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H: Limit point versus limit and implications for convergence
I understand that similar questions to this have been asked here (1) and here (2) but I believe my question is different because with regards to (1) I am not asking about the set $X$ that contains the points of my sequence and with regard to (2) because I ha... |
H: How do I integrate this expression: $\int{2x\,dx\over x^3+x^{2/3}}$?
How do I integrate this expression:
$\displaystyle\int{2x\,dx\over x^3+x^{2/3}}$?
AI: Let $u=x^{1/3}$, then $u^3=x$ so $dx=3u^2 du$ and now your integrand is a rational function. |
H: Two problems with prime numbers
Problem 1. Prove that there exists $n\in\mathbb{N}$ such that in interval $(n^2, \ (n+1)^2)$ there are at least $1000$ prime numbers.
Problem 2. Let $s_n=p_1+p_2+...+p_n$ where $p_i$ is the $i$-th prime number. Prove that for every $n$, there exists $k\in\mathbb{N}$ such that $s_n<k... |
H: Finding integer solutions to $m$ and $n$
How can we find the positive integer solutions to the variables $m$ and $n$, if we know $r$, that satisfy the equation:
$$r = \frac{\sqrt{3(m-n)^2 n^2}}{2},$$
where $m$ and $n$ are coprime, and $0 < n < m$.
AI: Let's suppose $r\ge 0$ (because of the square root) then squarin... |
H: Pi-subgroup in group
Let $\mathbb{P}$ is a set primes numbers, $\pi \subseteq P$ and $\pi
^{\prime }=P-\pi $
Let $O_{\pi }\left( G\right) =\left\langle N~;~N\trianglelefteq G\text{ and }%
N\text{ is }\pi \text{-subgroup}\right\rangle $
Is it true that $O_{\pi }\left( G\right) =1\Longrightarrow G$ is $\pi
^{\prime }... |
H: Affine transformations of measures and the resulting densities
Define the transformation of a measure $v(X)$ for $X \subseteq {\mathbb R}^d$ by function $A: {\mathbb R}^d \rightarrow {\mathbb R}^d$ as
$$
A(v)(X) = v(A^{-1}(X))
$$
Then the density function $v(x)$ for $x \in {\mathbb R}^d$ is not transformed in the s... |
H: Existence of Analytic Functions Based of f(1/n)
I have the following question, and I don't even know where to begin:
Do functions $f$ or $g$ exist which are analytic at the point $z=0$ and satisfy the conditions: $f(\frac{1}{n})=f(\frac{-1}{n})=\frac{1}{n^2}$, and $g(\frac{1}{n})=g(\frac{-1}{n})=\frac{1}{n^3}$?
AI:... |
H: What is $|X|$ and $1n$?
I'm doing a course in calculus but I'm quite young so haven't learnt all the mathematical terms one should perhaps no before learning calculus, and these 2 terms keep coming up and I have no idea in the world of what they are:
"$|X|$" and "1n".
I am really sorry that this probably seems ridi... |
H: A question about integral operator
I have a question: Prove or disprove that: for every $f\in L^{1}\left(\mathbb{R}\right)$, $$\sup\left\{ { \int_{\mathbb{R}}\frac{\sqrt{n}}{\sqrt{\left|x-y\right|}\left(1+n^{2}\left(x-y\right)^{2}\right)}f\left(y\right)dy:n\in\mathbb{N}}\right\} <\infty,$$ for Lebesgue almost every... |
H: Abelian subgroup of a group of order $2002$
Another unsolved question from my studying for quals -
Show that if $G$ is a group of order $2002=2\cdot 7 \cdot 11 \cdot 13$, then $G$ has an abelian subgroup of index $2$.
I know it has to do with the direct product of the normal subgroups of $G$, but I'm not sure ho... |
H: Local Path Connectedness and Cones
I am trying to prove the following:
Let $X$ be a topological space. $CX$ is locally path connected if and only if $X$ is locally path connected, where $CX=X\times I/(X\times\{0\})$
I cannot seem to make much headway in either direction of the problem.
