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> **Proposition.** Complete ordered fields are Archimedean.
>
>**Proof.** Let $\mathbb{F}$ be a complete ordered field and consider $x\in\mathbb{F}$. We must show that there is an integer $N$ such that $x<N.$ If not, the monotone sequence $1,2,3,\ldots$ is bounded above by $x$ and converges by the monotone sequence p... |
Given:
- $ \Omega \subseteq \mathbb{R}^n $.
- $ f: \Omega \rightarrow \mathbb{R}^n $ and $ g: \Omega \rightarrow \mathcal{L}(\mathbb{R}^n, \mathbb{R}^n) $ are two $ C^1 $-class functions on $ \Omega $.
- Define $ F: \Omega \times \Omega \rightarrow \mathbb{R}^n $ as $ F(x, y) = g(x)(f(y)) + f(x) $.
(a) Show that ... |
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators of $I$, $u$ and ... |
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators of $I$, $u$ and ... |
How can one prove/disprove this argument for an arbitrary number $n$ of subspaces?
[![Generalization Problem][3]][3]
A proof of this statement should involve induction over $n$, but this approach eludes my intuition.
Or is there a counterexample that disproves the claim?
----------
Note: This is an... |
Is the bound $\Gamma(\alpha, x) \leq e^{-x} (x + \alpha )^{\alpha - 1}$ valid for all $x > 0$ and $\alpha \geq 0$? |
Recently, I saw this infinitely nested radical whose value seems $2$. How will you prove it? $\displaystyle\sqrt{1^2+\sqrt{2^2+\sqrt{4^2+\sqrt{8^2+\sqrt{16^2+\sqrt{32^2+\cdots}}}}}}=?$ |
Recently, I saw this infinitely nested radical whose value seems $2$. How will you prove it? $\displaystyle\sqrt{1^2+\sqrt{2^2+\sqrt{4^2+\sqrt{8^2+\sqrt{16^2+\sqrt{32^2+\cdots}}}}}}=2$ |
Recently, I saw this infinitely nested radical whose value seems $2$. How can we prove it? $\displaystyle\sqrt{1^2+\sqrt{2^2+\sqrt{4^2+\sqrt{8^2+\sqrt{16^2+\sqrt{32^2+\cdots}}}}}}=2$ |
What do you call a function that takes an ordered set S of reals to a real and has the property that f(S) = f({f({S-{x}),x}) for all x in S? Ex summing a set of numbers. You can take out one number sum the rest then sum the partial sum with the number that was removed to get the same result. |
When I was reading a textbook about functional analysis, I notice a theorem:
> In metric space, if every continuous functions on a closed set $A$ is bounded, then $A$ must be compact.
And the first step is to suppose that if $A$ is not compact in metric space, then $\exists \delta>0$ and closed balls $B_n$ in $A$... |
Say I have 5 matrices call them
$$A,B,C,D \text{ and } E.$$
Let $A,B$ and $C$ be $r\times k$ matrices which are fixed. Let $D$ and $E$ be $k\times k$ matrices which are free but fixed to be hermitian. I wish to know whether it is possible to provide general conditions on when there exists a solution to the matrix eq... |
I was studying polynomial rings over commutative rings from the book Topics in Algebra by I.N Herstein.
From there what I understood was, that:
If $R$ be a commutative ring with a unit element then $R[x]$ is defined to be the set of all symbols $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ where, $a_0,a_1,...,a_n\in R$ and... |
Is the bound $\Gamma(\alpha, x) \leq \tfrac{3}{2} e^{-x} (x + \alpha )^{\alpha - 1}$ valid for all $x > 0$ and $\alpha \geq 0$? |
Picture below is from [do Carmo's Riemannian Geometry][1], seemly, the author think that the 2.2 Index Theorem implies the 2.9 Corollary. I don't see that at all. Did the author make a mistake?
