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H: Maximal ideal in a quotient ring
Consider the ring $A=\mathbb{C}[x,y]/(y^2-x^4+5x-4)$, and consider the ideal $\mathfrak{m}=(y, x+2)$. Is $\mathfrak{m}$ maximal?
My sketched solution: consider an arbitrary element of $A$ and a representative $p(x,y)\in \mathbb{C}[x,y]$ for this element. Then $p(x,y)$ will be of th... |
H: Equality constrained norm minimization
In these slides (5) the dual function for the norm minimization problem:
$$
\min_x \|x\| \quad \mbox{s.t.} \quad Ax =b
$$
is defined as:
$$
g(v) = \inf_x (\|x\| - ν^\intercal Ax + b^\intercal ν)
$$
what I don't understand is why the signs are reserved. The Lagrangian is accord... |
H: Calculate "x % slower/faster"?
Ok, this might sound a bit stupid, but I find these kind of statements "x % slower/faster" pretty confusing.
Let's say I have algorithm A and algorithm B. Algorithm A takes 50 seconds to complete a task and algorithm B takes 25 seconds.
Now I could make the following statements:
"B i... |
H: Finding constant speed
Lets say someone is flying at constant speed from place $X$ to place $Y$ and back.
Going to X takes 5 hours (with the wind) and coming back from X to Y takes 6 hours.
Lets assume that the wind is at a constant speed, then how long would it take for a piece of paper being propelled by the wind... |
H: Why the contour integral of $f(z)=\frac{i-1}{z+i}$ is not zero although it should be because $f(z)$ is analytic?
Why the contour integral of $\,\displaystyle{f(z)=\frac{i-1}{z+i}}\,$ is not zero although it should be because $f(z)$ is analytic? I have used contour $z=\gamma(t)=2e^{it}$, where $0\leq t\leq\pi$.
AI:... |
H: Existence of infinite discrete family of open sets in a non compact topological space
How can I prove if a Hausdorff topological space $X$ is not compact, then there exist a countably infinite discrete family of open sets in $X$.
AI: I don't think this is true. Note that any Hausdorff space $X$ which has an infini... |
H: Mathematical function for four corelated attributes
I have $4$ attributes $A,B,C,D$
each of them takes value between $[0,1]$
The more $A$ and $B$, the more the function value is.
The more $C$ and $D$, the less the function value is.
if C or D equals "one" the function value is one.
How can I model this function.
I ... |
H: Proof that $e^x$ is a transcendental function of $x$?
Let a function $f(x)$ be algebraic if it satisfies an equation of the form $$c_n(x)(f(x))^n + c_{n-1}(x)(f(x))^{n-1} + \cdots + c_0(x)=0,$$ for $c_k(x)$ rational functions of $x$, and let $f$ be called transcendental if it is not algebraic. Is it possible to use... |
H: How many numbers less than $x$ are co-prime to $x$
Is there a fast way , or a direct function to give the count of numbers less than $x$ and co-prime to $x$ .
for example if $x$ = 3 ; then $n = 2$ and if $x$ = 8 ; then $n = 4$.
AI: The relevant function is the Euler Totient function and the link contains several wa... |
H: reformulation of square residues for odd numbers?
If you look at the table of square residues:
http://en.wikipedia.org/wiki/Quadratic_residue#Table_of_quadratic_residues
you will find that for x^2 mod N where N is odd, we have N different residues but with simetry in half:
Example: N=25
mod 25
1,4,9,16,0,11,24,14,... |
H: How to solve $100x +19 =0 \pmod{23}$
How to solve $100x +19 =0 \pmod{23}$, which is $100x=-19 \pmod{23}$ ?
In general I want to know how to solve $ax=b \pmod{c}$.
AI: Hint $\displaystyle\rm\,\ mod\ 23\!:\,\ x\,\equiv\, \frac{\color{brown}{-19}}{4\cdot \color{#0A0}{25}}
\,\equiv\,\frac{\color{brown}4}{4\cdot\color{#... |
H: Finding all $x$ for $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$
I'm trying to find all $x$ for the inequality $\frac{2x - 13}{2x + 3} \lt \frac{15}{x}$. In order to do this, I want to factor one side so that I can find all values where $x$ determines the term to equal $0$.
$$\frac{2x - 13}{2x + 3} \lt \frac{15}{x} \i... |
H: Complex analysis: Radius of convergence of power series
$$
\sum_{n=1}^\infty \frac{\cos n \theta}{(\sqrt{13})^{n+1}}x^n
$$
Find the radius of convergence for the above series. I have learnt to use the root test and ratio test but neither of them seem to work. I have problems manipulating.
