text stringlengths 83 79.5k |
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H: Humphreys' Lemma 4.3 for Cartan's Criterion
I'm having a trouble with this proof (see bottom). At the last fourth line it says
$$tr(xy)=0 \implies \sum_{i=1}^na_if(a_i)=0$$
but $$tr(xy)=tr(sy)+tr(ny)=\sum_{i=1}^na_if(a_i)+tr(ny)$$
how do we know $tr(ny)=0$? The endomorphism $n$ is nilpotent and $y$ is diagonalizabl... |
H: Why are check digits for UPCs computed modulo 10, whereas for ISBNs, modulo 11?
Recently I was reading a mathematics book and encounterd the fact that UPC codes have a check digit at the end that is computed modulo 10, whereas ISBN numbers have a check digit that is computed modulo 11.
What disturbs me is the fact ... |
H: Maps into products
Theorem: Let $f:A\to X×Y$ be given by the equation
$$f(a)=((f_1(a),f_2(a)).$$
Then f is continuous if and only if the functions
$f_1:A\to X$ and $f_2:A\to Y$ are continuous.
how to prove $\Leftarrow$ this direction.I wanna prove like this if we take an open set in $X×Y$ then get an inverse image ... |
H: Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in $ E^{*} $.
Suppose $ \pi: E \rightarrow E $ and $ \pi^{*}: E^{*} \rightarrow E^{*} $ are dual mappings. Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in ... |
H: sigma fields measures
I am studying basics measure theory, and am stuck with the following problem:
Let Ω = {(i, j) : i, j ∈ {1, . . . , 6}}, F = P(Ω) and define the
random variables $X_1$(i, j) = i and $X_2$(i, j) = j.
Is X1 + X2 measurable with respect to either σ($X_1$) or σ($X_2$)?
The given answer to this is... |
H: Log simplification
Question Image
Now this is the question of asymptotic analysis in Algorithms; My question is $H(n)= n^\frac{1}{\log n}$, First i am taking log of this expression then can I apply power rule here and it will become $\frac{1}{\log n} \log n$ and Here we will have 1 only as answer.
AI: Your image ha... |
H: Derivative of $p\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits_x^\infty {{{\mathop{\rm e}\nolimits} ^{ - \frac{{{u^2}}}{2}}}du} $?
I found a result,
I have,
$p\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits_x^\infty {{{\mathop{\rm e}\nolimits} ^{ - \frac{{{u^2}}}{2}}}du} $
$y = - p\left( x \rig... |
H: Bounded partial derivatives on convex set implies uniform continuity
Let $f(x,y):\mathbb{R^2} \rightarrow \mathbb{R}$, where $A \subset \mathbb{R^2}$ is convex, $f_x, f_y$ are bounded on $A$. How does one actually show that $f$ is uniformly continuos on $A$? I imagine that I have to use Mean Value theorem like that... |
H: Trying to understand the chain rule for partial derivatives
So I've been studying the chain rule for partial derivatives recently and I'm having an extremely difficult time wrapping my head around it as I'm having an incredibly hard time understanding the formulation of the chain rule for partial derivatives in my ... |
H: Picking colored balls randomly.
I created this post yesterday but I have a further enquiry that I will post here.
In 13 balls we have: 5 Blue 4 Red 4 Green
We randomly select 6 balls without replacement, what is the probability of having 2 blue, 2 red and 1 green? (The color of the last ball does not matter)
I thi... |
H: Is it true that if an ideal $I$ of ring $R$ can be denoted as the product of ideals $J$ and $K$ then $I \subseteq J$ and $I \subseteq K$?
I just proof-read a proof of someone, and in the proof the assumption is used that if $I$ is an ideal of a ring $R$ such that $I = JK$ for some other ideals $J$ and $K$, then $I ... |
H: Find a matrix $A$ such that $B=AA^T$
I'm trying to write a fokker-planck equation as an SDE. I know my diffusion matrix, $D(\mathbf{x})$, where $D = \frac{1}{2}\sigma \sigma^T$.
How can I find this sigma, to then use in the SDE?
Edit: I don't think the fokker-planck bit is relevant to the question, it's just there ... |
H: integral calculation mistake
I try to solve this:
$
\int_{0}^{\pi /2}\sin x \cos x\sqrt{1+\cos^{2}x } dx
$
This is what I do:
$ \cos x = t; -\sin x dx = dt; -\sqrt{1-u^{2}} dt $
$ \frac{-\sqrt{1-t^{2}}*t*\sqrt{1+t^{2}}}{-\sqrt{1-t^{2}}} dt $
$-\int_{0}^{1} t * \sqrt{1+t^{2}} dt$
$1+t^2 = a; 2tdt = da; tdt = da/2$
... |
H: New Criteria of Isomorphism + Abelian Group
The following questions were suggested by my friend, while we were studying fundamental group theory. We had no exact ideas of the way to approach the problems.
