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H: A linear holomorphic function
If $f$ is a holomorphic function on a simply connected open domain $\Omega$, and $f$ is a linear function on the boundary $\Omega$, i.e. $f=az+b$ on $\partial\Omega$. Then, can I say that $f$ is also linear function on $\Omega$?
AI: Since $f(z)=az+b$ on the boundary, we know $g(z):=f(z... |
H: Does this logic have the downward Skolem-Löwenheim theorem?
Let $\mathcal L_Q$ denote the logic obtained from adding the quantifier $\newcommand{\almost}{\forall^\infty}\almost$ to the usual first-order logic, where the semantic interpretation of $\almost x\varphi$ is "All but finitely many $x$ satisfy $\varphi$", ... |
H: Find a vector $y$ such that $g(x)=\langle x,y \rangle$ for all $x \in V$
$V$ is an inner product space and $g: V \rightarrow F$ is a linear transformation.
Find a vector $y$ such that $g(x)=\langle x,y \rangle$ for all $x \in V$
$V=P_2(R)$ with $\langle f,h \rangle=\int_{0}^{1}f(t)h(t)dt$, $g(f)=f(0)+f'(1)$.
I know... |
H: Does there exist a function $f : X \to Y$ such that $f \in Y$?
Does there exist a function $f : X \to Y$ such that $f \in Y$?
I think this is related to Russell's paradox, but I'm not exactly sure how.
Added Later: As Brian points out, given any function $f : X \to Y_0$, we can just add $f$ to the codomain, tha... |
H: What law of algebra of proposition is happening here?
I'm preparing for a test tomorrow and going over some reading material, and I came across this problem that was worked out. So far I think I'm following each step of logic, but I've hit a wall with this part:
(p $\land$ ($\lnot$(r $\land$ q))) $\lor$ (($\lnot$p... |
H: Concept about series test
I have five kind of test here
1. Divergent test
2. Ratio test
3. Integral test
4. Comparison test
5. Alternating Series test
And a few questions here.
1. Are test 1,2,3,4 only available for Positive Series? and alternate series test is only for alternating series?
2. To show $\sum_{n=1}^... |
H: Computing integral of $2$ - form on a torus
I am looking at problem 16-2 of Lee's Smooth Manifolds, second edition.
Problem 16 - 2: Let $\Bbb{T}^2 \subseteq \Bbb{R}^4$ be the two torus defined as the set of points $(w,x,y,z)$ such that $w^2 + x^2 = y^2 + z^2 = 1$, with the product orientation determined by the st... |
H: How would I graph this?
How would I graph this: $t^2+3t=40$? I tried factoring $(t-5)(t+8)=0$ but I am not sure how to graph it because the function equals zero. I know how to do it if it is $y=t^2+3t-40$. I am probably overlooking the obvious, any help?
Thanks
AI: What you have is an equation: you can think of its... |
H: How to prove this symmetric polynomial equations?
I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}%
\frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1.
I tried to multiply the left of equation with $\Pi _{1\leq i<j\leq
n}(x_{i}-x_{j})$, and get $\Pi _{1\leq i<j\leq n}(x_{i}-x_{j})\... |
H: Singleton subset of a metric space
I am currently working through chapter two of Principles of Mathematical Analysis (ed. 3) by Walter Rudin. My question comes from pages 30-31.
I know that a metric space must satisfy the definition:
Definition: A metric space is an ordered pair, $(X,d)$, where $X$ is a set (whose... |
H: Digits in a large power of two
I am trying to find the answer to: 2^34359738368. As to be expected every calculator and computer program I have used has crashed.
To be honest I don't even want to know the exact answer, I just want to really roughly know the number of digits the answer has.
