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H: What will be expected value of smallest element of chosen set
We are given a set $X = \{x_1,x_2,\ldots,x_n\}$ where $x_i = 2^i$. A sample $S$ (which is a subset of $X$) drawn by selecting each $x_i$ independently with probability $p_i = \frac{1}{2^i}$. What will be the expected value of the smallest number in sampl... |
H: What's the limit?
Suppose $a>0$. Let $x_1=\sqrt{a}$, and define $x_{n+1}=\sqrt{a+x_n}$ for $n\ge 1$. I've already used induction to show that $x_n<1+\sqrt{a}$ for all $n$ and that $\{x_n\}$ is an increasing sequence. I know that it is bounded and must converge by the monotone convergence theorem. I'm just not sure ... |
H: Proof about some recursion
Let $f(n)=\overset{n}{\underset{k=1}{\Sigma}}\lceil log_{2}k\rceil$. Prove that
$$f(n)=n-1+f(\lceil\frac{n}{2}\rceil)+f(\lfloor\frac{n}{2}\rfloor)$$
for all $n\geq1$.
Hint i've got for this: divide $\overset{n}{\underset{k=1}{\Sigma}}\lceil log_{2}k\rceil$ for sums for odd and even $k$'s.... |
H: Finding the change in radius that induces an $11\%$ drop in gravitational force between two bodies.
How come differentials only estimate the answer and don't give an exact answer that you might get when you calculate the real figure using other methods?
Example: Let $F = Gm_1m_2/r^2$
Let $r=8$. What is the change i... |
H: Find another recursive algorithm that is equal to a series
I have the following sequence:
$$
y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots
$$
Now I have the following recursive algorithm which is equal to the sequence:
$$
y_0 = \log{6} - \log{5}
$$
$$
y_n = \frac{1}{n} - 5y_{n-1}, n = 1,...
$$
I have to find an... |
H: Evaluating a sum with binomial coefficients: $\sum_{k=1}^n {n \choose k} \frac{1}{k^r} a^k b^{n-k}$
I have come across the following sum evoking the binomial theorem:
$$\sum_{k=1}^n {n \choose k} \frac{1}{k^r} a^k b^{n-k},$$
where $r > 0$ is a positive real constant and $a,b \in \mathbb{R}$ are arbitrary real numbe... |
H: Calculating the number of triangles
I am trying to calculate how many triangles that can be found in an equilateral triangle with $2n$ lines starting at the bottom angles and ending at the opposite side, such that equally many lines start/end of either side.
This is rather hard to explain, so I drew that first 5 te... |
H: Find the Laplace transform?
Find the Laplace transform of $f(t) = 1 + (1 - t)u_1(t) + (t-2)u_3(t)$.
Obviously each term of the function must be of $f(t - c)u_c(t)$ or be clearly transformable. Thus we have for our first term simply $\frac{1}{s}$. Our second term has been shifted by $f(t) = -t \to f(t - 1) = 1 - t$.... |
H: Negating "If no one is absent, then if the weather permits, we will study outside"
I am a beginner; please help solve this.
Write the negation of the statement:
"If no one is absent, then if the weather permits, we will study
outside."
AI: First, write the statement as a logical statement (with appropriate notatio... |
H: Linear Algebra problem, old Berkeley exam.
I came across this problem:
Let $G$ be the group of $2 \times 2$ matrices with determinant $1$ over the four-element field $\bf F$. Let $S$ be the set of lines through the origin in $\bf F^2$. Show that $G$ acts faithfully on $S$. (The action is faithful if the only eleme... |
H: Is this polynomial solvable? $\sum_{i=0}^{k} p_{i}x^{k-i}$
I'm given $p_{0},...,p_{k} \in \mathbb{R}$ and I only know, that $p_{0}=-1$. We create a polynomial using these values like this :
$\sum_{i=0}^{k} p_{i}x^{k-i}$
Can I tell for sure, that no roots exist? I got class notes that said so, yet I don't see why no... |
H: How to prove that for a sheaf functor F, F(empty set) = terminal object?
The empty set is covered by an empty family of sets. Therefore any two sections of F(empty set) are vacuously equal, so the F(empty set) can be at most a singleton. Now, why does this mean that it is in fact the terminal object, as Vakil claim... |
H: Equivalence relations, Cosets
Let G be a group and for elements a,b (elements of)G let a R b mean that there exists an element x(element of)G such that a=xbx^(-1). Show that R is an equivalence relation on G.
