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H: Find the poles and residues
Find the poles and residues of $\frac{z \ln(z)}{(z^2+1)(z-c)}$, where $c$ is a real positive constant.
I've found the poles to be $z=i$, $-i$ and $c$. These are simple poles. How do I now calculate the residues?
AI: For simple poles it is very easy. For example, and choosing the main ... |
H: Simpler way to display mathematical model?
I am developing a mathematical model for the power usage (watts) of a computer device against its percentage of CPU load.
So far I have:
P(x) when x <= 25% = 8.5 + x*0.46
P(x) when 25% > x <= 50% = 20 + x*0.08
P(x) when 50% > x < 75% = 24 + x*0.01
P(x) when x >= 75% = 25 +... |
H: computation of more random variables
Suppose $U=X+Y$ and $V = Y+ Z$ and $X,Y,Z$ are independent $\operatorname{Bern}(1/3)$ distribution (so $P(X=1)=1/3$ ,$P(X=0)=2/3$ ) what is P(U=V)=?
answer given by teacher:
the chance is equal to $P(U=V)=P(X+Y=Y+Z)=P(X=Z)=P(X = Z = 1)+
P(X = Z = 0) = P(X = 1)P(Z = 1) + P(X = 0... |
H: Probability of getting red ball at ith step
An urn has r red and w white balls that are randomly removed one at a
time. Let $R_i$ be the event that the $i$th ball removed is red. Find
$P(R_i)$
I started with calculating $P(R_1) = \frac{r}{r+w}$. Next,
$$P(R_2) = P(R_2|R_1)\cdot P(R_1) + P(R_2|W_1)\cdot P(W_1... |
H: Complex Integration Problem. Please help.
Please help me with this one.
Calculate the integral:
$$\int_0^{2\pi} \frac{\mathrm{d}t}{a\cos t+b\sin t+c}$$
as $\sqrt{a^2+b^2}=1<c$.
I'm working on it for quite a while and somehow I can't manage to solve this problem.
AI: First use trigonometric identity: $$a\cos(x)+b\si... |
H: Complex Conjugate (FourierSeries)
Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and \begin{equation} f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*} \end{equation} be its complex Fourier series. What is its relationship to the Fourier series of $f(\overline{x}),\overline{f(x... |
H: Matrix with linearly dependent rows
Let $A$ be a matrix with linearly dependent rows.
What can we infer about the the solutions of:
$$Ax = 0$$
$$Ax = b$$
AI: In the first case, you have infinite solution. In particular, said that $n$ is the size of $A$ and $r < n$ is the rank of $A$, then there exists a linear su... |
H: Alternating sign for unequal numbers
How to model a function which satisfies following condition:
$$
f(x) = \begin{cases}
1 & x \in 3,7,11,\ldots\\
-1 & x \in 1,5,9,\ldots
\end{cases}
$$
The first result can be generated using $4n+1$ and the second with $4n-1$ for $n \in \mathbb{N}$. Can this be modelled... |
H: Ways of expressing permutations as products of transpositions
Determine whether the following permutation is even or odd and write
it as a product of transpositions in two different ways.
$(1527)(3567)(273)$
So far, I have the following:
$(1527)(3567)(273) = (15)(12)(17)(35)(36)(37)(27)(23) = (57)(51)(56)(53)$... |
H: Show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3\cdot4^{n+2} +5\right)$ using congruences
Using congruence theory, show that $ \forall n \in \mathbb N$, $9\mid\left(10^n + 3 \cdot 4^{n+2} +5\right)$. The proof is quite simple with induction, but how can it be proved with congruences?
AI: HINT:
As $10\equi... |
H: Truth and Definability Lemmas
I'm slightly confused about truth and definability lemmas (sometimes called forcing theorem A and forcing theorem B) of forcing. I've been using Kunen's new text and from his remarks in the matter I think it should be understood as a schema in the meta-theory as follows:
Let $\varphi... |
H: Proving that two spans are equal
How can we prove that $span(A)$ and $span(B)$ are equal? Say we have a set of vectors $A(v_{1}, v_{2})$ and $B(w_{1},w_{2})$, i was thinking to prove that A was contained B and viceversa, but how to do it though? This is where I am lost (note: I can't use Gram-Schmidt process).
AI: ... |
H: Minimum value of $ f(x) = x\log_2x +(1-x)\log_2(1-x) $
What is the minimum value of the following function for $ 0<x<1 $ ? Here the base of logarithm is 2 .