Assuming $CX$ is locally ... |
H: Proving $|A^A|=|2^A|$ for infinite $A$.
Possible Duplicate:
Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?
How can one prove that $|A^A|=|2^A|$ for infinite $A$? (summary of proof or providing link with proof will also suffice.)
Thanks!
AI: We have $|2^A|\leq |A^A|$ and by Cantor-Bernstein theor... |
H: What is a conformal mapping with $\infty$ in its image?
Suppose we have a bijection $f$ between two open sets in $\mathbb{C}\cup\{\infty\}$, for example $U=\{|z|<1\}$ and $\Omega$, with $\infty\in\Omega$. Let $f(0)=\infty$.
What do we mean by saying that "$f$ is a conformal mapping" in this case? What conditions a... |
H: Original article on the Grothendieck group
Is there someone who knows the title of the original publication of Grothendieck on the construction of the Grothendieck group?
Thanks in advance.
AI: What seems to be clear is that the origin of the Grothendieck group is Grothendieck's work on Riemann-Roch's theorem aroun... |
H: If $a^2$ divides $b^2$, then $a$ divides $b$
Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$.
Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out why it's true. Would appreciate a solution.
AI: By the fundamental theorem... |
H: how to prove optimality of this greedy algo
I need some suggestions on how to prove the below greedy algorithm is optimal.
Problem: There are $n$ fires on a road. Each fire $i$ is given as an interval where it starts and ends $[s(i), f(i)]$. An extinguisher can cover a fire of length $m$. Find the minimum number of... |
H: How to obtain more than two abcissas for one ordinate?
I'm reading a book on trigonometry and the author shown that some of the examples contain one abcissa for one ordinate and he showed that a parabola has two abscissas for every positive ordinates, I got curious:
Is it possible to have more than two abscissas ... |
H: Complex analytic functions and their zeros.
I am having trouble with the following statement found in a textbook:
"Let $U$ be a connected open set. Let $f$ be a complex analytic function on $U$ and not constant.
Either $f$ is locally constant and equal to $0$ in a neighbourhood of a zero $z_0$, or $z_0$ is an isol... |
H: A question on Lie sub-group
Well, definition of Lie subgroup what I know is, a Lie subgroup of a Lie group $G$ is an abstract subgroup $H$ which is an immersed submanifold via the inclusion map so that the group operations on $H$ are $C^{\infty}$.
Could any one make me understand the following with an example?
"Bec... |
H: Given a tetrahedron, how to find the outward surface normals for each side?
Say I have a triangle in $3$D space. I can get the surface normal by calculating the vector cross product of two of the edges.
But, lets say I make this a tetrahedron. How can I work out the outward surface normal for each side?
AI: For a ... |
H: Which Interpolation should I use to create a curve?
I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two points within the curve itself.
For instance, if I have a set of ... |
H: From $\sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1)$ conclude that $\sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1)$
I'm reading a book on analytic number theory. It asks me to prove:
$$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$
and conclude, via integration, that:
$$ \sum_p \frac{1}{p^s... |
H: Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?
Desired future direction: Dynamical System(Chaos), PDE
More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core areas (e.g. Analysis, Al... |
H: Finding direction of steepest descent on a hyperplane subject to orthogonal constraint
Given a linear objective function $f(\vec{x})=\sum_ia_ix_i$, the direction in which $f$ varies the greatest is known to be $\vec{\nabla}{f}$.
Now given a non-zero vector $\vec{v}$, I am interested in finding the direction $\vec{d... |
H: Bounding an expression involving digamma function
Let $\psi$ be the digamma function. I have a conjecture that
$$\frac ax > \log(x) - \psi(x)$$
holds for all $x > 0$ if (and only if) $a \ge 1$. I do not know how to prove it. Please help.
AI: Let
$$f(x) = \frac{1}{2}\coth\left(\frac{x}{2}\right) - \frac{1}{x}.$$
The... |
H: Clarification on the Hausdorff property
If $X$ is a Hausdorff space then for points $a,b \in X$ there are disjoint open sets $U$ and $V$ such that $a \in U$ and $b \in V$. So, take a set of points $\{a_1, \ldots , a_n\}$ and another point $x$. Then for each $a_i$ there are disjoint open sets $U_i$ and $V$ contain... |
H: How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?
