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H: If $S$ consists of units then $S^{-1}R \cong R$
I want to show that if $S$ consists of units then $S^{-1}R \cong R$.
Can you tell me if my proof is correct?
Since $S$ consists of units, $S$ is zero-divisor free and hence $f: R \to S^{-1}R$, $r \mapsto \frac{r}{1}$ is injective. So we have an isomorphism $h: R \to f... |
H: How do I "learn" more difficult algebra?
I do not really understand where I was suppose to learn this kind of stuff. I am always told that my algebra knowledge is the biggest reason I am so bad at math. But I do not understand where I was suppose to learn it. I started math with pre-algebra and I learned basically ... |
H: does there exist a continuous function
$A_1=\{ \text {closed unit disk in plane}\}$ $A_2=\{(1,y):y\in \mathbb{R}\}$ $A_3=\{(0,2)\}$
We need to confirm: there exist always a continuous real valued function $f$ on $\mathbb{R}^2$ such that $f(x)=a_j $ for $x\in A_j$ $j=1,2,3$
$1$. Iff atleast two of these number are e... |
H: Choosing the sign of the separation constant for a vibrating string
Suppose we have this PDE problem
$$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$
$$\psi(0,t)=\psi(L,t)=0$$
It represents the vibrations of a string tightly stretched between two points.
The standard techni... |
H: completeness under different metric
would any one tell me whether $C[0,1]$ is complete under these metrics
1.sup norm i mean $\|f\|_{\infty}$
2.$\|f\|_{\infty,1/2}=\|f\|_{\infty}+|f(1/2)|$
3.$\|f\|_{2}=\sqrt{\int_0^1|f|^2dx}$
Under supnorm I know it is complete,I am not sure about the other two.
AI: for the second... |
H: Solving (or estimating) $x$ in $\tau=\log_x\left(\frac{x+1}{2}\right)$
How would one find a real value for $x$ that satisfies
$$\tau=\log_x\left(\frac{x+1}{2}\right),$$
given $0 < \tau < 1$ and $\tau \neq \frac{1}{2}$
(PS I'm not that good with math, so if this is impossible, please explain it to me like I'm 5). I ... |
H: When weak convergence implies moment convergence?
Given a sequence $(\mu_n)_n$ of probability measures on $\mathbb R$, which converges weakly to a probability measure $\mu$, when do we have
$$
\tag{1} \lim_{n}\int x^kd\mu_n(x)=\int x^k d\mu(x) \qquad \forall k\geq 0\;?
$$
Is "$\mu$ has compact support" a sufficient... |
H: How to compute sample variance from sample moments
Given $X_1, \dots , X_n$ i.i.d. and the two sample moments
$$M_1 = \frac{1}{n} \sum_{i = 1}^{n} X_i = \bar{X}$$
and $$ M_2 = \frac{1}{n} \sum_{i = 1}^{n} X_i^2$$
how can I compute:
$$ S^2 = \frac{1}{n} \sum_{i = 1}^{n} (X_i - \bar{X})^2$$
such as:
$$S^2 = f(M_1, ... |
H: Sum of Gaussian processes
I would like to prove that the sum of Gaussian processes is also Gaussian, to be precise, $M_t=W_t+W_{t^2}$, where $W_t$ is standard Wiener process. That is kind of obvious, but I am looking for some more rigorous, as short as possible proof, other than just saying that it is the sum of tw... |
H: Historic literature
I'm wondering if it's advantageous to read the original works of Gauss, Jacobi, Cauchy and others (in particular, Jacobi).
Many people say that it's worth it to read original (not translated) works of literature - you have the author's own diction and get a better feel of his/her cogitation. I ... |
H: Computing: $L =\lim_{n\rightarrow\infty}\left(\frac{\frac{n}{1}+\frac{n-1}{2}+\cdots+\frac{1}{n}}{\ln(n!)} \right)^{{\frac{\ln(n!)}{n}}} $
Compute the following limit:
$$L =\lim_{n\rightarrow\infty}\left(\frac{\frac{n}{1}+\frac{n-1}{2}+\cdots+\frac{1}{n}}{\ln(n!)} \right)^{{\frac{\ln(n!)}{n}}} $$
I'm looking for a... |
H: Equivalent definition of exactness of functor?
