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H: Simplifying easy expression
$\frac{u-2}{u^{1/2}}=u^{1/2}-2u^{-1/2}$
How does this work?
I know that it adds up, but I'm just not sure about the mechanics at play.
Are there any similar cases? Thank you.
AI: It’s a matter of splitting the fraction into a difference of two fractions, each of which is then simplified ... |
H: Eigenvalues of the product of two symmetric matrices
This question may seem a little dumb, but I really googled around with no success.
Let $A$ and $B$ be two square matrices of size $n$ whose (real) eigenvalues are denoted by
$$\lambda_1(A) \leq \lambda_2(A) \leq \cdots \leq \lambda_n(A)$$
$$\lambda_1(B) \leq \l... |
H: Learning Complex Geometry - Textbook Recommendation Request
I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in learning complex analytic & complex algberaic geometry both.
C... |
H: Let $f,g \in {\mathscr R[a,b]}.$ If $\int^{b}_{a}f^2=0,$ then $\int^{b}_{a}fg=0.$
Let $f,g \in {\mathscr R[a,b]}.$ If $\int^{b}_{a}f^2=0,$ then $\int^{b}_{a}fg=0.$
I have shown that $2|\int^{b}_{a}fg|\leq t\int^{b}_{a}f^2 + \frac{1}{t}\int^{b}_{a}g^2, t>0.$ Hence how do I show $\int^{b}_{a}fg=0?$ Thank you.
AI: HIN... |
H: Finding the shortest distance between a point and a circle
The question is "Find the shortest distance from the origin of the graph of the circle $x^2-14x+y^2-18y+81=0$ ".
I found the circle in the following form: $(x-7)^2+(y-9)^2=7^2$
Then I found the line that connects the origin $(0,0)$ and the center $(7,9)$, a... |
H: Cotangent bundle of a $n$-dimensional differentiable manifold is a $2n$-dimensional manifold
How to prove that cotangent bundle of a $n$-dimensional differentiable manifold is a $2n$-dimensional manifold? Detailed explanation is welcome. Thanks in advance.
AI: I suppose you are given a definition of tangent space t... |
H: If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:
If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then:
$$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) \left(1+\dfrac{1}{x_2}\right)\left(1+\dfrac{1}{x_3}\right)\left(1+\dfrac{1}{x_4}\right)\left(1+\dfrac{1}{... |
H: How do I calculate the marginal probability density function of Y?
Let $\displaystyle{{\rm f}\left(x, y\right) = \left\{x\quad \mbox{if}\quad 0< y < {1 \over x}\right\}\ \mbox{and}\ 0}$ otherwise.
I need to calculate the marginal pdf of $Y$. I know I need to integrate out $x$, but I'm having a hard time seeing w... |
H: Subset, Not a subset, And Elements
Question $1$) Write $\subseteq$ or $\not\subset$:
$\Bbb N\underline{}\Bbb Q$
$\Bbb N\underline{}\wp(\Bbb R)$
$\varnothing\underline{}\Bbb Z$
$\sqrt2\underline{}\Bbb R$
$\Bbb Z \cup [-1,1]\underline{}[-2,2]$
Question $2$) Write $\subseteq$ or $\in$:
$(3,5)\underline{}[3,5]$
$[-1,4]... |
H: Equivalent definitions of quasi-projective algebraic sets.
I'm trying to prove this equivalence which are also the definition of quasi-projective algebraic sets:
$X\subset \mathbb P^n$ is an open subset of its closure $\Leftrightarrow$ $X\subset \mathbb P^n$ is an open subset of a closed subset of $\mathbb P^n$.
Th... |
H: Lebesgue measure - $\alpha$ be defined for arbitrary subsets of $X$
I'm trying to solve the following question:
Let $X$ be a set and let $\alpha$ be defined for arbitrary subsets of $X$ to $R$ and satisfy
$0\leq\alpha(E)\leq\alpha(E\cup F)\leq \alpha(E)+\alpha(F)$,
when $E$ and $F$ are subsets of $X$. Let $S$ be th... |
H: Finite calculus: Apply difference operator to generalized falling factorial $(ax+b)^{\underline m}$
The $m$th falling factorial power of $x$ is defined as
$x^{\underline m}:=x(x-1)...(x-m+1),$
and the difference operator as
$\Delta f(x) := f(x+1)-f(x).$
One fundamental statement in finite calculus is the identity
$... |
H: Prove that x is in the boundary of A iff x is an accumulation point of the complement of A given x is an isolated point of A
The complete question is the following: "Let A be a subset of metric space X and let x be an isolated point of A. Show that x is in the boundary point of A iff x is an accumulation point of $... |
H: Is a group cyclic with its generators
A group (S, $\odot$) is called cyclic if there exists g $\in$ S such that for every a $\in$ S there exists an integer n such that a = g $\odot$ g ... $\odot$ g (n times). If such a g $\in$ S exists, it is called a generator.
