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H: Evaluating an integral with ${\mathrm{d}t}$ in the numerator, $\int \frac{\mathrm{d}t}{\cos(t)^2}$.
How do I solve an integral with a differential on top? E.g.: given this integral to evaluate: $$\;\int \frac{dt}{(\cos(t))^2}\;\;?$$
What does it even mean when there's a differential?
AI: To address your notational ... |
H: Stumped by a notation.
I'm reading through http://cr.yp.to/papers/primesieves.pdf and came across the following notation on p. 1:
For example, a squarefree positive integer $p \in 1 + 4\Bbb Z$ is prime if and only if the equation $4x^2 + y^2 = p$ has an odd number of positive solutions $(x,y)$.
What I'm confused... |
H: How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?
How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?
I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ?
These are just linear operators acting on a hilbert space.
AI: Here's a hint: bas... |
H: Modular exponentiation pattern question
I once found this problem
Find the form of all $n$ such that
$$3 \cdot 5^{2n+1}+2^{3n+1}=0 \pmod{17}$$
I started by writing the residues $\pmod{17}$ of $3 \cdot 5^{k}$ and $2^k$
$$3 \cdot 5^1+2 \equiv 0\pmod{17},\,2^1+15\equiv 0 \pmod{17}$$
$$3 \cdot 5^2+10\equiv 0\pmod{17},\... |
H: If polynomial with rational number is injective on rationals then it is injective on reals?
Let $p:\Bbb{R}\to\Bbb{R}$ is polynomial with rational coefficients. If restriction of $p$ to $\Bbb{Q}$ is injective, then $p$ is injective?
I conjectured that $p$ is monotonic, but I don't know how to prove this conjecture. ... |
H: Proof of Zorn's Lemma using the Axiom of Choice. Why is $\mathscr U$ a tower?
I am reading a proof of Zorn's Lemma using the Axiom of Choice in Halmos' classical text, and I fail to see how to prove$\mathscr U$ satisfies the third condition of the definition of a tower. I will transcribe the relevant parts:
Let $X... |
H: Problem with Free Index in Einstein Summation Notation
From http://www.physics.ohio-state.edu/~ntg/263/handouts/tensor_intro.pdf:
Rules of Einstein Summation Convention — If an index appears (exactly) twice, then it is summed over and appears only on one side of an equation.
A single index (called a free index) ... |
H: Vectors Angles from $[0,2\pi]$
Given two vectors $V_1 = (x_1, y_1)$ and $V_2 = (x_2, y_2)$. How to calculate the angles between them in the range of $[0, 2\pi]$?
I know the $\cos\theta$ similarity equation could present a $\theta$ in the range of $[0, \pi]$.
AI: You're right that the cos way only gets you the unsig... |
H: Cluster point of a sequence $\{x_n\}$ is the limit of some subsequence - Axiom of Choice?
In a metric space, a cluster point of a sequence $\{x_n\}$ is the limit of some subsequence. The only proof that I know works like this:
Construct a sequence $\delta _k \to 0$. For each $\delta _k$ find a point $x_{n_k}$ in th... |
H: Understanding prime factors method of finding a LCM
I am able to understand why the LCM of prime numbers is just their product; but for non primes we get their prime factors and choose the 'maximum occurrence' of every prime factor and multiply them, I lose my understanding here, why choose the 'maximum occurrences... |
H: About divergence theorem
Consider the portion $S$ of the sphere $x^2+y^2+z^2=4$ with $z\ge -1$. Calculate the integral $$\iint_{S} (x^3, y^3, z^3)\cdot \vec{n} dS$$
a) Using directly a parametrization
Well, what are the steps that I need to follow to parametrize this? Can I use spherical coordinates?
AI: $$\begin{a... |
H: prove the limit inferior of $(x_n)$ where $n \in\mathbb{N}$
The problem states let $(x_n)$ be a bounded sequence for each $n \in\mathbb{N}$. Let $t_n=inf\{x_k: k\geq n\}$. Prove that $(t_n)$ is monotone and convergent.
After a little research because I was confused what limit inferior really was this was the proof ... |
H: How to prove that the cosine of the angle of two vectors is preserved after rotating the vectors with the same angle?
How can we with linear algebra prove that if two vectors in 2 dimensions form an angle $\phi$ then by multiplying those two vectors with the same rotation matrix the cosine of the formed angle will ... |
H: Projection, canonical immersion/submersion - are they equivalent, and are they open maps?
