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H: Line integral about a circle
Here is the question: Evaluate$$\int Pdx+Qdy$$ where $$P(x,y)=\frac{y+x}{x^2+y^2}$$ and $$Q(x,y)=\frac{y-x}{x^2+y^2}$$ about the circle $$C: x^2+y^2=a$$ oriented clockwise. I tried finding $P_y$ and $Q_x$ and I got $0$. However, that is not the answer. I know it is because the function ... |
H: Show that the tangent plan pass through the origin
Show that all the tangent plans to the conic surface $z = xf(\frac{y}{x})$ at the point $M(x_o,y_o,z_o)$, where $x_o \neq 0$, pass through the origin of the cordinates
First, I've found the tangent plan at this generic point $M$ of the surface:
$z - z_o = (x-x_o)(f... |
H: Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?
I've come across the Fourier transform being defined as:
$$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$
But this convention is not present in the Wikipedia article. The one given there, un... |
H: Convert double integral from cartesian coordiantes to polar coordiantes
I have the integral $$\int_{-3}^3 \int_0^\sqrt{9-x^2} (x^2 + y^2)^{3/2} {dy}{dx}$$
I cannot solve this in it's current form so I realize that the limit is a circle ${x^2} + {y^2} = 9$ using this I attempted to convert the integral to polar coor... |
H: Fundamental group of the topological space obtained by identifying the four vertices of a square
The task is: Compute the fundamental group of the topological space obtained by identifying the four vertices
of a square.
So we identify the vertices with the same letter. Can we say something about the orientation... |
H: Evaluating the surface integral $\iint_\Sigma \mathbf{f} \cdot d \mathbf{a}$ where $\mathbf{f}(x,y,z)=(x^2,xy,z)$
Evaluate the surface integral $\iint_\Sigma \mathbf{f} \cdot d \mathbf{a}$ where $\mathbf{f}(x,y,z)=(x^2,xy,z)$ and $\Sigma$ is the part of the plane $6x+3y+2z=6$ with $x,y,z\geq 0$.
I changed the fun... |
H: Mayer-Vietoris sequence for the figure eight
On my professor's solutions for my last algebraic topology homework, he gets the following Mayer-Vietoris sequence for the figure eight space (the wedge of two circles):
$0\to H_{2}(X)\to 0\to \mathbb{Z}\oplus \mathbb {Z}\to H_{1}(X)\to \mathbb{Z}\overset{\varphi_{*}}{\t... |
H: How would I find the residue of $\text{sech}$ and $\coth$ at their poles?
I thought I had understood this, but I'm now lost when trig. functions are introduced and I don't know how to continue. I attempted to apply the $\lim_{z \to a} (z-a)f(z)$ on it, but that didn't take me far.
AI: The residues of $\coth$ are si... |
H: Build regular grammar from regular expression
Is there an algorithm for creating a regular grammar directly from a regular expression? All the discussions and notes I found so far go through an intermediary step of creating an FA for the reg ex and then the regular grammar from the FA. E.g., for a reg ex $a(b|c)*d$... |
H: How to show that the set of three primes whose sum is a fixed integer is an integral?
Let $I_F = \int_0^1 F(\alpha)^3 e^{-\alpha n} d\alpha$, where $F(\alpha)=\sum_{p\leq n} e^{\alpha p}$, $n$ is an integer. It is said that $I_F$ is the number of $(p_1, p_2, p_3)$ such that $p_1, p_2, p_3$ are primes and $p_1+p_2+p... |
H: $\frac{1}{n}\sum_{i=1}^nZ_i \rightarrow \int_0^1f(x)dx$
PROBLEM
Let $X_1,Y_1,X_2,Y_2,...$ be a sequence of independent random variables, all of which distributed uniformly on $[0,1]$. Let $f: [0,1] \rightarrow [0,1]$ be a continuous function. Define $Z_i = 1_{f(X_i)>Y_i}$.
$a)$ Show that almost surely $\frac{1}{n}... |
H: Probability density problem
Suppose that on each day that I cycle to work, there is a probability 0.33 that I get wet because of rain. Suppose I cycle to work on 12 days, then what is the probability that I get wet in more than 2 days?
Give your solution accurate to 4 decimal places.
i dont know how to solve the di... |
H: If $f(x) < g(x)$, prove that $\int_a^b f(x) dx < \int_a^b g(x)dx.$
1) Let $f$ and $g$ be Riemann integrable functions on $[a,b]$. Suppose that $f(x) < g(x)$ for each $x\in [a,b]$. Prove that $\int_a^b f(x) dx < \int_a^b g(x)dx.$
Basically my idea was to break the integral down into partitions and show the inequalit... |
H: Proving that $2^n$ is greater than a binomial expression
This is from a friend's textbook. There is a really obvious counting argument, but it is a calculus, not a combinatorics, textbook, and the answer probably involves messing up with algebraic equations.