$ f(x) = x\log_2x +(1-x)\log_2(1-x) $
AI: Changing the base you have $\log_2x=\frac{\ln x}{\ln 2}$. So you have
$$
f(x)=\frac{1}{\ln 2}\left(x\ln x+(1−x)\ln(1... |
H: Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts
$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$
None of these fractions are equal to 0.We need to show that,
$(a+b+c)(x+y+z)=ax+by+cz$
This question comes from a chapter that wholly deals with factoring homogeneous cyclic polynomials.I multiplied the three sides of the first equ... |
H: Vector subspace clarification
We need to check if the following sets are vector subspaces:
$$ S_1=\{(0,0),(1,2),(2,1) \};V=(\Bbb Z_3)^2 \\
S_2=\{P\mid\exists x \in \Bbb R :P_{(x)}=0 \};V=\Bbb R_n [x]$$
For $S_1$ I just add: $u+v=(1,2)+(2,1)=(3,3)=(1,1) \notin S_1$ so this is not a vector subspace.
For $S_2$, P ... |
H: Translating from informal logical notation to formal logical notation
While introducing formal logical notation, the book I'm reading says the following:
"$\forall x$ in $D$. $P(x)$" can be written as "$\forall x (x$ in $D \rightarrow P(x)$".
"$\exists x$ in $D$ such that $P(x)$" can be written as "$\exists x(x$ i... |
H: Modular Arithmatic
I have been struggling with modular arithmetic, and I would like to try and finally grasp the concept.
In particular, solving problems like $7^{30}$ mod 49.
I know I will have to use Fermat's Theorem and from it we will have $7^{31-1}=1$ mod 31
But from here I am stuck.
Thanks for the help!
AI: ... |
H: Uniform convergence of $f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ on $ [0,1]$
Is $f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ uniform convergence on $ [0,1]$?
Of course we have $f_n \rightarrow \cos(x)$ but how prove that $$\operatorname{sup} \left| \cos \left( \sqrt{x^2 + \frac{1}{n} }\righ... |
H: Relationships of orthogonal subspaces
I'm struggling with the following problem:
Let $U_1$ and $U_2$ be supspaces of $V$. What is the relation between $(U_1+U_2)^\perp$ and $U_1^\perp \cap U_2^\perp$; and between $(U_1 \cap U_2)^\perp$ and $U_1^\perp + U_2^\perp$?
I would say that if $U_1^\perp = U_2$ then $U_1+U_2... |
H: Is it wrong to write a linear system as below?
Suppose we have the following linear system
\begin{align}
2x_1+3x_2+4x_3&=1\\
2x_2-3x_3&=6\\
0&=0.\\
\end{align}
Is it wrong to write the above linear system by including all the zero coefficients as below
\begin{align}
2x_1+3x_2+4x_3&=1\\
0x_1+2x_2-3x_3&=6\\
0x_1+0x_2... |
H: Explain/illustrate Goedel's theorems and possible implications to non-mathematicians
I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P Smith's Introduction to Gödel's th... |
H: About blocks on permutation groups
I have the following situation:
$G$ is a permutation group, transitive on $\Omega$, and $B$ is a non-trivial block.
(for every $x\in G$, a block $B$ verifies either $B\cdot x=B$ or $B\cdot x\cap B=\emptyset$)
Then I have to prove that $\Sigma=\lbrace{B\cdot x\;/\;x\in G}\rbrace$ i... |
H: Klein's j-invariant is automorphic w.r.t. the modular group $SL^\pm(2, \mathbb Z)$ (solution verification)
I want to solve the following problem from Ahlfors' text. I think I got the solution right and I would like to verify it.
Show that the function
$$J(\tau)=\frac{4}{27} \frac{(1-\lambda +\lambda^2)^3}{\lambd... |
H: How to deal with series of the form $\sum^{\infty}_{n=1}f(n)^{(-1)^n}$
I have no idea on how to prove series like $$\sum^{\infty}_{n=1}\left(\frac{1}{n^x}\right)^{(-1)^n}$$ for natural $x$ -divergent or convergent. By intuition it seems they diverge .
AI: Marking $k=n/2$ the sum can be written as (without $n=1$, wh... |
H: Is it valid to apply an operation to coordinates on a graph? Ex: $2(a,b) = (2a, 2b)$?