Is the group $\mathbb{Z}^{*}_{13}$= {1, 2 ... 11, 1... |
H: Differentiating under integral for convolution
I have a function $f\in L^1(\mathbb{R}) $ and $g(x)=\dfrac{1}{2\sqrt{\pi t}}e^{-\frac{(at+x)^2}{4t}}$, where $a,t\in\mathbb{R}$, $t>0$. I want to show that $$\dfrac{d}{dx}\int_{-\infty}^\infty f(y)g(x-y)dy=\int_{-\infty}^\infty f(y)\dfrac{d}{dx}g(x-y)$$ Leibniz doesn't... |
H: Is this horse proof by induction okay?
Let $P(n)$ be the statement "all horses in a set of n horses are of the same colour."
Basis Step: Clearly, $P(1)$ is true.
Inductive Hypothesis: Suppose that $P(k)$ is true for some arbitrary integer $k\geq 1$; that is, all horses are of the same colour.
Inductive Step: We now... |
H: Approximating the Indicator Function Using Continuous Functions
How do you approximate the indicator function $1_{A}(x)$ (which equals 1 if $x\in A$, 0 if $x$ not $\in A$ using continuous functions?
AI: EDIT: I realized I was assuming that you want to approximate a simple function $\Sigma c_i \chi_{E_i}$ by a conti... |
H: Conclusion about limit definition of e^a for a sequence of real numbers {a_n} converging to a?
I have seen this fact used in several demonstrations, but have never seen a proof of it.
I believe the statement is: If $\{a_n\}$ is a sequence of real numbers such that $a_n \rightarrow a$ finite, then $(1 + \frac{a_n}{n... |
H: Compact metric spaces is second countable and axiom of countable choice
Why we need axiom of countable choice to prove following theorem:
every compact metric spaces is second countable?
In which step it's "hidden"?
Thank you for any help.
AI: Usually the proof would go like this:
For every $n$ define $\mathcal U_n... |
H: Inverse eigenvalue of a linear transformation
T is a linear transformation, and $\lambda$ is N eigenvalue of T. How do I prove that $\lambda^{-1}$ is an eigenvalue for $T^{-1}$?
I know for a matrix, I can use the fact that $Av=\lambda v$, but how does a linear transformation work?
AI: If $\lambda\ne0$ (and we know... |
H: How to show $|(1-z)e^z| \geq e^{-|z|^2}$ for all $|z| \leq 1/2$.
I'm trying to prove the above inequality, but keep running into difficulties. It seems breaking up the RHS into $e^{-x^2}e^{-y^2}$ doesn't help much, and similarly writing LHS $\geq |e^{z}| - |z||e^{z}|$ isn't useful. Any help would be appreciated.
AI... |
H: Help with limit of trigonometric function
$$\lim_{x \to 0}\frac{x \csc 10x}{\cos20x}$$
I'm unsure of how to solve this. I think if I were to simplify it, it would be:
$$\lim_{x \to 0}\frac{x \csc 10x}{\cos20x} = \lim_{x \to 0}(\frac{1}{x\sin10x})\div(\frac{1}{\cos20x})$$
$$\implies\lim_{x \to 0}\frac{1}{x\s... |
H: Show that for any $n \in \mathbb{Z}, n^3$ is congruent to 0,1,-1 modulo 9.
Having a little difficulties with this one. Tried thinking of going down the line of even/odd proofs, but couldn't get anywhere.
AI: any integer has at least one of these representations:
$$3k, 3k+1, 3k-1$$
in cubic power
$$27k^3, 27k^3 + 2... |
H: Existence of differential form on a manifold
I have a fundamental question about the existence of differential forms on manifolds.
A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a smooth function.
Given any smooth function $f$, is $f(x_1,...,x_n)dx_i$ a $1... |
H: Showing a matrix can't be factored into unit lower triangular matrix and upper triangular matrix
I'm trying to show the following matrix cannot be factored into the product of a unit lower triangular matrix and an upper triangular matrix.
$$\pmatrix{ 2 & 2 & 1 \\ 1 & 1 & 1 \\ 3 & 2 & 1}$$
I'm trying different row o... |
H: Quotients of the Ordered Square
Let $[0,1]^2$ be the ordered square; i.e. it has the order topology given by the dictionary order. This is a first countable compact space. Let $\Delta=\{(x,x)\mid x\in [0,1]\}$. Then is the quotient space $[0,1]^2/\Delta$ first countable?
