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H: How Many Ways to get N by adding k integers ?
I know the number of ways getting $ n $ by adding k integers is $ \binom{N-1}{k-1} $ . Now I have applied this theory to the following problem but could not get the correct answer . Why ?
I have N coins having values 1,2,3, … N. I have to select a subset of
exactly K... |
H: $\int_{-\infty}^{\infty} \frac{1}{2\pi} \exp\{ -\frac{1}{2} ((y-x)^2 + x^2) \} dx$
$$\int_{-\infty}^{\infty} \frac{1}{2\pi} \exp\{ -\frac{1}{2} ((y-x)^2 + x^2) \} dx$$
What I thought of doing was expand the inside
$$\int_{-\infty}^{\infty} \frac{1}{2\pi} \exp\{ -\frac{1}{2}y^2 + xy -x^2 \} dx$$
then I can take out... |
H: showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
I studied the cyclotomic extension using Fraleigh's text.
To prove that Galois group of the $n$th cyclotomic extension has order $\phi(n)$( $\phi$ is the Euler's phi function.), the writer assumed, without proof, that $n$th cyclo... |
H: Inverting a quartic equation of state
I have the following equation (which is an adaptation of the Beattie-Bridgeman Equation of State):
$$ P = \frac{RT}{V} + \frac{B}{V^2} + \frac{C}{V^3} + \frac{D}{V^4} $$
This is a function of the form $P = f(V)$ as R, T, B, C and D are all constant with respect to P and V.
From... |
H: Deduce that $\sum \frac {a_n}{r_n}$ diverges.
Suppose $a_n>0$ and the series $\sum a_n$ converges. Put
$r_n=\sum_{n=m}^\infty a_n$.
Prove that $$\frac {a_m}{r_m}+\dots+\frac {a_n}{r_n}>1- \frac {r_n}{r_m},$$ if $m<n$, and deduce that $\sum \frac {a_n}{r_n}$ diverges.
My problem is to deduce that $\sum \frac {a_n}{r... |
H: Can this situation arise for a subset of a group?
Let $(G, *)$ be a group, let $e_G$ be the identity in $G$, and let $H$ be a non-empty subset of $G$. Can it happen that $(H, *)$ be a group in its own right (with the same binary operation) but with an identity element $e_H$ that is different from $e_G$? Of course, ... |
H: Simple group with order $\geq n!$ cannot have subgroup of index $n$.
My problem is as seen in the title:
For positive integer $n>1$, prove that a simple group with order $\geq n!$ cannot have subgroup of index $n$.
Could anyone give me some hints on how to approach this?
AI: If $[G:H]=n$ then $G$ acts on $n$ cose... |
H: Finding a trigonometric function for a problem
The question:
At Dolphin Bay the depth of the water at the end of the jetty is 6 metres at high and 4 metres at low tide. High tide occurs at 11am and low tide occurs at 5pm.
a). Using the information given find a trigonometric function which models the depth of the wa... |
H: Closure of globe
Is it true that $\overline{K}(x,r) = \mbox{cl}K(x,r)$ ?
I suppose that it is true. In euclidean space I can see that it is true. But maybe in other metric spacies we can find counterexample? If not, how can I prove that fact?
AI: There are numerous counterexamples. Take the metric $d$ on $\Bbb R$ ... |
H: The ring $\ \mathbb{Z}[\zeta]\ $ where $\ \zeta \in \mathbb{C}\setminus \{ 1 \} \ $ such that $ \ \zeta^3 = 1 \ $
Some exercise I have to make states that this is a ring.
$$ \mathbb{Z}[\zeta] \ := \ \{a+b\zeta: a,b \in \mathbb{Z} \} \qquad \text{where $\zeta^3=1$ and $\zeta\neq 1$}$$
I see that this is a ring wi... |
H: from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$
How can you go from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$? I mean how to prove that $1-\sin x = 2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$?
AI: Use $\displaystyle \sin x=\cos\left(\frac\pi2-x\right)$ a... |
H: Proving $\sin 6\theta \cos4\theta + \cos4\theta\sin2\theta = {(\cos 2\theta \tan 8\theta)\over \sec 8\theta}$
I need to prove the following:
$$\sin 6\theta \cos4\theta + \cos4\theta\sin2\theta = {(\cos 2\theta \tan 8\theta)\over \sec 8\theta}$$
How would I do this?