As the title says, is it valid to do something like $2(a,b)$ where $(a,b)$ are points on a graph, such that $(a,b)$ becomes $(2a,2b)$ ? or is this not valid because coordinates cannot be changed using an operation?
AI: The $xy$-pl... |
H: Solution of a system of linear equations
I tried to solve a system of three linear equations in three unknowns. After a series of elementary row operations the augmented matrix of the system of linear equations becomes
$$\left(\begin{array}{ccc|c}1 & 0 & 0 & 3 \\0 & 1 & 0 & 2 \\0 & 0 & 1 & 3\end{array}\right).$$
Ne... |
H: Calculating $\phi(x^y)$
I know how to compute $\phi(x)$ like $\phi(21)$ or $\phi(7)$ but how can I compute $\phi(x^y)$. Specifically how can I compute $\phi(5^{20})$?
AI: You can use the general formula: if |$d =gcd(m,n)$ then
$$\phi(mn)=\phi(m) \phi(n)\frac{d}{\phi( d)}$$
In particular, if $m|n$ then $d=m$ an you... |
H: Question from a conservation law example in Evans' PDE book
I'm trying to fill in some details in an example given in Evans' PDE book, chapter 3.4, example 1 on page 139. Starting with an initial-value problem for Burgers' equation:
\begin{equation}
\begin{cases} u_{t}+\left(\frac{u^{2}}{2}\right)_{x}=0 &\mbox{in ... |
H: Can anybody give me a proof of binomial theorem that doesn't use mathematical induction?
I have seen the proofs of Binomial theorem that use induction, but I would like to know if there is any other way to prove the theorem (apart from the combinatorial way that is already there).
AI: $(a+b)^n = a^n + b^n + \text{ ... |
H: This problem of probability
If every time I go out I have 1% of chance of finding a coin of the street, if I go out 100 times, what are the chances that I will find at least 1 coin on the street?
At first glance I thought that I would have to sum the chances of finding a coin every time I go out, but that would be... |
H: Riemann Integral Estimation
I have been having difficulties with the following problem:
Show that $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin(x)}{x(5+x)} dx < \frac{\pi}{10}.$
My attempt at a solution:
I know that $-1 \leq \sin(x) \leq 1$, and the denominator of the problem has a domain of $(-\infty, -5) \cu... |
H: Vector count and polynomials
How many vectors there are in: $(\Bbb Z_p)_n[x] \ \{n\in\Bbb N, \ p \ is \ a \ prime\}$
I really doubt that we need to use Gauss' formula since we haven't covered that, so there must be a simpler way to count all the vectors in mod(p) up to the nth polynomial but I'm clueless.
Any hint... |
H: A coin is flipped 8 times: number of various outcomes
A coin is flipped eight times where each flip comes up
either heads or tails. How many possible outcomes
a) are there in total?
b) contain exactly three heads?
c) contain at least three heads?
d) contain the same number of heads and tails?
AI: First, you need to de... |
H: If every subset of X is open then there are no limit points in X
Let $(X,d)$ be a metric space. Then if every subset in $X$ is open there are no limit points in $X$.
This is probably a very simple question, but I just can't seem to get anywhere with it.
AI: Hint: $x$ is a limit point if every open neighborhood of $... |
H: A distribution $u=\frac{1}{x}$
I am interested in finding a distribution $u \in \mathcal{D}'(\mathbb{R})$ such that
$u=0$ on $(-\infty,0)$ and $u=\frac{1}{x}$ on $(0,\infty)$.
This is exercise 1.4 in Friedlander. Hints or help is welcome!
Thanks.
AI: Hint:
$\log|x|$ is locally integrable and $(\log|x|)'=1/x$. |
H: A way to solve matrices within equation
Solve for X
$$2X+X^t=B$$
Where $X,B, $are matrices, and $ X^t$ stands for transpose.
I was trying to work with indices but it doesn't seems to work...
AI: Note taking the transpose of the equation $2X+ X^T = B$, we get that
$$2X^T + X = B^T$$
Solving which gives us
$$2(B-2X) ... |
H: Normal Subgroups and Internal Direct Products
Why are Normal Subgroups important?
Why are Internal Direct Products important?
I'm studying abstract algebra and I have always wondered about its relevance and usefulness.
Does anyone could help me please?
AI: Normal subgroups are important for the same reason that fac... |
H: What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour?
This sounds more like a brain teaser, but I had some kink to think it through :( Suppose you're parking at a non-parking zone, the probability to get a parking ticket is 80% in 1 hour, what is the probability ... |
H: Real roots of $3^{x} + 4^{x} = 5^{x}$
How do I show that $3^{x}+4^{x} = 5^{x}$ has exactly one real root.