I am very confused with the concept of projection with the introduction of immersion and submersion. By local immersion/submersion theorem, for a simmersion/submersion $f$, there is is a canonical immersion/submersion local... |
H: Show that if $I$ is an interval and $f:I\to\mathbb R$ is continuous on $I$, then $f(I):=\{f(x):x\in I\}$ is an interval.
Show that if $I$ is an interval and $f:I\to\mathbb R$ is continuous on $I$, then $f(I):=\{f(x):x\in I\}$ is an interval.
I dont't know how to start, can someone please give me ideas? Thanks.
AI... |
H: Stalk of the sheaf of regular functions on a subvariety
Suppose $Y$ is a subvariety of a variety $X$ (according to Hartshorne this means if $X$ is quasi-affine or quasi projective then $Y$ is a locally closed subset of $X$, c.f. exercise 3.10, chapter 1). Now given $i : Y \to X$ the inclusion map, I am trying to f... |
H: Prove that: $a^2+b^2+(1-a-b)^2\ge \frac {1}{3}$
Where $a$ and $b$ are any given real number. I have tried solving it using partial derivative.
$$
s=a^2+b^2+(1-a-b)^2$$
$$\frac{\partial s}{\partial a}=2a-2(1-a-b) \tag{1}$$
$$\frac{\partial s}{\partial b}=2b-2(1-a-b) \tag{2}$$
for maxima both (1) and (2) are 0..from ... |
H: A sub-problem on real number construction
Let we have a Dedekind cut set $\alpha$ and $w$ be a positive rational number. How to prove that there exists an integer $a$ such that $aw \in \alpha$ ? I am able to prove using Archemedian property that there exists a natural number $b$ such that $bw$ does not belong to $\... |
H: pigeonhole principle homework question
These are Homework question. They are pigeon hole principle questions and I have a very hard time with these unless I have worked on a similar problem before.
Q.1. Prove that if we select 87 numbers from the set $S = {1,2,3,....,171}$ then there are at least two consecutive nu... |
H: $\mathbb{Z}$ as a free product of two groups
My question is that can the group $(\mathbb{Z},+)$ be written as a free product of two (non-trivial) groups?
Thanks
AI: Suppose $\mathbb Z \cong G\coprod H$, the free product of $G$ and $H$. But $\mathbb Z$ is abelian while if $g\in G$ and $h\in H$ are non-trivial elemen... |
H: {0,1,2,.....9} are randomly assigned to the vertices ${x_0,x_1,...x_9}$ of a decagon.Prove there are 3 consecutive vertices whose sum is at least 14.
This is a homework Question and has to do with Pigeonhole principle. Could use a hint.
Q. The numbers ${0,1,2,.....9}$ are randomly assigned to the vertices ${x_0,x_1... |
H: Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant
How can I shows that the summation of $1/k^2$ from k=1 to n is bounded above by a constant?
I could bind it by the geometric series from k=0 to n and add 1 to $(1/k^2)$ to get the ratio, r, and get $A_0 (1/(1-r)) $. So is that going to be summ... |
H: Solve the following equation: $x^4- 2x^2 +8x-3=0$
Solve the following equation:
$$x^4- 2x^2 +8x-3=0$$
We get 4 equations with 4 variables. But that is too difficult to solve.
My try:
Let $a,b,c,d$ be the roots of the equation.
$$a+b+c+d=0$$
$$\sum ab = -2$$
$$\sum abc = -8$$
$$abcd = -3$$
Is there any other meth... |
H: Algebra and Substitution in Quadratic Form―Einstein Summation Notation
Schaum's Outline to Tensor Calculus ― chapter 1, example 1.5 ―――
If $y_i = a_{ij}x_j$, express the quadratic form $Q = g_{ij}y_iy_j$ in terms of the $x$-variables.
Solution: I can't substitute $y_i$ directly because it contains $j$ and there's ... |
H: $ \lim_{ \varepsilon \rightarrow 0^+ } \int_{|x| \geq \varepsilon} \frac{ \varphi(x) }{x}dx = - \int_{-\infty}^\infty \phi'(x) \ln(|x|) dx$
How do I prove that
$$ \lim_{ \varepsilon \rightarrow 0^+ } \int_{|x| \geq \varepsilon} \frac{ \varphi(x) }{x}dx = - \int_{-\infty}^\infty \phi'(x) \ln(|x|)dx $$
for all $ \va... |
H: Name of integration techniques
Consider following integrations :
$$\int \sec^3 x\ dx,\ \int \cos\ x\ e^x\ dx $$
These can be calculated by integration by parts.
