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H: How to evaluate $ \lim_{(x,y)\to(0,0)}\frac{y^2\sin^2x}{x^4+y^4} $
$$
\lim_{(x,y)\to(0,0)}\frac{y^2\sin^2x}{x^4+y^4}
$$
I believe the limit doesn't exist but I'm not sure how to prove/evaluate correctly. A hint would be great. Thanks.
AI: Write
$$\mathop {\lim }\limits_{(x,y) \to (0,0)} \frac{{{y^2}{x^2}}}{{{x^4} +... |
H: Composition of exponential function with discontinuous function
Suppose $f(x)$ be a real discontinuous everywhere function. Then is $\exp(f(x))$ also discontinuous?
AI: Let $g=\exp(f)$. Since $f$ is real, then $g$ is a positive function. But if $g$ were continuous at some $x$, then $f=\ln(g)$ would be continuous th... |
H: proof that the series converges?
I just need to make sure that I do it correctly
Thanks in advance,
AI: You can use the fact $\ln(x) < x$ which makes it easier
$$ \int_{0}^{1}\frac{\ln(x)}{1+x^2}dx < \int_{0}^{1}\frac{x}{1+x^2}dx=\frac{1}{2}\ln(1+x^2)|_{x=0}^{1}=\dots. $$
Note: Just note that, $ \frac{\ln(x)}{1... |
H: Domain, Co-Domain, and Linearity of Linear Systems homework check.
I am asked to find the domain, co-domain, and to determine whether of not the transformation is linear. I'm not sure if I am doing this properly, so I figured I would ask as my textbook doesn't have solutions.
If someone could explain how to determi... |
H: Relationship between $\int_a^b f(x)\,dx$ and $\int_a^bxf(x)\,dx$
I was working a problem, and came to a point where it would help greatly if there was a relationship between the following two expressions:
1) The numeric value of $\int_a^b f(x)\,dx$, and
2) The numeric value of $\int_a^bxf(x)\,dx$.
That is, if I kno... |
H: How to prove that this sequence converges to zero?
Given a natural number $k$ and a real number $a$ such that $|a|<1$, defines a sequence $(x_n)_{n\in\mathbb{N}}$ by
$$x_n = \frac{a^nn!}{k!(n-k)!}.$$
Show that $x_n\rightarrow 0$.
AI: Note the following:
$$
\lim_{n\to\infty}\left|\frac{x_{n+1}}{x_n}\right|
=\lim_{n\... |
H: If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues
Let $T:V\to V$ be a linear operator. If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues.
I have been working on this proof for a few days and I am not sure what direction to really ... |
H: Telephone Number Checksum Problem
I am having difficulty solving this problem. Could someone please help me? Thanks
"The telephone numbers in town run from 00000 to 99999; a common error in dialling on a
standard keypad is to punch in a digit horizontally adjacent to the intended one. So on a
standard dialling ke... |
H: What test should I use to classify this series?
Classify
$$\sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}$$
as absolutely convergent, conditionally convergent or divergent.
I am not sure whether I should use integral test or comparison test.
Do you know what's the best test to classify this series?
Thanks in advance.... |
H: Equivalence classes - on $\mathbb{N}^2$
Let $R$ be the relation on $\mathbb{N}^2$ defined by
$(a,b)R(c,d)$ if $2a + 3b = 2c + 3d$
Write $4$ elements in the equivalence class of $(1,2)$
So I think I need to find all the pairs $(a,b)$ with $2a + 3b = 2(1) + 3(2) = 8$
But no other positive integers other than $(1,2)$ ... |
H: Mathematical symbol for "has"
Just out of curiosity, I was wondering if there was a symbol for "has" so intead of saying $x \in A$, we could say something like "$A$ has $x$", they both mean the same thing but I was just wondering if there was another way to say it.
Thanks!
AI: I believe I have seen $\ni$ used for t... |
H: Standard Matrices for Linear Transformation
I'm not able to find an explanation for finding the standard matrix for a linear transformation of equations. For example, if I have;
$$w_1=2x_1-3x_2+x_4$$
$$w_2=3x_1+5x_2-x_4$$
Would the standard matrix just be:
$$
\begin{bmatrix}
2 & -3 & 0 & 1 \\
3 & 5 & 0 & -1
\end{bm... |
H: Compactness Proof
I have a doubt respect this theorem
A compact subset $M$ of a metric space is closed and bounded.