Is there a trick to doing... |
H: Simplify Boolean Algebra
How do I simplify the following expression with Boolean Algebra? Please show what you used to simplify so I can understand.
$$ABC + AB'C' + ABC' + A'B'C'$$
AI: First I want to group the elements that are similar. This will allow me to start reducing the expression.
$$ABC + AB'C' + ABC' + A... |
H: why I always thought polynomials as a function
As I've started studying Polynomial Ring on my own, I would like to verify/ask the concept/questions occurred to me. I've noticed over some ring the polynomials are of little/no interests as a function and all we're concerned about is the components obtained using $x^n... |
H: Proving $C([0,1])$ Is Not Complete Under $L_1$ Without A Counter Example
I'd like to show that $C([0,1])$ (that is, the set of functions $\{f:[0,1]\rightarrow \mathbb{R} \, \textrm{ and } \, f \, \textrm{is continuous} \}$ is not a complete mertric space under the $L_1$ distance function:
$$
d(f,g) = \int_0^1 |f(x)... |
H: Prove a relation related to sets
In a city, among each pair of people, there can be exactly one of k different
relationships (relationships are symmetric). A crowd is a set of three
people in which every pair have the same relation. Let $R_k$ denote the
smallest number of people in the city such that the city alway... |
H: What does apostrophe as a suffix denote?
I was just curious as to what "$'$" denotes; i.e. $x' = y$, as in $x'(t) = x(t)$ which has the solution $x(t) = c_1\;e^t$.
I've found out that it has something to do with differential equations, but I can't seem to find any information specifically on "$x'$".
If someone coul... |
H: Prove this following inequality
Show that, for all integers m > 1,
$\frac {1}{2me}$ < $\frac {1}{e}$ - $(1-\frac{1}{m})^m$ < $\frac {1}{me}$
AI: Here's one part:
We have $\left(1+\frac1{km}\right)^{km}\to e$ as $k\to\infty$, hence for $k$ big enough the error in
$$ \left(1-\frac1m\right)^m\cdot e\approx\left(1-\fr... |
H: $\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$
I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure.
For a sequence $\{a_n\}$, there exists a limit $a$ such that $\lim_{n\to\infty} a_n=a$ if and only if for any $p\in \Bbb N$, $... |
H: Proving equivalences between prime counting functions.
If we have that: $$\theta(x)=\sum_{p\leq x}\log p,$$ and $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ Where $\Lambda(n)=\log p $ if $n=p^m$ and $\Lambda(n)=0$ in another case.
How can I prove that :
1) $\theta(x)=\psi(x)+O(\sqrt{x})$
2) $\pi(x)=\frac{\psi(x)}{\log x}+O... |
H: kill rate of insecticide differential equations
A field of wheat teeming with grasshoppers is dusted with an insecticide having a kill rate of 200 per 100 per hour.What percentage of the grasshoppers are still alive 1 hour later?
I did not understand the units.What does 200 per 100 per hour means?Hence I could not ... |
H: What mathematical objects permit "taking of limits"?
Background
I have been reading a lot of abstract algebra recently (at the level of Artin/Dummit & Foote/Herstein Topics in Algebra for those of you familiar with these books). I have noticed that many of the abstract objects like groups and rings lack a certain ... |
H: Expressing $x^{2}-y^{2}-y + i \cdot (2xy+y)$ in terms of $z$.
How to express the following expression $$x^{2}-y^{2}-y + i \cdot (2xy+y)$$ in terms of $z$.
$x^{2}-y^{2} + 2xyi = (x+iy)^{2}=z^{2}$ but what about $iy -y$.
AI: $$ x^2-y^2+i(2xy) +iy-y= z^2 + (i-1)\Im(z) =z^2 + \frac1{2i}(i-1)(z-z^*) $$ |
H: Is my row calculation of row echelon form correct?
I was directed by a community member to a resource on how to calculate the row echelon form of a matrix here. The resource says:
First we wish to put A into reduced row echelon form. There are several ways to do those (and
thus several matrices P), but here is on... |
H: Proving that $\mathbb{Z}[i]$ is a noetherian ring
Claim: the ring $\mathbb{Z}[i]$ is a noetherian ring
My proof
1) $\mathbb{Z}[i]$ is a finitely generated $\mathbb{Z}$-module.
2) $\mathbb{Z}$ is a noetherian ring.
3) Every finitely generated module over a noetherian ring is a noetherian module, hence $\mathbb{Z}[i]... |
H: A smooth function instead of a piecewise function
I want to find a smooth function approximating f(x) as best as possible:
\begin{equation*}
f(x) =
\begin{cases}
x & \text{if } x \le a,\\
a & \text{if } x > a.
\end{cases}
\end{equation*}
as a smooth function ($a$ is a positive constants and x is a positive real nu... |
H: What does the letter U mean in math?
What does the letter U mean in the following expression:
$$
\bigcup_{\alpha} A_\alpha \;?
$$
It doesn't look like logical OR.
