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TITLE: Sum and product of linear recurrences
QUESTION [6 upvotes]: Given
$a_n = \alpha_1 a_{n-1} + \cdots + \alpha_k a_{n-k}$
and
$b_n = \beta_1 b_{n-1} + \cdots + \beta_l b_{n-l}$ are linear recurrences with complex coefficients, how can I find linear recurrences for $a_n + b_n$ and $a_n b_n$?
REPLY [8 votes]: It will be convenient to have all of the terms on the same side of the equation, so I will write your recurrence relation as
$$
\tag{$\star$}
a_n+\alpha_1a_{n-1}+\ldots+\alpha_k \,a_{n-k}=0,
$$
and similarly for $b_n$.
The characteristic polynomial of the recurrence $(\star)$ is $T^k+\alpha_1 T^{k-1}+\ldots+\alpha_k$. Every solution to $(\star)$ is a linear combination of sequences $n\mapsto r^n n^k$, where $r$ is a root of the characteristic polynomial and $k$ is a non-negative integer that is smaller than the multiplicity of $r$. Let $f_a(T)$ and $f_b(T)$ be the characteristic polynomials of your two recurrences.
The sequence $c_n:=a_n+b_n$ is a linear combination of sequences of the form $r^n n^k$, where $r$ is a root of either $f_a(T)$ or $f_b(T)$. The roots of the product $f_a(T)f_b(T)$ are the roots of $f_a(T)$ and the roots of $f_b(T)$, so the sequence $a_n+b_n$ satisfies the recurrence whose characteristic polynomial is $f_a(T)f_b(T)$. Explicitly:
$$
c_n+ (\alpha_1+\beta_1)c_{n-1} +(\alpha_2+\alpha_1\beta_1+\beta_2)c_{n-2}+\ldots+(\alpha_k\beta_\ell)c_{n-k-\ell}=0.
$$
Let $d_n:=a_nb_n$. Then $d_n$ is a linear combination of terms $(r_1r_2)^n n^{k_1+k_2}$, where $r_1$ (resp. $r_2$) is a root of $f_a$ (resp. $f_b$) of multiplicity greater than $k_1$ (resp. $k_2$). So we need to find a polynomial whose roots are products of a root of $f_a$ times a root of $f_b$. One way to come up with such a polynomial is using the resultant.
The resultant $\mathrm{Res}_x(f(x),g(x))$ of two polynomials $f(x)$ and $g(x)$ is a quantity that depends polynomially on the coefficients of $f$ and $g$, which is $0$ if and only if $f$ and $g$ share a common root. The resultant can be computed as the determinant of the Sylvester matrix.
For $T$ a complex number, the polynomials $f_a(x)$ and $x^\ell f_b(T/x)$ share a common root if and only if $T$ is the product of a root of $f_a$ and a root of $f_b$. So $d_n$ satisfies the recurrence whose characteristic polynomial is $\mathrm{Res}_x(f_a(x),x^\ell f_b(T/x))$. Explicitly, this polynomial is
$$
\mathrm{det}\left[\begin{array}{cccc}
1&\alpha_1&\alpha_2&\cdots&\alpha_k&0&0&\cdots&0\\
0&1&\alpha_1&\cdots&\alpha_{k-1}&\alpha_k&0&\cdots&0\\
\vdots&&\ddots&&&&&&\vdots\\
0&\cdots&0&1&\alpha_1&&&\cdots&\alpha_k\\
\beta_\ell&\beta_{\ell-1}T&\cdots&\beta_1 T^{\ell-1}&T^\ell&0&&\cdots&0\\
0&\beta_\ell&\cdots&\beta_2 T^{\ell-1}&\beta_1 T^{\ell-1}&T^\ell&0&\cdots&0\\
\vdots&&\ddots&&&&&&\vdots\\
0&\cdots&0&\beta_\ell&\beta_{\ell-1}T&&&\cdots&T^{\ell}
\end{array}\right].
$$
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In the southeast Valley Zack Shepherd’s name may not be the first one mentioned surrounded by neighboring 2019 Division I quarterbacks Spencer Brasch (Higley), Gunner Cruz (Casteel) and Devin Larsen (Queen Creek). But make no mistake, the Williams Field three-year starters prep resume’ simply stands alone among the quartet.
All Shepherd has done since becoming the Black Hawks’ starter before his sophomore season is guide WF to their first state championship in program history, win 24 of his 28 starts under center and throw 40 touchdowns in his first two seasons competing against some of the state’s best defenses.
Bomb Squad❄️
— Zack Shepherd (@zshep12) August 1, 2018
However, it’s what he’s done off the field from a leadership standpoint which pleases his head coach the most.
“He’s been a transformation,” Steve Campbell said to Sports360AZ.com on Friday. “[He and I] talked every year about that. When he was going to be a sophomore and going to be the quarterback…some of the older guys took him under their wing. This year he’s really took it upon himself to say, ‘these guys are looking to me as a three-year starter.'”
A student of the game, Shepherd has mastered the Black Hawks’ pro-style offense
So far, it’s worked. Despite a Week Zero loss in southern California to perennial power Norco High the Black Hawks (2-1) figure to be right back in the 5A playoff mix in early November thanks to the sometimes overlooked, but rarely outplayed, 6-foot-1, 185-pound surgeon between the white lines on Friday nights.
After a few amazing conversations with @Coach_Sets I’m excited to say I’ve received an Ivy League offer to further my education and play football at Brown University! Can’t wait to get to Providence and see the campus! @BrownUFootball pic.twitter.com/QSVGyxz1Is
— Zack Shepherd (@zshep12) May 9, 2018
“He can make every throw that you measure a quarterback [by],” Campbell said. “His knowledge of the game and what he brings on the field is something probably most high school quarterbacks don’t do.”
Just keep being you Zack Shepherd.
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Toxic subcultures are thriving in the medical profession, often putting patients at risk. Everyone who works in the NHS has met these people, but dealing with the problem is a huge management challenge.
A senior manager recently described to me the struggle in his trust to stop surgeons bullying anaesthetists and admin staff. He got to the point of having to spell out that he would personally walk someone off the site, no matter how senior, if it happened again.
