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H: In how many ways can you bet thirteen different football matches to get exactly 12 right
In how many ways can you bet thirteen different football matches to get exactly 12 wins?
My attempt
First I notice that there is only $1$ way to bet $13$ different matches to get $13$ wins.
Second, I notice that ther are $3^{13... |
H: Why it is not possible to construct a set function that is defined for all sets of real numbers with the following 3 properties?
I was reading page 30 of Royden and Fitzpatrick" real analysis ", fourth edition and the book said: " it is not possible to construct a set function that is defined for all sets of real ... |
H: Proving $g: \mathbb{Z} \to \mathbb{R}$, $g(x)=2x+3$ is one to one.
Working on the book: Daniel J. Velleman. "HOW TO PROVE IT: A Structured Approach, Second Edition" (p. 242)
We can define a function $g: \mathbb{Z} \to \mathbb{R}$ by the rule that for every $x \in \mathbb{Z}$, $g(x) = 2x + 3$.
Assume $a,a' \in ... |
H: continuity and norm from a Hilbert space to a Hilbert space
Let $H$ be a Hilbert space. let $e$ be a normed vector of $H$. Let $A$ be the application from $H$ in $H$ défined by :
[
x \mapsto x-2\langle x, e\rangle e
]
Show that $A$ is linear continuous, of a norme inferior to 1
Show by a certain choice of $x$ that... |
H: Making intuition rigorous that integral of some positive function on set should be monotone in the Haar measure of the set
Let $\mathcal{M}$ be a compact Riemannian manifold with geodesic distance function $d$ and $\Omega$ its volume measure.
Pick some $A,B\subseteq\mathcal{M}$ such that $\Omega(A)\ll\Omega(B)$, bu... |
H: If R is a ring, and A has all the sets in R and it's complements, is A an algebra? (Halmos Measure Theory question)
The question I have is related to problem 4.5 in chapter 1 of Halmos' text. Some definitions related to the question are the following. If $X$ is a set then a ring $\textbf{R}$ is a non-empty class of... |
H: Best method to find how many solutions are there to the equation $a + b + c = 21?$
The positive integers $a, b, $ and $c$ are such that $ a + b + c =21$.
$a = 5, b = 5, c = 11$ is a solution. $a = 5, b = 11, c = 5$ is another solution (i.e. order matters).
How many different solutions are there?
I can see a very sl... |
H: Is there a bijection between $\mathcal{P}(\mathbb{N})$ and $\mathcal{P}(\mathbb{N} \times \mathbb{N})$?
I tried using the Cantor-Schröder-Bernstein theorem. I defined $f_1 \colon \mathcal{P}(\mathbb{N}) \to \mathcal{P}(\mathbb{N} \times \mathbb{N})$ as $\{a_1, a_2, a_3,...\} \mapsto \{(a_1,a_1), (a_2, a_2), (a_3,a_... |
H: What are the ordered pairs when A = {1, 3, 5, 15, 18} and R be defined by xRy if and only if x|y.
I just wanted to confirm I understand correctly:
When trying to find the pairs for:
A = {1, 3, 5, 15, 18} and R be defined by xRy if and only if x|y
First I determine the factors:
x|y 1 is a factor of 3
x|y 1 is a fact... |
H: Hausdorff Property for a Covering Space of a Manifold $E\to M$.
I want to show that if $\pi : E \to M$ is a topological covering map and $M$ is a manifold then $E$ is a manifold. I was reading this post which helped me for the second-countability. The OP says it is simple to show that $E$ is Hausdorff but I don't s... |
H: The intuition of the round value of $e^{i \pi}$
I can see the value of $e^{i\pi}$ is $-1$, this value is round without decimals. The $e$ value can be defined as the value (v) that the derivative function of $v^{x}$ is still $v^{x}$. And the $\pi$ value is defined as the value of circumference over diameter of which... |
H: Intuitive proof for distributive property of dot product using $\overrightarrow{u}\cdot\overrightarrow{v} = u_{x}v_{x} + u_{y}v_{y}$
I understand the intuitive way to think of the distributive property using $\overrightarrow{A}\cdot\overrightarrow{B} = AB\cos\theta$:
Then, this makes me wonder if it's possible to ... |
H: Does Hilbert's theorem 90 hold for local rings?
Let $R$ be a local ring (commutative with 1). Let $G$ be a finite subgroup of $\text{Aut}(R)$ preserving the maximal ideal. Then it seems to me that we also have:
$$H^1(G,R^\times) = 0$$
Is this correct? (The classical Hilbert theorem 90 states this when $R$ is a fiel... |
H: Conservation of energy in three dimension
I'm trying to derive the conservation of energy in 3D from the equation $\vec{F}=m\vec{a}$.