By considering $(1+x)^n$ for suitable $x$, show $$2^n > ... |
H: Calculation of coefficients of a Fourier series
Calculating the Fourier series of a periodic function I need to evaluate these integrals:
$$1) \int_{-\pi}^{\pi}dt\left(\cos^{-1}(\alpha t-1)+2(1-\alpha t)\sqrt{\frac{1}{2}\alpha t-\frac{1}{4}\alpha^2t^2}\right)\sin(t)$$
$$2) \int_{-\pi}^{\pi}dt\left(\cos^{-1}(\alph... |
H: Does $ \log(x)^{x^a}$ eventually dominate $x^k$?
Does $ \log(x)^{x^a}$ eventually dominate $x^k$ for all $a\gt 0$ and for all positive integers $k$?
And if so, how does one prove this?
Thanks a lot for your help.
AI: $$\begin{align}
\log\bigl((\log x)^{x^a}\bigr)&=x^a\log\log x\\
\log(x^k)&=k\log x
\end{align}$$
Fo... |
H: Predicting data in many dimensions
I have two matrices deriving from one matrix of the original data. One is the training, the other is the validation set. Each matrix has rows= examples, columns = featuers. The proportions are 65% vs 35% respectively.
Given that the data is in many dimensions and it is not possibl... |
H: the probability of rolling a die
There is a die with six faces numbered consecutively from 1 to 6. What is odd about it, is that the probability of rolling the face with number k on it is c*(q^k), where c is a constant, and q = 0.9.
What is the expected value of a roll of the die?
i could not get the constant c fir... |
H: solving an equation by fixed point theorem
This is captured from a chapter talking about completeness of metric space in Real Analysis, Carothers, 1ed. And I have been confused by an application of fixed point theorem:
The definition of fixed point theorem is:
An application of it is showed as following:
Why doe... |
H: Equations with exponents
I can't remember how to solve equations that have exponent and a variable in them. This is somewhat embarrassing, because this used to be really easy for me. I know that logarithms are involved I just can't remember how they are involved. Would anybody be able to help me out? Here is the eq... |
H: Sets and expectations
Imagine two sets $A = \{1, 2, \dots, a\}$ and $B = \{1, 2, 3, \dots, b\}$ with $a \leq b$.
Let $f$ be a uniformly independently distributed random map $f:A\rightarrow B$ and $F = \bigcup_{i=1}^a f_{i}$
If I pick different functions $f$ until I find one such that $|A| = |F|$ what is the expecte... |
H: deducing $\lnot B \implies \lnot A$ from $A \implies B$
One way how to prove a statement of the form $A \implies B$ is to presume that $A$ is true and deduce $B$. Lets have $A \implies B$ and lets assume that $\text{not}~B$ is true. $A$ is true or it is false (duh). If it were true, $B$ would also be true. However,... |
H: weight of heaviest box?
A shipping clerk has five boxes of different but unknown weights each weighing less than 100 kg. The clerk weights the boxes in pairs. The weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121 kg. What is the weight of the heaviest box?
The answer options are given as 60,... |
H: How to compute the Hessian Matrix
I want to compute the Hessian matrix of a 2-dimensional vector function.
\begin{pmatrix}x_1 + x_2 + x_3\\x_2^2 -x_1x_2\end{pmatrix}
Can anyone pleae explain how to compute this since I can find it nowhere..
AI: The hessian matrix of a vector function is a $(1,2)$ tensor whose entri... |
H: What is the focal width of a parabola?
I'm not wondering what the formula is—I already know that. For a parabola in standard form of $(x-h)^2=4p(y-k)$ I know that the focal width is $|4p|$.
But what does that mean, conceptually?