AI: Use $$2\sin A\cos B=\sin(A+B)+\sin(A-B),$$
$$2... |
H: Effect of Moving within the Feasible Region
$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point $\mathbf{x}\in\mathcal{F}$. If we move from $\mathbf{x}$ in such a direction... |
H: $(\frac{1}{\kappa})^2+(\frac{\dot{\kappa}}{\kappa^2\tau})^2=r^2$
Show that for a curve lying on a sphere of radius r with nowhere vanishing torsion, the following equation is satisfied:
$$\left(\frac{1}{\kappa}\right)^2+\left(\frac{\dot{\kappa}}{\kappa^2\tau}\right)^2=r^2$$
Please help me doing this. Honestly, I co... |
H: Integral of $\sqrt{1-x^2}$ using integration by parts
I was asked to solve this indefinite integral using Integration by parts.
$$\int \sqrt{1-x^2} dx$$
I know how to solve if use the substitution $x=\sin(t)$ but I'm looking for the Integration by parts way.
any help would be very appreciated.
AI: $$I=\int 1\cdot\... |
H: Frechet derivative of compact operator is compact
... this seems to be a well known fact as mentioned in this and in this manuscript.
However, I was not able to find a proof or to prove it by myself.
So my question is: How to prove this?
Any hint or reference is appreciated.
AI: Theorem. Let $\Omega$ be an open se... |
H: Finding a minimum value of an expression
I have this question to solve:
A city's temperature of y degrees Celsius on a day in February is given by $y = 16 + 8 \sin \left(\dfrac{\pi t}{12}\right)$ where t is the time in hours after 9am.
a). What is the minimum temperature?
b). At what time is the temperature a minim... |
H: Center of Heisenberg group- Dummit and Foote, pg 54, 2.2
Let $H(F)$ be the Heisenberg group over the field $F$ introduced in Exercise 11 of Section 1.4. Determine which matrices lie in the center of $H(F)$ and prove that $Z(H(F))$ is isomorphic to the additive group $F$.
Section 1.4 defines the Heisenberg group $... |
H: How to get value of x and y here?
$x^2=16$ $y =\sqrt{16}$
here I know that when we solve value of $x$ then we get two values $+4$ and $-4$
But why we don't' get two values of $y$.
Can you please explain this. Thanks for help.
AI: This is not an inequality but a system of equations, there are two numbers that squar... |
H: Dividing n balls into n buckets so 2 are empty.
We know that the balls and buckets are distinguishable, and n>2. I was searching the site for some clues but didn't really find any of them helpful. Any thoughts?
AI: I'm going to assume the question is, how many ways are there to distribute $n$ distnguishable balls a... |
H: $(p\lor \lnot q) \land (q \lor ¬r) \land (r \lor ¬p)$ is true $\iff$ $(p, q, r$ all have the same truth-values$)$
Explain why $(p\lor \lnot q) \land (q \lor ¬r) \land (r \lor ¬p)$ is True when p,q,and r have the same truth value and it is false otherwise. (Without using a truth table )
Please help me solve this
AI:... |
H: Dantzig's unsolved homework problems
From Wikipedia:
An event in George Dantzig's life became the origin of a famous
story in 1939 while he was a graduate student at UC Berkeley. Near the
beginning of a class for which Dantzig was late, professor Jerzy
Neyman wrote two examples of famously unsolved statistic... |
H: Positive definiteness of matrix?
Suppose $A$ is positive definite of order $k$. Let $X$ be of order $k \times n$. Is $W := X'AX$ necessarily positive definite?
Maybe it is something like a theorem of Sylvester, but I have been unable to find a theorem.
Thanks
AI: $W$ will be positive semidefinite, with (strict) po... |
H: why is $(2) \subseteq \mathbb{Z}[\zeta]$ a prime ideal?
Previously I asked something about the ring $\mathbb{Z}[\zeta]$ where $\zeta$ is a cube root of $1$ in $\mathbb{C} \setminus \{1 \} $. Could you provide me hints for showing that $(2)$ is prime
Research effort
I tried to find some isomorphim with $\mathbb{Z}[... |
H: Updating eigen decomposition for a matrix after some row changes
Let us say we have a matrix $A$ which has eigen decomposition
$$A=UDU^{-1}$$
If some of the rows of A are changed by multiplying a constant positive value, is there a simple way to update the eigen decomposition from existing $D$ and $U$?
Many thanks.... |
H: My conjecture on almost integers.
Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x \rfloor)^n$$ then $\Omega$ is an almost integer . The value of $n$ depends upon the dif... |
H: Translation matrix without homogeneous coordinates
I am looking to translate a vector, and I found this article on wikipedia, which states that I need to use homogeneous coordinates to do so.
I don't get the reason why, and I don't really see how the result of the translation matrix with homogeneous coordinates is ... |
H: Proof of Sobolev Inequality Theroem
I have a short question about the proof of Theorem 2 below. I have included Theorem 1's statement since it is used in the proof of Theorem 2.
Definition: If $1 \leq p < n$, the Sobolev Conjugate of $p$ is $p* := \frac{np}{n-p}$, note that p* > p.
Consider the following two Theo... |
H: Find the value of $x$ such that $2^x=10$
Given that $\log 5 = 0.7$ (to one decimal place), find the value of $x$ such that $2^x = 10$ (again to one decimal place)
I don't know what to do with the information that $10^{0.7} = 5$. Why is this information useful?
AI: We know,$$\log_{10}10=1$$
But $$\log_{10}10=\log_... |
H: Does this hold in general for inverse function?
Let $z=G^{-1}(y)$ be any number such that $G(z)=y$.
Now,
does $z>G^{-1}(y)$ imply $G(z)>y$ in general ?
does $z<G^{-1}(y)$ imply $G(z)<y$ in general ?
AI: It depends if $G(x)$ is an increasing function, decreasing function, or neither.
($\sin(x)$ is neither becaus... |
H: How does this set look like?
I have trouble to understand the following set:
$M_2 = \{ 1 \le \|x\|_2 \lt 2 \} \cup \{ \vec{O} \} \subset \mathbb{R}^2 $
On the left side there are scalars and on the right side there is the null vector. Can we have sets with elements of different dimension?
AI: As Prahlad mentioned i... |
H: If $f$ is continuous on $[a,b],$ $f(x)\geq 0 $ $(\forall x\in [a,b])$ and $\int^{b}_{a} f = 0,$ then $f = 0$ on $[a,b].$
Here is my proof:
Given $\epsilon>0, \exists$ partition $P = \{x_0 =a,...,x_n =b\}$ of $[a,b]$ such that $\sum^{n}_{i=1} (M_i-m_i)\Delta x_i < \epsilon,$ where $M_i = \sup\{f(x): x\in \Delta x_i\... |
H: If a function is Frechet differentiable, does the Frechet derivative equal the Gateaux derivative?
If a function is Frechet differentiable, does the Frechet derivative equal the Gateaux derivative?
AI: A function $f \colon U \to Y$, where $U \subset X$ is open, is Fréchet differentiable in $x \in U$ with (Fréchet) ... |
H: Two sets given : solve $A \cup X = B$
Do you know how to solve this problem? I have two sets and need to solve $A \cup X = B. $
Thanks a lot for your help.
AI: Note that if in fact there exists a set $B = A \cup X$, then $A\subseteq B$ AND $X\subseteq B$.
If $A$ and $X$ are disjoint, so that $A \cap X = \varnothing... |
H: Homeomorphism of closed intervals
One can prove that if $f: [a,b] \to [f(a), f(b)]$ is continuous and monotone increasing that then it is a homeomorphism. The only part one might have to think about at all is that $f$ is open but that can be shown easily by showing that $f$ is closed: If $C=[x,y]$ is closed in $[a,... |
H: How to get the roots of a quartic function when given a quadratic factor
We have the function $$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$ $x^2 -x - 6$ is a factor of this function. Find all the roots of the polynomial.
So we have $(x-3)(x+2)$, and since it is a quartic we need 2 more solutions. Intuitively I'd say you ... |
H: Integral of absolute value of Brownian motion
I know it is a really stupid question and it should be quite easy, but how can I show that $\int_0^{\infty}|B_t|\mathbb{d}t=\infty$ a.s. with $B_t$ being a standard brownian motion?
I just don't get it.
What I tried so far: I know that $\limsup_{t\rightarrow\infty}|B_t... |
H: Does almost sure convergence implies convergence of the mean?
I asked a slightly similar question here: Does Convergence in probability implies convergence of the mean?, but now I wish to examine a stricter scenario:
Let $\{X_n\}_{n=1}^\infty$ be a sequence of random variables converging a.s to a const $c$. Is it r... |
H: Borel-measurable functions $R\rightarrow R$
For $f: R\rightarrow R$ this two proposition are identical ($B$ is Borel set).