>[![enter image description here][2]][2]
>[![enter image description here][3]][3]
>[![enter image description here][4... |
Why Morse Index Theorem imply Jacobi Theorem? |
> My doubt is, how is the instantaneous velocity a rate of change? If
> you compute the rate of change, at an instant (i.e at an point),
> you'll see that the x axis is not changing.
From what I can gather, it's the use of *instantaneous* (instantaneous velocity) that's giving you a hard time. You are taking insta... |
Let $\mathcal{M}_A=\langle\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M}_A}, x^{\mathcal{M}_A}, <^{\mathcal{M}_A}\rangle$ be the standard model for the language of arithmetic $\mathcal L_A$. Define theory $\text{Th}(\mathcal{M}_A)=\{\alpha : \mathcal{M}_A\models\alpha\}$.
I have doubts about whether... |
Does Peano's axioms axiomize true arithmetic? |
Let $\mathcal{M}_A=\langle\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M}_A}, \times^{\mathcal{M}_A}, <^{\mathcal{M}_A}\rangle$ be the standard model for the language of arithmetic $\mathcal L_A$. Define theory $\text{Th}(\mathcal{M}_A)=\{\alpha : \mathcal{M}_A\models\alpha\}$.
I have doubts about wh... |
Given a linear inequality in $n$ variables:
$$\sum\limits_{i=1}^n c_i x_i \leq b$$
I want to estimate how many integer tuples $(x_1, x_2, \dots x_n)$ that satisfy that inequality. What is the best way to do this? Obviously, I'm looking for something much more efficient than simply counting up the points in the gr... |
>Show that if $x>1$, $\log_e\sqrt{x^2-1}=\log_ex-\dfrac{1}{2x^2}-\dfrac{1}{4x^4}-\dfrac{1}{6x^6}-\cdots$
$\log_e\sqrt{x^2-1}=\dfrac{1}{2}\log_e(x^2-1)=\dfrac{1}{2}[\log_e(x+1)+\log_e(x-1)]$
I know:
$\log_e(x+1)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}- \cdots$
$\log_e(1-x)=-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}- \cdots$... |
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ and $n\geq 2$ is an integer, then is it true that $$(I(R:_{Q(R)} I))^n=\sum_{f\in (R:_{Q(R)} I)} (fI)^n$$ ?
I am su... |
Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$? |
I am studying chapter 1 of Gathmann's notes about algebraic geometry and there is something that I don't understand in the proof of proposition 1.2.4, where Hilbert's Nullstellensatz is proved for $\mathbb{C}$. The part that I don't understand is as follows.
Let $\mathfrak{m}$ be a maximal ideal of $\mathbb{C}[x_1,x... |
Assume you have an audience of size A and some ad impressions to serve them denoted B. Assignment is independent for each ad.
**Question:** What's the probability that every audience member will be served at least one ad impression?
This is how I tried to solve it:
P(every person has at least 1 impression) =... |
When I was reading a textbook about functional analysis, I notice a theorem:
> In metric space, if every continuous functions on a closed set $A$ is bounded, then $A$ must be compact.
In the proof, it says that
> Suppose that if $A$ is not compact in metric space, then $\exists \delta>0$ and closed balls $B_n$ i... |
When is cdf $F_{X_1+\dots+X_n}(c)$ of sum of iid zero mean random variables decreasing in sample size $n$? |
I thought it was `homology` that is dual to `cohomology`. But reading `Bott & Tu` I am now confused. In the intro, page 2 and onwards, it briefly describes a dual point of view and the relation between homotopy and cohomology. It sounds like homotopy and cohomology are _dual_ to each other, although it is not precisely... |
I'm confused why my two approaches are leading to different answers.
Assume you have an audience of size A and some ad impressions to serve them denoted B. Assignment is independent for each ad.
**Question:** What's the probability that every audience member will be served at least one ad impression?
This is... |
Picture below is from [do Carmo's Riemannian Geometry][1], seemly, the author think that the 2.2 Index Theorem implies the 2.9 Corollary. I don't see that at all. Did the author make a mistake?
