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Assuming the definition of 'Closed Form' given in the table of:
Closed Form Wikipedia entry, what areas tend to have problems that are traditionally expressed in closed form?
EDIT: Given the comment below about only 'easy' problems having closed form solutions, does this change if analytic closed forms are considered?
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I came across this integral in a problem I was trying to solve yesterday: $$ \int_{0}^{b0}{\int_{0}^{b1}{\int_{0}^{b2}{\int_{0}^{b3}{\int_{0}^{b4}{\int_{0}^{b5}{\int_{0}^{b6}{\dfrac{(1-x_0)^6}{\sum_{n=0}^{6}2^n{6 \choose n}x_n^n(1-x_n)^{6-n}}dx_6 dx_5 dx_4 dx_3 dx_2 dx_1 dx_0}}}}}}} $$ where $b_n$ is the bound on the i... |
Let $F$ be a coherent sheaf on $\mathbb{P}^1$. How to show that there exists a unique exact sequence of the form
$$0\to\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus a}\to\mathcal{O}_{\mathbb{P}^1}^{\oplus b}\to F\to\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus c}\to\mathcal{O}_{\mathbb{P}^1}^{\oplus d}\to 0?$$
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I am solving exercise 1.3.36 from Linear Algebra (Fraleigh). It says:
Let $v$ and $w$ be column vectors in $\mathbb{R}^{n}$. What is the size of $vw^{T}$? What relationships hold between $vw^{T}$ and $wv^{T}$?
My solution:
Both $v$ and $w$ column vectors have size $n\times 1$.
The size of $w^{T}$ is $1 \times n$.
Now, ... |
What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$?
$$2^{-k} = \frac 1 n \implies
n = 2^k \implies \log_{2} n = k$$
Is my solution correct?
|
I have a biased voter on $\mathbb{Z}^d,$ where $d>0$ (I am mostly interested in the cases where $d>1$) with the bias parameter $\lambda$. In other words, let us have a process $X=(X_t)_{t \ge 0}$ on $S=\{0,1\}^{\mathbb{Z}^d}$ such that if $i,j \in \mathbb{Z}^d$ are two neighbouring sites such that $X_t(i)=1, X_t(j)=0,... |
We are trying to show that the map $f\mapsto|f|$ is a continuous (nonlinear) map from $W^{1,p}(\Omega)\to W^{1,p}(\Omega)$ for any bounded/open region $\Omega$ and for $p\in[1,\infty)$.
We have tried using the regular definition of continuity using the standard norm on Sobolev spaces for both $f$ and $|f|$:
$\displayst... |
What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is $2^33^3$ where $x, y,z\in \Bbb N$?
What I tried :
At least one of $x, y$ and $z$ should have factor $2^3$ and at least one should have factor $3^3$. I then tried to figure out the possible combinations but couldn't get the corr... |
In statistics, it is a common practice to say that "correlation does not mean causation", and mostly the proof for this is given by examples. While that is good for the intuition, it's not rigorous. Ideas such as a third variable which may be causing both is often cited, but again, that's an example.
Can someone give m... |
Let $n\geq 2$, and define
$$\phi(t) = \int_{S^{n-1}} \cos(t \omega_1) d\omega,$$
where $S^{n-1}$ is the unit sphere in $n-$dimensional euclidean space, and $d\omega$ denote the surface area on $S^{n-1}$.
What can we say about the asymptotic behavior of $\phi(t)$ when $t$ tends to infinity? Does it know in literature?
T... |
Suppose $\operatorname{Var}(X_i) = \sigma^2_i$ and let $S_n = \sum_{i=1}^n \sigma^2_i = O(1)$, then I need to find $\lim_{n \to \infty} S_n$. Do we have $S_n \to \infty$?
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Let's say that I have $P(A \mid B,C)$ - is it accurate to say that $P(A \mid C)$ can be found like this:
$P(A \mid C) = \sum_B P(A \mid B,C)$
I know the values of all $P(A \mid B,C)$ as well as $P(B)$ and $P(C)$
|
In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying :
$$\text{for all } x \in \Bbb R, \quad x \in \Bbb R_+ \;\text{ if and only if} \;\; \phi[x] $$
My question is : what happens if I replace $\Bbb R$ by $\Bbb... |
I can see, using a paper, that the number of different combinations of forumals (in the sense extensively discussed in the comments) that one proposition can have is $4$, and even that the number for two propositions is $16$.