$\forall A \in B\ f^{-1}(A) \in B$
if $A$ is open subset then $f^{-1}(A) \in B$
Is it true? And if it's then how can I prove it?
Thanks.
AI: Consider
$$
\Sigma:= \left\{A \in \mathcal{B}(\mat... |
H: How to place bets to get > 50% chance of winning?
Assuming we have a simple game of fair-coin throwing, there's a 50% chance to win and a 50% chance to lose.
Let's assume that we have a large amount of tokens and there is no cap to the amount placed in the bet, I have this strategy to sure-win:
Bet x amount.
If wi... |
H: Problem with approximation of semicontinuous function with continuous functions
Assume that a function $f: \mathbb R \rightarrow \mathbb R$ is a lower semicontinuous.
The Baire's theorem says that there is an increasing sequence of continuous functions $(f_n)$ which is pointwise convergent to $f$.
I known a pr... |
H: How to find the expectation of $\sqrt{x}$ and others like it
Assume a Bernoulli distribution with probability of success $p$. I understand how to find expectations for random variables in general, but how do I go about finding something like $E(\sqrt{x})$ or $E(x^6+1)$? All the examples I have seen refer to finding... |
H: Need help in understanding state transition diagram of a convolutional coder. How are the output bits calculated?
Have a look at the above figure. I am confused in how the output bits are calculated. e.g. according to my understanding a state transition from 00 to 10 (with input bit 1) should produce output 10 ins... |
H: Question on Cauchy Sequence definition?
Kaplansky defines a Cauchy sequence if for any $\epsilon > 0$ there exists sufficiently large $i, j$ such that $D(x_i, x_j) < \epsilon$ for some sequence $\{x_n\}$ in a metric space. The sequence need not converge to some specified limit point, just that $x_i$ and $x_j$ becom... |
H: Compactness and sequential compactness
Let $X$ be a nonempty set. If the cocountable topology $\tau$ is considered on $X$, then I want to find all compact and sequential compact subsets of $X$.
Definitely only finite subsets of $X$ are compact. But I am in confusion to find all sequentially compact sets. Is there a... |
H: Limit as x approaches ∞ using epsilon-delta
Let $f$ and $g$ be defined on $(a,\infty)$ and suppose $\displaystyle\lim_{x\rightarrow\infty}(f)=L$ and $\displaystyle\lim_{x\rightarrow\infty}(g)=\infty$. Prove that $\displaystyle\lim_{x\rightarrow\infty}(f\circ g)=L$.
I am having trouble proving this using $\epsilon-\... |
H: How can I prove a function is odd?
I have a problem of calculus and I can compute it.
The problem is
$\iiint_V x^5 y^7 z^9 dxdydz$, where $V = \{ (x, y, z) : x^2 + y^2 + z^2 \leq 1 \}$.
And the solution is 0. this solution is exactly right. If you want I can show whole my solution.
I solved it using by polar coor... |
H: Convergent subsequence
1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0,
{k ∈ N : x_k ∈ B(L; ϵ)}
is infinite. Show that (x_n) has a subsequence converging to L.
2) Suppose that S is not a compact subset of R. Explain why there is a sequence (y_n) in S so that no subsequence of (y_n) converges t... |
H: Joint Probability Distribution of a Gaussian Random Variable Correlated with a Gamma Random Variable
I want to know if the joint PDF of a Gaussian RV correlated with a Gamma RV can be found in closed form. The correlation is known.
AI: There are lots of ways to construct bivariate distributions from given marginals... |
H: Numerical integration problem
How can I solve integral equations of the form $$\int_{-3}^x e^{e^t}dt=3?$$ Is there for example Sage code for that kind of equations? Is there better method that evaluating numerically
$$\int_{-3}^{-1} e^{e^t}dt<2.4,$$
$$\int_{-3}^{0} e^{e^t}dt>4.2,$$
$$\int_{-3}^{-0.5} e^{e^t}dt>3.1,... |
H: Maclaurin series for $e^z /\cos z$.
I want to find the Maclaurin series for the function
$$f(z)=\frac{e^z}{\cos z}.$$
Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest singularity (not sure if this explanation is rigorous enough, but I can't think of anythi... |
H: Interested in a "more fundamental" proof for basic properties of the logical connectives
Starting with the classical propositional logic, is there a rather canonical way to prove that $$p\wedge q=q\wedge p$$ for the commutativity of the conjunction and analogously for the other properties and connectives, other tha... |
H: Determine whether $\langle f, g\rangle = \int_1^e {1 \over x} f(x)g(x)\,dx $ is a inner product.