This experience is borne out by General Medical Council research which laid bare five distinct problematic groups of healthcare professionals. Ironically, the study aimed to understand how doctors approach the task of building good workplace cultures that deliver high-quality care.
It reveals the signs of poor culture, such as cynical staff, blaming and shaming, a defensive attitude to performance data and a lack of mutual support. Other flags include a focus on the technical side of medicine while ignoring patients’ experiences, professional battles taking precedence over patient needs and lax implementation of protocols such as surgical checklists. Stress and burnout are also often endemic in teams with a poor working culture.
Five toxic subcultures emerged: divas, factional, patronage, embattled and insular. Divas are powerful, successful professionals who think the normal rules of behaviour only apply to others. They are typically ill-tempered bullies who disrespect managers and colleagues, and ignore protocols.
Teams dominated by patronage have high-status, well-connected specialists who exert huge power over their colleagues. They will often be doing exceptional work, but the downside is that the leaders are difficult to challenge and manage, and will happily compromise other parts of the organisation in their lust for glory.
In factional groups – let’s not call them teams – disagreement becomes endemic and people start to organise themselves around perpetuating conflict. Faction leaders try to drag everyone else in. Patients suffer.
Just how bad this can get has been exposed by the culture of tribalism and mistrust that was allowed to fester for years in the cardiac unit at London’s St George’s hospital. Complex heart surgery was suspended amid concerns that the conflicts were causing avoidable deaths.
Embattled teams feel overwhelmed by all the unmet need they see in their patients. Staff are stressed, anxious and burned out. Believing their cause is hopeless they reject offers of help because they don’t see how anything could make a difference. Some GP practices – flooded with patients and short-staffed – slip into this siege mentality.
Insular teams have become isolated from the professional mainstream, and have often lost sight of what a good service looks like. Old-style, single-handed GP practices easily fall prey to this trap. Insularity is a particular problem for secure mental health units; isolated behind barriers, they focus on incarceration rather than care and treatment.
Hospital managers know that turning around a rotten clinical culture often takes years and can feel quite brutal. Confronting inappropriate behaviour by powerful staff eats time and saps energy. Sometimes the cause of the problem simply needs to be forced out of the organisation.
But managers need to check their own behaviour before judging that of others. Rotten cultures take root when senior leaders lack the skill or courage to act quickly and decisively when they see the warning signs. Bullying by managers is guaranteed to bring out the worst in people. Many senior doctors have justifiable complaints about their treatment at the hands of local executives or staff from NHS Improvement and NHS England.
Demonstrating behaviour they want to see in others, putting effort into building relationships and supporting colleagues, understanding problems from the perspective of frontline staff, showing up when things are tough and responding positively to difficult news are just some of the simple leadership actions that make a profound difference.
Medical leadership itself requires massive investment. Too many doctors thrust into the top roles still find they are largely left to make it up as they go along because training and support is inadequate.
Divas and bullies are the symptoms of a rotten culture, but not always the cause.
• Richard Vize is a public policy commentator and analyst
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Smoked Peach Hot Sauce. Macaroni and Pimento Salad. Belle of Dayton Red Pepper Mayo. All of this and more can be found atop a Bad Dog Nice Taco hot dog or taco, which now can be delivered right to you for lunch!
“We’ve been getting a great response at night and on the weekends out at the cart,” Brian Johnson, Co-owner of Bad Dog Nice Taco, said. “We are really excited to be offering an expanded menu during lunch delivery that will include bacon tots, Nan sandwiches, and more!”
The expanded menu ranges from $3-$9. “The focus is to offer something unique that’s enjoyed by everyone from foodies to people just expecting a dog,” Lee Anne House, Co-owner and operator, said.
For those interested, lunch delivery will run from 11am – 2pm on Monday through Friday throughout the Greater Dayton downtown area. Preordered catering is available region wide. More information on the Bad Dog Nice Taco Facebook page.
Bad Dog Nice Taco Delivery Menu
At Bad Dog Nice Taco we love toppings, sauces, and condiments! On any of the dogs, tacos, tots or mac and cheese choose any three toppings or if you just want to choose one of our signature items, that’s okay too!
The Signature, Chef Made Toppings:
Roasted Jalapeno Crema, Belle Of Dayton , Red Pepper Mayo, Bbq Pumpkin Aioli, Spicy & Creamy Mustard, Pickle Relish, Sweet Red Slaw, Chimichurri, Veggie Salsa, Smoked Peach Hot Sauce, Roasted Jalapeno Salsa Verde, Green Sriracha, Whipped Feta, Macaroni And Pimento Salad, Toxic Beer Cheese
Add meat (chorizo or pulled chicken) to anything for $3
Add an over easy egg and bacon for $3 or just one or the other for $2
Add cheese (crumbled feta or greek yogurt cheese) for $2
Signature Bad Dogs $6
(Made with grass fed beef from keener farms, comes with chips)
1. Toxic Dog with toxic mustard, toxic beer cheese and bacon
2. Spice addict with pumpkin bbq aioli, carrot mango habanero sauce and crumbled feta
3. Veg head with whipped feta cheese, smoked peach hot sauce and chimichurri.
Customize your dog with any three chef made toppings – $6
Signature Nice Tacos $6 for 2
1. Chorizo with green sriracha, whipped feta and sweet red slaw
2. Chicken with Belle of Dayton mayo, smoked peach hot sauce and veggie salsa
3. Veggie with roasted jalapeno crema, chimichurri and pimento macaroni salad.
Customize your tacos with any three chef made toppings – $6 for 2
Bad Nacho: $3
Kettle Potato Chips covered in your choice of three toppings
Signature Crazy tater tots: $8
Bacon! Tots are tossed in bacon drippings, topped with more bacon and scallions then drizzled with Toxic Brewery creamy spiced mustard and Belle of Dayton mayo. $8
Customize your tots with any three chef made toppings – $6
Magic Mac – $6
We make macaroni and cheese the right way! We start with a bacon roux made with local whole milk from Stardancer Creamery. Sharp cheddar cheese and Parmesan finish it up right. Straight up mac for $6,
Customize your mac with any three chef made toppings – $7
House specialty mac and cheese (chorizo, chimichurri, feta and scallions) – $9.
