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H: Evaluating $\int (x^6+x^3)\sqrt[3]{x^3+2}dx$
I am trying to evaluate:
$$\int (x^6+x^3)\sqrt[3]{x^3+2} \ \ dx$$
My solution:
$$\int (x^5+x^2)\sqrt[3]{x^6+2x^3} \ \ dx$$
Let $$(x^6+2x^3) = t^3 \ \ \text{and} \ \ (x^5+x^2) \ \ dx = \frac{1}{2}t^2 \ \ dt$$
$$\frac{1}{2}\int t^2\cdot t \ \ dt = \frac{1}{2}.\frac{t^4}{4... |
H: Formally prove/disprove that $\sqrt{n}o(\sqrt{n}) = o(n)$
I'm wondering how to formally show that $\sqrt{n}o(\sqrt{n}) = o(n)$. The problem I'm having is that I don't really know how to formally resolve the multiplication on the LHS. It would be straightforward to show the result for $\sqrt{n}O(\sqrt{n}) = O(n)$ by... |
H: Does $N(z)=\pm 1$ imply $z$ is a unit in $\mathbb{Z}[\sqrt{10}]$?
I've been trying to prove that $\mathbb{Z}[\sqrt{10}]$ is not factorial. I did this by defining the norm $N(a+b\sqrt{10})=a^2-10b^2$.
I was able to show for myself that $N(z)=\pm 2$ and $N(z)=\pm 5$ have no solutions, and my idea is to show that $2,5... |
H: Name for a type of subgraph that comes from identification of vertices?
Is there a special name for the kind of subgraphs you get by taking some sequence of the following operation: Pick two vertices and identify them so all edges going to either vertex get sent to the new vertex.
AI: You don't get a subgraph when... |
H: How to find convergence region of $\sum_{n\geqslant 0, m \geqslant 0} x^n y^m \binom{n+m}{n}^2$
The following two series are special cases of Appell $F_3$ and $F_4$, namely:
$$
\mathcal{S}_1 = \sum_{n \geqslant 0, m \geqslant 0} \frac{x^n y^m}{\binom{n+m}{n}}
$$
and
$$
\mathcal{S}_2 = \sum_{n \geqslant 0, m \ge... |
H: Is it possible to have a point $P_1$ not $\chi$-semistable but $P_2$ $\chi$-semistable with these two points in the same orbit?
Let $G$ be a group acting on an affine variety $X\subseteq \mathbb{A}_{\mathbb{C}}^n$.
Suppose $P_1$ and $P_2$ are two points in $X$ such that $g\circ P_1=P_2$ for some $g\in G$. This me... |
H: How would I calculate the area of the shaded region of a circle with radius $6$ and length of chord $AB=6. $
How would I calculate the area of the shaded region of a circle with radius 6 and length of chord AB is 6.
AI: Hint: Join the center of the circle to the points A and B. You'll obtain a triangle. What type o... |
H: What is a good technique to decide step size in sub-gradient method for dual decomposition?
I am looking at the following paper to implement dual decomposition for my algorithm:
http://www.csd.uoc.gr/~komod/publications/docs/DualDecomposition_PAMI.pdf
On Pg.29 they suggest setting the step size for the sub-gradient... |
H: Does $\sum\limits_{n=1}^\infty \frac{n^n}{3^n n!}$ converge?
Test the convergence of the series $$\sum_{n=1}^\infty \frac{n^n}{3^n n!}$$
I know that if the nth term tends to $\infty$ then the series is divergent and if it is tends to 0 it is convergent . Also I'm familiar with some test e.g. Ratio test, d'Alember... |
H: Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$
The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has
(A) exactly two distinct real roots
(B) exactly three distinct real roots
(C) exactly four distinct real roots
(D) six distinct real roots
AI: You have:
$f(x)=x^6-5x^4+16x^2-72x+9$
$... |
H: Formula obtained by using Trignometric approximation for a triangle with a very small side
I am reading a paper on the force between hooft polyakov monopoles, but I am completely baffled by one of the 'elementary trignometric' equation they have got using an approximation. Consider a triangle say triangle ABC. The ... |
H: Find the approximate change in $y$ as $x$ increases from 2 to 2.02
Find the approximate change in $y$ as $x$ increases from 2 to 2.02
The equation of a curve is $y=4x^3-8x^2+10$
a)Find $\frac{dy}{dx}$
$\frac{dy}{dx}=12x^2-16x$
But I don't know how to answer below
b) "Find the approximate change in $y$... |
H: Question about proof of Going-down theorem
I have written a proof of the Going-down theorem that doesn't use some of the assumptions so it's false but I can't find the mistake. Can you tell where it's wrong?