I've got a limit which puzzle me several days. The question is
$$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$
Can you help me? Thank you in advance
AI: Intuitively, we have
$$\log\left( 1 + \frac{k}{n^2} \right) = \frac... |
H: Can one solve a recurrence that contains a function?
I'd like to solve a recurrence, so I've been reading about solving recurrences, and all the ones I've seen solved involve only previous terms of the recurrence, and constants. My recurrence is
$$t(n) = \frac{t(n-1)(1-\ln(n-1))}{\ln n}$$
and if I could get help so... |
H: How do I prove that any sequence converging to a non-zero value is bounded away from 0?
Please prove that any sequence converging to a non-zero value is bounded away
from 0.
AI: This is not true. Suppose $a_1 = 0$ and $a_n = 2$ for all $n > 1$. $\lim_{n \rightarrow \infty} a_n = 2$ but $\{a_n : n \in \mathbb{N}\}$... |
H: Attaching a topological space to another
I'm self-studying Mendelson's Introduction to Topology. There is an example in the identification topology section that I cannot understand:
Let $X$ and $Y$ be topological spaces and let $A$ be a non-empty closed subset of $X$. Assume that $X$ and $Y$ are disjoint and that ... |
H: Deducing formula for a linear transformation
The question I'm answering is as follows:
Let $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation such that $ T(1,1) = (2,1) $ and $ T(0,2) = (2,8) $. Find a formula for $ T(a,b) $ where $ (a,b) \in \mathbb{R}^2 $.
Earlier we proved that $\{(1,1), (0,... |
H: Computing the differential of the map $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$
Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map
$$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket.
Then why does $d\phi_{(B_1,... |
H: The differential of $\psi: GL_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ which sends $g\mapsto gAg^{-1}$
Suppose $\psi: GL_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ defined by sending
$$
g\mapsto gAg^{-1}.
$$
Then why is it that $d\psi:T_eGL_2(\mathbb{C})=M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ is defined to b... |
H: Summation of an Arithmetic, Parametric Sequence
I'm trying work on my ability to break complex patterns down,
and in this case I'm trying to model the denominators of Lacsap's Fractions:
I managed to get the sequence that represents the denominators:
But now I want to convert it to a series in order to get rid o... |
H: Number of solutions of $x^2_1+\dots+x^2_n=0,$ $x_i\in \Bbb{F}_q.$
Let $\mathbb F$ be field with $q$ element and $\operatorname{char}(\mathbb F)\neq2$. I want to know about the number of solutions of this equation:
$$x^2_1+x^2_2+\dots+x^2_n=0 \text{ where } x_i\in \mathbb F.$$
Any suggestion?
AI: Partial answer that... |
H: Function is continuous or not
If $f$ is a function defined on $[0, 1]$ for which
$f'(\frac{1}{2})$
exists but$f'(\frac{1}{2})$ is not an element in $[0,1]$
then f is
discontinuous at
$x=\frac{1}{2}$.
I need to find the statment is true or false and my conclusions are
the statement is true since ,if an inverse ... |
H: There is a stright formula for calculate n Power Sets P(A) of a A set with m members
By using Cantor's Theorem: if We have a Set A, with n members the Cardinality of the Power Set is $P(A)=2^n$.
There is Formula to calculate n Power Sets $P(...P(P(A))...)$ of a set A with m elements without doing the simple iterat... |
H: The double cover of Klein bottle
I try to find out all the double covers of Klein bottle. Since the Euler characteristic is multiplicative with respect to covering space, there are only two candidates, that is, torus and Klein bottle itself. It is not hard to construct a mapping from torus to Klein bottle.
My quest... |
H: Explicitness in numeral system
Prove that for every $a\in\mathbb{N}$ there is one and only one way to express it in the system with base $\mathbb{N}\ni s>1$.
Seems classical, but I don't have any specific argument.
