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H: Number of integer solutions of $2n^2=a^2+b^2$
For a given integer $n$, how many positive integer $(a,b)$ pairs exist which satisfy $2n^2=a^2+b^2$?
In particular, I'm looking for all $n$s where there are exactly 105 solutions. (One solution is $(n,n)$, and there are $2\cdot 52$ other solutions: $(a,b)$ and $(b,a)$ a... |
H: Primes, congruent, Fermat squares
I need some proof or procedure to solve the question:
For given n = 1 or > 1, We know by fermat, $p = x^2 + ny^2$ for an odd prime $p$. Now, can we proof the
(a) $(-n/p) = 1$ and $(p/q) = 1$ for every odd prime $q\mid n$?
(b) $p \equiv 1 \pmod{4}$ for $n \equiv 0, 1 \pmod{4}$?
(c... |
H: Showing $L=\{uw \mid \exists v:uv\in L_{1},vw\in L_{2}\}$ is regular
Let $L_{1,}L_{2}$ be regular languages and define $L:=\{uw \mid \exists v\in\Sigma^{*}:uv\in L_{1},vw\in L_{2}\}$.
I wish to prove that $L$ is regular using only closure properties
(such as $L_{1},L_{2}$ is regular then so is $L_{1}\cap L_{2},L1\c... |
H: Finding \alpha and \beta of Beta-binomial model via method of moments
I am looking for a laymen step by step of how the process of finding the 1st and 2nd sample moments located:
http://en.wikipedia.org/wiki/Beta-binomial_distribution#Maximum_likelihood_estimation
Also it's my limited understanding that k-th sampl... |
H: Do there exist vector spaces over a finite field that have a dot product?
I've stumbled in the german wikipedia over the question if a vector space over a finite field exists, that has a dot product.
Definition of dot product
A dot product over a $\mathbb{K}$-vector space V is a mapping $F : V \times V \rightarrow ... |
H: divisor of rational function on a surface
Help me please with this question:
Find divisor of rational function $f=(w_0,w_1)$ on a surface $X=\left \{ w_0w_1-w_2w_3 \right \}\subset \mathbb{P}^{3}$
How should I to take into account this surface?
Thanks a lot!
AI: I interpret your question as asking about the rationa... |
H: Analogy of Binary subtraction to decimal subtraction
Possible Duplicate:
How to do +, -, *, / with number in a base b?
Subtraction of numbers with arbitrary bases
This is a very basic question. But I come from non computer science background and is having trouble understanding this . Lets say I want to subtract... |
H: A homotopy question
I already asked about the interpretation of this problem here. Now I would like to ask about the solution. The problem is
Let $A\subseteq X$ be a contractible space. Let $a_0\in A$. Is the embedding $f: X\setminus A\to X\setminus\{a_0\}$ a homotopy equivalence?
I'm not very bright, and this is... |
H: Whether there is a non commutative k-algebra of dimension no larger than 3
Suppose $k$ is a field and $A$ is a $k$-algebra of dimension no larger than 3. If $A$ is semi-simple, then $A$ can be written as a direct sum of simple $k$-algebras. Further one can find $A$ is commutative by exhausting all the cases.
Withou... |
H: Lebesgue generalizations of Hilbert spaces?
Is an L[p] space a generalization of Hilbert spaces using Lebesgue integration?
And if this is the case, is it true that Holder's and Minkowski's Inequalities are generalizations of the Cauchy-Schwarz and triangle inequalities using Lebesgue measures?
AI: No. To start, $L... |
H: $\mathbb{Z}/n\mathbb{Z}[T]$ and zero divisors
I read the following in wiki, but I can't understand what is meant by "divisor" there.
Notice that $(\mathbb{Z}/2\mathbb{Z})[T]/(T^{2}+1)$ is not a field since it admits a zero divisor $(T+1)^2=T^2+1=0$ (since we work in $\mathbb{Z}/2\mathbb{Z}$ where $2=0$)
I unders... |
H: For what values of $n$ is $n^2+n+2$ a power of $2$?
Working on isometric paths in hypercubes, I came up with the following simple, yet (imo) interesting problem. For what natural numbers $n$ exists a natural number $t$ such that $n^2+n+2=2^t$? The first few terms are $n=0,1,2,5,90$, and these are all below one mill... |
H: 5 Points uniformly placed on a sphere
I have a sphere and I have to place some points on it, the most uniformly possible.