*Going-down*$^\prime$: Let $R,S$ be rings such that $S \subset R$ and $R$ is integral over $S$. Let $q_1$ be... |
H: A property of non-Archimedean metrics
I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers,
$p$-adic Analysis, and Zeta-Functions), and came across the exercise:
Prove that a norm $\|.\|$ on a field $F$ is non-Archimedean if and only if
$$\{x\in F : \|x\| < 1 \} \cap ... |
H: characterization of functions
I have a question that consists of the characterization of all functions $f(x)$ and all constants $k\in\mathbb{R}$ satisfying:
$$f:\mathbb{R}^+\rightarrow (0,1)$$
$$k-\int_4^x\frac{f(t)}{t}dt\leq\log(2)-\frac{1}{2}\log(x),\ \ \forall x\geq 4$$
Does someone have an idea about the second... |
H: Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$
I need to check if
$$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is appreciated. I ha... |
H: Transition probabilities for a nonlinear state space model
I am trying to compute the transition probabilities of the model given by
$X_{n+1} = f_{n+1}(X_n,W_{n+1})$ where $X_n's, W_n's$ are $R^k$ valued random variables for $n \geq 0$, $W_n's$ are independent and $f_n's$ are measurable. Also, define $\mathcal{F}_n... |
H: Category Theory usage in Algebraic Topology
First my question:
How much category theory should someone studying algebraic topology generally know?
Motivation: I am taking my first graduate course in algebraic topology next semester, and, up to this point, I have never taken the time to learn any category theory. ... |
H: $n$ points forming a convex $n$-gon
Suppose I am given a collection of $n$ points, any four of which form a convex quadrilateral. I wish to establish that these $n$ points form a convex $n$-gon.
I am thinking about using induction. The case $n=4$ is trivial. If the result is assumed for $n-1$ how do I establish it ... |
H: If $N\lhd H×K$ then $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially
I am thinking on this problem:
If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially.
I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ such that $([n,m],[n',m'])\neq 1$. B... |
H: General form of Integration by Parts
This is a question just out of interest to know the power of integration by parts. There are various level of integration by parts. What are some of the most general form of integration by parts?
I have encountered it very often in PDE's. I look forward to gaining more insights ... |
H: Advantage of accepting non-measurable sets
What would be the advantage of accepting non-measurable sets?
I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
AI: One correction to your question is that non-measurable sets actually proved to exist by Vitali in 1905, his ... |
H: 2 quick notation questions re: vectors and transformations
Is it customary to omit one pair of parentheses and write $T \begin{pmatrix} 1\\2\\1\\1 \end{pmatrix} $ instead of $T \begin{pmatrix} \begin{pmatrix} 1\\2\\1\\1 \end{pmatrix} \end{pmatrix}$ to indicate the image of $\begin{pmatrix} 1\\2\\1\\1 \end{pmatr... |
H: Contractible homotopy fibre for CW complexes, categorial construction of the homotopy inverse
Let $f:X\to Y$ be a map of topological spaces. Assume further that the homotopy fibre is contractible. We get a long exact sequence on the homotopy groups and if $X$ and $Y$ are connected $f$ is a weak equivalence.
If $X$ ... |
H: Down-sets in posets and directed sets
Let P be a poset and let us say that a subset A of P is a down-set if:
$$x \in A, y < x \implies y \in A.$$
A directed set is a poset P such that for every two elements, $a,b \in P$ we can find $c \in P$ such that $c \geq a $ and $c \geq b$. Now, I am trying to prove the follo... |
H: area between polar equation $r = \sin\theta$ and $r = \cos\theta$
Below is the exact question and answer from my textbook:
Find the area of the region enclosed between the two curves $C_{1}$
and $C_{2}$ where $C_{1}$ has the polar equation $r = \sin\theta$ and $C_{2}$ has the
polar equation $r = \cos\theta$.
an... |
H: How many points does Stone-Čech compactification add?