Link to original screen shot.
AI: It represents the union of all the sets $A_\alpha$. By the "union", what it basically means that:
$$ a \in \bigcup_{\al... |
H: Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?
Here's an excerpt from my notes:
Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$
Where $f$ is the pdf of the distribution we're sampling the $x$'s from. Caution: the likelihood function $\mathcal{L... |
H: Base change and ordinals
Problem. Define the operation base change from $k$ to $m$: to make the operation for natural number $n$ we should write $n$ in the base-$k$ numeral system and read this in the base-$m$ numeral system.
Let $n$ be a natural number. Make for $n$ base change from $2$ to $3$, then subtract $1$,... |
H: Proof that $\sum\limits_{n=1}^{\infty} z^{1/n}$ does not converge
I believe I found a proof for the divergence of this sum for any value of $z$ besides 0.
We can look on the telescopic series: $$\sum_{n=1}^{\infty}z^{1/(n+1)}-z^{1/n} = \lim_{N\rightarrow \infty} \left(z^{1/(N+1)}-z\right) = 1-z$$
If the sum in the ... |
H: How many 8-character passwords are possible using one of each of three types of character possible?
How many eight character passwords are there if each character is either an
uppercase letter A-Z, a lowercase letter a-z, or a digit 0-9, and where at
least one character of each of the three types is used?
... |
H: Looking for a way to find the proportional growth rate in time for any given notation
I am wondering if there is a straight forward way to illustrate the proportional growth rate in time (or space) for any given notation such as $O(n^2)$ or $O(logn)$?
My initial thought is that $O(n^2)$ would be equal to $O(n)*O(n)... |
H: Calculating rotation matrix of coordinate system from 2 known axis
In the image my main coordinate system is in the upper right corner. I measured $3$ points on a board and created a help coordinate system. V1 points directly to the origin of the help coordinate system. V2 and V3 lie on different axis of the help... |
H: Find a function that gives this sequence: $+1,+1,-1,+1,+1,-1,-1,+1,+1,+1,-1,-1,+1,-1,-1,...$
I start with a string $S_1=1$
then the $(n+1)$-th string is $S_{n+1}=\{ S_n,+1 ,-(S_n)\}$
if $S_j=\{s_1,s_2,s_3,..., s_i\}$
then $-(S_j)$ is defined as $-(S_j)=\{-(s_i), -(s_{i-1}),..., -(s_3), -(s_2), -(s_1) \}$
The sequ... |
H: For what values of $a$ the equation $(ax)^2-x^4=e^{|x|}$ have no solution
I am trying to find for what values of $a$ this equation have no solution.
the condition is $|a|<\sqrt{2}$
and the equation: $$(ax)^2-x^4=e^{|x|}$$
what I did so far is set ln on this equation:
$$2ln(ax)-4ln(x)=|x|$$
I would like to get some ... |
H: Symmetric matrix and inner product: $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$
If A is real, symmetric, regular, positive definite matrix in $R^{n.n}$ and $x,h\in R^n$, why is it $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$?
Is there some rule or theorem for this?
AI:... |
H: Source coding theorem - optimum number of bits?
The source coding theorem says that information transfer with variable length code uses less bits and is equal to the entropy of the distribution. It also says that there is no code that uses lesser number of bits ( or does it? Have I misunderstood this). My question ... |
H: What's the term for a value x that satisfies the constraint $f(x) = f$ for a function f?
I know that $x$ is called the fixed point of a function $f$ if it satisfies the constraint $f(x) = x$.
However, for a function $f$ if there exists some value $x$ such that $f(x) = f$ then what is the term for the value $x$ with... |
H: Integrals using Arctangens
We want to find $\displaystyle \int\dfrac{12}{16x^2 +1}$
I rewrote it to the form $ 3 \cdot \dfrac{1}{u^2 + 1} \cdot u' $ where $u=4x$.
I found out that the correction sheet does the same thing, but their next step leaves my puzzled:
$$ F(x) = 3 \arctan (4x) + C$$
Where did the $u' = 4$ g... |
H: Continuity of function given as a maximum
Let $f(x,y)$ is continuous in $[a,b]\times[c,d]$, and we define the function $g(y)$ as follows $$g(y):=\max_{x\in[a,b]}f(x,y),\quad\forall y\in[c,d].\tag{1}$$
The question is when we can conclude that $g\in C[c,d]$, or provide a counterexample to show $g$ is not a continuou... |
H: When I was said to factorize a polynomial in school what did I actually told to perform?