AI: Suppose that some positive integer $N$ has two representations base $s$. Then we can write $$N=... |
H: Difference between graph homomorphism and graph isomorphism
I am still not getting how graph isomorphism and homomorphism differ.
Can anyone show me two graphs that are homomorphic, but not isomorphic?
Also, Wikipedia says that when the function and the inverse function are both homomorphic, two graphs are isomorph... |
H: How to calculate the rotation matrix between 2 3D triangles?
I need to calculate the rotation matrix and the translation vector between 2 given triangles in Euclidean space.
This is really just a rotation and a translation, the lengths of sides of the triangle are equal.
Coordinates of the points are real. $P_1, ... |
H: Is it possible to have $D=\Bbb P$
Let $f:\Bbb R\to \Bbb R$ and $D=\{x\in \Bbb R: f $ is discontinuous at $x\}$.
My problem is : Is it possible to have $D=\Bbb P$ where $\Bbb P$ is the set of irrationals in $\Bbb R$.
I know the answer is negative, but, how to prove it??
My attempt: First, I proved that $\Bbb P$ is n... |
H: $f'+\lambda f$ acts as $f'$ without integral
Problem
Suppose that $f(x)$ is a differentiable function and $f'(x)$ is the derivative of $f(x)$. Without aids of integral, can we prove that $f'(x)+\lambda f(x)$ has intermediate property?
Intermediate property
A (real) function $f(x)$ having intermediate property means... |
H: How to call simple expressions in a paper which do not need any proof?
I am using an expression in a paper which doesnt need a proof at all such as
$$\sum_{i=k}^K\binom{K}{i}p^i(1-p)^{K-i}$$
is a decreasing function of $k$ and
$$\sum_{i=0}^{k-1}\binom{K}{i}p^{K-i}(1-p)^i$$
is an increasing function of $k$. However... |
H: Decomposition of Modules
Every finitely generated module $M$ over a principal ideal domain $R$ is isomorphic to a direct sum of cyclic modules. If $M=M_n(\mathbb Z)$, the matrices of order $n$ over the integers. What is the decomposition in sum of cyclic modules of $M_n(\mathbb Z)$?
AI: As was suggested above, the... |
H: One problem with Euler's function
Let $n=100$. Is $\phi(n)$ the smallest number $\lambda$ such that if $\gcd(a,n)=1$ then $a^{\lambda}\equiv_{n}1$ ? Generalize this observation.
I suppose the answer is no. I've read something about $\mathrm{ord}_n(a)$ but group theory rather goes beyond my discrete mathematics co... |
H: Holomorphism of $f$ and $f^n$
Given $f^3$ and $f^7$ holomorphic functions, I want to show that $f$ is holomorphic.
Can I say that if $f$ is not holomorphic then $f^3$ is not holomorphic which mean $f$ is must be holomorphic?
AI: As David pointed out, $f(z)=\frac{f^7(z)}{f^6(z)}$ is holomorphic at all points where $... |
H: Uniquely Decodable Code
Given a code $c:S\rightarrow T^*$ let $n_i$ denote the number of symbols in S that are encoded by strings of length $i$ in $T^*$
If we want to construct a uniquely decodable code for 12 symbols using binary words of length less than or equal to 4. Make a list of all the sets of parameters $... |
H: Confusion about chain of subspaces
This is part of Problem 11.1.6 of Dummit and Foote. The problem reads
Let $V$ be a vector space of finite dimension. If $\phi$ is any linear transformation from $V$ to $V$ prove that there is an integer $m$ such that the intersection of the image of $\phi^m$ and the kernel of $\... |
H: Help calculating combination of combinations
I have a problem which I thought was really easy to solve but now I am here =)
I need to construct a final combination of a content based on combinations of various sub-contents. A sub content is a group of items which can be varied in combinations. The example below lis... |
H: Stabilizer of a point and orbit of a point
I really need help with this topic I have an exam tomorrow and am trying to get this stuff in my head. But the book is not explaining me these two topics properly.
