Certainly, if we fix the genus $g$ of a curve $X$, we have $\# $Aut$(X) \leq 84(g-1)$.
Let $X$ be a hyperelliptic curve. Is there a bound on $\#$Aut$(X)$? (Note that I do not want to fix the genus!)
More generally, can $\#$Aut$(X)$ be bounded in terms of the gonality of $X$?
I am trying to argue that for distinct primes $p,q,r$ we have that
$$ \gcd (pq + qr+ pr ,pqr ) = 1 = g $$
and I am wondering whether people find the following argument convincing :
Consider the prime decompositions; for $g$ to be greater than $1$ we need at least one of $p,q,r$ to occur in the decomposition of $pq + qr+ pr$. Without loss of generality suppose it is $p$ ( not necessarily exclusively ). But then the remainder modulo $p$ must be zero. However the remainder is $qr$ which is not zero modulo $p$ as $p,q,r$ are all mutually coprime. ( using $x\mid (a+b)$ and $x\mid a$ implies $x\mid b$ ) So the result follows.
Here we are at the end of January and a merger between US Airways and American still has not been announced. This is downright silly since just about every party agrees that it’s the right thing to do. So what’s up? Well, I say “just about” every party, because from what I can see, it’s American’s current management team that is the lone holdout. And it’s that stubborn stance that seems to be keeping the right thing from 
Last week, Reuters put out a joke of an “exclusive” article saying that a merger between the two could come in the next two weeks. Why was it a joke? Apparently the airlines were in final negotiations with “the final price and management structure still to be resolved.” So the “only” two sticking points are the ones that have always been the issue. Maybe this was leaked to try and help build some more traction, but it didn’t really say anything new.
Now the latest “news” of the day is that American CEO Tom Horton may end up being the Chairman of the combined entities. There is some good and some bad to this kind of thing. The good is pretty simple to explain. If Horton is willing to settle for a Tilton-esque agreement where he can just sit in a fancy office and collect a huge paycheck for a couple of years, then that finally removes the last real barrier to a merger – the fight being put up by management.
On the other hand, if he insists on a more active role, then it’s a bad idea. There are very few supporters of Horton outside management ranks. Wall Street has been quite clear that Horton’s plans to date are unacceptable. In particular, the plan to grow the hubs by 20 percent is suicidal. As one analyst, Dan McKenzie, puts it, the growth plan “would be toxic for industry pricing and ruinous for shareholders….” The views throughout the financial community appear to echo that sentiment. If Horton has any kind of influence in the merged entity, then the money folks will not be happy. And that hurts the chances of the deal going through.
Other than Wall Street, we know the employees have lost all faith in the current management team. That happens to any leadership team that takes an airline through bankruptcy, as it should. That’s why it’s best to have a new team to take you out of bankruptcy. Delta did this well.
Delta Does Bankruptcy Right
When Jerry Grinstein took over at Delta, he started making deep cuts and then led the airline into bankruptcy. He wasn’t exactly hated since he was quite upfront with what had to happen and didn’t take excess compensation, but it was always meant to be a temporary job so there would be no residual friction. When he stepped down, the hard restructuring work had been done. Richard Anderson stepped in to lead the resurgence which included the Northwest merger. Today, Delta is the industry leader.
American could have a similar story if it wanted. Tom Horton has made a lot of necessary cuts that have angered people in order to get this airline in a position where it would even have a shot at competing. He should get credit for that, but now it’s time to step aside and let the people who know revenue take over. When you come out of bankruptcy, you need a new leadership team with a solid revenue plan that people can rally behind. Horton is clearly not the guy to lead that team.
When I talk about the leadership team, I think some people might picture US Airways management coming in and throwing American leaders out the window, but that’s not the case. There is clearly a need for American leaders to run a combined airline. The US Airways team has to know that. In fact, I bet there are people in high places at American who secretly want to see a new leadership team at the top, because it will provide more opportunity for them. This new leadership team needs to fill the key roles at the very top. The rest will be a mix of experience between the two sides.