In school we spend several hours factorizing polynomials. But now as I've started gainnning some knowledge on polynomial rings, it suddenly occurred to me that none of the books I practiced then, suggested the factorized form o... |
H: How do I show that a set is an element of a set in a Venn diagram?
More precisely, is there a difference between {p,s,r,q,t} and {{p,s},r,q,t} and if so, how would you show it using a Venn diagram?
I have in my notes a Venn diagram where p, s, r, q and t are all obvious elements of set A. C is a subset of A, fully ... |
H: Need help with $\int \dfrac{2x}{4x^2+1}$
We want$$\int \dfrac{2x}{4x^2+1}$$
I only know that $\ln(4x^2 + 1)$ would have to be in the mix, but what am I supposed to do with the $2x$ in the numerator?
AI: Hint: make the substitution $u = 4x^{2} +1$. Then $du = 8xdx$, and the integral becomes:
$$\frac{1}{4} \int \frac... |
H: Differentiation of inverse functions using graphs with conditions?
I was trying to differentiate this equation. And I got the answer but it matches none. Any help on how to solve this one. I tried by converting this function to
$y=tan^{-1}tan{\frac{x}2} $ and then using condition to differentiate but my answer is... |
H: Two roots of polynomial
If a polynomial with rational coefficients has a root $1 + \cos(2\pi/9) + \cos^2(2\pi/9)$, then the one also has a root $1+\cos(8\pi/9)+\cos^2(8\pi/9).$ How to prove it?
AI: Take $\omega=e^{i2\pi/9}$.
Let $\alpha=1 + \cos(2\pi/9) + \cos^2(2\pi/9)$.
Then $4\alpha = 6+2\omega+ \omega^2 + \omeg... |
H: Proving that if $N<10^{30}$ then $\sum_{n=1}^{N}\frac{1}{n}<101.$
So, I am asked to prove if $N<10^{30}$ then $$\sum_{n=1}^{N}\frac{1}{n}<101.$$ I am given the information that $2^{10}=1024$ and in the previous part of the question I proved that $$0\leqslant \sum_{n=1}^{N}\frac{1}{n}-\ln N\leqslant 1.$$
So I reason... |
H: Nontrivial solution
What's the trick to find the real numbers $ \lambda $ for which the following equation system has a nontrivial solution ?
$x_1 + x_5 = \lambda x_1 $
$x_1 + x_3 = \lambda x_2 $
$x_2 + x_4 = \lambda x_3 $
$x_3 + x_5 = \lambda x_4 $
$x_1 + x_4 = \lambda x_5 $
AI: The system can be written in matri... |
H: How to prove (global) uniqueness of solution to linear, first order ODE?
Consider the first order linear ODE
$F(x) + A x F'(x) + B = 0$, with initial condition $F(1) = 1$. Moreover, let $A,B \neq 0$, real, and $sgn(A) = sgn(B)$.*
*... if the latter matters.
Mathematica gives me the following solution, which is easy... |
H: Nonstandard structure of Presburger arithmetic
Let $\mathfrak {R}_A = (\Bbb {N}; 0, S,<,+)$. What can we say about ${}^{\ast}\Bbb N$, the universe of non-standard structure of the first order theory of $\mathfrak {R}_A$?
Firstly, because of the property every non-zero element must have a predessor and a successor, ... |
H: if $X$ is a vector field how can I find $Y$ such that $[X,Y]=0$?
Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I determine another vector field (non-colinear with $X$... |
H: Show that a group homomorphism $f$ is the identity.
Suppose that $f$ is a group homomorphism from $\mathbb Z_7\times\mathbb Z_7$ to itself satisfying $f^5 = \operatorname{id}$ (where $f^5=f\circ f\circ f\circ f\circ f$). Show that $f$ is the identity.
AI: Hint:
In a group $G$ (what group? Its order divides $42\cd... |
H: Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$
The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$
I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way:
First, note that $$\frac{1}{\cosh{(ax)}}=\frac{2e^{-ax}}{e^{-2ax}... |
H: $u ( x, y) = 2x (1− y)$ for all real $x$ and $y$
I am stuck on the following problem :
Let $u ( x, y) = 2x (1− y)$ for all real $x$ and $y$.
