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H: Finding a unique representation as a linear combination
ok, another problem suggested by my prof.
the vectors $u_1 = (1,1,1,1)$, $u_2 = (0,1,1,1)$, $u_3 = (0,0,1,1)$, $u_4 = (0,0,0,1)$, are a basis for $F^4$.
Find a unique representation of an arbitrary vector $(a_1, a_2, a_3, a_4)$ as a linear combination of $u_... |
H: $A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?
This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze.
Here is the question.
Let $V$ be the space of $n\times n$ matrices over a field $F$. Let $A$ be a fixed matrix... |
H: Why is $\frac{1}{\frac{1}{X}}=X$?
Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$
And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't reciprocal by definition the invert of the fraction?
AI: Maybe this will help you ... |
H: the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$
Let $\mathbb{Z}_3[i] =${$a+bi | a, b \in \mathbb{Z}_3$} . Show that the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$
how can I able to do this?can someone help?
AI: Consider the ring homom... |
H: Square roots and powers
This is a rather silly question.
In what order does one evaluate a combination of powers and fractional powers?
I have the question phrased:
$\sqrt{ 1/4^2 }$ OR ${ 1/4 }$? Which is greater?
I answered that it cannot be determined, because the ${ 1/4^2 }$ could be evaluated first and then t... |
H: What's the best way to calculate all of the points for a curve given only a few points?
I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.)
Given two end points and any number of control points (includin... |
H: Characterization of open subset of $\mathbb{R}$
Let $X$ be an open nonempty subset or $\mathbb{R}$. Prove that there exists a unique countable set of open intervals $\{(a_n,b_n)\}_{n=1}^{\infty}$ such that
(a) $\cup_{n=1}^{\infty}(a_n,b_n)=X$
(b) $(a_m,b_m)\cap(a_n,b_n)=\emptyset$ if $n\neq m$ ($a_n=-\infty$ and $... |
H: what % of remaining of remaining coats are full length
A garment supplier stores $800$ coats in a warehouse of which $15$% are full length coats.
If $500$ of shorter length coats are removed from warehouse what % of remaining of remaining coats are full length ?
AI: Step 1 : $15% $ of $800$ is 120 . Now out of $80... |
H: Upper bound to a series with ceiling
If we know that:
$$a_{i+1} \leq {a_i \over 2}$$
Then we can calculate an upper bound for every n:
$$a_{n} \leq {a_0 \over 2^n}$$
But what if we keep the elements integer, by taking the ceiling:
$$a_{i+1} \leq \lceil {a_i \over 2} \rceil$$
As long as $a_i > 1$, the series is decr... |
H: What is ratio of total number of people who read B newspaper
In a survey conducted with $95$ people on readership of two newspapers $A$ and $B$ , it was noted that $30$ people read both ,$20$ people read only $A$, $5$ Read only $B$ and balance $40$ read neither. What is ratio of people who read A to the total numbe... |
H: Differentiability of $f$ if $f^{-1}$ is differentiable
Suppose that $f$ is a one-to-one function and that $f^{−1}$ has a derivative which is nowhere $0$. Prove that $f$ is differentiable.
AI: Since $f$ is one-to-one, $f^{−1} $ is a function whose inverse is $f$.
Since $f^{−1} $ is differentiable, $f^{−1} $ is conti... |
H: Finding consecutive odd number
The negative of the sum of 2 consecutive odd numbers is less than -45 , which of the following may be one of the numbers?
$A)21 $
$B) 23 $
$C) 26 $
$D) 22 $
$ C) 24$
What will be logic to solve this problem.
AI: Let the consecutive odd numbers be $2n-1, 2n+1$ where $n$ is any intege... |
H: Permutations with exactly $k$ inversions
Let $I_{n,k}$ denotes the number of permutations of $\left\{1,..,n\right\}$ that have exactly $k$ inversions. Prove that:
$$\sum_k I_{n,k}x^k = \frac{\prod_{i=1}^n (1-x^i)}{(1-x)^n}$$
The only one fact I came up with is recursive formula $I_{n,k}=\sum_{i=0}^{}I_{n-1,i}$,... |
H: Rationale behind MLE of $f_{\theta}(x) = \frac{1}{\theta} I_{\{1, \dots,\theta\}}(x)$
Our probability density for $\theta \in \{1,\dots,\theta_0\}$ is $$f_{\theta}(x) = \frac{1}{\theta} I_{\{1, \dots,\theta\}}(x)$$
Let $X_{(n)}$ be the largest order statistic. Acooding to the solution, the likelihood function is $$... |
H: Finding gallons of milk for $2$%
How many gallons of milk that is $2$% butterfat must be mixed with milk that is $3.5$% butterfat to get $10$ gallons that are $3$% butterfat?