Proof by my lecture:
For every $x\in \bar{M}$ there is a sequence $(x_n)$ in $M$ such that $x_n\longrightarrow x \cdots$. My question is Why this?. I know that if $x$ is a boundary point then exists $... |
H: Rings of fractions and homomorphisms
Let the ring of fractions be denoted as $S^{-1}R$.
Proposition (from Robert Ash's textbook "Basic Abstract Algebra")
Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring homomorphism. If S has no zero divisors then f is a monomorphism, and we say that R can be e... |
H: Finding the smallest subset of a set of vectors which contains another vector in the span
Consider a set $S=\{ \underline{v_1},\dots , \underline{v_n} \} $ of vectors of dimension $d<n$. Suppose for some vector $\underline{b}$ that the solution space for the matrix equation
$\left[ \underline{v_1} \dots \underlin... |
H: Calculating the Roots of Sine
Aside from the obvious knowledge that the roots of $\sin x$ are all integer multiples of $\pi$, is there a formal, algebraic method to calculate the roots of trigonometric functions similar to the quadratic equation?
(e.g. roots of $\sin^2(ax) + \sin(bx) + c$ or some other non-trivial ... |
H: Is $\sum_{n=3}^\infty\frac{1}{n\log n}$ absolutely convergent, conditionally convergent or divergent?
Classify
$$\sum_{n=3}^\infty \frac{1}{n\log(n)}$$
as absolutely convergent, conditionally convergent or divergent.
Is it,
$$\sum_{n=3}^\infty \frac{1}n$$ is a divergent $p$-series as $p=1$, and
$$\lim_{n\to\infty}... |
H: How to prove that $Z[i]/7$ is a field?
I know that $Q[i]$ is a field since we can find the inverse for each $a+bi$ namely $\frac{a}{a^2+b^2} - i\frac{b}{a^2+b^2}$. However, I am not sure how to do so for something like $Z[i]/7$ since we can only have non-negative integers less than 7 as coefficients.
AI: Hint: the ... |
H: Totally ordered sets
Let T be a totally ordered set that is finite. Does it follow that minimum and maximum of T exist?
Since T is finite, I believe there exists a minimal of T. From that it maybe able to be shown that the minimal is the minimum but not quite sure whether it is the right approach.
AI: Claim: A tot... |
H: $m(\alpha E)=\alpha m(E)$ ? for every Lebesgue measurable set and $\alpha >0$
Let's consider a Lebesgue measurable set $ E \subset \mathbb R$. And let's consider a positive constant $\alpha>0$. I want to know if it's always true $m(\alpha E)=\alpha m(E)$. Clearly $m$ denote the Lebesgue measure.
At least in the ca... |
H: Calculate polyhedra vertices based on faces
I have some origami polyhedra which I know the type of faces it has and how they are connected (such as this torus) and I want to calculate the co-ordinates of the vertices to use as an input to script.
My question is how should I go about translating knowledge of the fac... |
H: Finite implies Quasi-finite
Please help if you know a proof or a good reference for the following fact (exercise 3.5, Hartshorne's text).
Fact. A finite morphism of schemes $f: X \rightarrow Y$ is quasi-finite.
Here, the definition of quasi-finite is taken as $f^{-1}(q)$ is finite for all $q \in Y$.
AI: Hartshorne ... |
H: Suppose G is a finite and simple group which acts transitively on S. Given that $k \equiv |S| > 1$, prove $|G|$ divides $ k!$
I think they key point here is to prove that $G$ must be isomorphic to a subgroup of $S_k$ then we would be done.
I am quite lost in trying to do so. Since $G$ acts transitively on $S$, can'... |
H: A translation and a negation in $\mathbb{C}$ generate the infinite dihedral group.
I'd like to show that the linear functions
$$ \varphi(z) = z+b, \;\;\; 0\neq b\in \mathbb{C}$$
$$ \psi(z) = -z+c, \;\;\; c\in \mathbb{C}$$
generate, under composition, a group isomorphic to $Dih_\infty$, the infinite dihedral group.
... |
H: Optimal Solution Set To Linear Programs
I have the following assignment question, and I am not quite sure how to proceed.
Q: Consider the following LP (P): $\max\{{c^Tx:Ax=b, x \geq 0}\}$, where $A$ is an $m$ by $n$ matrix. Prove or disprove the following.
i) Let $D$ be any $m$ by $m$ matrix. Then the following LP ... |
H: Is $\omega_1 ^\omega$ countably compact?
Give $\omega_1$ the order topology, and then $\omega_1 ^\omega$ the product topology.