But here for instance to calculate the latter example,
we meet
$$\ast\ \int e^x \cos \ x\ dx = e^x \sin\ x + e^x \cos\ x - \int e^x \cos \ x dx $$
No... |
H: How to properly evaluate the Riemann sum for the integral of $x^2$ for $x$ from $0$ to $3$
In my homework we start out with
$$\int_{x=0}^3 x^2 \, dx=\lim_{P: \Delta x \to 0} \sum_{i = 1}^n f(x_i) (\Delta x)_i$$
Where I take
$$P_i=[\frac{i-1}{n},\frac{i}{n}], x_i=\frac{i}{n}, (\Delta x)_i=\frac{3}{n}$$
So then
$$\in... |
H: Prove that a continuous function on a closed interval attains a maximum
As the title indicates, I'd like to prove the following:
If $f:\mathbb R\to\mathbb R$ is a continuous function on $[a,b]$, then $f$ attains its maximum.
Now, I do have a working proof: $[a,b]$ is a connected, compact space, which means that b... |
H: show that the function $z = 2x^2 + y^2 +2xy -2x +2y +2$ is greater than $-3$
Show that the function
$$z = 2x^2 + y^2 +2xy -2x +2y +2$$
is greater than $-3$
I tried to factorize but couldn't get more than $(x-1)^2 + (x+y)^2 +(y-1)^2 - (y)^2$.
Is there any another way to factorize or another method??
AI: HINT:
$$z =... |
H: Question about natural logarithm in the exponent of the e-function
I wonder which rule dictates that e^(-2x+ln(c)) is equal to e^(-2x) * c
I know that the logarithm naturalis is the "reverse-function" of the e-function but why isn't it e^(-2x) + c instead?
AI: Note that
$$ e^{-2x + \ln(c)} = e^{-2x}e^{ \ln(c)} =e... |
H: Simplify Triple Sum — Einstein Summation Notation
Schaum's Outline to Tensor Calculus — Chapter 1, Solved problem 1.5 —
Use the summation convention to write and state the value of $n$ necessary in:
$$g^{\LARGE{1}}_{11} + g^{\LARGE{1}}_{12} + g^{\LARGE{1}}_{21} + g^{\LARGE{1}}_{22} + g^{\LARGE{2}}_{11} + g^{\L... |
H: Star graph embeddings
This is an homework question which I'm struggling with:
Let $S = (V, E, w)$ a star graph, meaning, $S$ is a tree that all it's vertices are leafs except one.
I need to :
show that every weighted star has an isometric embedding into $\ell_1$.
find an example of a weighted star that cannot be e... |
H: A discrete space of cardinality $\aleph_0$.
How does a discrete space of cardinality $\aleph_0$ looks like? On finite sets I always get finite discrete spaces, countable sets (i.e. sets of cardinality $\aleph_0$) yields spaces of cardinality $> \aleph_0$, cause $2^{\mathbb N}$ is uncountable.
AI: It looks like the ... |
H: A simple inequality in calculus?
I have to solve this inequality:
$$\left(\left[\dfrac{1}{s}\right] + 1 \right) s < 1,$$
where $ 0 < s < 1 $.
I guess that $s$ must be in this range: $\left(0,\dfrac{1}{2}\right]$.But I do not know if my guess is true. If so, how I can prove it?
Thank you.
AI: The question states... |
H: Conditional probability with Bayes' Rule
On a practice exam from statistics I encountered a very difficult exercise I couldn't manage to solve:
In the tent next to you there is a family with two children. Early in the morning you see a boy coming out of the tent. What is the probability that the other child is a g... |
H: Simple probabilty / set problem I would love an explanation to.
I am currently studying som very elementary set theory in my algebra text book and I just arrived at this problem (I am a beginner in this so please bear with me):
Question: In a specific situation there are $100$ people. At least $70$% of these lose a... |
H: Stirling numbers combinatorial proof: $S(m,n)=\frac 1{n!} \sum_{k=0}^{n} (-1)^k\binom nk (n-k)^m$
This is a Homework Question.
I am required to give a Combinatorial proof for the following.
$$S(m,n)=\frac 1{n!} \sum_{k=0}^{n} (-1)^k\binom nk (n-k)^m$$
Hint given is : Show that $n!S(m,n)$ equals the number of onto ... |
H: If the * of morphisms (poly. maps) are equal, are the morphisms equal?