AI: Set up the following system
$$\left\{ {\begin{array}{*{20}{c}}
{.02x + .035y = .03 \cdot 10} \\
{x + y = 10}
\end{array}} \right.$... |
H: Comparing $\gamma^e$ and $e^\gamma$
How can I calculate without calculator or something like this
the values of $\gamma^e$ and $e^\gamma$
in order to compare them?
($\gamma$ the Euler-Mascheroni constant)
Note: the shape of this question lend from the beautiful question of Mirzodaler >>> here.
AI: Since $$γ^e=e^{e... |
H: Calculate the Laplace transform.
Can anybody help me with the answer of this question?
Find the inverse Laplace transform of $$f(t)=10$$
AI: The LT of a constant $a$ is
$$\hat{f}(s) = a \int_0^{\infty} dt \, e^{-s t} = \frac{a}{s}$$ |
H: Find $x$ in terms of $n$
A survey of $n$ people found that $60$ % preferred brand $A$. An additional $x$ people were surveyed who all preferred brand $A$. $70$% of all people surveyed preferred brand $A$ . Find $x$ in terms of $n$
AI: HINT:
So, initially $60$% of $n$ i.e., $n\cdot \frac{60}{100}=\frac{3n}5$ people ... |
H: How many coordinates are unreachable?
I wanted to know, if a man was to go from $(0,0)$ to $(46,46)$ moving only straight and up with the following constraints:-
If he walks right, he will walk atleast $4$ consecutive coordinates.
If he moves up, he will walk atleast $12 $ consecutive coordinates.
How many coord... |
H: Why does a first axiom space have to be $T_{1}$ in order for limit points to have a sequence converging to them?
My textbook General Topology by Pervin) says "If $x$ is a point and $E$ a subset of a $T_{1}$-space $X$ satisfying the first axiom of countability, then $x$ is a limit point of $E$ iff there exists a seq... |
H: Things that can happen to a differential equation
We have a list of things that can happen to a differential equation $y'(t)=f(t,y(t)), y(t_0) \in \mathbb{R}^n$ and $ f: G \rightarrow \mathbb{R}^n$ continuous.
That is given by
(i) $ b = \infty$
(ii) $\lim \sup \limits_{t \rightarrow b} ||y _{\max(t)}||=\infty$
(iii... |
H: Proof regarding factorials.
Suppose $a$ and $k$ are positive integers, then how would you prove(not intuitively) that:
$a!k! \leq (ak)!$
Although it is apparent that the inequality is correct, but how can I show this algebraically?
AI: Assume $a, k\geq1$.
\begin{align}
\frac {(ak)!} {k!} &= (ak)(ak-1)\cdots(k+1) \... |
H: How to convert a permutation group into linear transformation matrix?
is there any example about apply isomorphism to permutation group
and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix
AI: Hint:Prove that Permutatation matrices form a subgroup of ... |
H: A question on geometry?
I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ .
Help.
Thanks.
AI: Hints:
I'll do it with analytic geometry: place the vertices at
$$D=(0,0)\;,\;C=(x,0)\;,\;A=(a,b)\;,\;B=(\alpha,\beta)$$
Then we get that
$$AB^2+CD^2=(a-\alpha)^2+(b-\beta)^2+... |
H: property of the exterior derivative $d \circ d=$ for a $\mathcal C^\infty$ function
One of the properties of the exterior derivative is that $d\circ d=0$. We're trying to prove this for the case $f\in\mathcal C^\infty (U)$ on an open set $U\subset \mathbb R^n $.
The prove starts with the uniqueness of the exterior ... |
H: Does $f(z)$ exist such that $f'$ and $f''$ exist in $\mathbb{C}$ but $f'''$ does not?