Sides: $3
Dijon feta potato salad
Sweet red slaw
Pimento cheese macaroni salad
Sandwiches:
(Comes with chips)
1. Chicken and bacon croissant. Ashley’s bakery croissant with a marinated pan seared chicken breast, chopped thick cut bacon, white cheddar cheese, greens and Toxic brewery creamy spiced mustard. $8
2. Slow cooked pork naan. Pulled pork shoulder on soft naan bread, with pumpkin bbq aioli, greek yogurt cheese and zucchini pickles. This is a staff favorite! $8
3. Veggie grilled cheese. Pan seared bread with olive oil. Loaded up with squash, zucchini and thin sliced kale with melted white cheddar cheese and finished with Belle of Dayton mayo and cilantro basil chimichurri. $6
Salads:
1. Kale Caesar. Thin sliced crunchy kale with a lemon Caesar dressing and parm. $7
2. Mixed green salad with veggie salsa and sunflower seeds. Choice of dressing on the side: herbed creamy buttermilk, roasted jalapeno crema, sweet white wine vinaigrette and caesar. $6
add chicken($3), bacon($2) or cheese($2)
Soup of the day: $5
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imprint
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MADISON - Were they going to get off the field? Were they going to give their offense a chance to get back on the field?
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Hey everyone!
This week’s Monday Movie is about how you can use the LookAt constraint and the Path constraint to make objects follow each other in a “piston” like fashion. This is useful for mechanical rigs like hydraulics or characters. You can use this technique, and others like it, to create 3d characters that have their subtle animations (like moving parts) delegated away to constraints so that you can focus on the more important keyframes like positions!
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tanks
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Description
Improve your snare sound without having to spend hundreds on a new drum.
If you’ve got a cheaper (or older) snare drum, there are a few things you can do to tighten up the sound without needing to spend too much. Besides getting a new drumhead or adding dampening, see below for some of the options available to modify and improve your sound.
Snare wire upgrades
Cheap snare wires can rattle and buzz excessively, especially if they’re made with low-grade material or are not straight. Old snare wires can be loose, dull, or broken. Changing your snare wires will help control snare buzz, and can also improve the sound. There are a lot of different options – the number of wires, wire thickness, wire placement, and wire material will all play a role in your sound. 20-strand snare wires are most common. Less strands will mean less snare “snap” and less snare buzz – think about the amount of snare wire snap you want in relation to the amount of drum sound you want to hear. Thinner and lighter wires will be more sensitive (and also quieter) – if you’re playing very loud music you may want to consider heavier or thicker wires. Lighter and softer music calls for thinner and more sensitive snare wires. Snare wires are also available with the middle strands missing – this gives a dryer sound and will also reduce the amount of resonant snare buzz when other drums are hit – try these if you want to reduce sympathetic snare buzz.
Snare throwoff upgrades
A snare throwoff (also sometimes called a strainer) holds the snare wires in places and gives you control over their tension. Upgrading a cheap throwoff will give you much more control over snare wire tension, and allow for smoother and more even transitions. Tight tension gives a focused quick snap (and will also choke the resonance of the resonant drumhead), while medium and lower tensions allow for a nice fat or deeper punch with more buzz. Snare drums usually have one or two sweet spots on the throwoff where the snares give a thicker, fatter response. Having a good quality throwoff will allow you to achieve the sound you want. Most drum throwoff upgrades are designed to fit almost all snare drums (double-check the new throwoff’s screw placement matches your old one).
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If your tuning rods get stuck, are hard to turn, or are rusty, you’ll have a harder time getting consistent and even tuning. Before buying new parts, try using a lubricant to solve the problem. Use spray on grease, or a small amount of oil or vaseline to coat each tuning rod. If the problem persists (or if you want a cosmetic upgrade), you can buy new lugs and tuning rods. You’ll usually find the exact measurements and screw placements of lugs before buying them (check, for example, on the Amazon product page) – compare these to your own drum to make sure they will fit. Generally, the less contact each lug makes with the drum shell, the more the shell will be able to resonate. Smaller and lighter lugs will allow for the most resonance, while thick and heavy lugs will reduce the drum shell’s natural vibrations (choking the sound a little). Another quick and cheap upgrade is nylon or plastic tension rod sleeves (or washers) – these will remove metal-on-metal contact, make tuning smoother, and will stop your tuning rods from slowly coming loose when you play.
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TITLE: Compute $ \sum\limits_{k=0}^{m}(-1)^k {n \choose k} {n \choose m-k}$ and $\sum\limits_{A,B\subseteq X} |A\cup B|$ for some given finite set $X$
QUESTION [6 upvotes]: I've got a problem with deducing closed-form expressions for sums:
$1) \ \sum_{k=0}^{m}(-1)^k {n \choose k} {n \choose m-k}$
$2) \ \sum_{A,B\subseteq X} |A\cup B|$ where $|X|=n$
Can anyone help me?
In 1) I have no idea. I can use identity $\sum_{i=0}^{k} {n \choose i} {m \choose k-i}= {n+m \choose k}$ but I don't know if it will be useful.
In 2) I was thinking about this way: $\sum_{A,B\subseteq X} |A\cup B|=\sum_{i=0}^{n} {n \choose i}\cdot i \cdot 2^i$. Explanation: firstly I choose subset with $i$ elements. I can do this in ${ n\choose i} $ ways. It's cardinality equals $i$ and secondly I choose $2^i$ subsets of choosen subset. But it gives a wrong answer even for $n=1$.. Moreover it's still not closed-form expression..
REPLY [1 votes]: An alternative view of the first problem:
$\binom{n}{k} \binom{n}{m-k}$ counts the number of ways to put a total of $k$ red marks and $m-k$ blue marks on $n$ squares, where a square can get marked more than once. Consider the operation $\phi$ which takes the first square which has exactly one mark and switches that marked color. This operation is an involution (applying it twice returns you to the same marking) and changes the parity of $k$. So each coloring $C$ where this is defined cancels out with $\phi(C)$. The only non-cancelling colorings are those where $\phi$ does nothing. This can only happen if $m$ is even, $k=m/2$, and there are $m/2$ squares marked twice.