Questions
(1) Let $G$ and $H$ be groups such that $|H|=|G|$. If we can make a bijection $\phi:G\to H$ such tha... |
H: Why is this not an equivalence relation on real functions? $(\exists c\in\mathbb{R})(\forall x\in\mathbb{R})|f(x)-g(x)|=c$
Say we have the following relation on the set of all functions $\mathbb{R} \to \mathbb{R}$
$$(\exists c \in \mathbb{R})(\forall x \in \mathbb{R})|f(x) - g(x)| = c$$
I'm having trouble understan... |
H: Turning a fraction with repeating decimals into a mixed number: why doesn't this work?
Problem:
Turn $\frac{0.\overline{48}}{0.\overline{15}} $ into a mixed number.
My solution:
$0.\overline{15}$ goes into $0.\overline{48}$ 3 times, with a remainder of $0.\overline{48} - 3 x 0.\overline{15} = 0.\overline{48}-0.\ove... |
H: Space of meromorphic functions is not finitely generated
During a lesson of the course on Riemann Surfaces our lecturer made the following remark, saying this could be proved as an exercise:
The space $\mathcal{M}(X)$ of meromorphic functions on a compact Riemann surface, if not empty, is not finite dimensional (as... |
H: Linear algebra - diagonizable matrix: find matrix P and D such that A = PDP^-1
Provide a P and a diagonal matrix D such that A = PDP^-1
Given:
A=
\begin{array}{l}-1-5i&1+2i&1+7i\\-4-14i&3+6i&1+19i\\-6+4i&3-2i&5-5i\end{array}
λ=1−i, 2−3i, 4
The matrix P would be: ____
The matrix D would be: ____
So I'm struggling to... |
H: Truncated Fourier series
Let $f\in L^2[0,2\pi]$. Suppose that $\exists k\in\mathbb{N}$ s.t. the Fourier coefficients $a_n,b_n$ of $f$ vanish for $n\geq k$. In this situation, can we conclude that $f(x)=\frac{1}{2}a_0+\sum_{n=1}^{k-1}(a_n\cos nx+b_n\sin nx)$ for a.e. $x\in[0,2\pi]$? It seems that we cannot use the F... |
H: Existence of limits and colimits in a pointed category.
I am reading Mark Hovey's model category theory.
In the first chapter, on page $4$, we have a category $\mathcal C$ which has all small limits and colimits. He claims that the pointed category ${\mathcal C}_*$ has arbitrary limits and colimits. He says:
Indee... |
H: Why must this density function be greater than zero almost everywhere?
In Klenke's Probability book, in Example 8.31, he states
Why is it that $f(x)>0$ a.a.? There are several densities for which we have $f(x)=0$ outside a compact set... For example, we can see the densities of the Beta distribution as zero outsid... |
H: I have to calculate this dirac integral: $I=\int_{-1}^{0}\delta(4t+1)dt$
How can I evaluate the following integral?
$$ I=\int_{-1}^{0}\delta(4t+1) \, dt $$
Here is my working out so far:
\begin{align*}
I
&=\int_{-1}^{0} \delta(4t+1) \, dt \\
&=\int \delta \cdot (4t+1) \, dt \\
&=\int \delta \cdot 4\cdot \left(t+\... |
H: How do I find the real general solution of this third order ODE?
I have this inhomogenous ODE
$$y'''-y''+y'-y= 2e^{ \omega x} $$
where $ \omega \in \mathbb{R} $
I want to find the real (!) general solution to it.
The problem starts in the beginning:
The characteristic polynomial is $ \lambda^3- \lambda^2 + \lambda ... |
H: Sections on a finite union of principal open subsets in affine $n$-space
This is exercise 2.5.12 of Liu's Algebraic Geometry.
Let $k$ be a field. Let $X = \bigcup_{i=1}^rD(f_i)$ be a finite union of principal open subsets of $\mathbb{A}_k^n$. Show that $\mathcal{O}_{\mathbb{A}_k^n}(X) = k[T_1,\dots,T_n]_f$ where $... |
H: Confusion over a trigronometric function
I found 2 solutions
But they say it is:
It feels like there's a mistake in the shift here, I tested it on Desmos and the functions doesn't reflect the perihelion and aphelion years. Do you guys have a different understanding of the shift here? Thanks :)
AI: You're confus... |
H: Evaluate $\int_0^1 \ln{\left(\Gamma(x)\right)}\cos^2{(\pi x)} \; {\mathrm{d}x}$
I have stumbled across the following integral and have struck a dead end...
$$\int_0^1 \ln{\left(\Gamma(x)\right)}\cos^2{(\pi x)} \; {\mathrm{d}x}$$
Where $\Gamma(x)$ is the Gamma function.