Not really sure how to go about this one.
AI: You need to check the three conditions for an equivalence rel... |
H: Question About Dual Vector Spaces and the Adjoint Map
I'm having trouble understanding some things about this problem:
Suppose that $U \subset V$ is a subspace. Let $I : U \rightarrow V$ be the inclusion map. The adjoint map $I^* : V^* \rightarrow U^*$ has a kernel: $$U^{\bot} := \mathrm{ ker}\; I^* \subset V^*$$ ... |
H: Derivatives of $\frac{\csc x}{e^{-x}}$ and $\ln\left(\frac{3x^2}{\sqrt{3+x^2}}\right)$
I have tried to mainly ask thoughtful conceptual questions here, but now I am reduced to asking for help on a specific problem that I have been wrestling with for over an hour.
Disclaimer: I am not a lazy student trying to ge... |
H: For all integers a, b, c, if a | b and b | c then a | c.
Is this T or F? and most importantly, why?
I'll be using any answers for a basis or completely my other questions, since my understanding is still a little poor.
AI: HINT: This is just a matter of applying the definitions, which is always the first thing tha... |
H: Proving that the Union of Two Compact Sets is Compact
Prove if $S_1,S_2$ are compact, then their union $S_1\cup S_2$ is compact as well.
The attempt at a proof:
Since $S_1$ and $S_2$ are compact, every open cover contains a finite subcover. Let the open cover of $S_1$ and $S_2$ be $\mathscr{F}_1$ and $\mathscr{F}... |
H: Counting the Elements of a Finite Group
Let $G = \mathbb Z_6 \times \mathbb Z_4$, and find $[G:H]$ for:
a) $H = \{0\} \times \mathbb Z_4$
b) $H = \langle 2\rangle \times \langle 2\rangle$
AI: Determine the order of $G: \,|G|$, and the orders of $H$ in each case; then you can easily calculate the index of $H$ in $G... |
H: Percentage of Amounts
I'm studying and I'm not that sure how to answer this question.
Is $97.1%$ $=$ $650,000,000$?
I was going to find $2.9%$ of $650,000,000$ however this would be wrong as I would finding our the annum of the $2002$ population, if you know what I mean. I know this looks quite easy but thanks.
A... |
H: Sample Variance question
The random variable
$$ S^2=\sum_{i=1}^n \frac{(X_i- \overline X)^2}{(n-1)} $$
is called the sample variance.
A) Show that $ (n-1)S^2=\left[\sum_{i-1}^n ((X_i-\mu)^2 \right]-n(\overline X-\mu)^2 $
(Hint: Start with $(n-1)S^2=\sum_{i-1}^n ((X_i-\mu)+(\mu-\overline X))^2 $)
B) Use result from ... |
H: Vector Calculus: to get the parametric equations of the tangent line, why can't you just use the derivative of the position vector?
As you can see in the example, the author generates the parametric equations of the tangent line through the use of r(3) and r'(3).
Why couldn't the author just use the derivative of ... |
H: Example of two norms and ONE linear operator that is bounded and unbounded in a norm.
I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many examples.
AI: Assume $\varphi... |
H: How to sample points on a triangle surface in 3D?
To take random uniform points inside a triangle Triangle Point Picking method is used. But this is for 2D points, how can I take random points from a triangle that is defined by 3 arbitrary 3D points?
In other words, let's say I have 3 points in 3D space, and these ... |
H: What is this type of fixed point called?
After numerically solving the following differential equations:
$$v'(t)=\frac{-\frac{1}{3} v(t)^3+v(t)-\omega (t)}{\tau } $$ $$ \omega '(t)= \frac{a+v(t)}{\tau}$$
at $a=1,\tau=0.2$ and taking $t$ from 0 to 10, I get this phase space plot:
The fixed point $(-1,-\frac23)$ is... |
H: Volume of ellipsoid
How do I calculate the volume of the intersection of ellipsoid $x^2/36+y^2/49+z^2/49\leq 1$ and subspace $x/6+y/7+z/7\leq1$ ?