I would like to know how Stone-Čech compactification works with simple examples, like $(0,1)$, $\mathbb{R}$, and $B_r(0)$ (the open ball of $R^2)$. I've studied the one-point compactification and this is way more difficult to understand. All the texts I've found... |
H: The difference between $\frac{\partial^2 y}{\partial x^2}$ and $\frac{\partial y^2}{\partial x^2}$
The question is $y''=2y^3$.
I know I can substitute $y'=p$. My question is if I can seperate x and y and integrate both sides twice?
AI: $$\dfrac{d^2y}{dx^2}=\dfrac{d(\frac{dy}{dx})}{dx}$$
So you have
$$\frac{d(\frac{... |
H: Convergence of a function in the continuous functions metric space with infinite norm induced metric.
I know that $(C[0, 2], d_{\infty})$ is a complete metric space, being $C[0, 2]$ the set of continuous functions in the closed interval $[0, 2]$ and $d_\infty$ the distance metric induced by the infinite norm, i.e.,... |
H: Does an uncountable intersection of sets with probability one also have probability one? ; in connection with the ergodic theorem
Let $(\Omega, {\cal F},P)$ be a complete probability space and $T$ a mesure-preserving transformation on $\Omega$ that is ergodic. The point-wise ergodic theorem states that for any $f\... |
H: Evaluating $\int(2x^2+1)e^{x^2}dx$
$$\int(2x^2+1)e^{x^2}dx$$
The answer of course:
$$\int(2x^2+1)e^{x^2}\,dx=xe^{x^2}+C$$
But what kind of techniques we should use with problem like this ?
AI: You can expand the integrand, and get
$$2x^2e^{x^2}+e^{x^2}=$$
$$x\cdot 2x e^{x^2}+1\cdot e^{x^2}=$$
Note that $x'=1$ and t... |
H: Using a Bivariate Gaussian Distribution to Predict Range of Movement
I am currently attempting to use a bivariate normal distribution to identify the most likely range of movement for a blob in computer vision. This itself is not the problem, however; I do not understand how σ plays a role in finding discrete proba... |
H: N-points compatification
I know that Alexandroff compatification is unique, and if the Alexandroff compatification of two spaces are not homeomorphic, then the spaces can't be. Does uniqueness stand in n point compatifications? And what does homeomorphism (or not) between the compatifications tells us about the ori... |
H: Why is an alternating $2$-form decomposable if and only if its self-wedge vanishes?
Given a vector space $V$, and a $2$-tensor $w$ in the second exterior power $\Lambda^2 V$. Assume that $w \wedge w=0$. Why is $w$ decomposable?
Thanks for your help!
AI: There is a canonical form: there is a basis $e_1,\dots,e_n$ of... |
H: Whats the probability a subset of an $\mathbb F_2$ vector space is a spanning set?
Let $V$ be an $n$-dimensional $\mathbb F_2$ vector space. Note that $V$ has $2^n$ elements and $\mathcal P(V)$ has $2^{2^n}$.
I'm interested in the probability (under a uniform distribution) that an element of $\mathcal P(V)$ is a sp... |
H: Does this kind of matrix have a name?
Are these kind of matrices generally known in mathematics?
Do they have a name?
$$
\left[\begin{array}{rrr}
A & B \\
B & A \\
\end{array}\right]
$$
$$
\left[\begin{array}{rrr}
A & B & C \\
C & A & B \\
B & C & A \\
\end{array}\right]
$$
$$
\left[\... |
H: Multiple-choice question regarding $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^n \left| e^{\frac{2\pi ik}{n}} − e^{\frac{2\pi i(k-1)}{n}} \right|$
The limit
$$\lim_{n \to \infty} \sum_{k = 1}^n \left| e^{\frac{2\pi ik}{n}} − e^{\frac{2\pi i(k-1)}{n}} \right|$$ is
(A) $2$
(B) $2e$
(C) $2\p... |
H: Application of the Chebyshev inequality
was revising my stuffs for my stochastics exams and came across this question that I couldn't figure my way around..