What does that distance, $|4p|$, represent? If I were to graph the parabola, would that... |
H: Proof of $|\int^{b}_{a}fg|^2\leq(\int^{a} _{b}|fg|)^2\leq (\int^{b}_{a}f^2)(\int^{b}_{a}g^2), \forall f,g \in {\mathscr R[a,b]}.$
There is an exercise in an Analysis textbook that requires one to establish the
Cauchy- Schwarz Inequality: $|\int^{b}_{a}fg|^2\leq(\int^{a} _{b}|fg|)^2\leq (\int^{b}_{a}f^2)(\int^{b}_{... |
H: how to solve trigonometric inequalities?
how does one solve trigonometric inequalities? Is there a method to this or is every solution done ad hoc?
simple equations of the type: $cos3x \leq 0$ when: $0\leq x \leq 2π$
The attempt at a solution: equating $cos 3x = 0$ yields $$ π /6 + 2\frac13πk\leq x \leq 2π -π ... |
H: About $n$ consective integers such that each of them is a multiple of the element of a given set
Let $S(a)$ be a set of the multiples of an integer $a$.
Then, here is my question.
Question : Is the following true for any $n\ge 2\in\mathbb N$ ?
Supposing that any two of $n$ integers $a_1, a_2,\cdots,a_n$ are coprim... |
H: For what values of $\gamma > 0$ does $n^{\gamma} (\sqrt[n]{n} - 1)^2$ converge?
This is not for homework, but I would please just like a hint. The question asks
For what values of $\gamma > 0$ does $n^{\gamma} (\sqrt[n]{n} - 1)^2$ converge?
I did a couple of tests, and believe that $n^{\gamma} (\sqrt[n]{n} - 1)^... |
H: prove that $a_n$ is convergent if $\limsup a_n \cdot \limsup \frac1{a_n} = 1$
$a_n$ is a positive series, and I know that $\limsup a_n \cdot \limsup \frac1{a_n} = 1$.
Prove that $a_n$ is convergent.
What do I need to do?
AI: Can you show that $$\limsup\frac1{a_n}=\frac1{\liminf a_n},$$
(for any sequence $(a_n)$ suc... |
H: Existence of a norm
K - compact, convex subset of $ \Bbb R^n $
0 $\in$ int K
K is symmetrical to 0. I'm sorry, but i don't know how to write it properly. I mean: $ (x_1,x_2,...,x_n) \in K \Rightarrow (-x_1,-x_2,...,-x_n) \in K $
I need to prove that there exists only one norm determined by K such that K is in this ... |
H: Simplify functions involving modular arithmetic
In this question, the answer says that $f \circ g(x) = x$.
But I am unable to get this result. The expression I am able to get is that $$f \circ g(x) = 7(x\text{ mod } 3) + 57(x\text{ mod }7) \pmod {21}.$$ I am unable to proceed any further.
AI: Let $x \mod 3 = a$ an... |
H: hypothesis on bilinear form
Let $H$ an Hilbert space and $a:H\times H\to \mathbb{R}$ a bilinear form. Let $H_h\subset H$ a finite dimentional subspace and let $\{w_1,\ldots,w_n\}$ a basis of $H_h$.
What hypothesis must have on bilinear form such that the matrix $K=(a(w_i,w_j))_{i,j=1,\ldots,n}$ is an invertible ma... |
H: How many iterations does it take to cover a range with random values?
Let's say I have a random number generator that generates integers uniformly from 0 to n-1 (where n is some positive integer). What is the expected number of iterations after which all the values 0..n-1 will be generated? I did some simulations a... |
H: How do I prove that the range of log is $\mathbb{R}$?
For a real analysis course. In the first part of the problem, I proved that $\log xy = \log x + \log y$. Here $\log x$ is defined as $\int_1^x\frac1t\mathrm dt$. Since it's a two part problem, I am assuming that will come in handy.
I am not sure how to go about ... |
H: Inverse of a $4 \times 4$ matrix with variables
I missed my class on the inverses of matrices. I'm catching up well, but there's a problem in the book that got me stumped.
It's a $4 \times 4$ matrix that is almost an identity matrix, but whose bottom row is $a,b,c,d$ instead of $0,0,0,1$.
$$\begin{pmatrix}
1 &0 &... |
H: A quick question about the additive identity of a ring.
Is it always the case that the additive identity annihilates all elements under multiplication? I can't think of an example; my course essentially relies on integers, polynomials, and matrices for most examples of any given concept. I'm curious if there is an ... |
H: How do I find/predict the center of a circle while only seeing the outer edge?
Question
What formula would allow me to predict the center of this circle?
In addition, what attributes of this image must be detected in order
to predict the center? I figured understanding the math first will help me determine what p... |
H: Understanding the intermediate field method for the $\phi^4$ interaction
In Rivasseau's and Wang's How to Resum Feynman Graphs, on page 11 they illustrate the intermediate field method for the $\phi^4$ interaction and represent Feynman graphs as ribbon graphs. I had to read up about ribbon graphs as I've never hear... |
H: Mirror a function about y axis
I have a piecewise function from -1 to 0 in Maple, and I want somehow get a mirror piecewise function about y axis, just like here:
Is there any bult-in function for that?