AI: If the 36 were replaced with 49, and the 6 were replaced with 7, you have a plane cutting a sphere, and the volume of the resulting pieces of the sphere can be found... |
H: Solving $x^x=\frac{1}{\sqrt 2}$
The equation $$x^x=\frac{1}{\sqrt 2},x\in \mathbb R$$
has two obvious solutions $0.5$ and $0.25$
One can easily prove they are the only ones using differential calculus.
Is there any natural algebraic manipulation that would lead to finding these solutions ?
AI: One natural thing to ... |
H: Free product of the trivial group with another group
I'm new to the idea of a free product..
Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
AI: Yes, $1*G\cong G$. A proof would depend on which definition or constructio... |
H: Does $\cos(x+y)=\cos x + \cos y$?
Find the value using a calculator: $\cos 75°$
At first I thought all I need is to separate the simpler known values like this:
$\cos 75^\circ = \cos 30°+\cos45° = {\sqrt3}/{2} + {\sqrt2}/{2} $
$= {(\sqrt3+\sqrt2)}/{2} $ This is my answer which translates to= $1.5731$ by calculato... |
H: Uniform convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ on squarewith vertices $\pm6\pm6i$?
Can someone explain me how I can check whether the convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ is uniform on the (boundary of) square $A$ with vertices $\pm6\pm6i$?
AI: When we... |
H: A sequence $( a_n )$ is defined by $(a_1)=1$, and $a_{n+1}$=$\sqrt {a_n + 1}$, for $n \in \mathbb{N}$
How to prove that $( a_n )$ is an increasing sequence by induction?
$n\ge 1$
AI: *(for $n=1$) ${{a}_{2}}=\sqrt{2}>1={{a}_{1}}$ so $a_2>a_1$.
*now assume ${{a}_{n+1}}>{{a}_{n}}.$ we know $\sqrt{{{a}_{n}}+1}={{a}_{n+... |
H: proving the lim inf and the lim sup for any listing of rational in the interval (a,b)
Let $r_n$ be any listing of the rational numbers in the interval $(a,b)$. Establish with proof $\liminf_{n\to \infty} r_n$ and $\limsup_{n\to\infty} r_n$
Please forgive, I have no idea how to write the actual mathematical language... |
H: Intuitive understanding of work.
I am working on a problem that has to do with work.
I am also assuming that the acceleration due to gravity is $10m/s^2$.
A 15 kg crate is moved along a horizontal floor by a warehouse worker who's pulling on it with a rope that makes a 30 degree angle with the horizontal. The tens... |
H: Complex contour integral with residue theory
I need to calculate the following contour integral using residue theory.
$z \in \mathbb{C}$
$f(z)=\exp(-1/z) \sin(1/z)$
$\oint_C f(z) dz$
$C: \left | z \right |=1$
The difficult points I detected include only z=0.
But i'm a bit stuck at calculating the residue of f at z... |
H: Geometric meaning of minors
This is a bit silly question I found on another discussion forum. I know that determinants can be used to compute volumes of parallelepiped. I also know that determinants can be computed by linear combination of its minors. Is there any geometric meaning of minors or some proof/explanati... |
H: Exponential distribution: Finding the parameter
Please help me solve the following problem
Time of production of one electronic component is given with
exponential distribution with parameter λ. If the process lasts less than 3 hours, the component is working, otherwise it is defective. Events that component is wo... |
H: Subset of sets exrecise in discrete structures course
$\{\{b\}\}$ is a subset of $X=\{a,b,\varphi,\{a,b\},\{a,\{b\}\},\{c,\varphi\}\}$: true or false? Can someone explain why is it true/false?
AI: Is every element of $\{\{b\}\}$ an element of $X$?
That is: Is $\{b\}$ an element of $X$?
Well, we do have $b\in X$ and... |
H: What does "under divisibility" mean?
I'm currently working on this problem, and the term "under divisibility" is kind of ambiguous to me. I googled the term "under divisibility" and I couldn't find any definitions.
The solution also doesn't make sense to me either. It states that all numbers divide 0, but isn't it... |
H: If a function is continuous and converges finitely, then it is bounded
Suppose that $f$ is a continuous function on $[a,+\infty)$ and that there exists a finite limit $\lim \limits_{x \to +\infty} f(x)$, then $f$ is bounded.