Is it possible to find $f(z)$ defined on $\mathbb{C}$ such that $f'$ and $f''$ exist everywhere on $\mathbb{C}$ but $f'''$ does not?
I'm guessing no such $f(z)$ exists, but I don't know how to prove this. What I've done so far:
L... |
H: Ordered set partitions
Let $a_n$ be a number of ordered partitions of the set $\left\{1,\ldots,n\right\}$, which means that order of elements in block is not relevant, but order of blocks does matter. (so $a_n = \sum_k\left\{n\atop k\right\}k!$, if I'm not mistaken). Prove recurrence relation:
$$a_0=1; \ a_n=\su... |
H: Topology : R open domain and closed domain
I dont understand why we can consider the domain R and $\varnothing$ as close and open domains ?
I have no idea of how to demonstrates it and like to see how to do it?
Edit : My definition : an open space is circle where each point contained can be circled by a circle with... |
H: Let $g(x) = \int_{0}^{2^x} \sin(t^2)\,dt $. What is the $g'(0)$?
I'm kinda stuck in this exercise :
Let $\displaystyle g(x) = \int_{0}^{2^x}\sin{(t^2)\,dt}$. What is the $g'(0)$?
How do I approach this kind of thing? I was thinking about Riemann sums but does this have any application here?
AI: We will use the F... |
H: Laplace Transform of $100e^{-5t}\sin10t$
Can anybody help me with the answer of this question?
$$100e^{-5t}\sin10t$$
AI: We have:
$$\mathcal{L}(\sin 10 t) = \dfrac{10}{s^2 + 100}$$
We have:
$$\mathcal{L}(e^{-5 t} \sin 10 t) = F(s+5) = \dfrac{10}{(s+5)^2 + 100}$$
Lastly, we multiply by $100$, to yield:
$$\mathcal... |
H: Need to find function by the data
I have found interesting sequence, but I can't find its function.
Here are the input and output data:
0 30
1 26.7
2 24
3 21.8
4 20
5 18.5
6 17.1
7 16
8 15
9 13.3
10 12
11 10.9
12 10
13 9.2
14 8.6
15 8
16 7.5
17 6.7
18 6
19 5.5
20 5
21 4.6
22 4.3
23... |
H: Hartshorne exercise 1.6.4 : Is it true that $\mathcal{O}_{P,X} \cong \mathcal{O}_{\varphi(P),\Bbb{P}^1}$?
Let us work over a fixed algebraically closed field $k$ and consider a non-singular projective curve $X$ and $\varphi : X \to \Bbb{P}^1$ a non-constant morphism.
My question is: For $P \in X$, do we have an ... |
H: Question - Möbius inversion formula
I need your help in the next question:
Prove directly from the definition the Möbius inversion formula.
(Möbius function defined as follows:
μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.
μ(n) = −1 if n is a square-free positive integer wi... |
H: Criterion for sum/difference of unit fractions to be in lowest terms
Pick two nonzero integers $a$ and $b$, so $(a,b)\in (\mathbb{Z}\setminus\{0\})\times(\mathbb{Z}\setminus\{0\})$. We want to add the fractions $1/a$ and $1/b$ and use the standard algorithm. First carefully find the least common multiple of $a$ and... |
H: $\mathbb Q$-basis of $\mathbb Q(\sqrt[3] 7, \sqrt[5] 3)$.
Can someone explain how I can find such a basis ? I computed that the degree of $[\mathbb Q(\sqrt[3] 7, \sqrt[5] 3):\mathbb Q] = 15$. Does this help ?
AI: Try first to find the degree of the extension over $\mathbb Q$. You know that $\mathbb Q(\sqrt[3]{7})$ ... |
H: matrix calculus (differentiation of complex matrix)
I know that $f(x)=||Ax-b||_2^2$ (real matrix) has gradient $\partial f/\partial x=A^T(Ax-b)$.
Now suppose $A$ is complex, then how can I prove that $\partial f/\partial x=A^*(Ax-b)$?
AI: I assume that $x$ is real in both cases. For real-valued $A$ and $b$ you have... |
H: Are weak* topology and strong topology the same in $L^\infty$?
Let $(\Omega, \mathcal{F}, R)$ be a reference probability space.