It isn't hard to see that there is a co... |
H: Is $\mathbb{Z}_p$ a Finite Field?
Denote the integers modulo $p$, $\mathbb{Z}$ mod $P$, as $\mathbb{Z}_P$. Denote the set of integers equivalent to $n$ mod $P$ - the equivalence class of $n$ as $\overline{n}$.
We know that for any prime $p$, $\mathbb{Z}_P$ is a field. As a finite field contains a finite number of e... |
H: prove there's another subsequental limit
$a_n$ a sequence such that:
$\forall n \in \mathbb{N} :|{a_{n + 1}} - {a_n}| < 1$
$\{ 0,2\} \subseteq P({a_n}) $
Prove there's another subsequental limit, $L \ne 0,2$
I'll be glad for help here
AI: Let $B=(\infty, \frac{1}{2})$, $M=[\frac{1}{2}, \frac{3}{2}]$ and $T=(\fra... |
H: Prove that set of all points on a sphere is uncountable
Let $S=\{(x,y,z): x^2+y^2+z^2=4\}$ be the set of points on a sphere. Prove $S$ is uncountable.
Attempt: Basically, each coordinate is between $0$ and $2$, i.e. $0\le x \le 2, 0\le y \le 2, 0\le z \le2$. So if I prove that for some $a$ set $A=\{a \in \mathbb{... |
H: Hankel trasformation of acoustic wave equation
We consider a simplified version of acoustic wave equation
\begin{equation}
\frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 p=\frac{1}{r} \delta(r) \delta(z-z_0)
\end{equation}
where p is a complex quanti... |
H: Number of equivalence relations with a fixed size
How can I find the number of equivalence relations R on a set of size 7 such that |R|=29?
Any advice would be greatly appreciated! :D
AI: HINT: Let $A$ be the set of size $7$. Think about the partition corresponding to the equivalence relation $R$. Suppose that it ... |
H: what are the differences between alignment and colinear?
Im reading a chapter talking about orthogonal complement of dual space in optimization by vector space. And the author introduced a definition of Alignment as following:
where X* means the dual of X and < x , x* > denote a functional as following:
What are... |
H: Probability of a event in time
I consistently see the probability P(random variable at time t) = F(random variable before t-1) - F(random variable before t) or some form of this. How do I verify this?
(Sorry if the notation is goofy)
Edit: F is the cdf
AI: I assume here that $F$ is meant to denote some sort of cumu... |
H: how do I compute the eigenvectors for spectral clustering from a singular value decomposition?
I am implementing spectral clustering following A tutorial on spectral clustering. After preparing the Laplacian matrix $L^{n \times n}$, I compute the Singular Value Decomposition $U \Sigma V^{*}$. From $\Sigma$ I extrac... |
H: Finding positive integer solutions to $3^x + 55=y^2$
I think it must be finite, $y$ is always even, but I don't know how to continue.
edit: with $x,y\in\mathbb Z$
AI: Hint Modulo $4$ we have
$$(-1)^x \equiv 1 \pmod{4} \Rightarrow x =2k$$
Then
$$(y-3^k)(y+3^k) =55$$
Now all you have to do is check all possible facto... |
H: The Magic Hat Puzzle ...Vinnie's number is always one above or below Tommy's...
Imagine there is a hat sitting on the table. And there are two contestants. You, Tommy, will be one of the contestants, and we'll call the other one Vinnie.
You reach into the hat, and pull out a number. Then, Vinnie does the same. Now,... |
H: Price-Demand, Marginal-price and other financial jargon
So my book likes to assume that I already have a business degree while learning calculus so I need your help to clarify my book's questions.
It asks:
Price-demand equation.
The marginal price for a weekly demand of x bottles of shampoo in a drugstore is given ... |
H: Chern classes are not numbers, are they?
Let $X$ be a smooth projective algebraic variety, say over $\mathbb C$. Let $E$ be a rank $r$ vector bundle on $X$. We can associate with $E$ its Chern classes $c_i(E)$. When I read "$c_i(E)$", the first thing I (automatically) do is to think where it lives. And it lives in ... |
H: Probability of forming a 3-senator committee
If the Senate has 47 Republicans and 53 Democrats,
in how many ways can you form a 3-senator committee
in which neither party holds all 3 seats?