I'll use the following definition:
(Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences.
Now I'd like to prove the following (not entirely sure it's true but someone mentioned something like this to me some time ago):
Clai... |
H: Is the set of all probability measures weak*-closed?
Let $(\Omega,\Sigma)$ be a measurable space. Denote by $ba(\Sigma)$ the set of all bounded and finitely additive measures on $(\Omega,\Sigma)$ (see http://en.wikipedia.org/wiki/Ba_space for a definition). Is the set of all probability measures $\mathcal{M}_1(\Sig... |
H: How many $k+2$ letter groups in a $n$ letter string
Given an $n$ letter string of identical letters, how many $k+2$ letter words can be formed of adjacent letters?
By observing data I came up with n-(1+k), but I'm at a loss for a descent combinatorial explanation.
For example, if I had a 5 letter string and k=1 a... |
H: Proof that every element of A_5 is an involution or a product of two involutions?
It can be verified with brute force that the alternating group on 5 elements ($A_5$) has the property that every member is either an involution or can be written as the product of two involutions. Is there a simple proof of this fact... |
H: Convergence of a function series
Check whether function series is convergent (uniformly):
$\displaystyle\sum_{n=1}^{+\infty}\frac{1}{n}\ln \left( \frac{x}{n} \right)$ for $x\in[1;+\infty)$
I don't know how to do that.
AI: The series doesn't converge. Use integral test or Cauchy-condensation-test via monotonicity o... |
H: Taking fractions $S^{-1}$ commutes with taking intersection
Let $N,P$ be submodules of an $R$-module $M$ and let $S$ be a multiplicative subset of $R$. I think I proved $S^{-1}(N \cap P) = S^{-1}N \cap S^{-1} P$ but since my proof is not the same as the one given in Atiyah-MacDonald on page 39 I suspect there is so... |
H: Multiplicative Selfinverse in Fields
I assume there are only two multiplicative self inverse in each field with characteristice bigger than $2$ (the field is finite but I think it holds in general). In a field $F$ with $\operatorname{char}(F)>2$ a multiplicative self inverse $a \in F$ is an element such that
$$ a \... |
H: Proof of a test for series
I would like to prove that given three sequences ${a_n}, {b_n}\text{ and }{c_n}$ and knowing that:
They aren't necessarily of positive terms.
$a_n \leq b_n \leq c_n, \forall n \geq 1$
$$\text{If }\sum_{n = 1}^{+ \infty}{a_n}\text{ and }\sum_{n = 1}^{+ \infty}{c_n}\text{ are both converg... |
H: About the definition of Cech Cohomology
Let $X$ be a topological space with and open cover $\{U_i\}$ and let $\mathcal F$ be a sheaf of abelian groups on $X$. A $n$-cochain is a section $f_{i_0,\ldots,i_n}\in U_{i_0,\ldots,i_n}:= U_{i_0}\cap\ldots\cap U_{i_n}$; we can costruct the following abelian group (written ... |
H: Lower bound on Tail Probabilities
Inequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?
AI: Markov's inequality is also called the first moment method. What you want is the second moment ... |
H: Define a logical formula as another formula
I'm reading Dirk van Dalen's Logic and Structure and noticed that in many parts of his book he defines some formula to be an alias for another formula (he doesn't use the name alias, he just says that some formula will be defined as another formula).
Example Let $\phi$ be... |
H: Number of terms in an Arithmetic progression
1 and 20 are first and last terms of the arithmetic progression. If all the terms of this arithmetic progression are integers, then find the different number of terms that this arithmetic progression can have ?
AI: Here's an obvious generalization which may be interestin... |
H: Normal subgroups of $S_4$
Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?
In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?
AI: In any group, a subgroup is normal if and only if it is a union of conjugacy classes.
In $S_n$, th... |
H: Under what circumstances is the discrete metric space separable?