I know that from the assumption I can conclude that $f$ is uniformly continuous on that in... |
H: Second Borel-Cantelli without independence assumption
The second Borel-Cantelli lemma says
Let $(X,F,\mu)$ be a probability space. Let $A_1,A_2,\ldots\in F$ be independent, and let $A=\bigcap_{i=1}^\infty \bigcup_{j=i}^\infty A_j$ (i.e. the probability that infinitely many $A_i$'s occur). If $\sum_{i=1}^{\infty}\... |
H: Bound of Standard Normal Integral
Consider the Standard Normal Integral given by:
$$ I=\int_{-\infty}^{\infty} \frac{1} { \sqrt{2\pi} } e^{ \left( -z^2 /2 \right)} dz $$
In order to prove that it exists we note that the integrand is a positive continuous function which is bounded by an integrable function; that is:... |
H: Linear Algebra Projections and orthogonality
I am currently studying linear algebra and am getting very confused on orthogonality and projections. I am reviewing questions to better understand it but can not get a certain one. Here is the question:
$$
\vec{b} = \begin{bmatrix} 1 \\ 2 \\ 2\end{bmatrix} \quad\text{a... |
H: Disjoint union with limsup
For any sets $A_n,n\in\mathbb{N}$ consider
$$
A^+:=\limsup_{n\to\infty}A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k,~~~~~E_m:=\bigcup_{n\geq m}A_n.
$$
Show that the sets $E_m, m\geq 1$ can be written as a disjoint union
$$
E_m=A^+\uplus\biguplus_{n\geq m}(E_n\setmin... |
H: Radius ratio for four packed circles
Suppose we are given four circles $A,B,C,D$ in the Euclidean plane having radii $r_A,r_B,r_C,r_D$ such that $r_A=r_C,r_B=r_D$ and circles $A,C$ are tangent to each other and to $B,D$ but $B,D$ are only tangent to $A,C$. Suppose further that a given bounding square $E$ in the pl... |
H: Line Integration over a broken line
I have taught myself Line Integration and when doing some practice questions I came across the following:
Evalute the line integral $$\int_{C}(y^2\:dx+xy\:dy+zx\:dz)$$ Where $C$ is the broken line from $A(0,0,0)$ to $B(1,1,1)$ connecting $(0,0,0)$, $(0,0,1)$, $(0,1,1)$ and $(1,1... |
H: Dense subset in Hilbert space: a basic question
I am trying to solve the following problem:
Let $\bar{H}$ be a Hilbert space, $H$ - its dense linear subspace, $z_0\in \bar{H}-H$. Consider the following subspace $M\subset H$ defined as $\{x\in H:<x,z_0>=0\}$. Is is true that $Cl(M)=Span(z_0)^{\perp}$ ($Cl(M)$ is the... |
H: Expectation/ independence of random variables
Let $X,Y$ be two correlated variables and $Z\sim N(0,1)$ independent of $X,Y$. Consider the expectation:
$$E[f(X,Y)Z].$$
If $f(X,Y)$ and $Z$ are independent then clearly $E[f(X,Y)Z]=E[f(X,Y)]E[Z]=0$ but I guess this is not in general true. Nevertheless, I can argue as f... |
H: There are 6 cards with letters a, c, e, i, m, n in a box. Somebody picks cards in a random order.What is the probability of getting the word “cinema”?
This is a probability question.
There are 6 cards with letters a, c, e, i, m, n in a box. Somebody picks cards in a
random order. What is the probability of getting ... |
H: Unit length vectors that sum to zero
Let's say we have a collection of $n$ vectors in $\mathbb{R}^2$ where $n$ is odd. Suppose each vector has unit length and that the sum of the vectors is zero. Is it necessarily true that the vectors will correspond to the vertices of a regular $n$-gon?
AI: Hint: Take an equila... |
H: Diffusion-advection equation with time-variable coefficients
Is the fundamental solution (Green's function) of the 1D advection-diffusion equation
$$\frac{\partial{\phi}}{\partial{t}} = D(t)\frac{\partial{^{2}\phi}}{\partial{x^{2}}} - c(t)\frac{\partial{\phi}}{\partial{x}}-k(t)\phi$$
(where the diffusivity $D(t)$, ... |
H: How do i calculate the maxima in a Polynomial between two roots.
I'm writing a program that plots compaction curves for soil density tests.