Not sure if this is useful... |
H: Radius of Convergence of this Series
This is a question from a GRE math subject test practice material.
$$ \sum^{\infty}_{n=1} \frac{n!x^{2n}}{n^n(1+x^{2n})} $$
The set of real numbers $x$ for which the series converges is: $\{0\}$, $\{-1 \leq x \leq 1\}$, $\{-1 < x < 1\}$, $\{-\sqrt{e} \leq x \leq \sqrt{e}\}$ or ... |
H: a non-monotonic function on [0,1] with infinitely many points of discontinuity such that the function is bounded & Riemann integrable on [0,1].
could any one Give an example of a non-monotonic function on $[0,1]$ with infinitely many points of discontinuity such that the function is bounded & Riemann integrable on ... |
H: Similarity of matrices over $\Bbb{Z}$
Let $A$ be a $2 \times 2$ matrix over $\Bbb{Z}$ with characteristic polynomial $x^2 + 1$. Determine whether $A$ is similar to
$$\pmatrix{0 & -1 \\ 1 & 0}$$
over $\Bbb{Z}$.
I'm so used to working with matrices over fields $\Bbb{Q}$ and $\Bbb{C}$ that I'm not sure how it differs... |
H: Relationship between different sequences generated through modulo arithmetic.
I am unsure the formal mathematical terminology/notation for dealing with sequences generated from integer modulo arithmetic. So first off, could someone recommend a book that focuses on the sequences generated from arithmetic operations... |
H: about a not continuous function and its derivatives
I'll start with a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ whose partial derivatives exists at a point, but is not continuous at that point.
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by
$f(x)=\begin{cases}1, &\mbox{if}& x=0 &\mbox{or if}& y... |
H: Little-o vs. asymptotic equivalence
As big- and little-o notation are a little too technical to me, I prefer an expression with asymptotic equivalence ($\sim$). However, how does one "translate" this expression to an asymptotic equivalence? It's about the primorials, as found on Wikipedia (http://en.wikipedia.org/w... |
H: What type of graph problem is this?
Lets say I have four group
A [ 0, 4, 9]
B [ 2, 6, 11]
C [ 3, 8, 13]
D [ 7, 12 ]
Now I need a number from each group(i.e a new group) E [num in A,num in B, num in C, num in D], such that the difference between the maximum num in E and minimum num in E should be possible lowest.Wha... |
H: What is an irreducible character of a finite group?
Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b \in S_n$ [NOTE: this is not 100% accurate - see comments and answ... |
H: Solving the cubic polynomial equation $x^3+3x^2-5x-4=0$
How can I solve the cubic polynomial equation $$x^3+3x^2-5x-4=0$$
I simplified it to:
$$x(x^2+3x-5)=4$$
But I don't know where to go from here.
AI: The manipulation that you performed does not help. Use the rational root test: since the leading coefficient i... |
H: Is this a sufficient condition for a subset of a topological space to be closed?
Let $X$ be a topological space, and let $\{U_i\}$ be an open cover. If $Y$ is subset of $X$ such that $Y\cap U_i$ is closed in $U_i$ (for each $i$), does this imply that $Y$ is closed in $X$?
AI: Note that $Y^c\cap U_i = U_i \setminus ... |
H: distance between centers of two touching circles
Lets say we have two circles with radii 2 and 3 respectively.
If we then put Circle radius 2 on a flat surface, then the other circle on the flat surface so they touch once, what is the distance between the two centers?
AI: A the point of contact the circles share a ... |
H: What does $H(\kappa)$ mean?