For short, we use $\mathbb E[\cdot]$ to denote the expectation operator
$\mathbb E^{R}[\cdot]$ under probability $R$.
We consider the following two layers of function spaces.
The lower le... |
H: Electric field of finite sheet: Full analytical solution of integration?
I am trying to work out the integral
$$
\operatorname{E}_{z}\left(x,y,z\right) =
\alpha\int\int\frac{z\,\mathrm{d}x'\,\mathrm{d}y'}
{\left[\left(x - x'\right)^{2} + \left(y -y'\right)^{2} + z^{2}\right]^{3/2}}
$$ with the limits $$-\frac{a}{2}... |
H: How prove this $\frac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\frac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\frac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$
let $0<A,B,C<\dfrac{\pi}{2}$,and $A+B+C=\pi$,prove that
$$\dfrac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\dfrac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\dfrac{\sin{(C-A)}\sin{(C-B)}}{\si... |
H: Explain why the integral test can't be used to determine whether the series is convergent
I have a homework question on the integral test chapter of my book and I'm not sure i'm answering this correctly.
$$\sum_{n=1}^\infty = \frac{\cos \pi n}{\sqrt n } $$
Now I know that in order for me to apply the integral test ... |
H: How to show that a given set is a subspace
OK I just want to be sure I have done this correctly.
Given: $R^3$, are the following sets subspaces?
(a) $W_1$ = {($a_1$,$a_2$,$a_3$) $\in R^3: a_1 = 3a_2$ and $a_3 = -a_2$
Since the set you get when you plug in the values to the set you get (3$a_2$,$a_2$,$-a_2$), and t... |
H: Show that this set is linearly independent
Ñotation:
$V$ is a vector spaces of real functions $g:X\rightarrow\mathbb{R}$;
$\{g_1,...,g_m\}$ is a subset of $V$;
$\{x_1,...,x_n\}$ is a subset of $X$, where $x_i\neq x_j$ when $i\neq j$;
$v_i=\left (g_i(x_1),...,g_i(x_n)\right )$ for all $i=1,...,m$;
$\{v_1,...,v_m\}$ ... |
H: Help with anti-image matrix
First of all, I am very sorry but I don't know the mathematics terminology in English, so I'll try to explain as good as i can but i will probably do some mistakes since it's not my native language.
I have this problem: given the endomorphism $f_k(x,y,z) = (x+y+kz, x+ky, 2x+(k+1)y+kz)$ f... |
H: Product rule question about Alphabet
I am trying to understand the product rule and I have a simple example it says,
If I have a license plate with two English letters how many different plates
can be made?
The answer is 26^2
Now another question is in the same format but it asks how many plates can be made with... |
H: $G_1/H\cong G_2\implies G_1\cong H\times G_2$?
Lagrange's lets us write the deceptively tidy relation:
$$\left|\frac{G}{H}\right|=\frac{|G|}{|H|}$$
and from this we can do neat things like, in the proof of the Orbit-stabiliser theorem,
$$\frac{G}{\text{stab}(s)}\cong \text{orb}(s)\implies\left|\frac{G}{\text{stab}(... |
H: How to efficiently generate five numbers that add to one?
I have access to a random number generator that generates numbers from 0 to 1. Using this, I want to find five random numbers that add up to 1.
How can I do this using the smallest number of steps possible?
Edit: I do want the numbers to be uniformly distrib... |
H: Proving that a matrix is diagonalizable
Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix
$$ A =
\begin{bmatrix}
6 & -3 & -2 \\
4 & -1 & -2 \\
10 & -5 & -3 \\
\end{bmatrix}
$$
Prove that $ T $ is diagonalizable.
AI: Hints:
$$\det(xI-A)=\b... |
H: homeomorphism between the same spaces with different topologies
$\mathbb{R}^2$ with different topologies on it are homeomorphic as a topological space?
for example with discrete topology and usual topology, what I need is a continous bijection with inverse is continous, from usual to discrete any continous map is... |
H: $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group?
I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group.