In where, $E$ is the energy of curve in the variation. For example, assuming $f(s,t)$ is the variation of $\gamma (t)$, the... |
Let $F$ be a strictly increasing distribution function. For a given $\tau \in (0,1)$, suppose there exists $\epsilon_{\tau}$ such that $F(\epsilon_{\tau}) = \tau$. Considering a set of independent and identically distributed (i.i.d.) random variables $X_{1}, \cdots, X_{n}$, with their order statistics denoted by $X_{(1... |
How can P1 ⇔ P2 ⇔ P3, ... ⇔ Pn be True when P1 =/= P2 =/= P3 =/= Pn and all P are not false ?
For example: If P1 = T P2 = T P3 = F P4 = F would this be true?
T(P1) ⇔ T(P2) ⇔ F(P3) ⇔ F(P4) =
T(P1&2) ⇔ F(P3) ⇔ F(P4) =
F(P1,2,3) ⇔ F(P4) =
True
If so how can an evaluation for a series of compound bicondit... |
When can compound biconditional statements be true, when the values of the individual components have different Truth Values? |
Firstly, your problem is not well-defined. Here are a couple of ambiguities, for example.
1. Does each ad have to be served to exactly 1 person in the audience?
2. Can one person be served multiple ads?
If we assume the answer to both of these questions is yes, then your second approach is correct. The first h... |
I'm attempting to code math formulas from square 1 where i use addition to make subtraction, multiplication, division, -etc. etc. But that brung me to exponents, i felt it was simple enough, and i smacked in a repeat and a self multiplying script, and i got what i was looking for, but when i got to things like 2^0, i g... |
what is the definite formula for an exponent? |
I read in [this paper][1] that by doing Dehn surgery along every algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$) link $L$ in an integral homology sphere yields a homology sphere, the author says it's easy to confirm the result. Does this follow from a direct computing of $H_1$ of... |
Why unit surgery on knots yeilds homology sphere? |
I'm confused why my two approaches are leading to different answers.
Assume you have an audience of size A and some ad impressions to serve them denoted B. Assignment is independent for each ad. **ETA**: Each ad is served to one person only, and each person may get several ads. Assume $B \geq A$.
**Question:** ... |
$y_{0}=0$
$y_{1}=\frac{-1}{2}x^2$
$y_{2}=\frac{-1}{2}x^2 - \frac{1}{10}x^5$
$y_{3}=\frac{-1}{2}x^2 - \frac{1}{10}x^5- \frac{1}{80}x^8$
$y_{4}=\frac{-1}{2}x^2 - \frac{1}{10}x^5- \frac{1}{80}x^8- \frac{1}{880}x^{11}$
Now I am trying to figure out a general pattern. I can see that the powers follow an arith... |
Let $L$ be a framed link in an integral homology 3-sphere $M$. I read in [this paper][1] that if $L$ is algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$), then by doing Dehn surgery along $L$ yields a homology sphere, the author says it's easy to confirm the result. Does th... |
Why $\pm 1$ surgery on knots yeilds homology sphere? |
Given a linear inequality in $n$ variables:
$$\sum\limits_{i=1}^n c_i x_i \leq b$$
I want to estimate how many positive integer tuples $(x_1, x_2, \dots x_n)$ that satisfy that inequality. What is the best way to do this? Obviously, I'm looking for something much more efficient than simply counting up the points ... |
I understand that this question has been asked before on this forum but, my question is not on how to do the question but on why my approach using reduced sample space fails.
Consider this question, also asked [here][1]:
> When coin 1 is flipped, it lands on heads with probability .4; when
> coin 2 is flipped,... |
Firstly, your problem is not well-defined. Here are a couple of ambiguities, for example.