Under what circumstances is the discrete metric space separable?
Can anyone help me please?
AI: A space $X$ is separable if it contains a dense countable subset $D$.
Now that we know the definition we need to think about what it means for $D$ to be de... |
H: How to graph this?
I have a non-right triangle. I will call
the bottom or base edge $b$,
the top left edge $a$,
the top right edge $c$.
Let $ a=c+2 $ and $b=10$. How do I graph a curve where the graph $ x $ and $ y$ coordinates represent the vertex where edges $ a $ and $c$ meet?
AI: Taking the orig... |
H: Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$
One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?
$$\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$$
AI: The statement $\dfrac{\tan(x)-x}{x^3} \to c$ as $x \to 0$ is equ... |
H: $N$ is a matrix such that $N^3=0$
Given a $3\times 3$ matrix $N$ such that $N^3=0$, then which of the following are/is true?
$N$ has a non zero eigenvector
$N$ is similar to a diagonal matrix
$N$ has $3$ linearly independent eigenvector
$N$ is not similar to a diagonal matrix
Well, eigenvalues of $N$ are all zer... |
H: $f(1)=-3$ and $ f'(x)\geq7$ how small is $f(5)$?
Here is a test question in my textbook.
Suppose : $f(1)=-3$ and $f'(x)\geq7$ How small $f(5)$ can be possibly
:
a)$25$
b)$-21$
c)$28$
e)$31$
f) None of others.
I just have this : because $f'(x)>0$ then $f(x)$ increasing. So, for all $x$ greater than $1$... |
H: Matrix Multiplication and Function Composition
Given the vector space $F^n$ and two linear function $T,S:F^n \rightarrow F^n$ is it true that multiplying the representative matrices according to the standard basis of $T$ and $S$ is equivalent to the composition of their explicit formula's? I.E. Given a vector v in... |
H: Normal, Non-Metrizable Spaces
We know that every metric space is normal. We know also that a normal, second countable space is metrizable.
What is an example of a normal space that is not metrizable?
Thanks for your help.
AI: Edit: In light of the comments, I thought it prudent to give the precise definition of nor... |
H: $C \otimes A \cong C \otimes B$ does not imply $A \cong B$
Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$.
Today I proved both of the following:
$$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ and
$$ S^{-1} R \otimes M \cong S^{-1} M$$
Now I'm slightly ... |
H: Theorems/entailment notation
When defining a predicate logic system with natural deduction, we can define the syntatic entailment with the operator $\vdash$. Generally, I see authors using the formula $\vdash \phi$ to say that $\phi$ is a theorem of the logical system. However I was used to say that $\phi$ is a the... |
H: analysis limit question
Let f be an integrable function on $\mathbb{R}$. Show that $\lim_{t\rightarrow 0} \int_{\mathbb{R}}|f(x + t) -f(x)|dx = 0$.
I can make it work once it is shown to be true for $f\in C_c(\mathbb{R})$ but I am having trouble proving this case.
AI: If $f\in C_c(\Bbb R)$, then the support of $f... |
H: Is there a sequence that contains every rational number once, but with the "simplest" fractions first?
The Calkin-Wilf sequence contains every positive rational number exactly once:
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, ….
I'd consider 5/1 to be a "simpler" ratio than 8/5, but it app... |
H: Rigorous proof that $\frac{1}{3} = 0.333\ldots$
I'm a PreCalculus student trying to find a rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$, but I couldn't find it. I think (just think) that this proof would start by proving that
$\displaystyle\sum_{i=1}^{\infty}3\cdot10^{-i} = \frac{1}{3}$. My guesses ... |
H: Localization arguments in Dedekind domains
I am reading Serre's Local Fields, and have questions about the text. Specifically, pages 11 and 12.