Too Many Promises
Without a new team, what we get is a current leadership team with a highly questionable plan. The thought seems to be that if American promises enough bells and whistles, people will fly the airline. And it will need 20 percent more capacity to serve all this magical new demand. Not a good plan. First, the promises are lofty and expensive. Today is the first day that any passenger on American can experience those amenities with the introduction of the 777-300ER on the Dallas/Ft Worth to Sao Paulo route, but that is just one airplane. It will take years for the airline to consistently deliver a product that can live up to what is being promised. All this “new American” stuff including the new livery will result in the worst thing possible – overpromising and underdelivering for years to come.
But even then, does it make sense to put all of these amenities out there? Some, yes (like flat beds in biz, of course), but probably not all. It seems like the airline is going overboard with announcing its plans just to try to sway people who may have influence on whether a merger happens or not. If it doesn’t happen? With a 20 percent increase in capacity and no merger, the airline is not going to have the revenue base to support all these expensive promises anyway. (It may not have it afterwards either, which is why the promises should likely be scaled back.)
At this point, my assumption is that the only reason we haven’t seen the merger announced is because the current management team is fighting it. Everyone else seems to be in agreement that it’s the right way to go. If that’s true, then it will happen. There are a lot of great people at American and around the US who deserve just that. It will just take longer than it should. And there will be more pain involved in the process.
[Original child tantrum photo via Shutterstock]
Immigration reformers have the winds at their backs this week, buoyed by support not just from Republican senators, but an array of conservative interest groups who have signed on to their cause. But the reignited debate is also driving the opposition's hardliners out into the open, offering the clearest picture yet of the fault lines Congress must overcome to pass a bill.
Among lawmakers, most Republicans in Congress are taking a wait-and-see approach for now. An exception in the Senate is David Vitter (R-LA), who ripped into Sen. Marco Rubio (R-FL) on Wednesday as "amazingly naive" and "nuts" for supporting the Senate's bipartisan plan.
In the House, a handful of Republican members made clear immediately that they were not on board for anything resembling the reforms under discussion in the Senate and White House. And despite Republican efforts to neutralize attacks on a path to citizenship for undocumented immigrants as "amnesty," the dreaded a-word was at the center of their attacks.
"By granting amnesty, the Senate proposal actually compounds the problem by encouraging more illegal immigration," Rep. Lamar Smith (R-TX) said in a statement.
"It's very difficult for me to support something that allows that type of amnesty," Rep. Peter King (R-NY) told Newsday.
Rep. Lou Barletta (R-PA) has also taken on an early role as an outspoken critic of reform.
At least as troubling for national Republicans hoping to use immigration talks as a means to repair their political standing after 2012 is conservative media's reactions. While some major TV personalities like Sean Hannity and Bill O'Reilly have voiced support for a comprehensive bill, other major outlets and commentators are preparing for war. And already the conversation is drifting to arguments that could further alienate Hispanic voters.
One emerging meme on the right, for example, is that Hispanics will never vote for Republicans because their community is inherently contemptuous of capitalism and family values. This was a key argument in the National Review's editorial condemning the Senate's immigration framework on Wednesday:
"[I]f we are to take Hispanics at their word, conservative attitudes toward illegal immigration are a minor reason for their voting preferences. While many are in business for themselves, they express hostile attitudes toward free enterprise in polls. They are disproportionately low-income and disproportionately likely to receive some form of government support. More than half of Hispanic births are out of wedlock. Take away the Spanish surname and Latino voters look a great deal like many other Democratic constituencies."
Rush Limbaugh, who recently declared that "it's up to me and FOX News" to stop a bill from passing, told Rubio in an interview this week that he believed immigrants no longer came to America because they wanted "to become citizens of the greatest country on earth."
"I've seen a number of research, scholarly research data, which says that a vast majority of arriving immigrants today come here because they believe that government is the source of prosperity, and that's what they support," Limbaugh said.