AI: However, we can think about the code reflect(p, [pt_2d, pt_2d]) in Maple, It is easy to do that as follows. ... |
H: Three-Dimensional geometry + trigonometry question
Dear all: I read this question yesterday and it is driving me crazy! I shall offer a bounty to whoever gives a reasonable answer...
We have a straight pyramid with a square ABCD as its base and apex S. We're given the pyramid's height 8 and the angle 48 deg. betwee... |
H: Convergence question in measure theory
I have a convergence question in measure theory that requires assistance:
Let $1\leq p<\infty$. Suppose $f,\ f_n \in L^P$, and $f_n\to f$ in $L^P$. (i.e $(\int|f_n-f|^pd\mu)^{1\over p}\to 0$ as $n\to\infty$) Show that $\int|f_n|^pd\mu \to \int|f|^pd\mu$.
For $p=1$, $$\lvert\in... |
H: Prove that $ \sum_{1 \le t \le n, \ (t, n) = 1} t = \dfrac {n\phi(n)}{2} $
Problem: Prove that the sum of all integers $ t \in \{ 1, 2, \cdots, n \} $ and $ (t, n) = 1 $ is $ \dfrac {1}{2} n \phi (n) $, where $ \phi $ is the Euler Totient Function.
My proof:
Define the set $\mathcal{S}$ to be the set of all the el... |
H: Let $z \in \mathbb C$, $|z| = 1$. Assume the sequence $a_n = z^n$ is convergent. Prove $z = 1$.
Let $z \in \mathbb C$, $|z| = 1$. Assume the sequence $a_n = z^n$ is convergent. Prove $z = 1$.
The case $z = 1$ implies convergence of $a_n$ is easy to prove. It is also easy to prove that $z = -1$ implies divergence of... |
H: Show that there is no natural number $n$ such that $3^7$ is the largest power of $3$ dividing $n!$
Show that there is no natural number $n$ such that $7$ is the largest power $a$ of $3$ for which $3^a$ divides $n!$
After doing some research, I could not understand how to start or what to do to demonstrate this.
W... |
H: Probability of getting something with a low probablity
If there are 100 marbles in a bag (1 red one, 99 green ones), then the probability of picking the red one is 1/100. But if I do 100 trials then I believe it is likely that I will pick the red one at least once in those 100 trials. I'm curious as to what this p... |
H: Is this a typo in Hoffman and Kunze's linear algebra 2e?
On page 203 on the part about characterizing triangulability it looks like there's a typo in the indicies they sum over on eqn (6-12).
AI: I agree with Hoffman and Kunze. The $j$th column of the matrix should be the coefficients of $T\alpha_j$ with respect to... |
H: $P(P(\cdots(P(x))))$ and its integer solutions
Problem: Suppose that $P(x)$ is a polynomial with degree at least $2$ and integer coefficients. Let $Q(x)$ have the form $$ Q(x) = P(P(P(\cdots P(x) \cdots))) $$ for some finite number of nested $P$s. Prove that the equation $Q(t)=t$ can have at most $ \text {deg}(P) $... |
H: Integration of $\int_{0}^{1.7}[x^2]dx$
so we have got this problem $$I=\int_{0}^{1.7}[x^2]dx$$ where $[f(x)]$ is under greatest integer function so i thuought of this possible solution
$$I=\int_{0}^{1.7}[x^2]dx =\int_{0}^{1}[x^2]dx+\int_{1}^{1.4}[x^2]dx+\int_{1.4}^{1.7}[x^2]dx$$
$$=0+(1.4-1)+2*(1.7-1.4)=1$$ which i... |
H: Help me to prove that $|BA|\leq|B||A|$ holds
Given the norm $|A|= \sqrt{tr(A^*A)}$, where $tr$ is the trace of a linear operator, help to prove that $|BA| \leq |B||A|$ holds.