My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to \operatorname{SL}(2,\mathbb{R})$ is not su... |
H: Solve $g(g(x))=f(x)$
If $f(x)$ is a continuous and monotonically increasing function on an interval $(0,∞)$ and $f(x)>0$ for every $x>0$, then does there always exist a continuous and monotonically increasing function $g(x)$ on $(0,∞)$ so that for every $x\in (0,∞),g(x)\in (0,∞)$ and $g(g(x))=f(x)$?
If $f(x)=c~x^k~... |
H: Conditional probability with $5$ sided dice
Question:
In Brooklyn people play a fair dice with $5$ sides, numbered $1,2,3,4,5$.
Jack rolls the dice over and over again. What is the probability that the results $2$ or $4$ will come up before the result $5$?
What We want to understand
We saw 2 solutions which we didn... |
H: How to show all eigenvalues are positive?
Could you help me to show that the following matrix has all its eigenvalues positive?
$$H=
\begin{bmatrix}
\sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & \sum_{k=1}^ng_1(x_k)g_m(x_k)\\
\sum_{k=1}^ng_2(x_k)g_1(x_k) & \sum_{k=1}^ng_2(x_k)^2 & \cdots & \sum_... |
H: Confirm the meaning of Prime and Primitive in a Galois(2) polynomial.
Here it discusses primality (or more accurately irreducibility) and primitivity of polynomials in $G(2)$. More specifically it states that $x^6 + x + 1$ is irreducible and primitive.
But here I can divide $x^7 + 1$ by $x^6 + x + 1$ and get $x$ re... |
H: Express $y=|-x^2+1|$ as a piecewise function.
I'm unsure of how to start this problem. Any help would be greatly appreciated.
AI: As for real $z,$$$ |z|=\begin{cases} z &\mbox{if } z\ge0 \\
-z & \mbox{ otherwise } \end{cases}$$
$$|-x^2+1|=\begin{cases} -x^2+1 &\mbox{if } -x^2+1\ge0 \iff -1\le x\le 1 \\
-(-x^2+1)=x^... |
H: Rewrite fraction, probability generating function
I'm looking at an example from probability (generating function) where the following fraction came up:
$g_Y(t)=\frac{q+tp}{2-p-tq}=\frac{1}{q(2-p)} (-p(2-p) + \frac{1}{1-\frac{qt}{2-p}})=\frac{1-p}{2-p}+\frac{t}{(2-p)^2}+\frac{qt^2}{(2-p)^3}+ \dots$
$p+q=1$
From whi... |
H: What's the difference between $\frac{dy}{dx}$ and $dy$?
Ok, so I was doing a substitution problem and I realized that $dy = u\ dx + x\ du$ and not $\frac{dy}{dx} = u\ dx + x\ du$ and I was wondering what the difference was between those two. My first guess would be that $\frac{dy}{dx}$ means differential of $y$ wit... |
H: Recursive algorithm
I am trying to understand how this works. My instructor is teaching his first class, in summer on top of that, and only had 3 slides on this and had to rush over it. His example is:
Give a recursion algorithm for computing 0+1+2+...+n, where n is non-negative
His answer is
procedure add(n: non... |
H: How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?
I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a half-circle part ($\alpha_r$) and a path connecting ... |
H: Connectivity (graph theory) - proof
A graph $G$ has $n$ vertices. Prove that $G$ is connected if and only if we can find an injective sequence ($\ v_{1}...v_{n}$) such that for each $i>1$ there exists $j < i$ with $\ v_{i} v_{j}$ $\in$ E(G).
AI: Proof by induction. $n=1$ both things are vacuously true. $n=2$ both... |
H: Is $\sum_{k=4}^{\infty }{k^{\log(k)}}/{(\log(k))^{k}}$ convergent or divergent?
I came across this problem in a textbook, and the question is to investigate the convergence/divergence of the following series: $$\sum_{k=4}^{\infty }\frac{k^{\log(k)}}{(\log(k))^{k}}$$. I have no idea how to start solving this problem... |
H: How to prove $f( D(a,c) ) \le f [ D(a,b) + D(b,c) ] \implies f( D(a,c) ) \le f ( D(a,b) ) + f ( D(b,c) )$?
I am presently working on an exercise from Kaplansky, I. Set Theory and Metric Spaces (ex. 7, pg 70).
The question is as follows:
Suppose that: $f$ is concave, $f(0) = 0$, $f(x) > 0$ for $x > 0$, and $f$ is m... |
H: Help with graphing an inequality?