It gives me the definition of a stabilizer at a point where $\mathrm {Stab}_G (i) = \{\phi \in G \mid \phi(i)... |
H: Co-ordinate geometry problem
I've been on this for about an hour, researched everywhere, but I cannot find a viable solution.
Question: The straight line $mx = 5y + 4$ has the same gradient as the line $7x + 6y + 5 = 0$, Find the value of $m$.
I've tried:
$mx - 5y + 4 = 7x + 6y + 5$ but can't get it right :(.
Any... |
H: Is there any known formula for $A^n \mathbf{x}^n + A^{n-1} \mathbf{x}^{n-1} + \ldots +A \mathbf{x} + I$?
Good afternoon. My question is, if there is any known formula for the expression
$$A^n \mathbf{x}^n + A^{n-1} \mathbf{x}^{n-1} + \ldots+ A \mathbf{x} + I,$$ where $A$ is a matrix and $\mathbf{x}$ is a column vec... |
H: Determining a function through three equations
I have the following assignment question, and I'm having trouble even getting started:
Consider the set of functions $\mathcal{F}=\{f,g\}$, with $f:\mathbb{R}^2\to\mathbb{R}$, and $g:\mathbb{R}\to\mathbb{R}$, defined by $f(x,y)=xy$ and $g(x)=x+1$. Let $A=cl_\mathcal{F}... |
H: Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows
Earlier, I asked a question on MathOverflow
regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10... |
H: Show $f$ is primitive recursive, where $f(n) = 1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s
Let $f:\mathbb{N}\to\mathbb{N}$ be given by $f(n)=1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s, and $f(n)=0$ otherwise.
How would you go about showing such a function is primitiv... |
H: Convergence in $L^1$ problem.
Problem: Let $f \in L^1(\mathbb{R},~\mu)$, where $\mu$ is the Lebesgue measure. For any $h \in \mathbb{R}$, define $f_h : \mathbb{R} \rightarrow \mathbb{R}$ by $f_h(x) = f(x - h)$. Prove that:
$$\lim_{h \rightarrow 0} \|f - f_h\|_{L^1} = 0.$$
My attempt: So, I know that given $\epsilon... |
H: What is actually "relatively consistent"?
Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc.
However, why does this mean that ZF is consistent? What does "relatively consistent" actually mean... |
H: Base of shortest connection of two skew straight lines
During tutoring (12th grade, regular Math class), I had to explain how to find the two points $s$ and $s'$ that are the base of the perpendicular connection between two skew straight lines $g$ and $h$.
Where $g$ and $h$ are given respectively as:
$$ g \colon \m... |
H: Notation question, Stieltjes integral
I'm reading a signal processing textbook and they frequently write out formulas where if I have a Stieltjes integral:
$$\int_a^b f(t) \, dX(t)$$
They'll write an explicit formula for $dX(t)$, such as
$$dX(t) = g(t)$$
Is the best way to interpret this notation to read my integra... |
H: About absolute convergence and completeness in rings
EDIT: Let R be a commutative ring with unit ring and $I$ a maximal ideal in R. The completion of R with respect to $I$ is the inverse limit of the factor rings $R / I^k$ under the usual quotient maps.
A ring is said to be complete with respect to a maximal ideal ... |
H: What is the set operation corresponds to the last Venn diagram below?
Let the universal set be $U$ which is represented as the rectangular frame.
Set $A$, $B$ and $C$ are the 3 circles on the left top, right top and the bottom, respectively.
The first diagram represents the $U$. The second diagram represent the co... |
H: Does $\sum\ln(n) / n^3$ converge?
By Direct Comparison Test:
$$\ln(n)/n^3 < n/n^3 = 1/n^2 $$
Converges as that is a convergent a p-series
By n'th term Test:
$$(\ln(n))' = 1/n$$
$$(n^3)' = 3n^2$$
$$\lim_{n\to\infty} (1/n) / 3n^2 = \lim_{n\to\infty} (3n^2)/n =\lim_{n\to\infty} 3n = \infty$$
so by $n$'th term it d... |
H: Can I shift this integral along the complex plane?
