text stringlengths 83 79.5k |
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H: What is the formula of the sequence? And how to deduce the formula?
Does this sequence have a formula?
66 90 117 150 195 264 360 450 540 690 870 1,020 1,260 1,500 1,830 2,160 2,580 3,000 3,510 4,080 4,770 5,490
If it has, please tell me how to find a formula for this kind of sequence, are there any general ways?
t... |
H: Equivalent of $ u_{n}=\sum_{k=1}^n (-1)^k\sqrt{k}$
I'm trying to show that $$ u_{n}=\sum_{k=1}^n (-1)^k\sqrt{k}\sim_{n\rightarrow \infty} (-1)^n\frac{\sqrt{n}}{2}$$ when $n\rightarrow\infty$
How can I first show that $$u_{2n}\sim_{n\rightarrow \infty} \frac{\sqrt{2n}}{2}$$ and then deduce the equivalent of $u_{n}$?... |
H: Joint distribution gives two marginal
In the following exercise I got two different distributions for $Z.$ I want to know where my mistake is. Every hint or comment is appreciated.
The exercise goes as follows:
Let $(X,Y)$ be a random vector with values in $\mathbb{R}^2$ such that it has a joint density given by:... |
H: Residue theorem
The residue theorem
Let $\Omega\subseteq \mathbb{C}$ open, $f$ meromorphic on $\Omega$ and $A$ be the set of the poles of $f$. If $\Gamma$ is a cycle in $\Omega\backslash A$ with $\mathrm{ind}_{\Gamma}(\alpha)=0$ for $\alpha\in \mathbb{C}\backslash \Omega$, we have:
$$
\frac{1}{2\pi i}\int_{\Gamma}f... |
H: Poisson summation formula and Schwartz functions
I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on $\mathbb{R}$ (that is, a smooth function with all derivatives of $f(x)... |
H: Baire sigma-algebra
Halo everyone. I would like to enquire how do I solve this question which I extract from Cohn book on Measure Theory.
Let $X$ be a compact Hausdorff space, and let $C(X)$be the set of all real-valued continuous funtions on $X$. Then $B_{o}(X)$, the Baire $\sigma$-algebra on $X$ is the smallest ... |
H: Can one construct a "Cayley diagram" that lacks only an inverse?
My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct such a diagram?
Nathan Carter p. 24 Question 2.15. in ... |
H: $t > 2n^2 \implies t!>n^t$ for $n,t \in \mathbb{N}$
I have come across this in a proof:
If $t>2n^2$ then,
$$t!>(n^2)^{t-n^2}=n^tn^{t-2n^2}>n^t$$
Obviously, this is much help to determine the relationship between factorials and exponential, but I fail to see the motivation behind the initial assumption.
Is there a ... |
H: What are operators, commutators and anti commutators algebra?
What is the proof for the fact that the product of two operators is generally not commutative?
$$\hat A\hat {\vphantom{A}B}\not=\hat{\vphantom{A}B} \hat A.$$
What is the difference between $\hat A\hat {\vphantom{A}B}$ and $\hat {\vphantom{A}B} \hat A$?... |
H: How to Prove ($\mathbb{C}\langle x, y \rangle$, $\|\cdot\|$) is a Banach Space
Let $\mathbb{C}\langle x,y\rangle$ be the group ring of the complex numbers over the free group in $x,y$. Let $len : \langle x,y \rangle \rightarrow \mathbb{N}$ denote the standard word norm and let $\varphi=exp \circ len: \langle x, y \... |
H: Undefined natural logarithm?
The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental constant.
$$e=\lim_{n\to \infty}\left(1+\frac {1}{n}\right)^n.$$
$$\ln a=\log_{e} a.$$
I know that $\ln {(AB)}=\ln {(A}) + \ln {(B)}$ and $\ln {(A^B)}=B \ln {(A)}$.
Is there any differ... |
H: response of unit step input in harmonically oscillating system
As far as I've understood or misunderstood in constant coefficient second order differential equation
$$\frac{d^2y}{dt^2} + b \frac{dy}{dt} +cy = ef(t)$$
$b$, $c$ being constants,
$f(t)$ the input to the system,
$y$ being response of the system.