Let $C[1,e]$ be the set of continuous real-valued functions with domain $W:=[1,e]$.
Let $$\langle f, g\rangle = \int_1^e {1 \over x} f(x)g(x)\,dx $$
be a function.
Determine whether $\langle \cdot, \cdot\rangle$ is an i... |
H: How to integrate $\int_{0}^{\sqrt{3\pi}}\int_{y}^{\sqrt{3\pi}}\cos 7x^2\;dxdy$?
Evaluate the integral by reversing the order of integration:
$$\int_{0}^{\sqrt{3\pi}}\int_{y}^{\sqrt{3\pi}}\cos 7x^2\;dxdy.$$
I have tried a lot of times, but I did'n get something reasonable.
AI: $\newcommand{\angles}[1]{\left\langle #... |
H: Radius of Convergence of a Series
How would I find the radius of convergence of the following series?
$$
\sum_{n=1}^{\infty}\frac{(-1)^n}{n}z^{n(n+1)}
$$
The ratio test and root test are inconclusive, so I think I have to use the definition of the radius of convergence
$$\frac{1}{R}=\limsup|a_n|^{\frac{1}{n}}.$$
I... |
H: Find derivative of [(4-pi)/(4pi) x^2 - 10x + 100]?
How do I find derivative of this equation?
my attempt:
A(x) = $$\frac{4x^2+\pi x^2}{4\pi}-10x+100.$$
so how do I find the derivative of the first part? Like do I use quotient rule for the beginning of the equation??
please show full solutions :)
thanks
Sincerely... |
H: Three series of Kolmogorov
Let $X_n\geqslant 0$ be a sequence of independent random variables. The following are equivalent:
$i) \sum_{n=1}^{\infty}{ X_n} <\infty$ a.s
$ii)$ $\sum_{n=1}^{\infty}{ \mathbb P(X_n>1)} +\sum_{n=1}^{\infty}{ \mathbb E X_n1_{X_n\le 1}} <\infty$
$iii)$ $\sum_{n=1}^{\infty}{\mathbb E\left(\... |
H: Isomorphism between the real plane and a set of functions
I have been told that $\mathbb{R}^2$ is isomorphic to the collection of functions $2\to\mathbb{R}$. Does this statement make sense at all? And if yes, how does this isomorphism work?
Thanks
AI: Indeed, it is sensible. Here, $\textbf{2}:=\{0,1\}.$
Every eleme... |
H: In a topological vector space, show if $A$ and $B$ are bounded, then $A + B$ is bounded?
I get as far as this before I am stuck:
Pick any neighbourhood of $0$ and call it $U$. Then there exists $a, b$ such that $A
\subseteq aU$ and $B \subseteq bU$. So hence $ A + B \subseteq aU+bU$. This last part should probably... |
H: Error term for a cubic interpolation
I have a question on one interpolation problem. The problem is below.
For the given points, $x_0 = -1, x_1 = 0, x_2 = 3$ and $x_3 = 4,$ find the error term $e_3(\bar{x}) = f(\bar{x}) - p_3(\bar{x})$ for cubic interpolation of $f(x)=x^5 -5x^4.$ Give an upper bound on the absolute... |
H: Inverse Laplace Transform for $F(s) = (9s-24)/(s^2-6s+13)$
Find the inverse Laplace transform of $\displaystyle F(s) = \frac{9s-24}{s^2-6s+13}$. I have tried factoring out a $3$ from the top and putting it into the form of $\displaystyle\frac{b}{(s-a)^2+b^2}$ but I can't seem to do that with this equation. Any help... |
H: Product of affine varieties vs. product of (quasi-)projective varieties
Suppose that $F_1,\ldots,F_r\in k[T_1,\ldots, T_n]$ and $G_1,\ldots,G_s\in k[S_1,\ldots,S_m]$ where $k$ is an algebraically closed field. Clearly $X=V(F_1,\ldots,F_r)\subseteq\mathbb A^n_k$ and $Y=V(G_1,\ldots,G_s)\subseteq\mathbb A^m_k$ are tw... |
H: Application of Minkowski inequality for integrals
I have a question regarding Minkowski inequality for integrals:
Suppose $f:(0,\infty)\rightarrow\mathbb{R}$ is a function in $L^p$ with respect to Lebesgue measure on $(0,\infty),\ p\in(1,\infty)$.