I tried expressing $\Gamma(x)$ as $(x-1)!$ the... |
H: Hatcher Theorem 2.13 - is the subspace $X$ of its cone $CX$ a deformation retract of some neighborhood in $CX$?
Hatcher's Theorem 2.13 says
If $X$ is a space and $A$ is a nonempty closed subspace that is a
deformation retract of some neighborhood in $X$, then there is an
exact sequence $$\cdots \to \widetilde{H}_n... |
H: Limit $\lim_{x \to -2^- } \frac{a - e^\frac{1}{x+2}}{2e^\frac{1}{x+2} - 1}$
According to my friend you put $ x=-2-h$ directly and say that it becomes like $ e^{\frac{-1}{0}}$ is $-\infty$ , he says that this is justified because x is tending to that limit however from what I've learned you can't directly put $ x=0... |
H: If $A \in SL(d,\mathbb{Z})$ does the same hold for $A^{-1}$?
If I consider an element $A \in SL(2,\mathbb{Z})$, then I have that $A^{-1}\in SL(2,\mathbb{Z})$. I can see this because the inverse of $A$ is obtained by movin the coefficient of the metrix or changing their sign.
Does the same hold for an element of $SL... |
H: Fourier coefficients, $\sum_{n=1}^\infty(|a_n|+|b_n|)<\infty$
Suppose $f$ is absolutely continuous on $[0,2\pi]$ with $f'\in L^2[0,2\pi]$ and $f(0)=f(2\pi)$. I would like to prove
$$\sum_{n=1}^\infty(|a_n|+|b_n|)<\infty.$$
By using Parseval's identity, I have shown that
$$\frac{1}{\pi}\int_0^{2\pi}|f'|^2=\sum_{n=1}... |
H: Finding Mass of Object Given Density
I need to find the mass of an object that lies above the disk $x^2 +y^2 \le 1$ in the $x$-$y$ plane and below the sphere $x^2 + y^2 + z^2 = 4$, if its density is $\rho(x, y, z)=2z$.
I know that the mass will be $\iiint_R 2z$ $dV$, and I just need to determine the region $R$ whic... |
H: Definition of Wave map on manifolds
Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold.
The wave equation is defined as $g. \nabla^2 u$.
As far as I see, $\nabla^2 u \in \Gamma(T^{*}V \otimes T^{*}V \otimes TM)$, since $\nabla_{\partial_{\alpha}} u= \partial_{\alph... |
H: How can I construct these homeomorphisms?
From Rotman's Algebraic Topology:
If $X$ is a polyhedron and $x \in X$, there exists a triangulation $(K,h)$ of $X$ with $x = h(v)$ for some vertex $v$ of $K$.
I'm having difficulty figuring out how to work this out, or even understand how it's possible for a simple examp... |
H: Norm that comes from inner product and quadratic function
Let $Y$ be a normed real vector space with the norm $||.||$, I am trying to see that this norm comes from an inner product on $Y$ if and only if for any $y,y'\in Y$ the function $Q(t)=||y+ty'||^2$ is quadratic on $\mathbb{R}.$
Now the way I am trying to do t... |
H: Sum of six numbers from 1 to 4 divisible by 5 (and generalization.)
Find the probability that 6 positive integers from 1 to 4 are chosen such that their sum is divisible by 5.
In other words, you could have that $[1, 4, 3, 1, 2, 3], [2, 2, 3, 3, 2, 3], \text{and } [3, 2, 2, 3, 3, 2]$ are three separate sets. In mat... |
H: What is the derivative of $F[\mathbf{v}]=\mathbf{v}^T\mathbf{v}$?
How does one attack a derivative of this type?
$$
\frac{\partial }{\partial (\mathbf{v})} \mathbf{v}^T\mathbf{v}
$$
$$
\begin{align}
\frac{\partial }{\partial (\mathbf{v})} \mathbf{v}^T\mathbf{v}&=\left(\frac{\partial }{\partial (\mathbf{v})} \mathb... |
H: Is there a name for a group-like structure under a unary operation?
It seems to me that the set,
$$
S=\{ \sin, \cos, -\sin, -\cos \}
$$
forms something like a cyclic group under differentiation. But I understand groups to be defined to have a binary operation.
Is there a name for a group-like structure under a unar... |
H: Simple Moving Average Value
I'm trying to create a 7 Day Moving average column for my sales. I am having trouble in comprehending the notion of moving averages as I'm not sure which date should the Moving Average value be associated to.