As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence:
I've heard that if $\kappa$ is a strongly inaccessible cardinal, then $H(\kappa)$ (or sometimes $H_\kappa$) equa... |
H: Algebra over a Field
Let $\mathbb A$ be an algebra of dimension $k$ over the field $\mathbb F$. It is true that $\mathbb A$ is isomorphic to a subalgebra of the matrix algebra $M_k(\mathbb F)?$
AI: Yes, if the algebra $\mathbb{A}$ is associative then as Sean Eberhard points out the left-multiplication maps give lin... |
H: nonlinearity of PDE's
(1) $u_x + u_y = 0$
(2) $u_x + yu_y = 0$
(3) $u_x + uu_y = 0$
(4) $u_{xx} + u_{yy} = 0$
(5) $u_{tt} − u_{xx} + u^3 = 0$
(6) $u_t + uu_x + u_{xxx} = 0$
(7) $u_{tt} + u_{xxxx} = 0$
(8) $u_t − iu_{xx} = 0$
Can someone provide an explanation from definition of linearity why (3), (5), and (6) are N... |
H: Divisibility of consecutive natural numbers
I found this task followed by a hint,that I should try to apply Chinese remainder theorem to that:
Prove, that there exist 2012 consecutive natural numbers,which satisfy that every one of them is divisible by a cube of a natural number $\ge$ 2.
The problem is, I don't r... |
H: Why a quadratic equations always equals zero?
On evaluating quadratic equations, It always equals zero:
$$ax^2+bx+c=0$$
Why zero? Is it possible to use other number for another purpose?
AI: The value of c is a simple number with no variable. So you can move any value on the right side over to the left and it will ... |
H: Dimension and basis of kernel and image of linear transformation
From a bank of past prelim exams:
Let $V$ be the real vector space of real polynomials of degree at most $5$, and define $T: V \to \Bbb{R}^3$ by $T(f) = (f(0), f(1), f(2))$. Find the dimensions of the kernel and image of $T$, and find a basis for eac... |
H: How to show that $\sum\limits_{n=1}^\infty \exp(i\,nz)$ converges?
How do I show that $$ f(z) = \exp(i\, z) + \exp(i\, 2z) + \ldots + \exp(i\, nz) + \ldots $$ converges? Problem is taken from a Yahoo! Answers question: "Find the infinite sum of sin(n)/n?".
AI: For the series to converge, you have the following cond... |
H: does it come from cauchy integral formulae
Well I have to do the integral $$\int_{|z|=4}\frac{f(z) dz}{(z-2)(z-3)}$$
where $f(z)=\sin(\pi z^2)+\cos(\pi z^2)$
one breaks like: $$\int_{|z|=4}\frac{f(z) dz}{(z-2)(z-3)}$$ $$=\int_{|z-2|=1/2}\frac{f(z)/(z-3) dz}{(z-2)}$$ $$+\int_{|z-3|=1/2}\frac{f(z)/(z-2) dz}{(z-3)}$$
... |
H: In how many ways a number $\gt 5000$ can be formed using given digits without repeating?
In how many ways one can form a number greater than $5000$ when allowed only to arrange digits taken from $2,3,4,5,8$ without repeating any digit?
I would think it would be $2\cdot4\cdot3\cdot2$.
Would this be correct?
AI: We n... |
H: Yet another differential equation
Hello,
I would appreciate any help solving the following equation:
$$\begin{align}
y''[t] + \dfrac{d}{m}y'[t] + \dfrac{k}{m}y[t] = G \\
\end{align}$$
subject to:
$$t[0] = t0$$
$$t'[0] = 0$$
This is not HW, this is for a website where I try to simulate object attached to a spring fa... |
H: Contradiction?: Partitions/Equivalence classes vs. Bijective homomorphism
I am reading through Michael Artin's Algebra, and the following passages seem contradictory to me.
On page 55: "The elements of the image correspond bijectively to the nonempty fibres, which are the equivalence classes."
On page 56: Propositi... |
H: How do I "describe the region" of $\Bbb R^3$ represented by an inequality?
I don't really understand what these two questions are asking. Nor do I know how to start it.
1) Describe in words the region of $\mathbb {R}^3$ represented by: $x^2 + z^2 \le 9$.
2) Write an inequality to describe: The solid upper hemis... |
H: When does the product topology have a countable base?
Could any one tell me how to prove this one?
The product topology has a countable base if and only if the
topology of each coordinate space has a countable base and all
but a countable number of coordinate spaces are indiscrete
AI: The harder direction is t... |
H: Morse functions are dense in $\mathcal{C}^\infty(X,\mathbb{R})$.
In Shastri's Elements of Differential Topology, p.210-211, there is written:
Why do we get a Morse function $f_u$ on $X$? We know that for any $f\!\in\!\mathcal{C}^\infty(X,\mathbb{R})$, there is some $a\!\in\!\mathbb{R}^N$, such that $f_a(x)=f(x)\!+... |
H: find all the entire functions that satisfies a given condition
I have been struggling to find a solution for this problem:
Find all the entire analytic functions $f(z)$ (analytic in the complex plane) that satisfy the condition $|z^2f(z)-3+e^z|\leq3$ for all $z \in \mathbb{C}$.