If I have 4 points, placing them as vertices of a tetrahedron seems good.
If I have 6 points, placing them as vertices of a octahedron seems very good too.
How can I find a way (the best if it e... |
H: "Commutative" functions
Say a function is commutative if it remains unchanged under any permutation of its arguments. E.g. $f(0,1)=f(1,0)$. (Alternatively we could describe these as functions over multi-sets, or say that they are reflective about any hyperplane $x_i=x_j$). Some examples are sum, product and average... |
H: Stuck With The Differentiation Of A Inverse Hyperbolic Function
I'am suppose to show that $$\frac{\mathrm{d} }{\mathrm{d} x}[x \operatorname{cosh}^{-1}(\cosh x)] = 2x$$
And this is what i've tried.Upon differentiating the above function wrt $x$ using the product rule and applying the formula $\cosh^{-1}f(x) = \fra... |
H: Expressing $\mathbb{R}$ as the quotient of a disjoint union of unit intervals
I am trying to complete exercise 3.18(a) in Lee's Introduction to Topological Manifolds. The exercise is as follows:
Let $A \subseteq \mathbb{R}$ be the set of integers and let $X$ be the quotient space $\mathbb{R}/A$, where $A$ is colla... |
H: Given known reference points, how do I calculate the approximate height of a point from a photograph
I have a photo with three towers. I know that the base of all three towers are at the same elevation, and at the top they each comes to a point. Two of the towers are the same height, though they appear to be differ... |
H: orthogonal subspaces in a Hilbert space
Is it true that if $A,B$ are closed subsets of a Hilbert space $H$, such that $A\perp B$, we have $A+B+(A\cup B)^{\perp} =H$ ? What if $A,B$ are closed subspaces ?$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
AI: If $A$ and $B$ are arbitrary sets, it is not hard to ... |
H: $x,y,z$ independent and uniform random on $[0,1]$, $P(x \geq yz)$?
$x,y,z$ are independent and uniform random on $[0,1]$.
I know that $P(x \geq yz) = 3/4$ by triple integrating the triple density function.
However, is there a easier way to get this solution? Without triple integration.
AI: The conditional probabil... |
H: Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.
I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
AI: The simplest possible example of this would be $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, as thi... |
H: Prove that any family of disjoint 8-signs on the plane is countable
I try to answer the following question from Basic Set Theory by Shen and Vereshchagin :
1.
(a).Prove that any family of disjoint 8-signs on the plane is countable.(By an 8-sign we mean a union of two tangent circles of any size; the interior part ... |
H: A finite sum of reciprocals of complex numbers cannot be confined to a half-plane
Let $z_1,\dots,z_n$ be non-zero complex numbers such that $\sum_{k=1}^n \frac{1}{z_k} = 0$. Prove that for any line $ax+by=0$ passing through the origin, $z_1,\dots,z_n$ cannot all lie in either of the half-spaces $\{ ax+by<0\}$, $\{ ... |
H: Two complex series of functions
The serie of function
$$\sum_{n\in\mathbb Z}\frac{1}{{(z-n)}^2}$$
converges normally in $\mathbb C\setminus\mathbb Z$ and it defines a meromorphic simply periodic function. Now let be $\Lambda$ a lattice, then the serie
$$\sum_{\omega\in\Lambda}\frac{1}{{(z-\omega)}^2}$$
doesn't conv... |
H: Is this 3D curve a circle?
The following is a curve in $3$ dimensions:
$$\begin{eqnarray} x & = & \cos(\theta) \\ y & = & \cos(\theta - \pi/3) \\ z & = & \cos(\theta - 2\pi/3) \end{eqnarray}$$
Is the curve a circle?
If it is, what about this curve in $4$ dimensions?
$$\begin{eqnarray} x & = & \cos(\theta) \\ y & =... |
H: Which parentheses are implied by $\prod$?