1. Does each ad have to be served to exactly 1 person in the audience?
2. Can one person be served multiple ads?
If we assume the answer to both of these questions is yes, then your second approach is correct. The first h... |
Let $L$ be a framed link in an integral homology 3-sphere $M$. I read in [this paper][1] that if $L$ is algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$), then by doing Dehn surgery along $L$ again yields an integral homology sphere, the author says it's easy to confirm the result.... |
Why $\pm 1$ surgery on homology sphere again yeilds homology sphere? |
I am reading the following paper of Calderon and Zygmund [http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf][1]
On page 3 (252) they provide an estimate (2.3) which is following
$$ |K(x-x_{\nu}) - K(-x_{\nu})| \leq \frac{1}{|x_{\nu}|^k} \omega\left(\frac{A}{|x_{\nu}|}\right) $$
where they have said earlier tha... |
I am reading the following paper of Calderon and Zygmund [http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf][1]
On page 3 (252) they provide an estimate (2.3) which is following
$$ |K(x-x_{\nu}) - K(-x_{\nu})| \leq \frac{1}{|x_{\nu}|^k} \omega\left(\frac{A}{|x_{\nu}|}\right) $$
where they have said earlier tha... |
Let $\mathcal{M}_A=\langle\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M}_A}, \times^{\mathcal{M}_A}, <^{\mathcal{M}_A}\rangle$ be the standard model for the language of arithmetic $\mathcal L_A$. Define theory $\text{Th}(\mathcal{M}_A)=\{\alpha : \mathcal{M}_A\models\alpha\}$.
I am trying to show th... |
Do you have a probabilistic interpretation of this monster? If $Y$ is a random variable with density $g(y)=f'(y)$ I found that
$$h(t)=\frac{1-\int_0^tyg(y)dy}{(\phi-t)\int^1_tg(y)dy}-\frac{1-E(Y)}{\int^1_t(y-t)g(y)dy + (\phi-t)\int^1_tg(y)dy}.$$ This suggests to check the property when $g(y)dy$ is replaced by the Di... |
I guess you were able to piece together from Willie Wong's answer that you can obtain a global construction that yields the natural volume pseudo-form on any (pseudo)Riemannian manifold. The recipe is to
1. use $g$ to induce a smooth fibrewise inner product on the bundle $|\Lambda|^nTM^*$ of top pseudoforms.
2. ... |
How can one prove/disprove this argument for an arbitrary number $n$ of subspaces?
> Let $F$ be a field such that $2 = 1+1$ in $F$ has a multiplicative inverse. Prove or disprove that for subspaces $U_1, U_2, ... U_n$ of a vector space $V$ over $F$,
$$U_1 \cup U_2 \cup... U_{n-1} \cup U_n$$ is a subspace of $V$ i... |
We can consider the following **equivalent** objective:
$$\|Lx - y\|^2=x^T\color{blue}{L^TL}x-2y^TLx+y^Ty.$$
As $L^TL$ is a symmetric (positive semidefinite) matrix, it admits **the spectral decomposition**: $$\color{blue}{L^TL = Q^T \Lambda Q }$$ for some orthogonal matrix $Q$ with $Q^TQ=I$ and diagonal matrix ... |
Consider the functions f(x)=$\frac{1}{x^2}$, g(x)=$\frac{sinx}{x}$, and h(x)=$\frac{|x|}{x}$, all defined on ℝ\{0}, what do we set f(0) to be for f to be continuous on ℝ? Do the same for g and h.
I know for a function to continuous in a point, it has to fulfill:
$$ ∀ε>0 ∃δ>0: ||f(x)-f(a)||<ε, ∀x in A with ||x-a||... |
In 1 dimension if i am given two numbers a<b and function f that is positive for $x \in [a,b]$ i know that $\int_{a}^{b}f(x)dx$ is positive and $\int_{b}^{a}f(x)dx$ is negative.