David Morin, in his book "Introduction to Classical Mechanics With Problems and Solutions" p. 138-139, proves the conservation of energy in 1D in the following way:
I wanted to prov... |
H: Prove that $f_{2k} \cdot f_{(2k+4)} + 1$ is a perfect square, where $f_n$ is the $n$th Fibonacci number, $n\geq0$.
I found a pattern that makes me conjecture that this is also true for $f_{2k+4}$, as well as $f_{2k+2}$. I actually accidentally found this pattern while thinking about the proof for $f_{2k+2}$! But I ... |
H: Unboundedness of a continuous function
Given a function f on R is continuous and also it satisfies $|f(x)-f(y)|≥K|x-y|$ for all x,y in R. This function is one-one, which is clear but how to prove such functions are unbounded? Thanks in advance.
AI: Take $y=0$. Then $|f(x)-f(0)|\ge K |x|$. Then the triangle inequali... |
H: understanding the limits in calculation of expectation
My question is from the book Bertsekas, "Introduction to probability".
Let's say X is continuous first
I believe I should maniputate the given expression to look the like known definiton of Expecation.
$ E[X] = \int_{-\infty}^\infty x \times f_X(x) dx $
equiv... |
H: Prove that $\sqrt{2}$ must be in the open set strings which covers the $\mathbb{R}$
I read a book about the set theory and met with the following:
I really do not understand the notes in the parenthesis, is it means that the irrational number is very close to the rational number? And why $\sqrt{2}$ is so special i... |
H: Is there a tighter upper bound for $\sum_{k=1}^n|k\sin k|$ than $\frac12n(n+1)$?
Consider the sum
$$\sum_{k=1}^n|k\sin k|$$
An obvious upper bound for this is clearly $\sum_{k=1}^n k=\frac12n(n+1)$. But it seems that this upper bound is too "loose", and so I was wondering if it is possible to find a tighter upper b... |
H: Consider $v=v_1+v_2$, where $v_1 \in M$ and $v_2 \in M^{\perp}$
$M = span\{\begin{pmatrix}8\\0\\-6\end{pmatrix}, \begin{pmatrix}8\\6\\-6 \end{pmatrix}\}$
I am trying to calculate $v_1$ and $v_2$ when $v=\begin{pmatrix}2 \\ 4 \\ 6 \end{pmatrix}$.
I know that since $v_1 \,\in \, M$ and $v_2 \, \in \, M^{\perp}$, $v_1... |
H: Sum of $s_n=10-8+6.4-5.12+...$
I'm asked to find the sum for $s_n=10-8+6.4-5.12+...$ as $n\rightarrow \infty$. I discovered that the sum can be written as $$10\sum_{n=1}^{\infty}(-1)^{n-1}\left(\frac{8}{10}\right)^{n-1}$$
I know from the ratio/roots test the series indeed converges.
My problem is figuring out what ... |
H: Linear Least Squares with Monotonicity Constraint
I'm interested in the multidimensional linear least squares problem: $$\min_{x}||Ax-b||^2$$
subject to a monotonicity constraint for $x$, meaning that the elements of $x$ are monotonically increasing: $x_0 \leq x_1$, $x_1 \leq x_2$, ... , $x_{n-1} \leq x_n$.
I basic... |
H: Suppose $H
I read in some text the following statement:
Let $H$ be a subgroup of $G$. Denote $N=\bigcap_\limits{x\in G} xHx^{-1}$, then $N$ is the largest normal subgroup of $G$ contained in $H$.
It's easy to show $N<G$, since $H$ is a subgroup of $G$ any conjugate $xHx^{-1}~(x\in G)$ of $H$ is also a subgroup of... |
H: Parallelogram Inequality
Let M be a point inside parallelogram ABCD. Then prove that $MA + MB + MC + MD < 2(AB + BC)$
I tried this problem using Triangle Inequality but couldn't proceed. Please help.
AI: Let $PP'$ be parallel to $BC$ and $QQ'$ parallel to $AB$, both through $M$. Note that you can apply the triangl... |
H: Why is subspace $\mathcal{C}$ the intersection of the kernels of $n-d$ linear forms?
I was reading Waldschmidt's notes on Finite fields and error coding, where I came across the following statement, in section $\S 3.3$:
A subspace $\mathcal{C}$ of $F_q^n$ of dimension $d$ can be described by giving a basis ${e_1, ... |
H: Definition by Abstraction in Axiomatic Set Theory by Suppes
I am just starting self-studying Axiomatic Set Theory by Patrick Suppes. I have a doubt on the definition by abstraction.