As he did with Limbaugh, Rubio has been making the rounds with almost every major conservative media personality to pitch reform. But while they've offered him a respectful hearing and often praised him personally, many still appear skeptical of any legislation and could form the core of any organized opposition to a bill.
RedState's Erick Erickson, often a bellwether for the conservative grassroots, also opposed the Senate's plan, albeit for less conventionally partisan reasons than Limbaugh or the National Review. According to Erickson, one of his chief concerns was that the bill doesn't expand legal immigration to low-skill workers enough to discourage people from entering the country illegally. Rubio wrote a lengthy point-by-point rebuttal on Erickson's site arguing that a temporary worker program included in the Senate plan would address the issue.
On Jan. 11, five days before President Barack Obama unveiled 23 executive actions he intended to to take to reduce gun violence, Pennsylvania state Rep. Daryl Metcalfe (R) announced that he was working on a firearms measure of his own.
"I am currently having legislation drafted that is similar to firearm legislation recently introduced in Wyoming," Metcalfe wrote in a memorandum sent to his fellow House members. "My legislation would prohibit the enforcement of any new federal restriction, prohibition or registration requirement for firearms, magazines, and ammunition. My legislation would also require the state to intercede on behalf of Pennsylvania citizens against any federal attempt to register, ban or restrict the purchase or ownership of firearms and firearms accessories which are currently legal products."
Metcalfe is one of many. Over the last several weeks, he and Republican lawmakers nationwide have introduced bills addressing firearms, the Second Amendment, and federal power. The bills are a response to President Barack Obama and Democrats' renewed interest in gun control measures, which itself is a response to the December mass shooting at Sandy Hook Elementary School in Newtown, Conn.
At least 20 states have had some kind of firearm-related legislation recently introduced or currently pending: Texas, Utah, Arizona, Alaska, Oklahoma, Pennsylvania, Wyoming, Missouri, North Dakota, South Carolina, Tennessee, Virginia, Nebraska, New Mexico, Georgia, Mississippi, Alabama, Iowa, Indiana, and Montana.
The language in the bills differs from state to state (and some states have seen several different bills proposed), as do some of the particulars. But the message and general thrust of the bills are the same: the lawmakers are telling the federal government to back off on guns.
"Our government and our legislature is not in place to simply do the bidding of the federal government," Missouri state Rep. Casey Guernsey (R) told TPM in an interview. "That is not the function of a state. And there is an ability and a right, an inherent right, of states to protect [their] citizens when the federal government becomes overreaching."
On Jan. 15, Guernsey introduced HB 170, which among other things would make it a class D felony for any federal official or agent to enforce or attempt to enforce any federal law or regulation on a firearm owned or manufactured in the state that remains "exclusively within the borders of the state." It's an idea contained in a number of other bills.
Asked what he imaged it would look like if Missouri were to, for instance, prosecute an agent of the Bureau of Alcohol, Tobacco, Firearms and Explosives, Guernsey said that "hopefully, it would look like a very strong statement on the part of state government."
While these bills have come all at once, lawmakers and professionals who are tracking the bills said there is no publicly identified group organizing or coordinating the efforts. Jon Griffin, a policy associate with the National Conference of State Legislatures who has been keeping a list of recent firearms bills, told TPM the bills did not appear to be particularly uniform.
Oklahoma state Sen. Nathan Dahm (R), for one, told TPM that a constituent had pointed him to model legislation prepared by The Tenth Amendment Center, a think tank which supports "the principles of strictly limited government." (The center has prepared a model "2nd Amendment Preservation Act," which serves to "prevent federal infringement on the right to keep and bear arms," and maintains a map tracking which states have introduced similar bills.) Dahm said he liked the center's model bill, but he "wanted to add some other things."