Let, $... |
H: Tensor operation on a vector space
From the various definitions provided in the article https://en.wikipedia.org/wiki/Tensor, the tensor seems always to be defined, even in the more abstract forms, as a multilinear map, from a product of vector and dual spaces to the underlying field.
However, in applied mathematic... |
H: Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$
I read on Wikipedia that
$$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$
to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical background?
AI: It is a well known coincidence that
$$e^{\pi}-\... |
H: An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension
I'm having a hard time showing this:
If $K$ is an extension of $\mathbb{Q}$ with degree $m$ and $f(x)$ an irreducible polynomial over the rationals with degree $n$, such that $\gcd(m, n)=1$, then $f(x)$ is ... |
H: For System dependent on normally distributed parameter, are deviations added or variations?
Say, A and B are two normally distributed parameters with their variations being $\sigma^2_a$ and $\sigma^2_b$. Now for system C, which is linearly dependent on these parameters, is its $\sigma^2_c=\sigma^2_a+\sigma^2_b$, or... |
H: isomorphism or bijection?
I have a little problem and confusion. I will write something that is maybe wrong.
Let $\alpha$ and $\beta$ two ordinals and $f$ a bijection between $\alpha$ and $\beta$. $f$ is not necessary a isomorphism between $(\alpha,<)$ and $(\beta,<)$ (if it is then we have $\alpha=\beta$). But $f$... |
H: Primes of the form $n\pm k$
Given some arbitrary natural number $n$, can we always find a $k$ such that $n+k$ and $n-k$ are both prime? Has there been any work on finding an upper bound for $k$?
AI: Being able to find such a $k$ for any $n$ is equivalent to the Goldbach conjecture, since it would imply that any eve... |
H: Simple laplace transform
I am trying to find the laplace transform of this equation:
$$4-4t+2t^2$$
What I am doing:
$$\frac{4}{s}-\frac{4}{s^2}+\frac{4}{s^3}$$
$$\frac{4s^2-4s}{s^3}+\frac{4}{s^3}$$
$$\frac{4s^2-4s+4}{s3}$$
But I am getting the wrong answer, can you please tell me what I am doing wrong?
AI: Using th... |
H: A trigonometric series
Let $\alpha$ be a real number. I'm asked to discuss the convergence of the series
$$
\sum_{k=1}^{\infty} \frac{\sin{(kx)}}{k^\alpha}
$$
where $x \in [0,2\pi]$.
Well, I show you what I've done:
if $\alpha \le 0$ the series cannot converge (its general term does not converge to $0$ when $k \... |
H: Theorem of liouville
Consider two entire functions with no zeroes and having a ratio equal to unity at infinity. Use Liouville’s Theorem to show that they are in fact the same function.
My attempt
Consider $h(z) = f(z)/g(z)$.
First of all, $h$ is entire, since $f$ and $g$ are entire, and $g(z)$ is nonzero for all $... |
H: Evaluating $ \lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$
Evaluating
$$L = \lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$$
AI: Marvis showed that
$$
I_n = \int_0^1 \dfrac{nx^{n-2}}{x^{2n} + x^n + 1} dx = \int_0^1 \dfrac{dt}{t^{1/n}(t^2 ... |
H: Laplace transform of multiplication of two terms
I have the following expression to get its laplace transfer:
$$e^{2t}(3t-3t^2)$$
Is it ok to just calculate the transfer of each term then multiply the result? I calculated the expression above like this but it is different than the answer in my book:
$${\frac{1}{s-2... |
H: Recommendations for probability books
i do IT work, and the "it" thing these days is to throw the occasional probability question out there. The last time i stumbled on this, i'd just sat the GMAT and had probability somewhat down... still, it is the one region in maths that has always left me the most confounded.
... |
H: Explicit expression for eigenpairs of Laplace-Beltrami operator
In $R^n$, the Laplace-Beltrami operator is just the Laplacian, and its eigenstructure is well known. There are also explicit expressions for the eigenvalues/eigenvectors of the Laplace-Beltrami operator on the sphere.