Any ideas?
Thank you in advance.
AI: ... |
H: how to show that power series is analytic inside the radius of convergence?
Let $f(z) = \sum a_n z^n$ be a power series with radius of convergence $R$. How do we show that $f$ is analytic in the circular region of radius $R$?
AI: Lemma
Let $R$ be the radius of convergence of $f(z) = \sum_{n=0}^{\infty} a_nz^n$.
Let... |
H: Leading coefficient of polynomial with more than one variable
What is the leading coefficient of a polynomial with more than one variable, when two or more terms have the same degree but different coefficients?
For example:
$3x^2y^2 + 5xy^3$.
The degree is 4. Is the leading coefficient 3, 5, both, none?
Thanks.
AI... |
H: How many distinct degree 7 polynomials are there over the modular arithmeic modulo 7?
If it's infinite, is it countable or uncountable infinite?
I am a newbie to this topic... I don't know what modular arithmetic for polynomials means. Can someone please give me a link where I can learn?
AI: There are $8$ coefficie... |
H: How to find intersection of an ellipse and a line that passes through the foci
There are two lines, parallel to the $x$-axis, which pass through the foci and intersect the ellipse at four points. How can I find the points of intersection?
vertex: $(0,0)$
foci: $(0,10)$ and $(0,-10)$
co-vertices: $(20\sqrt2, 0)$ an... |
H: What is the locus of $\mathrm{arg}(\frac{z-a}{z-b})=\theta$?
This isn't quite homework, but it might as well be. I'm working through Needham's Visual Complex Analysis and I'm finding myself stuck on the following problem:
Explain geometrically why the the locus of z such that
$\mathrm{arg}(\frac{z-a}{z-b})=\thet... |
H: Why do $r=\cos[2\theta]$ and $r=\frac{1}{2}$ have 8 intersections?
I am not sure if the question is a duplicate. The conclusion from the title is clear, for example by viewing the wolframalpha graph at here.
But on the other hand I feel intuitively there should be only 4, since a line intersect with $y=\cos[\theta]... |
H: How to approximate the value of $ \arctan(x)$ for $x> 1$ using Maclaurin's series?
The expansion of $f(x) = \arctan(x)$ at $x=0$ seems to have interval of convergence $[-1, 1]$
$$\arctan(x) = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}-\frac{x^{11}}{11}+\mathcal{O}\left(x^{13}\right) $$
Does it mean t... |
H: Graph of the infinite sum $g(x)=\sum_{k=1}^\infty \sin^k(kx)$
For which real numbers $x$ does the infinite sum $\sin(x)+\sin^2(2x)+\sin^3(3x)+ \cdots+\sin^n(nx)+\cdots$ converge?
How does its graph look?
AI: Let $x=r\pi$ and consider the following cases:
$r=\frac{a}{2b}$: For infinitely many $n$ we have $n=(2m+1)b... |
H: Solving simultaneous equations with imaginary numbers
Consider the following simultaneous equation:
$$\begin{cases} 5z-(3+i)w=7-i\\ (2-i)z+2iw = -1+i \end{cases} $$
What is the simplest way to manipulate one of the equations so that a variable can be eliminated and the equation solved?
AI: You do it the same way yo... |
H: $G/H$ is a finite group so $G\cong\mathbb Z$
Let $G$ is an abelian infinte group such that for all nontrivial subgroups $H$ $$\forall H\leq G, \left|\frac{G}{H}\right|<\infty$$ Prove that $G\cong\mathbb Z$.
What I have done:
Clearly, it is enough to show that $G$ is a cyclic group. Moreover, I see as $G/H$ is a f... |
H: Calculate absolute values with unknown constant
I am to calculate all $x$ if $f(x) = g(x)$ and if
$$f(x)= |2x+2| + |3-2x|$$
$$g(x)= x + 14$$
How do I mix regular numbers with absolute values in such a sense? I thought I could calculate it like this:
$$|2x+2| + |3-2x| = 3x+2 + 3+2x = 3x+5$$
But then I realised that ... |
H: Subgroups of $\Bbb{R}^n$ that are closed and discrete
I am trying to prove that every closed discrete subgroup of $\Bbb{R}^n$ under addition is a free abelian group of finite rank. I have tried to do this by induction on the dimension $n$.