On Jan. 16, Dahm introduced SB 548, which reads, in part:
The Legislature of the State of Oklahoma declares that the 2nd Amendment guarantees an individual right to keep and bear arms free from infringement; that federal acts, laws, orders, rules, regulations, bans, or registration requirements regarding firearms constitute an infringement on the individual right, are not authorized by the Constitution of the United States and violate its true meaning and intent as given by the founders and ratifiers, and are hereby declared to be invalid in the State of Oklahoma, shall not be recognized by this state, are specially rejected by this state, and shall be considered null and void and of no effect in this state.
Any agent or employee of the federal government, or of a corporation providing services to the government, who enforces any measure found to be in violation of the act would face up to 5 years in prison and a $5,000 fine, or both.
Dahm's bill, along with a speech he gave at a Guns Across America Rally in Oklahoma City on Jan. 19, have earned the new lawmaker some welcome attention.
"I had people contacting me from as far away as Maine, California, Puerto Rico," Dahm told TPM. "Even overseas. People from England and as far away as Russia, saying that they understand how important it is to protect our Second Amendment rights for self-defense."
State lawmakers have tried using proposed legislation to protest perceived federal Second Amendment overreach in the past. A few years ago, Montana led the way on an effort to pass a measure known as the Firearms Freedom Act. The bill declares that firearms made and kept within state laws are beyond the reach of Congress' authority. The measure passed in Montana and a half-dozen other states, according to firearmsfreedomact.com, a website which supports the measure. But it has since been tied up in litigation, and is currently at the appellate level. Some of the recent bills appear to owe some of their ideas to Firearms Freedom Act.
In South Carolina, state Sen. Tom Davis (R) has introduced S 0224, a joint resolution to nullify in South Carolina "any presidential executive order restricting, abridging, or otherwise infringing upon a citizen's second amendment right to keep and bear arms." In Texas, Rep. John Otto (R) has introduced HB 553, which makes it an offense for any state employee, officer, or public servant to enforce any law or rule related to firearm banning, confiscation, taxation, or registration, as well as any limitations imposed on ammunition. In Wyoming, state Rep. Kendell Kroeker (R) is working with Rep. Allen Jaggi to update a law that was passed in 2010. The bill would increase the penalties on federal agents who try to enforce federal laws or regulations on firearms that remain within state lines. It also declares unenforceable any new federal laws or rules that attempt to ban or restrict semi-automatic weapons or magazines or that require any form of firearm registration. In an email, Kroeker said he and Jaggi received assistance from an organization called Wyoming Gun Owners when putting the bill together.
In Virginia, Delegate Bob Marshall (R) is among those trying a different approach.
"If you start saying you're going to start arresting federal agents per se, you're in big trouble," Marshall said. "People whip out the supremacy clause. I know a bunch of people don't like that, but that's going to happen. ... Ours, it says, 'OK, it's your show. You pay for it. We're not applauding.'"
Marshall's bill, HB 2340, which was presented Jan. 18, doesn't take federal laws head-on. Instead, it prevents any agency or employee of the state from assisting the federal government with investigations or prosecutions related to laws or regulations put in place after Dec. 31, 2012 that infringe "the individual Right to Keep and Bear Arms by imposing new restrictions on private ownership or private transfer of firearms, firearm magazines, ammunition, or components thereof." Marshall's idea is to withhold investigative resources and information from the federal government, which he believes would make new firearms measures unenforceable.
"This is a very simple tactic," Marshall told TPM. "Mahatma Gandhi, he brought down the British Empire in the Far East by doing this very thing.
Dahm, the Oklahoma state senator, has received a small measure of internet fame for his efforts. His speech at the Guns Across America Rally was uploaded to YouTube on Jan. 19, under the title: "If only ALL Senators Spoke Like this Senator! MUST WATCH!!" Dahm is youthful looking, with short, dark hair, and a flair for reciting constitutional language with a twang. At the end of his speech, he noted that the following Monday was not only Inauguration Day but also Martin Luther King Jr. Day. As a coda to his remarks, he offered two quotations from the civil rights leader.