Define $F(y)=\int_{(0,1)}f(xy)d\lambda(x),\ y>0$.
Show that $F\in L^... |
H: Group isomorphism between rationals with sum and rationals with multiplication
It is well known that the exponential function induces an isomorphism between the additive group of real numbers and the multiplicative group $\mathbb{R}_{>0}$. I was wondering if there exists an isomorphism between the additive group of... |
H: Permutations with prior positions
In how many ways $P,Q,R,S,T,U$ can be arranged such that $ P, Q$ should come before $T,U$ ?
Do we have to find the ways that $P$ and $Q$ can be placed in the first four position?
AI: We interpret the question as asking for the number of ways to arrange the letters so that $P$ is... |
H: Properties of the interior of a set
Consider $ \tau_1 , \tau_2$ two topologies defined on X.
I want to prove that these are equivalent:
(i) $
\tau_2 \subseteq \tau_1
$
(ii) $A\in X$, $(A^2)^\circ \subseteq (A^1)^\circ $
I have started by proving that (i) implies (ii) by definition but I don't know if this is corr... |
H: Sketching graphs : Most importaint points
I'm currently studying for a test which places a lot of emphasis on sketching graphs of certain functions, without anything but a ruler and a pencil. I mean tricky functions, for example:
$y = \sin^2(x)$
$y=\dfrac{1-x-x^2}{x^2}$
I always try to sketch these functions by ... |
H: Balls complete in metric space
I'm struggling with the following task, i'd be thankful if anybody could help me out with a solution.
Note $ (X,d)= (\mathbb{R}^n,d) $, $ \ R= S^{n-1}(0)= \{ x \in \mathbb{R}^n $: $ \ |x|=1 $} and $z \in $ R.
Then: $$ B_e(z)= \{x \in R: \ d_R(x,z)< \varepsilon \ \} \ \text{is comp... |
H: Probability density function with conditional distribution and correlation
I am unsure with question c/d/e/f but I will give my answers for all questions I have attempted.
The random variables $X$ and $Y$ have joint probability density function
$f_{X,Y}(x,y) = ke^{-(x+y)},\ x>0,\ y>0,\ 0<x<y.$
(a) Sketch the region... |
H: Composition of polynomials - is it a simple group?
I wouldn't be surprised if this can be found maybe even on Wikipedia but I'm not a native English speaker and unfortunately couldn't find this myself.
So for a set of polynomials
$F = \left\{ \, f(x) \mid f(x) = ax+b,\ a,b \in \mathbb{R},\ a \neq 0 \, \right\}$ is ... |
H: In the solution below how is (cosC+3√sinC)= 2sin(π6+C)?
You can use the Law of cosine and the area formula of a triangle to solve this problem. Suppose the three angles are A,B,C opposite to the sides a,b,c, respectively. Then
c2=a2+b2−2abcosC,S=12absinC
and hence
a2+b2+c2−4S3√==≥=≥2[a2+b2−ab(cosC+3√sinC)]2[a2+b2−2... |
H: x raised by the power of y equals infinity.
How many zeros are there in this equation?
10000^999
In my calculator it says infinity but that doesn't seem right.
AI: The answer is: $3996$.
$10000^1 - 4$ zeroes $(4 \cdot 1)$
$10000^2 - 8$ zeroes $(4 \cdot 2)$
...
$10000^{999} - 3996$ zeroes $(4 \cdot 999)$ |
H: Proof of a equality-topology
Consider he topological space $ (X,\tau)$ and $A \subseteq X$.
I want to prove that these are equivalent:
(i)$A\in \tau$ ($A$ is open)~
(ii) For any $B\subseteq X$, where $A\cap B=\emptyset$ then, $A\cap \overline B =\emptyset$
I have started in this way:
(i) $\to$ (ii)
$B\subseteq X$... |
H: If an element of a basis is a scalar multiple of an element of another basis, are the two bases considered distinct?
Suppose that we have the bases {(1,0) (0,1)} and {(1,0) (0,-1)}
We see that (0,-1) is a scalar multiple of (0,1). Are the two bases considered distinct?
Thank you
AI: The vectors in the two bases are... |
H: What is that sign in the context of vectors?
Suppose $v = (0, -5, 5, -6, -7)$ a vector.
I need to find $$\|v\|_1, \|v\|_2, \|v\|_9, \|v\|_\infty.$$
can you please explain me what does $\|v\|_i$ mean?