AI: Cauchy-Schwarz gives:
$$
\begin{split}
|BA|^2&=\mathrm{trace}((BA)^*(AB))=\sum_{i,j}|(BA)_{ij}|^2=\sum_{ij}\left|\sum_k b_{ik}a_{kj}\righ... |
H: Number of ways to rearrange a line of $n$ marbles
My friend challenged me to solve the following problem, and after having thought about it for a long time and not being able to find the answer, I decided to give up. His explanation which followed wasn't very clear, and I've already forgotten the answer, but I'm st... |
H: The integral $\int_0^{\infty } \frac{L_m(-x)}{e^{2 \pi x}+1} \, dx$
Could you expain the following sum I seen in a forum
$$\int_0^{\infty } \frac{L_m(-x)}{e^{2 \pi x}+1} \, dx=\sum _{n=0}^{\infty } \left(2^{-2 n-1} \left(2^n-1\right) \pi ^{-n-1} \zeta (n+1)\right) \binom{m}{n}$$where L in a Lagarre polinomium
AI: ... |
H: Good Probability Practice Problems
I'm looking for a good probability textbook with lots of worked out examples and problems to prepare for my course's final exam. I'm in an introductory probability class in college, and we've covered basic probability, combinatorics, and discrete and random variables. We're using ... |
H: Finding $\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$
Recently got this on a test:
$$\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$$
Because it's a freshman calculus course, I think we were expected to solve it like a physicist. Taking a look at the first few terms of the series:
$$\{\frac{1}{2},\frac... |
H: If all $D_v f(P)$s are same, what is it?
function $f$ is defined around a dot P in n-space. and $f$ is differentiable.
And an arbitrary unit vector $\mathbf{v}$
if directional differential coefficients($D_\mathbf{v}f(P)$) are all same, what is that? And how can I show that?
AI: When $f:\>{\mathbb R}^n\to{\mathbb R}... |
H: How to evaluate a limit with subtractions $\lim_{x \rightarrow -1}(\frac{3}{x^3+1}-\frac{1}{x+1})$?
I'm having trouble thinking of a way to solve this.
$$\lim_{x \rightarrow -1}\left(\frac{3}{x^3+1}-\frac{1}{x+1}\right)$$
AI: Because$$x^3+1=(x+1)(x^2-x+1)$$we get
$$\frac{3}{x^3+1}-\frac{1}{x+1}=\frac{3}{(x+1)(x^2-x... |
H: What do these characters mean in RDF/OWL Domain and Range logic?
Does someone know what these strange looking characters are? I would like to learn what they mean. Can you send me a reference/hyperlink so I can understand what they mean?
http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#owl_ObjectPro... |
H: f(X ∩ Y) = f(X) ∩ f(Y) for all non empty subsets X and Y of A, given f:A→B is 1-1?
Let A and B be nonempty sets and f:A→B be a 1-1 function. Then f(X ∩
Y) = f(X) ∩ f(Y) for all non empty subsets X and Y of A.
I believe this statement is true?
AI: If $f:A\to B$ is not one-one, we can find $x,y$ for which $x\neq ... |
H: If $f,g\in {\mathscr R[a,b]}$ and $\int^{b}_{a}f=\int^{b}_{a}g,$ then $\exists c \in[a,b]$ such that $f(c)=g(c). $
Could anyone provide some hint to the problem? Thank you.
AI: As Christian Blatter has pointed out, the claim is not true. The function which is the identity over $[-1,1]\setminus\{0\}$ and $1$ at the ... |
H: Asymptotics of the logarithmic integral
Problem
Given
$$
\gamma = \int_0^1 {1-e^{-u} \over u} du - \int_1^\infty {e^{-u} \over u} du,
$$
prove that
$$
\int_0^x {dt \over \log t} = \gamma + \log \log x + \sum_{k=1}^\infty {\log^k x \over k \cdot k!}.
$$
Hint: Let $u = \log t$.
Notes: $\gamma$ is the Euler-Mascheron... |
H: Help with differential equation problem
Could someone give any directions on this problem:
A ball is thrown into upright direction. Acceleration $a$ satisfies the following equation: $$a = -g$$
where $(g = 9.81 \frac{m}{s²}, a = s''(t))$. Solve the function for distance travelled $s = s(t)$, when at $t = 0$, the b... |
H: Vector space over local field
Let $L/K$ be an extension of number field and $\frak p$ a place of $K$ and $\frak P$ aplace of $L$ above.
My question: $L$ can be considered as a vector space over $K_\frak p?$
thanks !
AI: $L$ is an extension of $K$ (extension of number fields).
$L_\mathfrak{P}$ is an extension of $K_... |
H: Using Rouche's for function constant on a circle
Let $c\in\mathbb{R}$. A non-constant function $f(z)$ is holomorphic in $|z|<2$. Suppose $|f(z)|=c$ for all $|z|=1$. Show that $f(z)$ must have a root in $|z|<1$.
Here there is an answer using the maximum principle. Since the question deals with showing the existing ... |
H: Determine if the following series are convergent or divergent?