I have very little math background and I'm using Extreme Optimization's Math Library to curve fit my data(linear least squares) and plotting it as either a 2nd or 3rd degree po... |
H: How to tell an ideal of integer polynomial ring is principal?
I understand that uni-variate polynomial rings with coefficients in a field only have principal ideals. For example, $\mathbb{C}[x]$. But how can I tell if an ideal of integer polynomial ring is principal, please? For example, a textbook claims that "the... |
H: $NP$ problems not known to be in $P$ and not known to be $NP$-complete
I've read that solving Pell's equation is neither known to be in $P$ nor known to be $NP$-complete. What are other natural and important examples of such problems?
AI: They are called NP-intermediate problems (under the assumption of $P\neq NP$,... |
H: Prove that $\{f_n\cdot g\}\rightarrow f\cdot g$ in measure.
From Royden's Analysis book (4th Edition) Chapter 5 Section 2 Problem 7:
Let $E$ have finite measure, $\{f_n\}\rightarrow f$ in measure on $E$ and $g$ be a measurable function on $E$ that is finite a.e. on $E$. Prove that $\{f_n \cdot g\} \rightarrow f\cd... |
H: is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?
is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?
for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times [x_2-\epsilon,x_2+\epsilon]\times...$
is $V$ a compact neighborhood of $x$?... |
H: Abstract Linear Algebra for Statistics
Is abstract linear algebra required for a deep understanding of statistics? I'm a computer science major deciding between a linear algebra for applications class versus a very theoretical proof based linear algebra class (where they don't cover applied linear algebra).
I'm ve... |
H: Set in the Complex Plane
How can I describe the set:
$$ \left\vert z - {\rm i}\,\right\vert = 3\left\vert z\right\vert $$
It does appear quite unfamiliar.
Attempt: $$ \left\vert\frac{z-i}{z}\right\vert = 3 $$ so,
$$ \left\vert 1 - {\rm i}\,\frac{1}{z}\right\vert = 3 $$
But this seems to be even more difficult to v... |
H: Solving for x - Trig
Someone mind helping on this? I think have done the question correct but the system isnt accepting my answer.
AI: Using $\sin{2x} = 2 \sin{x} \cos{x}$, we see that your equation is equivalent to
$$2 \sin{x} \cos{x} = \cos{x}$$
or alternatively
$$\cos{x} \left(2 \sin{x} - 1\right) = 0$$
Now $\... |
H: if the improper integral $\int^\infty_a f(x)\,dx$ converges, then $\lim_{x→∞}f(x)=0$
I need to prove that:
$$\lim_{x→∞}f(x)=0$$
if
$$\displaystyle∫^∞_af(x)\,dx$$ converges.
I need a proof or an specific, and if possible simple, counterexample. Would really appreciate your help! Thank you in advance.
AI: This is fa... |
H: Prove: $\liminf_{n \to \infty} s_n \le \limsup_{n \to \infty} s_n$
I am looking over examples and the definitions for this section but I am still not familiar with all the tricks. I appreciate any help with proving this (from hints to maybe a solution. It is only a review so I just need something to look at and mak... |
H: How to tell an ideal of integer polynomial ring is principal?
I understand that uni-variate polynomial rings with coefficients in a field only have principal ideals. For example, $\mathbb{C}[x]$. But how can I tell if an ideal of integer polynomial ring is principal, please? For example, a textbook claims that "the... |
H: Prove that $|a| < b \iff -b
I know I have to prove both sides here so:
$\implies$:
Suppose $|a| < b$. Since $|a|$ is a positive number, we know $b$ is a positive number greater than $|a|$:
If $a$ is positive, $a < b$
If $a$ is negative, $a < b$
But for $|a| < b$, it has to be that $a > -b$.
$a < b$ and $a > -b$, ... |
H: Why use Euclidean distance in linear model?