$\omega_1$ is countably compact, but what about this product?
I attempted to prove it in two different ways, but each time something goes wrong. How about for arbitrary powers of $\omega... |
H: Counter examples on Categories
I'm reading Categories for the Working Mathematician by Saunders Mac Lane. At the section 5 from chapter 1, for a fixed category, he claims that every arrow with right inverse, is epic (right cancellable). He claims also that the converse is true in the category of Sets, but fails in ... |
H: Statement similar to Fermat's last theorm and Beal's conjecture: $A^x+A^y=A^z$
If $A^n+B^n=C^n$ is Fermat's last theorem and $A^x+B^y=C^z$ is Beal's conjecture , then what is $A^x+A^y=A^z$ ? Is there any conjecture like this? Just curious to know.
AI: Fermat's last theorem is not $A^n+B^n=C^n$, but rather
There ar... |
H: $1^k+2^k+\cdots+(p-1)^k\equiv 0 \pmod{p}$
How can I show this equation
$$1^k+2^k+\cdots+(p-1)^k\equiv 0 \pmod{p},$$
where $k$ is integer that $p-1\nmid k$ and $p$ is odd prime.
AI: Another way of presentation is too choose such $a$ that $a^k\ne 1(\mod p)$ and
note that $0,a,2a,... (p-1)a$ form complete residue syst... |
H: Find $c$ which makes $cA$ is an orthogonal projection on $A$
$A=\begin{pmatrix} 2&-1&-1\\-1&2&-1\\-1&-1&2\end{pmatrix}$
$c>0$ and $B=cA$. Find $c$ which makes $B$ is an orthogonal projection on $A$.
Hmmm.....I first find the orthogonal eigenvectors of $A$...
Am I going to the right way?
AI: Hint: If $B$ is a proj... |
H: Mathematical Induction with Inequalities
$ P(n) = n < 3^n - 4 $ for all $ n \ge 2$
Base case: $2 < 3^2 - 4$
$2 < 5$
Inductive step: Assume true for $n = k$, show true for $n = k + 1$
That is, assume $k < 3^{k} - 4$, and show $k + 1 < 3^{k + 1} - 4$
So,
(This is where I might be wrong)
$k + 1 < 3^k + 1 - 4$ (by IH) ... |
H: $\lim_{n\rightarrow\infty}n(\ln n)a_n=0$ implies $\sum a_n$ converges?
Is it true that
If $$\lim_{n\rightarrow\infty}n(\ln n)a_n=0,$$
then the series
$$\sum_{n=1}^\infty a_n$$
converges?
If so, I want to know the proof.
If not, I want to know the counter example.
AI: The series $$\sum_{n=4}^\infty \frac{1}{n\log n ... |
H: Order preserving maps
Suppose $f:X \to Y$ is order preserving. Let $A$ be a subset of $X$.
Does is follow that if $A$ is well ordered then $f(A)$ is well ordered?
AI: Yes. If $S$ is a nonempty subset of $f[A]$, let $T=\{x\in A:f(x)\in S\}$. Then $T$ has a least element $t$, and $f(t)$ is clearly minimal in $S$. |
H: difference between 2 prime numbers
We have to prove that if the difference between two prime numbers greater than two is another prime,the prime is $2$.
It can be proved in the following way.
1)$Odd -odd =even$.
Therefore the difference will always even.
2)The only even prime number is $2$.Therefore the difference... |
H: $|s_{2k}-s_k|<\epsilon$ implies $\{s_k\}$ converges?
$\{s_k\}$ is a sequence in $\mathbb{R}$.
$\forall\epsilon>0$, $\exists N\in\mathbb{N}$ s.t. $k\geq N$ implies $|s_{2k}-s_k|<\epsilon$.Then, does $\{s_k\}$ converge or not?
I need a proof. Thanks.
AI: It may not converge. For example, let $s_k=1$ when $k=2^n$ for ... |
H: Factor $x^5-1$ into irreducibles in $\mathbb{F}_p[x]$
I have to factor the polynomial $f(x)=x^5-1$ in $\mathbb{F}_p[x]$, where $p \neq 5$ is a generic prime number.
I showhed that, if $5 \mid p-1$, then $f(x)$ splits into linear irreducible.
Now I believe that, if $5 \nmid p-1$ but $5 \mid p+1$, then $f(x)$ splits ... |
H: Solving a set of two polynomial equations
For a given $a,b,c,d$ in $\mathbb{R}$, I want to prove that if
$$ac-bd=0 \quad \text{and} \quad ad+bc=0$$
then $a=b=0$ or $c=d=0$.
