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H: Under which conditions the rings $\mathbb{Z}_p[x]/(x^n+1)$ and $\mathbb{Z}_p[x]/(x^n-1)$ are fields? (for $p$ prime)
I'm looking for the necessary and sufficient conditions for any prime $p$ and any positive integer $n$ to make the quotient rings $\mathbb{Z}_p[x]/(x^n+1)$ and $\mathbb{Z}_p[x]/(x^n-1)$ not only ring... |
H: Does joint distribution affect Radon-Nikodym derivative?
Given two real-valued random variables $X, Y$ with distributions $\mu_X, \mu_Y$. Suppose $\mu_X<\!<\mu_Y$, then the Radon-Nikodym derivative $\frac{d\mu_Y}{d\mu_X}(\cdot)$ exists $\mu_X$-a.e. on $\mathbb{R}$.
It seems that joint distribution $(X, Y)$ does not... |
H: What is $dudv$ in the metric tensor?
In the definition of a $m$-dimensional Riemannian manifold $(M,G)$, if $(U;u^i)$ is a local coordinate system of $M$, the tensor field $G$ on $U$ can be written as
$$
G = g_{ij}du^i\otimes du^j\;\tag{1}
$$
I have seen in many places people write the "Riemann metric"
$$
ds^2 = g... |
H: Prove $\int_a^b f(x)\,\mathrm dx=\int_a^cf(x)\,\mathrm dx+\int_c^bf(x)\,\mathrm dx$ without the Fundamental Theorem of Calculus
Let $f(x)$ be a continuous function. Let $a,b,c$ be constants, with $a < c < b$. Prove that
$\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ $(*)$
In particular, I would ... |
H: represent $x^3 +2x$ as power series
Well it is pretty wierd for me to see this question, the function already is power series isn't it?
Am I missing the purpose of the excersize?
AI: A power series is an infinite series of the form $\sum_{n=0}^\infty a_n(x-x_0)^n$. If for some function $f$ and some set $D$,
$$
f(x... |
H: Is every (sufficiently nice) function a solution to an autonomous ode?
For whatever reason, I'm suddenly interested in whether or not one can characterized those functions which are solutions to autonomous ode. The first natural question to ask, then, is: are all $C^1([a,b])$ functions solutions to some autonomous ... |
H: If $\lim_m \mathbb{P}(\{\sup_{j\geq 1}{|S_{m+j}(t)-S_{m}(t)|}\leq \epsilon\})=1$ then $\mathbb{P}(\{S_n(t)\text{ is a convergent sequence}\}=1$
Let $(X_n)$ be a sequence of random variables in Probability space.
Take $S_k=\sum_{n=1}^{k}{X_k}$. We suppose that : for all $\epsilon >0$
$$\lim_m \mathbb{P}(\{\sup_{j\g... |
H: Closed form of $\sum_{n=1}^{\infty}a^n \cos(nt)$
I'm trying to find a closed form (if one exists) for
$$\sum_{n=1}^{\infty}a^n \cos(nt)$$
where $a \in (0, 1)$. I know that
$$\begin{matrix} \sum_{n=1}^{\infty} \frac{a^n}{n} \cos(nt) &= &\ln(a^2-2a\cos(t)+1) \\
\sum_{n=1}^{\infty} \frac{1}{n} \cos(nt) &= &2 \ln|2 \s... |
H: Proving continuity: asking to see if I have a counter-example
I was asked to prove the following:
Suppose $F_1, \dots, F_2$ are disjoint closed subsets of $\mathbb{R}$.
Prove that if
$$g: \cup_{k=1}^n F_k \to \mathbb{R}$$
is a function such that $g_{|F_k}$ is a continuous function for each
$k \in \{1,\dots,n\}... |
H: If T ◦ S = S ◦ T and v is an eigenvector of T, then v is an eigenvector of S
I have to prove or provide a counterexample to the following claim:
Let $V$ be a vector space, $T, S : V \to V$ linear transformations. If $T S = ST$ and $v$ is an eigenvector of $T$, then $v$ is an eigenvector of $S$.
I thought it's corr... |
H: Does there always exist a multiple keyhole avoiding finite number of points?
Let's $C$ be a circle in $\mathbb C$. Let $z_1,\ldots,z_N$ be points in the region enclosed by $C$ ($C$ itself is not contained in that region). A multiple keyhole avoiding $z_1,\ldots,z_N$ is a shape of the form below.
The whole shape sh... |
H: Polynomial quotient ring $\mathbb{Z}_m[x]/f(x)$ when $\mathbb{Z}_m$ is only a ring (not a field)?