EDIT: These are the ones for one proposition:
p
0 0 0 1 1
1 0 1 0 1
We could basically see ... |
As the title says, for what values of a intersects $y = ax$, $y = \sin x$ just one time?
I am not able to solve this problem, and I really want to know the answer.
|
Consider a real sequence $\{ a_n \}_{n \in \mathbf{N}}$. Now if I write $A = \lim_{n \rightarrow \infty }$ $ \sup$ $ a_n$ , what do I actually mean by this? I mean if $a_1 , a_2, . . . $ is my sequence then $ \sup$ $ a_n$ is a single value. Then what is the point of taking its limit? Am I interpreting $\sup$ $ a_n$ i... |
The plane passes through $(3,2,-1)$ and $(1,-1,2)$ and is parallel to the line $v=(1,-1,0) + t(3,2,-2)$
I know how to do this with a vector perpendicular to the plane and that passes through a point, but I'm clueless on this one
|
Here's my question:
Let $f$ be a function which has a derivative in $\Bbb R$ such that $f'(x)\geq0$ and $f''(x)\geq0$ for all $x \in \Bbb R$
Prove that if there is some $a \in \Bbb R$ such that $f'(a)>0$ then $\lim_\limits{x\to\infty}=\infty$.
I have a hint in the question: Mean value theorem
What I know:
The functi... |
I am studying for a qualifying exam and am having difficulty with this problem:
Let $\left( X, \mathcal{M}, \mu \right)$ be a measure space and assume $f_n \geq 0$ such that $\int f_n = 1$ for all $n$. Show that $$\limsup_n \left( f_n(x) \right)^{\frac{1}{n}} \leq 1 \text{ for a.e. } x. \;\;\;\;\;\;\;\;\;\; (*)$$
A... |
Say I know this function
$$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$
where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal value is $1$, minimal value is $0$, and for every argument less than $0$ or greater than $1$ it is $0$. ($m(x)$ is know... |
There is no format more popular in the world than the representation of Integers: 32-bit and 64-bit strings are used by basically every single computer in existence and there's no practical reason to change that. But I'm curious to know what kind of different, creative representations integers and natural numbers could... |
Let $G$ be a finite set and $\cdot$ a binary operation on $G$ such that:
The operation $\cdot$ is associative;
$\forall x, y, z \in G, (x \cdot y = x \cdot z) \implies y = z$ and $(y \cdot x = z \cdot x) \implies y = z$
Show that there exists $e \in G$ such that for all $x \in G, e \cdot x = x$.
Show that we also hav... |
Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we have an upper bound on the Hessian, i.e. $\nabla_i \nabla_j f \leq Cg_{ij}$, then we have the following estimate for $f$... |
I'm currently trying to get into PDE's and as part of a course i'm focusing on proofs on existence of solutions to the Navier-Stokes Equations. Although existence of solutions has been proved for 1D and 2D my major problem is that i can't find one or more references/books/papers which proof the existence for the same N... |
In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering equations of the form $$\frac{{\rm d}X_t}{{\rm d}t}=F\left(t,X_t,\dot B_t\right)$$ where $\left(\dot B_t\right)_{t\ge 0}$ is a "... |
Here's Prob. 10, Sec. 19 in the book Topology by James R. Munkres, 2nd edition:
Let $A$ be a set; let $\{X_\alpha \}_{\alpha \in J}$ be an indexed family of spaces; and let $\{ f_\alpha \}_{\alpha \in J}$ be an indexed family of functions $f_\alpha \colon A \to X_\alpha$.
(a) Show there is a unique coarsest topology $... |
This might even be a notational nuisance, but here it goes.
Let $\kappa$ be a cardinal, $X_\alpha\subseteq\kappa$ for all $\alpha<\kappa$. As you know, the diagonal intersection of $\{X_\alpha\}_\alpha$ is the set
$$
\triangle_\alpha X_\alpha := \{\xi\in\kappa : \xi\in \textstyle\bigcap_{\alpha<\xi} X_\alpha\}.
$$
Now,... |
I am trying to evaluate a limit, but without much luck.
I keep getting infinity as the answer, but both Wolfram and my textbook state otherwise.