AI: Hints:
Let $\displaystyle f(x) = \biggl(\frac{3}{5}\biggr)^{x} + \biggl(\frac{4}{5}\biggr)^{x} -1$
Note : $f'(x) < 0$ and $f(2)=0$. Apply Rolle's Theorem. |
H: Any nonabelian group of order $6$ is isomorphic to $S_3$?
I've read a proof at the end of this document that any nonabelian group of order $6$ is isomorphic to $S_3$, but it feels clunky to me.
I want to try the following instead:
Let $G$ be a nonabelian group of order $6$. By Cauchy's theorem or the Sylow theorem... |
H: $\varphi\colon M\to N$ continuous and open. Then $f$ continuous iff $f\circ\varphi$ continuous.
$\varphi\colon M\to N$ continuous and open. Then $f\colon N\to P$ continuous iff $f\circ\varphi\colon M\to P$ continuous.
"Proof:" Let $A\subset P$ open then exists $U\subset M$ such that $(f\circ \varphi)[ U]=f(\varph... |
H: Saturated ideal
Let $k$ be a field, let $I \triangleleft k[X_1,\dots,X_n]=S$ be an ideal and fix $f \in S$.
The saturated ideal of $I$ is $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \ s.t. \ f^mg \in I \}=\displaystyle\bigcup_{i \geq 1} I:f^i$.
Prove that $I^{sat}=I:f^m \Leftrightarrow f^m=f^{m+1}$.... |
H: Show that $\int_{0}^{1} \sqrt{x}\,e^{-x^2}dx \leq \frac{\pi}{6}$
I need to prove that the following inequality holds:
$$\int_{0}^{1} \sqrt{x}\space e^{-x^2}dx \leq \frac{\pi}{6}$$
No progress on it, yet. Any suggestion is welcome. Thanks.
AI: If Cauchy-Schwarz inequality is in one's toolkit, one can write
$$
\left(... |
H: How many 4-element subsets of a given set contain no consecutive integers
How many 4-element subsets of the set S = {1,2,3...,10} contain no consecutive integers?
We went over a problem like this is class. The lecturer said the answer is the number of integer solutions to
$b_0 + b_1 + b_2 + b_3 + b_4 = 9$
Where f... |
H: Finite ordinal to the power a = a
Let $\alpha>\omega$ be an ordinal such that
$2^\alpha$ = $\alpha$.
Then $\alpha$ is an epsilon number?
I have tried many different ways, but i can only work with the left side of $\alpha$(e.g, I have proved such ordinals satisfy $\omega$$\alpha$ = $\alpha$ and etc), but i think it'... |
H: Is a sphere a closed set?
The unit sphere in $\mathbb{R}^3$ is $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \}$. I always hear people say that this is closed and that it has no boundary. But isn't every point on the sphere a boundary point? Since for every point $x$ on the sphere, any open ball in $\mathbb{R}^3$ (defined by $\... |
H: Determine $\alpha$ for which this vector equation takes place.
Assume $ABCD$ is a parallelogram. $O$ is the intersection of the diagonals and $M$ an arbitrary point in the same plan. Determine $\alpha$ for which the following relation takes place:
$\overrightarrow{MA}\cdot \overrightarrow{MC} = ||OM||^2+\alpha\cdot... |
H: Rabinowitz trick and saturated ideals
Let $k$ be a field and $I\trianglelefteq k[X_1,\dots,X_n]=S$ an ideal, generated by $\{f_1,\dots,f_s\}$. Fix $f \in S$ and let $Y$ be a new indeterminate. Let $\tilde{I}=(f_1,\dots,f_s,1-fY)\trianglelefteq S[Y]$.
Let $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \... |
H: If $d_1, d_2$ are metrics of $X$, is it true that $d_1 +d_2 $, $d_1 - d_2$, $d_1\cdot d_2$, $\sqrt d_1$ are metrics on $X$?
If $d_1, d_2$ are metrics of $X$, is it true that $d_1 +d_2 $, $d_1 - d_2$, $d_1\cdot d_2$, $\sqrt d_1$ are metrics on $X$?
Here is my attempt:
If we take $d_1 = d_2 $ = standard metric on ... |
H: Quotient field of a domain
Let $A$ be a commutative domain and $K=Quot(A)$, its field of fractions (quotient field).