Looking at this question, and I'm unsure of how to do it properly. I'm trying to graph y≥3x+4 if x<2 and y≥-3.
Any help would be greatly appreciated!
AI: Using Geogebra, which is a really neat piece of software |
H: $\left(-\frac{1}{2}\right)! = \sqrt{\pi}?$
I recently learned that $\left(-\frac{1}{2}\right)! = \sqrt{\pi}$ but I don't understand how that makes sense. Can someone please explain how this is possible?
Thanks!
AI: In order to extend the factorial function to any real number, we introduce the Gamma Function, which ... |
H: Euler graph and cycles
Prove that the graph $G$ is an Euler graph if and only if the set of its edges
can be divided into separate non-empty subsets, each of which induces simple cycle in $G$.
AI: Each connected component of a graph $G$ is Eulerian if and only if the edges can be partitioned into disjoint sets, eac... |
H: Number of generators of the maximal ideals in polynomial rings over a field
Hi I'm trying to prove the following
If $K$ is a field (not necessary algebraically closed) then every maximal ideal of $K[x_{1},\dots,x_{n}]$ is generated by exactly $n$ elements.
I know that if $K$ is algebraically closed by Hilbert bas... |
H: Testing convergence of $\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$
Can anyone help me to prove whether this series is convergent or divergent:
$$\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$$
I tried using the ratio test, but the limit of the ratio in this case is equal to 1 which is inconc... |
H: Simple question about closure
Let $p, A, B$ be open sets such that $p \subset A$, $\bar A \subset \bar B$ and $p \cap A\neq \emptyset$. Does $p \cap B \neq \emptyset$ holds? I need this for a proof.
AI: As $p \cap A$ is non-empty and open, and as $p \cap A \subset A \subset \overline{A} \subset \overline{B}$, it mu... |
H: I don't understand this proof of the AM-GM inequality?
The proof uses this lemma which I understand:
$\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ such that $x-E \ge y+E$, then $(x-E)(y+E) \gt xy$ . $\;$Hence, by ... |
H: Does setting derivative to zero suffice always for minimization of convex functions?
I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real.
Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for $X$ in the minimization of $Tr... |
H: Acceleration to velocity with coordinate $x$
Can someone please help me in finding the formula used to get the answer to this question?
An object moves along $x$-axis. In any coordinate $x$, the acceleration is $a=x^4$ (SI units). If the object goes from rest in $x=1$m, what velocity it will get at $x=2$m?
\begi... |
H: Proof of equivalent definitions of continuity of a function
Let $X$ and $Y$ be metric spaces, and $f : X\rightarrow Y$ a function
I have to prove:
(1) $f : X\rightarrow Y$ is continuous
(3) $\forall\,F \subset Y closed: f^{-1}(F) \, is\,closed$
from $(3) \Rightarrow (1)$
I know, for a continuous function:
$\forall... |
H: Given a linear transformation matrix, T, find the equation for the curve that T transforms a circle into.
Given the linear transformation matrix:
$$T=\pmatrix{2&-3\\1&1}$$
Find the equation for the curve that $T$ transforms a circle with equation $x^2+y^2=6$ into.
What I know:
My basis is going to be $[1,0]^T$ and ... |
H: Hausdorff space and compact subspaces
Let $X$ be Hausdorff and $A,B \subseteq X$ be disjoint compact subspaces of $X$. Prove that there are $U$ and $V$ open disjoint sets in $X$, $A\subseteq U$ and $B\subseteq V$.
I know that: $A$ and $B$ are closed in $X$.
But I have no idea how to prove statement using that $X$ i... |
H: The form of maximal ideal in the real polynomial ring $\mathbb R[x,y]$
Every maximal ideal of the real polynomial ring $\mathbb R[x,y]$ is of the form $(x-a, y-b)$ for some $a,b \in \mathbb R$. True or false? Any suggestions?
AI: Hint: How about the ideal $(x, y^2+1)$? |
H: How to find exponent coefficients in a sum of exponents?