However in 2d and above the region of integration is often given simply as a set of numbers: $\iint_{D}f(x)dxdy$
Am i to assume the D in... |
How do i determine the "sign" of a region of integration in multivariable integrals? |
How to solve this ode of the distribution? |
Consider the functions $f(x)=\frac{1}{x^2}$, $g(x)=\frac{\sin x}{x}$, and $h(x)=\frac{|x|}{x}$, all defined on $ℝ\setminus\{0\}$, what do we set $f(0)$ to be for $f$ to be continuous on $ℝ$? Do the same for $g$ and $h$.
I know for a function to continuous in a point, it has to fulfill:
$$∀ε>0 \; ∃δ>0: \|f(x)-f(a)\... |
It is a common knowledge that a function is $c$-strongly convex if $f - c\|x\|^2_2$.
However, can we replace $\|x\|_2$ with any norm $\|x\|_p$? I strongly suspect this holds, but from looking at several references online (see p 26 of [this][1]), it might also be true that this condition ONLY holds for $p = 2$.
I... |
Is it true that a function is $c$-strongly convex if $f - c\|x\|^2_p$ is convex for ANY norm $\|x\|_p$? |
In one of my recent answers to an optimization question (see [here][1]), I came across the following equation:
$$\sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}=1$$
where $a_j>0, \lambda_j \ge 0, j=1,...,n$. After doing some numerical checks, I realized that this equation always has **two real roots** (try it yourse... |
Suppose we are measuring the heights of different people. Our measures will be in $cm$, but the variance will be in $cm^2$. If we were measuring the time to complete a race course, our measures would be in seconds, but the variance would be $seconds^2$
What do those units mean? Why are they there? I'm not asking abo... |
What does it mean for the variance to be squared? |
We can consider the following **equivalent** objective:
$$\|Lx - y\|^2=x^T\color{blue}{L^TL}x-2y^TLx+y^Ty.$$
As $L^TL$ is a symmetric (positive semidefinite) matrix, it admits **the spectral decomposition**: $$\color{blue}{L^TL = Q^T \Lambda Q }$$ for some orthogonal matrix $Q$ with $Q^TQ=I$ and diagonal matrix ... |
Consider the following equations:
$$\left\{ \matrix{
{z_1} + {z_2} = 9 + 5i \hfill \cr
\overline {{z_1}} + 2{z_2} = {z_1} + 10 + 2i \hfill \cr} \right.$$
Assuming ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$, substitute them to equations
$$\left\{ \matrix{
{x_1} + i{y_1} + {x_2} + i{y_2} = 9 ... |
I want to prove this statement:
$a\geq1$ and $n\geq 2\alpha log(2\alpha)$ then $n \geq \alpha log(2\alpha)$
It seemed pretty simple but I couldn't do too much. My first attempt was
$$ log(n) \geq log ( 2\alpha log(2\alpha) ) $$
But this attempt didn't look promising. Could you give me a hint? |
Problem:
(i) Show that no matter how we color all the points of the plane R2 in 3 colors, there always exist two points at distance 1 with the same color.
(ii) Is it true that for every positive integer k, when all the points of R2 are k-colored, there always exist two points that are at distance 1 and have the s... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
Is my intuition here regarding generic sets used in forcing correct? |
Consider the following equations:
$$\left\{ \matrix{
{z_1} + {z_2} = 9 + 5i \hfill \cr
\overline {{z_1}} + 2{z_2} = {z_1} + 10 + 2i \hfill \cr} \right.$$
Assuming ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$, substitute them to equations
$$\left\{ \matrix{
{x_1} + i{y_1} + {x_2} + i{y_2} = 9 ... |
Let $U=(0,\infty)\times (0,2\pi]$
Define a function $F:U \rightarrow \mathbb R^2$ by
$$F(r,\theta)= (r\cos(\theta),r\sin(\theta)).$$
Here all $4$ partial derivatives of component functions exist and are continuous. So $F$ is continuously differentiable with non zero Jacobian $r$ at $(r,\theta)$. So by inv... |
Problem:
(i) Show that no matter how we color all the points of the plane $\mathbb{R}^2$ in $3$ colors, there always exist two points at distance $1 $with the same color.