Prove that $K$ is a f.g. $A$-module if and only if $A=K$.
AI: Let $\{\frac{1}{a_1},\ldots,\frac{1}{a_n}\}$ generate $K$ as an $A$ module, with each $a_j\in A$, and let $x$ be an element of $K$. Then... |
H: random graphs without cycles
Recall that a closed walk (in a undirected graph) is a cycle if its vertices are pairwise distinct.
Does there exist random constructions of bipartite graphs without cycles with high probability?
AI: If a graph has no cycles then it is clearly bipartite. Moreover a graph without
cycles... |
H: A few questions on the different aspects of differentiation from $\mathbb{R}^2 \to \mathbb{R}$
Given the function $$f(x,y) = \begin{cases} \frac{x^3y^2}{x^4 + y^4} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ How would you prove (or disprove) the following statements:
$f$ has all partial derivatives at $... |
H: How many Euler tours exist in a given graph?
A Euler tour is defined like that:
Let $G = (V, E)$ be a graph and $C$ a circuit in $G$.
$C$ is called Euler tour $\Leftrightarrow$ every edge $e \in E$ is exactly once in the circuit.
If a graph $G$ has at least one Euler tour $C$ that starts with $v \in V$, can $G$ hav... |
H: how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?
What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant?
Polynomial solutions don't seem to work, because the LHS will always have higher degree than the ... |
H: Help with$ _nP_0 + _nP_1 +_ nP_2 + _nP_3 + .. + _nP_x =$?
Can someone give me a formula to calculate $ _nP_0 + _nP_1 +_ nP_2 + _nP_3 + .. + _nP_x$ ?
I need a simple formula to calculate this.
MY actual question is
In a certain programming language identifiers must meet the following requirements: - the first cha... |
H: A tricky sum to infinity
I try to solve the following tricky limit:
$$\lim_{x\rightarrow\infty} \sum_{k=1}^{\infty} \frac{kx}{(k^2+x)^2} $$
For some large values, W|A shows that its limit tends to $\frac{1}{2}$ but not sure how to prove that.
AI: ETA: These bounds are wrong, as $\frac{kx}{(k^2+x)^2}$ is not monoton... |
H: Integral of $\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$
I am trying to find $$\int_\sqrt{2}^2 \frac{dt}{t^2 \sqrt{t^2-1}}$$
$t = \sec \theta$ $dt = \sec \theta \tan\theta $
$$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \sqrt{\sec^2 \theta-1}}$$
$$\int_\sqrt{2}^2 \frac{dt}{\sec ^2 \theta \tan^2 \theta}$$
$$\int_\sqrt... |
H: Generating function for Banach's matchbox problem
Here's the description for Banach's matchbox problem from Concrete Mathematics EXERCISE 8.46 (edited)
Stefan Banach used to carry two boxes of matches, one containing $m$ matches and the other one containing $n$ matches. Whenever he needed a light he chose a box at... |
H: question about real numbers
My question is:
Solve $(x-a)(x-b)(x-c)=0$ where $a,b,c$ belong to real numbers.
By observing, I found out that $x$ can be $a$ or $b$ or $c$.
AI: In the real numbers, if a product $xy$ is zero (look in your textbook to find this, usually near the "factor theorem"), then one of the terms ... |
H: How to prove spherical harmonics are orthogonal
A lot of texts and derivations eg here simply say:
"The Spherical Harmonics are orthonormal, so:
$$
\int{ Y_l^m Y_{l'}^{m'} } = \delta_{ll'}\delta_{mm'}
$$
And if you try any (l,m) pair you will find this always works out.
But how do you prove they are orthonormal for... |
H: Convergence of $\sum\limits_{n = 1}^{+\infty} e^{-\sqrt{n + 1}}$
I need to show, using the comparison test, that $\sum\limits_{n = 1}^{+\infty} e^{-\sqrt{n + 1}}$ converges, but I can't come up with a larger convergent series.
Thanks.