The definition of polynomial quotient ring requires $\mathbb{Z}_m$ to be a field.
But if $m$ is not prime, then $\mathbb{Z}_m$ is just a ring (not a field). So under what conditions $\mathbb{Z}_m[x]/f(x)$ can still be... |
H: Find infimum of the sets of number $x + \frac{1}{x} $
Let $A = \{ z = x + \frac{1}{x} : x > 0 \} $ and $B = \{z = 2^x + 2^{1/x} : x > 0 \} $
I want to find $\inf A $ and $\inf B $.
Proof.
Clearly, by AM-GM inequality one has $x + \dfrac{1}{x} \geq 2 $ and $2^x + 2^{1/x} \geq 2 \sqrt{2^{x+1/x} } \geq 2 \sqrt{2^2} = ... |
H: Solving "Square" PDE
I'm trying to find all solutions to:
$$f_{xx}+2f_{xy}+f_{yy} = 0$$
I noticed that if we let $D = \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$ then we are looking for the kernel of $D^2$. I'm not sure where to go from here though.
AI: Make a change of variables:
$$u=x-y\\v=x+y$$
T... |
H: Compute an $n\times n$ determinant with factorial and powers of $x$
Compute
$$
D_{n}=
\begin{vmatrix}
1 & 0 & 0 & 0 & 0 & \ldots & 1\\
1 & 1! & 0 & 0 & 0 & \ldots & x\\
1 & 2 & 2! & 0 & 0 & \ldots & x^{2}\\
1 & 3 & 3\cdot2 & 3! & 0 & \ldots & x^{3}\\
\ldots & \ldots & \ldots & \ldots & \ldots & \ddots & \ldots\\
1 ... |
H: A question relate to dominated convergence theorem
given that：$g(x)=\frac{1}{xln(x)},x > 1$. $f_n=c_n\mathbf{1}_{A_n}$, where $c_n \geq 0$, $A_n$ is measurable and $\subset [2, +\infty)$. $|f_n|\leq g$ ($f_n$ is dominated by $g$.)
$\forall x, \lim_{n \rightarrow \infty}f_n(x) = 0$.
the question is：do we have$\int^... |
H: If $\Delta \varphi =0$ then $\nabla \times \nabla \varphi =0$.
I was looking into a question about vector functions $\varphi$ satisfying the Laplace equation $\Delta \varphi =0$. I found that this answer to the question was given in most places I looked for, but my problem with it is that it doesn't use the hypothe... |
H: What is the set of $\sup\{x\geq 1/2: (x-\frac{1}{2})^2+1\leq \log 3\}$?
What is the set of
$$\sup\{x\geq 1/2: (x-\frac{1}{2})^2+1\leq \log 3\}$$
It seems that for any $x\geq 1/2$, $\sup {(x-\frac{1}{2})^2+1}=1\leq \log 3$. So what is the sup?
AI: I am editting my answer since the OP made a mistake and edited his q... |
H: Prove that a non-trivial commutative ring $R$ is a field if and only if $0$ and $R$ are the only ideals of $R$.
Prove that a non-trivial commutative ring $R$ is a field if and only if $0$ and $R$ are the only ideals of $R$.
Proof Let $R$ be a field. Let $a$ be a non-trivial ideal of $R$. Let $x$ be in $a - \{ 0\}$.... |
H: Giving away $500$ pencils: $1$ on the first day, $2$ on the second day, etc. How long will it take?
I'd like to know the name of this subject so that I discover how to solve the problem on my own.
I suppose that's something like "$(n-1) + (n-2) + \cdots$" but I'm not sure about it. =\
I have 500 pencils and I want... |
H: Fiber preserving map that is not a bundle map
I used to think every map that is fiber preserving is a bundle map in the sense of the definition below:
Definition Let $E\xrightarrow{\pi}M$ and $F\xrightarrow{\rho}N\ $ be vector bundles and $g:M \rightarrow N\ $ a diffeomorphism between $M$ and $N$. Then, we define t... |
H: Math formulation for # of outcomes of 2 indistinguishable dice
I'm not sure why the equation for finding this is $(6 * 6)/{2!} + 6/3$ I understand that out of the 36 outcomes when they are distinguishable, we need to divide them by 2 to get rid of duplicates like (1,2) and (2, 1). But why are we adding $6/3$ back?
... |
H: Proving postulate about a property fo spherical vectors
Assume we have $X, Y$ constant unit vectors of $\mathbb{R}^3$
I postulate that the maximum of the function:
$(V \cdot X) (V \cdot Y)$
I reached by the halfway vector between $X,Y$ i.e at the vector $V_0 = slerp(X,Y, 0.5)$
To try to prove it I tried finding the... |
H: Relation between the radius of $n$ identical circles and the radius of an enclosing tangent circle
$n$ small circles are tangent to each other and tangent to the big circle. Here's a figure for $n=4$:
Asking hints of how to find the reason between the radius of small circles into the big circle.