Question: Are there any other no... |
H: Coin Toss Probability Question (Feller)
I'm working out of Feller's "Introduction to Probability and its Application (Vol I.)" textbook and I'm stuck on a coin toss problem. I'll list the full problem and show where I'm having trouble.
A coin is tossed until for the first time the same result appears twice in succ... |
H: Partial derivative of trace of an inverse matrix
I have the following vector function $f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$ where $\operatorname{diag}(\mathbf{x})$ is the diagonal matrix with values from $n\times 1$ vector $\mathbf{x}$ on the diagonal, and $\mathbf{A}$... |
H: Help Understanding Proof of Replacement Theorem?
Sorry if this is a trivial question.
The book is Linear Algebra Done Right by Axler, page 25-26.
Theorem: In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of ve... |
H: Orientation and simplicial homology
I'm reading Chapter 2 of Hatcher's Algebraic Topology, and I just can't figure out the computations of the boundary homomorphism for the examples provided. To provide some context, reproduced the figure for the torus from the book below:
As I understand it, to compute $\partial ... |
H: If $K = \mathbb{F}_p(\alpha)$ where $\alpha^n \in \mathbb{F}_p$ and $n$ is the minimal such $n$. Does this imply that $[K : \mathbb{F}_p] = n$?
If $K = \mathbb{F}_p(\alpha)$ where $\alpha^n \in \mathbb{F}_p$ and $n$ is the minimal such $n$. Does this imply that $[K : \mathbb{F}_p] = n$?
If not, is there a condition... |
H: Evaluating $\int \frac{x+4}{x^2 + 2x + 5}dx$
I have not encountered a problem like this before.
$$\int \frac{x+4}{x^2 + 2x + 5}dx$$
I do not know how to factor the bottom so I am not sure what to do.
AI: Note that $(x^2+2x+5)' = 2x+2$. So we can break up the integral into two integrals, one that can be solved by su... |
H: Cute Determinant Question
I stumbled across the following problem and found it cute.
Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant:
$$\left|
\begin{matrix}
2 & 3&0&2&8 \\
3 & 1&8&8&2\\
8&6&4&6&9\\
0&6&3&2&7\\
6&1&9&0&2... |
H: Existence of the Pfaffian?
Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes.
If $n$ is even, my book claims that the determinant is the square of a polynomial function of the entries, and Wikipedia confirms this. The ... |
H: Why does $\mathrm{ord}_p(n!)=\sum_{i=1}^k a_i(1+p+\cdots+p^{i-1})$?.
Suppose
$$
n=a_0+a_1p+\cdots+a_kp^k\qquad 0\leq a_i<p
$$
is the base $p$ (for $p$ a prime) representation of an integer $n$. I'm trying to prove to myself that
$$
ord_p(n!)=\sum_{i=1}^k a_i(1+p+\cdots+p^{i-1}).
$$
I know the formula that $\displa... |
H: Identification of $T_v(T_pM)$ with $T_pM$
In some passages of Do Carmo's Riemannian geometry book he identify $T_v(T_pM)$ with $T_pM$, my question: How one see $T_pM$ as a manifold? who is the atlas? What is the expression of a vector $x \in T_v(T_pM)$ in local coordinates?
AI: Since $T_pM$ is a real vector space, ... |
H: Measure-theoretic view of expectation of a Bernoulli sequence
Problem:
I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of measure theory to a tangible concept. I have been working wi... |
H: Find $f$ such that the following integral equation is satisfied.
Find $f$ such that the following integral equation is satisfied:
$$\int_0^x \lambda f(\lambda) ~d\lambda= \int_x^0(\lambda^2 + 1)f(\lambda) ~d\lambda + x$$
I attempted it in the following way:
$\int_0^x \lambda f(\lambda) ~d\lambda= -\int_0^x(\lam... |
H: Block Determinants
This is a nice question I recently found in Golan's book.
Problem: Let $A,B,C,D$ be $n\times n$ matrices over $\mathbb{R}$ with $n\ge 2$, and let $M$ be the $2n\times 2n$ matrix \begin{bmatrix}
A & B \\
C & D\\
\end{bmatrix}
If all of the "formal determinants" $AD-BC$, $AD-CB$, $DA-CB$, and ... |
H: How to calculate these summations?