Which is correct?
$$ \prod_a ab = \left[ \prod_a a\right]b $$
or
$$ \prod_a ab = \prod_a \left[ ab \right] $$
I'd say the latter, but with $\sum$ we have
$$\sum_a a + b = \left [ \sum_a a \right ] + b $$
Do they work differently, or did I guess wrong? Also, are these things... |
H: Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
Suppose $F$ is a nondecreasing and right continuous function, and the sequence $\{x_n\}_{n\geq1}$ converges to $x$. Then $\liminf\limits_{n\to\infty}F(x_n)\leq F(x)$.
How can I prove this?
AI: Hints:
For every $z\gt F(x)$, there exists $y\gt x$ such that $F(u)\... |
H: Show a function is holomorphic
Let $\phi: [0,1] \rightarrow \mathbb{C}$ be a continuous function. For all $z \in \mathbb{C} \setminus [0,1] $ define $f(z) = \int_0^1\frac{\phi(t)}{t-z} \ dt$. Prove that f is holomorphic on $\mathbb{C} \setminus [0,1]$.
I can't express $f$ as a composition of holomorphic functions, ... |
H: Proof by induction for Division
I was kind of lost with the following example of induction:
$$ (11^{n+1} + 12^{2n-1}) \mathbin{\%} 133 = 0 $$
It shows the following steps to solve it: (I excluded base proof for n = 1)
$$ 11^{(n+1)+1} + 12^{2\cdot(n+1)-1} = $$
$$ = 11 \cdot 11^{n+1} + 12^{2} \cdot 12^{2n-1} = $$
$$ ... |
H: How to generate equiprobable numbers with a random generator?
Is it possible to emulate a 2 sided coin flip (50/50) with a random number generator which outputs equiprobably the numbers 1, 2 and 3 ?
If yes, how ?
If not why ?
Is there a theorem?
Can it be expanded to any numbers x ("sides" of the coin) and y (ran... |
H: Combining primes
A friend felt the need to explore wikipedia and stumbled across "prime notation". I've been working with primes for awhile (cryptography programmer) and have never seen a notation for "combining prime" until today.
I'm not sure if this is an accepted notation or of it is some wikipedia art. I've s... |
H: Create Graph In Sage
I want define new graph in sage. Let $V$ be vector space over finite field $GF(q)$. The graph's vertices are 1-dimensional subspace from $V$ and $n-1$ -dimensional subspace from $V$ and two vertices are adjacent if and only if direct sum of two subspace is $V$.
I have trouble with define this g... |
H: Prove inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq 6$
Let $a,b,c>0$.
What is the proof that:
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq 6$$
AI: Take
$$ \frac{a}{b}+\frac{a}{b}+\frac{b}{c}\geq 3\frac{a}{ (abc)^\frac{1}{3}}$$
$$ \frac{b}{c}+\frac{b}{c}... |
H: Calculate the first derivative without the function?
I am a complete newbie when it comes to advanced mathematics, and am trying to learn calculus on my own. I wanted to know - is it possible to calculate the first derivative if you don't know the function that created a curve, but you DO have all of the points alo... |
H: In eigenvector calculations, why does $A-\lambda I$ need to be singular?
I have this confusion.
If $A$ is a matrix, then its eigenvectors are given by $(A-\lambda I)x = 0$. Since $x$ is not equal to zero, $A-\lambda I$ needs to be singular. I don't understand why $A-\lambda I$ needs to be singular. I thought $A-\la... |
H: order of a meromorphic function and relation
could any one help me to prove the following?
Let $f$ be meromorphic at $p$, whose Laurent series in local coordinate is $\sum_{n} c_n(z-z_0)^n$. The order of $f$ at $p$ denoted by $ord_p(f)$, is the minimum exponent actually appearing(with non zero coefficient) in the L... |
H: Formula for number of lines you can draw through $n$ points
So I've got a homework question I'm stuck on. It's asking me to develop a formula that when given $n$ points, it gives the number of straight lines that can be drawn through those points.
For example, the first two questions were "How many lines can be dr... |
H: What are the subobjects of a manifold?
Categorically a subobject of an object $a$ of some category $A$ is an object $a'$ with a monic morphism to $a$, ie $a'\to a$, upto isomorphism.