"Martin Luther King Jr. said, 'Injustice anywhere is a threat to justice everywhere' -- we are not going to stand for this injustice," Dahm told the receptive crowd. "Another quote that he said was, 'Nothing in all the world is more dangerous than sincere ignorance and conscientious stupidity.' Now, I'm willing to give the President the benefit of the doubt, and believe that this is just sincere ignorance and not conscientious stupidity. But I'm here to say, no matter what it is, we in Oklahoma will not stand for it. The Second Amendment reigns supreme. 'Shall not infringe' means 'shall not infringe.'"
The escalating dispute between China and Japan over an archipelago of rocky islets known as Diaoyu in China and Senkaku in Japan has curbed Japanese exports into China, forcing Tokyo to fast-track moves to distance itself economically from its larger neighbor.
In a recent report, the World Bank said the Japanese economy appeared to be contracting “in part because of political tension with China over the sovereignty of islands in the region.”
But while these sovereign disputes hurt the bottom lines of Japan’s export-driven companies and push some to hedge their China exposure by increasing sales to other Asian economies, the size of the bilateral trade should secure the flow of Japanese investments into China over the long term.
In the short term, however, there is pain. Trade between Asia’s two largest economies fell 3.9 percent last year to $329.45 billion, the first drop in three years, as Japan slipped from being China’s fourth-largest trade partner to its fifth largest.
The deteriorating trade relations come as Japan gets ready to implement both fiscal and monetary stimulus measures to revive its struggling economy.
Japanese automakers are among those hardest hit by the souring relations. Nissan saw sales in China, the world’s biggest auto market, fall 5.3 percent to 1.18 million units; Toyota reported a 4.9 percent drop to 840,000 units; and Honda posted a 3.1 percent decline to 598,576 units.
The tensions appear to have hurt China as well. Foreign direct investment in China fell for the first time in three years in 2012. Most of the FDI into China comes from 10 Asian countries and economies, including Japan, and this fell 4.8 percent last year to $95.74 billion, official figures show.
“Over the long term, the shockwave that the dispute has sent across the Japanese business community will expedite a move to hedge the risk,” said Michal Meidan, a London-based analyst for global political risk consultancy Eurasia Group.
But, she went on, “I think Japanese investments in China are going to continue, probably going to pick up slightly, because that still is a major market for the Japanese.”
Jonathan Holslag, a research fellow at the Brussels Institute of Contemporary China Studies, agrees.
“The recent turmoil has added impetus to an agenda that already existed for diversifying commercial relationships,” Holslag told The Financialist.
On Jan. 16, Shinzo Abe embarked on his first overseas trip since being elected prime minister for the second time in December. Rather than make China his first stop, as he did in 2007 during his last premiership, he headed to Vietnam, Thailand and Indonesia instead.
“The growing dependence of Japan on China…is not a very desirable evolution so there is a lot of thinking going on to try to diversify away from China, to conclude commercial partnerships with the countries of Southeast Asia and with India and some places as far as the Middle East and Africa,” Holslag said.
While Abe hoped the trip would result in stronger relations with Southeast Asia’s other major economies, it does not mean Japan is turning its back on China — far from it.
Nevertheless, the dispute over the islands will continue to cause political and economic headaches for China and Japan, with neither acting to defuse the tensions.
Abe warned recently that there was “no room for negotiations” with China over the islands.
“My resolve to defend our waters and territories has not changed at all,” the hawkish Abe said, according to The Daily Yomiuri, shortly after announcing the first increase in Japanese defense spending in more than a decade.
The Chinese also have taken a hard line.
Last week, an editorial in the state-controlled Global Times warned its readers to “prepare for the worst” and said the Chinese military “shouldn’t be hesitant to take military revenge” in response to Japanese provocations.
A mixture of historical animosity, self-serving politics and energy security is fueling the dispute. As the US increases its strategic engagement in the Asia-Pacific region, China is eager to use the spat with Japan as an opportunity to show off its strength and boost its influence in the region.
But energy and the control of potentially large hydrocarbon reserves are at the core of the dispute which ensures lasting tensions between Asia’s economic giants.