The solution says that:
You can choose one Democrat, one Republican, and one more senator from either party. We can make the... |
H: Why is $S = \{(x,y) \in \mathbb{R}^2 \mid \text{$x$, $y$ even integers}\}$ not a subspace of $\mathbb{R}^2$?
Let $S = \{(x,y) \in \mathbb{R}^2 \mid \text{$x$, $y$ even integers}\} \subset \mathbb{R}^2$.
Why is this not a subspace of $\mathbb{R}^2$? $0$ seems to be in it ($2(0) = 0$), $x+y$ seems to be in it, and $k... |
H: Define a relation $\sim$ on $\mathbb{N}$ by $a\sim b$ if and only if $ab$ is a square
(a) Show that $\sim$ is an equivalence relation on $\mathbb{N}$.
(b) Describe the equivalence classes [3], [9], and [99].
(c) If $a\sim b$, which attributes of $a \text{ and } b$ are equal?
For (a) I have to show that $\sim$ is re... |
H: The $\sigma$-algebra of a class.
We've got the following definition
Let $\mathcal C$ be a class of subsets of $\Omega$. We say that $\sigma(\mathcal C)$ is the $\sigma$-algebra generated by $\mathcal C$ if satisfies that:
1. $\mathcal C\subseteq \sigma(\mathcal C)$.
2. If $\mathcal C\subseteq \mathcal A$, wi... |
H: Prove that relation $R$ on a set of functions is an equivalence relation
Let set $S$ be the set of all functions $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$. Define a realtion $R$ on $S$ by $(f,g)\in R$ iff there is a constant $M$ such that $\forall n (\frac{1}{M} < \frac{f(n)}{g(n)}<M). $ Prove that $R$ is an equiv... |
H: Prove that $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a norm for a complete space
Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space.
I can't prove the completeness of this space. Hope someone can help me. Than... |
H: Evaluate $\int_{\partial \mathbf{D}} f(z) dz$ for some meromorphic $f$.
This is for homework, so just hints please! The question asks
If $f$ is a meromorphic function in $\mathbb{C}$ that satisfies $|f(z) z^2| \leq 1$ for $|z| \geq 1$, then evaluate $\int_{\partial \mathbf{D}} f(z) dz$ (where $\mathbf{D}$ represe... |
H: Why can't polynomials have negative exponents or division by a variable
Why can't:
$$2x^{-3} - 3x$$
or
$$\frac{1}{2x}$$
be polynomials too?
Why have a definition that excludes these algebraic forms?
AI: Polynomials are defined as they are for a few distinct reasons: (1) because polynomials as functions have certain... |
H: Finding polynomal function with given zeros and one zero is a square root
I've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients.
When I tried it, I got this:
\begin{align}
(x+1)(x-(1-\sqrt{3}))=& x^2 - x(... |
H: Changing one coefficient in a set of linear equations
Consider a set of linear equations described by $A\vec{X}=\vec{B}$ is given, where $A$ is an $n\times n$ matrix and $\vec{X}$ and $\vec{B}$ are n-row vectors. Also suppose that this system of equations have a unique solution and this solution is given.
Imagine a... |
H: Minimizing the sum of a product
I am having a hard time coming up with a function to represent a word problem.
The product of three values equals 192. one of the values is twice another. What is the minimum value of their sum. Given all three values are greater than 0.
So far I have came up with:
$ABC=192$
$A=2B$
$... |
H: How to calculate $E[X^2Y^5]$ given density functions for $x$ and $y$
Let $X$ and $Y$ be random independent variables within the limits $[0, 1]$ with the following density functions:
$f_X(x) = 0.16x + 0.92$ such that $x$ is within the parameters $[0, 1]$
and $f_Y(y) = 1.41y^2 + 0.53$ such that $y$ is within the para... |
H: Negative curvature compact manifolds
I know there is a theorem about the existence of metrics with constant negative curvature in compact orientable surfaces with genus greater than 1. My intuition of the meaning of genus make me think that surfaces with genus greater that 1 cannot be simply-connected, but as my k... |
H: How to calculate the mod value of a rational/irrational value?
We have a course in network security this semester and we are being taught RSA algorithm.
I came across a typical math problem that I was unable to solve here.
$$D*E \equiv 1 \mod{\phi(n)}$$
This became $$D \equiv E^{-1} \mod{\phi(n)}$$ How do you solv... |
H: What's the symbolic definition of the maximum value of a domain?
Lets say we have a domain S
Maximum value of domain S = {S | ? ? ? ? ? ? }
How could one define the possible maximum value of a set of values, symbolically?