The method of least squares is the most basic method in statistical linear models. For the simplest linear model$$Y_i=\beta_0+X_i\beta_1+\epsilon_i$$we are looking for $\beta_0$ and $\beta_1$ that minimizes the Euclidean distance $\sum\limits_{i=1}^n|Y_i-\beta_0-X_i\beta_... |
H: Domain of this function
The function is: $$y=\frac{x+6}{15-\sqrt{{x^2}-64}}$$
The domain is: $(-\infty,-17) \cup (-17,-8] \cup [8,17) \cup (17,\infty) $
The value inside the square root can't be negative, so I set up:
$$x^2-64 \geq0$$
$$(x-8)(x+8) \geq0$$
Then I found the roots, $x=8$ and $x=-8$, which are the o... |
H: Finding the limit $ \lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)} $
I have been working on this question for a while now:
$$
\lim_{x\to a^+} \frac{\cos(x)\ln(x-a)}{\ln(e^x-e^a)}
$$
I know the answer is $\cos a$, that's what the solutions say. But I don't understand how it became to that. I've tried solving i... |
H: Show the closure of $\mathbb{Q}$ in $\mathbb{R}$ is $\mathbb{R}$.
I know this question has been asked on here before, but the answers did not contain a formal proof which I am trying to do.
To show $\operatorname{cl}\mathbb{Q}$ = $\mathbb{R}$, I want to do it by showing two inclusions. Clearly we know $\operatorn... |
H: Evaluate the limit $\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right)$
Evaluate
$$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$
This sequence looks extremely horrible and it makes me crazy. How can we evaluate this?
AI: Use Stirling's approximation:... |
H: How many 7-character passwords can be generated with...Stipulations?
Characters can only be (a-z), (A-Z), and (0-9) w/stipulation that first character must be a lower case letter, the last character must be an upper case letter, and of the 5-characters in the middle there must be at-least ones digit (0-9).
How many... |
H: Prove that $Tu(x)$ is a contraction. $Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$
I want to show that $Tu(x)$ is a contraction where
$$Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$$ and
$$g(x,y) = \begin{cases} x(1-y) & 0\leq x\leq y\leq 1, \\
y(1-x) & 0\leq y \leq x \leq 1. \end{cases}$$
I have
$$\begin{align*... |
H: Proof that the gamma function is an extension of the factorial function
I've already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And I´m sorry for my language, I am Spanish, so th... |
H: Continuous functions from one topological space to another topological space.
Question: Let $X=\lbrace 1,2,3,4,5 \rbrace$ with topology $\lbrace \emptyset,X,\lbrace 1 \rbrace,\lbrace 3,4 \rbrace,\lbrace 1,3,4 \rbrace\rbrace,$ and let $Y=\lbrace A,B\rbrace$ with topology $\lbrace \emptyset,Y,\lbrace A\rbrace\rbrace.... |
H: Linear Algebra - Determine if the following 2 subspaces add up to $\mathbb R^3$
$W_1=\{ (0,-b,b)|\ b\in \mathbb R$}
$W_2=\{(a_1, a_1+a_2,a_2)\;|\ a_1,a_2\in \mathbb R\}$
Is $\mathbb R^3=W_1+W_2$?
I know that if it is true, then $\mathbb R^3\subset W_1+W_2$. How do I determine whether the 2 subspaces span $\mathbb R... |
H: Is there an easy way to factor polynomials with ugly numbers?
Say for example I have the polynomial $x^2-8x-153$. It is not easy to immediately see the factors of this are $9$ and $17$. Is there a method to find these factors easily when the terms in the polynomials are this big? The reason I ask this is because I ... |
H: Finding coordinates of closest approach
Given two lines $l_1=\mathbf E_1+k\mathbf E'_1$ and $l_2=\mathbf E_2+\mu\mathbf E'_2$ in 3D, there exists a shortest distance between the two lines. How does one find the coordinates of the points $P$ on $l_1$ and $Q$ on $l_2$, such that $P$ and $Q$ are the points where the d... |
H: Sending each basis element $a,b,c,d$ to $y-x$.
Hatcher P99 last paragraph:
Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail.
So I am confused: $x,y$ are just vertices, how can they subtract?
http://www.m... |
H: Find the remainder when $2^{47}$ is divided by $47$
So I am trying to work out how to find remainders in several different way.
I have a few very similar question,
1.) Find the remainder when $2^{47}$ is divided by 47
So i have a solution that says
$$2^{24} \equiv 2$$
$$2^{48} \equiv 4$$
$$2^{47} \equiv 2$$ Since ... |
H: If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers
The title statement can be proven using the contrapositive, note that $x$ odd or $y$ odd means that at least one of $x\cdot y,x+y$ is odd. Is there a way to prove the statement directly?