1) Consider a Dedekind domain. We want to show that all fractional ideals are invertible. Serre claims that because the image of a fractional ideal under every localization... |
H: Proof of Turan's theorem
I'm following the proof of Turan's theorem on $\text{ex}(n,K^r)$ in Diestel's Graph Theory book (click to see the page) and something bothers me:
Since $G$ is edge-maximal without a $K^r$ subgraph, $G$ has a subgraph
$K=K^{r-1}$. By the induction hypothesis, $G-K$ has at most
$t_{r-1}(... |
H: How do mathematicians think about high dimensional geometry?
Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more.
How do mathematicians think about higher dimensions? Can intuitions about the meaning of dot-product, a... |
H: Relations of language/theory/signature
Say that the language of the first order logic is the collection of symbols that can be used in the formulas + the grammar (the rules that specify how they can be combined)?
1) However, the signature of the system can include additional symbols to be used in the formula. So, i... |
H: proving facts about $\alpha$-Hölder-continuous functions
I am studying myself some facts about $\alpha$-Hölder-continuous functions but I don't get any further by proving the following:
$(1)$ $\forall\alpha\in ]0,1]$ is $C^{0,\alpha}$ dense in $C^0(D)$ concerning the uniform norm and $D\subset\mathbb R^n$.
$(2)$ $\... |
H: Commutator map and the derived series
Let be $G$ a solvable group, let
$$ G=G_0\supset G_1\supset\cdots\supset G_k=1$$
be the derived series for $G.$ Is clear that $G_ {k-1}$ is abelian.
Now take $b\in G_{k-1}$ e $a\in G_{k-2}$ my question is how to see that:
$$a^{-1}b^{-1}ab \in G_{k-1}~~~?$$
this is a crucial... |
H: The subset of elements of order dividing $k$ in an abelian group is a subgroup
Suppose $G$ is an abelian group and $k$ is a natural number.
Prove $H = \{ g \in G : g^k = 1 \}$ is a subgroup of G.
I know I need to show that $1_G \in H$, existence of inverse element in group, and closure,
but how?
AI: $1 \in H$ since... |
H: Differential equation of $y = e^{rx}$
I am trying to find what values of r in $y = e^{rx}$ satsify $2y'' + y' - y = 0$
I thought I was being clever and knew how to do this so this is how I proceeded.
$$y' = re^{rx}$$
$$y'' = r^2 e^{rx}$$
$$2(r^2 e^{rx}) +re^{rx} -e^{rx} = 0 $$
I am not sure how to proceed from here... |
H: $f\colon M\to N$ continuous iff $f(\overline{X})\subset\overline{f(X)}$
Possible Duplicate:
Continuity and Closure
$f\colon M\to N$ is continuous iff for all $X\subset M$ we have that $f\left(\overline{X}\right)\subset\overline{f(X)}$.
I only proved $\implies$.
If $f$ is continuous then for any $X\subset M$,
$$X... |
H: Proving Continuity in Several Complex Variables
So I don't have a whole lot of experience in general proving continuity for multivariable functions, and I want to make sure I'm going about things correctly.
Prove that the function $B(z,w):=\int_0^1 t^{z-1}(1-t)^{w-1}dt$, for $z,w\in \mathbb{C}$, is continuous.
So I... |
H: Cooley-Tukey Algorithm?
Why does the Cooley-Tukey Fast Algorithm take $O(n \log n)$ time? The book derives this from the fact that evaluation takes time:
$T(n) = 2T(n/2) + O(n)$
and then uses the Master Theorem to arrive at the Big O Notation Time. Could someone explain how the above equation was derived?
How I see... |
H: Random variable with mean $\mu$ and variance $\sigma ^2$
I have never taken probability theory, and I wonder whether one can express some random variable $X$ with mean $\mu$ and variance $\sigma ^2$ in terms of $\mu$ and $\sigma$ only. Or at least something close to it.
Thank you for your help.
AI: No - the mean ... |
H: $g^{k}S=S \Rightarrow g$ has finite order
Let $G$ be a group (not necessarily finite), $g \in G$, $g \ne 1$. Suppose that $S \subseteq G$, $S$ is finite, $1 \in S$, and $gS=S$. It follows that $g^{k}S=S$ for all $k \in \mathbb{N}$. Does it also follow that the order of $g$ is finite?