AI: If $S$ is an (ordered) set, we write the maximum value of $S$ as $$\max S.$$ If $S = \{s_1... |
H: How to study the convergence of this series?
Is convergent series?
$$\sum_{n=1}^∞{{1}\over{n^2+2\sqrt{n}-21}}.$$
AI: HINT: How does $\dfrac1{n^2+2\sqrt{n}-21}$ compare with $\dfrac1{n^2}$ when $n\ge 121$, say?
The key idea here is that the $n^2$ term is the dominant term in the denominator, so for large $n$ the de... |
H: If a function is positive on a set of measure greater than zero, is the Lebesgue integral of that function greater than zero?
Suppose we have a set $A \subset \mathbb{R}^n$ such that $f(x) > 0$ for $x \in A$ and $m(A) > 0$.
Does it follow that $\int_A f > 0$?
Obviously if there is some kind of lower bound on $f(x)... |
H: To show this $R$ is a PID
I am studying for a qual, and I can't quite figure this one out. Any hints or suggestions for which theorems to use? It might be a very simple problem that I'm just not seeing the solution:
Suppose that $R$ is a Noetherian integral domain and every finitely generated torsion-free $R$-modul... |
H: When do two functions differ by a constant throughout an interval (Fundamental Theorem of Calculus)
I'm reading the proof of the Fundamental Theorem of Calculus here and I don't understand the following parts (at the bottom of page 2):
I don't know how to conclude that $G(x)-F(x)=C$ for a $x \in [a,b]$.
How do I p... |
H: Cauchy Goursat Theorem
If $C$ is the positively oriented unit circle |$z$| = 1, then is it true that
$\int_C\!Log(z+3)\, \mathrm{d}z$ = $0$
Why or why not?
Is is true because its analytic right?
AI: Yes, this is only not analytic on $s:=${$Re(x)\leq3,y=0$}. Here you are using the principle branch of Log. |
H: A noetherian ring $R$ which is commutative integral domain but not a PID?
I am looking for an example of a ring $R$ which is a commutative and Noetherian integral domain but not a PID. Thanks.
AI: In the polynomial ring $\mathbb{Z}[x]$, the ideal $$I = \langle 2, x\rangle$$ is not principal. |
H: Calculation of $\int_{0}^{\pi}\frac{1}{(5+4\cos x)^2}dx$
Calculation of $\displaystyle \int_{0}^{\pi}\frac{1}{(5+4\cos x)^2}dx$
$\bf{My\; Try}::$ Using $\displaystyle \cos x = \frac{1-\tan^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}$
Let $\displaystyle I = \int_{0}^{\pi}\frac{1}{\left(5+\frac{4-4\tan^2 \frac{x}{2}}{1+\tan... |
H: Completeness of a normed vector space
This is captured from a chapter talking about completeness of metric space in Real Analysis, Carothers, 1ed.
I have been confused by two questions:
What does absolutely summable mean in metric space? Does it mean the norm of xi(i=1,2,3,...) that belongs to norm vector space ... |
H: Contour integrals
Evaluate $\int_C\dfrac{\mathrm{d}z}{z^2-1}$ where
a) $C$ is the clockwise oriented circle $\left|z \right| = 2$;
b) $C$ is the anti-clockwise oriented square with sides on $x= \pm2$ and $y= \pm2$;
c) $C$ is the clockwise oriented circle $\left|z-1 \right|= 1$.
So for this I would set $z = x+iy$ a... |
H: Open subsets of the closure
I want to prove that every open subset of a topological subpace is an open subset of its closure.
Let $Y$ be a topological space and $X$ a subspace of $Y$. If $U$ is an open subset of $X$, we have $U=U\cap \overline X$, thus U is an open subset of $\overline X$ also.
Am I right?
Thanks i... |
H: Prove that $\int_0^1 f(x^2)dx\geqslant f\left(\frac{1}{3}\right)$
Let $f:[0,1]\to\mathbb{R}$ be twice differentiable. Suppose $f''(x)\geqslant 0$ for all $x\in[0,1]$. Prove that
$$\int_0^1f(x^2)dx\geqslant f\left(\frac{1}{3}\right).$$
I am thinking of using Taylor's Theorem to expand $f(x^2)$ at $\frac{1}{\sqrt{3}}... |
H: Counting - puzzle question
Suppose that you have infinitely many one dollar bills (numbered
1, 3, 5, . . . ) and you come upon the Devil, who is willing to pay
two dollars for each of your one-dollar bills.