When $A$ is either a Topological or Differential manifold, what are the subobjects of a manifold, and are they the same as submanifol... |
H: Integration over a sphere
While I am aware that when integrating over a ball in $\mathbb{R}^n$, we have
$\int_{B(0,R)}f(x)dx=\int_{S^{n-1}}\int_0^Rf(\gamma r)r^{n-1}drd\sigma(\gamma)$
I cannot figure why it is true that
$\int_{S(0,r)}f(x)d\sigma(x)=\int_{S(0,1)}f(y)r^{n-1}d\sigma(y)$
I know this is very trivial bu... |
H: Differential equation involving function defined in terms of an integral
Let $$F(y)=\int_0^\infty e^{-2x}\cos(2xy)\, dx$$ for $y\in \mathbb{R}$. Show that $F$ satisfies $F'(y)+2yF(y)=0$.
I've shown that I can differentiate under the integral sign, and this gives $F'(y)=-2\int_0^\infty xe^{-2x}\sin(2xy)\,dx$, but ... |
H: Equivalent definitions of linear function
We say a transform is linear if $cf(x)=f(cx)$ and $f(x+y)=f(x)+f(y)$. I wonder if there is another definition.
If it's relevant, I'm looking for sufficient but possibly not necessary conditions.
As motivation, there are various ways of evaluating income inequality. Say the ... |
H: Show that exists $\delta >0$ such that:$d(x,y)\geq \delta$
This is the problem:
Let $X$ be a metric space with metric $d$ and $K\subset X$ compact and $F\subset X$ closed and $K\cap F=\varnothing$.
Let $x\in F,y\in K$.
Prove that there exists $\delta>0$ such that for every $x\in F$ and $y\in K$, $d(x,y)>
\delta.$
M... |
H: Using adjacency matrix to calculate the number of hamiltonian paths
I heard that adjacency matrix can be used to calculate the number of k-length paths in a graph. Can't this be used as a way to calculate the number of hamiltonian paths?
AI: The adjacency matrix does not calculate the number of $k$-length paths in ... |
H: Manipulating Exponents
I'm doing my homework and there are a couple of things that I am having trouble grasping. All my homework asks is that I simplify the exponents. For example: 6^5 * 6^3 = 6^8
There are 2 problems I am unsure on what to do. They have to do with multiplying fractions:
(7^3)^5/6 = 7^5/2 is the a... |
H: Silly problem with joint variation.
So, I was recently revisiting some old competition math problems and came to the jarring (and embarrassing) realization that I apparently don't understand joint variation as well as I thought. By way of example, if we have $4 apples = 3 oranges$ and $3 apples = 2 pears$, we could... |
H: How to evaluate $(-2\sqrt2)^{2/3}$?
What is the exact value of this expression?
$$ \left ( -2 \sqrt 2 \right )^{2 \over 3} $$
Isn't $2$ one of the answer? Wolfram gets $-1+i \sqrt 3$. Is root multivariable function for complex number?
AI: Let $x = (-2 \sqrt 2)^{2 \over 3}$.
$x^3 = (-2\sqrt{2})^2 = 8$.
Let $\omega \... |
H: How to normalize a range of values taking the difference into consideration
Given a set of numbers
e.g.
{1, 2, 3, 4, 5} or { 50, 100} or {50000, 50001}
I want to normalize these into a range with a min and max e.g. 2 >= x <= 50
My current algorithm is
$$
((range_{max} - range_{min}) / (x_{max} - x_{min})) * (x - x... |
H: Expanding a basis of a subspace to a basis for the vector space
I'm not really sure how to extend a basis. I'm trying to do the following question.
Consider the subspace $ W = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = -x_4, x_2 = x_3\}$ of $ \mathbb{R}^4$. Extend the basis $\{(0,2,2,0),(1,0,0,-1)\}$ of $W$ to... |
H: Find $\lim_{x\to 0}x^{\frac{1}{x}}$
I tried to solve $$\lim_{x\to 0} x^{\frac{1}{x}}$$
I tried doing it like this:
$$\lim_{x\to 0} x^{\frac{1}{x}}= \lim_{x\to 0} e^{\ln(x^{\frac{1}{x}})} = \lim_{x\to 0} e^{\frac{\ln(x)}{x}} = \exp\left(\lim_{x\to 0} \frac{\ln(x)}{x}\right)$$
Then solving $\lim_{x\to 0} \frac{\ln(x)... |
H: Two subgroups being conjugate
As with the element-wise conjugation action, for $H$ and $K$ subgroups of $G$ in the same $G\text{-orbit}$ under this action, say that $H$ and $K$ are conjugate.