The limit problem goes as follows:
$\lim_{n \to \infty} n^{2/3}({\sqrt{n+1}}+{\sqrt{n-1}}-2{\sqrt{n}}) $
/* EDIT */
This is my original work, I wonder why it has failed to yi... |
Introduction:
Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98.
Exercise:
Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p\in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.
I think it's pretty hard to prove theorems... |
Let $f \in \tau(V)$, $V$ a variety in $\Bbb A^n$. Define
$G(f)=\{(a_1,\ldots,a_{n+1})\in \Bbb A^{n+1} \mid (a_1,\ldots,a_n)\in V$ and $a_{n+1}=f(a_1,\ldots,a_n)\} $
Prove that $G(f)$ is an affine variety.
Now first I thaught $G(f)=V(f(x_1,\ldots,x_n)-x_{n+1})$ but this is not true then we can express it as $G(f)=(f^{-1... |
Let $\{a_n\}$ be a sequence of real numbers bounded from above, $A\in \Bbb R$.
Given any $\epsilon>0$,
a)$\exists n_0 \in \Bbb N$ such that $ a_n<A+\epsilon$ for all $n\ge n_0$.
b)$\exists k\ge n_0$ such that $a_k>A-\epsilon$.
If the sequence satisfies the above two properties, show that $A=\limsup_\limits{n\to\infty}a... |
Let ${a_n}$ and ${b_n}$ be two real sequences such that $a_n\leq b_n$ for all $n$. Is it true that $\lim \sup a_n \leq \lim\inf b_n $? Outline the proof if so.
|
It is the problem 1.2.3 of Karlin's book An introduction to stochastic modeling:
A population having $N$ distinct elements is sampled with replacement. Because of repetitions a sample of size r can have fewer than r different elements. Let $S_r$ be the sample size necessary to have r different elements. Show that
$$\B... |
Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1).
I understand that this can be solved using induction, and I've set up my base cases. However, I'm not sure how I can prove that if the proposition is true for k, it must also be true for k+1.
Any help would be app... |
So this a fundamental assumption in mathematics. Can someone explain informally what it actually is please.
My guess is that its when we say in proofs that "Let $x \in X$". But I am not sure.
|
Given the volume formula of unit hypersphere
$V\left(n\right) = \left(2/d\right)\left[\pi^{d/2}\,/\,\Gamma\left(d/2\right)\right]$:
How do one prove that the max volume occurs when $d = 5\, ?$.
I try the ratio $V\left(n\right)/V\left(n - 1\right)$ and see when this is smaller than $1$ but ends up with something like
$... |
Given the matrix characteristic polynomial coefficients. Is there a quick way to determine the rank of the matrix?
|
Consider the dynamical system defined by
$$\overset{\circ}{x} = x\cdot g(x),$$
where
$$g(x) = \frac{r}{\alpha - x};\quad r,\alpha\in\mathbb{R}^+.$$
I am asked for a biological interpretation of this system, with $x(t)$ representing a population with a growth rate of $g(x)$. Everything would be easier if $g(x)$ were con... |
Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that
$F(x, 0) > 0$ for all $x \in X$
$F(x, 1) < 0$ for all $x \in X$
Then, for each $x \in X$, let $t_0(x) \in [0, 1]$ be the smalles... |
Picking randomly a number with two digits that can not be divided by $10$
$A)$ Find the expectation value and the variance of the number
$\color{gray}{\text{This part is for tommorow:}}$
- $B)$ Find the expectetion value and the variance of the digits multiply- $C)$ Find the probability using Chebyshev's inequality ... |
I've stumbled upon this question in my discrete math book:
Prove
$$
R = \{(x,y) \in N \times N \ | \ 2x \mid y^2 \}
$$
is transitive.
I tried thinking about it having to do something with division by 4 but couldn't proceed from there.
This is a question from really early stages of university studies so I think the solu... |
I'm pretty sure I did this right but I'd just like to check to make sure - I've been presented with the following problem:
Given a non-homogenous system of equations with 4 variables so that the vectors: $(0,1,2,3),(1,2,3,4),(1,2,4,5),(5,8,3,1)$ are solutions to the system, find the general solution of the system of eq... |
famous Savitch theorem states:
For any function $f\in\Omega(\log(n)), \text{NSPACE}(f(n)) \subseteq
> \text{DSPACE}((f(n))^2).$
Why we need an assumption that $f\in\Omega(\log(n))$?