Just to give some context, Suppes defines a set as
$y $ is a set $\leftrightarrow \exists x\in y\ \vee\ y=0$ (empty set)
I found th... |
H: Dictionary ordering in $\mathbb{R}^2$ is not complete.
Show that the order "$\leq$" on $\Bbb{R}^2$ defined by,
$(a,b)\leq(c,d)$ if ($a<c$) or, $(a=c$ and $b\leq d)$
is not complete.
Hint: Use the set $E=\{(\frac1 n, 1-\frac1 n): n\in \Bbb{N}\}$.
Can any one help me with this? How can the set $E$ be used to show th... |
H: Proof feedback $ f $ is differentiable at $c$ it is also continous at $c$
Hi I have done my own proof of this theorem, and it's likely wrong as it's different to the proof in the book which is also very simple.
I would love some feedback in what wrong assumptions I am making. I think it may be that I'm bounding x-c... |
H: combination of points in the open unit disk also lie in the unit disk
Suppose $a,b$ are points in the open unit disk $\{z\in\mathbb{C}:|z|<1\}$. Then, does the combination $$\frac{(1-|a|^2)b+(1-|b|^2)a}{1-|ab|^2}$$ also lie in the unit disk (open)?
I think yes, but am unable to prove. The triangle inequality does n... |
H: Prove that the limit of a convergent subsequence has to be greater than or equal to 4.
So, here's the problem:
Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that all the terms of the sequence belong to the interval $[4,9)$. Then, prove or disprove the assertion that there exists a convergent s... |
H: Interval of convergence of $\sum_{n=1}^\infty\frac{(x-2)^n\prod_{i=1}^n(n+i)}{n^n}$
$$\sum_{n=1}^\infty\left(\frac{\prod_{i=1}^n(n+i)}{n^n}\cdot(x-2)^n\right)$$
I am here to ask another part of this power series given above. I am to find the "interval of convergence". I have the radius of convergence, which is $e\o... |
H: Are there 3 out of 17 students A,B,C such that the hieght,id and phone number of B is in between A and C
In a clasroom there are 17 students such that no 2 students are at the sam hieght. Are there 3 students A,B,C such that the hieght,id and phone number of B is in between A,C
meaning:
$A_{hieght} > B_{hieght} > C... |
H: How to solve $\sum_{k=1}^{2500}\left \lfloor{\sqrt{k}}\right \rfloor $?
I was trying to solve $\sum_{k=1}^{2500}\left \lfloor{\sqrt{k}}\right \rfloor $ using Iverson's brackets but I can't get the bounds right. I think I'm also missing something.
Here's what I did:
$ m = \left \lfloor{\sqrt{k}}\right \rfloor$
$\su... |
H: Maps passing to Quotient Topology
Let $f:X\rightarrow Y$ be a continuous map between topological spaces and let $q_X:X\rightarrow Z$ and $q_Y:Y\rightarrow W$ be quotient maps. Then, is there a unique map $F:Z\rightarrow W$ making the following diagram commute:
$$
F\circ q_X = q_Y\circ f?
$$
AI: The short answer is... |
H: A function which integrates to 0 over any set of measure 1
Problem:
$f$ is integrable on $[0,2]$, for any measurable set $E \subseteq [0,2]$, when $m(E) = 1$, $\int_E fdm = 0$ prove $f = 0$ $a.e.$ on $[0,2]$
I think the following theorem might be of help, but I'm not sure.
Theorem. If $f$ is integrable on $[a,b]$... |
H: What is the fundamental error in my reasoning?
What is fundamentally wrong in writing $(-a)^{1/2}$ as $((-a)^{2})^{1/4}$ when $a$ is positive and thus equating it to $a^{1/2}$?