Does
$$\int_{-\infty}^\infty \text{e}^{\ a\ (x+b)^2}\ \text dx=\int_{-\infty}^\infty \text{e}^{\ a\ x^2}\ \text dx\ \ \ \ \ ?$$
hold, even if the imaginary part of $b$ is nonzero?
What I really want to understand is what the phrase "By analogy with the previous i... |
H: Puzzle : There are two lengths of rope ...
You have two lengths of rope. If you set fire to the end of either of them, the rope will burn in exactly one hour. They are not the same length or width as each other. They also are not of uniform width (they might, for example, be wider in the middle than at the end), ... |
H: Two-Sample Confidence Interval for Normal Distributions
Let's say I have two independent random samples $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$ from normal distributions with real, unknown means $\mu_x$ and $\mu_y$ and known standard deviations $\sigma_x$ and $\sigma_y$.
How would I go about deriving a $1... |
H: Need help to show $R/I$ is not necessarily flat over $R$
Let $R$ be a ring with unit and $I$ an ideal in $R$. I want to show that $R/I$ is need not be flat over $R$, but I do not know how to come up with a counter-example.
Any hint is appreciated.
AI: Consider $R = \mathbf Z$ and take the ideal $I = 2\mathbf Z$. No... |
H: (ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed.
Let $X$ be a metric space.
Let $\{p_n\}$ be a sequence in $X$.
Let $E$ be a set of all subsequential limits of $\{p_n\}$.
How do i prove that $E$ is closed in ZF?
Is there a well-ordering of convergent subsequences? I can't t... |
H: Is the inverse to a monoidal equivalence also monoidal?
Let ${\cal C,D}$ be two categories, and let
$$
F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C},
$$
be an equivalence of categories. Let us now further assume that ${\cal C}$ can be endowed with a monoidal structure $\otimes$, and that ${\c... |
H: Multiplicative group of integers modulo n definition issues
It is easy to verify that the set $(\mathbb{Z}/n\mathbb{Z})^\times$ is closed under multiplication in the sense that $a, b ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $ab ∈ (\mathbb{Z}/n\mathbb{Z})^\times$, and is closed under inverses in the sense that $a... |
H: Collatz-ish Olympiad Problem
The following is an Olympiad Competition question, so I expect it to have a pretty solution:
For a positive integer $d$, define the sequence:
\begin{align}
a_0 &= 1\\
a_n &=
\begin{cases}
\frac{a_{n-1}}{2}&\quad\text{if }a_{n-1}\text{ is even}, \\
a_{n-1}+d &\quad\t... |
H: What is the difference between $d$ and $\partial$?
After seeing the following equation in a lecture about tensor analysis, I became confused.
$$
\frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds}
$$
What exactly is the difference between $d$ and $\partial$?
AI: As mentioned $d$ means total and $\par... |
H: Instant insanity question
My question is regarding the necessary conditions that a graph must fulfill to satisfy instant insanity problem.
Now take for example the left, right, front and back face colors of the four cubes be(the colors are red, black, green and white):
Front Back Right Left
R W ... |
H: Are monomorphisms of rings injective?
Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
AI: Yes, $f$ is injective:
Let's assume the map of the underlying sets is NOT injective. Then there are distinct $x,y\in R$ such that $f(x)=f(y)$.
Next, consider the monomorphisms $g_1,g_2\colon \mathbf{Z... |
H: Sobolev spaces for functions mapping to $\mathbb R^n$
Hi I would like to know if there is a definition of Sobolev space as a set of mappings $f:U\subset \mathbb R^n \rightarrow \mathbb R^m$ (instead to $\mathbb R$). In particular what should be $H^{1,2}(U)_{\mathbb R^m}$. Does it make sense to define it as the se... |
H: Proving a continuity inequality
I am new to the mechanics of the $\epsilon$-$\delta$ definition of continuity of a function at a point, and so I am having trouble selecting my $\delta$ for the following question:
Suppose a function $f$ is continuous at a point $c$ and $f(c) > 0$. Prove that there is a $\delta > 0$... |
H: How to find the maxium number of edge-disjoint paths using flow network
Given a graph $G=(V,E)$ and $2$ vertices $s,t \in V$, how can I find the maximum number of edge-disjoint paths from $s$ to $t$ using a flow network? $2$ paths are edge disjoint if they don't have any common edge, though they may share some comm... |
H: In the definition of a group, is stating the set together with the function on it redundant?