I am able to prove this in a long and cumbersome way, and I'm sure that there is a better way to prove that.
Many thanks.
Gil.
AI: Multiply $a... |
H: Proof that the Dirichlet function is discontinuous
I think I don't understand how it works.. I found some proofs.. okay, let's see:
Well I'd like to show that the function,
$$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$$
is discontinuous.
Now with epsilon-delta-definition:
Let'... |
H: Can a topological space satisfying the first axiom of countability, which is not Hausdorff but every sequence has a unique limit, exist?
My textbook (Principles of General Topology, by Pervin) says "A topological space $X$ satisfying the first axiom of countability is a Hausdorff space iff every convergent sequence... |
H: Existence of limit of a function of a complex variable.
Say that I have a function $f(z)=f(x+iy)=f(x,y)$ and I want to investigate whether the limit at a particular point, say $z=0$ exists.
I recall that within the domain of real numbers, I checked whether the "right" and "left" limits were the same.
Now is it poss... |
H: Proof about Holomorphic functions in the unit disc
We want to prove the following:
If $f$ is a holomorphic function on the unit disc $\mathbb{D}$ s.t. $f(z) \neq 0$ for $z \in \mathbb{D}$, then there is a holomorphic function $g$ on $\mathbb{D}$ such that $f(z)=e^{g(z)}$ for all $z \in \mathbb{D}$.
For such a $g$... |
H: How can I solve this differential equation?
Consider the differential equation
$x^2y'' + a\,x\,y' + b\,y = 0 \text{ where } y = y(x) \text{ and } a,b \in R$
Using the change of variable $u = \ln(x)$, how can I transform the differential equation in the form of?
$Z'' + \alpha Z'+ \beta Z = 0 \text{ where } Z = Z(u)$... |
H: Vector analysis: $(\vec v \cdot \vec \nabla) \vec v=(\vec \nabla \cdot \vec v) \vec v$?
If I know that $\vec \nabla \cdot \vec v=0$, can I say that:
$$( \vec v \cdot \vec \nabla )\vec v=\underbrace{(\vec \nabla \cdot \vec v)}_{=0} \vec v=0 $$ ?
Note: this is a question I asked in Physics StackExchange but as it is ... |
H: Can this logic about locating a point uniquely without using "-" be challenged?
I was just exploring a possibility of locating point without using "-" ( negative ) sign.
( Actually negative sign confuses me a lot when understanding the basics of coordinate geometry )
So, here is the line where actually
c1 = -1
c... |
H: Proving a limit exists for the next multi-variable function:$f(x,y)=\frac{x^3+y^3}{x^2+y^2}$.
Proving a limit exists for the next multi-variable function: $f(x,y)=\frac{x^3+y^3}{x^2+y^2}$.
I know it's pretty much the basics, But i do not undetstand how to prove whether a limit exists.
What i did so far was:
Let $x... |
H: A Catalan-like counting of walks of length $n$ on $\mathbb{Z}$
I would like to count the number of walks of length $n$ on $\mathbb{Z}$ starting at $0$, where in each step you move either one left or one right, such that you never land on a negative integer (i.e. you can't go left more times than you go right at on ... |
H: How to prove that $\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = \Phi$?
Consider:
$$\frac{1}{n+1} = {\sqrt{n\over n+1}}$$
How could one prove that $n$ is of such form that:
$$\frac{1}{n+1} = {\sqrt{n\over n+1}}\implies n = {\sqrt{5\,}-1 \over 2} \implies n = \Phi$$
where $\Phi$ denotes the golden ratio?
Edit:
I n... |
H: Derivative of $\frac{\cos(y)^4}{x^4}$ using the quotient rule
I have to find the derivative of $\frac{\cos(y)^4}{x^4}$ in order of $x$. Using the quotient rule I'm making $$\frac{(\cos(y)^4)'x^4-(x^4)'\cos(y)^4}{x^{4^2}}$$
This gives me $\frac{0-4x^3\cos(y)^4}{x^{16}}\Rightarrow -\frac{4\cos(y)^4}{x^{13}}$ but Wolf... |
H: understanding probability distribution notation
Assume that $\mu$ is a probability distribution on $[n]$, let $A\subseteq[n] $ be a probability event. What does it mean : $Pr_\mu[A]$ ?