My attempt: Since $S$ is finite... |
H: Finding direction vector
Can someone please explain how the direction vector was found in problem $2$ of this worksheet?
Below is an image of the problem $2$ of the worksheet.
AI: You want to know the tangent line to the ellipse at the given point along the plane $y=2$, so go ahead and plug in $y=2$ to obtain $4x^... |
H: Cauchy-Schwarz Inequality proof (for semi-inner-product A-module).
I am reading a proof of the following Cauchy-Schwarz Inequality and I don't understand one part of the proof:
Theorem:
Let $A$ be a $C^*$-algebra and let $E$ be a semi-inner-product $A$-module. Then
$$
\langle x,y \rangle ^* \langle x,y \rangle \leq... |
H: Deriving parameterization for hyperboloid
I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived. I would appreciate it if either someone could explain to me how such a parameteriz... |
H: Dimensionality of null space when Trace is Zero
This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216). I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me. Part (a) requires the read... |
H: circles and linear fractional transformations
I'm realizing how little (in some respects) I know about circles. Here's something that emerged out of something I was fiddling with. My question is whether this is
"well known" in the way that $229\times983=225107$ is "well known" (don't publish it unless you're pub... |
H: Finding the fixed point of a function
Let $p:A \times B \to \mathbb{R}$ be a nonnegative real-valued function on $A \times B$,
where $A$ and $B$ are arbitrary set.
Assume $f:A \to B$ and $g:B \to A$ are such that
\begin{align*}
f(a) &= \operatorname*{arg\,min}_b~p(a,b) \\
g(b) &= \operatorname*{arg\,max}_a~p(a,b... |
H: Is there any closed-form expression to calculate each element of the inverse of a matrix?
Considering a generic square matrix $A=(a_{i,j})$ we want to compute its inverse $A^{-1}=\left[a^{(-1)}_{i,j}\right]$.
Is there a way to express each $a^{(-1)}_{i,j}$ using a closed form expression?
AI: The $ij$ entry of $A^{-... |
H: Find the imaginary part of this sum
Let $$S = e^{i\alpha} + \frac{e^{i3\alpha}}{3} +
\frac{e^{i5\alpha}}{3^2} + \cdots$$
Find Im$(S)$ and show that it is equal to the sum
$$I = \sin(\alpha) + \frac{\sin(3\alpha)}{3} + \frac{\sin(5\alpha)}{3^2} + \cdots$$
So, I found that $S = \frac{3(3e^{i\alpha} - e^{-i\alpha})... |
H: Infinite products - reference needed!
I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to read Davenport's Multiplicative Number Theory, and the treatment of L-functions... |
H: Why a connected subspace of a locally connected space X is locally connected if X is the real line?
Why a connected subspace of a locally connected space X is locally connected if X is the real line?
Is this true if X is an arbitrary locally connected space?
Thanks for your help
AI: This is true if $X=\mathbb{R}$ b... |
H: Relation between defining polynomials and irreducible components of variety
I've been puzzled about some basic facts in (classical) algebraic geometry, but I cannot seem to find the answer immediately:
Let $V=V(f_1,\ldots,f_n)$ be a variety over some field $k$, and let $n > 1$. Suppose that $V$ turned out to be re... |
H: Find all functions with $f(x + y) + f(x - y) = 2 f(x) f(y)$ and $\lim\limits_{x\to\infty}f(x)=0$.
Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions:
(a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$;
(b) $\lim\limits_{x\to\infty}f(x) = 0$.
I found... |
H: Laplace transform having this unusual property in convolution?
Here is the problem
Solve $y'(t) = 1 - \int_{0}^{t} y(t - v)e^{-2v}dv$
The solution sets $\mathcal{L}(y) = Y(s)$ and does the following
Notice that in step 1, they have $$Y(s)\dfrac{1}{s+2}$$
Are they implying $$\mathcal{L}(y(t) * e^{-2t}) = \mathcal{... |
H: Name of this angle?