To generalize on this st... |
H: Continuity of an identity linear transformation between two spaces.
$V$ is a vector space of all continuous complex valued function on $ J = [a,b]$. Let $X_1 = (V,\|\|_{\infty})$ where $\|x\|_{\infty} = \max_{t \in J}\{x(t)\}$ and $X_2 = (V,\|\|_2)$ where $\|x\|_2 = <x,x>^{\frac{1}{2}}$. The the identity mapping $x... |
H: How many ways are there to distribute $18$ different toys among $4$ children,(a) without restriction?(b) if $2$ children get $7$ toys each and $2$…
How many ways are there to distribute $18$ different toys among $4$ children?
a) without restrictions
b) if $2$ children get $7$ toys each and $2$ children get $2$ to... |
H: Counter example of Zorn's lemma when we only take countable chains
I was learning Zorn's lemma yesterday and I couldn't find any example, where Zorn's lemma fails when we only require that every countable chain in P has a maximal element in P. Does anyone know an example?
AI: Consider the collection of countable su... |
H: Greatest Lower Bound in $\mathbb{R}$ as a corollary of the LUB?
I can assume as fact that $\mathbb{R}$ is an ordered field in which every non-empty subset that is bounded above has a least upper bound.
My question is whether I can also assume as fact that every non-empty subset that is bounded below also has a grea... |
H: Topology on cartesian product and product topology.
Let X and Y be sets. Does it have every topology on cartesian product X$\times$Y must be product topology ?
AI: It goes a bit further: one can show that for virtually any pair $X,Y$ of sets there are topologies on $X \times Y$ which cannot be expressed as the pro... |
H: construct x =ab by using compass alone, if a and b are given segments.
I found the problem in the book "What is mathematics?".
The following is a description of Mohr's constructions.(Macheroni problem)
9) Find $x = ab$, if $a$ and $b$ are given segments.
I found the question didn't give a segment denotes 1.I... |
H: Generating function of recurrence sequence
I have the following recurrence sequence
$$
a_{1} = 0\,,\quad
a_{2} = 1\,,
\qquad\qquad
a_{n}
=
{2 + 2\left(n - 2\right)\, a_{n - 2}
+
\left(n - 2\right)\left(n - 1\right)\, a_{n - 1}
\over
n\left(n - 1\right)}
$$
It starts form
$\displaystyle{%
\left\lbrace%
0,\ 1,\ ... |
H: Definition of semi-ring homomorphism
I need the definition of semi-ring homomorphism. Thanks in advance!
AI: The definition should depend on your purposes, but I think that this should be the best definition:
$$f(a+b) = f(a) + f(b)$$
$$f(ab) = f(a)f(b)$$
$$f(0) = 0$$
$$f(1) = 1$$
Unlike with rings, you cannot deduc... |
H: Can't solve this problem $lim_{x \to 0}\frac{3^{5x}-2^{7x}}{\arcsin\left(2x\right) - x}$(Without using L'Hospital's rule)
I don't see the way to solve this limit.
$$\lim_{x \to 0}\frac{3^{5x}-2^{7x}}{\arcsin\left(2x\right) - x} $$
My attempt is
1) Divide the numerator by $3^{5x}$
$$\lim_{x \to 0}\frac{3^{5x}-2^{... |
H: Proof of Continuous compounding formula
Following is the formula to calculate continuous compounding
A = P e^(RT)
Continuous Compound Interest Formula
where, P = principal amount (initial investment)
r = annual interest rate (as a decimal)
t = number of years
A = amount after time t
The above is specific to c... |
H: Can this expression be reduced to a difference quotient?
Setting up an equation I've come into this factor:
$\displaystyle \lim_{h\rightarrow0}\frac{1-\frac{f(x+h)+f(x-h)}{2f(x)}}{h}; \quad f\in \mathcal{C}^\infty$
To me this looks more or less like a derivative, but I've not been able to reduce it to a common diff... |
H: Topology Questions
So I am just wanting to make sure I had these right. For the most part I think I have a solid understanding of both problems 1 and 2. Both the last 2 problems, I think I understand them, but I am not 100% sure.