It is easy to determine a coefficient 'c' of exp(c*x), just log it and find slope. Or if it's
exp(c1*x) + exp(c2*x)
then after log from 0 to the right of left we would find 'c1' and 'c2'. But what if we have more terms? For example, such a sum
exp(x) + exp(1... |
H: A trouble about the Ekeland variational principle
I have a trouble in the proof to $EVP$ theorem:
About the existence of the $\lim (\varphi(y_n))$ ?
Any hints would be appreciated.
AI: By construction, the sequence $\varphi(y_n) + 2^{1-n}$ is non-increasing. It is also bounded from below (since $\varphi$ is bound... |
H: Constructing a functions with Gelfand Naimark
If $X$ and $Y$ are compact Hausdorff spaces, show that for any algebra homomorphism
$$
F:C(Y) \to C(X)
$$
there exists a continuous function $f:X\to Y$ such that
$$
F(\phi)=\phi \circ f, \forall \phi \in C(Y)
$$
The spaces are compact Hausdorff, so presumably one shoul... |
H: Does $\frac{\mathrm d}{\mathrm dx} \ln(x)=\frac{\mathrm d}{\mathrm dx} \ln|x|$?
For some time, I've seen different solutions for the same problems.
Let $f$ be any continuous function, differenciable on its domain such that,
$$\int \frac{\mathrm d f}{f}=\ln(f)$$
but some authors say,
$$\int \frac{\mathrm d f}{f}=\ln... |
H: Formula for the $nm$th cyclotomic polynomial when $(n,m) = 1$
Let $n,m$ be coprime. I want to find a formulae for $\Phi_{n\cdot m, \mathbb Q}$. I conjecture that because $$d \mid nm \implies d \mid n \lor d \mid m,$$ that
$$
\Phi_{n\cdot m, \mathbb Q} = \Phi_{n, \mathbb Q} \cdot \Phi_{m, \mathbb Q} = \prod_{d|n,~(... |
H: Suppose that a $3\times 3$ matrix $M$ has an eigenspace of dimension $3$. Prove that $M$ is a diagonal matrix.
How would I go about this? I realise that having dimension 3 means that the solution to $(A-\lambda I)\mathbf b = \mathbf 0$ has 3 free parameters, which would in turn mean that $(A-\lambda I)$ is the zero... |
H: if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$?
If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$.
Is it true that $B \times A/B \cong A$?
I ask because I was watching an online lecture from a course in abstract algebra at Harvard extension school.
An... |
H: Equivalence relation help
If $|A| = 30$ and the equivalence relation $R$ on $A$ partitions $A$ into (disjoint) equivalence classes $A_1$, $A_2$, and $A_3$, where $|A_1| = |A_2| = |A_3|$, then what is $|R|$?
AI: We have $R = A_1 \times A_1 \cup A_2 \times A_2 \cup A_3 \times A_3$, so $|R|=|A_1\times A_1|+|A_2\times ... |
H: Develop five terms in the Taylor series
Develop five terms in the Taylor series around $x_0=\pi$ for the function $f(x)=\cos\left({x\over3}\right)$
$f^0(x)=\cos\left({x\over3}\right) \Big|_\pi $
$f^{'}(x)=-\sin\left({x\over3}\right) {1\over3} \Big|_\pi$
$f^{''}(x)=-\cos\left({x\over3}\right) {1\over9} \Big|_\pi$
$f... |
H: How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?
Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions?
Thanks.
AI: Suppose that, for every $x\in\Bbb R$ we have $$a_1\sin x+a_2\sin 2x... |
H: Why is the length R cosine theta?
Why is the length described as R cosine theta (the top where the Sphere is sliced off)? I've been staring at the geometry for quite a bit & can't figure.
Thanks
AI: By SOHCAHTOA (a special case of the law of sines that only works for right triangles), the cosine of an acute angle o... |
H: problem about continuity and limits
Let $f\colon\mathbb R \to \mathbb R$ be a continuous function.
Suppose that $\lim_{x \to +\infty} f(x) = \lim_{x \to -\infty} f(x) = +\infty$.
Prove that $f$ has a minimum, i.e., $\exists x_0 \in \mathbb R: \forall x \in \mathbb R f(x) \geq f(x_0)$:
My solution:
Suppose $\exists... |
H: question about epsilon, delta limit definition
Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$.