Data
+--------+-------+----+
| date | sales | MA |
+--------+-------+----+
| ... |
H: Given two $3$-distinct-digit-natural numbers. Prove that the probability that at least one of both is a multiple of $10$ is $16/81\simeq 0.197530..$ .
Given two $3$-distinct-digit-natural numbers. Prove that the probability that at least one of both is a multiple of $10$ is $$16/81\simeq 0.197530..$$
My observati... |
H: Proving/Disproving there are always two uncountable sets whose intersection is uncountable.
I have been trying to prove the following:
Let $\mathcal{C}$ be an uncountable family of uncountable subsets of $\mathbb{R}$. Either prove or disprove that there are always two sets in $\mathcal{C}$ whose intersection is an ... |
H: Structure constants in Poisson algebras
I am currently studying Poisson algebras. Regarding the structure constants of a Poisson algebra, How can it be defined for Poisson algebras?
AI: If your Poisson algebra $A$ is generated as an algebra by $x^1,\ldots,x^n$, then the Poisson bracket of arbitrary elements of $A$ ... |
H: Solving System of ODEs Using Matrix Diagonalisation
I have been given the matrix $A = \begin{bmatrix} -3 & -2 & 2 \\ 0 & 2 & 0 \\ -4 & -1 & 3 \\\end{bmatrix}$.
I firstly needed to find the matrix $P$ that diagonalises it, so I found the eigenvalues of $A$, the corresponding eigenvectors and then constructed $P$ wit... |
H: Basic question about the definition of the homology of a spectrum
The general definition of the homology of a spectrum $E$ with coefficients in an abelian group $G$ is $$H_*(E;G):=\pi_*(E\wedge HG)$$
and I always see people using the equality $$H_*(E;G)=\mathrm{colim}_nH_{*+n}(E(n);G)$$
and say that it is easily se... |
H: Elementary operation on determinant, but actually basic algebra
If $\begin{vmatrix} -1 & a & a \\ b & -1 & b \\ c & c & -1 \end{vmatrix} =0$ then what's the value of $$\frac{1}{1+a}+\frac{1}{1+b} +\frac{1}{1+c}$$
I just expanded the Determinant, to get
$$ab+bc+ac+2abc=1$$
Which further leads to $$\frac{1+a}{a}+\fr... |
H: Support of a measure and Lebesgue decomposition
Let $X=[0,1]^n$ endowed with the Euclidean norm and $\mathcal B$ the Borel $\sigma$-algebra on $X$.
Let $\lambda$ be the Lebesgue measure and $\mu$ be a finite measure on $(X, \mathcal B)$ with full support (following wikipedia's definition).
Let $\nu_{ac}$ and $\nu_{... |
H: Prove there exists $\alpha \ge 0$ s.t $\int_0^\alpha f(x)dx =\int_0^\infty g(x)dx$ given that $f,g\ge 0$, $F(x)$ diverges and $G(x)$ converges
This is one of the problems we got as an assignment:
if $f(x),g(x)$ are two integrable functions on $[0,t]$ for any $0<t\in \Bbb{R}$.
and suppose that:
$f(x)\ge 0,\ g(x)\ge... |
H: Can a set with a cardinality $\mathbb R^{\mathbb R}$ be ordered?
$\mathbb Z$ has a natural order. $\mathbb R$ has one too. $\mathbb R \times \mathbb R$ can be ordered by first comparing the left index and then, if left-equal, comparing the right index. That scheme for ordering can be extended to $\mathbb R^{n}$ for... |
H: Is there a polynomial of degree 2 such that $f (0) = 1, f (2) = 2$ and $f (3) = 2$?
I was wondering if there exist a polynomial whith these.
I've trying, but since I only started to see polynomials I don't how to get to the correct answer
Any help?
AI: Yes, there is. Write your polynomial as $ax^2+bx+c$ and substi... |
H: Prove that we can have a graph with 15 vertices that every vertex is exactly connected to 5 another vertices
Imagine a city that has 15 public phones. Is it possible to connect them to each other with some cables in case that every phone must connect to exactly 5 another phone.
I tried to draw this graph with 15 ve... |
H: What is the sum of the products of digits of all three digit numbers?
How do I proceed? All approaches are welcomed.
AI: We can break the numbers in brackets of $100$. For numbers from $101-199$, the product of two digits will be multiplied by $1$. Similarly, for $201-299$, they will be multiplied by $2$. So on an... |
H: Is $a-a=0$ defined or can it be proved by using any axioms?
Following is a partial proof for the trichotomy of integers from Terence Tao's book Real Analysis:
Lemma 4.1.5 (Trichotomy of integers).