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TITLE: $f(f(x))=f(x),$ for all $x\in\Bbb R$ suppose $f$ is differentiable, show $f$ is constant or $f(x)=x$
QUESTION [4 upvotes]: $f(f(x))=f(x),$ for all $x\in\Bbb R$ suppose $f$ is differentiable, show $f$ is constant or $f(x)=x$
Clearly, $f'(f(x))f'(x)=f'(x)$. This implies for each $x$, $f'(f(x))=1$, or $f'(x)=0$. But this is not enough.
REPLY [0 votes]: I'm not able to follow the argument given in the existing answer. It may be perfectly valid, but I don't understand what's being said at several points. So let me give my own answer.
Suppose $f : \mathbb{R} \to \mathbb{R}$ is a differentiable function such that
$f(f(x)) = f(x).$
Differentiating, we have
$$f'(f(x))f'(x) = f'(x)$$
which we can rewrite as
$$f'(x)[f'(f(x)) - 1] = 0$$
So for each $x \in \mathbb{R}$ we have either $f'(x) = 0$ or $f'(f(x)) = 1$.
Since $f$ is differentiable, it is continuous. Since $\mathbb{R}$ is nonempty and connected, so is its continuous image $Y := f(\mathbb{R})$.
Let $y \in Y$, say $y = f(x)$. We must have either $f'(y) = 0$ or $f'(f(y)) = 1$. But since $y = f(x)$,
$$f(y) = f(f(x)) = f(x) = y.$$
So in all we have either $f'(y) = 0$ or $f'(y) = 1$ for each $y \in Y$.
Derivatives of differentiable functions need not be continuous, but they at least satisfy the intermediate value property. If $f'(y_1) = 0$ and $f'(y_2) = 1$ for some points $y_1, y_2 \in Y$, it would necessarily be the case that there was some point $y_3$ in between them with $f'(y_3) = 1/2$. Since $Y$ is connected, though, this point $y_3$ would necessarily be in $Y$ as well, contradicting the fact that $f'$ only takes on the values 0 and 1 on $Y$.
It follows that either $f'(y) = 0$ for all $y \in Y$ for $f'(y) = 1$ for all $y \in Y$.
In the former case, we have $f'(f(x)) = 0$ for all $x$, so in particular $f'(f(x)) \neq 1$ for all $x$, so we must have $f'(x) = 0$ for all $x$. But this means that $f$ is constant.
In the latter case, we have $f'(y) = 1$ for all $y \in Y$. First, since $f'$ is nonzero at a point, $f$ is nonconstant, so $Y$ must be an interval of positive length. Second, since $f'(y) = 1$ on this interval, there is some constant $C$ for which $f(y) = y + C$ for all $y \in Y$. That is, for all $x$, $f(f(x)) = f(x) + C$. But $f(f(x)) = f(x)$, so this says that $f(x) = f(x) + C$, i.e. $C = 0$. In other words, $f(y) = y$ for all $y \in Y$.
I want to claim in this case that $Y$ is necessarily all of $\mathbb{R}$. Suppose not. Write $a = \inf Y$ and $b = \sup Y$. Since $Y$ is an interval of positive length, we have in particular $a < b$. By assumption, one or both of these must be finite. Assume without loss of generality that $a$ is not $-\infty$.
Since $f(y) = y$ at least on the interval $(a, b)$ and $f$ is continuous, we must have $f(a) = a$. For $x < a$ we have $f(x) \geq a$ since $a$ is the infimum of the range of $f$. For $a \leq x < b$ we have $f(x) = x$.
But $f$ is supposed to be differentiable, so let's consider its derivative,
$$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
The limit from the right is
$$\lim_{h \to 0^+} \frac{f(x + h) - f(x)}{h} = 1$$
On the other hand, if we consider the limit from the left,
$$\lim_{h \to 0^-} \frac{f(x + h) - f(x)}{h}$$
the numerator is nonnegative (since $f(x+h) \geq a$) while the denominator is a negative number. So the limit, should it exist, could at most be zero, and is definitely not equal to one.
This is a contradiction, so it must be the case that $a = -\infty$. Similarly, $b = +\infty$. So the $f(y) = y$ for all $y \in Y$, but $Y$ is all of $\mathbb{R}$ so $f$ is just the identity function.
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You can always tell the masters from those just trying to be a master. There are several in music that come to my mind, Sandi Patti, George Strait, Rosemary Clooney....all masters and the list could go on. It is amazing how a duet between one of the masters and one of those trying to be a master reveals the truth of pure talent. There is a scene in the movie of "Hello, Dolly!" that has a duet between Barbara Streisand and Louis Armstrong. This is a perfect scene of two masters. The entire movie is worth seeing just to experience the two minute duet between the two of them.
I want to be known as a person that serves a song, a story, and a life. I don't want to try to be a master, but to be a master. The difference is in the heart. A person trying to be a master is using talent to drive ambition, yet the master allows the ambition to rest and lets the talent shine forth. The talent is the God given gift that reflects the creative spirit of a generous God. Even if the person is not willing to recognize that God.
A Christ-follower allows the "mastery" of the God given gift to be guided, nurtured, and molded by the Holy Spirit. That is why we can not only be servants of the song, but of our story, and our life. We serve in humility so that He can make us into a great song, great story, and great life. It is the submission that is so difficult. 1 Corinthians 12:4 states, "Now there are varieties of gifts, but the same Spirit." Thus, we make it our goal and we strive for His direction. He is willing and able to guide us through His Spirit. It is also so very worth it.
"Lord, peal all away that which keeps me from being a master of the gifts that you have given me. Help me to relax and serve the gift with truisms of life and not the ambitious things I make it. You are my God and I bow to your direction in all that I am. You are also my master, so train me as you see fit. I ask for this through the power and wisdom of Jesus. Amen."
Serving together, you know I love ya - Don
3 comments:
Wow! These are some great thoughts, and I appreciate the shared wisdom. From where I see things, you are a master in a couple of areas: meeting people where they are and taking them to Jesus (like Andrew), total integrity (being exactly who you say you are). Your musical and leadership talents are always used in service for God and His kingdom. Praise the Lord! (And thanks for your friendship!)
I needed to hear that today. God's is so good thank you for being so supportive. Faith, you are another person I have seen that serves the song...you know I love ya, Don
I am not sure if this is relevent, however, when I became ill and lost everything in my eyes (ability to drive and work) What do I do now God?
I prayed and one day at Church "people whom are ill and can not leave the house pray" the pastor said.