“They will give you a long, historical explanation of their sovereignty claim. But the idea that there are vast resources under the East China Sea just off their coast is a tremendous motivation for the intensity of their territorial dispute,” Sheila Smith, a senior fellow at the Council on Foreign Relations in Washington, D.C, told National Geographic late last year.
While Sino-Japanese relations are likely to remain tense, the strong economic interests linking the two countries — and, in particular, Japan’s large investment in China — should ensure neither side overplays its hand.
It won’t be an “either-or issue,” Tony Nash, managing director at IHS Consulting in Asia, told Bloomberg earlier this month.
“Firms will stay in China, and they will invest in Southeast Asia and other places. It’s hard for Japanese exports to move totally away from China and it’s hard for Chinese OEMs (original equipment manufacturers) to move totally away from Japanese components,” said Nash.
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Not sure if this was asked before. I'm not a math genius/person like most of you here. I just have this question that i've always wondered about and though.. let me ask the math people how it works.
Anyways, the scenario is.
While travelling in a car doing 120km/h. There is a fly in the car flying in the air (in one spot just hovering). Is the fly flying at 120km/h ? or is it just hovering? Why is that? If the fly is not moving himself at 120km/h , why does it not smash into the back window with the forward momentum of the car?
Just a interesting one i've always wanted to know.
Thanks
Use Lagrange's method to find the maximum value of $<A$*x*, x*$>$ subject to condition $<$*x, x*$> =1$ and $<$*u_1, x**$>=0$ where **u_1 is a non zero vector in $N_{(s_1^2I_n-A^TA)}, s_1$ is the largest singular value of A and $A=A^T \in \mathbb{R}^{n*n})$.
Can anybody please explain me how to proceed with this question?
Studying some Newtonian mechanics, I've encountered this differential equation :
$y'+a y^2=b$
where $a,b$ are constants.
how could we solve it ? (I trying to get an algebraic solution)
I have to take derivative of the following expression (For example):
$$\frac{\partial \left( -x^T QBL y \right)}{\partial L}$$
where $x,y$ vectors and $Q,B,L$ are matrices with arbitrary (proper) dimensions (this is not necessarily to be a quadratic form - think the most general case). This is only one case I came accross, in general I want to become comfortable for this type of derivatives. Can somebody point a reference?
And also if one show how this derivative can be taken, I will be very happy. Thanks in advance!
Let $\pi:X\longrightarrow C$ be the minimal good resolution of the curve singularity (C,o) with exceptional set $E$, where $C:=\{x_1x_2(x_1^{a_1}+x_2^{a_2})=0\}\subset \mathbb C^2$. Let $\bar C_i$ be the strict transform of $\{x_i=0\}$ for $i=1,2$. Then $\bar C_1$ and $\bar C_2$ intersect distinct ends of $E$ if $E$ is not irreducible??
Confusion regarding stochastic integral.Let
$Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion.Now $dY_t=W_tdt$.So from this expression one can conclude that $dY_t \cdot dY_t=d[Y_t,Y_t]$ where $[\ ]$ is the qudratic variation prcess and it must be zero as $dt \cdot dt =0$.But people apply Fubini-Stochastic integral exchange trick and see $Y_t=\int_0^t(t-u)dW_u$.So by applying Ito isometry one can see $E(Y_t^2)=\int_0^t(t-u)^2du \neq 0 .$Can someone tell me where I'm going wrong ?Why this discrepancy ..??
Use Lagrange's method to find the point on the line of intersection of the two planes $a_1x+a_2y+a_3z+a_0 = 0, b_1x+b_2y+b_3z+b_0 = 0$ which is nearest to the origin. you may also assume that the tow planes really intersect, but you should explain where this enters into the calculation.
I tried to solve this and I reached up to this point:
assuming $g=a_1x+a_2y+a_3z+a_0 = 0, h=b_1x+b_2y+b_3z+b_0 = 0$ and using Lagrange's multiplier method $L(x,y,z,\lambda,\mu) = f -\lambda g -\mu h$ and proceeding by taking the $gradient L =0$; I got the plane $2x^2+2y^2+2z^2 =0$. But the question need a point on the line of intersection of the two planes. How can I find this point?