Let $x$ be an integer. Then
exactly one of the following three statements is true:
(a) $x$ is zero;
(b) $x$ is equal to... |
H: Differential Equations Integrating y by x
This may be a bit of a silly questions, but when solving a differential equation by finding an integrating factor, is it possible to integrate a function of y and x by x? I understand that in multi variable calculus the y would be treated as a constant, but I am not sure wh... |
H: Controlling the convergence of a series
I have a sequence of real numbers $(a_n)_{n \in \mathbb N}$ such that each $a_n$ is positive and the $a_n$s decrease monotonically with limit zero. Is there any way to control the convergence of the series
$$\sum_{n=0}^{\infty} e^{in\varphi} a_n$$
for $\varphi \in \mathbb R ... |
H: Find all possible pairs of digits $(a,b)$ such that the six-digit number $5a4bb2$ is divisible by $9$
Find all possible pairs of digits $(a,b)$ such that the six-digit number $5a4bb2$ is divisible by $9$
I tried to answer this question, but when I answered $(1,3)$, $(3,2)$, $(5,1)$, I got a 3/6 on the question. C... |
H: Which items to buy if the best one will always be stolen?
The Problem:
A store sells $N$ items. Each item $i$ is priced at $p_i \ge 0$ and you value the $i$th item at $x_i \ge p_i$. You can carry at most two items.
To complicate matters, when you leave the store you will be attacked by a bully who will steal the it... |
H: Rank of matrix $M$
Let $M$ a matrix over $M(m,k,\mathbb{K})$ and matrix $B$ over $M(m,l,\mathbb{K})$. What is the sufficient condition for $\operatorname{rank}(\lbrack M\mid B \rbrack ) = \operatorname{rank}(M)$?
AI: (Assuming I've understood your notation correctly)
You need each column of $B$ to be a linear combi... |
H: Finding $|f(4)|$ given that $f$ is a continuous function satisfying $f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+1\forall x,y\in\mathbb{R}$
The question is simply to find $|f(4)|$ given that $f$ is a continuous function and satisfies the following functional equation $\forall x,y \in \mathbb{R}$.
$$f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+... |
H: How can I say a set has measure $1$?
Suppose $(\Omega,\mathscr{E},\mathbb{P})$ is a measure space such that $\mathbb{P}(\Omega)=1$.
Suppose $A_i \in \mathscr{E}$ for $1 \leq i \leq n$ where $n \in \mathbb{N}$.
Suppose $\mathbb{P}(\bigcup_{1 \leq i \leq n} A_i)=1$.
Can I conclude that there exists $1 \leq i \leq n$ ... |
H: How to solve this parametric logarithmic limit of sequence?
Trying to figure out how to solve this limit:
$$\lim_{n\to \infty} \frac{ln(n^a+1)}{ln (n)} , with \ a \ \in \Re $$
This is what I tried so far:
$ \lim_{n\to \infty} \frac{ln(n^a+1)}{ln (n)} = \lim_{n\to \infty} \frac{ln(n^a(1 + \frac{1}{n^a}))}{ln (n)} =... |
H: Combination of reflection symmetries in $\mathbb{E^4}$
Is the combination between point reflection (https://en.wikipedia.org/wiki/Point_reflection) symmetry and hyperplane(or axial using Hodge duality) reflection symmetry (https://en.wikipedia.org/wiki/Reflection_symmetry) in $\mathbb{E^4}$ possible for a given ori... |
H: Prove that $4^n-3^n\gt 2n^2$ for all $n\ge 3$
I found this problem in a textbook, I confirmed it works when n = 3, and followed up with the inductive step,
$$4^{n+1}-3^{n+1}\gt2(n+1)^2$$
but I'm stuck at
$$4^n\cdot4-3^n\cdot3\gt2n^2+4n+2$$
Induction is a new thing for me, so please excuse any mistakes, thanks.
AI: ... |
H: Vic can beat Harold by $1/10$ of a mile in a $2$ mile race. Harold can beat Charlie by $1/5$ of a mile in a $2$ mile race. Very Confused.
Vic can beat Harold by $1/10$ of a mile in a $2$ mile race. Harold can beat Charlie by $1/5$ of a mile in a $2$ mile race. If Vic races Charlie how far ahead will he finish?