Thank you.
|
I have a PDE that I have attempted to solve using the method of 'separation of variables'
$$u_t = (1+2t)u_{xx} \,\,\,\, 0 \leq x < \pi, t \geq 0 $$
With initial and boundary conditions:
$$u(0,t) = 0 $$
$$u(\pi, t) = \pi $$
$$ u(x,0) = 0$$
Using separation of variables, we have
$$u(x,t) = X(x)T(t)$$
$$\frac{T'}{(1+2t)... |
I made a matlab code to find numerical solutions to the Schrödinger equation independent of time, and it works very good, the eigenfunctions always are what they should be, the problem is in the eigenvalues, they never make any sense.
To prove that my code work I have to use it to calculate the first 10 eigenvalues of ... |
If for the polynomial $P(x)$ is true that $$P(x) = (x^4+x^3-3x^2+4x-4)\cdot q(x) + (2x^3-5x^2+7x-3)$$ find $P(2)$
I assumed that the polynomial $(x^4+x^3-3x^2+4x-4)$ has $(x-2)$ as one of its factors, but it turns out that this is not true.
By the equation we got that $P(2)=16\cdot q(2) + 7$ but I don't know what I s... |
Rosenthal (3.6.5) Let $(\Omega, \mathcal{F}, P)$ be a probability triple such that $\Omega$ be countable, and $\mathcal{F}=2^{\Omega}$. Prove that it is impossible for there to exist a sequence $A_1, A_2, \dots \in \mathcal{F}$ which is independent, such that $P(A_i) = 1/2$ for each $i$. [Hint: first prove that for ... |
I have a data set
x y
--------------------
505.000000 77.000000
507.000000 88.000000
509.000000 99.000000
509.000000 110.000000
511.000000 121.000000
511.000000 132.000000
511.000000 143.000000
513.000000 154.000000
513.000000 165.000000
513.000000 176.000000
515.000000 187.000000
515.000000 198.000000
51... |
I'd appreciate if you could please check to see if my proof is valid.
Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$.
Solution:
$q^2 > p^2/5 \iff q > p/\sqrt{5} \iff 0<p/q<\sqrt{5} $.
$\implies E = \mathbb{Q}\cap [0, \sqrt{5})$.
(i) Supremum. Since E is bounded, there exists an $M\ge ... |
I have to find all functions so that
$$ f\left(\frac{x}{f(y)}\right) = \frac{x}{f(x\sqrt{y})} $$
I have no idea how to solve this one. Any help would be appreciated!
|
Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in terms of volume forms.
It made me wonder if the notion of orientability can be generalized to spaces with slightly less re... |
I've been working on an algorithm that includes a step in which it has to calculate the angle between two 2D lines.
I've done some research and I've found this formula:
$\tan \theta = \pm \frac{m_2 - m_1}{1 + m_1m_2}$,
in which $m_1$ and $m_2$ are the slopes of the lines written in the form $y = mx + b$.
I also found t... |
I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly.
Also, I am having troubles justifying htat $\frac{n!}{k!} = n(n-1)(n-2)....k(k+1)$
|
I often read the following notation:
$\propto$.
How is this sign called and what is the definition of it?
|
So I have been unable to solve the following equation:
$y'= \sqrt{1-y^2}$, $\:\:y(0)=-1$
I have tried z-substitution to no avail, and was wondering if anyone had any ideas on how I should go about solving this. I do know that the final answer is non-unique:
$y=1$ and $y=-\cos(x)$.
Thank you!
|
I have two vectors $x_{1}$ and $x_{2}$ and some function $f(x, k)$. So for example function $f$ can be evaluated at some point say $f(x_{1}, 10)$. Then I have an integral written $$\int_{x_{1}}^{x_{2}}f(x,k)\ dx$$. I don't know how to calculate an integral written in a such form. So for example if $x$ is $2$-dimensiona... |
I am trying to fit some data in Matlab to a Hill function of the form $y = \dfrac{1}{1+(K/r)^n}.$ I have data for $r,y$ and I need to find $K,n$.
I have tried following the approach shown here in Demo $3$ at this link:
http://www.math.ubc.ca/~keshet/MCB2012/SlidesDodo/DataFitLect3.pdf
but I'm getting wrong results. Ple... |
In 2-phases simplex method what kind of operations must be done to get the canonical form tableau?