Edit:
I'm basically asking if there is anything wrong with this operation like multiplying $1$ and $2$ with $0$ and equating it to "prove" ... |
H: Class of 16 Participants Answering 6 Questions in Subsets of 4, Each One in A Different Combination With All Pairings Covered
16 Participants are arranged into 4 groups of 4. The participants work together on a question within their groups. Next the groups are rearranged into another 4 groups of 4, where they work ... |
H: Computation of the integral of the product of a double exponential and an exponential
I would like to compute the following integral:
$$\int_0^t\exp\left(\frac{\alpha^2}{2\lambda}e^{-2\lambda s}-\lambda s\right)ds\space\space\space(1)$$
An integral near from this one is:
$$\int_0^t\exp\left(\frac{\alpha^2}{2\lambda... |
H: Let $A=\begin{bmatrix}1 & 2\\-1 & 1\end{bmatrix}$. Find a Matrix $B$ s.t for any $u,v \in \Bbb{R}^2, (u,Av)=(Bu,v)$
Let $A=\begin{bmatrix}1 & 2\\-1 & 1\end{bmatrix}$. Find a $B\in \Bbb{M}_2(\Bbb{R})$ such that for any $\textbf{u,v} \in \Bbb{R}^2, (\textbf{u},A\textbf{v})=(B\textbf{u},\textbf{v})$ or prove that no s... |
H: On classification of groups of order $p^5$
Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ?
Edit 1 : I need complete classification (not upto isoclinism), and also in finitely presented form . I found that with increase in value of prime $p$, number of gr... |
H: How do I integrate an unknown function of a variable e.g. $a(t)$?
How would I integrate equations of the following form:
$$\frac{d a(t)}{dt}=ka(t)$$
where $k$ is constant.
I have the feeling that this is quite simple, but I seem to be stuck.
My initial thought was that I could do:
$$a(t) da(t) = k dt \iff \frac{a^... |
H: In a math competition with $8$ students and $8$ problems, if each problem is solved by $5$ students, then two students together solve all problems.
Eight students are entered in a math competition. They all have to solve the same set of $8$ problems. After correction, we see that each problem was correctly resolved... |
H: Expected Number of Distinct Numbers in N trials from a set.
Given the set of numbers from 1 to n: { 1, 2, 3 .. n } We draw n numbers randomly (with uniform distribution) from this set (with replacement). What is the expected number of distinct values that we would draw?
I came across this question on Brainstellar. ... |
H: Spreading tickets in a lottery actually diminishes your chances?
Here is the scenario: There is a lottery running for $n$ terms, which means that it is repeated. In each term, there are a total of $T$ tickets and one prize. You currently own $t$ tickets and your dilemma is to either use all of your tickets in one g... |
H: Balls in connected metric spaces
Let $(X,d)$ be a connected metric space and let $x\in X$. I was wondering if it can happen that there exists $\delta>0$ such that $B_\epsilon(x)=X$ for any $\epsilon>\delta$ and $B_\epsilon(x)\neq X$ for every $\epsilon<\delta$.
If we remove the hypothesis that $X$ is connected, the... |
H: Fish weight - Normal distribution
If the weight of fish distributes by Normal distribution, with $\mu = 900$ and ${\displaystyle \sigma ^{2}}\ = 150^2$. What is the probability that out of $10$ fish that were chosen randomly, at least $2$ and at most $9$ are weighted not less than $667.5$?
So I assumed by the cen... |
H: Uncountably many disjoint dense subsets in $\Bbb{R}$
Show that, there are uncountably many disjoint dense subsets in $\Bbb{R}$.
I know $\Bbb{Q}$ is a dense subset in $\Bbb{R}$. But other than this and disjoint to $\Bbb{Q},$ I have no idea.
Please help me. Thank you.
AI: Consider an equivalence relation $\sim$ on $\... |
H: Prove $\lim_{h\rightarrow0}m(E\Delta(E+h)) = 0$ for measurable set $E$ with finite measure
Here is my attempt:
Define $f_n=\chi_{E\Delta(E+ \frac{1}{n})}$. Then $f_n$ decreses with regard to $n$. Since $$m(E\Delta(E+\frac{1}{n})) = \int_\mathbb{R}\chi_{E\Delta(E+ \frac{1}{n})}dm,$$ it suffices to show $$lim_{n\to\i... |
H: What's the differentiation of $2x(\frac{dx}{dt})$ with respect to the variable $t$? Is it $2(\frac{dx}{dt})^2+2x(\frac{d^2x}{dt^2})$?
What's the differentiation of $2x\left(\frac{dx}{dt}\right)$ with respect to the variable $t$?
Is it $$2\left(\frac{dx}{dt}\right)^2+2x\left(\frac{d^2x}{dt^2}\right)?$$
AI: Let $g(t)... |
H: Defining the Polar set
For a subset P of R^n (real numbers) the polar set is defined by:
$$
P^*:= \{ y\in \Bbb R^n\mid y\cdot x \leq 1 \text{ for all } x\in P \}.
$$
Can someone break the definition into plain english as im struggling to understand the notations used and such a sets structure.
The best guess ive g... |
H: What does this colon notation mean?