How to find the values of these kind of summations:
$$\large\sum_{i=0}^6(6-i)\;\ast\;\sum_{j=1}^6(7-j)\;\ast\;\sum_{k=2}^7(8-k)\;\ast\;\sum_{\ell=3}^8(9-\ell)$$
AI: Use that $$\begin{align}\sum_{t=a}^b(c-t)&=\left(\sum_{t=a}^bc\right)-\left(\sum_{t=a}^b t\right)\\\\&=\left(\sum_{... |
H: $\{1,1\}=\{1\}$, origin of this convention
Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements?
Like $\{1,1,2,3\} = \{1,2,3\}$.
I have looked over a few books and it didn't mention such thing. (Wikip... |
H: A formula with only one $0$ that evaluates to a given integer
Just found this math puzzle online:
Original puzzle: Using one zero (and no other numbers) and
mathematical (including trigonometric and advanced) operations, how
can you achieve a final result of 6?
Advanced one: Using only trigonometric functions ... |
H: Lagrange's method to find min/max question
I have a doubt when using this method to find min/max of a function. When I can find two (or more solutions) of system equation, so I can easily know it has max and min.
But, the problem is: if the system equation I solve just have only one solution. So, I can know it max ... |
H: If $X$ has an exponential distribution, prove the hazard function is constant
$X$ has an exponential distribution, $Pr(X>0)=1$, p.d.f is $f$, c.d.f is $F$.
$h(x)=\frac{f(x)}{1-F(x)}$ for $x>0$.
Prove that $h(x)$ is constant for $x>0$.
AI: We have $f(x)=\lambda e^{-\lambda x}$ (for $x\ge 0$). For the cumulative dens... |
H: How do I solve a certain characteristic system?
I am studying PDEs and have the following (seemingly simple) problem:
Find a surface that passes through the curve $$x^2+y^2=z=1$$ and is orthogonal to the family of surfaces $$z(x+y)=c(3z+1)\qquad(c\in\Bbb R)$$
After writing down the orthogonality condition (assumi... |
H: Combinatorics: endless series
I have the following problem:
In an urn, you have 1 blue and 9 white balls. You pull out one ball a time; if it is the blue one, you win. If it it is white, you throw it back in and pull again. Imagine 2 people are playing this game. Who has a better chance to win, the person who goes... |
H: Proving an interesting feature of any $1000$ different numbers chosen from $\{1, 2, \dots,1997\}$
Assume you choose $1000$ different numbers from the group $\{1, 2,
\dots,1997\}$.
Prove that within the $1000$ chosen numbers, there is a couple which
sum is $1998$.
I defined:
pigeonholes: possible sums.
pigeons:... |
H: Triangles inside a square
I have a question with a figure of Triangle inside a square. The base of the triangle is on the base of the square and the peak of the triangle touches the top of the square.It then asks the ratio of the area of triangle to the area of the square.
According to the book the answer is 1/2.... |
H: Examples of infinite groups such that all their respective elements are of finite order.
I am in need of examples of infinite groups such that all their respective elements are of finite order.
AI: Here is one. Let $(\mathbb{Q},+)$ denote the groups of rational numbers under addition, and consider it's subgroup $(\... |
H: Laplace transform of a product
I tried to solve the product below:
$$3t\sin(6t)$$
but it seems that getting the transform of each and multiply the result is not leading to a correct answer:
$$\frac{3}{s^2}\frac{6}{s^2+36}$$
How does one solve such transforms?
AI: We know that if $ L(f(t))=F(s)$ so $ L(t.f(t))=-F’(s... |
H: Explain a statement about math induction base.
I was reading an article in wikipedia about math induction:
http://en.wikipedia.org/wiki/Mathematical_induction
And there is a sentence:
"Note that the first quantifier in the axiom ranges over predicates rather than over individual numbers."
It is told about the axiom... |
H: If $\gcd(a,35)=1$ then show that $a^{12} \equiv 1 \pmod{35}$
If $\gcd(a,35)=1$, then show that $a^{12}\equiv1\pmod{35}$
I have tried this problem beginning with $a^6 \equiv 1 \pmod{7}$ and $a^4 \equiv 1 \pmod{5}$ (Fermat's Theorem) but couldn't get far enough. Please help.