AI: This is $\mu(A)=\sum\limits_{k\in A}\mu(\{k\})$. |
H: Expected number of coin tosses before $k$ heads
If the probability of getting a head is $p$, how do you compute the expected number of coin tosses to get $k$ heads?
I thought this might be the mean of the negative binomial distribution but this gives me $pk/(1-p)$ which is $k$ for $p=1/2$ which can't be right.
AI... |
H: How to interpret $(V^*)^*$, the dual space of the dual space?
Suppose $V$ is a real vector space.
Then $V^*$, its dual space, is the vector space of linear maps $V\to \mathbb R$
How then do I interpret $(V^*)^*$, the dual space of the dual space?
AI: The space of linear maps $\ell : V \rightarrow \mathbb{R}$ is its... |
H: Where's the error in this $2=1$ fake proof?
I'm reading Spivak's Calculus:
2 What's wrong with the following "proof"? Let $x=y$. Then
$$x^2=xy\tag{1}$$
$$x^2-y^2=xy-y^2\tag{2}$$
$$(x+y)(x-y)=y(x-y)\tag{3}$$
$$x+y=y\tag{4}$$
$$2y=y\tag{5}$$
$$2=1\tag{6}$$
I guess the problem is in $(3)$, it seems he tried to divi... |
H: Lipschitz functions are compact or not?
For a constant $\alpha,\lambda$ I want to determine whether $K=\{f:[a,b]\to \mathbb{R}: |f(x)-f(y)|\le \lambda|x-y|^\alpha\}$ in fuctions space with suprimum norm($\|f\|=\sup_{x\in [a,b]}{f(x)}$) is compact or not. I proved that $K$ isn't bounded. Please tell me is the prove ... |
H: A picky question on set theory
I just came to this math statement:
Let A,B,C be sets. Then: (AxB)xC = Ax(BxC)
My question is, why is it so?
I mean,
(AxB)xC = { ((a1,b1),c1), ((a1,b2),c3), ... }
and
Ax(BxC) = { (a1,(b1,c1)), (a2,(b3,c2)), ... }
Is there any "hidden" assumptions on this?
AI: You're correct that they... |
H: $1=2$ | Continued fraction fallacy
It's easy to check that for any natural $n$
$$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$
Now,
$$1=\frac{1}{2-1}=\frac{1}{2-\cfrac{1}{2-1}}=\frac{1}{2-\cfrac{1}{2-\cfrac{1}{2-1}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-1}}}}=\ldots
=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\f... |
H: What 's the differece between $\cot(x)$ and $\arctan(x)$?
I know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$
What is the difference between these two function?
Is $\cot(x)$ the reciprocal function of $\space \tan(x) \space$ and $\arctan(x)$ i... |
H: Characterisation of norm convergence
Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$):
We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and
we have $x_n \xrightarrow{\sigma} x$ (weak convergence) as well as $\|x_n\| \to \|x\|$.
Proo... |
H: Unicity (or not) of the solution of an integral equation
Given the integral equation:
$$\int_0^a f(x)\left[ \frac{d^2}{dx^2}f(x) \right]dx=a$$ with the condition:
$$\lim_{x\to\infty}f(x)=0$$ how can I find its solution?
Is the solution (if any) the only one possible?
AI: Take any function $g$ defined on $[0,a]$, an... |
H: Probability question enough data?
So,I have this probability question:
In average, 15 clients visit a store in a hour.What is the probability that :
a) None of the clients buys
b)12 clients buy
c)less than 20 clients buy.
But I dont think I have enough data to answer this :/ I thought about using Poisson distribut... |
H: proving existence of a crossing point
Given are two cars, that travel the distance from city A to city B in the same time.
We have to show that there is (at least one) point in time $t_0$, when the two cars have exactly the same speed.
I approached this as follows:
let $f_1(t)$ and $f_2(t) : [0,1] \rightarrow \mat... |
H: Existence of eigenvalues for self-adjoint maps in finite-dimensional inner product spaces
For a finite-dimensional inner product space over $\mathbb{C}$, it is clear that every linear transformation is diagonalisable. In my lecture notes, the lecturer claims that:
For a finite-dimensional inner product space, ever... |
H: Autocovariance of moving average process
Let $\epsilon_t\text{ ~ i.i.d.}(0,1)$, and $X_t=\epsilon_{t}+0.5\epsilon_{t-1}$. I need to find its autocovariance function.