Given a planet and a point $P$, is there an existing name for the angle $\theta$ as seen in the diagram below? If not, what would you call it? ("Angle of elevation"?)
Thanks!
AI: I think you want the "angular radius" of the planet, although people more commonly use the "angular diameter" instea... |
H: Permuting 15 books about 2 shelves, with at least one book on each shelf.
From Discrete and Combinatorial Mathematics: An Applied Introduction:
Pamela has 15 different books. In how many ways can she place her books on two selves so that there is at least one book on each shelf? (Consider the books in each arrange... |
H: Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$
Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$.
It is clear that
$$
0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0
$... |
H: Why should a topological space itself be open?
For convenience, let $X$ be our space.
Specifically, can anyone name a few desirable properties or theorems that would fail if $X$ weren't required to be open? More generally, is there a part of topology that would completely fall apart?
It seems to me that we mainly w... |
H: Show that this function is not increasing on any interval containing $0$:
$$f(x) = \begin{cases}x + 2x^2\sin\left(\frac1x\right),& x\ne 0\\0,& x = 0\;.\end{cases}$$
I am having a tough time answering this question in a rigorous mathematical way, here is what I have tried:
I have proved in a previous part of this qu... |
H: How to resolve Skolem's Paradox by realizing what can be said of a set is relative to what is in the domain of some model?
I apologize in advanced if I'm hopelessly confused...
Skolem's Paradox, I suppose, can be put like this:
$M$ is a countable model of ZFC and $M$ implies the existence of uncountable sets.
I sup... |
H: Matrix with no eigenvalues
Here is another problem from Golan.
Problem: Let $F$ be a finite field. Show there exists a symmetric $2\times 2$ matrix over $F$ with no eigenvalues in $F$.
AI: The solution is necessarily split into two cases, because the theory of quadratic equations has a different appearance in chara... |
H: Infimum of a union
I have a set $X$ and a function
\begin{equation}
f: X \rightarrow \mathbb{R}
\end{equation}
and I am interested in the value
\begin{equation}
\inf\limits_{x \in X} f(x) \,.
\end{equation}
I can represent $X$ as
\begin{equation}
X = \bigcup\limits_{i \in I} X_i \,,
\end{equation}
where the index s... |
H: Linear dependence of linear functionals
Problem: Let V be a vector space over a field F and let $\alpha$ and $\beta$ be linear functionals on $V$. If $\ker(\beta)\subset\ker(\alpha)$, show $\alpha = k\beta$, for some $k\in F$.
A proposed solution is in the answers below.
AI: If $\alpha$ is the zero functional, we a... |
H: complex polynomial satisfying inequality
Each of the polynomial of the form $p(z)=a_0+\dots+a_{n-1}z^{n-1}+z^n$ satisfies the inequality
$\sup\left\{\,|p(z)|\,\big\vert\,|z|\le 1\,\right\}\ge 1$
Is this statement true or false that we have to find. well MMP says that sup will be attained at $|z|=1$ so when $|z|=1$... |
H: Finding the Laplace transform of $f(x)=|\cos(x)|$
I have function $f(x)=|\cos(x)|, x≥0$ and like to derive its Laplace transform. I am told that $f(x+\pi)=f(x)$. Help me please.
AI: Your function is periodic ($T=\pi$) so you can easily use the formula:
$L(f(x))=\frac{1}{1-e^{-sT}}\int_{0}^{T}e^{-sT}f(x)dx$
Note t... |
H: Question on Topological vector space 1
I have numbered this question as (1) because I will be posting series of questions where I don't understand. I hope its allowed.
I want to prove the following :
If $X$ is a topological vector space then : If $A\subset X $ then
$\bar A = \cap(A+V)$, where $V$ runs through... |
H: Finite set of matrices closed under multiplication
The following problem is from Golan's linear algebra book. I have been unable to make headway.
Problem: Let $n\in\mathbb{N}$ and $U$ be a non-empty finite subset of the $n\times n$ matrices over $\mathbb{C}$ which is closed under the multiplication of matrices and... |
H: Simple harmonic motion and trigonometry
Here's the question I would like help with:
A particle is moving in simple harmonic motion according to $x=6\sin
\left (2t+\frac{\pi }{2} \right )$. Find the first two times when the
velocity is maximum, and the position then.