AI: It's probably the p-norm:
$$\|v\|_p = \left(|v_1|^p + | v_2|^p + \cdots \right)^{1/p}$$
Where for $p\to\infty$, ... |
H: Convolution of functions with compact support
I have a question regarding convolution with compact support:
Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support.
Show that $f*g$ (convolution integral of $f$ and $g$) has compact support.
Kindly advise in proceeding the ... |
H: Fixed Point Iteration is Not Converging to the Desired Root
What is the fixed point for the following function?
$f(x) = 2sin(\pi x) + x = 0$ between $[1,2]$
I expressed $f(x)$ as $x=g(x)$ such that $g(x)$ can be $- 2sin(\pi x)$ or $\frac{1}{\pi}sin^{-1}(\frac{-x}{2})$. However, in both cases, the method is not cove... |
H: Exponent Upon Exponent?
I can't understand this. Can you please make a clearer explanation?
AI: Write $y=x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$. Then
$$y = x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = x^{(x^{x^{x^{\cdot^{\cdot^{\cdot}}}}})} = x^y$$
The book is saying that if $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}=2$ then we can sti... |
H: Determining which Probability Distrubitive Function to use
I have two questions that I am to solve as practice. I am having difficulties with determining which probability function to use.
The firstquestion is:
Todd decides to keep buying a lottery ticket each week until he has 4 winners (of some prize). Suppose 30... |
H: Open cover with no finite subcover
Let (x_n) be a sequence, let $L$ ∈ R, and for each ϵ>0,
{k ∈ N : x_k ∈ B($L$; ϵ)}
Suppose S is not a compact subset of R. There is some ϵ_L > 0, such that
{k ∈ N : x_k ∈ B($L$; ϵ_L)} is finite.
Question: Let O:={B($L$,ϵ_L): $L$ ∈S}. Explain why $O$ is an open cover for S that ha... |
H: To show that $\Lambda^pL(V\rightarrow W)$ and $L(\Lambda^pV\rightarrow W)$ are not necessarily isomorphic
Let $V$ and $W$ be two vector spaces. Use $L(V\rightarrow W)$ to represent the vector space of linear map from $V$ to $W$.
It is proved that $\Lambda^p(V^*)\cong (\Lambda^pV)^*$, where $\Lambda$ is the exteror ... |
H: Why does this whole term go away with the $dx$?
I'm learning how to take indefinite integrals with U-substitutions on khanacademy.org, and in one of the videos he says that: $$\int e^{x^3+x^2}(3x^2+2x) \, dx = e^{x^3+x^2} + \text{constant}$$
I understand that the differential goes away, but not how the whole $(3x^2... |
H: Convergence of the arithmetic mean
Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with limit $a \in \mathbb{R}$. Show that the arithmetic mean given by: $$s_n:= \frac{1}{n}\sum_{i=1}^n a_i \tag{A.M.} $$
also converges to $a$.
I have read: arithmetic mean of a sequence converges but unfortunately the answer... |
H: Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $
Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $
$f(t)=L^{-1}\left\{\frac{7s-22}{s^2-6s+13}\right\}$.
I was trying to break $F(s)... |
H: Basic number theory question, proving something is an integer.
Let $a$ and $n$ be two non-zero natural numbers that are relatively prime.
Show that there exists $b \in \mathbb Z$ such that $ab \equiv 1\pmod n$.
So $(a,n)=1$ and we know there exists $\alpha, b \in \mathbb Z$ such that $$\alpha a + b n =1$$ and $$b=... |
H: how to prove this is a metric given the following conditions
I need help wrapping my head around the concepts of metrics and how to prove that something is a metric. For example, prove that if $p_1$ and $p_2$ are metrics in $X$, then $p_1 + p_2$ and $\max\{p_1, p_2\}$ are also metrics. Are the functions $\min \{p_1... |
H: Is $\mathbb{Q}[α]=\{a+bα+cα^2 :a,b,c ∈ \mathbb{Q}\}$ with $α=\sqrt[3]{2}$ a field?
I'm making some exercises to prepare for my ring theory exam:
Is $\mathbb{Q}[α]=\{a+bα+cα^2 :a,b,c ∈ \mathbb{Q}\}$ with
$α=\sqrt[3]{2}$ a field ?