I know that $E(X_t)=0$, $E(\epsilon_{t})=0$. Let's say, that $s=t+1$:
$Cov(X_t,X_s)=E[(\epsilon_t+0.5\epsilon_{t-1})(\epsilon_{t+1}+0.5\epsilon_{t})]=... |
H: Questions about the Space of Matrix Coefficients
Apologies in advance for the basic question: In reading up on representation theory, I came across a confusing definition for the $M(\rho)$, the space of matrix coefficients of a representation $(G, \rho, E)$:
Let $\rho_{ij}(s)$ be a matrix representation of $\rho(s... |
H: Removing principal part to get analytic function in disk
I'm working on this problem:
Suppose $f$ is analytic on the disk $|z| < 2$ except at $z = 1$, where $f$ has a
simple pole. If
$$f(z) = \sum_{n=0}^{\infty}a_nz^n$$
is the Taylor series expansion for $f$ around $z = 0$, prove that $\lim\limits_{n\to \infty} a_n... |
H: Do spaces satisfying the first axiom of countability have monotone decreasing bases for every point?
I'm facing some problems with this. Every proof I've read assumes this, although it is not obvious to me as of now.
First axiom of countability as defined in my book- for every point $x\in X$, there is a countable ... |
H: Definition of submatrix
Let $A$ a matrix, I need the definition of sub matrix of $A$. Thanks in advance.
AI: Try our dear friend Wikipedia.
Elaboration, as requested. Let $A\in M_{m,n}$ have $m$ rows and $n$ columns. Set $S\subseteq \{1,2,\ldots, m\}$, $T\subseteq \{1,2,\ldots, n\}$, such that both $S,T$ are none... |
H: How do I know if the linear system has a line of intersection?
I was wondering how can I determine if there is a line of intersection with any matrix?
For example, if I have the following matrix:
$$\left(\begin{array}{rrr|r}
1 & -3 & -2 & -9 \\
2 & -5 & 1 & 3 \\
-3 & 6 & 2 & 8 \\
\end{array} \right)$$
What does th... |
H: First fundamental form and angle between curves
On surface, for which $$ds^2=du^2+dv^2$$
find the angle between lines $v=u$ and $v=-u$.This exercise is related to the first fundamental form. I think I need to find the angle between curves with parametrization: $$u(t)=t, v(t)=t$$ and $$u(s)=s, v(s)=-s$$ therefore $$... |
H: recurrence criterion for random-walk like Markov chain
Suppose we have a random-walk like Markov chain, i.e. state space is the set of all integers from $-\infty$ to $\infty$, the transition probability $P_{ij}$ is nonzero only when $j=i+1$ or $j=i-1$.
The difference from the random walk is that the transition prob... |
H: Find the values of the constants in the following identities $A(x^2-1)+B(x-1)+c = (3x-1)(x+1)$
I'm stuck on a basic question regarding identities.
$A(x^2-1)+B(x-1)+C = (3x-1)(x+1)$
I've managed to substitute $x$ for $1$ to work out C is $4$. However, I'm unsure how to work out A and B respectively.
AI: For $x=-1$... |
H: Littlewood's isoperimetrical problem
Please consider the following self-contained excerpt from Chapter $1$ of Littlewood's A MATHEMATICIAN'S MISCELLANY. I have two questions:
1) How is the second (weak) inequality derived?
2) How does the result follow from the two (weak) inequalities?
AI: I don't think the part st... |
H: On projective dimension of quotients of polynomial rings
Let $A$ be a commutative ring, $B=A[X]/(X^2)$, and $C=B/(x)$. (Here $x$ denotes the residue class of $X$ modulo $(X^2)$.) Why the projective dimension of $C$ is infinite ?
AI: I'm assuming you ment projective dimension as a $B$-module.
Have you tried finding ... |
H: how to know of the number of real roots?
Let $ax^4 +bx^3 +cx^3 +dx + e = 0$ with $a,b,c,d,e\in\mathbb R$. I would like to know, how can I determine the condition for the polynomial to have exactly three distinct real solutions.
one has to be a double root, or is there any other possibility.
I need help please.
Than... |
H: Probability that the first digit of $2^{n}$ is 1
Let $a_{n}$ be the number of terms in the sequence $2^{1},2^{2},\cdots ,2^{n}$ which begins with digit 1.