Let $X_1,\ldots,X_n$ be independent, identically distributed random variables with $E(X_1) =a$ and
$$S_n = \frac{1}{n} \sum_{i = 1}^n X_i$$
Using the Chebych... |
H: Double-Well Delta Potentials - Schrödinger Equation
Page 177 on Davies' book- Spectral theory of diff operatrs contains the following computation problem:
Calculate the negative eigenvalues and the corresponding eigenfunctions of the following operator: $H:= -\frac{d^2 }{dx^2 } -\delta_{-r} -2\delta_{r} $ .
The b... |
H: multiple choice summation problem
Let
$$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}.$$
Then
(A) $X < 1$
(B) $X > 3/2$
(C) $1 < X < 3/2$
(D) none of the above holds.
I assume that the answer is the third choice $1<X<3/2$. I integrate out $1/x$ in the interval $(1001, 3001)$... |
H: Proving $A\cap(B-C)$ is equal to $(A\cap B)-(A\cap C)$
Prove that: $A\cap (B-C) = (A\cap B)-(A\cap C)$.
Tried to prove this by using Algebra of Classes. Then used idempotent property in $A$ as well as double negation in $A$ but still it didn't work.
AI: If an element is in the left side, then it is:
in A
in B
n... |
H: What is $\frac{dy}{dx}|_{y=-1}$ for $(xy^3 + x^2y^7)\frac{dy}{dx} = 1$ given that $y \left(\frac{1}{4}\right)=1$
Suppose a solution of the differential equation
$$(xy^3 + x^2y^7)\frac{dy}{dx} = 1$$
satisfies the initial condition $y \left(\frac{1}{4}\right)=1$ . Then the value of $\dfrac{dy}{dx}$ when $y = −1$... |
H: Difference between Norm and Distance
I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book.
I understand the difference between two concepts when i'm working on non-euclidean space, but is there any even slight difference bet... |
H: Group inclusion- Quotiens Inclusion
Given a group $G$ , and two subgroups $ G_1, G_2 $ such that $ G_1 \subseteq G_2 $ ,
is it true that $ G/G_2 \subseteq G/G_1 $ ?
Thanks in advance !
AI: What you probably had in mind is the following:
$$G/G_2\cong\left(G/G_1\right)/\left(G_2/G_1\right)$$
by the 2nd or 3d. isomo... |
H: Cardinality of the complex numbers in ZF
As you all know, cardinality of $\mathbb{R} = 2^{\aleph_0}$ can be proved in ZF, since cardinality of $\mathbb{N} \times \mathbb{N} = \aleph_0$ can be proved in ZF.
I know that the statement 'For any infinite set $A$, $|A\times A|=|A|$ is weaker than A.C.
I wonder if there i... |
H: Antisymmetric's Opposite (If existant)
I am learning of Equivalence Relations and for something to be one is has to be:
Reflexive (i.e., $aRa$)
Symmetric (i.e., $aRb$ $\Rightarrow$ $bRa$)
Transitive (i.e., $aRb$ & $bRc$ $\Rightarrow$ $aRc$)
Where we let $R \subseteq A\times A$ be a relation on a non-empty set $A... |
H: Period of a finite binary sequence
Let $G:N\to\{0,1\}$, and let $L$ be some period of $G$, so that $G(i+kL)=G(i)$. What's the best a good way to find the smallest period of $G$? I mean an algorithm that takes ($G$,$L$) and outputs the smallest period.
AI: Let $G[m,n]$ denote the string formed by the values of $G$ b... |
H: Bound on unit vectors
could someone help me with this simple problem. As always with homework, hints are specially welcome.
Let $v=(v_1,v_2)$ be a two-dimensional unit vector with complex coefficients. If $|v_1|<a$ and $|v_2|<a$ then $|v_1|+|v_2|\geq \frac{1}{a}$.
AI: I think I got it.
\begin{equation}
|v_1|+|v_2|\... |
H: A random walk on $\mathbb{Z}$ with a twist
I am trying to decide whether the following random walk is recurrent or not. Intuitively, I think it is - but I am not familiar with techniques of proving it.