How to determine if the following series are convergent or divergent? I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
$$\sum_{k=1}^\infty \ln(1+ \sqrt{\frac 2k})$$
$$\sum_{k=1}^\infty\displaystyle \s... |
H: If $A$, $B$, and $C$ are sets, the only way that $A\cup C = B \cup C$ is if $A=B$
If $A$, $B$, and $C$ are three sets, then the only way that $A\cup C$ can equal $B\cup C$ is $A = B$.
I believe this statement is false and here is why:
Let $A=\{1\}$, $B=\{2\}$, and $C=\{1,2,3,4\}$. In this scenario $A\cup C=\{1,2,... |
H: Basis of eigenvectors of a linear transformation
Let $\mathbb R_n[x]$ the vector space of polynomials with degree less or equal $n$ and we consider the linear transformation $f$ defined by
$$\forall P\in \mathbb R_n[x]\quad f(P)=(x^2-1)P''+2xP'$$
I proved that $f$ has the spectrum
$$\mathrm{sp}(f)=\{k(k+1),\ k=0,\... |
H: Joint distribution proof
I am trying to study for an exam and I am kind of lost on how my professor came to a particular result on his practice exam.
Let $W$ be an exponentially distributed random variable with $\lambda = 2$
Prove that $P(W > 5 | W > 2) = P( W > 3)$
I made it as far as re-writing the problem as
$$... |
H: Find characteristic polynomial of $\,A^2$ if the characteristic polynomial of $\,A$ is $\,t^4 -t$
$A \in M_{4\times4}(\mathbb{R})$. The characteristic polynomial of $A$ is $P_A(t)=t^4-t$. I have to find the characteristic polynomial of $A^2$ and $A^4$.
So I know that due to the Cayley–Hamilton theorem that $P_A(... |
H: Derivative of integral with time varying domain
Let $f:\mathbb{R}^p \rightarrow \mathbb{R}$ be a smooth function. Let $A(t) \subset \mathbb{R}^p$ be varying with time $t$. Is there a nice expression for
$$\frac{d}{dt}\int_{A(t)}f(x) dx$$
?
AI: We have $A(t) = \bar{B}(tc,r)$. Let $\phi(\delta) = tc+\delta$. Then $\... |
H: Three consecutive integers with power of 5 mod 11
Let $(n - 1)$, $n$ and $(n + 1)$ be three consecutive integers, and $(n - 1)^5 \equiv n^5 \equiv (n + 1)^5 \equiv a \pmod{11}$, what are the possible values of $a$?
I know the facts that $3^5 \equiv 4^5 \equiv 5^5 \equiv 1 \pmod{11}$ and $6^5 \equiv 7^5 \equiv 8^5 \... |
H: find a degree and splitting field for $x^4-2$ over $\mathbb{Q}(i)$
let $K=\mathbb{Q}(i)$ and let $f=x^4-2$. Find the splitting field, its degree and the basis.
My solution
First I find roots of the polynomial $x_{1,2}=\pm\sqrt[4]{2},\hspace{2mm}x_{3,4}=\pm i \sqrt[4]{2}$ and I notice that the polynomial $f=x^4-2$ i... |
H: Evaluating product $\prod_{n=2}^\infty\left(1-\frac{1}{n^2}\right)$
I'm reading about infinite products in complex analysis, where there is a theorem like
The product $\prod_{n=1}^\infty\left(1+a_n\right)$ converges absolutely iff the series $\sum_{n=1}^\infty|a_n|$ converges.
Then an exercise is to show that $\p... |
H: Application of differentiation, modeling bacteria
https://www.dropbox.com/s/defhs0u02yuqtyw/differentiation%20bacteria.jpg
I basically don't understand the first sentence of the question.
A good explanation and a a complete solution would be appreciated
Edit: Ok since people asked for my work here it is
a)
innit... |
H: Find a generating function for $a_r = n^3$
What is the generating function for $a_r = n^3$? I computed an answer, just wanted to double check my answer.
AI: Here is how you advance. Assume
$$ F(x) = \sum_{r=0}^{\infty} a_r x^r \implies F(x)=\sum_{r=0}^{\infty} r^3 x^r $$
$$ \implies F(x)= (xD)(xD)(xD)\sum_{r=0}^{\... |
H: Geodesics: a (for me) "mysterious" property related to affine parameters.