In this step(phase 2 of 2-phases method) after the remotion of artificial variables columns of auxiliary problem and copy of coefficents of objective functions, I don't know what are the operations performed to get canoni... |
The solution given shows $a = 2r + 6s$, $b = -3r - 7s$.
When I attempted to solve it myself, I put the two equations into a coefficient matrix, and row reduced, which gave me $a = r + 3s$, $b = -2r - 4s$.
How was the conclusion given in the solution reached?
|
I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. I'll phrase the question in terms of the definition which I like the most. Feel free to answer with a different definitio... |
Can someone help me to continue? Let $p \in [1,\infty)$, $l_{p}^{+}=\{((x_{n})_{n \ge 1}) \in l_{p}|x_{n}\ge 0, \forall n\ge 1 \}$. I am trying so show that $l_{p}^{+}$ has empty algebraic interior. I thought to prove it like that, but I dont know if I am on the right way. Suppose that it would be non empty. So there ... |
I need solve this recursion:
$$T(n) = T(\alpha n)+T(\beta n)+\gamma n$$
I know that for $\alpha +\beta< 1$ solution is $O(n)$
How is for $\alpha + \beta = 1$ and $\alpha + \beta > 1$?
|
How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like?
I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a commutative ring $R$. In the first case this is the set of the prime ideals and in the second the set of the maximal ideal... |
Let $X$ be an uncountable set and let $E$ be the collection of all singleton subsets of $X$.
Find the smallest ring generated by $E$.
Attempt:
Let $R$ be the ring generated by $E$.
Let $A_{1}$ and $A_{2}$ be any elements in $X$. Let $R_{1}$ be the smallest ring containing $A_{1} \setminus A_{2}$, $A_{2} \setminus A_... |
That is, say we had the following:
define a set, $A$, as:
$A = \{x,y,z\}$
If we had a function which only takes the elements of $A$ as its inputs, and returns "true" if $x$ is an input and false if $x$ is not an input, then is this technically considered a boolean-function?
I have always seen boolean-functions defined... |
I have proved the following theorem in an earlier part of the question:
Let $p,q \geq 1$ be such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that:
$$\|fg\|_1 \leq \|f\|_p \|g\|_q$$. I proved this statement using Young's Inequality. I am now asked to show the following:
If instead, $p,q,s \geq 1$ be such that $\frac{1}{p... |
Let $\mu^*:\mathcal P(\mathbb R^2)\longrightarrow \mathbb R$ defined by $$\mu^*(E)=\inf\left\{\sum_{i=1}^\infty m(T_i)\mid E\subset \bigcup_{i=1}^\infty T_i\right\}$$
where $T_i$ are triangles and $m$ is the Lebesgue measure.
1) Show that $\mu^*$ is an exterior measure.
2) Which measure is given by $\mu^*$ ?
My work
I ... |
It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies
$$d^\ast(A):=\limsup_{n\to +\infty} \frac{\lvert\{a\in A : a \leq n\}\rvert}{n}>0. $$
Fix $\epsilon\in (0,\frac{1}{2}]$. Is there any hope that there is an ultrafil... |
If $G$ is an abelian group, every subgroup $H$ of $G$ is normal.
I searched for non-abelian finite groups $G$ , such that every subgroup is normal and
GAP showed only the groups $G'\times Q_8$ , where $Q_8$ is the quaternion-group
of order $8$ and $G'$ is an abelian group with the additional property that $C_4$ is not ... |
what is an example of $f \in L^1(\mathbf{R})$ such that $\sum_{n=1}^\infty f(nx)$ converges a.e. but is not in $L^1(\mathbf{R})$?
Context: This question appeared on an old qualifying exam.
I tried something like $\sin(x)/x$ but that didn't work. Nothing else is coming to me.
|
The problem is the following:
Given the space of the polynomials $V=P(x)$ of degree $3$ prove the following:
The set $U$ which is defined as the set of the elements for which $P(5)=0$ is a subspace of $V$.
Find a basis for $U$.
If it was asked that $P(x)$ has $2$ also as a root then show that the new subset $W$, whic... |
I am trying to figure out the below question:
15. For which values of the constants $a$ and $b$ is the matrix
$$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$
invertible? What is the inverse in this case? See Exercise 13.