I was reading a paper and found the following, "$k$-th entry of sorted $S((i,:))$". I don't understand what does this colon notation mean.
Suppose we have $$S=\begin{pmatrix}1.01330\dots &1.00958\dots &0.96263\dots &0.35814\dots &0.75399\dots \\ 0.59616\dots &0.79699\dots &0.5663... |
H: skew-diagonalizing an anti-symmetrc matrix
Let's assume that i have a (real) $2N\times2N$ anti-symmetric matrix
$B=\left\{ b_{ij}\right\} $ with the property that
$BB^{T}=\boldsymbol{1}$
where $\boldsymbol{1}$ is the identity matrix.
Is it true that I can always find an orthonormal matrix $U$ such that
$\tilde{B}=U... |
H: Prove that a particle will never pass through the centre of a sphere under a condition.
Question: A particle was fired inside of a sphere. There was no gravity acting on the particle, no air resistance and each time it hit the inside of the sphere, it reflected without losing any velocity. If the particle doesn't p... |
H: Sobolev Spaces inner product
The inner product in Sobolev space defined as
$$ \langle u,v\rangle _{H^m(\Omega)} = \int_{\Omega}\sum_{\alpha=0}^m \sum_{\beta=\alpha} D^{\beta}uD^{\beta}v d\Omega $$
where
$$D^{\alpha}u = \frac{\partial^{\alpha}u}{\partial^{\alpha_1}x_1\partial^{\alpha_2}x_2...\partial^{\alpha_n}x_n} ... |
H: Intersection of two disconnected sets
I have prove or disprove that the intersection of two disconnected sets is a disconnected set. I tried finding a counter example on the real line with the usual metric,but I can't seem to find a counter example.I want to proceed to prove that it is the case that the intersectio... |
H: Shouldn't I be able to use both rads and degrees in complex exponentials?
Up until now, I've been using rads and degrees interchangeably, simply using the $^{\circ}$ symbol to signify degrees, and then using the correct trigonometric function, so that:
$$sin(90^\circ)=sin(π/2)$$
I would think that the same line of ... |
H: If all elements of Lie algebra are nilpotent , is the Lie algebra nilpotent?
Suppose $\mathfrak{g}$ be a Lie algebra over $\mathbb{F}$. Then $\mathfrak{g}$ is
nilpotent if and only if, for all $x \in \mathfrak{g}$, $\mathrm{ad}~ x$ is a nilpotent linear operator on $\mathfrak{g}$.
This is Engel's theorem
My doubt ... |
H: Prove that $ \sum_{k=0}^{n}\left\lvert x-\frac{k}{n}\right\rvert\binom{n}{k}x^k(1-x)^{n-k} \le \frac{1}{2\sqrt{n}} $
I have to prove that $$\sum_{k=0}^{n}\left\lvert x-\frac{k}{n}\right\rvert\binom{n}{k}x^k(1-x)^{n-k} \le \frac{1}{2\sqrt{n}} $$ where $n\in \mathbb{N}$ and $x \in [0,1]$
I have already proven that $\... |
H: How do i find the normal?
Find the equations of the tangent and normal to $y = x^2$ at the point $H(2, 4)$.
I've already found the equation of the tangent which is $y=4x-4$
but I'm not sure how to approach the actual question, which is finding the normal.
AI: By definition a normal line is the line that touches a... |
H: Finite extensions
I'am trying to solve the following exercise in Dummit & Foote(Chapter 13 Section 2 Question 13),
Suupose $F={Q}(\alpha_1,\ldots, \alpha_n)$ where $\alpha_i^2\in Q$ for each i. Prove that $2^{1/3}$ is not in F.
Q denotes the set of rational numbers.
According to a solution that I read, it says $[Q(... |
H: Representing a Rayleigh distribution by Gamma distribution
Rayleigh distribution is formulated as $$P(x\mid\sigma)=\frac{x}{\sigma^2}\exp\left(-\frac{x^2}{\sigma^2}\right) \,,\tag{1}$$ where $\sigma^2$ is the variance. $Z$ is a complex variable specified as $Z=a+jb$.