Consider a Riemannian manifold $(M,g)$ with the Levi-Civita connection $\nabla$. If $D_t$ is the covariant derivative along curves descending from $\nabla$, a geodesic is a curve $\gamma: I\subseteq\mathbb R\longrightarrow M$ such that $D_t\g... |
H: Ideals in commutative noetherian rings with unique prime ideal
Let $R$ be a commutative noetherian ring with $1$ having only one prime ideal $\mathfrak{P}$. It follows that $\mathfrak{P}^n = 0$ for some integer $n$. Can we say that every proper ideal in $R$ is a power of $\mathfrak{P}$?
AI: Hint:
No. Take a look at... |
H: Countably infinite set of real numbers with a complement that is infinite but not countably infinite
How can I show that if a set of real numbers is countably infinite, then its complement is infinite but not countably infinite?
Thanks a lot in advance!
AI: Let the set you're looking for be
$$
\{a_1,a_2,a_3,\ldots\... |
H: Probability, that when we send a $0$ down the network we will get back a $0$
We can send a $0$ or a $1$ over a network of $1,2...$ nodes. Unfortunately on each node with probability $p$ the message is not made different, and with probability $1-p$ the message is XOR'ed. Find recurrence relation that determines with... |
H: Product $\prod_{n=0}^\infty(1+z^{2^n})$
I want to prove that $\prod_{n=0}^\infty(1+z^{2^n})=(1-z)^{-1}$ for all $|z|<1$.
By multiplying $1-z$ to both sides, the equation becomes $$(1-z)(1+z)(1+z^2)(1+z^4)\ldots=1$$
Multiplying the first pair on the left yields
$$(1-z^2)(1+z^2)(1+z^4)\ldots=1$$
And then
$$(1-z^4)(1... |
H: System of differential equations (in X, Y): expression of Y.
I want to find an expression for the function $Y(t)$ by the following system
\begin{align}
\frac{d X (t)}{dt} & =-\alpha Y \\[6pt]
\frac{d Y (t)}{dt} & =\sigma \beta Y^2-(\beta+\gamma) Y - \sigma X Y+X
\end{align}
The equation must contain only $Y$ and no... |
H: How to check if the series $\sum_{n=1}^{\infty} n\cdot \sin(\frac{1}{n})$ is convergent
or divergent? $$\sum_{n=1}^{\infty} n\cdot \sin(\frac{1}{n})$$
Which test I should use?
Thank you so much for your help!
AI: The sine function has slope $1$ where it crosses the axis at the origin. Therefore it lies above the l... |
H: Naming general objects in more than 3 dimensions
In a paper I am writing, I need to talk about a general "object" formed by the points of a connected set in an $n$-dimensional euclidean space. I have found some suggestion here, but none fit my needs. The "object" I am concerned with do not have any specific shape r... |
H: Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$
So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group.
My thoughts are as follows:
Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is easily shown that $\alpha$ satisfies $\alpha^4-4\alpha^2+2=0$.
Define... |
H: Spanning Trees of the Complete Graph minus an edge
I am studying Problem 43, Chapter 10 from A Walk Through Combinatorics by Miklos Bona, which reads...
Let $A$ be the graph obtained from $K_{n}$ by deleting an edge. Find a formula for the number of spanning trees of $A$.
So how I approached this problem was by c... |
H: Counting six-letter strings over $\{a,b,c,d,e\}$ containing a single $a$
Consider all strings whose letters belong to the set:
$A = \{ a, b, c, d, e\}$
How many strings of length $6$ are there that contain exactly one $a$?
Attempt:
Since we are only using $\frac{4}{5}$ letters for the rest of the string,
There are... |
H: Prove a summation inequality by induction: $\sum_{i=1}^n \frac{3}{4^i} < 1$
I was having trouble proving by induction with this problem.
$$\sum_{i=1}^n \frac{3}{4^i} < 1$$ for all $n \geq 2$
I went to see my professor and he said try proving this equality $$\sum_{i=1}^n \frac{3}{4^i} < 1 - 1/4^n $$
Where did he... |
H: Let $A$ be an orthogonal $n\times n$ matrix. Show that $\|A\vec x\|=\|A^{-1}\vec x\|$ for any vector $x$ in $\mathbb R^2$
Let $A$ be an orthogonal $n\times n$ matrix. Show that $\|A\vec x\|=\|A^{-1}\vec x\|$ for any vector $\vec x$ in $\mathbb R^2$
I want to show that $\|A\vec x\|=\|A^{-1}\vec x\|=\|\vec x\|$
I tri... |
H: Solution Verification: Given $|A\cup B|=45, |A|=30,$ and $|A\cap B|=7,$ find $|B|$
Given |A∪B|=45, |A|=30, and |A∩B|=7, find |B|.