Here is my working. I then let $x=0$ and d... |
H: Question about generating functions
I have question about generating functions.
I need to make this equation:
$(\frac{1}{1+x})^n\centerdot(1+x)^{2n} = (1+x)^n$
in this form:
$\sum\limits_{i=0}^{k}(-1)^iD(?,?)\binom{?}{?} = \binom{n}{k}$
How can I do this?
Thanks in advance
AI: The coefficient of $x^k$ in $(1+x)^n$ ... |
H: Finding Big-Theta
I need to use the Master Theorem to find $\Theta(f(n))$ if $f(n)=f(n/2)+3n$ and $f(1)=3$
I don't know how to use the MT in this case, can anyone provide help?
AI: The Master Theorem concerns relations of the form
$$f(n) = a f(n/b) + g(n)$$
which fits your problem with $a=1$, $b=2$ and $g(n)=3n$.
B... |
H: Bat and ball calculations
How would you work this out in MS-Excel?
A bat and ball cost a dollar and ten cents. The bat costs a dollar more than the ball. How much does the ball cost?
The answer is that the bat costs $1.05 and the ball costs $0.05.
Source: http://gizmodo.com/5918045/why-smart-people-are-actually-du... |
H: Expression for $n$-th moment
I stumbled upon an expression in an article of statistics for an $n$-th moment with $X$ being a random variable over $[0, \infty)$.
$$\mathbb{E} X^{n} = \int^{\infty}_{0} nz^{n-1}\; \text{Pr}(X > z) \; \text{dz}$$
Could someone enlighten me on why the above is true? It indeed works for ... |
H: Characterization of an element being algebraic over $\mathbb{Q}$.
Let $Aut(\mathbb{C}/\mathbb{Q})$ be the set of field automorphisms of $\mathbb{C}$ over $\mathbb{Q}$ (in short, all field automorphisms of $\mathbb{C}$). Let $x$ be an element of $\mathbb{C}$ such that the set $\{\sigma(x)|\sigma \in Aut(\mathbb{C}/\... |
H: Research in algebraic topology
I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future:
What else should I study after completing homology(basic), cohomology(basic) and homotopy theory(basic)?
After completing Hatc... |
H: Is there any difference between the absolute values operators $|z|$ and $\|z\|$?
Is there any difference between the absolute values operators $|z|$ and $\|z\|$ where $z=a+ib$?
AI: Usually, no. But if you see both notation used in the same discussion, it is possible that the author intended to define $\|z\|$ to be ... |
H: How to solve motion question?
A particle is moving at $x=3\cos\left(2t\right)$. Find the expression for velocity in terms of $x$.
I'm not sure where to start.
AI: All right, here's a slightly cleaner version of previous answers. We have
$$x=3\cos 2t,\frac{dx}{dt}=-6\sin 2t$$
Again, we use the trig identity
$$\sin^... |
H: Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
AI: The main thing is presumably that $AA^T$ is symmetric. Indeed $(AA^T)^T=(A^T... |
H: Prove: if $A(x)$ is divisible by $(x-a)^m$, then $A'(x)$ is divisible by $(x - a)^{m-1}$ [Derivative decrements multiplicity]
I have this question in my textbook:
If the polynomial $A(x)$ is divisible by $(x - a)^m$, then it's derivative is divisible by $(x - a)^{m - 1}$. Prove this.
I have really no clue on how to... |
H: What was the book about birds and sets?
Possible Duplicate:
KY Birds…which book is that from.
Does anyone know the title of the book that taught sets (I think) through examples with the birds?
AI: Possibly you’re thinking of Raymond Smullyan’s To Mock a Mockingbird, though its topic is combinatory logic, not set... |
H: Composition of Analytic Functions
I have a basic question in my mind and wish to consult your ideas:
Suppose $\Omega_1$ and $\Omega_2$ are regions, $f$ and $g$
are nonconstant functions defined in $\Omega_1$ and $\Omega_2$,
respectively, and $f(\Omega_1) \subset \Omega_2$. Define $h=g \circ f$.