I tried doing... |
H: Net based on a filter
Assume a topological space X, t. I read a proposition in my textbook:
A filter A $\to$ y if and only if every net {$s_a$}, a $\in$ A, based on A also converges to y.
What are the net elements $s_A$ in this case? Are they elements of the topological space X or are they elements of the filter A?... |
H: If positive integers $a$, $b$, $c$ satisfy $\frac1{a^2}+\frac1{b^2}=\frac1{c^2}$, then the sum of all values of $a\leq 100$ is ...
I'm struggling to solve the following problem. I would like hint (just a hint, not a full solution please) on how to solve it:
The positive integers $a$, $b$, and $c$ satisfy $$\dfrac... |
H: Let $f: X \to Y$, $\mathcal{C}$ be an arbitrary collection of subsets of $Y$, then $f^{-1}(\sigma(\mathcal{C})) \subset \sigma(f^{-1}(\mathcal{C}))$
I have already read the solution to my question at https://math.stackexchange.com/a/1530785/81560, but unfortunately, the notation is confusing.
I am trying to prove:... |
H: Find the stationary points of $f(x,y)=5y\sin(3x)$
Given the function $f(x,y)=5y\sin(3x)$ find the stationary points.
I found $f_x=15y\cos(3x)$. Solving $f_x=0$, I got $y=0,x=\frac{(2n+1)\pi}{6}$
Similarly, $f_y=5\sin(3x)$. Solving $f_y=0$, I got $x=\frac{n\pi}{3}$
I thought my stationary points would be $(\frac{(2n... |
H: Let G be a group of order p^n suppose that H is a normal subgroup of G
Prove that there exists a normal subgroup H' a subset of G such that [H':H] = p
AI: The number of subgroups of $H$ with index $p$ is congruent to $1\pmod p$. We look at the orbits of these subgroups under conjugation by elements in $G$; of cours... |
H: If $A = [-5, 3)$ and $B = (1, \infty)$, is $3$ a member of $A \cap B$?
If $A = [-5, 3)$ and $B = (1, \infty)$, what is $A \cap B$?
Since $3$ is not a member of $A$, do we include it in the intersection?
AI: This question might make more sense if you try to visualise it. When you plot it on the number line, it wi... |
H: How to solve $vv''+v'^2-6t^2=0$
Hi, I have been trying to solve this equation for a week now.
However, I keep ending up with the same result everytime, it may be because my knowledge on this chapter is not great.
I am unable to figure it out, please help.
Below is the question and the conditions.
$$
\begin{cases}... |
H: Monotonically increasing/decreasing functions
I am having some trouble with this. I can see that the last two are true by testing them with a few functions, but I don't know how to prove them. Also, I am not too certain on the first two. Can I get some pointers on how to do this? Thank you.
Definition: Let $A,B\su... |
H: isomorphism between a field and non-field rings
I think that once you have a field and a ring which is not a field, you can conclude that there is no isomorphism between these two.
Is it right? if not, is there an example? if true, can someone give a rigorous proof?
AI: This is correct, and the intuition should be... |
H: Question on proving continuity
My goal is to show that $(\vec{y})^{-1}$ is continuous, where $\vec{y}(x)= (x, \frac{1}{x})$ on the domain $x > 0$.
Now clearly $\vec{y}$ itself is continuous and it is bijective; I first thought to use the theorem that states if a function $f$ has a compact domain and is continuous ... |
H: Determine the largest size n of a problem that can be solved in time t , assuming that the algorithm to solve the problem takes f(n) microseconds.
For each function f(n) and time t in the
following table, determine the largest size n of a problem that can be solved in time t ,
assuming that the algorithm to solve t... |
H: How can it be physically justified that the energy is conserved for the solution of wave equation with Neumann boundary condition?
I have learned that the IBVP: $$u_{tt}-c^{2}u_{xx}=0,$$ $$u(x,0)=f(x),\,u_{t}(x,0)=g(x),$$ $$u_{x}(0,t)=u_{x}(L,t)=0$$
has the unique solution $\displaystyle u(x,t)=\frac{1}{2}(A_{0}+B_... |
H: $M$ is a simple $A\oplus B$-module implies $M$ is either a simple $A$-module or a simple $B$-module?