What is the " $0<$ " part that's sometimes included in the definition, I'm a bit confused about that. Is it ... |
H: The solutions of $x^2+ax+b=0\pmod n$ in $\mathbb Z_n$
For every positive integer $n\ge 6$ which is not prime, there exist integers $a$ and $b$ such that the congruence equation $x^2+ax+b=0 \pmod n$ has more than two solutions modulo $n$. I have no idea to prove the above statement. Any suggestions?
AI: Hint: $(x-u)... |
H: A question about Baby Rudin Theorem 2.27 (a)
Theorem 2.27: If $X$ is a metric space and $E \subset X$, then $\bar E$ (the closure of $E$)
is closed.
The proof says: If $p \in X$ and $p \not \in \bar E$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighborhood which does not intersec... |
H: Why does the Tower of Hanoi problem take $2^n - 1$ transfers to solve?
According to http://en.wikipedia.org/wiki/Tower_of_Hanoi, the Tower of Hanoi requires $2^n-1$ transfers, where $n$ is the number of disks in the original tower, to solve based on recurrence relations.
Why is that? Intuitively, I almost feel that... |
H: Why does derivation use lim? Alternative method possible!
Okai, so we were learning about Newtons method of differentiation and I came to questions why Isaac Newton or Leibniz use the following function.
$${f(x+h) - f(x)\over h}$$
$h$ is the distance at the X axis of the point we wish to find.
However, this equatio... |
H: Proof of Cauchy-Schwarz inequality, why does this work?
My books says to prove that the following inequality is true, and to use it to prove Cauchy-Schwarz: $$(a_1x+b_1)^2+(a_2x+b_2)^2+(a_3x+b_3)^2+\dots+(a_nx+b_n)^2 \ge 0$$
This is easy to prove because by the trivial inequality each term on the LHS is $\ge 0$. Ho... |
H: Infinitesimal $SO(N)$ transformations
An infinitesimal $SO(N)$ transformation matrix can be written :
$$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$
Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric.
I've started with the orthogonality condition like following:
$$R^TR=\boldsymbol 1$$
$$\implies... |
H: Show $G$ and $H$ are isomorphic
Let $G$ and $H$ be finite abelian groups of the same order $2^n$. If for each integer $m$, $$\left|\left\{x\in G\mid x^{\large 2^m}=1\right\}\right|=\left|\left\{x\in H\mid x^{\large 2^m}=1\right\}\right|$$ then $G$ and $H$ are isomorphic. How to show the above statement? Thanks.
AI:... |
H: Differentiability of a function at a point to prove it differentiable everywhere on the given condition.
A simple question which I came across recently. Just wanted to confirm if my logic on it is right....
Suppose $f(x)$ is a function and it's given that it's differentiable everywhere except possibly at $0$, but f... |
H: Probability: Two people get multiple choice questions
I was wondering... If two people do multiple choice test, and the questions are taken from a pool of question (the size of the pool is unknown). Both people must be given 30 questions. They both end up with 15 of the same questions (and the other 15 different... |
H: Does $a^3 + 2b^3 + 4c^3 = 6abc$ have solutions in $\mathbb{Q}$
Does $a^3 + 2b^3 + 4c^3 = 6abc$ have solutions in $\mathbb{Q}$?
This is not a homework problem. Indeed, I have no prior experience in number theory and would like to see a showcase of common techniques used to solve problems such as this. Thanks
Edit Ap... |
H: Products in a Set
Let:
$$S := \{1,2,3,\dots,1337\}$$
and let $n$ be the smallest positive integer such that the product of any $n$ distinct elements in $S$ is divisible by $1337$. What are the last three digits of $n$?
I'm having a bit of trouble with this problem: the context is that my prof. gave this as a 'extra... |
H: From a deck of 52 cards, the face cards and four 10's are removed. From these 16 cards four are choosen.
From a deck of 52 cards, the face cards and four 10's are removed. From these 16 cards four are chosen. How many possible combinations are possible that have at least 2 red cards?
My solution I'm not sure how ... |
H: is this subset a subspace - redux
OK, I have been bothering people here with this for days and with luck I finally have this. People have helped a lot here so far. (Doing these examples is I hope helping me learn the proofs, but I want to know that I am doing this right).
Let W be a subset of vector space V. Is it... |
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