AI: Since $\gcd(a,7) =\gcd(a,5) = 1$, f... |
H: What is a fast way to evaluate:$\int P(x)e^{ax}dx$
Undoubtedly, this question is so easy but I'd like to ask it. We know that the way in which the indefinite integrals like $\int P(x)e^{ax}dx$ and $\int P(x)\sin(bx)dx$ wherein $P(x)$ is an arbitrary polynomial of $x$ and $a, b\in \mathbb R$ are evaluated is Integra... |
H: Cohomology of sheaves : reference-request
I need a good reference book where I can learn the cohomology of sheaves through the approach of Čech cohomology. The Hartshorne's book, for example, doesn't help me a lot because he choose the "derived functors approach".
AI: The number one account is still in Serre's leg... |
H: Epsilon-delta proof of continuity
Prove that $f(x,y,z)=x^4+y^4+z^4$ is continuous on point $(x,y,z)=(0,0,0)$ with epsilon-delta
I prove this so:
if $$\lim_{x,y,z \to 0,0,0} f(x,y,z) = f(0,0,0)$$ then that function is continuous
$$\lim_{x,y,z \to 0,0,0} x^4+y^4+z^4 = 0^4+0^4+0^4=0$$
But how to prove this with $\e... |
H: Continuous root map of the coefficients of a polynomial
I have a set of polynomials $P_t(z)= z^n+ a_{n-1}(t)z^{n-1}+\cdots+ a_0(t)$ which depends on a real parameter $t \in [a,b]$ and where $a_{n-1}(t),\ldots, a_0(t)$ are real continuous functions.
May I say that there exists a continuous map $\theta(t)$ such that... |
H: What does the following statement mean?
If $\mu(A) <\infty $, then from almost everywhere convergence follows the convergence in the measure .
I don't understand what the "convergence in the measure" means. Waiting for your explanation.
I am trying to understand proposition 2 from the following link :
http://med... |
H: $p$ an prime number of the form $p=2^m+1$. Prove: if $(\frac{a}{p})=-1$ so $a$ is a primitive root modulo $p$
Let $p$ be an odd prime number of the form $p=2^m+1$.
I'd like your help proving that if $a$ is an integer such that $(\frac{a}{p})=-1$, then $a$ is a primitive root modulo $p$.
If $a$ is not primitive roo... |
H: what does that statement mean about the relation?
what does this mean about P?
$$\forall x \exists y (p(x,y) \rightarrow p(y,x)) $$
i know that
$$\forall x \forall y (p(x,y) \rightarrow p(y,x)) $$
means that P symmetric
but what does the first statement means?
and what does the last statement mean about the rela... |
H: Simplify this expression with nested radical signs
My question is-
Simplify:
$$\frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}$$
AI: $$
\begin{align} & {}\quad \frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}\\[10pt]
& =\frac {1}{\sqrt{ 12-2... |
H: How to formally justify that $\int o(x) \, dx\sim o(x^2)$?
I'm trying to evaluate the following limit:
$$\lim_{x\to 0}\frac{\sin\left(\int_{x^3}^{x^2}\Bigg(\int_0^t g(s^2) \, ds\right) \, dt\Bigg)}{x^8}$$
for $g:[-1,1]\to\mathbb{R}$ differentiable function such that $g(0)=0$, $g'(0)=1$.
I developed $g$'s taylor exp... |
H: $m!n! < (m+n)!$ Proof?
Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$
Given hint:
$m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$
It looks simple but I'm baffled and can't reach the correct proof.
Thank you.
AI: Notice that $m!n!$ and $(m+n)!$ both have the sa... |
H: There exists a unique isomorphism $M \otimes N \to N \otimes M$
I want to show that there is a unique isomorphism $M \otimes N \to N \otimes M$ such that $x\otimes y\mapsto y\otimes x$. (Prop. 2.14, i), Atiyah-Macdonald)
My proof idea is to take a bilinear $f: M \times N \to N \otimes M$ and then use the universal ... |
H: Asymptotics for sum of binomial coefficients from Concrete Mathematics
Concrete Mathematics EXERCISE 9.25:
Supposing
\[ S_n = \sum_{k=0}^n \binom{3n}k \]
Prove that
\[ S_n = \binom{3n}{n}\left(2-\frac4n+O\left(\frac1{n^2}\right)\right) \]
This sequence also appears in OEIS A066380
I have been trying to unde... |
H: Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
How would you prove or disprove that the function given by
$$f(x,y) = \begin{cases} \dfrac{x^2y}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$
is continuous at $(0,0$)?