Prove that $$\log2 -\frac{1}{n}<\frac{a_{n}}{n}<\log2\text{ (log base is 10)}$$
Note: This is only a part of the question.The actual question is:Prove that th... |
H: Closure and pathwise connected
On $\mathbb R^2$ I consider the following set $G:=\{(x,\sin(\frac{1}{x}))| x\in (0,1]\}$
Well this set is pathwise connected (as a graph of a contionous function on an interval). The function is also oscillating on the left hand side so it makes sense to write the closure as $\bar{G}... |
H: first order ordinary differential equation
How can I solve this ODE of first order:
\begin{align} y^{'}= y^{2}+x, & \text{where } y^{'}=\frac{dy}{dx} \end{align}
Is there any exact mehod to solve it ?
Thank you.
AI: Make the change of function $\displaystyle y=-\frac{v'}{v}$. This transforms your differential equa... |
H: A formula to find the organs's value from $1$ to $100$.
We have a variable named NUMBER
This variable, can hold ANY number from $1$ to $100$.
Let's mark that number as $X$.
We know that:
$x(1) = 1000$ Coins
$x(100) = 250$ Coins
I need to write down a formula that will automatically find the coins amount of the vari... |
H: Sum of products of elements in matrix form.
Suppose I have two matrices $\textbf{A}$, and $\textbf{B}$ as follows:
$\begin{array}{c=c}
\textbf{A}
=
\left[ \begin{array}{ccc}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}\\
\end{array}\right]
\end{array}$
, and
$\begin{array}{c=c}
\te... |
H: Least Squares Plane using Matricies
For a Least Squares solution to a 2D set of coordinates, the formula is:
$X^T\,X\,\vec b = X^Ty$ (where $X^T$ denotes $X$ transpose)
(for: $y = B_0 + B_1x + B_2x^2$)
where:
$X$: $b$: $y$:
[$1$ $x_1$] [$B_0$] [$y_1$]
[$1$ $x_2$] [$B_1$] [$y_2$]
[$1$ $x_3$] [$B... |
H: Simple Combination of cards
I'm a bit ashamed to ask such a simple question, but my math skills are a bit rusty to say the least. Here's the big deal:
I have 10 cards, 5 black and 5 white. How many combination can I make with those cards while using all of them?
-Obviously permutating 2 same-color cards won't creat... |
H: Some doubts about predicate calculus
I am studying the predicate calculus in First Order Logic and I have some doubt about this argument. In my book I find that a formula in the predicate calculus is built from
Literals constructed from
Predicate Symbols: $P = \{p,q\}$
Constants: $A = \{a,b\}$
Variables: $X ... |
H: correct expansion of a sum using multiple indexes
I have looked for a similar posting but haven't found anything... but then I am also a bit unsure of how to search because I've never posted a math question before. In my introductory finite element method course the prof was introducing multiple index notation and ... |
H: How does one solve this Sum?
How does one compute,for integer $a,b\ge0$,$$\sum_{i=0}^b (-1)^{(b-i)} \dfrac{1}{a+b-i}\dbinom{b}{i}$$
AI: $\displaystyle(1-x)^{b}=\sum_{i=0}^{i=b}\dbinom{b}{i}(1)^{i}(-x)^{b-i}=\sum_{i=0}^{i=b}\dbinom{b}{i}(-1)^{b-i}x^{b-i}$
Multiplying by $x^{a-1}$ we have,
$\displaystyle x^{a-1}(1-x)... |
H: Measurability from Inner Measure
Let $E$ have finite outer measure. Show that there is a $G_\delta$ set $G$ which contains $E$ and has the same outer measure. Then show $E$ is measurable if and only if $E$ contains an $F_\sigma$ set $F$ of the same outer measure.
$\textit{Proof:}$ Choose and open set $\mathcal{O}_k... |
H: how to determine the following set is countable or not?
How to determine whether or not these two sets are countable?
The set A of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$.
The set B of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$ that are eventually 1.
First one is easier to det... |
H: Find the values of the constants in the following identitity $x^4+4/x^4 = (x^2-A/X^2)^2+B$
A step by step solution would be preferred for the following question :
Find the values of the constants in the following identitity $x^4+4/x^4 = (x^2-A/X^2)^2+B$.
so far I managed to substitute $x$ for $1$ to get :
$x^4+4/x^... |
H: Why is $k \rightarrow A \rightarrow A / I$ and isomorphism of rings if $I \subset A$ is maximal?
Let $k$ be a algebraically closed field, $A$ a finitely generated $k$-Algebra and $I \subset A$ a maximal ideal. Let $\varphi: k \rightarrow A$ be a ring homomorphism.
Why is this combination
$$k \rightarrow A \rightarr... |
H: How to find the integral of $\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$
$$\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$$
What is the method to find an integral like this?