Base case: We claim that a subgroup $H \subset \Bbb{R}$ has a least posit... |
H: Wilson's theorem related problem
prove $$18! \equiv -1 \pmod{437} $$
I do not want full solution to the above problem but if anybody can tell me how we can approach to it, I will really appreciate that.
AI: hints :
$437=19\cdot 23$ (as proposed by Sean)
Wilson's theorem :-)
$19\cdot 20\cdot 21 \cdot 22=... |
H: Proving $n^4 + 4 n^2 + 11$ is $16k\,$ for odd $n$
if $n$ is an odd integer, prove that $n^4 + 4 n^2 + 11$ is of the form
$16 k$.
And I went something like:
$$\begin{align*}
n^4 +4 n^2 +11
&= n^4 + 4 n^2 + 16 -5 \\
&= ( n^4 +4 n^2 -5) + 16 \\
&= ( n^2 +5 ) ( n^2-1) +16
\end{align*}$$
So, now we have to prove that... |
H: General question about solving equations involving a definite integral
Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\mathbb{C}$ is the unknown variable, and the ex... |
H: Does a fundamental solution set exist for homogeneous first-order difference equations?
When $A$ is diagonalisable, $\vec{x}_{k+1}=A\vec{x}_k$ implies that
$\vec{x}_k = c_1\lambda_1^k\vec{v}_1 +...+c_n\lambda_n^k\vec{v}_n$
because an eigenbasis exists and any $\vec{x}$ can be decomposed into
a linear combina... |
H: Spotting maximum, minimum, maximal and minimal elements in a poset
Let $(\mathbb N^* \times \mathbb N^*, \varphi)$ be a poset defined as follows:
$$\begin{aligned} (a,b)\varphi(c,d)\Leftrightarrow ab<cd \text{ or } (a,b) = (c,d)\end{aligned}$$
Check if $\varphi$ is a total order and determine maximum, minimum, maxi... |
H: Proof for length of period in simple modulo $N$ sequence.
I am looking for a concise proof that the length of the smallest period of the sequence
$$f[n] = a n \pmod N $$
is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that $f[n]$ is periodic with $N$, but how do I know that it is not p... |
H: Find the largest prime factor
I just "solved" the third Project Euler problem:
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
With this on Mathematica:
Select[Divisors[600851475143], PrimeQ]
It will first give me a list with all the divisors of 60085... |
H: Is the set $\{f: \int_{\mathbb{R}} f \leq h\}$ closed in $L^2(\mathbb{R})$?
If the sequence $\{f_n\}$ in $L^2(\mathbb{R})$ converges to $f$ in $L^2(\mathbb{R})$ and $\int_{\mathbb{R}}f_n(t)dt\leq h$ for some $h>0$ and all $n\in \mathbb{N}$; does $f$ satisfy $\int_{\mathbb{R}}f(t)dt\leq h$?
AI: Not necessarily. Try ... |
H: Evaluating this determinant
I am asked to find the following determinant $$D = \begin{vmatrix} 1 & 2 & \cdots & n \\\ n+1 & n+2 & \cdots & 2n \\\ \vdots & \vdots & \vdots & \vdots \\\ n(n-1)+1 & n(n-1)+2 & \cdots & n^2 \end{vmatrix} \ $$ where $n\in\mathbb N$. Any help. Thank you.
AI: Step 1: Substract line 1 t... |
H: Number of elements of order $7$ in a group of order $28$
Given a group $G$ with order $28 = 2^2 \cdot 7$. Sylow-Theory implies that there is a exactly one $7$-Sylow-Subgroup of order $7$ in $G$, and $1$ or $7$; $2$-Sylow-Subgroups.
Where to go from here concerning the number of elements of order $7$?
AI: If $x$ is ... |
H: Integral of function over displaced sphere
Let $F$ be a vector field defined in $\mathbb{R}^{3}$ minus the origin.
$$F(r)=\frac{r}{|r|^{3}}$$ where $r=(x,y,z)$.
Let $S$ be the sphere of radius $1$ and center $(2,0,0)$.
Compute $\int_{S}{F\cdot n \ dS}$ where $n$ is the vector normal to the surface of $S$.