AI: Observe that
$$
\left| ... |
H: Area of Square - Comparing squares
The question is:
If the area of a parallelogram $JKLM$ is $n$ and if length of $KN$ is $n+(1/n)$, then find the length of $JM$. (The answer is $n^2 /( n^2+1 )$.)
How would i go about solving this problem ?
AI: The area of a parallelogram (or see on Wikipedia) is the base time... |
H: Evaluating $\int_{0}^{1} \frac{x^{2} + 1}{x^{4} + 1 } \ dx$
How do I evaluate $$\int_{0}^{1} \frac{x^{2} + 1}{x^{4} + 1 } \ dx$$
I tried using substitution but I am getting stuck. If there was $x^3$ term in the numerator, then this would have been easy, but this one doesn't.
AI: Hints:
Try dividing the numerator ... |
H: Wedge product of 1-Forms
I'm trying to write down the wedge product of 2 1-forms on an n-dimensional Manifold.
$\alpha = \alpha_1 dx^1 + \alpha_2 dx^2 + \cdots + \alpha_n dx^n$
and
$\beta = \beta_1 dx^1 + \beta_2 dx^2 + \cdots + \beta_n dx^n$
I know how to do this for the 2 and 3 dimensional case. But I'm having a ... |
H: Confusion about unique isomorphism $M \otimes N \to N \otimes M$
This is a follow up question to my previous question here.
I'm confused about the following: in Atiyah-Macdonald they state that there exists a unique isomorphism $M \otimes N \to N \otimes M$, $m \otimes n \mapsto n \otimes m$.
I'm not sure why AM wr... |
H: Prove that $4^{2n} + 10n -1$ is a multiple of 25
Prove that if $n$ is a positive integer then $4^{2n} + 10n - 1$ is a multiple of $25$
I see that proof by induction would be the logical thing here so I start with trying $n=1$ and it is fine. Then assume statement is true and substitute $n$ by $n+1$ so I have the f... |
H: Finding the maximal value of a function on a ellipse
How would you find the maximal value of $$f(x,y) = x - y^2$$ on $K = \left\{ (x,y) : \frac{x^2}{4} + \frac{y^2}{9} = 1 \right\}$?
AI: Because $\frac{x^2}4 + \frac{y^2}9 = 1$, we know $y^2 = 9\left(1 - \frac{x^2}4\right)$. Substitute this into $x-y^2$ should give ... |
H: Prove $\binom{p-1}{k} \equiv (-1)^k\pmod p$
Prove that if $p$ is an odd prime and $k$ is an integer satisfying $1\leq k \leq p-1$,then the binomial coefficient
$$\binom{p-1}{k} \equiv (-1)^k\pmod p$$
I have tried basic things like expanding the left hand side to $\frac{(p-1)(p-2).........(p-k)}{k!}$ but could... |
H: Determinant called Grammian
Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if
$$ \det\left( \begin{array}{ccccc} f_1 & f_2 & f_3 &… &f_n \\ f'_1 & f'_2 & f'_3 &... &f'_n \\ ⋮ & ⋮ & ⋮ &⋮ &⋮ \\ f_1^{(n-1)} & f_2^{(n-1)} & f_3... |
H: Simplify these expressions with radical sign 2
My question is
1) Rationalize the denominator:
$$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$
My answer is:
$$\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{18}$$
My question is
2) $$\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$$
My answer is: $$\... |
H: Why is the following map well defined?
Let $H\leq G=\operatorname{Gal}(K/F)$ ($K/F$ is a finite galois extension), why is the following map well defined:
$\varphi:G/H\to\Gamma_F(K^H,K)$ defined by
$\sigma H\mapsto\sigma|_{K^H}$ ,where $\Gamma_{F}(K^H,K)$ denotes all homomorphisms from $K^H$ to $K$ that fixes $F$.