AI: Perhaps the easiest, though it requires some knowledge of the complex exponential function, is to substitute $$\sin x=\frac{e^{ix}-e^{-ix}}{... |
H: Use the residue theorem to evaluate
$$
\int _ {|z|=2} \frac { dz} {(z-4)(z^3-1)}
$$
What I've done now is the following.
$f$ has isolated singularities at $z=4$, $1$, $\exp(\pi i/3)$, $\exp(-\pi i / 3)$
$$
\int _ {|z|=2} \frac { dz} {(z-4)(z^3-1)} = 2 \pi i (\operatorname{Res}(f;1) + \operatorname{Res}(f;\exp(\pi i... |
H: proving $\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+\frac{1}{5\cdot 6}+\cdots+$ Without Induction
i proved that:
$$
\begin{align}
& {} \quad \frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+\frac{1}{5\cdot 6}+\cdots+\frac{1}{(2n-1)\cdot 2n} \\[10pt]
& =\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\text{ for }n\in \... |
H: $\mathbb F_9 = \mathbb F_3(i)$ Question about squares
Is it true that $\forall a,b \in \mathbb F_9$: $a \cdot b$ is a square $\iff$ $a \cdot \overline b$ is a square ?
AI: Since your group of units is cyclic, you can argue very similarly to the case of $F_p$, and the "Legendre symbol" analogue here is still a homom... |
H: Is there any "superlogarithm" or something to solve $x^x$?
Is there any "superlogarithm" or something to solve an equation like this:
$$x^x = 10?$$
AI: Yes - it's called the Lambert W Function. Scroll down and take a look at Example 2. |
H: Contradiction to Continuity of Integration?
For each of the two functions $f$ on $[1,\infty)$ defined below, show that $\lim_{n \rightarrow \infty} \int_1^n f$ exists while $f$ is not integrable over $[1,\infty)$. Does this contradict the continuity theory of integration?
(i) Define $f(x) = (-1)^{-n} / n$, for $n \... |
H: Solving Recurrent Relation
$a_{1}=\dfrac{3}{5}$ , $~$ $a_{n+1}=\sqrt{\dfrac{2a_{n}}{1+a_{n}}}$ $~$ $(n\geq 1)$
Find the closed form of $a_{n}$
AI: Let $b_n = 1/a_n$. Then
$$b_{n+1}^2 = \frac{1+b_n}{2}$$
Sorry if this is kind of a deux ex machina, but you might be able to recognize that the above recurrence fi... |
H: Why is this J(all the linear combinations of two polynoimials in F[x]) an ideal of polynomial field F[x]?
Suppose $a(x)$ and $b(x)$ are two non-zero polynomials in the polynomial field $F[x]$ have a gcd $d(x)$ can be expressed as a "linear combination":
$$
d(x) = r(x)a(x) + s(x)b(x)
$$
where $r(x)$ and $s(x)$ are i... |
H: Is there an easy way to get to a paper, given a citation?
This question isn't about math per se, but I hope it will be of general interest to people studying math so I feel reasonably comfortable asking here. Let me start with an example: Today I had the following citation from a paper:
W. Hurewicz, On Duality Th... |
H: Find values of the constants in the following identity: x^4+Ax^3 + 5x^2 + x + 3 = (x^2+4)(x^2-x+B)+Cx+ D
Another question on identities:
$$x^4+Ax^3 + 5x^2 + x + 3 = (x^2+4)(x^2-x+B)+Cx+ D$$
How can I find the coefficients for this?
I've got as far as multiplying out the brackets to get:
$$x^4+Ax^3 + 5x^2 + x + 3 =... |
H: Does $((x-1)! \bmod x) - (x-1) \equiv 0\implies \text{isPrime}(x)$
Does $$((x-1)! \bmod x) - (x-1) = 0$$
imply that $x$ is prime?
AI: Yes. This is known as Wilson's theorem.
It's not very practical as a primality test, because the amount of calculation it requires is more than even the obvious tests. |
H: $a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$.
I want to find a $a\in \mathbb{C}$ for which $[ \mathbb{Q}(a) : \mathbb{Q}(a^3) ] = 2$.
Any ideas?
Thanks.
AI: Hint: Let $a$ be a primitive cube root of unity. |
H: Analytic extension for a a function defined in $\mathbb{N}$
I would like to know if it is possible to extend analytically any function of the type $f:\mathbb{N} \to \mathbb{C}$ to all complex plane. If it isn't possible, what should I assume to do so? If
Just an example: the function number of divisors of $n$.
EDI... |
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