(ii) Is it true that for every positive integer k, when all the points of $\mathbb{R}^2$ are $k$-colored, there always exist two points that ar... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
Consider the following equations:
$$\left\{ \matrix{
{z_1} + {z_2} = 9 + 5i \hfill \cr
\overline {{z_1}} + 2{z_2} = {z_1} + 10 + 2i \hfill \cr} \right.$$
Assuming ${z_1} = {x_1} + i{y_1}$ and ${z_2} = {x_2} + i{y_2}$, substitute them to equations
$$\left\{ \matrix{
{x_1} + i{y_1} + {x_2} + i{y_2} = 9 ... |
We can consider the following **equivalent** objective:
$$\|Lx - y\|^2=x^T\color{blue}{L^TL}x-2y^TLx+y^Ty.$$
As $L^TL$ is a symmetric (positive semidefinite) matrix, it admits **the spectral decomposition**: $$\color{blue}{L^TL = Q^T \Lambda Q }$$ for some orthogonal matrix $Q$ with $Q^TQ=I$ and diagonal matrix ... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
I am trying to understand how a joint distribution is formed when two regular conditional distributions are involved.
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let there be the three random variables $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$, $Y:(\Omega, \mathcal{A})\r... |
I want to prove this statement:
$a\geq1$ and $n\geq 2\alpha log(2\alpha)$ then $n \geq \alpha log(2n)$
It seemed pretty simple but I couldn't do too much. My first attempt was
$$ log(n) \geq log ( 2\alpha log(2\alpha) ) $$
But this attempt didn't look promising. Could you give me a hint? |
I want to prove this statement:
$\alpha \geq1$ and $n\geq 2\alpha log(2\alpha)$ then $n \geq \alpha log(2n)$
It seemed pretty simple but I couldn't do too much. My first attempt was
$$ log(n) \geq log ( 2\alpha log(2\alpha) ) $$
But this attempt didn't look promising. Could you give me a hint? |
We can assume that $\lambda_1 \le \lambda_2 \le \ldots \le \lambda_n$. The function
$$
f(x) = \sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}
$$
is continuous on the open interval $(\lambda_n, \infty)$ with
$$
\lim_{x \to \lambda_n +} f(x) = +\infty \, , \, \lim_{x \to \infty} f(x) = 0 \, .
$$
It follows that $f... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
In one of my recent answers to an optimization question (see [here][1]), I came across the following equation:
$$\sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}=1$$
where $a_j>0, \lambda_j \ge 0, j=1,...,n$. After doing some numerical checks, I realized that this equation always has **exactly two real roots** (try i... |
Problem:
(i) Show that no matter how we color all the points of the plane $\mathbb{R}^2$ in $3$ colors, there always exist two points at distance $1 $with the same color.
(ii) Is it true that for every positive integer k, when all the points of $\mathbb{R}^2$ are $k$-colored, there always exist two points that ar... |
We shall bring in probability to simplify the problem.
For the moment, consider Mary and Jane as blue balls, John as a yellow ball, and the other $3$ as red balls.
P(blue balls together) $=P(A)= \frac{5}{\binom62} = \frac13$
P(yellow ball) at ends $= P(B) = \frac26 = \frac13$
$P(A\cap B) = \frac13\fra... |
We can assume that $\lambda_1 \le \lambda_2 \le \ldots \le \lambda_n$. The function
$$
f(x) = \sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}
$$
is continuous on the open interval $(\lambda_n, \infty)$ with
$$
\lim_{x \to \lambda_n +} f(x) = +\infty \, , \, \lim_{x \to \infty} f(x) = 0 \, .
$$
It follows that $f... |
Let $X$ be a complex Kahler manifold of dimension $k$. For a smooth $(k,k)$form $\omega$, it is for sure $\partial$ and $\bar\partial$-closed since it is of top degree. Assume that $\omega$ is a difference of two probability measures.