The Devil is very particular, however, about the
order in which the bills are exchanged. The contract stipulat... |
H: If $lim \int f_n dx$ exists and $<\infty$, can we switch limit and integral
A common example that we cannot switch limit and integral is $$f_n=1_{[n,\infty]}$$$lim\int f_ndx=\infty$, while $\int \lim f_ndx=0$. Thus we have Dominated Convergence Theorem. In my knowledge, such examples all deal with some integral whi... |
H: Is the converse of Lagrange's Theorem true for the permutation group $S_5$?
Is the converse of Lagrange's Theorem true for the permutation group $S_5$?
That is, if $n\mid |S_5|$, then is there a subgroup of $S_5$ with order $n$.
Since $|S_5|$ = 5! = 120, then any subgroup must have length of some divisor of 120.
I'... |
H: How to find the inverse laplace transform of [F(s)/s]^n
Let $F(s)=\mathcal{L}\{f(t)\}$, we have $\frac{F(s)}{s}=\mathcal{L}\{\int_o^tf(x)dx\}$. How to find $\mathcal{L}^{-1}\left\{\left(\frac{F(s)}{s}\right)^n\right\},\text{ for}~ n\in \mathbb{N} $
AI: A related problem. Here is a start for the case $n=2$,
Let
$... |
H: What is the explicit formula for the sequence representing the number of any triangles in a triangular grid?
What is the explicit formula for the sequence representing the number of any triangles in a triangular grid, like those below?
Context
Counting only up-facing triangles of size one, we get triangular numb... |
H: Prove that if $2^{4\times5^k}=x\times5^{k+3}+a,0
Let $$2^{4\times5^k}\equiv a \pmod {5^{k+3}},\\2^{4\times5^k}\equiv b \pmod {5^{k+4}},$$
and $0<a<5^{k+3},0<b<5^{k+4},$ prove that $a=b.$$(k>1)$
This is equivalent to this: if $2^{4\times5^k}=x\times5^{k+3}+a,0<a<5^{k+3},$ then $5\mid x.$
ADD: A similar problem:
Prov... |
H: How to find general solution of an L-R-C series circuit with inductance L = 1/5 henry, resistance R=4 ohms, and capacitance c=1/520 farad?
Consider an L-R-C series circuit with inductance L = 1/5 henry, resistance R=4 ohms, and capacitance c=1/520 farad. Find the general form of the charge on the capacitor if this ... |
H: Is the homomorphic image of a PID a PID?
$R$ is a ring which is a PID [i.e., $R$ is an integral domain in which every ideal is generated by a single element] and we are given with a map $f:R\to S$ which is a homomorphism, i.e.
$f(a + b) = f(a) + f(b)$ for all a and b in $R$,
$f(ab) = f(a) f(b)$ for all a and b in R... |
H: Confusion regarding boundedness & Equicontinuity
It is given that $ |f_{n} ' (x) | \le \frac {1}{x^{\frac {1}{3}}} \forall 0 \lt x \le 1$ , where {$f_{n}$} is a sequence of real valued $C^{1}$ function on $[0,1]$ and each {$f_{n}$} has a zero in $[0,1]$ . Now to prove that the sequence has a uniformly convergent su... |
H: Extreme value theorem - condition on continuity for boundedness
According to my math professor, the extreme value theorem is stated as: If $ f: [a,b] \to \mathbb{R} $
is continuous then $f$ is bounded, and the maxima and minima are obtained for some $x$ belonging to the domain.
My intuition tells me that the condi... |
H: Finding a recursive definition and computing $B(10)$
For $n \geq 1$, let $B(n)$ be the number of ways to express $n$ as the sum of $1$s and $2$s, taking order into account. Thus $B(4) = 5$ because $4 = 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1 = 2 + 1 + 1 = 2 + 2$.
(a) Compute $B(i)$ for $1 \leq i \leq 5$ by showing al... |
H: Show that $\sum_{i=1}^{n}x_if'(x_i)=f'(\xi)$
Let $f:(a,b)\to\mathbb{R}$ be differentiable and let $x_1,\dots,x_n\in (a,b)$. Suppose $x_i>0$ for all $i$ and $\sum_{i=1}^{n}x_i=1$. Show that there is $\xi\in(a,b)$ s.t.
$$\sum_{i=1}^{n}x_if'(x_i)=f'(\xi).$$
Clearly by mean value theorem there is $\xi\in(a,b)$ s.t.
$$f... |
H: Prove by minimum counterexample that $2^n>10n$ for $n>5$
Prove by minimum counterexample that for all integers $n>5$ the statement $2^n>10n$ is true.