What can we say abo... |
H: An inequality involving integrals
Let be $f:[0,1] \longrightarrow R $, $f$ is an integrable function such that:
$$\int_{0}^{1} f(x) \space dx = \int_{0}^{1} xf(x) \space dx=1$$
I need to prove that:
$$\int_{0}^{1} f^2(x) \space dx\geq4$$
AI: Note that if $h(x)=-2+6x$ then $$\int_0^1 h(x)\, dx = \int_0^1 xh(x)\, dx ... |
H: Why is the following % profit answer wrong
The question is:
A merchant buys an old carpet for $25 dollars.He spends 15 dollars to have it returned to to good condition and then sells it for 50 dollars . What is percent profit of his total investment ?
The answer is 20% and i get 40% . What am i doing wrong ?
He... |
H: Linear algebra: orthogonal projection?
(a) Find the orthogonal projection of $(-1, 0, 8)$ onto the normal vector to the plane $x-2y+z=0$.
Is this question saying to find the orthogonal projection in other words? The way the question is phrased "onto the normal vector to the plane" is confusing me..
(b) Find the dis... |
H: Length of the side of a discrete equilateral triangle from area
Firstly I haven't practised any mathematics in a long time, I understand that this might be pretty basic for math.stackexhcange, but I cannot seem to find any answers on the internet anywhere!
I've come across this problem at work, where basically if y... |
H: A question about similar triangles.
Please, I would like help in solving this problem: The sides of a triangle measure 2,3 and 4 cm respectively. The perimeter of a similar triangle is 36 cm. I want to find the length of each side of the second triangle.
I did this by trial and error and I got the sides to 8, 12... |
H: Is it possible to divide an equilateral triangle into 12 congruent triangles?
Can you divide an equilateral triangle into exactly 12 congruent triangles?
interesting question i haven't yet been able to work on.
The sides can be of any length.
AI: Here is a hint for one way to do it: |
H: A question on symmetric matrix and application of Spectral theorem.
Today in the class Prof. applied spectral theorem and wrote $A$ a semidefinite positive matrix as
$A=\sum \lambda_i v_i\times v_i$ , where $v_i$ are the eigenvectores and $\lambda_i$ are corresponding eigen values.
I think $"\times"$ should be so... |
H: Method for sketching $y = (1 – 3t + 2t^{2})e^{3t}$
I am doing some examination practice, and I've faced the following question:
Another particular solution which satisfies $y = 1$ and $\frac {dy}{dx} = 0$ when $t = 0$, has equation $$y = (1 – 3t + 2t^{2})e^{3t}$$
For this particular solution draw a sketch graph... |
H: Is there a word to describe the set of permutations of each member of the powerset of a set?
Just what it says on the tin:
For a set, X, is there a word to describe the union of sets of permutations of each member of the powerset of X?
AI: Your phrasing is a little unclear. If $X$ is our set and $\mathcal{P}(X)$ is... |
H: Visibility of the surface of a sphere
If you are $N$ radii above a sphere, what fraction of the hemisphere below you can you see?
The answer is so nice that it prompted another question: is there an intuition behind it, in the sense that one might have guessed it before going into the details of the computation? I... |
H: For set $X$ of integers, why is the square of the sum of its elements equal to the sum of pairwise products?
title pretty much says it all:
$\sum_{i \in X} \sum_{j \in X} ij = (\sum_{i \in X}i)^2$
I'm trying to find out why two ways of writing the same formula are identical, and this is what it comes down to. I fin... |
H: $|G|=12$ and it is isomorphic to $A_4$?
During reading a book, I have faced to this problem telling:
$G$ is a group of order $12$ such that $Z(G)$ has no element of order $2$ . Then $G≅A_4$.
Obviously, this group is not abelian and I think some information about $S_4$ is involved here because of the desired deduc... |
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