For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.
My attempt:
First, we can see that any such solution must satisfy:
$f(z)=f(a^kz)$ for all $k \in \mathbb{N} $.
If $|a|<1$:
The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\in \mathbb{N} $. Thus $f$ must be constant.
If $|a|=1$:
For all $a \neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant.
My quesions are:
Am I correct with my consclusions?
Also, I'm stuck in the case where $|a|>1$. Any ideas?
Thanks
Let $(\Omega, \cal{A}, \mathbb{P})$ be a probability space and $X$ a random variable on $\Omega$. Let, also, $f:\Omega\to\mathbb{R}$ be a Borel function. Then:
$X$ and $f(X)$ are independent $\Longleftrightarrow$ there exists some $t\in\mathbb{R}$ such that $\mathbb{P}[f(X)=t]=1$, that is $f(X)$ is a degenerate r.v.
The only thing that I could make out is that if $X$ and $f(X)$ are independent, then
$\mathbb{P}[f(X)\in B]=0$ or $1$ for every Borel subset of $\mathbb{R}$, since $\sigma(f(X))\subseteq \sigma(X)$ and hence, $f(X)$ is independent of its self. Suppose, now, that $\mathbb{P}[f(X)\leq x]=0$ for all $x\in\mathbb{R}$. Then:
$\mathbb{P}[f(X)\in\mathbb{R}]=\mathbb{P}[\bigcup_{n=0}^{\infty}[f(X)\leq n]]\leq\sum_{n=0}^{\infty}[f(X)\leq n]=0$ which obviously is a contradiction since $\mathbb{P}[f(x)\in\mathbb{R}]=1$.
However, I don't know ow to prove this and my attempt isn't likely to become a complete solution.
Any help would be appreciated.
Thanks in advance!
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let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,...,n^2\}$.
let $T$ be the set of all permutation of $\mathbf{A}$ which swap two adjacent entries. adjacent means one is above/below/right/left of the other (and both are neighbors).
Let $S$ be the permutation subgroup generated by $T$.
Show that $S$ acts intransitively on $\mathbf{A}$.
Every metric space is associated with a topological space in a canonical way. According to this source, this amounts to a full functor from the category of metric spaces with continuous maps to the category of topological spaces with continuous maps.
Is it possible that there exists another way of obtaining a topological space from a metric space that is equally deserving of the label "canonical"? Perhaps something that no one has thought of yet? To say it in a different way, is there a sense in which the aforementioned functor is unique?
Obviously this is a soft question, as I have neglected to specify what it means for a map to be deserving of the term "canonical." For this reason, let me motivate the question a little.
At some point in an introductory work on analysis, the author will define the meaning of the expression "the topological space associated with (or induced by) a metric space." I'd like to know if this definition is in some sense "the unique correct definition," or whether it is only "one of many possible."
How does one find the Fourier coefficients of the following functions $f:[-\pi,\pi]\rightarrow \mathbb{R}$
$f(t)=t$ for $-\pi < t < \pi$ and $f(-\pi)=f(\pi)=0$
$f(t)=-1$ for $-\pi < t < 0$, $f(t)=0$ for $t=-\pi,0,\pi$ and $f(t)=1$ for $0<t<\pi$ , $2\pi$ periodically extended to functions $\mathbb{R}\rightarrow \mathbb{R}$
What I have tried:
The coefficients for a function defined on $-\pi < x < \pi$ :$$a_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(kx)dx ; $$ $$b_k=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(kx)dx$$$$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x) dx $$
i think $$a_k+ib_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)e^{ikx}dx$$
for 1) : $\int x cos(kx) dx$= $$\frac{1}{k}sin(kx)x -\frac{1}{k} \int sin(kx)dx = \frac{1}{k}(xsin(kx)+\frac{cos(kx)}{k^2})$$ so: $$ a_k = \frac{1}{\pi}\int _{-\pi}^{\pi} xcos(kx)dx = \frac{(k\pi sin(k\pi))+ cos(k \pi) + k\pi sin(-k\pi)-cos(-k\pi)}{k^2}=0 $$
for $$\int x sin(kx)dx = \frac{-kxcos(kx)+sin(kx)}{k^2} $$ : $$\Rightarrow b_k= \frac{1}{\pi}\int_{-\pi}^{\pi} x sin(kx)dx = \frac{(-k \pi cos(k\pi)+sin(k \pi) - (k\pi cos(kx)+sin(-k\pi)}{\pi k^2}) = \frac{2 sin (\pi k)-2k\pi cos(kx)}{\pi k^2}$$
and $a_0= \int_{-\pi}^{\pi} x dx = 0$
is this correct for 1) ?