AI: Since $\mathrm e^u=\sum\limits_{k=0}^{+\infty}\frac{u^k}{k!}\gt\frac{u^4}{4!}... |
H: A problem about an endomorphism of the vector space $\mathrm{Mat}_2(\mathbb R)$
Let $f_A: \mathrm{Mat}_2(\mathbb R)\rightarrow \mathrm{Mat}_2(\mathbb R)$ be such that $f_A(X)=AX$ be an endomorphism of vector spaces. Clearly if $A$ is invertible, then $f_A$ is invertible and ${(f_A)}^{-1}=f_{A^{-1}}$. How can I pro... |
H: Finding $\int \frac {dx}{\sqrt {x^2 + 16}}$
I can not get the correct answer.
$$\int \frac {dx}{\sqrt {x^2 + 16}}$$
$x = 4 \tan \theta$, $dx = 4\sec^2 \theta$
$$\int \frac {dx}{\sqrt {16 \sec^2 \theta}}$$
$$\int \frac {4 \sec^ 2 \theta}{\sqrt {16 \sec^2 \theta}}$$
$$\int \frac {4 \sec^ 2 \theta}{4 \sec \theta}$$
$... |
H: A problem concerning Series
I meet a problem in a textbook( Page 125 in Partial differential Equation written by Fritz John).
Find sequences $a_k$, $b_k$ for which the series $\sum\limits_{k=1}^{\infty}k(a_k^2+b_k^2)$ diverges, while the series $\sum\limits_{k=1}^\infty(\left|a_k\right|+\left|b_k\right|)$ converges... |
H: Continuous function on metric space
I'm trying to show:
Let $(X,d)$ be a metric space and let $A, B$ be nonempty subsets, which are also closed and disjoint. Let $\rho_A:X\to \mathbb{R}$ be such that $\rho_A=d(x,A)$ and $\rho_B:X\to \mathbb{R}$ be such that $\rho_B=d(x,B)$, with $x\in X$ (distance from one point t... |
H: Question about Continuity of Path Integrals
I have this continuous function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined on an open set $\Omega$.
I also have a family of identical smooth curves up to translation $z_{t}:[a,b]\rightarrow\mathbb{C}$ in $\Omega$ and $t \in [0, T]$, where the parameter $t$ specifies some... |
H: discrete random variables transformation
I have two discrete random variables $X$ and $Y$, let $P_X$ and $P_Y$ be the PMF of the random variables, if $Y=X^2$ ,I want to know the PMF of Y in terms of PMF of X ?
I know how to do it with continuous RVs
$P_Y(y)=((P_X(sqrt(y))+P_X(-sqrt(y)))/(2*sqrt(y))$
is there a di... |
H: Solving $|x-2| + |x-5|=3$
Possible Duplicate:
How could we solve $x$, in $|x+1|-|1-x|=2$?
How should I solve:
$|x-2| + |x-5|=3$
Please suggest a way that I could use in other problems of this genre too
Any help to solve this problem would be greatly appreciated.
Thank you,
AI: Well, there are a couple of ways... |
H: Solving for $x$ in terms of $y$
For some reason, I'm having a hard time solving for $x$ in this equation: $$x^2=y,-2\lt x \lt 3.$$ I could use some help. Thanks.
AI: Without the constraint $-2 < x <3$, the solution to $x^2 = y$ is given by $x = \sqrt{y} \text{ or } - \sqrt{y}$ whenever $y \geq 0$. The constraint $-... |
H: second derivative does not exist at specified points
Would any one give me an example or hint how to construct a function whose second derivative does not exist at some specified points say at n number of points. for first derivative I have the modulas function( i.e $|x|$), I need an example for second.
AI: Really ... |
H: Asymptotic Analysis of trignometric functions
I am new to Asymptotic analysis so please bear with me and i apologize if the following question is not well formed or is trivial. I am trying to figure out Asymptotic behavior of the following two functions.
$$\phi_1(x) = 1 - \cos\left(1-\cos\left(x\right)\right)$$ and... |
H: Prove that every real vector space has infinitely many vectors
I can't seem to wrap my brain around this one, so I figured someone here could point out the connection I'm not making. I've been asked to prove that every real vector space other than the trivial one (V = {0}) has infinitely many vectors. This is int... |
H: A way to split this integral/norm
My last question hasn't got any replies so I'll try another..
Is there a way to split the following integral ($g$ is arbitrary) $$\int{f^2g}$$ so that I instead have an expression involving the $L^2$ norm on $f$ and either $L^2$ or $L^\infty$ norm on $g$? In particular, I want some... |
H: Why and when in the computation of an area or volume, into the integral we use function f = 1?
Sorry if this is something really easy..