I know that $Gs_o = \lbrace gs_o\ :\ g ∈ G\rbrace$ can be one orbit for G-orbits, but I am not just getting what "same $g\... |
H: Ellipse with non-orthogonal minor and major axes?
If there's an ellipse with non-orthogonal minor and major axes, what do we call it?
For example, is the following curve a ellipse?
$x = \cos(\theta)$
$y = \sin(\theta) + \cos(\theta) $
curve $C=\vec(1,0)*\cos(\theta) + \vec(1,1)*\cos(\theta) $
The major and minor a... |
H: Intuition on the sum of first (n-1) numbers is equal to the number of ways of picking 2 items out of n.
While going through an equation today i realized that sum of first (n-1) numbers is [n*(n-1)/2] which is equal to combinations of two items out of n i.e [n!/((n-2)! * 2!)]. I need some intuition on how these two ... |
H: Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the number $e$, so it's specific to the 11 $e$'s I used in the tower.
AI: You can readily check th... |
H: Rational solutions for arbitrary p
Let p be a prime, and by experimenting with various p, how can we guess a necessary and sufficient condition for x^2 + y^2 = p to have rational solutions.
And how to prove the guess?
One thing I see is maybe using Sylvester's determinant, by how and why?
AI: Let $p$ be a prime nu... |
H: A nontrivial subgroup of $G$ contained in every other nontrivial subgroup.
This question is from a collection of past master's exams.
Let $G$ be a group with a subgroup $H$ as described in the title. I'd like to show that $H$ is in the center of $G$.
My intuition is to choose some element $x\in H$ (and thus in e... |
H: Definition of Geodesic - Distance between two points on the same latitude
I do have a problem understanding the concept of geodesics. As I understand it a geodesic is the shortest distance between two points on any manifolds. Let's consider a spherical earth as depicted in this figure from the Wikipedia. As I read ... |
H: Partial recurrence relation for the number of permutations in $S_n$ which have a square root.
I ran into this problem the other day. The proof is supposed to be done by exhibiting an explicit bijection between two sets, without using induction, recurrence, or generating functions.
Denote by $\omega(n)$ the number... |
H: What does it mean for an expression to hold "identically"?
As an example, the following expression $$\sin2x=2\sin x\cos x$$ is a trigonometric identity. Because it is an identity we can replace $x$ by $ax$ and differentiate with respect to $a$ to get $$2x\cos2ax=2x\cos^2ax-2x\sin^2ax.$$ Then, let $a=1$ to obtain an... |
H: Finding range of a linear transformation
Define $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ by $T(x,y,z) = (2y + z, x-z)$. Find $\mbox{ker}(T)$ and $\mbox{range}(T)$
I could find the kernel easy enough, and ended up getting $\{(-2x, x, -2x) : x \in \mathbb{R}\}$ but I don't really know how the get the range. In this... |
H: Evaluate $\lim_{x\to \pi/2}\frac{\cos x}{x-\pi/2}$ in terms of $(\cos x)'$
Calculate $$\lim_{x\to\frac{\pi}{2}}\frac{\cos x}{x-\frac{\pi}{2}}$$ by relating it to a value of $(\cos x)'$.
The answer is available here (pdf) at 1J-2.
However, I can't seem to make sense of what is actually being done here.
AI: Applying ... |
H: Is this problem correct that $HG'=G$?
Here, I have the following homework:
Let $G$ is a finite $p-$group and let $H$ be a subgroup of it such that $HG'=G$. Prove that $H=G$ ($G'$ is the commutator subgroup).
I have tried to show that $G\subseteq H$ by taking an element in $G$ but this way seems to be weak here. I... |
H: Rearrangement of sequences with limit $0$
Is it true that every real sequence that converges to zero has the property that every rearrangement of it also converges to zero?
I have a proof in mind, but I'm not 100% sure it's correct (although I'm pretty sure), so I just want a yes/no answer.