In section 1.6 of Lectures on Algebraic Categorification by Mazorchuk, it is mentioned that if $A$ and $B$ are finite dimensional unital $k$-algebras, ($k$ a field) and $M$ is simple as an $A\oplus B$-module, then $... |
H: Likelihood of probability of heads = 0.5 given that the coin yields 4 heads in 9 tosses.
Suppose I have a coin.
The probability of heads is unknown.
I toss the coin 9 times and get 4 heads.
What is the likelihood that probability of heads(Ph) is 0.5
What I have tried till now
Likelihood of Ph being 0.5 given I ha... |
H: Find the sum of all coefficients of the polynom $p(x) = (x ^4 − 3x^3 − x + 1)^9$
A polynomial $$p(x) = (x ^4 − 3x^3 − x + 1)^9$$ is written in standard form. Find the sum of all its coefficients ?
AI: For any polynomial $p$ the sum of all the coefficients is simply $p(1)$. So the answer is $(1-3-1+1)^{9}=-2^{9}$.
(... |
H: Proving $|a+b|\le |a|+|b|$ from $-|a|\le a \le |a|$
In Spivak Calculus chapter 1, question no. 14, it is asked to prove the aforementioned inequality. However, the way I proved it is unnecessarily long. Can someone critique it for me, and mention an alternate shorter proof method if any?
My proof:
$-|a|\le a \le |a... |
H: Question in a theorem related to topic weak topology
I have done a course on topology and I am self studying concepts that were not taught in class from the book Foundations of Topology ( C Wayne Patty).
I got struck upon this theorem.
My question is in line 4 of the theorem - How does author wrote $< x_n>$ Uni... |
H: Finding the integral curves of a vector field
I have been stuck on this question for very long now
Given the vector field
$$
F(x, y) = (5x + 2y, 3x + 7y)
$$
And the ODE system:
$$
x' = 6x + 9y\\
y' = 1x + 4y.
$$
Determine the integral curves to $F$ by finding the general solution to the ODE system.
My problem:
I ha... |
H: How to re-express a continuous function with a singularity (but with well-defined limit) to avoid the singularity?
I have a function $\frac{x^{1-\gamma}-(xy)^{\frac{1-\gamma}{2}}}{y - x}$ that has a singularity at $y=x$ but the limit as $x \to y$ is well defined, namely, $\lim_{x \to y}\frac{x^{1-\gamma}-(xy)^{\fra... |
H: Find the common ratio of the progression.
A geometric progression has 625 as the first term. The product of its first 3 terms is equal
to the product of its first 6 terms. Find the common ratio of the progression.
AI: Write the terms of the progression as $a_n = 625 r^n$. This gives that our first term, $a_0$, is $... |
H: Combination definition question
I am relatively new to math.
I am confused as to the formula given is this Wikipedia page on
combination, where it states in the first paragraph
$$\displaystyle C^n_k={n\choose k}=\frac{n(n-1)…(n-k+1)}{k(k-1)…1},$$
which can be written using factorials as $\displaystyle\frac{n!}{k!(n... |
H: What is this tensor operation called?
I'm wondering if the following operation has a name:
$$
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{pmatrix} ? M =
\begin{pmatrix}
aM & bM & cM \\
dM & eM & fM \\
gM & hM & iM \\
\end{pmatrix}
$$
In other words the operat... |
H: Rudin POMA Chapter 1 exercise 18
Question:
If $k \geq 2$ and $x \in R^k$ ,prove that there exists $y \in R^k$ such that $y \neq 0$ but $x \cdot y = 0$. Is this also true if k = 1?
Proof:
$$|x+y|^2 = (x \cdot y)(x\cdot y) = |x|^2 + |y|^2 +2(x\cdot y)$$
$$\text{As} \space (x \cdot y)=0$$
$$\text{Then} \space|x|^2 +... |
H: An example showing subspace of compact space is not compact.
I had to give an example showing subspace of compact space is not compact.
For this I first picked example of compact space from the book Topology by James R. Munkres section 26, solved example 2.
$X= \{0\}\cup\{\frac{1}{n} | n\in \mathbb N \}$.
This spa... |
H: Show that, the spiral and the half straight line in $\Bbb{R}^2$ are homeomorphic.
We are given a spiral say the set $A=\{(e^x\cos x,e^x\sin x)|x\in\Bbb{R}\}$ and $B=\{(y,y)|y\ge0\}$. We have to find a homeomorphim $h:A\to B$.
Intuitively by looking at their graphs in $\Bbb{R}^2$, it seems to be homeomorphic as you ... |
H: Why can I cancel in modular arithmetic?