My understanding is that a matrix is invertible when the determinant is no... |
Given two sets $x = \{ a_1, a_2, a_3, a_4 \}$ and $y = \{ \emptyset, x, \{ a_1, a_2, a_3\}, \{ a_3 \}, \{ a_3, a_4 \} \}$, where $y$ is a topology defined on $x$.
How could we construct a continuous function $f:x\rightarrow y$ without $f(a_i) = a_i$, i.e. fix points ?
|
We know that the algebra and coalgebra axioms are given via following commutative diagrams (algebra $A$ and coalgebra $C$ are over a field $\mathbb{K}$):
I am now trying to show that the dual of a coalgebra $C^*$ has an algebra structure given by $$\mu_{C^*} := \Delta^* \circ \lambda \quad \text{and} \quad \eta_{C^*}... |
$$
\langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r)
$$
This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group.
EDIT In fact, these are called extended triangle groups, by G. Jones and D. Singerman in Maps, hypermaps and triangle groups...
What about the fol... |
This is more a question of the methadology one should use to solve these type of questions:
Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that
$$X \setminus V = U \cap X $$ Prove that $$ V = X\cap (Y \setminus U)$$
The easiest approach I found to prove these solutions is to simply draw a ven... |
The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same structure is a $2$-generated group : $(12)(34)$ and $(23)(5678)$ is a possible set of generators.
How can I prove that the g... |
Prove for every $x \neq 0$ in the domain $(-1, \infty )$ that:
$$ln(1+x) < x $$
My prove:
$$ln(1+x) < x \iff $$-$$ e^{ln(1+x)} < e^x \iff $$-$$ 1+x < e^x $$
and since Taylor's series of $e^x$ is : $$ e^x = 1+x + R_n2(x) $$
$$R_n2(x)= \frac{e^c}{2}x^2 >0 $$
$$\implies 1+x+R_n2(x) > 1+x $$
is this okay?
|
In this answer on MathOverflow, it is claimed that
$$\chi(X\setminus Z)=\chi(X)-\chi(Z)$$
holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd codimension, this may fail. Why is this? Is this point discussed in some standard reference?
|
This is completely for my own curiosity.
Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians.
I know that this is strange because usually integers are only used in degrees. I know that there will be convergent sub series of $n$'s to any rational number in $[-1,1]$. Would we ever get an ... |
Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is it true that the stopped process $X^{T_n}$ remains $L^p$ integrable?
The answer is clearly yes for the discrete time ca... |
I was looking at Why $\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$?. And there it was proved that $$\sum_{i=1}^n \frac{1}{i} =\mathcal O(\ln(n))$$ My question is that does this also stand for $$\sum_{i=1}^n \frac{1}{i^2}$$ If so why?
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In the definition of the reduced $C^*$ crossed product associated with an action of a discrete group $G$ on a $C^*$-algebra $A$, one can begin with any faithful representation of $A$ on a Hilbert space (which exists by the GNS construction), go on to represent the twisted group algebra $AG$ on $\ell^2(G)\otimes H$, and... |
Given an $(x, y, z)$ system I create a new system $(x', y', z')$ by applying two rotations $\theta$ and $\phi$.
In the new system the $(x',y')$ plane, i.e.: the $z'=0$ plane, can be written as:
$$ (sin\theta\,sin\phi)\,x + (-cos\theta\,sin\phi)\,y + (-cos\phi)\,z = 0$$
I need to find out the smallest range of $\theta$ ... |
Show that $$\lim_{\epsilon \to 0^{+}} \left\langle \frac{\epsilon}{x^{2}+ \epsilon} ,\phi \right\rangle =\langle \delta,\phi\rangle $$
where $\phi\in D(\mathbb{R}) $ and $ \frac{\epsilon}{x^{2}+ \epsilon} \in D' (\mathbb{R}) $
What I tried is this:
$ \frac{\epsilon}{x^{2}+ \epsilon}$ is an integrable function henc... |
Wolfram says, there are only two roots, but $\sqrt{i}$ already gives two roots. So if we express them in Cartesian form we can take square roots of them separately and end up with four roots.
$$\sqrt{i}=e^{\frac{i\pi}{4}}=\frac{1}{\sqrt2}+i\frac{1}{\sqrt2}$$
But also
$$\sqrt{i}=e^{\frac{-3i\pi}{4}}=-\frac{1}{\sqrt2}-i\... |
Let $f$ be a smooth function :
can someone tell me why we have :
$\lim\limits_{\epsilon \rightarrow 0}\frac{1}{\epsilon}\int_{t}^{t+\epsilon}f(s)ds=f(t)$
thank you very much !