AI: Yes. A sequence conv... |
H: Elementary rectangles in product measurable spaces
I am working with showing certain equivalence between two probability spaces, one of them being a countable product space of finite spaces, i.e.
$$
(\Omega,\mathscr F) = \prod_{k=0}^\infty \left(X,2^X\right)
$$
so clearly $\mathscr F$ is generated with measurable... |
H: For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?
Let $m \in \mathbb N$. Find a necessary and sufficient condition for
$m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$.
I have to find for which $m$ the quotient ring is an integral domain. I do not know how to us... |
H: Having Trouble With A Simple Equation Involving Differentials
Given that $xyz = c$, show that $dz = -z\left(\frac{dx}{x}+\frac{dy}{y}\right)$. I'am not sure how to get started. How do i differentiate this equation and wrt what? I really would like to try this before posting it but unfortunately i'am unable to even ... |
H: Rearranging the spectral theorem
The spectral theorem for selfadjoint compact operators $L$ with infinite range says that $$Lx=\sum_{k=1}^{\infty} \alpha_k \langle x,f_k \rangle f_k, $$ where the $f_k$'s form an orthonormal system and the $\alpha_k$'s are real nonzero eigenvalues, such that they tend to zero. Now m... |
H: Dimensions of symmetric and skew-symmetric matrices
Let $\textbf A$ denote the space of symmetric $(n\times n)$ matrices over the field $\mathbb K$, and $\textbf B$ the space of skew-symmetric $(n\times n)$ matrices over the field $\mathbb K$. Then $\dim (\textbf A)=n(n+1)/2$ and $\dim (\textbf B)=n(n-1)/2$.
Shor... |
H: In a polynomial of $n$ degree, what numbers can fill the $n$?
Until now, I've seen that the $n$ could be filled with the set $\mathbb{N}_0$ and $-\infty$ but I still didn't see mentions on other sets of numbers. As I thought that having 0 and $-\infty$ as degrees of a polynomial were unusual, I started to think if... |
H: The variance in the average of a set of normally distributed random variables
I have a set of $M$ normally distributed random variables, $r_i$, each with an associated mean $u_i$, but the same variance $\sigma^2$. What is the variance of the average of these $M$ random variables: $\frac{\sum_{i=1}^{M} u_i}{M}$? H... |
H: Probability for 3 balls of different colours
I got this question in an exam a while ago and I wasn't sure how exactly I could solve it:
An urn contains 3 red balls, 3 blue balls and 3 green balls. Three are drawn at
random from the urn. What's the probability they're all of a different colour?
It is not specified... |
H: Find the probability of one expression
Could you please help me to calculate the probability as follows
\begin{align}
\rho=\mathsf{Pr}\left\{\frac{X_1}{X_2} >\frac{Y_1+a}{Y_2 + a}\right\}
\end{align}
where $X_1,X_2, Y_2$, and $Y_2$ are independent exponential distributed random variables with mean $\Omega_{X_1},... |
H: Resources for IMO
I am seeking an online resource or any book where I can find the questions of International Mathematical Olympiad questions chapter wise eg Number Theory problems grouped together.
Like this site where we can get all the programming problems of SPOJ classified under different sections.
I have goog... |
H: Checking Uniform Convergence
Question:
Prove that the sequence of functions $f_n(x) = x^n$ converges uniformly to zero on any interval of the form $[0,\mu]$ if $\mu < 1$.
My Work:
Recalling the definition of uniform convergence, we say that $f_n(x) \to 0$ uniformly if $$\forall\epsilon>0 \ \exists N \ \forall n ... |
H: Why is this sequence of functions not uniformly convergent?
For the following sequence of functions and its limit function, we can see that $f_n(x)$ is clearly pointwise convergent
$$f_n(x) = x^n\text{ }\forall x\in[0,1]\text{ and }\forall n\in\mathbb N^*\\
f(x) = \begin{cases}0&\text{if } x\in[0,1)\\1&\text{if } x... |
H: why don't we define vector multiplication component-wise?