Problem 1: Let $A= [0,8)$ be a subspace of $(\mathbb{R}, U)$. Which of the followin... |
H: Show $\mathcal D_D$ is Dynkin system
Let $\mathcal D\subset\mathcal P(\Omega)$ be a Dynkin system and $D\in\mathcal D$. Then $\mathcal D_D=\{A\subset\Omega|A\cap D\in\mathcal D\}$ is a Dynkin system.
It's clear that $\Omega\in\mathcal D_D$. But I'm not sure how to show that
i) for $A\in\mathcal D_D$ we also have $... |
H: 2 regular graphs and permutations
I have found this question on MSE before but I didn't find the answer satisfactory and it is so old I doubt anyone is still following it.
Let $f_{n}$ be the number of permutations on $[n]$ with no fixed points or two cycles. Let $g_{n}$ be the number of simple, labeled two regular... |
H: I want to prove that $m||v||_1\le ||v||$
Let $(V,||\cdot ||)$ a finite-dimensional real normed space, and $\{v_1,...,v_n\}$ a basis. We define $||v||_1=\sqrt{x_1^2+...+x_n^2}$, where $x_1,...,x_n$ are the coordinates of $v$.
I already proved that there exists $w\in S:=\{v\in V:||v||_1=1\}$ such that $$||w||\le ||v|... |
H: Ramification of a Galois extension
I understand that an extension of number field $L/K$ is unramified if every non-zero prime ideal of $\mathcal{O}_K$ is unramified in $L$ (where a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ is ramified if it has $e_i > 1$ for some $i$ when you write it as the decomposition of pr... |
H: Integrals of trignometric functions
Question is to Prove that :
$$\int_0 ^{2\pi} \frac{d\theta}{a+b\sin \theta}=\frac{2\pi}{\sqrt{a^2-b^2}} \text{for}\ a>b>0$$
using residue theory.
What i have done so far is :
I transformed functio of $\theta$ as function of complex entity $z$ with $z=e^{i\theta}$.
Then, we have ... |
H: How can $|x|= -x$, when $x<0$?
Why is the following true?
$$
|x|=
\begin{cases}
x,&x\ge 0\\
-x,&x<0
\end{cases}
$$
Isn't the modulus of a number always positive? According to the above formula $|-4|=-4$ because $4<0$, which is incorrect.
Please explain this to me. Thank you.
AI: No, the formula says that $|-4|=-(-4... |
H: How to show that all roots of $(11+v)q^3-18q^2+9q-2$ have their absolute value less than 1.
The equation is $(11+v)q^3-18q^2+9q-2=0$, where $v>0$
I need to show that either absolute value of all the roots is not greater than one or there exists a root $q: |q|>1$.
Using Weierstrass theorem I showed that there is a r... |
H: Isolated Points [Confusion Regarding Size of Neighborhood]
I'm learning (general?) topology and am having trouble understanding the definition for Isolated Points. The definition from Wikipedia says:
"A point $x$ of a topological space X is called an isolated point of a subset $S$ of $X$ if $x$ belongs to $S$ and t... |
H: Did I solve this limit problem correctly? $\lim_{x \to 3}\left(\frac{6-x}{3}\right)^{\tan \frac{\pi x}{6}}$
Need to solve this limit
$$\lim_{x \to 3}\left(\frac{6-x}{3}\right)^{\tan \frac{\pi x}{6}}$$
When I put $x$ in this expression I have indeterminate form $1^{\infty }.$
So I choose the way which was describ... |
H: Disjoints dense sets
How can we define three disjoints dense sets? Then I have to define four and five disjoints dense sets and finally - infinity disjoints dense sets. Could you give me some hints? Thanks in advance!
AI: HINT: Consider sets of the form $x+\Bbb Q=\{x+q:q\in\Bbb Q\}$ for some $x\in\Bbb R$. |
H: What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
Evidently, $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ has order $4$, but I think it's infinite.
The four cosets are listed as $(0,0) + \langle (2,2) \rangle$, $(0,1) + \langle (2,2) \rangle$, $(1,0)+ \lan... |
H: Is every Boolean algebra a separative partial order?
A partially ordered set $\langle P,\leq\rangle$ is separative iff it satisfies the following condition:
\[
\neg x\leq y\Rightarrow\exists z(z\leq x\wedge z\bot y)
\]
where:
\[
x\bot y\iff\neg\exists z(z\leq x\wedge z\leq y).
\]
If $\langle B,\leq\rangle$ is a com... |
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