Now ... |
H: Permutations on $[2n]$ with relative ($\!\!\bmod n$) restrictions
Question: Let $\mathfrak{S}_{2n}$ be the permutations on $[2n]=\{1,2,\ldots, 2n\}$. Let $$\mathcal{J}_n=\{\sigma\in \mathfrak{S}_{2n} \mid \sigma(i) \not\equiv \sigma(i+n)\mod n, \text{ for all $i\in [n]$}\}.$$
Prove that $$|\mathcal{J}_n|=\sum_{k=0}... |
H: Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if $a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$
Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if
$a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$
The book tells me to use $x^k - ... |
H: If $\hat{f}(k)=0$ for all $k <0$, then $f(x)\geq0$ for all $x$
I just started learning about the Fourier series, is this statement true or false?
Looking at $\mathcal {R}(-\pi,\pi).$
If $\hat{f}(k)=0$ for all $k <0$, then $f(x)\geq0$ for all $x$.
AI: Notice that $\widehat{f}(k)=\frac{1}{2\pi}\int^\pi_{-\pi}e^{-iky}... |
H: Associative algebras with commutative multiplication?
I.e. the bilinear map/product is not only associative, but also commutative. I am looking for examples of unital associative algebras, so they should be a vector space and a ring, not a vector space and a rng.
One example, inspired by When is matrix multiplicati... |
H: How to find the domain and range for the composition $g\circ f$, i.e. $g(f(x,t),t)$?
I have the following:
\begin{align}
\frac{\partial }{\partial t} f(x,t)&=g(f(x,t),t) \tag 1\\
f(x,0)&= x \tag 2
\end{align}
where $g:\mathbb R^{n+1}\to \mathbb R^n$
The domain and range for $f$ is not stated, but I assume $f:\mat... |
H: Why is it called numerical integration when we numerically solve differential equations?
This has been bugging me literally for years.
When numerically simulating a system of differential equations (e.g., with Runge-Kutta or Euler methods), we are using the derivative to estimate the value of the function at the ne... |
H: Finding points $X=(x,y)$ and directions where directional derivative of $f(x,y)=3x^2+y^2$ is maximum. The points are to be taken on $x^2+y^2=1$
Let the directional vector be $d=(a,b)$ so that $a^2+b^2=1$
Directional derivative of $f$ along $d$ is $ f'(X,d)=\nabla f.d=(6x,2y).(a,b)$
$=6xa+2yb=6\sin \theta\sin\phi+2\... |
H: Polynomial interpolation of a polynomial
Let's say we start with a polynomial like
$$ f(x) = a_{1} x^n + a_{2} x^{n-1} + \cdots + a_{n} $$
then we take $2n$ points over this function, and we try to find the polynomial interpolation, using those $2n$ points (and so we will try to find a polynomial of grade $2n-1$)... |
H: Finding extreme values using chain rule in multivariate function
We are given the function
$$
F(x,y) = (x^2 + y^2)^2 - 2(x^2 - y^2)
$$
with the condition $F(x, y(x)) = 0$ for
$$
y: (0, \sqrt{2}) \to \mathbb{R}, x \mapsto y(x)
$$
The objective is to compute all extreme points and classify them into max/min.
So f... |
H: Find all $a\in\mathbb{N}$ such that $3a+6$ divides $a^2+11$
Find all $a\in\mathbb{N}$ such that $3a+6$ divides $a^2+11$
This problem has stumped me. I don't even know where to begin solving it. I know the solutions will be all $a$ such that
$$\frac{a^2+11}{3a+6}=k$$
with $k\in\mathbb{Z}$
But I really don't know how... |
H: Is the unit ball of a dense set dense?
Let $(X,||\cdot||)$ be a normed vector space, and let $Y \subset X$ be a dense subset of $X$. Does it follow that $\{y: y \in Y, ||y|| \leq 1\}$ is dense in $\{x: x \in X, ||x|| \leq 1\}$?
AI: Let $B$ stand for the unit ball. Then the answer is yes if $closure(interior(B)) = B... |
H: Prove an inequality $\ln(1-1/x)<2/(1-2x)$
I need some help to prove this inequality:
$$\ln(1-1/x) < \frac{2}{1-2x}$$
with
$$x > 1$$
I did plot the curve of $\ln(1-1/x)-2/(1-2x)$ and it's always in minus.
Many thanks in advance!
AI: Let $$f(x)=\ln(1-1/x)-\frac{2}{1-2x} \implies f'(x)=\frac{1}{x(x-1)(1-2x)^2}>0, if ~... |
H: Weak limit of non-negative functions is non-negative (without Mazur)
Let $\Omega \subseteq \mathbb R^2$ be compact subset.
Suppose that $g_n \ge 0$ lie in $L^1(\Omega)$ and that $g_n$ converges weakly in $L^1$ to $g$.
Is there a way to prove that $g \ge 0$ a.e. on $\Omega$ without using Mazur's lemma?
I guess what ... |
H: Find the probability of not owning $x$ or $y$ or both
Bit confused as to how to work this out.