I was just wondering why we don't ever define multiplication of vectors as individual component multiplication. That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 \rangle$ to be $\langle a_1a_2, b_1b_2 \rangle$? Is the ... |
H: What's the limit of this expression: $\lim\limits_{M \to \infty}1/(\sum_{i=0}^{\infty}\frac{M!}{\left(M+i\right)!}x^{i})$
I have a question about a limit: Assume $x$ is a positive real constant $(x>0)$, then what's the limit of the following expression?
$$
\lim_{M\rightarrow\infty}\frac{1}{\sum_{i=0}^{\infty}\frac{... |
H: Homomorphism in algebraic systems
In the problems involving two algebraic systems, for eg.,$\langle S,*\rangle$ and $\langle P,\bigoplus\rangle$ where the sets $S=\{a,b,c\}$ and $P=\{1,2,3\}$. Here we have to check whether they both are isomorphic or not. While solving, they take values as $g(a)=3$, $g(b)=1$ and $g... |
H: Help solving differential equation
I want to solve the following differential equation:
$y[t]$ : vertical position (height) of the object at time t
$y_c$ : height of the ceiling
$y_e$ : equilibrium point, the height at which the mass will stop at the end of its movement.
$a[t]$ : acceleration at time t
$t$ : ... |
H: Weak convergence in Sobolev spaces involving time
Let $$Lu:=-\sum_{i,j=1}^n(a^{i,j}(t,x)u_{x_i}(t,x))_{x_j}+\sum_{i=1}^nb^i(t,x)u_{x_i}(t,x)+c(t,x)u(t,x)$$ and the associated bilinear form $$B[u,v;t]:=\int_U\sum_{i,j=1}^na^{i,j}(t,x)u_{x_i}(t,x)v_{x_j}(t,x)+\sum_{i=1}^nb^i(t,x)u_{x_i}(t,x)v(t,x)+c(t,x)u(t,x)v(t,x)d... |
H: Finding the sum of this alternating series with factorial denominator.
What is the sum of this series?
$$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
AI: Hint: We have
$$e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\cdots.$$
Multiply both sides by $x$ and differentiate. |
H: Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$.
I encountered a problem in a book that was designed for IMO trainees. The problem had something to do with divisibility.
Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$.
Can somebody give me a hint on this pr... |
H: Removing term from RHS of equation
Suppose that we have the following equation:
$g = c\left( {\frac{{a + {\sigma ^2}}}{{b + {\sigma ^2}}}} \right)$
Is there anything that can be done to remove the $\sigma ^2$ term from the RHS of the equation so that the RHS $\rightarrow ca/b$, and the LHS of the equation has $\sig... |
H: Noncommutative rings, finding $a$ and $b$ such that every term in the sum $(a+b)^n = a^n + a^{n-1}b + \ldots$ is distinct
This question was inspired by the binomial theorem for rings. For commutative rings, we have the identity
$$(a+b)^n = \sum_{k=0}^n {n \choose k}a^kb^{n-k}$$
which does not hold for non-commutat... |
H: Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers. Is it possible using induction?
AI: I recommend to read the answer from here. section Weighted AM-... |
H: Question about topology definition
I am reading a topology definition:
Let $X$ be a set and let $\tau$ be a family of subsets of $X$. Then $\tau$ is called a topology on $X$ if:
Both the empty set and $X$ are elements of $\tau$
Any union of elements of $\tau$ is an element of $\tau$
Any intersection of finitely m... |
H: Number of basis representations of Boolean vectors
Suppose I have a basis $v_1, ..., v_n$ of boolean vectors and a boolean vector $v$ that was constructed by iterated $v = v_i$ $OR$ $v_j$ statements. I want a way to figure our how many unique combinations $v_I$, $I = \alpha_1,...,\alpha_k$, $\alpha_i \neq \alpha_j ... |
H: Explain the arithmetics of this trigonometrical expression
$$(\frac{d_1}{2} + \frac{d_1}{2} \cdot \cos(2x))+(\frac{d_2}{2} \cdot \sin(2x)) = \frac{d_1}{2}.$$
Can someone explain how the arithmetics works here?
(My teacher has only noted that "filtration removes $\cos(2x)$ and $\sin(2x)$")
AI: As has been pointed ou... |
H: Question about measurable sets $E_n$ such that $\lim_{n\rightarrow \infty}L^N(E_n) = 0$.