Let $t,s:X\rightarrow Y$ be polynomial maps between affine varieties and $t_*,s_*:k[Y]\rightarrow k[x]$ be their images under the representable contravariant functor. We've learnt that for any $\tau:k[Y]\rightarrow k[x]$ there exists a $t:X\righ... |
H: Solution of $\tan x^3 = -\frac 32 x^3$?
How to solve $\tan x^3 = -\frac 32 x^3$? Could you give me advice?
AI: Given tan(x^3)=(-3/2)x^3
Implies : x^3=tan^-1((-3/2)*x^3)
x^3 is a curve that always lies the 1st quadrant and the 3rd quadrant, as the function returns a positive number for a positive input and a negativ... |
H: A simple inequality in about integer part of numbers?
This question follows A simple inequality in calculus?.
I have to solve this inequality in about $s$:
$$\left(\left[\dfrac{r}{s}\right] + 1 \right) s \le 1,$$
where $ 0 < s < 1 $ and also $ 0 < r < 1 $.
That inequality is in about Computer Programming probl... |
H: Small arguments of $f$ vs large arguments of $\widehat{f}$
Say I know the behavior of $f:\mathbb{R} \to \mathbb{R}$ in the vicinity of 0. Are there any results linking that to the behavior of its Fourier transform $\widehat{f}(\xi )$ for large values of $|\xi |$?
I did not assume that $f$ belongs to some certain f... |
H: ODE phase portrait and vector function interpretation
I do not quite remember how to plot a vector function (or maybe I do). Consider the ODE:
\begin{equation}
x' = \begin{pmatrix}1&1\\-1&1\end{pmatrix}x
\end{equation}
I have found the general solution:
\begin{equation}
x(t) = c_1e^t\begin{pmatrix}-\sin(t)\\ \cos(t... |
H: Substitution for $\int \frac {dx} {ax^2 + bx + c}$
I'm looking for the substitution that makes easier to solve integral containing quadratic polynomial in denominator (!) when such polynomial cannot be broken into parts (if it can, then it's possible to use partial fraction decomposition). Example:
$$\int \frac {dx... |
H: Move Point A along a line
Sorry, can't post images if my rep is below 10, and can't post more than 2 links. I removed the http section so it won't count as a link. I hope this isn't against forum rules, I'm not hurting anyone.
I checked other questions, like this one (A line moving along the hypotenuse of a rig... |
H: Wikipedia example Cauchy's Integral formula
I do not really understand the wikipedia example that illustrates the usage of Cauchy's integral formula. enter link description here
The exact point I do not get is, how they argue that one can split this up into two integrals. There argument is, that this is given by Ca... |
H: Taking the derivative of $x^{\sin(e^x)}$
How am I suppose to take the derivative of $f(x)=x^{\sin(e^x)}$?
What should I make $u$ equals?
I tried to make $u=\sin(e^x)$ and $u=e^x$ but they didn't work.
AI: Hint: Here, you want to (start off) by using logarithmic differentiation - that is, take the derivative of $\ln... |
H: Smallest value of n for two algoritms with a certain running time
If one algorithm has a running time of $100n^2$ and another of $2^n$; how can I find the smallest value of $n$ such that the former is faster than the latter?
I could do:
$100n^2 < 2^n$ then $\ln(100n^2) < n\ln(2)$ but how do I simplify the left sid... |
H: Taking the derivative of $f(x)=x^{e^{e^x}}$
How can I take the derivative of $f(x)=x^{e^{e^x}}$?
How do I apply the chain rule? Thanks for the help!
AI: $f(x)= e^{e^{e^x}\ln x}$ (smooth on $(0,\infty)$, so $$
f^\prime(x)= \left(\frac{d}{dx}e^{e^x}\ln x\right)\cdot e^{e^{e^x}\ln x}
$$
Only the first term is somehow... |
H: Circles passing through three given points
How many such circles exist which pass though three given points in 2 dimensions?
Is it one unique circle? or possibly more than one?
Is there any proof?
AI: If the three points are not colinear, you can prove that there exists an unique circle by proving that it's center... |
H: Dependence of vectors : before and after linear transformation
I have a pretty simple question that confused me:
V is a vector space of a finite dimension.
$T: V \to V$ is a linear transformation.