If the probability of $x$ is $0.35$
and the probability of $y$ is $0.6$
and the probability of $x$ and $y$ is $0.26$
How do I go working out the probability of not owning $x$ or $y$ or both?
AI: Let's note the following V... |
H: Calculating number of items in a summed series - non-repeating connections between points
I would like to calculate the number of possible connections in a set of points, and although I can express the idea in a mathematical formula, I don't know how to "reduce" it to a working calculation.
Let's say I have points ... |
H: If a divides $b-1$ and a divides $c-1$ then a divides $bc-1$
I am wondering if the proof I did for this problem is correct.
We know a divides b-1
so $b-1=a(t)$ for some integer t
also a divides c-1
so $c-1=a(r)$ for some integer r, so by this logic
so then $b=a(t)+1$ and $c=ar+1$
So then
$b(c)-1=(at+1)(ar+1)-1$
$bc... |
H: Show that if $X$ and $Y$ are independent with the same exponential distribution, then $Z= |X - Y|$ has the same exponential distribution
$$P\left(Z\le z\right)=P\left(\left|X-Y\right|\le z\right)=P\left(-z\le X-Y\le z\right)=P\left(Y-z\le X\le Y+z\right)$$
This means that, because $\space f(x,y)=f(x)f(y) \space$ as... |
H: Problem from Discrete Mathematics and its application for Rosen section 4.4
This exercise outlines a proof of Fermat’s little theorem.
a) Suppose that a is not divisible by the prime p. Show that no two of
the integers 1 · a, 2 · a, . . . , (p − 1)a are congruent modulo p.
b) Conclude from part (a) that the produc... |
H: For $f \in L^1_{\text{loc}}(\Bbb{R}^d)$ the average $x \mapsto \frac1{\lambda(B(x,r))} \int_{B(x,r)} f(y)\,dy$ is measurable
Let $f \in L^1_{\text{loc}}(\Bbb{R}^d)$. For $x \in \Bbb{R}^d$ and $r > 0$ define the average of $f$ over the ball $B(x,r)$ as
$$(A_rf)(x) := \frac1{\lambda(B(x,r))} \int_{B(x,r)} f(y)\,dy.$$... |
H: Has the given statement regarding Set Theory been correctly stated in the form of a logical statement using logical symbols?
Let's assume that we have a statement, stated in words as follows:
If $A \subseteq B \implies$ for any given value of $x$ such that $x \in A$, it will imply that $x \in B$
Now, let's assume... |
H: Exercises identifying types of differential equations
I am preparing for a test and want to know if I can identify the different types of differential equations. Are there any tests online? I have searched but couldn't find any exercises of this type.
AI: There is a big number of different families of differential ... |
H: Find intersection between line and ellipsoid
I want to find points $\space P(x,y,z) \space$ where a line intersect an ellipsoid with $$P = P_{1}+t(P_{2}-P_{1})$$
Here is where I stuck:
The ellipsoid can be described as:
$$\frac{(x-x_{3})^{2}}{a^{2}} + \frac{(y-y_{3})^{2}}{b^{2}} + \frac{(z-z_{3})^{2}}{c^{2}} = 1$$
... |
H: If $A\neq\emptyset$ there not exist the set $S$ whose element are all sets equipotent to $A$
Statement
If $A\neq\emptyset$ there is no set $S$ containing all sets equipotent to $A$.
My text suggests to prove that if $S$ was a set then $\bigcup S$ would be the set of all set that is not a set but unfortunately I don... |
H: Explain a confusing bound for the integral of a decreasing function.
I am reading a solution of an exercise. In the solution, it says the following:
Consider $g(x,t):=\frac{x}{(1+tx^{2})t^{\alpha}}$, where $x\in (0,\infty)$, $t=1,2,3,\cdots$ and $\alpha>\frac{1}{2}$. Then, since for fixed $x$, $g(x,t)$ is decreasi... |
H: Geometry problem I am having trouble to solve
Prove that $AC = \sqrt{ab}$
$a$ is $AB$; $b$ is $CD$; the dot is the origin of the circle. ABCD is a trapezoid, meaning AB || DC.