Based on my school, the cancellation law for modular arithmetic is as stated:
For all integers $a$, $b$, $c$, $n$, with $n > 1$ and $a$ and $n$ are coprime, if $ab$
$≡ ac $( $mod$ $n$), then $b ≡ c$ ( $mod$ $n$ ).
Apparently, the proof for this was to multiply both sides by... |
H: What kind of manifold doesn't have a specific rank distribution?
For example, I know that $2$ dimensional sphere doesn't have rank $1$ tangent distribution for it vanishes at some point by the hairy ball theorem. Is there any criterion for it?
AI: If a closed oriented manifold has a distribution of rank $1$ then it... |
H: Decide if a function is Riemann Integrable
I have the following function: $f:[-\pi,\pi]\rightarrow [-\pi,\pi]\ \ f(x,y) = \left\{\begin{matrix}
\frac{xy}{x^4+y^2}, & (x,y) \neq (0,0)\\
0,& (x,y) = (0,0)
\end{matrix}\right.$ I need to decide if the function is or it is not Riemann Integrable. I am trying to do t... |
H: Explain a section of Euclid's Theorem on an infinite number of prime numbers.
I'm trying to understand Euclid's Theorem, using proof by contradiction, which states:
There are an infinite number of prime numbers.
In the book it has the following explanation: We assume that there are a finite number of prime number... |
H: A line up question where the position and the number itself should not equal in an arrangement
There are $n$ numbers, from $1$ to $n$. Suppose that the numbers are arranged randomly, and asks there is no arrangement where the position and the number itself are equal. How many arrangements totally?
For example, if $... |
H: Dirac comb Fourier transform
I have $ h(t) = a \delta (t) + \delta (t-T) + a \delta (t - 2T) $ and I found that the Fourier transform of this impulse response is $ H(f) = a + e^{-i 2 \pi f T } + a e^{-i 2 \pi f 2 T } $ is this transform correct ? because my book obtained the same result but after wrote it as $ (1 +... |
H: Describe all continuous and linear tranformations $f:(\mathbb{R}^{2},\mathscr{F}(x,y)) \rightarrow (\mathbb{R},|\cdot|)$
Let $\mathscr{B}=\{(-a,a)\times \mathbb{R}:a>0\}$. It's easy to see that $\mathscr{B}$ is the filter of neighborhoods of the origin in a topology compatible with the linear structure of $\mathbb{... |
H: Proving $f(x,n)=\lfloor x \lfloor x \lfloor x \lfloor x ...(\text{n times})\rfloor\rfloor \rfloor \rfloor $ is increasing for $x>0$
$$f(x,n)=\lfloor \underbrace{x \lfloor x \lfloor x \lfloor x \dots\rfloor\rfloor \rfloor \rfloor}_{\text{$n$ times}}$$
with $ n \in \mathbb{N}$. Can we prove this function is increasin... |
H: Determinant and norm of the cross product
I assume the vectors are in Euclidean space.
I know that the determinant of a vector family is the area/volume of the associated parallelogram/parallelepiped. But I also read that the norm of the cross product was the same area. So do we have $\det(u,v)=\|u\wedge v\|$? How ... |
H: Simultaneous probability distribution
Just starting to learn probability and would appreciate if someone could explain how this type of questions are done.
I have the following simultaneos probability function:
p-X,Y(x,y) 0 1 2
0 0.1 0.2 0.1
1 0.2 0.3 0.1
What is P(Y>0∣X... |
H: Calculate a line integral
I need to calculate a line integral: $\int_\gamma (2x+1)dx+(3xy+1)dy$, where $\gamma$ is the surface of the traingle $OAB$ where $O(0,0)\ \ \ A(3,-3)\ \ \ B(-6,-6) $. I tried to solve this by parametrizing the lines $AB$, $AO$ and $OB$, but it turns out to be a hard to compute integral, so... |
H: Short exact sequence of vector spaces
Suppose $V$ is a finite-dimensional vector space, and $U,W$ its subspaces. Is it possible to construct the following short exact sequence? $$0\rightarrow U\cap W\xrightarrow{f}U\oplus W\xrightarrow{g}U+W\rightarrow 0 \text{ (the direct sum here is external)}$$
I thought we coul... |
H: Find the Laurent series for $f(z)=\frac{1}{z(1-z)}$
I am having difficulties finding Laurent series of the above function, around these two domains
a) $1<|z|$
b) $1<|z-1|$
For a) I do
$$ \sum_{n=0}^\infty \frac{1}{z^{n+1}} - \frac{1}{z} $$
Is this correct? And how can I solve b)?
AI: Since$$f(z)=\frac1z\cdot\frac... |
H: Characterization of absolute value : how to tackle such problems?