A paper said that Gamma distribution $\Gamma(k,\... |
H: Frobenius norm involving Kronecker Product
Consider $ J = ||\mathbf{G} - ( \mathbf{B} \otimes \mathbf{X} )||_F^2 $, where $\mathbf{G}$ and $\mathbf{B}$ are complex matrices, and $||.||_F$ is the Frobenius norm. Find the derivative with respect to $ \mathbf{X} $
Note: My question is related to this post: Derivative ... |
H: Finding an algebraic number $z \in \mathbb{C}$ with Galois group over $\mathbb{Q}(\sqrt{5})$ equal to $\mathbb{Z}/7\mathbb{Z}$
Let $K = \mathbb{Q}(\sqrt{5})$. I'm trying to find an algebraic number $z \in \mathbb{C}$ such that $K(z)/K \cong \mathbb{Z}/7\mathbb{Z}$. How to go about this? The only thing I could think... |
H: Evaluate $\frac{1}{1 \cdot 2 \cdot 3}+\frac{2}{4 \cdot 5 \cdot 6}+\frac{3}{7 \cdot 8 \cdot 9}+ \cdots$
Evaluate $$\frac{1}{1 \cdot 2 \cdot 3}+\frac{2}{4 \cdot 5 \cdot 6}+\frac{3}{7 \cdot 8 \cdot 9}+ \cdots$$
I see this is the same as $$\frac{1}{3} \sum_{i=0}^{\infty} \frac{1}{(3i+1)(3i+2)} $$
$$\frac{1}{3} \sum_{i=... |
H: Finding a counter-example for Gaussian-periods for non-primes
I need to give a counter-example against the following theorem:
Suppose $H \subset \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is a subgroup. Then we have $\mathbb{Q}(\zeta_n)^H = \mathbb{Q}(\eta_H)$, with $\eta_H = \sum_{\sigma \in H} \sigma(\ze... |
H: Characterization of invariants of a matrix
The coefficients of the characteristic polynomial of an $n\times n$ matrix $A$, which are invariant with respect to similarity transformations $T A T^{-1}$ (where $T\in GL(n)$), are given by polynomials of traces of powers of $A$.
However, these invariants do not completel... |
H: Mittag-Leffler condition for De Rham complexes
I am currently looking at a proof of De Rham's theorem.
Denote by $\Omega^p(M)$ the space of $p$-forms. At some point, one considers a manifold $M=\bigcup U_i$ where each $U_i$ is an open subset, such that $U_i\subset U_{i+1}$ and $\overline{U_i}$ is compact.
Then by c... |
H: Uniform continuity of composition?
If we consider $f\circ u$ and know that
f is bounded and uniformly continuous
u is bounded and continuous
does this imply that $f\circ u$ is bounded and uniformly continuous?
It is clear that it is bounded.
But it is not clear to me whether it is uniformly continuous. Surely it ... |
H: Proof that $f(x)=x|x|$ is differentiable on $\mathbb{R}$
I want to show that
$$f(x) = x|x|$$
is differentiable for all reals.
My approach would be:
Since $ \forall x \in \mathbb{R}$ such that $x < 0$ we have $f(x) = -x^2$ which is differentiable.
also $\forall x \in \mathbb{R}$ such that $x > 0$ we have $f(x) = x^2... |
H: The radius of a sphere is measured as 5 cm ± 0·1 cm. Use differentiation to find the volume of the sphere in the form Vcm^3 ± bcm^3.
I have tried:
V = 4/3πr^3
If the radius is 5 ± 0·1, then V = 4/3π(5 ± 0·1) ^3
giving V = 555.65 or V = 492.81
The provided solution is (524 ± 31) cm^3 but I am not sure how you derive... |
H: Solve for $y$ in $\frac{dy}{dx}-\frac{3y}{2x+1}=3x^2$
I saw a challenge problem on social media by a friend, solve for $y$ in $$\frac{dy}{dx}-\frac{3y}{2x+1}=3x^2$$
I think this is an integration factor ODE
$$\frac{1}{{(2x+1)}^{\frac{3}{2}}} \cdot \frac{dy}{dx}-\frac{3y}{{(2x+1)}^{\frac{5}{2}}}=\frac{3x^2}{{(2x+1)}... |
H: Finding column space - why does this algorithm work?
When I want to find the column space of a matrix, I can row reduce it to echelon form and choose only the columns corresponding to the columns without the free variables in the reduced row echelon form matrix. I have proved it mathematically, but I fail to see wh... |
H: What is the inverse of a matrix with the following structure?
I have come across a $N \times N$ square matrix of the following structure
\begin{bmatrix}
m_1 & m_2 & m_3 & \ldots & m_{N-1} & m_N \\
1 & -1 & 0 & \ldots & 0 & 0 \\
0 & 1 & -1 & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 ... |
H: Proving angles in a circle are equal
$A$, $B$, $R$ and $P$ are four points on a circle with centre $O$.
$A$, $O$, $R$ and $C$ are four points on a different circle.