Let $E,E_n \subset \mathbb{R}^N$ be measurable such that $E_n \subset E, E$ is bounded domain, $E_{n+1} \subset E$ and $\lim_{n\rightarrow \infty}L^N(E_n) = 0$. Are there $K_n$ compact such that $E_n \subset K_n \subset E$ and ... |
H: How to calculate the Mahalanobis distance
I am really stuck on calculating the Mahalanobis distance. I have two vectors, and I want to find the Mahalanobis distance between them. Wikipedia gives me the formula of
$$
d\left(\vec{x}, \vec{y}\right) = \sqrt{\left(\vec{x}-\vec{y}\right)^\top S^{-1} \left(\vec{x}-\v... |
H: Confusion about pointwise vs. uniform convergence
I'm confused about expressions like $$L=\sum_{k=1}^{\infty} R_k, \quad \quad (1)$$where $L:H\rightarrow H$ and $R_k:H \rightarrow H$ are continuous linear operators in a Hilbert space $H$. We assume $\sum_{k=1}^{\infty} R_k x$ is convergent every $x\in H$.
The thing... |
H: Foam-like graphs
What's the "official" name of a connected planar graph consisting entirely of polygons (cycles), glued together at edges, e.g. - among other things - without "end vertices" (of degree 1) and without edges not belonging to an inner face. Maybe something like "planar foam graph"?
Has this concept be... |
H: How do I solve a Continued Fraction of solution to quadratic equation?
I know that it is possible to make a CF (continued fraction) for every number that is a solution of a quadratic equation but I don't know how.
The number I'd like to write as a CF is:
$$\frac{1 - \sqrt 5}{2}$$
How do I tackle this kind of proble... |
H: How do I find clockwise angle of a point from the negative x-axis?
Title pretty much says it. How can I rotate a point around the origin and find the clockwise angle of the point from the negative x-axis? I tried using the atan(height/width), but that gives me the angle in the specific quadrant, not from the nega... |
H: How to determine if this is a field?
A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$).
The question is:
True or False: The ring $R$ must be a field.
I thought that if $R$ was a field it had to be a finite field in this case (because $R$ is a finite ring). And to be able to be a finite ... |
H: How do I solve this limit scenario?
If $f$ is continuous at $5$ and $f(5)=2$ and $f(4)=3$, then $\lim_{x \to 2} f(4x^2-11)=2$
Is the above statement true or false with an explanation for the answer?
I'm not sure how to approach this.
AI: Theorem: if $g$ is continuous at $x_0$ and $f$ is continuous at $g(x_0)$, then... |
H: Derivation of formula for pressure gradient (fluid mechanics)
There is a derivation of a formula in my textbook which I don't fully understand. Most of the formulas I encounter I understand without difficulty, but if someone can help me understand one step of the following derivation, I would be very grateful:
Con... |
H: How many ways can you arrange the letters PLAY without repetition if the first letter can't be A and the second can't be Y?
How would you do this sort of problem in general, for example, if the the letter sequence was longer and there were more conditions?
One way I used to solve this problem was to enumerate the p... |
H: Sequence of $C^1[0,1]$ functions $(f_n) \to f$ but $f \notin C^1[0,1]$
Question:
Give an example of a sequence of continuously differentiable function $(f_n)$ on $[0,1]$ so that $f_n \to f$ uniformly, but $f$ is not differentiable at all points of $[0,1]$.
My Thoughts:
Would a Fourier Series be a correct answer t... |
H: Open Sets - example
I have the following exercise:
Prove that $$A=\{(x,y)\in \mathbb{R}^{2} \mid x >0\}$$ is a open set.
I try to solve that exercise with the help of definition, so :
To prove that $A$ is open, we show for every point $(x,y) \in A$ there exists an $r>0$ such that $D_{r}(x,y)\subset A$. Now I m... |
H: Differential system on the torus 2
I've got the following question related to my previous post and I was suggested to write a new question, so there you are.
The question is the following: prove that if $f$ is smooth, periodic and with zero average, then the solution to the system on $\mathbb T^2$
$$\begin{cases}\f... |
H: Showing that a function is bijective
Show that the function $f$ defined by $$f(x):=\frac{x}{\sqrt{x^2+1}}\;,$$ $x$ is an element of the reals, is a bijection of the reals onto $\{y:-1<y<1\}$.
So we need to show that it is 1-1 and onto.
I begin by trying to prove that it is 1-1:
$$\frac{x}{\sqrt{x^2+1}} = \frac{y}{\... |
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