If I am not mistaken here is how I am reading the scenario:
B must have 22 elements. The 7 that it shares with A, and then 15 of its own unique elements.
23 unique to A, 7 shared by... |
H: dividing a unit in several different ways
There is a king who wants to divide his kingdom between his infinity of daughters.
Suppose he wants to divide the kingdom evenly. It seems that under such conditions, each of the daughters gets an infinitely small piece of the kingdom.
Suppose on the other hand that the kin... |
H: $\mathbb Q[x]/(x^2+1)$ is not isomorphic to $\mathbb Q[x]/(x^2+2)$
I found an argument online that the two fields are not isomorphic but I can't make sense of the argument. You find it here.
These are the things I'm confused about when it comes to the argument.
It says that the field $\mathbb Q[x]/(x^2+1)$ conta... |
H: Proof by induction; simplify when adding k+1th term. Understanding induction.
I want to prove:
$$(-\frac{1}{2})^0 + (-\frac{1}{2})^1 + \cdots + (-\frac{1}{2})^k + (-\frac{1}{2})^{k+1}
=
\frac{2^{k+1}+(-1^k)}{3\cdot2^k} + (-\frac{1}{2})^{k+1}$$
How do I simplify the last bit, $\frac{2^{k+1}+(-1^k)}{3\cdot2^k} + (-\f... |
H: finding $\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $ radius of convergence
find this power serie radius of convergence and the area where it converges.
$\sum_{n=0}^\infty(\frac{(n+1)}{n})^{n^2}(z-2)^2 $
my attempt: a) $L=lim sup|an|^\frac{1}{n} \quad $$L=Lim sup(\frac{n+1}{n})^{\frac{n^2.1}{n}}$ = $Lim_... |
H: dividing an octave to $7$ instead of $12$
Usually an octave is divided into $12$ parts based on the harmonic series(basic zeta function).
how can I calculate the frequency of a note if I divide the octave into $7$ parts?
$N_1=A_4(440Hz)$
$N_8=N_1*2=A_5(880Hz)$
$N_2....N_7???$
Thanks guys!
AI: If you want equal mu... |
H: How to check if the series $\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$ is convergent
or divergent??
I tried few tests, but I didn't success to discover if the series is convergent or is divergent...
$$\sum_{n=0}^{\infty} \sqrt{n+1}-\sqrt{n}$$
Thank you!
AI: Let $S_n$ be the sequence of partial sums:
$$S_n = \sum_{k=... |
H: Prove integral of sequence of function uniformly converges
Show that if $f_n \to f$ uniformly on $[a,b]$ and $f_n$ is integrable for each n then $\int_{a}^{x}f_n(t)dt\to \int_{a}^{x}f(t)dt$ uniformly in $x$ on [a,b].
I know how to prove the question $\int_{a}^{b}f_n(x)dx\to \int_{a}^{b}f(x)dx$ uniformly in $x$ on [... |
H: Prove the Jordan lemma i.e. $\int e^{-R\sin{\theta}}< \pi/R$
In complex variables my instructor wrote on the board "Jordan's Lemma", and then, somewhat imprecisely,
$$\int e^{-R\sin{\theta}}< \pi/R \;\;\;\; \text{ e.g. } \int \frac{s \sin{x}}{x^2 + 2x + 2}.$$
I have searched for a reference for this result without ... |
H: How to calculate equivalence relations
How can I calculate how many equivalence relations can be defined on a given set? For example:
How many possible equivalence relations can be defined on S =
{a,b,c,d}?
AI: Okay, so equivalence relations effectively partition $S$ into subsets where each element in a given s... |
H: Is $L^2(\Omega)$ the only $L^p$ hilbertian space?
I've started today studying Hilbertian spaces, and all of the examples seen in class were about the space $L^2(\Omega)$, where $\Omega$ is a limited domain in $\mathbb{R}^N$ $(N \geq 1)$.
Online I didn't manage to find anything related to my doubt regarding other $p... |
H: Prove that a bounded sequence contains all of it's accumulation points
Homework question:
Let {$x_n$} be a bounded sequence. Prove that the set of all subsequential limits of {$x_n$} is closed.
Any help would be appreciated.
(I edited this so that anyone who looks at this in the future will not get confused by my ... |
H: Solution Verification: Maximum number of edges, given 8 vertices
Suppose a simple graph G has 8 vertices. What is the maximum number of
edges that the graph G can have?
The formula for this I believe is
n(n-1) / 2
where n = number of vertices.
8(8-1) / 2 = 28. Therefore a simple graph with 8 vertices can ha... |
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