... |
H: Most Probable Sum
Possible Duplicate:
Probability of dice sum just greater than 100
A fair dice is rolled and the outcome of the face is summed up each time. We stop rolling when the sum becomes greater than 100. Which of the following is most probable sum?
103
102
100
101
All have equal probability
How best ... |
H: When does this sequence start repeating itself?
Given the sequence $a_j = b^j \mod q$, where $1 < b, q < 2^n$, how can I prove that the sequence starts repeating itself at some term $a_k$ where $k \leq n$?
I have been looking at this problem for hours and am completely stuck on how to do it :/ Any help would be app... |
H: How to prove $641|2^{32}+1$
Possible Duplicate:
To show that Fermat number $F_{5}$ is divisible by $641$.
How to prove that $641$ divides $2^{32}+1$? What the technical way will be for this question? I want to teach it to my students. Any help. :-)
AI: In light of Peter's comment:
we have:
$2^2=4$,
$2^4=16, 2^8=... |
H: Additional insights when converting sums to products
Given some sum, $$\displaystyle\sum \ln x_i = k $$ We have $$\ln \prod x_i = k$$
I've always found this relation to be really interesting. I saw it used once in a linear algebra proof but I haven't seen it since. Are there any other interesting uses of this tr... |
H: Cardinality of the set of ultrafilters on an infinite Boolean algebra
Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
AI: It can be at least those two options:
Example I:
Consider the algebra $\{A\sub... |
H: Question on uniform intergrability
Consider a probability measure $m$ over $W \subseteq{R^m}$, so that $m(W) = 1$.
Consider a function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with compact $X \subset \mathbb{R}^n$, such that the following proposition holds true.
For any $\epsilon > 0$ there exists $c > 0$ s... |
H: Primitive element theorem - why any finite and separable extension is simple
I have it in my lectures notes that the claim:
Let $K/F$ be a finite and separable extension then $K$ is a simple extension of $F$ follows immediately from the theorom : Let $K/F$ be a finite extension, then it is simple iff $K/F$ have a f... |
H: Why can't a model "say" of itself that it is countable?
Why can't a (standard?) model of ZFC "say of itself" that it is countable?
That is, why is there no bijection $f$ ∈ between and $\omega^$?
(I've read that it fails regularity, or even without regularity we get Cantor's paradox. But a direct answer to the qu... |
H: Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?
I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are probably other ways to do ... |
H: Determining all Sylow $p$-subgroups of $S_n$ up to isomorphism?
I'm trying to understand a classification of all Sylow $p$ subgroups of $S_n$.
Let $Z_p$ be the subgroup of $S_p$ generated by $(12\cdots p)$. Then $Z_p\wr Z_p$ has order $p^p\cdot p=p^{p+1}$, and is isomorphic to a subgroup of $S_{p^2}$.
Define indu... |
H: Finding what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to
I am doing an exercise that asks me to find what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to.
I am allowed to only use the groups $D_n,S_n,\mathbb{Z}_n$ and the direct sums ) where $S_n$ is the permutatin gr... |
H: Evaluating a sum to infinity
I'm looking for a way that allows me to work out the following sum:
$$\sum\limits_{k=1}^{\infty} \sin^2\left(\frac{1}{k}\right)$$
Any hint/suggestion is welcome. Thanks.
AI: It may be too much to ask for a closed form.
We find an equivalent series that converges very fast.
We have
$$... |
H: Prove that the intersection of all subfields of the reals is the rationals
I'm reading through Abstract Algebra by Hungerford and he makes the remark that the intersection of all subfields of the real numbers is the rational numbers.
Despite considerable deliberation, I'm unsure of the steps to take to show that th... |
H: Comparison Test about the series $ \sum_{n=1}^\infty \frac{a^n}{n^b} $
When does this series converge?$$ \sum_{n=1}^\infty \frac{a^n}{n^b} $$
I want to know the condition of a and b.
AI: Hint: I assume we are working over the reals.
For $|a|\lt 1$, use Ratio Test.