My random walk is the following: on each point $i$, I can turn to $i-1$ with probability $\frac{1}{3}$, to $i+1$ w... |
H: Combinatorial Interpretation of Fractional Binomial Coefficients
My question is a bit imprecise - but I hope you like it. I even strongly think it has a proper answer.
The binomial coefficient $\binom{\frac{1}{2}}{n}$ is strongly related to Catalan numbers - the expression $(1-4x)^{\frac{1}{2}}$ appears when calcul... |
H: How to compare a sum of uniform RVs with a uniform RV?
Let $(X_n)_{n\geq1}$ be a sequence of i.i.d. $\sim \text{Uni}([0,1])$ distributed random variables. I want to show that
$$\mathbb{P}\big(n^2 X_{n+1} < \sum\limits_{k=1}^n X_k ~\text{for infinitely many } n \in \mathbb{N} \big)=1 $$
This cries out for Borel Can... |
H: The Vector Space over another Vector Space
Is it possible to consider a vector space over another vectorspace instead over a field as usual, where came into play that we need a field? And in such a vector space, the vector could be represented as tuples from $V^k$, where V is the vector space from which the other s... |
H: Proof of Chebyshev Inequality
I was going through the proof of the Chebyshev Inequality here .
And I seem to be facing some trouble in the approximation stage. I can't seem to follow how $\epsilon$ has been approximated to $(t-\mu)$.
AI: It is an inequality. The text in that document breaks up the flow slightly.
It... |
H: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$
I need to evaluate this integral: $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$.
I've tried $t=\log(x+1)$, $t=x+1$, but to no avail. I've noticed that:
$\int_0^1 \frac{\log(x+1)}{1+x^2} dx = \int_0^1\log(x+1) \arctan'(x)dx =\left. \log(x+1)\arctan(x) \right|_{x=0}^{x=1... |
H: The meaning of matrix powers
If a matrix can represent a system of equations, what is the meaning of the square of that matrix? It represents another system? What is the relation with the final system and the first?
AI: I'm not sure if this clarifies or obscures.
Write your set of equations as:
$$Ax = b$$
Where $b$... |
H: functions over dependent random variables
Say we have a set of identically distributed integer-valued random variables: $\{ A_i \}_{i=1}^n$, such that they are not independent. Say we have another set of identically distributed integer-valued random variables $\{ B_i \}_{i=1}^n$, such that they are not independent ... |
H: Find a Jordan Canonical Matrix Similar to a real matrix A
If A is a matrix, Find a Jordan Canonical matrix similar to A:
$
c(x)=\text{det}(xI-A)=(x-3)^{5}(x-2)^{4}
$
The information given about A is:
$
\text{rank}(A-3I)=7
$
$
\text{rank}(A-3I)^{2}=5
$
$
\text{rank}(A-3I)^{3}=4
$
$
\text{rank}(A-3I)^{4}=4
$
$
\text{... |
H: inequality on inner product
Let $x \in \Bbb R^n$ and $Q \in M_{n \times n}(\Bbb R)$, where $Q$ is hermitian and negative definite. Let $(\cdot,\cdot)$ be the usual euclidian inner product.
I need to prove the following inequality:
$$(x,Qx) \le a(x,x),$$
where $a$ is the maximum eigenvalue of $Q$.
Any idea?
AI: So, ... |
H: Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign
I want to prove (or disprove) the following statement:
If $A$ is a square matrix with non-negative elements that has spectral
radius less then $1$, then any matrix obtained from $A$ by arbitrarily
changin... |
H: Degree of Hessian surface invariant under linear transformations?
Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det (\frac{\partial f}{\partial x_i\partial x_j}))$ of $f... |
H: Proving that if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible then $g(x)$ have multiple roots iff $g'(x)=0$
I am going over my lecture notes in my Field theory class and I saw
this following statement without a proof: if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible
then $g(x)$ have mu... |
H: Intuitive interpretation of limsup and liminf of sequences of sets?
What is an intuitive interpretation of the 'events'
$$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$
and
$$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$
when $A_n$ are subsets of a measured space $(\Omega, F,\mu)$.
... |
H: Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$
how to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$?