The two circles intersect at the points $A$ and $R$.
$CPA$, $CRB$ and $AOB$ are straight lines.
Prove that angle $CAB$ = angle $ABC$.
Not really sur... |
H: Calculate $\int_D\sin(\frac{x\pi}{2y})dxdy$
Calculate $\int_D\sin(\frac{x\pi}{2y})dxdy$ $D=\{(x,y)\in\mathbb{R}^2:y\geq x,y\geq 1/\sqrt{2},y\leq(x)^{1/3}\}$.
I've calculate the limits of the integral $2^{-1/6}\leq x\leq1,1/\sqrt{2}\leq y\leq1$ and after doing integral of x first i got stuck
AI: First, your region o... |
H: Under what conditions is $A^T \Sigma A$ positive (semi-)definite for $\Sigma$ p.s.d?
Consider a positive (semi) definite, symmetric matrix $$\Sigma \in \mathbb{R}^{k\times k}$$ and a matrix $$A \in \mathbb{R}^{k \times m}$$.
Under what conditions is $$B = A^T \Sigma A$$ positive (semi) definite?
I had the followin... |
H: Equivalence of EVT consequence
I was getting over EVT whics states that if a real-valued function $f$ is continuous on the closed interval $[a,b]$ then $f$ must attain a maximum and a minimum, each at least once. Then there is one consequence which states following: If $f:[a,b]\rightarrow\mathbb{R}$ is continuous... |
H: Prove that any countable cartesian product of countable sets is countable
I want to prove that infinite (yet countable) cartesian product of countable sets is countable.
Here's what I tried:
Step 1:
I proved that for 2 countable sets $ A_1,A_2 $ , the product $ A_{1}\times A_{2} $ is countable.
Step 2:
I proved by ... |
H: Proof that this function is an isomorphism
If $K = (k_1, k_2, k_3, k_4)$ contains the basis of the linear function $f: R^4 \rightarrow R^4 $ with
$f(k_1) = k_4 , f(k_2) = k_1 + 2k_2 , f(k_3) = 2k_1 + k_2 + k_3 , f(k_4) = 2k_2 - k_3$
show that f is an Isomorphism.
So it got to be bijective to be an Isomorphism, i wo... |
H: Linear independence between functions
I'm trying to solve some exercises on the linear independence between functions and in most of them we use the "trick" of deriving. I would like to know why, if there is a theorem, a proposition or a simple consideration about function that I missed in order to explein this met... |
H: Contour integral of non analytic function
Let $a,b\in \mathbf{C}$ with $|b|<1$. I want to calculate
$$\int_{|z|=1} \frac{|z-a|^2}{|z-b|^2}\frac{dz}{z}\, .$$
I'm not sure what tools I can use here since the function is not analytic so (I think) this rules out Cauchy's integral formula, etc.
AI: Note that when $|z| =... |
H: Epsilon-delta on a function with restricted range
On James Stewart's Calculus Early transcendental it says:
The definition of limit says that if any small interval $(L - \epsilon , L + \epsilon)$ is given around $L$,
then we can find an interval $(a - \delta, a + \delta)$ around a such that $f$ maps all the points ... |
H: Statistical sampling and random variables?
I'm studying statistical sampling and there is a point which is not very clear to me. Let us discuss the following example.
Suppose we would like to study the heights of 3.000 students in a given school. Let us do this taking 80 samples of size 25 each. For each sample the... |
H: Taking second partial derivative of spherically symmetrical wavefunction $\psi(r)$ with respect to $x$ only
I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, when discussing the spherical coordinates $x = r \sin(\theta) \sin(\phi)$, $y = r \sin(\theta)\sin(\phi)$, $z = r \cos(... |
H: Question regarding the correlation of limit points and convergent sequences
I've been told, that:
$(i)$ $M$ is a closed set
$(ii)$ For every convergent sequence $a_{n}$ in $M$, $\lim \limits_{n \to \infty } a_{n} \in M$
are equivalent. I understand that $M$ being a closed set implies that every limit point is in ... |
H: How to refer to the single element inside a unit set?
The other similr questions/answers refer to advanced set theory notation or defining exclusive functions for this.
[I have answered here my own question with a middle ground solution.]
If sets were ordered, I could just do $s_1$ and get the first, also only elem... |
H: Inequality from Geometric Series
My Calculus II textbook shows the following example:
I do not understand how they got -2 from solving the inequality $\left|\left(\frac{x}{2}\right)^2\right| < 1$. Doesn't it result in an imaginary part when you square -4 in the last step?