Then a function v ( x, y), so that
$f ( z) = u ( x, y) + iv (x, y)$ is analytic, is
(a) $x^2-(y-1)^2$
(b) $x^2+(y-1)^2$
(c) $(x-1)^2-y^2$
(d) $(x-1)^2+y^2$
... |
H: Find a point through which every surface tangent to z=xe^(y/x) passes
Find a point through which every plane tangent to the surface
$$
z=xe^{\frac{y}{x}}
$$
passes.
It's not a homework. I know, that I need a normal vector and the point of tangency to find a tangent plane.
AI: Equivalently; if $F=z-x\exp(y/x)=0$ the... |
H: Confidence intervals, Coefficient of variance & box plots
here's the background:
I've stochastically modelled 3 techniques of culling a badger population over a ten year period. It quite nicely gives me the mean expected final population at the end of the 10-year period but I'd like to meaningfully show the spread ... |
H: Space complexity of the segmented sieve of Eratosthenes
It's commonplace to say that without compromising on the time complexity of $O(n\log\log n)$, the space complexity of the sieve of Eratosthenes can be reduced to $O(\sqrt{N})$ using a segmented version of the sieve.
This is true, however we can do slightly bet... |
H: Isomorphism between $\mathbb{C} X \otimes \mathbb{C} X$ and $\mathbb{C} (X \times X)$
Let $G$ be a finite group and $X$ a finite set. Denote by $\mathbb{C}X$ the vector space of functions from $X$ to $\mathbb{C}$. In a book I found the following statement:
$\varphi : \mathbb{C} X \otimes \mathbb{C} X \rightarrow \m... |
H: Remainder of a complex function
Dividing $f(z)$ by $z-i$, the remainder is $1-i$ and by dividing $z+i$ the remainder is $1+i$, then what is the remainder when $f(z)$ is divided by $z^2+1$?
I just started solution using division algorithm, but I struck at beginning. I am not getting any idea. Please give some hints.... |
H: Prove unique existence of solution for a Cauchy problem
Let $f$ be :
$f: [t_0,t_1] \times [x_0 - b, x_0+ b] \longrightarrow \mathbb{R}$
continous and such that $(f(t,x_2) - f(t,x_1))(x_2-x_1) \leq 0$ $\forall t\in [t_0,t_1] $ and $\forall x_1,x_2 \in [x_0-b, x_0+b]$. Prove that the Cauchy problem:
$x' = f(t,x)$
... |
H: Covariance of sums of random variables
I need some help understanding an excercise.
Let $X_1, X_2, X_3 \sim N(-2,3).$
(right here there is an ambiguity about the second parameter: is it $\sigma$ or $\sigma^2$ ?)
First they calculate the variance
$$\sigma^2\left(\sum_{i=1}^3 i X_i\right) = \sum_{i=1}^3 i^2 \sigma^... |
H: complex equation to be solved
I need to find all solutions to the complex equation $e^{1/z} = \sqrt{e}$
Then I need to show that all these solutions are on the circle $|z-1|=1$
Using the fact that $e^{2\pi i}=1$, I solved the equation to find $z = \frac{2}{1-4ik\pi}$ but that is not what's in the back of my book.
A... |
H: Linear Transformation - Distribution laws proof
How to prove that?
$f\circ(g+h) = f\circ g + f\circ h \ \text{ and }\ (f+g)\circ h = f\circ h + g\circ h$
Thanks in advance
AI: First, let $V,W$ be vector spaces over the same field of scalars. Also let $f,g,h : V \to W$ be linear transformations. To prove your first ... |
H: When do we need parenthesis to change order of operations?
A few questions about order of operations:
$1$) In an expression such as $1+3+3^2$, it is legal to simplify to $4+3^2$, a violation of the grade school order of operations. In this case, are we adding an implicit set of parenthesis and then simplifying? i.e... |
H: Does there exist such a closed subspace of normed linear space
let $(X,|| || )$ be a norm linear space. And $M$ be a closed subspace of norm linear space .does there exist a closed subspace $N$ such that $X=M \oplus N $ . I know such an subspace $N$ exist .but i am not conform about such an $N$ is closed or not .