I know how to prove
$ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $
in this way:
$ \tan{20^0} = \sqrt{3}.\tan{50^0}.\tan{10^... |
H: Describe all the compact subsets of this space
Consider the topological space $(X,\mathscr{U})$, where $X=\mathbb{R}^2$ and the topology $\mathscr{U}$ is generated by the collection of sets $\{(0,0)\}\cup \{I_a\}$ where $I_a$ are the open intervals on the rays departing from the origin.
We then make a quotient of... |
H: Can this function be rewritten to improve numerical stability?
I'm writing a program that needs to evaluate the function $$f(x) = \frac{1 - e^{-ux}}{u}$$ often with small values of $u$ (i.e. $u \ll x$). In the limit $u \to 0$ we have $f(x) = x$ using L'Hôpital's rule, so the function is well-behaved and non-singul... |
H: Concept check with Probablity. Classic birthday problem
The solution to the problem is
(a) 365 days for the sample space.
(b) $$\frac{365 \times 1 \times 1}{365^3} = \frac{1}{365^2}$$
I understand (a), there are 365 days in a year....
But I don't understand the reasoning of (b).
To compute, I think it's easier to... |
H: How many rolls until probability of a 5 is at least 1/2?
Problem: Jak rolls two die and wants the probability of rolling at least a 5 to be $\frac{1}{2}$. How many should Jak roll?
basically, I got two answers (same), but different approaches. Can someone tell me why one of them could be wrong?
Solution 1
The chanc... |
H: Triangle Inside Circle
If the radius of the circle is equal to the length of the chord $AB$, what is the value of $x$?
How would I solve this problem ?
AI: Without Trigonometry:
Let $O$, be the center of the circle. In $\triangle OAB$, $AB=OA=OB=$ radius implying $\triangle OAB$ to be an equilateral triangle. Th... |
H: Map Surjective on a Disk
I've got another question from a student that has stumped me: Let $D^{n+1}$ be the $n+1$-disk, with boundary sphere $S^n$. Suppose $f:D^{n+1}\longrightarrow \mathbb{R}^{n+1}$ is a map such that $f(S^n)\subseteq S^n$. Furthermore, suppose that $f|_{S^n}$ has nonzero degree. Show that $f(... |
H: Efficient method to evaluate the following series: $\sum_{n=1}^\infty \frac{n^2\cdot (n+1)^2}{n!}$
How do I calculate the infinite series:
$$\frac{1^2\cdot 2^2}{1!}+\frac{2^2\cdot 3^2}{2!}+\dots \quad?$$
I tried to find the nth term $t_n$. $$t_n=\frac{n^2\cdot (n+1)^2}{n!}.$$
So, $$\sum_{n=1}^{\infty}t_n=\sum_{n=1}... |
H: Expanding fractions as powers of $z$
I'm reading through complex functions in Boas' book, and there's a part when discussing Laurent series where she says:
"Now, for $0 <|z|<1$, we expand each of the fractions in the parenthesis in powers of $z$."
The equation she refers to is the following:
$$f(z) = \frac {4}{z} ... |
H: what is the meaning of 100% of 100%?
When we say a% of b, we mean 'a' parts out of the 100 parts of 'b'.
e.g. 1% of 200 means divide 200 in 100 equal parts and select 1 out of it, i.e. 2
So,
what does it mean when we say any percentage of any percentage?
like, what is the significance of 100% of 100%?
AI: 100 perc... |
H: Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
I need help proving the following statement:
For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradictio... |
H: What is the answer for the $\lim\limits_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}$?
Let $t\in (0,\pi)$ and $n$ change in natural numbers. I am wondering what is the answer to the following limit.
$$\lim_{n\rightarrow \infty} \frac{\sin(nt)}{\sin(t)}.$$
Thank you.
AI: Well $\ \frac {\sin(nt)}{\pi t}\to \delta(t... |
H: General relationships of variables in expression
I am asked to determine how certain modifications to the variables in Coulomb's equation will effect the resultant force:
$$F=k\frac{Q_1Q_2}{r^2}$$
The question asks me what will happen with $Q_1$ doubles, and I determine $F$ is doubled. Then I am asked what happens... |
H: Finding the Laurent series and pole for $f(z)=\frac {z^2+z+1}{(z-1)^2}$
How do I
(a) find the Laurent series for the following:
$$ f(z)=\frac {z^2+z+1}{(z-1)^2}$$
(b) Find its pole and its order.