In a problem we are asked to show that there existe a unique function $\alpha:\mathbb{C}\rightarrow\mathbb{R}$ such that :
$\alpha(x)=x$ for all real $x\geq0$
$\alpha(zw)=\alpha(z)\alpha(w)$ for all $z,w\in\mathbb{C}$
$\alpha$ is bounded on the unit... |
H: Projection of measure zero set in $\mathbb{R}^2$.
Let $E\subset \mathbb{R}^2$ be a set of measure zero in $\mathbb{R}^2$. Let $E_x,E_y$ be the projection of E on the $x$-axis and $y$-axis respectively.
Is it true that atleast one of $E_x$ and $E_y$ is measurable and measure zero in $\mathbb{R}$?
AI: No. Consider th... |
H: Reference for "If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$"?
Could you please give me a good reference as to the below-quoted real analysis result?
If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$
AI: By definition of a limit, for all $\v... |
H: Intuitive, possibly graphical explanation of why rationals have zero Lebesgue measure
I know that rationals, being a countable set, have zero Lebesgue measure. I think one way to prove it is to find an open set containing rationals that has measure less than $\epsilon$ for every $\epsilon >0$ fixed. You can do it b... |
H: Simplification of formula: $\cos{2\theta} + \sin{2\theta}\tan{\phi}$
So I'm trying to understand a certain proof/derivation of a formula in physics, but I seem to struggle with the following simplification to the final result.
The given formula is:
$$ 0 = \cos{2\theta} + \sin{2\theta}\tan{\phi} $$
And it is simpli... |
H: $E\int_0^T|X(t)|^2dt=0$ then $X=0 dP\times dt$
$X:\Omega\times [0,T]\to \mathbb{R}$ - process measurable with respect to the product sigma field such that $E\int_0^T|X(t)|^2dt=0$. Prove that then $X=0 dP\times dt$ Can anyone prove it?
AI: Let $A=\{(\omega,t): X(t))(\omega)=0\}$. By hypothesis we have $\int_0^{T} |X... |
H: How to prove this set of functions has only one element
Let $B= \{z \in \mathbb{C} : |z|< 1\}$ and $\overline{B}= \{z \in \mathbb{C} : |z|\leq 1\}$
Let $C(\overline{B},\mathbb{R})$ be the set of continuous functions from $\overline{B} \rightarrow \mathbb{R}$
and $g$ a continuous function define as following $g: \ov... |
H: System of higher order Differential Equations
Sorry asking out of interest but is it possible for there to be a system of ODEs
like
$X'' + X' = AX$
if so could one give an example of system which could be represented
as this
AI: If $\mathbf A = (a_{ij})_{ij} $ and if $\mathbf X = [x_1,\dots,x_n]^{\text{T}}$, we ca... |
H: Why curve has Jordan measure equal $0$
There is a theorem that curve $y=f(x)$, where $f:\ [a;b]\rightarrow \mathbb{R}$ and $f$ is a continuous function on this interval, then it's Jordan measure is equal $0$. Why is this so?
I have been taught that Jordan measure is an extension of the notion of size (length, area,... |
H: How many numbers of 4 digits have at least three specific digits
The question I tried to solve goes like this:
How many 4-digit numbers have at least one digit $2$, one digit $3$ and one digit $4$?
I saw it as simple as this (because it's exactly 4 digits):
The number will have a $2$, a $3$, a $4$ and $X$.
So the... |
H: Definition of isomorphism of graded rings
After searching through some literature I got a bit confused what one has to check to conclude that two graded rings are isomorphic (as graded rings).
Suppose that $R$ and $S$ are graded rings, then a homomorphism of graded rings is a ring homomorphism
$$f:R\rightarrow S$$... |
H: Godel's completeness theorem and Consistency of ZFC
Is it necessary to assume the consistency of $\sf ZFC$ when we prove the completeness theorem? When we use the completeness theorem to show that Peano Arithmetic is consistent, the assumption of consistency of $\sf ZF$ is necessary?
I mean, whenever one says that ... |
H: What is the homogenous parametric equation of a line given two points?
In the case we have two non-homogeneous points $\vec{A}=(x_1,x_2)$ and $\vec{B}=(y_1,y_2)$, we can write the vector equation of the line with a real parameter $\lambda$:
$r: \quad \vec{X}=\vec{A}+\lambda(\vec{B}-\vec{A})$
Or
$r: \quad \vec{X}=(1... |
H: Find the function does describe the the percentage of the area that each circle overlaps
I saw this question, yesterday and it got me thinking, what function does describe the the percentage of the area that each circle overlaps.