Attempt: Let $S$ be a set of counterexamples, $S=\{n \in \mathbb{Z_+}: 2^n \le 10n, \space n>5 \}$. Let $m \in S$ be the smallest element of $S$, so $m>5$. Then, $m... |
H: Using Schröder-Bernstein theorem to show same cardinality
Use the Schröder-Bernstein theorem to show that $(0,1)\subseteq \Bbb R$ and $[0,1]\subseteq\Bbb R$ have the same cardinality.
Firstly I'm not even entirely sure about what the syntax even means. The elements of subset (0,1) and [0,1] are also included in th... |
H: contour integrals complex analysis 2
Evaluate $\int_C\!\frac{2z-1}{z^4-2z^2+1}dz$ where $C$ is the circle |$z$|=$10$ oriented clockwise.
I have a exam tomorrow and need to understand this, can someone please help.
AI: Use the residue theorem. Look for the poles (in this case zeros of the denominator) inside $|z|=1... |
H: Triple integral using spherical coordinates
The following function is given: $$\iiint_{x^2+y^2+z^2\leq z} \sqrt{x^2+y^2+z^2}dx\,dy\,dz$$
And I have to calculate this integral using spherical coordinates. The substitutions are standard, I think, but I am having a problem with the limits. $$0\leq\phi\leq\pi$$$$0\leq\... |
H: Show that $y^2-3xy+2x^2=1$ is a solution of the differential equation $4x-36+y'(2y-3x)=0$
I want to show that the given equation($1.$) is a solution of the differential equation($2.$)
$y^2-3xy+2x^2=1$
$4x-3y+y'(2y-3x)=0$
I need to put the derivative of $1.$ in $2.$?
thanks.
AI: Hint: The first equation actuall... |
H: Calculus and line integrals
What's the difference between $\int_C f\,ds$ and $\int_C F \cdot dr$?
And is $\int_C P \, dx+\int_C Q\,dy$ just notation for $\int_C f\,ds$?.
I am referring here to Section 13.2 from James Stewart's Essential Calculus and trying to understand this very basic thing. It seems like there ... |
H: Orthogonality of Haar wavelet functions
I'm reading about wavelets and I bumped into the follwing:
$\text{Haar wavelet is a step function}\; \psi(x), \text{which takes values 1 and -1, when}\; x \;\text{is in the ranges}\; [0, \frac{1}{2}) \;\text{and}\; [\frac{1}{2}, 1).$
$\text{Dilations and translations of the H... |
H: Line integrals of given curves
This question has an integral $$\int(x^4+4xy^3)dx+(6x^2y^2-5y^4)dy$$to be evaluated on the parametric curve $$C:(-(t+2)\cos(\pi t^2), t-1)$$I took the partial derivatives of the terms in the bracket and subtracted them to get $0$. However, this is not the right answer. I don't know an... |
H: Infinite dimensional euclidian space with the product topology metrizable?
Let $\mathbb{R}^{\omega}$ be the space of real sequenes with the product topology.
Is $\mathbb{R}^{\omega}$ metrizable?
AI: Hint: As $\mathbb{R}$ is homeomorphic to $(0 , 2^{-n} )$ for all $n \geq 1$, it follows that $\mathbb{R}^\omega$ (wit... |
H: Prove that $\sqrt[3]{p}$, $\sqrt[3]{q}$ and $\sqrt[3]{r}$ cannot be in the same arithmetic progression
My cousin (he doesn't speak English well so I am writing on his behalf) is trying to do the following problem:
Let $p$,$q$, $r$ be different primes (let's assume $p<q<r$). Show that $\sqrt[3]{p}$, $\sqrt[3]{q}$ an... |
H: Are all metric translations isometries
Let $(M, d)$ be a metric space. I define a translation of $M$ to be a function $f$ from $M$ to $M$ such that $d(x, f(x)) = d(y, f(y))$ for all $x$ and $y$ in $M$. My conjecture is that every translation on $M$ is an isometry under the same metric.
Can anyone prove this, or gi... |
H: Show that $(y-2x+3)^3=(y-x+1)^2$ is a solution of the differential equation $(2x-4y)dx+(x+y-3)dy=0$
I want to show that
$$(y-2x+3)^3=(y-x+1)^2$$
is a solution for:
$$(2x-4y)dx+(x+y-3)dy=0$$
what I did so far is:
$$\frac{dy}{dx}=\frac{4y-2x}{x+y-3}$$
any suggestions?