And, how does one find the Fourier coefficients of a piecewise given function? Take the coefficients for each boundary and sum them up e.g. : $$a_0 = \int_{-\pi}^{0}-1 dx + \int_{0}^{\pi}1 dx $$
and analogous for $a_k$, $b_k$ ?
Consider triples (p,q,r) of prime numbers p, q and r such that (p+1)(q+1)=(r+1). here are some exmaples : (2,3,11), (3,7,31). how to prove these triples are infinitely?! I define two integer numbers n and m or (n,m) to be Isomorph iff F(n)=F(m). F(n) is sum of divisor of n. for example (6,11) , (10,17) , (14, 23), (21, 31).If for prime numbers p,q,r have :(p+1)(q+1)=(r+1) so (pq,r) are isomorph and so it maybe a conjecture: there are infinitely pairs of isomorph numbers!
I want to prove that $Th(\mathbb{Q},+,0)$ has quantifier elimination. Please point out where i go wrong or how to finish my reasoning:
We can bring the formula in disjunctive normal form and distribute the existential quantifier. So we are left with a conjunction of literals. Now the basic formula's can be brought to 'additive normal form' (I don't know if this is usual terminology), so $m_0 x_0 + ... + m_{n-1} x_{n-1} + m x = 0$ or an inequality, where $x$ is the variable from the existential quantifier.
But then?
Is it correct that a Borel probability measure $\sigma$ on the complex unit circle $\mathbb{T}$ is symmetric (i.e. $\sigma(A)=\sigma(\overline{A})$ for every Borel set) iff $\hat{\sigma}(n)=\hat{\sigma}(-n)$ for every $n\in\mathbb{Z}$?
Some closure on the story of how Yandex — the Russian search giant — built a social discovery app that relied on Facebook interconnection to gather data, and then found Facebook blocking its service within hours of launching. Today Yandex said that after discussions with the social network, Facebook has finally, terminally said that the app violates its Platform Policies, and so it will not reinstate the ability to use Facebook’s Graph API. As a result, Yandex is planning to pulll the app from the app store and put it on hold for now.
The full statement from Yandex:
“We discussed the issue with Facebook and it was confirmed that Facebook views the application Wonder as something that violates the Facebook Platform Policies (section I.12) and that the access to Facebook’s Graph API will not be restored.
“According to Section I.12, no data obtained from Facebook can be used in any search engine or directory without the company’s written permission. The reason behind Facebook’s decision to revoke our access to their data appears to be that they do consider Wonder to be a search engine, while our understanding of what it is differs from this view.
“Wonder’s functioning, in its current state, as well as the quality of user experience it provides, largely depends on the access to Facebook’s Graph API. Since this access was revoked, we decided to put our application on hold for the time being. We will be considering partnership opportunities with other social networks and services to offer our users a richer internet experience via Wonder.”
The emergence of Wonder, and a week before the blockage of Voxer, has kicked off a new level of scrutiny about how Facebook allows other apps to appropriate its data: in effect, the company has said that it is not sharing with apps that don’t share back.
More to come.
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