In that diagram it is given that the distance between the center of the circles is $1... |
H: Let $G$ be a nonabelian group of order $p^{3},$ where $p$ is a prime. Show that $G$ has exactly $p^{2}+p-1$ distinct conjugacy classes.
Problem
Let $G$ be a nonabelian group of order $p^{3},$ where $p$ is a prime. Show that $G$ has exactly $p^{2}+p-1$ distinct conjugacy classes.
Attempt
Let $G$ be a nonabelian ... |
H: Countability of the set $(0,1)$
I am trying to prove that the set $(0,1)$ is uncountable from "A First Course in Analysis by Yau". I have a question about a particular step.
In the text, the result is proved by contradiction. It is supposed that the set $(0,1)$ is countable, which it is then written that there must... |
H: Prove that for all sets $A$ and $B$ $A\subseteq B$ implies $A\cap B=A$.
In the next proof we use the following lemmas:
For $A$ and B sets, $A \subseteq B$ implies $A \cup B = B$.
For all sets $A$, $A \cap A = A$.
For all sets $B$, $B \cap \varnothing = \varnothing $.
Assume that $A⊆B$. As a result, $A \cup B = B$. ... |
H: Sum of complex roots' fractions
According to this:
If $\omega^7 =1$ and $\omega \neq 1$ then find value of
$\displaystyle\frac{1}{(\omega+1)^2} +
\frac{1}{(\omega^2+1)^2} +
\frac{1}{(\omega^3+1)^2} +
... + \frac{1}{(\omega^6+1)^2}=?$
First I try like
$\displaystyle\frac{1}{\omega+1} +
\frac{1}{\omega^2+1} +
\fra... |
H: Why this approximation for $\pi$ is so accurate?
Berggren and Borwein brothers in "Pi: A Source Book" showed a mysterious approximation for $\pi$ with astonishingly high accuracy:
$$ \left(\frac{1}{10^5}\sum_{n={-\infty}}^\infty e^{-n^2/10^{10}}\right)^2 \approx \pi.$$
In particular, this formula gives $42$ billion... |
H: Taylor Series of $\ln(1+x^3)/[\tan(x)-x]$
Hi i want to find the Taylor series of the function $$\frac{\ln(1+x^3)}{[\tan(x)-x]}$$ at $0$ until n=3
i have $\ln(1+x^3)=x^3-\frac{x^6}{2}+\frac{x^9}{3}+o(x^9)$ and $\tan(x)-x=\frac13 x^3+o(x^3)$
by devision I found that $$\frac{\ln(1+x^3)}{\tan(x)-x}=3-\frac{3}{2}x^3+o(x... |
H: Existence of a continuous linear functional with $f(x_0)=||x_0||$ and $ \sup\{|f(x)|:x\in X,x\neq 0\}=1$
Let X be a linear, real or complex normed space and let $x_0\in X,x_0\neq0$. Show that: It exists a continous
linear functional $f:X\rightarrow\mathbb{K}$ where $\mathbb{K}=\mathbb R,\mathbb C$ s.t :$$ f(x_0)=|... |
H: Show a continuous function $f$ attains a maximum value in $\Bbb{R}$.
"Let $f:\Bbb{R}\to \Bbb{R}$ be a continuous function such that $f(0)=2$ and $\lim_{x\to{-\infty}}f(x)=0$, and $f$ is decreasing when $x\geq 3$ . Show $f$ attains a maximum value in $\Bbb{R}$."
Lets call the maximum value $M$. When $x\to \ -\infty$... |
H: Interval of Existence for initial value problem: $\dot x = x^2$, $x(0) = x_0$
I am currently reading through a chapter of a book called 'Differential Dynamical Systems' by James D. Meiss and the initial value problem shown in the title is an example given in Chapter 3 (numbered 3.22) which is solved by separation o... |
H: Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges
Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges.
If $(a_n)$ converges, the sequences is a Cauchy sequence. Which means:
$\forall \epsilon >0 \,\,\,\exists N \in \mathbb{N}\,\,\,m,n>N:\left|\frac{(-1)^{n-1}}{2n-1}-\frac{(-1)^{m-1}}{2m-1}\right|<\epsilon$
$\left|\... |
H: Expected number of times a result comes up: Proof of formula
If I am flipping a fair coin, and want to know the expected number of heads after $n$ flips, then I immediately know the answer is $\frac{n}{2}$. Similarly, with a fair die, the number of $5$'s that come up after $n$ rolls is on average $\frac{n}{6}$. Pre... |
H: Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval?
