on this space that is 1 if the $i$th entry is $... |
H: Is the number of primes congruent to 1 mod 6 equal to the number of primes congruent to 5 mod 6?
(I know that there's an infinity of primes congruent to 5 mod 6, but I don't know if there is an infinity of primes congruent to 1 mod 6.)
But what I'd really like to know is whether or not the number of primes ≡1 mod 6... |
H: Definition of operation that creates matrix from two vectors
I'm looking for something similar to Kronecker's product that creates matrix based on two vectors I'm thinking about either:
$$
(1,2,3) @ (4,5,6) =
\begin{bmatrix}
(1, 4) & (2, 4) & (3, 4) \\[0.3em]
(1, 5) & (2, 5) & (3, 5) \\[0.3em]
(1, 6) & (2, 6) ... |
H: L'Hôpital's Rule for $ \frac{\infty}{\infty-\infty} $?
I'm currently trying to find this improper integral:
$$ \int^{\infty}_{-\infty}\frac{1}{\sqrt{x^{2}+1}} dx $$
I started off by splitting it into a proper integral, and then into the sum of two integrals:
$$ = \lim_{a\rightarrow\infty} \int^{a}_{-a}\frac{1}{\sqr... |
H: The logic behind partial fraction decomposition
In the general case of any function would be interesting but my question is concerning the general case of polynomials with integer powers. I can use the method of partial fractions in the simple case required for an introductory course on integration, but I'm not sur... |
H: Probability of hitting a bullseye in five shots
The probability of a target shooter hitting the bullseye on any one shot is 0.2.
a If the shooter takes five shots at the target, find the probability of:
i) missing the bullseye every time
ii) hitting the bullseye at least once
b) What is the smallest nu... |
H: Hilbert's Original Proof of the Nullstellensatz
Does anyone have a link to Hilbert's Original Proof of the Nullstellensatz, or know a book where it's printed? I'd be interested to see what it was like. I only really know the Noether normalisation and Zariski proofs. While these are both good, it would be nice to ha... |
H: 3D Projection (Perspective). Not what I would expect...
I have some code based on the 3d perspective projection formula I found on wiki.
http://en.wikipedia.org/wiki/3D_projection
It works great... mostly :)
I recently started making the code do z ordering/culling... but I was getting weird results. After a lot of ... |
H: A simple depreciation problem
Question 2: The share price of a public company fell 4% on each day of a week, from Monday to Friday. What is the price of a share at the end of the trading day on Friday, if the share was valued at $20 on the Sunday of the same week?
I could do this the long way by calculating the dep... |
H: Determing the order when moving to the other side of the equation.
Sorry if the title is vague. But I don't really know the terms I'm looking for (Nor do I know any tags that go with it). I usually end up being confused with the little(and perhaps trivial things) in math. The following for example:$$\mathbf{a} \cdo... |
H: Reconciling the classical formulation of a surface integral with a general integral over a manifold
So I was just brushing up on some calculus when I came across a problem. I was trying to perform a surface integral I found online through the more general formulation of a differential form on a manifold. This led... |
H: Is the fundamental theorem of calculus independent of ZF?
By the fundamental theorem of calculus I mean the following.
Theorem: Let $B$ be a Banach space and $f : [a, b] \to B$ be a continuously differentiable function (this means that we can write $f(x + h) = f(x) + h f'(x) + o(|h|)$ for some continuous function $... |
H: How to prove that if $\det(A)=0$ then $\det(\operatorname{adj}(A))=0$?
How to prove that if $\det(A)=0$ then $\det(\operatorname{adj}(A))=0$?
I have been trying to solve this but I can't use $$\det(A^{-1})=\det \Big(\frac{1}{\det(A)} \operatorname{adj}(A) \Big)$$ because $\det(A)=0$ and $\frac{1}{0}$ is not allow... |
H: about a good book - Vector Calculus[by Jerold E. Marsden, Anthony J. Tromba ]
I start reading Vector Calculus by Jerold E. Marsden, Anthony J. Tromba and I want to know if there is a book with the answers of the exercises. I like a lot this book, it seems to be made for a beginner but also for those who want to as... |
H: Axioms of set theory and logic
Zermelo–Fraenkel set theory is the most common foundation of mathematics with eight axioms and axiom of choice (ZFC):
http://plato.stanford.edu/entries/set-theory/ZF.html
But one can see that the statement of ZFC depends on first-order logic symbols and definitions, for example $=, \i... |
H: Exponential of the differential operator
I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and
$$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$
(ref. Dan Piponi's answer)
So I was curious as to whether the differential equatio... |
H: Extending the length of a curcumference by 1 meter
I once heard the following statement:
Take a piece of string and measure the length around a ball. Now add 1 meter to the string and stretch it out evenly around the ball. Obviously the diameter/radius will increase. Not sure by how much though.
Do the same thing f... |
H: Reference: Finite Groups and Geometry
The examples $C_n, D_n, A_4, S_4, A_5$ are the first nice examples of groups to relate with 2-D and 3-D Euclidean geometry. These groups can be investigated by studying action on sphere as described in M. Artin's book on Algebra. Next, Fano plane is another nice object to relat... |
H: Definition of $\operatorname{span}S$ and equivalence to $S$
I'm getting a little confused with the notation for spanning sets. In our notes we have the following definition.
Let $S$ be a vector space over the field $F$. Then:
$$\operatorname{span}S = \left\{\sum\limits_{i=1}^n a_iv_i:a_i \in F, v_i \in S\right\}$$
... |
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