I became a prayer warrior as they would call me.God bless you and I enjoy reading your blogs:)
| 67,989
|
Meredith Brooks
Professional Education
- Board Certification: Pediatric Anesthesia, American Board of Anesthesiology (2013)
- Residency: Brigham and Women's Hospital Harvard Medical School MA (12/31/2007)
- Fellowship: Lucile Packard Children's Hospital CA (12/31/2009)
- Fellowship: Stanford Pain Management CA (12/31/2010)
- Fellowship: Lucile Packard Children's Hospital CA (12/31/2008)
- Board Certification: Pain Medicine, American Board of Anesthesiology (2011)
- Board Certification: Anesthesia, American Board of Anesthesiology (2010)
- Medical Education: Johns Hopkins University School of Medicine MD (2003)
- Internship: Alameda County Medical Center CA (06/24/2004)
| 32,000
|
TITLE: Finitely many ideals of given norm in rings of integers
QUESTION [4 upvotes]: I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$.
I know there are "standard proofs" (e.g. How many elements in a number field of a given norm?), but I'm just wondering would a induction proof on the norm be valid:
(i) True for $n=1$.
(ii) For $n=k+1$, if $n$ is prime then it's either rammified, inert or split in $K$ which would mean there are 1, 0 or 2 ideals of norm $n$ respectively. If $n$ is composite then it can be factored into prime ideals of smaller norms (since $\mathcal{O}_k$ is a Dedekind domain), where there are only finite ideals of smaller norm, hence it follows by induction.
Thanks.
REPLY [2 votes]: Your argument is more or less correct. Be careful, however - there could be a lot more than two prime ideals above a given prime (but there is a finite number). Also, induction is really awkward and not necessary here.
A much cleaner proof is just to use unique factorization directly: an arbitrary ideal of $\mathcal O_K$ can be written as $\wp_1^{r_1}\cdots\wp_t^{r_t}$, and the norm of this is $p_1^{r_1f_1}\cdots p_t^{r_1f_t}$. If you require that this equals $n$ for some $n$, then there are finitely many possibilities for $t$, for the $\wp$'s, and for the $r$'s.
| 118,210
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WASHINGTON — President Trump is about to score a big win, but not the kind he promised.
The luxury hotel builder who campaigned as a champion of the “forgotten” middle class is poised this week to sign his first major bill — a tax overhaul that could deliver its biggest benefits to the wealthy and big corporations.
The middle class, too, is expected to receive a tax cut under the plan that House and Senate Republicans are slated to give final approval Wednesday morning — but unlike the corporate provisions, their breaks are temporary and add up to much less than those millionaires would enjoy.
Yet Trump is eager for a win — and so are congressional Republicans, who are facing a potential electoral buzzsaw in the suburbs next year, and are desperate to rally their base and donors with a signature achievement.
So GOP lawmakers, including those from the battleground Philadelphia suburbs, have hitched their fortunes to the tax measure, hoping that its reality turns out more rosy than most analysts and voters expect.
Every House Republican from outside Philadelphia, except for Rep. Frank LoBiondo, a New Jerseyan who is retiring, supports the measure, and Sen. Pat Toomey (R., Pa.), has been one of its most vocal advocates.
Polls, however, say that the public doesn’t believe the bill will help working families.
“They’re starting out with the most negative reaction to any tax bill in the past couple of generations,” said Patrick Murray, director of the Monmouth University Polling Institute, which found that 47 percent of the public opposes the bill while 26 percent supports it.
Two tax increases have even polled better. Said Murray, “That’s a big hole to dig out of.”
Similarly, an NBC/Wall Street Journal poll released Tuesday found that 63 percent of Americans think the tax plan will mostly help corporations and the wealthy, against 22 percent who believe it was designed to help everyone.
Republicans argue that the bill has been victimized by exaggerated liberal critiques and a compliant media, and that people will warm up to the measure once they start seeing more money in their paychecks next year (though most won’t see the full effects until they fill out their taxes in 2019, after next year’s midterm elections).
“If you look at the actual facts that are in the bill, there’s going to be a lot of economic growth, wages are going to be lifted, I think the stock market has already responded,” said Rep. Ryan Costello (R., Pa.), one of the region’s most vulnerable incumbents. Over time, “people will see the positive benefit of it.”
A family of four with $80,000 income, he said, could see a $2,000 tax cut. That could be the difference between taking a family vacation or not, buying a new car, extra Christmas presents, or more nights out to eat, he said.
Republicans also argue that future Congresses will extend individual tax breaks, much as lawmakers did for nearly all of the tax cuts passed under President George W. Bush. That would add benefits to the middle class that aren’t being counted now, but also spike the bill’s price tag.
“I know people are going to see dramatic improvements that are going to change their view of this,” said Rep. Tom MacArthur (R., N.J.).
He is the only lawmaker from New Jersey backing the bill. Every other one criticized a new $10,000 cap on the deduction for state and local taxes — a provision that could also hit some homeowners in Philadelphia’s western suburbs who pay high property taxes and the city’s wage tax.
Voters are more pessimistic than even the most critical tax analysts.
For example, while the nonpartisan Tax Policy Center projects that 80 percent of taxpayers will get a cut next year, the Monmouth poll found that 55 percent of those making less than $50,000 expect to pay more.
The dour expectation likely reflects a perception among middle-class families that the tax bill is not for them.
“This is a tax cut in the short run for the middle class, but it’s not nearly as big of a tax cut as it is for corporations, small business owners or the very wealthy,” said J. Richard Harvey, a former senior accountant in the Reagan administration’s Treasury Department who now teaches at Villanova’s Charles Widger School of Law.
Added Joel Slemrod, who worked as an analyst for President Ronald Reagan’s Council of Economic Advisers and now teaches economics at the University of Michigan, “Any tax cut aimed at business income, as opposed to employee income, is going to tend to be skewed toward high-income people.”
To be sure, there is help for the middle class, at least initially. The Tax Policy Center predicted that people making between $49,000 and $86,000 a year would get a tax cut averaging $930 next year. But by 2027 they’d pay a little more, on average, than they do now. Several other analyses, including from the Penn Wharton Budget Model, projected broadly similar results.