Therefore, we have
\begin{align*}
\frac{x^... |
H: What is the central algebra?
I' m studying over Brauer group.
But I have a problem in starting point. What is the central algebra?
Defination of center of $\mathbb{k}$-algebra S as following ( Noncommutative Algebra, Farb & Dennis, p:86,1991)
$ Z(S) = \{x\in S : x.s=s.x \, \forall s\in S\}$ that is, Z(S) is just t... |
H: True /false question about compact set in real analysis
This question was a true /false question in my real analysis quiz today and I am unable to provide a reason for it.
Question is : A closed and bounded subset of complete metric space is compact.
I think it is not compact but only because any open cover may not... |
H: A Simple Application of Mean Value Theorem?
If $f:\mathbb{R}\to \mathbb{R}$ is continuous, $a>0$ and $\int_{-a}^a f(x)dx=0$, prove that there exists some $\xi\in (0,a)$ such that
$$\int_{-\xi}^{\xi}f(x)dx=f(\xi)+f(-\xi)$$
A natural idea is to consider the function $g:[0,a]\to \mathbb{R}$, where
$$g(y)=\int_0^y \lef... |
H: Find the domain of convergence for the series as well as the sum $S(x)$.
The given series: $$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x}$$
Here is what I did:
$$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x} \le \frac{1}{n(n+1) \cot^n x} = \le... |
H: Does continuous extension exists under specific conditions
I was unable to solve this problem asked in my exam of Topology and need help.
True or False : Does A continuous function on $\mathbb{Q} $ Intersection [0, 1] can be extended to a continuous function on [0, 1] .
I couldn't think of what theorem or counte... |
H: Open and Closed subset of $\mathbb{R}^n$
Let $A,B\subseteq \mathbb{R}^n$, define $A+B=\{a+b : a\in A,b\in B\}$ Then which of the following is/are true?
$(1)$ If $A$ and $B$ are open, then $A+B$ is open.
$(2)$ If $A$ open and $B$ close, then $A+B$ closed.
$(3)$ If $A$ is closed and $B$ open, then $A+B$ is open.
$(4)... |
H: why is it called $f^{-1}(x)$?
Why do we name the inverse function $f^{-1}(x)$? Is it nonstandard to say $f^{0}(x)=x$, $f^{1}(x)=f(x)$, $f^{2}(x)=f(f(x))$, $f^{\infty{}}(x)=f(f(f(⋯f(x)⋯)))$?
Some (all) functions can have a $f^{1/2}(x)$?
Like if $f(x)=x+1$ then $f^{1/2}(x)=x+1/2$ so $f^{1/2}(f^{1/2}(x))=(x+1/2)+1/2=x... |
H: Online Calculator for Complex Calculus - path in. C z3(z 1− 1)2 dz; |z − 2| = 5tegrals
Does anyone know of an online calculator/tool that allows you to calculate integrals in the complex number set over a path?
I've searched in the standard websites (Symbolab, Wolfram, Integral Calculator) and none of them has this... |
H: Lower Bound for the square root of the sum of squares
I am looking for a good lower bound for the square root of the sum of squares:
Let's say we have some known parameters : $x_i > 0$ where i $\in [1,...,n]$
I am looking for a good lower bound of this term (a sum or something similar but not a one square root term... |
H: standard normal law is rotation invariant
Prove that the standard normal law is rotation invariant, i.e.,x≝ UX for any matrix unitary matrix U
i've no idea hoe to prove it, any suggestions?
AI: If an $n$-dimensional variable $X$ has PDF $(2\pi)^{-n/2}\exp(-\tfrac12x^Tx)$, $Y:=UX$ has PDF $(2\pi)^{-n/2}\exp(-\tfrac1... |
H: Tournament championship proof
Let's define a tournament as a contest among $n$ players where each player plays a game agains each other player and there are no draws. Now let me define a tournament champion.
A tournament champion is a player $c$ where, for each other player $p$ in the tournament, either
$c$ won hi... |
H: Prove asymptotic equivalence of $\text{li}(n)$ and $n/\ln(n)$
The prime number theorem, PNT, states that the prime counting function $\pi(n)$ is asymptotically equivalent to Gauss' first approximation:
$$\pi(n) \sim \frac{n}{\ln(n)}$$
We know this means that
$$\lim_{n \rightarrow \infty}\frac{\pi(n)}{n/\ln(n)} \rig... |
H: Prove that a linear operator $T$ on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of $T$.
(a) Prove that a linear operator $T$ on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of $T$.
(b) Let $T$ be an invertible linear operator. ... |
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