My initia... |
H: How do I show there isn't an order isomorphism b/w the two sets $\{1, 2, 3,...\}$ and $\{1, 2, 3, ..., \omega \}$
That is, how can I prove there isn't a bijection $f$ from one set to the other such that
$f(x) < f(y)$ iff $x < y$?
AI: Suppose that $f:\{1,2,3,\dots\}\to\{1,2,3,\dots,\omega\}$ is an order-preserving b... |
H: When should I make a colon in a statement?
As an example, take a look at the Cauchy's convergence test:
$\forall_{\varepsilon>0} \exists_{n_0\in\mathbb{N}} \forall_{m,n \geq n_0} \colon \left|a_m-a_n \right|<\varepsilon $
or
$\forall_{\varepsilon>0} \colon \exists_{n_0\in\mathbb{N}} \colon \forall_{m,n \geq n_0} \c... |
H: Irreducible polynomial over field of order p
Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$.
I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.
AI: The first theorem of the following paper provides the proof:
http://www... |
H: Making a logo with processing, can't figure out a coordinate...
I'm struggling to make a simple logo, I have no deep knowledge of Inkscape so I'm doing it with a little bit of processing.
The problem is I can't figure out how to determine one certain point's coordinate, considering I want this point to have certain... |
H: Limit of expression involving exponentials.
For $0 \le s < 1$, $t \ge 0$ let
$$G(s,t) := \frac{e^{-t} s}{\sqrt{1-(1-e^{-2t})s}}$$
For $\lambda > 0$ compute the limit of $G(e^{-2\lambda e^{-2t}},t)$ as $t \rightarrow \infty$.
AI: Well,
$$
G\left( \mathrm{e}^{-2 \lambda \exp(-2t)}, t\right) = \frac{ \exp(-t) \cdot... |
H: Calculate the edge length of a cube, which is inserted in a hemisphere
How can I calculate the edge length of a cube, which is inserted into a half-sphere with a radius $a$?
AI: This is an elaboration on @J.M.'s comment, and assumes we want the largest such cube (obviously we could put smaller cubes inside the half... |
H: A cube is divided into two cuboids
A cube is divided into two cuboids. The surfaces of those cuboids are in the ratio $7: 5$. Calculate the ratio of the volumes.
How can I calculate this?
AI: Without loss of generality we can let the sides of the original cube be $1$.
The larger of the two cuboids has four of its s... |
H: Image of a strip under the map $e^z$
Please help me in finding the image of the strip $\lbrace (x,y) \in\mathbb{C} : mx-\pi<y<mx+\pi,-\infty<x<\infty\rbrace $ under the mapping $w=e^z $ ?
Where $m$ is any real number.
AI: $${\mathbb C}\setminus \{0\}$$
as for any complex $(a,b)\neq0$, you can find $\rho>0$ and $\th... |
H: Precedence of concatenation: Is $5/7y$ equal to $(5/7)\times y$ or $5/(7\times y)$?
Possible Duplicate:
Do values attached to integers have implicit parentheses?
What is 48÷2(9+3)?
What is “multiplication by juxtaposition”?
What is the precedence of the concatention operator when used for
multiplication?
If ... |
H: How to derive distributivity from Boolean algebra laws
Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws:
$a \wedge b = \bot \Leftrightarrow a \le \neg b$
$\neg\neg a =a$
I'd like to show distributivity, namely, $a\wedge(b\vee c) = (a \wedge b) \ve... |
H: Reducing simple trigonometric expression
I have an expression: $\sin(\pi/3) - \sin(\pi/6)$.
I have it reducing to $3 / \sqrt{2} - 1/2$,
but my text has $3 \sqrt{3} - 3$.
How does my eq become their eq?
Edited the eq
AI: Starting with their work $$\dfrac{\Delta y}{\Delta t} = \dfrac{\sin(\dfrac{\pi}{3}) - \sin(\dfr... |
H: Periodic Function - Repeating Pattern Problem
For the following question:
A necklace is made by stringing N individual beads together in the repeating pattern of Red Bead , Green Bead , White , Blue and Yellow Bead. If necklace begins with a RED Bead and ends with a white bead , then N could be A)$16$ B)$32$ C)$4... |
H: Elementary Number Theory $(a+b,\frac{a^p+b^p}{a+b})=1 \text{ or } p$
Assume $(a,b)$=1, and let $p$ be an odd prime, prove that
$$(a+b,\frac{a^p+b^p}{a+b})=1 \text{ or } p$$.