As said in the title, why when we want to compute the area or the volume there is something like this:
$$A(D_1) = \iint 1 \ dx\,dy\text{ or }\iiint 1 \, dx \, dy\ ?$$
And when does... |
H: tough series involving digamma
I ran across a series that is rather challenging. For kicks I ran it through Maple and it gave me a conglomeration involving digamma. Mathematica gave a solution in terms of Lerch Transcendent, which was worse yet. Perhaps residues would be a better method?.
But, it is $$\sum_{k=1}^... |
H: Why is the ring of holomorphic functions not a UFD?
Am I correct or not? I think that a ring of holomorphic functions in one variable is not a UFD, because there are holomorphic functions with an infinite number of $0$'s, and hence it will have an infinite number of irreducible factors! But I am unable to get a con... |
H: How many ways can the sum of 4 positive integers equal 12?
I am currently doing research in Combinatorial Geometry and I have been able to reduce a quite complicated problem relating to extending the Newton-Gregory problem of kissing spheres to a simple number theory problem and then checking every case to see that... |
H: Prove: The weak closure of the unit sphere is the unit ball.
I want to prove that in an infinite dimensional normed space $X$, the weak closure of the unit sphere $S=\{ x\in X : \| x \| = 1 \}$ is the unit ball $B=\{ x\in X : \| x \| \leq 1 \}$.
$\\$
Here is my attempt with what I know:
I know that the weak closure... |
H: Sum of sets of measure zero
Let $A$ and $B$ be two subsets of $\Bbb R$ of measure zero. Is it true that the Minkowski sum $A+B = \{ a + b \mid a \in A, b \in B \}$ has measure zero as well? I think so but I can't prove it. The usual trick with the convolution $\mathbf 1_A \star \mathbf 1_B$ does not seem to lead to... |
H: How to eliminate Q(x,y) in system of two PDE
Analytical problem is simple but I am not sure is it possible in this kind of system. I will give my idea. We can apply on first equation $ \frac{\partial }{\partial x} $ and then that it is possible to substitute from second equation $ \frac{\partial Q}{\partial x} $ i... |
H: how to find this limit $\lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} = -2$ without using L'Hôpital's rule
I am looking for simple trigonometric or algebraic manipulation so that this limit can be solved without using L'Hôpital's rule
$$ \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ... |
H: Three points on a line
Given $\triangle ABC$. On the side AB externally is constructed square ABPQ. On the side AC internally is constructed square ACMN. AH is the altitude. If $O_1$ and$O_2$ are the centers of the two squares, prove that $O_1, O_2 $ and $H$ are collinear.
AI: Let $A'$ be the intersection of the al... |
H: (Un-)Countable union of open sets
Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well.
Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets.
My question is, does thereby follow that $\bigcap_{i \in \mathbb{N}} A_i$ and $\bigcup_{i \... |
H: Functions and parameters
I have a question which came up studying a certain (system of) PDE:
If I can write a function $f(x,y,z)$ as $g(x-y,z)$ as well as $h(x^2-z, y)$ with $g$, $h$ continuous and differentiable, does it follow that $f$ is constant? I can't imagine it does but I can't find any counterexample..
Th... |
H: The poles and residues of $f(z)=\frac{\sin z}{z^2}- \frac{\cos z}{z}$
Let $f(z)=\frac{\sin z}{z^2}- \frac{\cos z}{z}$ then
$f$ has a pole of order $2$ at $0$,
$f$ has a simple pole at $0$,
$\int_{|z|=1}f(z)=0$ anticlockwise,
residue of $f$ at $0$ is $-2\pi i$,
I am not able to figure out which of the statement ... |
H: given vector a, how to find vector b such that inifinitive norm of $a -b$ is smallest
I'd appreciate any hint for the following problem: Given a vector $a=(a_1, a_2,\ldots,a_n)$, how can I find a vector $b=(b_1, b_2,\ldots,b_n)$ such that $b_1 \leq b_2 \leq \cdots \leq b_n$ and the infinitive norm of the vector $a-... |
H: Length of curve in metric space
Let $(X, d)$ be a metric space, and $\gamma: [a,b]\to X$ be a curve. For any partition $P=\{a=y_0<y_1<\cdots<y_n=b\}$, one can associate to it the lengh of the "inscribed polygon" $$\Sigma(P)=\sum_id(\gamma(y_i), \gamma(y_{i+1}))$$
Then we define the length of the curve to be
$$L(\ga... |
H: $n\times n$ matrix with char poly $x^{n-2}(x^2-1)$
Let $A$ be an $n\times n, (n\ge2)$ matrix with char poly $x^{n-2}(x^2-1)$ Then which of The following is true?