I suppose finding the Laurent series would make it easy to find the latter, but I think there's a short cut to fi... |
H: Proof of $(A - B) - C = A - (B \cup C)$
I have several of these types of problems, and it would be great if I can get some help on one so I have a guide on how I can solve these. I tried asking another problem, but it turned out to be a special case problem, so hopefully this one works out normally.
The question ... |
H: Are there any elegant methods to classify of the Gaussian primes?
Out of curiosity, are there any relatively quick classifications of all the Gaussian primes, the primes in $\mathbb{Z}[i]$?
I found a classification here, but the process comes off as rather tedious. No doubt the end classification is nice, but is th... |
H: Prove that $ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$
Prove the following integral inequality:
$$ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$$
This is a high school problem. So far i did manage to prove that the integral is bigger than $1.462$ by using Taylor expansion, namely:
$$1.462\le 1.4625=\int_0^1 1+x^2+\frac{x... |
H: Non-aleph infinite cardinals
I'm now confused with a concept of $\aleph$.
1.$\aleph$ is a cardinal number that is well-ordered in ZF.(Defined as an initial ordinal that is equipotent with). Does that mean $\aleph_x$ in ZF may NOT be equal to $\aleph_x$ in ZFC?
2.I don't know how to define $\aleph$ in ZF.
Here's wha... |
H: How to solve this equation involving $()^x$?
I have the equation:
$\left (\sqrt{3+2\sqrt{2}} \right )^x- \left (\sqrt{3-2\sqrt{2}} \right )^x=\frac{3}{2}$
I wrote the left side of the equation as square roots.
$(1+\sqrt{2})^x-(1-\sqrt{2})^x=\frac{3}{2}$
How do I found out the final solution? Thank you very much!
P.... |
H: Evaluating $\lim\limits_{z \to 0} \frac{z\cdot \cos(z)}{\sin(z)}$
I'm reading Boas' chapter on functions of a complex variable, and she's talking here about finding residues. However, I don't understand the evaluation of the limit(below). How is it that we can take $\cos(0)$ out of the limit while $\sin(z)$ and $z... |
H: Can a subset of a group, which does not contain the identity element, be a group
Let $G$ be a group. Let $1 \in G$ be the identity element of $G$. Let $S \subset G$, with $1 \notin S$. Is it possible for $S$ to be a group, with some other element playing the role of the identity in $S$?
AI: A subset $S$ of $G$ can ... |
H: Count all degree 2 monic irreducible and not irreducible polynomials
There's this exercise that really has kept me stuck for a day by now, will you please help me figure out:
let's consider polynomials in $\mathbb Z_3$:
characterize degree 2 not irreducible monic polynomials. How many are they?
characterize degree... |
H: Chain rule for Hessian matrix
Given $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ smooth and $\phi \in GL(n)$. What is the Hessian matrix $H_{f\circ \phi} = \left(\frac{\partial ^2 (f\circ \phi)}{\partial x_i\partial x_j}\right)_{ij}$?
AI: Denote $H_g(x)$ the Hessian matrix of a function $g$. Denote $g=f\circ \phi$.... |
H: Example of non-finitely generated $R$-algebra
By definition, an $R$-algebra is a ring homomorphism $f: R \to S$. For example, if $R=\mathbb Z$ and $S= \mathbb Z / n \mathbb Z$ then the projection $k \mapsto k \mod n$ is a ring homomorphism so that $\mathbb Z / n \mathbb Z$ is a $\mathbb Z$-algebra. I think the poin... |
H: Inverse function theorem in Banach space to prove short time existence of PDE (explanation of statements)
Let $$X = C^{k+2, \alpha}(S(T)),$$ $$Y = C^{k, \alpha}(S(T)),$$ where $S(T) = S^1 \times [0,T]$. Don't think of $T$ as fixed, but varying. So these Banach spaces contains functions with different time intervals... |
H: Possibly false proof in AM
Here is the excerpt of the book where I suspect a mistake (page 66):
Where they say "The restriction to $A$ of the natural homomorphism $A^\prime \to k^\prime$" I think we don't want a restriction. We start with the quotient map $\pi: A[x^{-1}] \to A[x^{-1}] /m$ where $m$ is a maximal id... |
H: Trigonometric equation inversion
I am trying to invert the following equation to have it with $\theta$ as the subject:
$y = \cos \theta \sin \theta - \cos \theta -\sin \theta$
I tried both standard trig as well as trying to reformulate it as a differential equation (albeit I might have chosen an awkward substitutio... |
H: Another question about a proof in Atiyah-Macdonald
I have a question about the following proof in Atiyah-Macdonald:
1:Why is $\Omega$ infinite? Are all algebraically closed fields infinite?