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I'm having a little problem with a statistics problem I am working on. I'm not really sure where to start to prove the two statements. Any help would be greatly appreciated.
Let $x$ and $y$ be jointly distributed numeric variables and let $z = a + by$, where $a$ and $b$ are constants.
Show that $\text{cov}(x, z) = b\,... |
Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e.
$$\underset{s\rightarrow\infty}{\lim}\phi_{s}(x)=\mathbf{1}_{K}(x)$$
I have read elsewhere on this site that "As $\phi_s$ approaches the indicator of $K$ its gradient converges... |
I've been trying to verify the condition above but I get stuck on the passage :
$$(P \land (P \land \lnot Q)) \lor (P \land Q)$$
I don't know how to simplify it since there are two ands and a not Q.
I thought I might use absorption law, but it states that the signs should be different ( in other words, it should be an... |
I am supposed to show that there exists an $x$ in the interval $[0,\pi]$ such that $f(x)=f(x+\pi)$ by considering another function $g:[0,\pi] \to \mathbb{R}$ defined by $g(x)=f(x)-f(x+\pi)$.
Should I use a concrete example by considering a trig function?
|
Let be the domain $R= \left\{ (x,t) | 0 < x < l, \ t>0 \right\}$ and let be;
\begin{eqnarray}
u_{t}-a(t)u_{xx} &= &f(x,t), \ a>0\\
u(x,0)&=&u_{0}(x), \ x \in (0,l)\\
u(0,t)= 0,& \ u(l,t)=0;
\end{eqnarray}
take $ m > 1 $ an integer number, define the mesh $x_{j} = (j-1/2) \Delta x$, where $\Delta x = l/m$. how to build ... |
Let $A$ be a positive definite $d \times d$ matrix, and define $A(x,x)=x^TAx$. Let $x$ be a point such that $\vert x^T\xi\vert^2\leq \xi^T A\xi$ for all $\xi\in\mathbb{R}^d$. Is this somehow equivalent to
$A^{-1}(x,x)=x^TA^{-1}x\leq 1$?
|
my theory: Given a list of points on a 2 dimensional plane, and the degree of each point, there should correspond only one way to arrange the edges between points so that the final graph is a mesh of only triangles, and the degree of every point is correct.
MY question is, is my theory true, and if so, if there is a w... |
What is the delta in the image? How do I check it is a basis? Why are $i$ and $k$ the same, and $j$ and $l$? Need some clarification.
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I was looking at The accuracy from left to right and that from right to left of the floating point arithmetic sums which was asking for the accuracy of $$\sum_{k=1}^n\frac{1}{k^2}$$ from right to left, and I was wondering how summing from different directions gives you different accuracy?
Edit: I am specifically lookin... |
Let $f:[0,\infty)\to\mathbb{R}$ differentiable and suppose that $$\lim_{x\to\infty}f'(x)=L.$$ How can I prove that $$\lim_{x\to\infty}\frac{f(x)}{x} = L\;?$$
I have solved some similar problems using the Mean Value Theorem, and I am trying to use it again in this one, but nothing works. For example, I tried to apply th... |
I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function').
I also know the 'Zeta Function' $\zeta(s)$ is defined as:
$$\zeta (s) = \sum_{n=1}^{\infty} {1\over {n^s}}$$
But my question is why and how was this even derived?
I've studied an... |
The unnormalized negative entropy of $\mathbf{x} \in \mathbb{R}^n_+$ is
$$
g(\mathbf{x}) = \sum_i (x_i \log(x_i) - x_i)
$$
it is stated that $g(\mathbf{x})$ is strongly convex with respect to 1-norm, i.e.
$$
\sum_i\left(x_i \log\left(\frac{x_i}{y_i}\right) + y_i - x_i\right) \geq \frac{\lambda}{2} \left(\sum_i |x_i - y... |
I am unable to follow one if the steps described in this question. My question is, can someone describe the part I do not understand?
The full question and solution is below :
Find the value for $x$ for which the series $U=\sum_{n=1}^\infty (2n-1)x^n$ converges.
$$(2n-1)(x^n)$$
Using the ratio test:
$$ \begin{align}
R&... |
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