A... |
H: Prove by induction that $ 1^2+2^2+...+(n-1)^2
In Apostol's «Calculus I» on page 33 there is the following proof by induction:
To prove:
$$
1^2+2^2+...+(n-1)^2<n^3/3<1^2+2^2+...+n^2
$$
Solution:
Consider the leftmost iequality first, and left us refer to this formula as $A(n)$. It is easy to verify this asser... |
H: Square roots of complex numbers
I know that the square root of a number x, expressed as $\displaystyle\sqrt{x}$, is the number y such that $y^2$ equals x. But is there any simple way to calculate this with complex numbers? How?
AI: If you represent the complex number $z$ in polar form, i.e.
$$z=re^{i\theta} = r(\co... |
H: how you show that $[\frac{a}{n} ]^2=1$, where $a \in \mathbb{Z}$ and $n$ is odd integer?
$[\frac{109}{1925} ]=[\frac{109}{5} ]^2[\frac{109}{7} ][\frac{109}{11} ] =
[\frac{4}{5} ]^2[\frac{4}{7} ][\frac{-1}{11} ] = (?)^2[\frac{2^2}{7} ][\frac{-1}{11} ]
= (?)^2\cdot 1 \cdot (-1)^{\frac{11-1}{2}}=1 \cdot 1 \cdot (-1) ... |
H: "Center" of a spherical triangle
I have a very deficient background in geometry, so I come across questions like these and I'm not sure how to verify my intuition.
Consider three points in $\mathbb{R}^3$, given by position vectors, lying on a sphere centered at the origin. These define both a spherical triangle and... |
H: Essential singularities of $\exp(z)$
Could you explain to me why $z = \infty$ is an essential singularity of function $\exp(z)$ ?
What about $z$ = 0, is it essential singularity?
AI: If $z\to\infty$ along the positive real axis then $\exp(z)\to\infty$.
If $z\to\infty$ along the negative real axis then $\exp(z)\to0... |
H: $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
Let $f_1 : [a,b] \rightarrow \mathbb{R}$ be an integrable function.
Let's define a sequence $(f_n)$, $ \ \ f_n : [a,b] \rightarrow \mathbb{R}$ as $f_{n+1}(x):= \int_a ^x f_n(t)dt$.'
Prove that $\sum_{m=1} ^{\infty} f_m(x)$ is... |
H: Find the smallest natural number which is 4 times smaller than the number written with the same digit but in the reverse order.
The question says "Find the smallest natural number which is 4 times smaller than the number written with the same digit but in the reverse order."
I tried to solve it in this way:
New num... |
H: Max frequency of a signal?
Having
$$ f(x) = \cos(x) + \sin(10x)$$
How Can I know which is the max frequency of this signal?
I need it to set the right Nyquist frequency ($2\cdot\max\text{frequency}$)
I can use Matlab if it's needed
AI: The $\sin(10x)$ term has a frequency of $\frac {10}{2\pi }$ because $x$ must in... |
H: Need to calculate my final marks - help
I don't know the formula to calculate my marks for a subject. The first tests accounts for 38% of the overall grade whereas the final exam accounts for 62%. Let us say that on the first test I got a mark of 80% and the second one 60%. What is my final grade and how do I calcu... |
H: When does a PDE solve a variational problem?
I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in $f_0$, the $f$ that minimizes $J$. Solutions are non-degenerate if ext... |
H: How to prove that operator is not compact in $L_2 (\mathbb{R})$
I have the operator
$(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be compact on $L_2 (\mathbb{R})$.
AI: Recall that if $g$ is $L^1$... |
H: Non-integer bases and irrationality
I read somewhere:
When it comes to properties like prime, irrational, rational,
divisible by 2, etc., nothing changes when you change base.
But I'm not sure about the rational/irrational one.
If you use non-integer bases, even integer numbers can become ugly expressions (See ... |
H: Why if $G_n$ is a subgroup of G then G is abelian?
I found this exercise, that says:
Let $G$ be a group and let $G_n=\{g^n : g\in G\}$. Under what hypothesis about $G$ can we show that $G_n$ is a subgroup of $G$?
Apparently the answer is that G has to be abelian, but I don't see why.
Reference: Fraleigh p. 59 Que... |
H: How to describe the one point compactification of a space
In my Topology course we defined the one point compactification of a Hausdorff space $\left(X,\tau\right) $ to be a compact Hausdorff space $\left(Y,\tau^{'}\right)
$ such that $X\subseteq Y$, $\tau\subseteq\tau^{'}$ and $\left|Y\backslash X\right|=1
$.