In the definition of a group (or other similar structures)
$$(G,*)$$
as being the pair of a set $G$ together with binary operation
$$*:G\times G\rightarrow G$$
isn't the first component of the pair basically just redundant... |
H: Parameters giving maximal-length Collatz-like sequence
In a recent question the following recursive sequence was considered:
$$
a_{n+1} = \cases{\frac{a_n}{2} & $a_n$ is even \\ a_n +d & $a_n$ is odd}, \quad a_1 = d + 1
$$
where $d$ is an odd positive integer. Such a sequence will eventually reach $1$, and let ... |
H: Is a vector of coprime integers column of a regular matrix?
Given a vector $a_1=(k_1,\ldots,k_n)^T$ of coprime integers. Are there $a_2,\ldots,a_n \in \mathbb{Z}^n$ such that that the matrix $A := (a_1,\ldots,a_n) \in \mathbb{Z}^{n \times n}$ is regular, i.e. $A \in GL_n(\mathbb{Z})$ ?
In case $n=2$ this is true b... |
H: Showing a certain form is exact
I'm trying to solve the following:
Let $f: S^{2n - 1} \rightarrow S^n$ be a smooth map, and let $\omega$ be an n-form on $S^n$ such that $\int_{S^n} \omega = 1$. Show that $f^*\omega$ is exact, and if $f^*\omega = d\alpha$, then $\int_{S^{2n-1}} \alpha \wedge d\alpha$ is independent... |
H: What kind of problem is picking sets of sets of things that don't overlap?
I was working out some combinations for a game, when I hit upon a problem that seems unsolvable. I'm wondering if some existing mathematics (probably in graph theory or combinatorics) covers this particular class of problem. To express it ... |
H: Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?
Thank you.
AI: First, note that it is enough to find a bijection $f:\Bbb R^2\to \Bbb R$, since then $g(x,y,z) = f(f(x,y),z)$ is automatically a bijection fr... |
H: Limit of series $1 - 1/9 - 1/15- 1/21-\cdots$
In looking at a paper online I came across the following proposition:
$$1 - \frac{1}{9} - \frac{1}{15}- \frac{1}{21}-\cdots = 0$$
After wasting a lot of time, I rewrote it,
$$1 -\left(\frac{1}{9} + \frac{1}{15}+\cdots\right)= 1 - \left(\frac{1}{3}\sum_{k = 2}^\infty\fr... |
H: Characters of affine algebraic groups and the determinant
Let $G$ be an affine algebraic group (i.e. a $k$-variety which is also a group and the group multiplication and inversion are morphisms of varieties). A character of $G$ is a morphism of algebraic groups $\chi:G\to k^\times$.
You may assume for the following... |
H: Solving a recurrence of polynomials
I am wondering how to solve a recurrence of this type
$$p_1(x) = x$$
$$p_2(x) = 1-x^2$$
and
$$p_{n+2}(x) = -xp_{n+1}(x)+p_{n}(x).$$
I am wondering, how could one solve such a recurrence. One way would be to pretend $x$ is fixed and solve it using the well known method for linear... |
H: A Countable Set of Real Numbers
Let $S$ be a subset of $\mathbb{R}$. Let $C$ be the set of points $x$ in $\mathbb{R}$ with the property that $S \cap (x - r, x + r)$ is uncountable for every $r > 0$. Show that $S - C$ is finite or countable.
Thanks for any help.
AI: Instead of considering arbitrary neighborhood $(x ... |
H: If $a + b \equiv 0 \pmod p$, and $a + b \equiv 0 \pmod q$, why does $a + b \equiv 0 \pmod {pq}$?