$A^n=A^{n-2}$,
$r(A)=2$,
$r(A)$ is atleast $2$,
there exist non zero vector $x,y$ such that $A(x+y)=x-y$,
Well, I can only see that 1 is true from the c... |
H: analytic map defined on $D=\{z:|z|<1\}$
Let $f$ be an analytic map defined on $D=\{z:|z|<1\}$ such that $|f(z)|\le 1\forall z\in D$. Then which of the following statements are true?
There exists $z_0\in D$ such that $f(z_0)=1$.
The image of $f$ is an open set.
$f(0)=0$
$f$ is constant.
Well first I wonder how can... |
H: Can every polynomial be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 +\ldots + a_n (x-x_0)^n$
Can every polynomial of degree $n$ be written as $a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \ldots + a_n (x-x_0)^n$ while $x_0$ is an arbitrary but given real number and all $a_k$ can be freely chosen? Is there a proof for this... |
H: which of the set is open or closed
$X=\{C[0,1],||_{\infty}\}$
$F=\{f\in X : f(1/2)=0\}$
$G=\{g\in X : g(1/2)\neq 0\}$
I need to find which one is open and which one is closed set in $X$.
Well $F$ is closed I guess, as if say $h(x)$ be a limit point of $F$ so there exist sequence $s_n(x)\in F$ such that $s_n(x)\righ... |
H: History of Modern Mathematics Available on the Internet
I have been meaning to ask this question for some time, and have been spurred to do so by Georges Elencwajg's fantastic answer to this question and the link contained therein.
In my free time I enjoy reading historical accounts of "recent" mathematics (where, ... |
H: Long division in integration by partial fractions
I am trying to figure out what my book did, I can't make sense of the example.
"Since the degree of the numberator is greater than the degree of the denominator, we first perform the long division. This enables us to write
$$\int \frac{x^3 + x}{x -1} dx = \int \left... |
H: Determining validity of an argument
I would like to know whether the following argument is valid.
Some amphibians live in the water
All fish live in the water
Therefore, some fish are amphibians.
AI: Venn diagrams are sometimes helpful in seeing what’s going on. The conclusion Some fish are amphibians would fit th... |
H: Axiom of Choice (for example in the Snake Lemma)
If we have to make a choice, but in the end it doesn't matter what choice we made, did we really make a choice to begin with?
More explicitly, somewhere in the standard diagram-chasing proof of the snake lemma for $R$-modules (see http://mathworld.wolfram.com/SnakeLe... |
H: Integral of$\int_0^1 x\sqrt{2- \sqrt{1-x^2}}dx$
I have no idea how to do this, it seems so complex I do not know what to do.
$$\int_0^1 x\sqrt{2- \sqrt{1-x^2}}dx$$
I tried to do double trig identity substitution but that did not seem to work.
AI: Since you tried to use a trig identity, I'll use one in this solution... |
H: Integral of $\int \frac{5x+1}{(2x+1)(x-1)}$
I am suppose to use partial fractions
$$\int \frac{5x+1}{(2x+1)(x-1)}$$
So I think I am suppose to split the top and the bottom. (x-1)
$$\int \frac{A}{(2x+1)}+ \frac{B}{x-1}$$
Now I am not sure what to do.
AI: But $$\frac{5x+1}{(2x+1)(x-1)}\ne\frac{5x+1}{2x+1}+\frac{5x+1... |
H: To show if function $f(x), 0\leq x<\infty$ is convex then $a_{n}=f(n)$ is convex sequence.
How can we show that if the function $y=f(x)$ is convex in $0\leq x<\infty$, then the points $a_{n}=f(n)$ form a convex sequence.
AI: Is it true that $$f(n+1)\leq\frac{1}{2}\left(f(n)+f(n+2)\right)\,\,?$$Hint: take $\,\,x_1:=... |
H: Proving that two lines are parallel
Say we have two circles that intersect at points $X$ and $Y$. We also have a point $A$ on one of those circles, call it the "first circle", with a tangent line $T$ at the point $A$. Lines are extended from $A$ to $X$ and $Y$ until they intersect with the other circle, the "second... |
H: probabilities for numbers with uniformly random decimal digits
An urn contains 10 pebbles numbered from 0 to 9. Three balls are drawn in succession from the urn with replacement. After each a draw a number is associated to the pebble and registered. If the numbers associated with the pebbles is for example 015 is d... |
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