Mor... |
H: Modeling the coin weighting problem
Suppose we have $n$ coins with weights $0$ or $1$ and a scale for weighting them. We would like to determine the weight of each coin by minimizing the number of weightings.
The book that I am reading states that the above problem can be modeled in the following way. We say that... |
H: can the following sum be simplified
For $n \ge 3$, define
$$f(n) = \sum_{k=3}^n {n \choose k}{k-1 \choose 2}.$$
Is there a closed form expression for $f$?
AI: Note that
$$\binom{n}k\binom{k-1}2=\frac12\binom{n}k(k-1)(k-2)\;,$$
where the $(k-1)(k-2)$ looks like the coefficient of the second derivative of $x^{k-1}$. ... |
H: What is $\mathrm{rank}(A)$ if it is known that $A^4 = 0$?
Given $A \in M_{5 \times 5} (\mathbb F)$, what are the options for $\mathrm{rank}(A)$ if it is known:
(I) $A^4 = 0$
(II) $A^3 = 0$
(III) $A^2 = 0$
Now, I am very new to Jordan Forms and this is related, but I have no clue whatsoever on the relationship bet... |
H: How to evaluate $\lim_{n \to \infty} \sqrt[n]{\frac{1}{n^2}} $.
I was trying to find the radius of convergence of the power series
$$\Sigma \frac{2^nz^n}{n^2}$$
and with the ratio test, found that the radius of convergence is $1 \over 2$.
However, I am practicing on finding limits and I would like to know how to p... |
H: Sum of alternating sign squares of integers stuck with proof by induction
Note that
$$
A(1):1=1\\A(2):1-4=-(1+2)\\A(3):1-4+9=1+2+3\\A(4):1-4+9-16=-(1+2+3+4)
$$
Let us set up the $A(k)$:
$$
A(k)=1-4+9-…+(-1)^{k+1}k^2=(-1)^{k+1}(1+2+…+k)
$$
Setting up $A(k+1)$:
$$
A(k+1)=1-4+9-…+(-1)^{k+1+1}(k+1)^2=(-1)^{k+1+1}(1+2+…... |
H: Conditional independence regarding fourth event
Let's two events $S1$ and $S2$ are conditionally independent given the event $A$, i.e.,
$P(S_1|S_2,A) = P(S_1|A)$ and $P(S_2|S_1,A) = P(S_2|A)$
If $B$ is an arbitrary event, does the following probability hold?
$P(S_1|S_2,A,B) = P(S_1|A,B)$?
AI: No. We can ignore $A$... |
H: General Integral Formula
I know how to find the integral below, but I would like to know if there is any clever or general formula for the integral, since my method involves simple polynomial division...
$\int \frac{1}{1+\sqrt[n]x}dx$
Thanks.
AI: $$\int \frac{1}{1+\sqrt[n]x}dx = \sum_{k=0}^{\infty}(-1)^k \int x^{\f... |
H: Is thi set of vectors, $\{(2, 1), (3, 2), (1, 2)\}$, is linearly dependent or independent?
Given a set of vectors
S = $\left\{
\begin{bmatrix} 2 \\ 1 \end{bmatrix},
\begin{bmatrix} 3 \\ 2 \end{bmatrix},
\begin{bmatrix} 1 \\ 2 \end{bmatrix}
\right\}
$
Find out if the vectors are linearly dependent or independent
... |
H: do you need a Noether ring for Noetherian Theorem?
just wondering if a Noetherian ring has any relation to the conservation law of Noether's Theorem? I thought I read the universal enveloping algebra can fall under a Noetherian ring, and was wondering if a Noetherian ring implies Noether's Theorem?
AI: They are ju... |
H: Minimum number of k length paths over n vertices
(excuse my lack of Math Theory)
I have $N$ number of vertices and can make paths of up to $K$ length. How do I figure out the minimum number of paths to form the complete graph of $N$ vertices.
What is a complete graph?
A complete graph is $N$ vertices that each v... |
H: Realizing groups as symmetry groups
We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts by symmetries on the universal cover. Does anyone have any examples of (non-Abeli... |
H: Error in understanding the theorem about the invertibility of an element(coset) of a quotient ring
There's a theorem in Abstract Algebra which states that:
An element of a quotient ring $\mathbb{Z}/\langle n \rangle$ or $\mathbb{Z_n}$ that is a coset $\overline{a}$ is invertible iff $a$ and $n$ are relatively prime... |
H: Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges.