My attempt at solving:
According to this rule,
$$MA^2 = MB \cdot MC$$
I can apply this rule and say that $DA^2 = b\cdot DE$. If I manage... |
H: Distribution of $\frac{2X_1 - X_2-X_3}{\sqrt{(X_1+X_2+X_3)^2 +\frac{3}{2} (X_2-X_3)^2}}$ when $X_1,X_2,X_3\sim N(0,\sigma^2)$
Given that $X_1, X_2, X_3 $ are independent random variables form $N(0, \sigma^2 )$, I have to indicate that the statistic given below has a $t$ distribution or not.
\begin{equation}
\frac{... |
H: Find a convergent sequence with $\sum \limits_{n=0}^{\infty} a_n = \sum \limits_{n=0}^{\infty}a_n^2$
If $(a_n)_{n\in N_0}$ and $a_n>0$, find a convergent sequence $a_n$ with $\sum \limits_{n=0}^{\infty} a_n = \sum \limits_{n=0}^{\infty}a_n^2$ , whereas $\sum \limits_{n=0}^{\infty} a_n$ and $\sum \limits_{n=0}^{\i... |
H: Is there any $C^\infty$ monotonically non-decreasing function $f$ which satisfies the conditions below?
As stated in the above title, is there any $C^\infty$ monotonically non-decreasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f((-\infty, -2]) = \{-1\}, f([2, \infty)) = \{1\}$ and $f(x) = x $ n... |
H: Use Chebyshev's inequality to find the parameter
We roll a symmetrical die $200$ times. $X$ is a random variable representing the number of the 6 face appearing. Using Chebyshev's inequality find $c>0$ so that the probability $$Pr(X\in(a-c, a+c))$$ is at least $0.85$.
My attempt:
$$Pr(a-c<X<a+c)\geq0.85$$
$$1-Pr(a-... |
H: What's the name for $r=\cos^3\theta$ (alternatively $x^2+y^2=x^{3/2}$)?
The curve appeared while solving this question.
I tried to look up both $r=\cos^3\theta$ and $x^2+y^2=x^{3/2}$,
and even clicked almost all the links here on wolfram.com .
This image is for $r=4\cos^3\theta$.
Thanks in advance.
AI: its in the ... |
H: Are there more numbers in this sequence?
Are more numbers in this sequence that starts with $2$,$4$,...?
A number is in this sequence if all its factors add up to the same number as the product of the prime of its digits.
so 2 is in this list because $1$+$2$=prime($2$), and $4$ is in this list because $1+2+4$ =Pri... |
H: Why the integrals and derivatives do not kill each other in case of Thomae function?
The following is the Thomae function:
$$ f(x) = \begin{cases}
\frac{1}{q} & if \quad x = \frac{p}{q}, p \in \mathbb{Z}, q \in \mathbb{N},\text{ gcd(p, q) = 1 } \\
0 & if \quad x\in \mathbb{Q}^c \quad or \quad x=0... |
H: Does the alternating composition of sines and cosines converge to a constant?
Let $f(x) = \cos(\sin(x))$ and let $c(f, n)(x)$ denote the function $\underbrace{f\circ f\circ...\circ f}_{n \text{ times}}$. For example, $c(f, 1)(x) = f(x)$, $c(f, 2)(x) = f(f(x))$ and so on.
My question is: does $c(f, n)(x)$ approach a... |
H: Extending edge colourings
Suppose that $\Gamma$ is a connected locally finite graph with a uniformly bounded degree, i.e. there is a $d \in \mathbb{N}$ such that for every $v \in V\Gamma$ we have $\mathop{deg}(v) \leq d$. Using de-Bruijn Erdos theorem, such graph has an edge colouring using at most $d+1$ colours.
M... |
H: Gauss sums and Dirichlet characters
Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement:
Does anyone know how to prove this, or at least can give me a reference? Moreover, I'm also confused by the variable $a$, which appears in (5.4) and in (... |
H: $\int_{0}^{4a} f(x)\;dx=4\cdot \int_{0}^{a} f(x)\;dx$?
Let $f:\mathbb{R} \longrightarrow \mathbb{R_+}$ be a integrable function. If for some $a \in \mathbb{R}$ we have
$$f(4a-x)=f(x),\; \forall \; x \in \mathbb{R}$$
then is true that
$$\int_{0}^{4a} f(x)\;dx=4\cdot \int_{0}^{a} f(x)\;dx?$$
I think it's true and I c... |
H: Diffeomorphism from $\mathbb{R}^m\to\mathbb{R}^n$
I have a question about diffeomorphism between $\mathbb{R}^m$ and $\mathbb{R}^n$.
From this page of the internet we have the following definition:
Let $U\subseteq\mathbb{R}^m$ and $V\subseteq\mathbb{R}^n$. A function
$F:U\to V$ is called a Diffeomorphism from $U$ t... |
H: Show that the orbits are given by ellipses $\omega^{2}x^{2}+v^{2}=C$, where $C$ is any non- negative constant.
I am working through a text book by Strogatz Nonlinear dynamics and chaos . In chapter 5 question 5.1.1 (a). I have answered the question but would like to check if I have performed the integration step pr... |
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