If $(a+bα+cα^2)(a'+b'α+c'α^2)=1$, then (after quite some calculation and noticing t... |
H: Write a Mathematica program to find the four known factorions
so i have come up with code that will find the four factorion numbers. I have a bunch of ideas of how it needs to be done theoretically but its still not quite clicking for me not too mention i am new to mathematica so the syntax is a bit confusing. So c... |
H: Order of Double Coset
I am working on a homework problem (so don't just give me the answer) from Herstein's Topics in Algebra, which goes as follows:
If $G$ is a finite group, show that the number of elements in the
double coset $AxB$ is $$\dfrac{o(A)o(B)}{o(A\cap xBx^{-1})}$$
It makes sense to me, but my attem... |
H: Showing the parametrically representation of hyperbolic paraboloid. And how to find the curves $u$ and $v$ be constant.
Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$
Find the curves $u$ is constant and $v$ is constant.
I guess I need to use th... |
H: Expected Number of Cars in a Parking Space
Another expectation problem:
There is a parking space of length 4. Cars come and randomly choose any
position to park over the interval [0, 4]. Each car occupies a space of
length 1. Calculate expected number of cars that can park.
I am trying to solve it using indicator ... |
H: A set of formulas that classifies two-element structures
Give a set of formulas $\Gamma$ such that for any structure $\mathcal{A}=\langle A;-;-\rangle$ it holds that $\mathcal{A} \models \Gamma$ if $A$ has exactly two elements.
AI: The formula you mention the comments works, but should be parenthesised as follows:
... |
H: Twin prime "test" via congruence
I decided to try getting a test for a "twinness" of a prime via Wilson's theorem.
Wilson's theorem says that integer $n > 1$ is a prime iff $$(n-1)! \ \equiv -1 \pmod n $$
Now, if both $n$ and $n+2$ are prime, we get two equations:
\begin{cases}
(n-1)! \ \equiv -1 \pmod n & (1)\\
(n... |
H: Express $\sin\frac{\pi}{8}$ and $\cos\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$
I've been trying with no success expressing this functions.
a) $\sin\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$
b) $\cos\frac{\pi}{8}$ with $\cos\frac{\pi}{4}$
I've tried formula of the double angle ($\sin$ and $\cos$) and the ecuation $\cos^2... |
H: Poisson Process Notation: P(N(h) = 2) =P(N1(h) = 1; N2(h) = 1)
This is from the book: Stochastic Processes by Sheldon Ross
Does the ";" sign mean conditional probability in the second part of the equation?
AI: $$P(A,B)=P(A;B)=P(A\cap B){}$$ |
H: Determine the value of h such that the matrix is the augmented matrix of a consistent linear system.
Determine the value of h such that the matrix is the augmented matrix of a consistent linear system.
$$
\begin{bmatrix}
1 && h && 4 \\
3 && 6 && 8
\end{bmatrix}
$$
I'm entirely unsure how to go about solving thi... |
H: I need help figuring out how this sequence converges.
$A_n = ( 1 + \frac 2n ) ^ n$;
I know the end result is convergent at $e^2$, but how do I figure that out? I've started setting it up as $An = (1 + (2/n))^{(n/2)2}$ as someone has suggested but i don't understand how that results in $e^2$.
AI: The suggestion tha... |
H: I just proved that $ℂ$ is not a field. What is the mistake in my reasoning?
What is the mistake in my reasoning?
Consider $(X^2+1)$ in $ℝ[X]$. Then $(X^2+1)⊂(X^2,1)$. Because if $f \in (X^2+1)$. Then $f=(X^2+1)g=X^2g+1⋅g$. So $f \in (X^2,1)$. Therefore $(X^2+1)$ not a maximal ideal. And therefore $ℝ[x]/(X^2+1)≅ℂ$ i... |
H: Discrete Math - Combinatorics - Trinomial Coefficients question
Let $k,l,m,n \in Z \geq 0$ be such that $n=k+l+m$. The trinomial coefficient ${n \choose k,l,m}$ is given by the rules:
for $k+l=n$, ${n\choose k,l,0} = {n \choose k,0,l} = {n \choose 0,k,l} = {n\choose k}$
${n\choose k,l,m} = {n-1 \choose k-1,l,m} + ... |
H: Any Subset of a Set Containing No Accumulation Point is Closed
Let $(X,\tau)$ be a topological space. Suppose that $B\subseteq X$ has no accumulation point. That is, for any $x\in X$ there exists some $U\subseteq X$ such that $x$ is in the interior of $U$ and $B\cap U\cap\{x\}^c$ is empty.
Claim: If $C\subseteq B$,... |
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