A key question is if that $930 is enough to make them fans of the bill. Former President Barack Obama enacted a payroll tax cut worth a similar amount — but few people noticed, Murray said.
Meanwhile, people in the top 20 percent would get a $7,460 tax cut next year, the Tax Policy Center found.
Democrats also blasted the bill for including a repeal of the Affordable Care Act mandate that everyone carry health insurance — a move that Republicans say will free people to make their own health decisions, but that nonpartisan congressional analysts say will raise insurance prices and lead to 13 million fewer people having health care by 2027.
Still, the measure will give Trump support for his boasts of monumental achievement in his first year. And his fellow Republicans have long supported tax cuts — and now see the bill as a way to show they have done something for their supporters.
“If you go into the election year and you don’t have any big legislative wins,” said Jim Gerlach, a former Republican congressman from Chester County, “then the body of the electorate is going to say, ‘We put you in charge and you didn’t get stuff done, so why should we keep you in charge?’”
| 148,061
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In-what-way-do-socioeconomic-factors-effect-the-attitude-of-women-towards-their-dressing-style-in-India
Answer by Balaji Viswanathan:
Continuing the discussion from here: Why are North Indian city girls more stylish in comparison to South Indian girls?
- Where they live: In big cities, women tend to be freer when it comes to dress options. Unlike rural areas, where everyone knows everyone else, in cities we become "invisible" entities. This gives women extra boldness in trying out new things without worrying about what the neighbor things [in many instances they might know the neighbor at all]. This is true of all parts of the world. Rural areas forces one to be more conservative as the community influences a big part of you.
- Whether they are immigrants: In India's case, migrants into the cities often dress more freely. For instance, a girl from Manipur or Haryana staying in Mumbai might find more freedom to dress modern more than a local Marathi woman, while at her hometown she might have to worry about mother-in-law, parents relatives, etc.
- The diversity of the cities: In diverse cities, people come across more types of dresses worn by others. There might be multiple business travelers from abroad who might influence the dress choices of the locals. The people around you make you & if you live in a very diverse environment, your dressing might tend to be quite creative.
- Wealth & options: Even if a girl in Ranchi has the freedom and wants to dress modern, maybe there are not enough shopping options to buy the skirts & trousers she wants. There might be just 1 shopping mall in the whole city & that might have Gucci or Prada. Or she might not be wealthy enough to buy some of the dresses. Thus, she goes to the local tailor who gives her the standard salwar set.
- Cultural pride: In India's case, the surrounding culture has a big impact on what you wear. Even in metropolitan Europe it is the same. For instance, you won't go to a Opera in Amsterdam wearing a Jean/T-Shirt. One must buy a Dress/Suit/Tuxedo for wearing to an opera or a ball or a classical concert. In the same way, you don't wear a short skirt if you are going to a Chennai temple. How hard these cultural rules are influenced depends on how proud people are about their culture. The French & Italians are very proud of their culture and have a certain way of dressing to special occassions. In a few other places, people might not care. In many Indian towns where the heritage still retains, the rules get enforced more.
In what way do socioeconomic factors effect the attitude of women towards their dressing style in India?
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| 340,476
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OK, that was bad. But, was it bad enough to generate these comments from Utah Republican Party Chairman Dave Hansen (via today's Political Cornflakes)?
"Helen Thomas has clearly lost any semblence of rational thinking and has let her bigoted views of Israel and Jews take over. Yes Helen, its time for you to retire. You might be more comfortable living out your remaining years in a white supremacist enclave in northern Idaho."
Maybe Ms Thomas can take a few of Mr Hansen's party members with her. Namely Senator Buttars, Representatives Wimmer and Sanstrom, and Mike Lee and Tim Bridgewater, too. Their comments have been just as bad, if not worse, than those of Ms Thomas.
I'm sure they'll all love it up there with those folks.
-Bob
(P.S. - I know that Senator Buttars is in the hospital with health issues. I hope he gets better soon.)
7 comments:
I don't know Helen Thomas.
I have met the others you have mentioned and do not believe they deserve that.
Immigration
We need to remove and not create any rewards and incentives for immigrants to come here and be here illegally.
If you think that is racist, you might as well add 85% of the population of the US to that list.
Bob,
You're so completely "out there," you might as well be a satellite.
To even remotely compare the list of people you have to Helen Thomas' genocidal wishes for an entire class of people is OUTRAGEOUS! While you continue to defend her, keep in mind that you're saying more about you than her.
Frankly, it's sickening.
Helen Thomas wanted the Jewish people to "go back to where they came from." Germany,Poland, and the United States.
As far as I know, no Jews in the United States circe 1940's was a victim of Genocide.
The other people I mentioned in my post want Latinos to "go back where they came from." Yes, they say that it's just "illegals" that they are concerned about, but they are willing to allow legal immigrants of Latino decent to be harassed by authorities. Let's not mention the United States Citizens of Latino decent that they want to have lose their citizenship based on the sins of their parents.
I agree that illegal immigration is a problem. However, none of the proposals by the people mentioned address illegal immigrants, other than abusing Latinos.
Bob,
The difference here is that the illegal immigrants aren't the victims of a never-ending terrorist campaign to wipe them off the map.
Everyone trying to blame Israel for their defense of the homeland has proven themselves to be revisionist historians; the borders have been settled (much like our own), and even if the UN were to partition another state for Israel to inhabit, just how long would it be until these terrorists surrounded them again with their wish to destroy them?
This is very much like the situation we face with our southern border. The folks who are trying to reclaim our land as their own have openly disputed settled treaties and land purchases. The movement towards "Aztlan Reconquista" is a very disturbing parallel with what's occurring in Israel. Most of us (emphasis on MOST) want to see this situation not get out of hand.
There is a very strong anti-American sentiment amongst the folks coming here over the last few years. One only need attend a few of these protests to see for themselves the signage demanding that WE go back to Europe.
The people to whom you referred are simply mirroring the concerns of the electorate. The vast majority of us want an end to this occupation before we end up being in the same situation as Israel faces on a daily basis.
They absolutely have the right to defend themselves and their right to exist in their homeland, as do we. Further still, if you read their history, they inhabited the Holy Land long before the U.N. became involved in the partition.