I thought of letting $p=2k+1,k\in\mathbb{Z}$,
then use the identity
$$a^{2k+1}+b^{2k+1}=(a+b)(a^{2k}-a^{2k-1}b+a^{2k-2}b^2-\ldots+b^{2k})$$
S... |
H: What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?
So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Th... |
H: Is $\text{rk}L=\text{rk}L^*L $ true for finite rank operators?
Let $L$ be a compact linear operator in an infinitedimensional space that has finite rank. Do the equations $$\text{rk}L=\text{rk}L^*L\ \text{and} \ \text{rk}L^*L=\text{rk}R,$$ where $R$ is the (unique) root of $L^*L$ have to hold ? If they do, why is... |
H: Dedekind Cuts in Rudin' analysis - Step 4
This question refers to the construction of $\mathbb{R}$ from $\mathbb{Q}$ using Dedekind cuts, as presented in Rudin's "Principles of Mathematical Analysis" pp. 17-21.
More specifically, in the last paragraph of step 4, Rudin says that for $\alpha$ a fixed cut, and given $... |
H: How not to prove the Riemann hypothesis
I remember reading somewhere that there is a (probably a family of) quick false proof of the Riemann hypothesis that starts by using complex logarithms in a bad way, then does some elementary calculations and out pops the result. A perusal of the General Mathematics section o... |
H: Prove: Full Rank and a solution os linear system
I'm studying for my exam of linear algebra.. I want to prove the following corollary:
Given $A \in{R^{n\times n}}$, there is a solution $x$ to $Ax = y$ for all $y$, if and only if $A$
has rank $m$ (full row rank).
I know that the rank of a matrix is the maximum n... |
H: Free Associative Algebras
let $X=\{X_0, X_1,\ldots \}$ be a finite or countable set, $K$ a field, and $K\langle X\rangle$ the free associative algebra generated by $X$.
It is known that all associative algebras, generated by a finite or countable number of
elements, can be represented as the factor ring $K\langle... |
H: A very simple question about a Harris' exercise
In the book of Harris Algebraic Geometry, a First Course, exercise 2.29, it is asked to show that the composition of $1\times \nu: \mathbb{P}^1\times\mathbb{P}^1 \to \mathbb{P}^1\times\mathbb{P}^2$ where $\nu$ is the Veronese embedding with the Segre embedding $\sigm... |
H: integrate two curves
I just started Calculus II and have been given the following problem:
Find the area inside the curves: $$y = 5 - 2x^2\\y = 3x^2 \\ x = -4 \\x = 4$$
I graphed the two curves on my calculator to get a rough idea of the problem but noticed that the x values aren't where the two curves cross.
I'm... |
H: $H\vartriangleleft G$ and $|H|\not\equiv 1 (\mathrm{mod} \ p)$ then $H\cap C_{G}(P)\neq1$
Let $G$, a finite group, has $H$ as a proper normal subgroup and let $P$ be an arbitrary $p$-subgroup of $G$ ($p$ is a prime). Then $$|H|\not\equiv 1 (\mathrm{mod} \ p)\Longrightarrow H\cap C_{G}(P)\neq1$$
What I have done... |
H: Maximum intersecting subsets
There are n elements. What is the maximum number of subsets chosen at any one time so that every pair of subsets from collection intersect?
AI: This problem is classical, and has a surprisingly trivial solution. Let $[n]=\lbrace 1,\ldots, n\rbrace$. Then $\mathcal{P}([n])$ (the set of a... |
H: How to find the time at which a function is maximized given two component functions
I haven't touched calculus in around a year and a half. However, this semester, I have to take Intro to Electrical Engineering as a requirement for my Computer Engineering major, and the first part of this class seems quite heavy on... |
H: Convergence in $\sup$ norm $\Rightarrow$ Cauchy in $\sup$
Let $(f_n)$ be a sequence of bounded functions on a set $E \subseteq \mathbb R$ and suppose that $f$ is a bounded function such that $\|f_n - f\|_{\infty} \to 0$ as $n \to \infty$. Prove that $(f_n)$ is a Cauchy sequence in the $\sup$ norm.
My Thoughts
Meth... |
H: What is the meaning of $\mathbb R^+$?