2: How does the existence of $\xi$ follow from $\Omega$ being infinite?
Thanks.
AI: 1.) Yes, algebraically closed fields are ... |
H: What is the limit for the following sequence of products?
$\lim\limits_{n \to \infty}$ $\displaystyle\frac{q \cdot n +1}{q \cdot n} \cdot \frac{q \cdot n +p+1}{q \cdot n +p} \cdot \ldots \cdot \frac{q \cdot n +n \cdot p +1}{q \cdot n + n \cdot p}$ , for $q > 0, p \geq 2$ .
Thank a lot !
AI: Introducing the paramete... |
H: probabilty of random points on perimeter containing center
related question: probablity of random pick up three points inside a regular triangle which form a triangle and contain the center
What is the probability that a (possibly degenerate) triangle made by three randonly chosen points on the perimeter of an n-go... |
H: Value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$, $P(1)=10$, $P(2)=20$, $P(3)=30$
What will be the value of $P(12)+P(-8)$ if $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$
provided that $P(1)=10$, $P(2)=20$, $P(3)=30$?
I put these values and got three simultaneous equations in $a, b, c, d$. What is the smarter way to... |
H: $(B/m)[x] = B[x]/M$?
Assume $K$ is a field and $B$ is a subring of $K$ and $x \in K$. Let $m$ be a maximal ideal of $B$. Let $m^e$ denote the extension of $m$ in $B[x]$. Let $M$ be a maximal ideal in $B[x]$ containing $m^e$.
Can someone explain to me why $(B/m)[\bar{x}] = B[x]/M$ where $\bar{x}$ is the image of $x... |
H: Equation involving prime numbers
Given the equation:
$$p^2+\phi=q$$
where $p$ and $q$ are prime numbers and $\phi$ a constant, it seems the equation doesn't have solutions for $\phi=1,2,3$, but it has solutions for $\phi=4$. Is it possible to show why?
Or maybe, there are solutions that I am not able to find also f... |
H: The "need" for cohomology theories
In many surveys or introductions, one can see sentences such as "there was a need for this type of cohomology" or "X succeeded in inventing the cohomology of...".
My question is: why is there a need to develop cohomology theories ? What does it bring to the studies involved ?
(I h... |
H: Finite measure spaces with a total closed set
Endowing $R$ with a finite borel measure. How to find a closed set with its total measure and every closed subset of it has minor measure?
AI: Let $\mathcal O:=\{O\subset \Bbb R, m(O)=0, O\mbox{ open}\}$ and $S':=\bigcup_{O\in\mathcal O}O$. $S'$ is open hence separable,... |
H: Projection matrices
I have found these two apparently contradicting remarks about projection matrices:
A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection matrix.
$P$ is a projector if $PP = P$. Projectors are always positive which implies that they are alway... |
H: How to transform data distributed around zero to make it closer to normal?
I have data that ranges continuously from $-1$ to $+1$, with lots of zeros in the middle. I want to transform the data to a normal distribution. How would I do this? My normal approach with data containing zeros is to $+1$ then transform ($\... |
H: show that $x^2+y^2=z^5+z$ Has infinitely many relatively prime integral solutions
How to show that this equation:
$$x^2+y^2=z^5+z$$
Has infinitely many relatively prime integral solutions
AI: The number $z^4+1$ is a sum of two relatively prime squares. Let $z$ be the sum of two relatively prime squares. Then the pr... |
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