Helen's comments were treated as an insult to everybody who knows their history, and her desire to uproot them and start over with no land is utterly appalling and is indeed hateful antisemitism. Furthermore, it emboldens our own enemies (radical Islam), which most would agree is the wrong message we want to send.
Our efforts to hold our elected leader's feet to the fire in terms of the illegal occupation from foreign nationals to buy 'votes' for a party quickly losing favor with the American people is treason. We refuse to allow these traitors/pols to become as corrupt as the government/oligarchs in Mexico, which everyone knows is a wealthy country.
We celebrate the positions that our candidates are taking, and we want this problem dealt with.
The people have spoken, and it's time we're finally heard.
Helen Thomas is a national treasure, and she will be missed. It's sad she is retiring after a YouTube "macaca moment." For 50 years, it often seemed like she was the only real journalist in the White House press room, usually filled with sycophants.
rmwarnick,
I agree with you on the "macaca moment." But is it not fair to ask how she should receive a pass for her poorly-received opinions?
What if someone said we should send blacks back to Africa? It would and SHOULD be horrible. The idiot saying such a thing would and SHOULD lose their job.
We have to accept accountability for the things we say, whether we be liberal or conservative. Wouldn't you agree?
98 percent of Utah's elected Republicans are racist. That's why the Utah legislature is going to outlaw watermelon and make it a "States Rights" issue.
| 118,162
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TITLE: Closed analytic form of $\int x^x\,\mathrm{d}x$
QUESTION [2 upvotes]: I have been struggling to find a closed analytic form for the indefinite integral:
$$
\int x^x\,\mathrm{d}x
$$
After a number of failed attempts I am thinking that only a numerical solution exists given bounded limits of integration.
Main attempt is integration by parts but this yields more and more integrals of form $\int x^{x+i}\,\mathrm{d}x$ (along with other complicated terms involving $\ln{x} $) for values of $i=1,2,3\ldots$.
Using Mathematica and testing various domains of $x$ I see that for $x < 0$ definite integrals are complex values. For $x\geq0$, definite integral is bounded only by bounded upper limit of integration.
Question: is there an analytic form for this integral. If not, is there a good way to prove that no such analytic form exists?
REPLY [0 votes]: Given the references in comments about the Sophomore Dream function, I am assuming that the answer to my question is that "No, there is no closed form analytic function whose derivative is $x^x$.
I had never heard of this "Sophomore's Dream" function before. I came across this particular problem having already solved the derivative of $x^x$ by two different methods. One by a trick and the other by treating the $x^x$ function as $f(x,y)=y^x$ and then taking the derivative $df$ evaluated at the point $(x,x)$. This second method I am not even sure if it is a valid general method though I came up with the same answer.
Having done the derivative, I naturally attempted the integral of $x^x$ and failing in my attempt.
| 183,787
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\section{Background on the geodesic flow}\label{s=background}
Let $M$ be a Riemannian manifold. As usual $\langle v,w\rangle$ denotes the inner product of two vectors and $\nabla$ is the Levi-Civita connection defined by the Riemannian metric. It is the unique connection that is symmetric and compatible with the metric.
The covariant derivative along a curve $t \mapsto c(t)$ in $M$ is denoted by $D_c$, $\frac D{dt}$ or simply $'$ if it is not necessary to specify the curve; if $V(t)$ is a vector field along $c$ that extends to a vector field $\widehat V$ on $M$, we have
$$
V'(t) = \nabla_{\dot c(t)} \widehat V.
$$
Given a smooth map $(s,t) \mapsto \alpha(s,t)$, we let $\frac D{\partial s}$ denote covariant differentiation along a curve of the form $s \mapsto \alpha(s,t)$ for a fixed $t$. Similarly $\frac D{\partial t}$ denotes
covariant differentiation along a curve of the form $t \mapsto \alpha(s,t)$ for a fixed $s$. The symmetry of the Levi-Civita connection means that
$$
\frac D{\partial s}\frac{\partial\alpha}{\partial t}(s,t) = \frac D{\partial t}\frac{\partial\alpha}{\partial s}(s,t)
$$
for all $s$ and $t$.
The curve $c$ is a geodesic if it satisfies the equation $D_c \dot c(t) = 0$. Since this equation is a first order ODE in the variables $(c,\dot c)$, a geodesic is uniquely determined by its initial tangent vector. Geodesics have constant speed, since we have $\dfrac d{dt}\langle \dot c(t), \dot c(t)\rangle = 2\langle\dot D_c\dot c(t), \dot c(t) \rangle = 0$ if $c$ is a geodesic.
The Riemannian curvature tensor $R$ is defined by
$$
R(A,B)C = (\nabla_A\nabla_B - \nabla_B\nabla_A - \nabla_{[A,B]})C.
$$
The sectional curvature of the $2$-plane spanned by vectors $A$ and $B$ is defined by
$$
K(A,B) = \frac{\langle R(A,B)B,A \rangle}{\|A\wedge B\|^2}.
$$
The action of the Levi-Civita connection extends to covectors and tensors in such a way that the product rule holds. In particular
$$
(\nabla_W R)(X,Y)Z = \nabla_W\left(R(X,Y)Z\right) - R(\nabla_WX,Y)Z - R(X,\nabla_WY)Z - R(X,Y)\nabla_WZ.
$$
Similarly the second derivative $\nabla^2_{X,Y}T$ of a tensor $T$ is defined by the product rule formula
$$
\nabla_X(\nabla_YT) = \nabla^2_{X,Y}T + \nabla_{\nabla_X Y}T.
$$
We will use this later in the case $T = R$. If $T$ is a vector field $Z$,
a short calculation using the symmetry of the Levi-Civita connection yields
$$
\nabla^2_{X,Y}Z - \nabla^2_{Y,X}Z = R(X,Y)Z.
$$
\subsection{Vertical and horizontal subspaces and the Sasaki metric}
The tangent bundle $TTM$ to $TM$ may be viewed as a bundle over $M$
in three natural ways shown in the following commutative diagram:
\[
\begin{xy}
\xymatrixcolsep{4pc}
\xymatrixrowsep{3pc}
\xymatrix{
TTM\ar[r]^{D\pi_M}\ar[d]^\kappa\ar[dr]^{\pi_{TM}\circ\pi_M}&TM\ar[d]^{\pi_M} \\
TM\ar[r]^{\pi_M}&M \\
}
\end{xy}
\]
The first is via the composition of the natural bundle projections $\pi_{TM} :TTM\to TM$ and $\pi_M : TM \to M$. The second is via the composition of the derivative map $D\pi_M : TTM \to TM$ with $\pi_M$.