The information that's been given in question:
$\operatorname{Im} T = \ker T$
I want to know if what I'm doing is righ... |
H: Primary decomposition example
I want to find the primary decomposition of $(x^2, xy^2)$ as an ideal of $k[x,y,z]$ where $k$ is some field. My guess is $(x^2, xy^2) = (x) \cap (x^2, y^2)$ however I am not 100% certain if $(x^2, y^2)$ is a primary ideal.
My approach to see this was to use the fact that $I$ is primar... |
H: Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots.
Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots.
My Attempt to the solution
we know to ha... |
H: $I$ semisimple + $R/I$ semisimple $\implies$ $R$ semisimple
Let $R$ be a (not necessarily commutative) ring with unit. Let $I\subset R$ be an ideal that in turn is a ring with unit. Is there a theorem that says something like $I$ semisimple and and $R/I$ semisimple implies $R$ semisimple?
AI: Let $a$ be the unit of... |
H: Measurable cardinals as sets
A philosopher said that measurable cardinals are the largest possible sets. Is this true? Are those sets at all? I mean, cardinals measure size of sets and for example $2=\{\{\},\{\{\}\}\}$ but can we represent measurable cardinals similarly? And is it true that those can not exist beca... |
H: $1, e^{ix}, e^{-ix}$ are linearly independent
Consider the space of all functions $f: \mathbb{R}\longrightarrow \mathbb{C}$.
Prove that $\{1, e^{ix}, e^{-ix}\}$ are linearly independent vectors.
AI: Recall the definition of linear independence for vector $u,v$ is
$$C_1u + C_2 v = 0\iff C_1=C_2=0.$$
Now consider
$$... |
H: Partial fraction integration $\int \frac{dx}{(x-1)^2 (x-2)^2}$
$$\int \frac{dx}{(x-1)^2 (x-2)^2} = \int \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}\,dx$$
I use the cover up method to find that B = 1 and so is C. From here I know that the cover up method won't really work and I have to plug in va... |
H: Need help with derivation of conditional expectation
The following is taken from the book "Mathematical Statistics for Economics and Business":
\begin{align*}
E\left.\left( \left[ Y-h(x) \right]^2\ \right\vert\ x\right)
=& E\left.\left( \left[\,Y-E(Y|x)+E(Y|x)-h(x)\,\right]^2\ \right\vert\ x\right)\\
=& E\left( \l... |
H: Dimension of $\mathbb{F}^n$
Let $\mathbb{F}$ be a field. Consider the vector space of $\mathbb{F}^n$ over $\mathbb{F}$ for some positive integer $n$.
Is the dimension of $\mathbb{F}^n$ necessarily $n$?
AI: Yes. It is true. The following vectors are a basis vectors:
$e_{1} = (1,0,...0), ..., e_{k} = (0,...0,1,0,..,0... |
H: $\dim (V_1+V_2) \geq \dim(V_1 \cap V_2) +2$
I am trying to solve this problems from an old qual exam. I know a way to
prove this but I feel like its too long for something that looks pretty simple.
Can anybody suggest a cleaner way? for example that uses the dimension formula
for subspaces $ \dim (V_1) + \dim(V_2)... |
H: if $f_n\to f$ uniformly in $[a,b]$ then $f\in BV$
While preparing to test in calculus I found the question above:
Let $f_n(x)\to f$ uniformlly on $[a,b]$. Prove or give counterexample: if $\forall n\in\mathbb N f_n(x)\in BV$ and $\exists M>0$ s.t $\displaystyle V_a^b f_n\le M$ then $f\in BV$
I am really ashamed t... |
H: Complex integral - winding number
i want to find $$\frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz$$
well $0$'s winding number is $2$, so $\frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz=2$
but when I explicity calculate the integral I get
$$\frac{1}{2\pi i}\int _\gamma \frac{1}{z}dz=\frac{1}{2\pi i}(\log(z)\lvert ^{z=-1}_{... |
H: Number of Regions in the Plane defined by $n$ Zig-Zag Lines
Fellows of Math.SE, I have been scratching my head at a solution to an exercise in Donald Knuth's Concrete Math. Here is the problem:
Here is the solution (I hid it in case someone wants to solve this on their own)
Given $n$ straight lines that define $L... |
H: How many times can a $4^{th}$ degree polynomial be equal to a prime number?
If $f(x)$ is a $4^{th}$ degree polynomial with integer coefficients, what is the largest set ${x_1, x_2, x_3, ...x_n}$ (where $x_i$ are integers) for which $|f(x_i)|$ is a prime number?