For $|a|\gt 1$, terms don't go to $0$, or Ratio Tes... |
H: To define a measure, is it sufficient to define how to integrate continuous function?
Let me make my question clear. I want to define a measure $\mu$ on a space $X$. But instead of telling you what value I assign for some subset of $X$ (measurable sets that form a $\sigma$-algebra), I tell you that for each $f$ con... |
H: Reductions for regular languages?
To reason about whether a language is R, RE, or co-RE, we can use many-one reductions to show how the difficulty (R, RE, or co-RE-ness) of one language influences the difficulty of another. To reason about whether a language is in P, NP, or co-NP, we can use polynomial-time many-o... |
H: I can't find differences between $P(1+r)^n$ and $P(2.71828)^{rn}$
They told me $P(1 + r)^n$ can be used to calculate money interest, example:
You invert $15,000.00, 20% interest annual, for 3 years:
$15,000(1 + 0.20)^3$ = 25,920
And that $P(2.71828)^{rn}$ can be used to calculate population growth, example:
We h... |
H: Condition for frame of $L_2$
Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions
$$
f_{m,k}=f^m(x-2^k)
$$
Under which conditions $\{f_{m,k}\}$ would be a frame?
(A function $f\in L_2(R)$ is said to generate a frame $\{f_{m,k}\}$ of $L_2(\mathbb... |
H: Difference between sample mean and true mean of a gaussian
Assume I have a gaussian distribution $\mathcal{N}(\mu, C)$ with mean $\mu$ and covariance $C$. I'm drawing $n$ random numbers from this distribution. Let $m$ be the mean of these numbers. Is there some formula that gives the probability that the distance $... |
H: Is "algebraic-variety" a relative concept?
Let A, B be two NON-isomorphic finitely generated k-algebras, is it possible they isomorphic as abstract commutative unitary rings? (any concrete examples?)
If the answer to the above is possibly yes, then what assumptions should one add to preserve the NON-isomorphicity?
... |
H: Converting from Spherical to Rectangular
I need to convert $\rho \sin\phi=2\cos\theta$ in to rectangular form.
Attempt: I tried using those nice properties :
$$x=\rho\sin\phi\cos\theta \\y=\rho\sin\phi\sin\theta\\z=\rho\cos\phi$$ and $\rho^2=x^2+y^2+z^2$ and $\cos\phi=\frac{z}{\sqrt{x^2+y^2+z^2}}$. I cannot find ... |
H: Evaluating $\int \sqrt{5 + 4x - x^2}dx$
$$\int \sqrt{5 + 4x - x^2}dx$$
I am pretty certain what I need to do to this problem is complete the square and turn it into a trig subsitution but I have no idea how to complete the square with a $-x^2$ or really with this problem at all, I just can't make it work.
I tried t... |
H: Every $2k$-regular contains a 2-factor
I need to prove that given a graph which is $2k$-regular, I can find a 2-factor. Meaning, There is a sub-graph of the above graph, which contains all vertices, and is 2-regular.
I must say I have no idea where to start with this. so help would be greatly appreciated :)
Thanks.... |
H: Evaluating $\int \sqrt{x^2 + 2x}dx$
$$\int \sqrt{x^2 + 2x}dx$$
I have no clue what to do on this problem. It is in the trig substitution chapter so I know I have to use that somehow. I know that I can not complete the square because both terms are positive and will not give me a difference of squares.
I know u subs... |
H: Family of Self-Adjoint Operators that are Multiplications on a Common $L^2(\mu)$?
Suppose that $H$ is some (complex) Hilbert space and that $\{T_\alpha: \alpha \in I\}$ is some collection of bounded self-adjoint operators on $H$. A version of the spectral theorem states that for each $\alpha$, there exists a measur... |
H: Induction troubles proving a formula.
I'm having a tough time proving the following formula. Suppose $a,b\in R$ a ring. Define $a^{(0)}=a$, $a^{(1)}=[a,b]\equiv ab-ba$, and then $a^{(k)}=[a^{(k-1)},b]$. Then
$$
\sum_{i=0}^k b^iab^{k-i}=\sum_{j=0}^k\binom{k+1}{j+1}b^{k-j}a^{(j)}.
$$
I wanted to do this with inductio... |
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