Put a bit more coherently, given $p$ and $q$ as distinct prime numbers, and thus $(p,q)=1$, if
$$p^{(q-1)} + q^{(p-1)} \equiv 1 \pmod p$$
and
$$p^{(q-1)} + q^{(p-1)} \equiv 1 \pmod q,$$
why does that lead to
$$p^{(q-1)}... |
H: How to show that this is an Algebraic System
Define $\oplus$ on $\mathbb{R} \times \mathbb{R}$ by setting
$$(a,b) \oplus (c,d)=(ac-bd, ad+bc).$$
How to show that $(\mathbb{R} \times \mathbb{R}, \oplus)$ is an algebraic system. I don't understand the difference between algebraic structure andalgebraic system I read ... |
H: (ZF)subsequence convergent to a limit point of a sequence
Arthur's answer; (ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed.
Let $\{p_n\}$ be a sequence in a metric space $X$.
Let $B=\{p_n|n\in\mathbb{N}\}$ and $p$ be a limit point of $B$.
It first seemed obvious that there ... |
H: Determine the nature of a critical point (Lagrange multipliers)
Let $F \colon \mathbb R^2 \to \mathbb R$ be the function
$$
F(x,y):=xye^x + ye^y - e^x+1
$$
and denote with $C$ the set of zeroes of $F$, i.e. $C:=\{(x,y) \in \mathbb R^2 : F(x,y)=0\}$. Let also $f \colon \mathbb R^2 \to \mathbb R$ be a function... |
H: Topology of Branch Cuts and Elliptic Integrals
In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces.
On page 4, they discuss the integral
$$\phi(x) = \int_0 ^x \frac{dt}{\sqrt{1 - t^2}}$$
In order to define it on all of $... |
H: Showing truncation is a ring homomorphism
Let $R$ be a commutative ring with $1$ and let $n$ be any positive integer. Denote the ring of formal power series by $R[[x]]$ and define a map as follows:
$f: R[[x]] \rightarrow R[x]/(x^{n})$
by sending $g$ to $g+(x^{n})$.
It is clear this map preserves sums, but why prese... |
H: Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$
In my attempt to prove that $\Gamma'(1)=-\gamma$, I've reduced the problem to proving that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$.
Where $\gamma$ is the Euler-Mascheroni constant, and $\log$ denotes the n... |
H: Questions about $su(2)$.
Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?
AI: The form of the question is somewhat iffy, but may be intended to ask a question that has a sense that I understand: since most representations are on comple... |
H: Does there exist $m,a\in \mathbb{N}$ such that $m(m+1)=a^7$?
Let $m$ be a natural number. Can $m(m+1)$ be written as a seventh power of a natural number? If it is true, is it possible to generalize?
I can use only Euclidean Division. I am not suppose to use the Fundamental theorem of arithmetic.
I think the answer... |
H: Show the existence of a polynomial $f$ such that $T^{-1} = f(T)$ for an invertible linear operator $T.$
Came across this question and have had difficulty approaching it so any help is greatly appreciated.
Let $T$ be an invertible linear operator on a finite dimensional vector space $V$ over a field $F$. Prove that... |
H: 3D graphing in matlab
I'm having trouble graphing using Matlab, and was hoping someone could help. I just want to hand the program a function like $z=x^2-3xy+2$, or whatever, and have Matlab generate a 3D graph. From the tutorials I've seen online, it seems like this is difficult. Can I not just hand the program... |
H: What is the importance of integral domains?
What is the importance of integral domains?
In abstract algebra
Thanks for your help
AI: Definition:
An Integral domain is a commutative ring( with unity) $(R,+,\cdot)$ for which the following property holds
If $a,b \in R$ such that $a,b \neq 0$, then $ab\neq 0$.
To see ... |
H: Can you construct a field with 6 elements?
Possible Duplicate:
Is there anything like GF(6)?
Could someone tell me if you can build a field with 6 elements.
AI: $\def\x{\otimes}$There is not. Suppose $\langle F, +, \x\rangle$ is a field where $F$ has six elements. Then $\langle F, +\rangle$ is an abelian group;... |
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