Things I have tried:
I tried to see how I can restric... |
H: What is the probability of getting the sum of 5 or at least one 4 when you roll a dice?
I just want to know if my method is right:
P(Sum of 5 or At least one 4) = 2+3, 3+2, 4+1, 1+4 [+] (4+1,4+2,4+3,4+4,4+5,4+6)*2
So that will be 4+12/36
Ans: 16/36
am i right here?
AI: Note that each of $(4, 1)$ and $(1,4)$ appear ... |
H: Determining the effective tax rate in a tax on tax situation
There are taxation situations where the taxable amount includes the tax calculated on the taxable amount (e.g. this is a recursive calculation, as follows)...
Iteration Taxable Amount Tax per iteration
0 $100,000,000.00 $5,000,000.00
1 ... |
H: Using $\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+2)\ldots(z+n)} \right]$ to prove the Weiestrass product.
I was searching the web quite thoroughly in the last two days. I was in paralytically
looking for a rigorous proof using
$$
\Gamma(z) = \lim_{n \to \infty} \left[ \frac{n! n^z}{z(z+1)(z+... |
H: How can I calculate this determinant?
Please can you give me some hints to deal with this :
$\displaystyle \text{Let } a_1, a_2, ..., a_n \in \mathbb{R}$
$\displaystyle \text{ Calculate } \det A \text{ where }$ $\displaystyle A=(a_{ij})_{1\leqslant i,j\leqslant n} \text{ and }$ $\displaystyle \lbrace_{\alpha_{ij}=0... |
H: a question on the minimal prime divisors of an ideal
This question is motivated by the second part of Step 1 in the proof of Theorem 14.14 in Matsumura's Commutative Ring Theory, p. 112.
Let $k$ be an infinite field and $Q$ a homogeneous ideal of $k[x]=k[x_1,\cdots,x_s]$. Suppose that $\operatorname{dim} k[x] / Q ... |
H: Integrating $\int^{e^3-1}_{0}\frac{dt}{1+t}.$
How can I integrate $$\int^{e^3-1}_{0}\frac{dt}{1+t}.$$
I tried to make $u=1+t$ which means that $du=dt$ but it's not giving me anything useful, but instead made things more complicated. Maybe I did something wrong, but can someone tell me the correct way of solving th... |
H: Two cards are drawn without replacement. Find the probability the second card is a jack given the first is not a jack.
My calculations:
I got $\frac{4}{51}$ because there are $4$ jacks in a deck and if we didn't have a jack then there are still $4$ left out of $51$ because we already chose one card.
Is this right?
... |
H: Simplyfing Probability equation
I was solving a homework problem, and I had obtained a formula for the required probability in the question. What I wanted to ask could it be more simplified?
$$P = \sum_{i=0}^{a}( \frac{a!}{(a-i)!} * \frac{(s-i-1)!}{s!})$$
AI: The answer is in fact $P=1/(s-a)$. Here's why.
Rewrite ... |
H: $u$-substitution for integrating $\int\frac{\log|x|}{x\sqrt{1+\log|x|}}\,dx\;\;?$
How can I integrate $$\int\frac{\log|x|}{x\sqrt{1+\log|x|}}\,dx\;\;?$$
I'm not sure what I should put equal to $\,u.$
Can someone give me a hint on how to solve this question? I don't need a full solution, I want to try it on my own.
... |
H: Why does Wolfram Alpha say that $\sqrt{1}=-1$?
Why does Wolfram Alpha say that $\sqrt{1}=-1$?
Is this a mistake or what? Can anyone help?
Thanks in advance.
AI: Did you read the results carefully? It does give $1$ as the answer:
It then helpfully mentions that $-1$ is also a 2nd root of $1$:
which is perfectly c... |
H: Given $B \in M_{n\times n}(\mathbb R)$ is invertible and $B^2+B^4+B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$.
Given $B \in M_{n\times n}(\mathbb R)$ is invertible and $B^2+B^4+B^7=I$, find an expression for $B^{-1}$ in terms of only $B$.
I don't know where to start. Thanks in advance.
AI: You ... |
H: Need help with geometry of surfaces
I am a third year maths student. I am self-studying a course on surfaces. I have some questions and would really appreciate it if people can help me.
What exactly is a connected sum? According to my lecture notes, for two closed (compact) surfaces, if we remove a closed disc fro... |
H: Raising a rational integer to a $p$-adic power
Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$?
Under what conditions would this be an element of $\mathbb{Z}_p$?
Any help would be very much appreciated.