Could we prove that $\omega$ is further $\partial$ or $\bar\partial$-exact? If no... |
We can assume that $\lambda_1 \le \lambda_2 \le \ldots \le \lambda_n$. The function
$$
f(x) = \sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}
$$
is continuous on the open interval $(-\lambda_1, \infty)$ with
$$
\lim_{x \to (-\lambda_1)+} f(x) = +\infty \, , \, \lim_{x \to \infty} f(x) = 0 \, .
$$
It follows that... |
A variable line passing through a fixed point $P(x_0,y_0)$ intersects the circle $x^2+y^2=a^2$ at some point $Q$. If all the perpendicular bisectors of the variable line segment $PQ$ are tangent to a curve $C$, determine $C$ if (i) $P$ is inside the circle (ii) $P$ is outside the circle (Note: in both cases, only selec... |
In one of my recent answers to an optimization question (see [here][1]), I came across the following equation:
$$\sum_{j=1}^n \frac{a_j}{(\lambda_j+x )^2}=1$$
where $a_j>0, \lambda_j \ge 0, j=1,...,n$. After doing some numerical checks, I realized that this equation always has **two real roots** (try it yourse... |
I want to prove this statement:
$\alpha \geq1$ and $n\geq 2\alpha \log(2\alpha)$ then $n \geq \alpha \log(2n)$
It seemed pretty simple but I couldn't do too much. My first attempt was
$$ \log(n) \geq \log ( 2\alpha \log(2\alpha) ) $$
But this attempt didn't look promising. Could you give me a hint? |
$G/ \sim$ is the set of $\sim$-equivalence classes in $G$ and $|G/ \sim|$ is the cardinality of $G/ \sim$. $|A| \leq |B|$ means that there is an injective function from $A$ to $B$. $|A| < |B|$ means that $|A| \leq |B|$ and $|A| \neq |B|$.
If $\pi:G \to G/ \sim$ is defined by $\pi (x) = [x]_{\sim}$, then $\pi$ has an... |
How much choice is needed to prove that $|G| \nless |G/ \sim|$ for any equivalence relation $\sim$ on set $G$? |
I am reading the following paper of Calderon and Zygmund [http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf][1]
On page 3 (252) they provide an estimate (2.3) which is following
$$ |K(x-x_{\nu}) - K(-x_{\nu})| \leq \frac{1}{|x_{\nu}|^k} \omega\left(\frac{A}{|x_{\nu}|}\right) $$
where they have said earlier tha... |
Here is my question:
> Let $f:W\to X$ be a smooth $\mathbb{P}^1$-fibration of smooth projective varieties over $\mathbb{C}$ where $\rho(X)=1$ and $X$ is Fano. How can we get a a vector bundle $\mathscr{E}$ of rank $3$ such that $f$ is just the composition of an embedding $W\subset\mathbb{P}_X(\mathscr{E})\to X$ and ... |
How to embed a $\mathbb{P}^1$-fibration into a suitable projective bundle? |
I apologise in advance for my vague and borderline senseless questions. I'm just trying to get a good hold on these things and I was having a tough time putting my thoughts into words.
I have two questions.
1. $\nabla f (\vec{v})$ shows the direction of steepest ascent of f at v. I feel like the jacobian is a gener... |
Let us consider two bangles. On these bangles let us add $8$ pearls on each bangle, the sequence of colors of pearls are same on both bangle. But the sequence of labels on pearls is not same. Now assuming the pearls as vertices of graph and circumference of bangle as edges of a graph, these bangles would correspond to ... |
Is $\mathcal{B}_n:= \{(a, b): a, b \in \mathbb{R}, a<b\} \cup \{(a, b) \cup \{n\}: a, b \in \mathbb{R}, a<b\}$ where $n \in \mathbb{N}$ is a basis for a topology on $\mathbb{R}?$ How can I prove this one? In our proof for the second condition, we have a problem in the case if both $B_1$ and $B_2$ are in the form $ \{... |
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