AI: $(y-2x+3)^3=(y-x+1)^2$
differentiating this ... |
H: Show that $7$ is irreducible in $\Bbb Z[i]$
I have to show that $7$ is irreducible in $\Bbb Z[i]$.
To show irreducibility I have to show that it's not a unit. This is simple to just show exhaustively.
I'm having trouble with the second part which is to show that if it factors into $a.b$ that either $a$ or $b$ is a ... |
H: Uniform continuous distributions - question with a square RV
Question:
Jack wants to build a wooden cylinder, He decided to choose it's radius (Y) randomly s.t $Y\sim U[0,1]$.
a. What is the probability that the radius is in a closed interval $[\alpha,\beta]$? what is Y's density function?
b. What is the prob. that... |
H: A question regarding lines between points.
On pg.13 of Lang's "Second Course in Calculus", the following is asserted:
Let $P=(2,1)$ and $A=(-1,5)$. Then the parametric equation of the line through $P$ and in the direction of $A$ gives us $x=2-t, y=1+5t$.
Shouldn't the equations be $x=2+3t, =1-4t$?
Thanks in advan... |
H: $\mathbb{F}_{p}A$-module
Yesterday I was introduced to the definition of a module in a course: "Homological Algebra". I'm doing a project involving $p-$groups and in the text I got from my supervisor they use the word: $\mathbb{F}_{p}A$-module.
What does $\mathbb{F}_{p}A$-module mean? The problem is that the text ... |
H: For $j \in \{0,...,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true?
For $j \in \{0,...,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true?
I mean $\dfrac{n!}{(n-j)!(j+1)!}$ doesn't have to be an integer. I need this inequality in another exercise, so Is it provable?
AI: Hint: Compare
$$
(n-j)!(j+1)!=1\cdot \ldots\cdot (n-j)\cdot 2 \cdot ... |
H: Field extension $F\subseteq L_1$ and $F\subseteq L_2$ and $[L_1L_2:F]<[L_1:F][L_2:F]$.
I'm searching for an example of field extensions $L1$, $L2$ of $F$ for which $[L_1L_2:F]<[L_1:F][L_2:F]$.
Infact I'm trying prove the problem below. So any hint can be helpful.
Let $K$ be a finite extension of $F$. If $L_1$ and ... |
H: Geometric Solution for Equation with Complex Numbers
Given we have two complex numbers $z_1$ and $z_2$ with $|z_1| = |z_2|$.
How can it be shown geometrically, that $\frac{z_1+z_2}{z_1-z_2}$ is purely imaginery?
AI: Consider the quadrilateral with vertices $0, z_1, z_1+z_2, z_2$. The condition $\lvert z_1\rvert = \... |
H: Infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$
Recently, I encountered a problem about infinite series.
So my question is how to know whether the infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$ is convergent?
AI: To see whether $\sum_2^\infty 1/(n \log n)$ converges, we can use the integr... |
H: solving two simple line integrals
First one is : $$\int_\gamma e^zdz,\quad \gamma(t)=\pi ti,\quad t\in[-1,1]$$
my attempt:
$z=\gamma(t)=\pi ti \quad dz=\pi idt \quad -1\le t\ \le1, $ then
$$\int_\gamma e^zdz=\int_{-1}^1e^{\pi ti}\pi idt=\pi i\int_{-1}^1e^{\pi ti}dt=\pi i|_{-1}^1\frac{e^{\pi ti}}{\pi i}=e^{\pi i}-e... |
H: Vector Space Dimension
Let $A,B$ be $n\times m$, $s\times m$ matrices respectively, and let $$V=\{X\in \mathbb{F}^{m\times n};\ B X A=0\}.$$ Suppose that $$rank(A)=r,\ rank(B)=m.$$ Show that $dim V=m(n-r)$.
I have no idea.
AI: Note that from rank-nullity, we have
$$m = \mathrm{rank}(B) + \mathrm{nullity}(B) = m + \... |
H: proof of succession convergence
Prove that the following sequence is convergent and calculate its limit:
$x_1=2$ and
$x_{n+1} = \sqrt{3 + \frac{x_n^2}{2}}$
I get a limit of 2.4, but by calculating several sequence terms, I can see that it converges to 2.
AI: The sequence $\{x_n\}$ is bounded between $2$ and $\sqrt... |
H: Solution of an equation in a certain field
Let $F$ be a certain field.
Prove or disprove that the following statements:
The equation $X^3=0_F$ has only one solution.
The equation $X^3=1_F$ has only one solution.
Suppose F is finite, then the equation $X^3=1_F$ has only one solution.
I'm pretty sure all of them a... |
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