AI: Assume that $d\equiv... |
H: Is this theorem in Rudin's Real and Complex Analysis wrong as stated?
I have forgotten all of my measure theory and will now ask a very dumb question.
Consider the following theorem, which I have produced word for word from Rudin's Real and Complex Analysis, third edition. It is theorem $7.13$ in chapter $7$ and c... |
H: Sequence of natural numbers
Numbers $1,2,...,n$ are written in sequence. It's allowed to exchange any two elements. Is it possible to return to the starting position after an odd number of movements?
I know that is necessarily an even number of movements but I can't explain that!
AI: Hint: Define an inversion of a ... |
H: rotate graph of function by 180
suppose that we have graph of function
$$f(x)=1+x\cos(x)$$
and we should rotate it by $180$ degree,question is what is a function which describe new graph?answer is $$f(x)=x\cos(x)-1$$but i can't understand why it is so?as i know rotation by $180$ is equal to instead of $x$,put ... |
H: Matrix representation
I wonder if the following term
$$
\ln\det\left(I+\text{diag}\left(d_1,\ldots,d_n\right)A\right),
$$
where $I$ is an identity matrix, $d_1,\ldots,d_n$ is a sequence of binary numbers, (taking the values $0$ and $1$), and $A$ is some symmetric matrix, can be rewritten in linear form as function... |
H: find minimum of given function
today my relative asked a problem,which had strange solution and i am curious, how this solution is get from such kind of equations. let say function has form
$f(x)=a\sin(x)+b\cos(x)$
we should find it's minimum,we have not any constraints or something like this,as i know to find m... |
H: Derivative of norm-infinity of vector
So I know that $\frac{dX}{dX} = \mathbb{I}$ where $X \in \mathbb{R}^n$ and $\mathbb{I} \in \mathbb{R}^{n \times n}$ is the identity matrix.
Now, what is the following derivative?
$\frac{d|X|_\infty}{dX}$
where $|X|_\infty$ is the norm-infinity, i.e. $|X|_\infty = max(X)$ is a ... |
H: How to prove that $C(a)=C(a^3)$ when $|a|=5$
I am having trouble proving the following result:
"Suppose $a$ belongs to a group and $|a|=5$. Prove that $C(a)=C(a^3)$."
($C(a)$ denotes the centralizer of $a$)
The typical way to do this would be to show that $C(a) \subseteq C(a^3)$ and $C(a^3) \subseteq C(a)$. The f... |
H: Solution to $y'' - 2y = 2\tan^3x$
I'm struggling with this nonhomogeneous second order differential equation
$$y'' - 2y = 2\tan^3x$$
I assumed that the form of the solution would be $A\tan^3x$ where A was some constant, but this results in a mess when solving. The back of the book reports that the solution is simpl... |
H: Existence of prequantization on a simply connected manifold
Let $M$ be a simply connected manifold. Then when, $M$ has a unique pre-quantization and when there is no pre-quantization on $M$.
AI: The following holds for prequantizations of arbitrary symplectic manifolds:
Existence: A prequantization of $(M, \omega)... |
H: Show that the theory of densely linear orders and the theory of discrete linear orders are incompatible
I'm trying to prove that the theory of dense linear orders and the theory of discrete linear orders are incompatible by showing that their union is inconsistent. Does anyone know how to do this? Thank you for you... |
H: Parseval's Theorem Proof
Parseval's Theorem states that: If $$f(x)=\sum^\infty_{-\infty}c_ne^{inx}, g(x)=\sum^\infty_{-\infty}\alpha_ne^{inx}.$$ Then, $$\lim_{N\to\infty}\frac{1}{2\pi}\int^\pi_{-\pi}|f(x)-s_N(f;x)|^2dx=0,\frac{1}{2\pi}\int^\pi_{-\pi}f(x)g(x)dx=\sum^\infty_{-\infty}c_n\alpha_n.$$
where $f$ and $g$ ... |
H: In how many ways I can put $2$ red balls and $3$ green balls in $5$ boxes?