Intuitively speaking, I first thought that if the series $\Sigma a_n$ is divergent then
$$\lim_{n \to \infty} a_n \ne 0$$
therefore it was clear that $\Sigma \frac{a_n}{1+a_n} $ would be divergent, but when I thought about it there are cases... |
H: Find $f^{(1001)}(0)$
I am to find the value in 0 of 1001th derivative of the function $$f(x) = \frac{1}{2+3x^2}$$
How should I approach this kind of problem? I tried something like : $$\frac{1}{2+3x^2} = \frac{1}{2}\cdot\frac{1}{1-(-\frac{3}{2}x^2)}= \frac{1}{2}\sum_{n=0}^{\infty}\left(\frac{3x^2}{2}\right)^n$$ and... |
H: Dividing factorials is always integer
Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
AI: I'm guessing you would like a noncombinatorial proof, as it ${n \choose k}$ is used to count and therefore must be an integer. If that's the case, here's an idea:
Hint: $$\frac{n!}{r!(n-r)!} = \f... |
H: find the dimension of vector subspace $S_p$ of $M_3(\mathbb{R})$
I need to find the dimension of vector subspace $S_p$ of $M_3(\mathbb{R})$ we have fixed a matrix $P$ which is singular and $S_p=\{X: PX=0\}$ I defoned a linear map from $M_3(\mathbb{R})\to M_3(\mathbb{R}), X\to PX$ but not able to get the $\dim Ima... |
H: any subgroup of $(\mathbb{Q},+)$
Any subgroup of $(\mathbb{Q},+)$ is _______
cyclic and finitely generated but not abelian and normal,
cyclic and abelian but not finitely generated and normal,
abelian and normal but not cyclic and finitely generated, or
finitely generated and normal but not cyclic and abelian... |
H: How to calculate surface area of a curved plane?
could anyone explain how to calculate the surface area of a curved plane? I am trying to calculate the surface area of a "vaulted" ceiling that is 24' long, 7' wide, and the height of the curve is 4' at the mid-point so that I can figure out how much paint I need to... |
H: An operator on $H\times H$, with $H$ Hilbert
Let $(H, \langle \cdot,\cdot\rangle_H)$ a Hilbert complex space and consider $H\times H$ with the inner product
$$\langle (u,v),(z,w)\rangle_{H\times H}\ =\ \langle u,z\rangle_H + \langle v,w\rangle_H.$$
Let, $A\in\mathcal{L}(H,H)$ and define the operator $B:H\times H\r... |
H: How to understand ideals in $F$, which is a finite commutative ring with $1$
I do not fully understand about ideals in finite rings, and I have to choose the correct answer to the following:
If $F$ is a finite commutative ring with $1,$ then
(i) Each prime ideal is a maximal
ideal.
(ii) $F$ has no nontrivial maxima... |
H: Isomorphism including a tensor product
I whould like to konw why the following map is an isomorphism :
$ K[X_1,...,X_n]/(f_1,...,f_r) \otimes K [Y_1,...,Y_n]/(g_1,...g_k)
\simeq K[X_1,...,X_n, Y_1,...,Y_m] / (f_1,...,f_r,g_1,...g_k) $ which $ K $ is an algebraically closed field.
Here is my suggestion :
We put : $... |
H: Solving equation with absolute value signs
Can someone see why there is only get one solution when solving following equation in this way:
The equation $|x+1|+|2x-3|=|x-5| $
$$|x+1|+|2x-3|=|x-5| $$
$$\pm (x+1) \pm(2x-3)=\pm(x-5)$$
$$\pm x \pm 1 \pm 2x \mp 3 = \pm x \mp 5$$
$$\pm x \pm 2x \mp x \pm 1 \pm 5 \mp 3=0 $... |
H: Area moment of Inertia and Center of Gravity
Can someone please explain to me once and for all, why is the moment of inertia of a body $A$ Is calculated as:
$$I_x = \int_A y^2 dA ,\quad I_y= \int_A x^2 dA .$$
I searched a lot google for a summary and derivation, but couldn't find any good one that explains in detai... |
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