For a function $f$ that maps set $A$ to $B$,
$f\colon\mathbb R^+\to\mathbb R^+$, $f(x) = x^2$ is injective.
$f\colon\mathbb R\to\mathbb R$, $f(x) = x^2$ is not injective since $(- x)^2 = x^2$.
what is the difference between $\mathbb R^+$ and $\mathbb R$?
Additionally, what is... |
H: Is the statement "1/3 of the natural numbers are divisible by 3" true? Is anything similar to it true?
If we're talking about a finite set of the natural numbers, like those between 1 and 500 or 1 and a million, it seems to me that the fraction of numbers in that finite set that have a factor of 5 approaches $1/5$ ... |
H: Solving this linear system based on the combustion of methane, has no constants
I recently discovered that I could solve a chemical reaction using a linear system. So I thought I would try something simple like the combustion of methane.
where x y z and w are the moles of each molecule
x $CH_4$ + y $O_2$ = z $H_2$O... |
H: Binomial/Tensor Identity
Let $k$ be a a field and consider the space $k[x] \otimes_k k[x]$. I would like to verify the equation
$$ \sum_{k=0}^{m+n} {m+n \choose k} x^k \otimes x^{(n+m)-k}= \sum_{i=0}^n \sum_{j=0}^m{n \choose i}{m \choose j} x^{i+j}\otimes x^{(n+m)-(i+j)}$$
however I am not getting very far with thi... |
H: Quadratic Equation Error
For the floating point system $(B, t, L, U) = (10,8,-50,50)$ and for the quadratic equation:
$ax^2 + bx + c$, I need to show error that arises in various cases and how to fix those.
$$a=10^{-30}$$
$$b= -10^{30}$$
$$c=10^{30}$$
I think there is cancellation error when plugging into quadrati... |
H: Numerical Analysis
I am trying to determine some numerical difficulties that arise from a couple problems, and a good way to re-write them to avoid those errors.
For instance, I have:
1) $\sqrt{x+\dfrac{1}{x}} - \sqrt{x-\dfrac{1}{x}}$ where $x\gg 1$
I think that since these two terms approximately equal each other... |
H: Why do we define the group law on elliptic curves only for Weierstrass forms and $O$ an inflexion point?
In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at infinity is de... |
H: machine learning project ideas
I am interested about playing with machine learning algorithm and time series analysis. Is there website/resource with a comprehensive list of sample projects/proposals one may be interested about?
AI: You may want to read this and look at the two terms of projects and additional proj... |
H: Why are "irrational numbers" not named "nonrational numbers"?
Possible that I'm misunderstanding the concept of irrational numbers, but seems like the term nonrational would be much more clear. Why is "irrational" more clear than "nonrational"?
UPDATE: Just to be clear, it would be true to say the terms “irrational... |
H: How to evaluate $ \int_c (\sin z)/ z^6$?
Evaluate:
$$ \int_c {\sin z\over z^6} \, dz$$
Where, $c$ is a circle of radius $2, |z| = 2$.
Don't understand why ${\pi i \over 5!} $ (Answer from book).
$$ {\sin z \over z^6} = {1 \over z^6}\left [ z-\frac{z^3}{6}+\frac{z^5}{120}-\frac{z^7}{5040}+\frac{z^9}{362880}-\frac{z... |
H: The difference between convergence in $L^{\infty}$ and almost uniformly
I am reading these notes http://terrytao.wordpress.com/2010/10/02/245a-notes-4-modes-of-convergence/ by Terry Tao. I have a question about the difference between convergence in $L^{\infty}$ and convergence almost uniformly.
Is the difference th... |
H: Reed-Solomon Code calculation
I have a Reed-Solomon Code which can correct t=2 errors. The generator polynomial is $p(X) = X^3 + X + 1$ and $p(a) = a^3 + a + 1 = 0$ this means $a^3 = a + 1$
What is the degree of generator polynomial of this code?
Are the following code words valid, why or why not?
$(1, a+1, 1, 0,... |
H: Convergence demonstration?
I got the following assigned as homework:
"Demonstrate that the following series are convergent:"
$$\sum_{k=0}^N a^k\\ \sum_{k=0}^\infty a^k \\ \sum_{k=0}^\infty ka^k \\\sum_{k=0}^\infty k(k-1)a^k\\ \sum_{k=0}^\infty k^2a^k$$
I know most of these converge when $|a| < 1$, but I'm not sure ... |
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