I have $5(N)$ boxes and some balls. Here's the description:
Red Balls $= 2 (k1 = 2)$
Green Balls $= 3 (k2 = 3)$
--------------------------
| | | | | |
--------------------------
How many ways do I have to put in t... |
H: About writing a countable family of sets in terms of pairwise disjoint sets
I was wondering if the following statement is true:
Let $\mathfrak{a}$ be a countable collection of sets. Does this imply that there is a countable collection $\mathfrak{b}$ of pairwise disjoint sets such that every set in $\mathfrak{a}$ is... |
H: Smallest $N$ for which we can guarantee the approximation error of an alternating series
What is the smallest value $N$ for which we can guarantee that the approximation error of the alternating series
$$S=\sum_{n=1}^\infty\frac{(-1)^n}{n^{7/2}}$$
by the partial sum, $$S_N=\sum_{n=1}^N\frac{(-1)^n}{n^{7/2}}$$ is at... |
H: Consider the series $\sum_{n=0}^\infty \frac{(x-3)^n}{3^n \sqrt{n+1}}$
$$\sum_{n=0}^\infty \frac{(x-3)^n}{3^n \sqrt{n+1}}$$
Its interval of convergence is of one of the forms $(a,b)$, $(a,b]$, $[a,b)$ or $[a,b]$.
What is $a$?
What is $b$?
I know the interval on convergence is $|x-3|<3$ but i am not sure how to ch... |
H: Quadratic Operator Notation?
I am dealing with functions that are linear combinations of:
$[x_1^2, x_2^2... x_n^2, x_1x_2, x_1x_3... x_n-1x_n]$
spanned over a column.
All these functions obey the law:
$F(aX) = a^2F(X)$ for constant values a.
Is there a notation for handling these? Similar to matrix notation for lin... |
H: Prove that $B$ is a basis of $R_n$ iff $\mathbf A$ is invertible
Let $A \in \mathbf M_{n\times n}(R)$ and let $\{v_1, \ldots, v_n\}$ be a basis of $R_n$.
Prove that $B = \{\mathbf Av_1,...,\mathbf Av_n\}$ is a basis of $R_n$ if and only if $A$ is invertible.
My idea is that $B$ is basis so $B$ is linearly indep... |
H: A strengthening of Raabe's test: $\sum a_n$ diverges if $\frac{a_{n+1}}{a_n} \geq 1 - \frac{1}{n} - \frac{A}{n^2}$ for $A>0$
The usual form of Raabe's test says that if $a_n>0$ and if for large $n$, $\frac{a_{n+1}}{a_n}\leq 1-\frac{p}{n}$ for $p>1$, then $\sum a_n < \infty$. A proof I've seen of this relies on the ... |
H: Prove that the number of ways to put $n$ distinct balls into $n$ distinct boxes is $n^n$
Prove that the number of ways to put $n$ distinct balls into $n$ distinct boxes is
$n^n$
The only approach I've come up with is to start with $1$ ball in each box, count the permutations, then take a ball out of one of the ... |
H: Partial fractions to integrate$\int \frac{4x^2 -20}{(2x+5)^3}dx$
$$\int \frac{4x^2 -20}{(2x+5)^3}dx$$
I can't use the coverup method that I learned since making anything zero in this makes everything zero. I would probably just use random test points because I don't have any other tricks memorized. Is there some sp... |
H: Solving for the integrating factor in a Linear Equation with Variable Coefficients
So I am studying Diff Eq and I'm looking through the following example.
Solve the following equation:
$(dy/dt)+2y=3 \rightarrow μ(t)*(dy/dt)+2*μ(t)*y=3*μ(t) \rightarrow (dμ(t)/dt)=2*μ(t) \rightarrow (dμ(t)/dt)/μ(t)=2 \rightarrow
(d/... |
H: I feel very stupid. Will someone walk me through a step-by-step in plain english of this Big-O problem?
Prove that $n^2 + 2n + 3$ is $O(n^2)$. Find values for $C$ and $k$ that prove that they work.
Edit: In particular, I don't at all understand how to find C and k.
I asked a similar question but every response we... |
H: What is the pattern in this series: $\frac{r}{2} + \frac{4r^2}{9} + \frac{9r^3}{28} + \frac{16r^4}{65} + \dotsb$
A book gave me the following series, and asked for which $r\in \mathbb{R}$ does it converge:
$$\frac{r}{2} + \frac{4r^2}{9} + \frac{9r^3}{28} + \frac{16r^4}{65} + \dotsb$$
I feel dumb because I can't eve... |
H: Optimizing $x^2+y^2$ from two given equations?
What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to:
$$2x^2+5xy+3y^2=2$$ and
$$6x^2+8xy+4y^2=3$$
Note: Calculus is not allowed. I tried everything I could but whenever I got for example or $x^2+y^2=f(y)$ or $f(x)$ the function $f$ would always be a ... |
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