Does the existence of the integral $\int_0^\infty f(x)dx$ imply that f(x) is bounded on $[0,\infty)$ when f(x) is continuous in this same interval ?
edit
I'm trying to ... |
H: Find value for 3-dimensional function f such that f is continuous at (0,0,0)
Let $f$ be a function $f: \mathbb{R}^3\setminus M \to \mathbb{R}$ where $M:= \{(x,y,z)^t \in \mathbb{R}^3\; |\; x = y = 0,\; z\in \mathbb{R}\}$ and
$$f(x,y,z) = \frac{xz^2+y^3}{x^2+y^2}.$$
Examine if there's a possible value $m = f(0,0,0)$... |
H: value of $g(3)$ in functional equation
If $$\int^{2x+\cos(\pi x)}_{0}g(t)dt=x^2$$for all real $x$ and $g(x)$ is continuous function. Then $g(3)$ equals
What i try: using Newton leibnitz formula
$$\frac{d}{dx}\bigg(\int^{2x+\cos x}_{0}g(t)dt\bigg)=\frac{d}{dx}(x^2)$$
$$g(2x+\cos \pi x)(2-\pi\sin(\pi x)=2x$$
When i... |
H: Given only the lengths of the three sides of a triangle, can we find if the triangle is impossible?
I was recently trying to implement Hero's Formula and I was surprised to find it declare that the perimeter of a triangle with sides of length 3, 4, and 9 is a complex number. This suggests that given the lengths of ... |
H: Maximum $n$ such that $e2^{1 - {k \choose 2}}{k \choose 2}{{n-2} \choose {k-2}} \leq 1$, as $k \to \infty$.
Using:
${k \choose 2} \leq \frac{k^2}{2}$, and
${{n-2} \choose {k-2}} \leq \frac{k^2}{n^2}{n \choose k} \leq \frac{k^2}{n^2} \left(\frac{ne}{k}\right)^k$,
we have $e2^{1 - {k \choose 2}}{k \choose 2}{{n-2... |
H: Order of the sum of two meromorphic functions
I haven't been around complex analysis at some time now and I need it for Riemann surfaces and I a bit confused about something. I know that if I have a meromorphic function $f$ with order $k$ at $p$ and a meromorphic $g$ with order $n$ at $p$ I will have that $ord_p(f+... |
H: Detail in the proof that $\mathbb Z[\sqrt {10}]$ is not a UFD
We want to show that $2$ is irreducible in $\mathbb Z[\sqrt {10}]$, so we assume: $$2=(a+b\sqrt{10})(c+d\sqrt{10}),\quad a,b,c,d \in \mathbb Z$$
While there exists an automorphism, call it $f$, sending $\sqrt {10} \mapsto -\sqrt {10}$, we can also write... |
H: Graphs of $e^{-y} = x$ and $y = \frac 1x$: Why do they seem so similar on $[0, 1]$, but one converges while the other diverges?
First things first, in the above graphs the one in blue is just a rotated version of ex from where the question did really form. I rotated it just in order to make comparisons easier.
I... |
H: write permutation as product of 3-cycles
Let $\sigma$ be the permutation $\sigma=(1234)(35)(2653)^2(56)$. I've got to say whether $\sigma$ is a product of 3-cycles or not.
My attempt was to simplify $(2653)^2=(36)(25)=(25)(36)$. Is this correct?
At the end I'd get $\sigma=(63)(5)(241)$, which shows that it can't b... |
H: checking uniform convergence of series $\sum_{n=1}^\infty x^n$
I have a doubt in a question in which I need to check the uniform convergence of the series given by:
$$\sum_{n=1}^\infty x^n$$ on (-$1,1$)
Now if the series is uniformly convergent,then its sequence of partial sums (s$_n$) is uniformly convergent.
I ha... |
H: If $X$ is path connected then $X$ has not isolated point.
Lemma
If $X$ path connnected then for any $x_0$ there exist a path such that connect $x_0$ with any other point $X$ of $X$.
Statement
If $X$ is path connected then $X$ has not isolated point.
Unfortunately I can't prove the statement, but I'm sure that it ... |
H: Joint distribution and covariance of Poisson process and waiting time
Hi I am having a trouble solving for this problem where I have to find
1) Joint distribution of $W_{1}$, $W_{r}$ for $r\geq2$.
2) $\operatorname{Cov}(W_{1},W_{r})$ for $r\geq2$.
[Notation explanation: $W_{r}=\min(t:N_{t}\geq r) $ is waiting t... |
H: $\sup(a + B) = a + \sup B$
I believe my proof of this simple fact is fine, but after a few false starts, I was hoping that someone could look this over. In particular, I am interested in whether there is an alternate proof.
For a real number $a$ and non-empty subset of reals $B$, define $a + B = \{a + b : b \in B\... |
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