The third involves a map $\kappa: TTM \to TM$, often called the {\em connector map}, which is determined by the Levi-Civita connection. If $\xi \in TTM$ is tangent at $t = 0$ to a curve $t \mapsto V(t)$ in $TM$ and $c(t) = \pi_M( V(t))$ is the curve of footpoints of the vectors $V(t)$, then
$$
\kappa(\xi) = D_cV(0).
$$
The {\em vertical subbundle} is the subbundle $ker(D\pi_M)$. It is naturally identified with $TM$ via the map $\kappa$. The {\em horizontal subbundle} is the subbundle $ker(\kappa)$. It is naturally identified with $TM$ via the map $D\pi_M$ and is transverse to the vertical subbundle. If $v \in T_pM$, we may
identify $T_vTM$ with $T_pM \times T_pM$ via the map $D\pi_{M} \times \kappa : TTM \to TM\times TM$.
Each element of $T_vTM$ can thus be represented uniquely by a pair
$(v_1,v_2)$ with $v_1\in T_pM$ and $v_2\in T_pM$.
Put another way, every element $\xi$ of $T_vTM$ is tangent to a curve $V\colon (-1,1) \to TM$ with $V(0) = v$. Let
$c = \pi_M\circ V\colon (-1,1) \to M$ be the curve of basepoints of $V$ in $M$. Then $\xi$ is represented
by the pair
$$
(\dot c(0), D_cV(0)) \in T_pM\times T_pM.
$$
These coordinates on the fibers of $TTM$ restrict to coordinates on $TT^1M$.
Regarding $TTM$ as a bundle over $M$ in this way
gives rise to a natural Riemannian metric on $TM$, called the {\em Sasaki metric}. In this metric, the
inner product of two elements $(v_1,w_1)$ and $(v_2,w_2)$ of $T_vTM$ is defined:
$$
\langle (v_1,w_1) , (v_2,w_2) \rangle_{Sas} = \langle v_1 , v_2 \rangle + \langle w_1 , w_2 \rangle .
$$
This metric is induced by a symplectic form $\omega$ on $TTM$; for vectors $(v_1,w_1)$ and
$(v_2,w_2)$ in $T_vTM$, we have:
$$
\omega((v_1,w_1) , (v_2,w_2)) = \langle v_1,w_2 \rangle - \langle w_1,v_2 \rangle.
$$
This symplectic form is the pull back of the canonical symplectic form on the cotangent bundle $T^*M$ by the map from $TM$ to $T^*M$ induced by identifying a vector $v \in T_pM$ with the linear function $\langle v, \cdot\rangle$ on $T_pM$.
Sasaki \cite{Sasaki} showed that the fibers of the tangent bundle are totally geodesic submanifolds of $TTM$ with the Sasaki metric. A parallel vector field along a geodesic of $M$ (viewed as a curve in $TM$) is a geodesic of the Sasaki metric. Such a geodesic is orthogonal to the fibers of $TM$. If $v \in T_pM$ and $v' \in T_{p'}M$, we can join them by first parallel translating $v$ along a geodesic from $p$ to $p'$ to obtain $w \in T_{p'}M$ and then moving from $w$ to $v'$ along a line in $T_{p'}M$. If $v'$ is close to $v$, we can choose the geodesic so that its length is $d(p,p')$.
It follows easily from Topogonov's comparison theorem \cite[Theorem 2.2]{CheegerEbin} that
$$
d_{Sas}(v,v') \asymp d(p,p') + \|w - v'\|,
$$
as $v' \to v$, where the rate of convergence is controlled by the curvatures of the Sasaki metric in a neighborhood of $v$. The notation $a\asymp b$, here and in the rest of the paper, means that the ratios
$a/b$ and $b/a$ are bounded from above by a constant. In this case the constant is $2$.
\subsection{The geodesic flow and and Jacobi fields}\label{SSJfields}
For $v \in TM$ let $\gamma_v$ denote the unique geodesic $\gamma_v$ satisfying $\dot\gamma_v(0) = v$. The geodesic flow $\varphi_t : TM\to TM$ is defined by
$$
\varphi_t(v) = \dot\gamma_v(t),
$$
wherever this is well-defined. The geodesic flow is always defined locally. Since the geodesic flow is Hamiltonian, it preserves a natural volume form on $T^1M$ called the Liouville volume form. When the integral of this form is finite, it induces a unique probability measure on $T^1M$ called the {\em Liouville measure} or {\em Liouville volume}.
Consider now a one-parameter family of geodesics, that is a map $\alpha:(-1,1)^2\to M$ with the property that
$\alpha(s,\cdot)$ is a geodesic for each $s\in (-1,1)$. Denote by $J(t)$ the vector field
$$
J(t) = \frac{\partial \alpha}{\partial s}(0,t)
$$
along the geodesic $\gamma(t) = \alpha(0,t)$.
Then $J$ satisfies the {\em Jacobi equation:}
$$
J'' + R(J,\dot\gamma)\dot\gamma = 0,
$$
in which $'$ denotes covariant differentiation along $\gamma$.
Since this is a second order linear ODE,
the pair of
vectors $(J(0), J'(0)) \in T_{\gamma(0)}M \times T_{\gamma(0)}M$ uniquely determines the vectors
$J(t)$ and $J'(t)$ along $\gamma(t)$. A vector field $J$ along a geodesic $\gamma$ satisfying the Jacobi equation is called a {\em Jacobi field}.
The pair $(J(0),J'(0))$ corresponds in the manner described above to the tangent vector at $s= 0$ to the
curve $V(s) = \frac{\partial \alpha}{\partial t}(s,0)$. To see this, note that $V(s)$ is a vector field along the curve $c(s)=\alpha(s,0)$, so $V'(0)$ corresponds to the pair
$$
(\dot c(0), D_c\frac{\partial \alpha}{\partial t}(s,0))
= (J(0), \frac D{\partial s}\frac{\partial \alpha}{\partial t}(s,0))
= (J(0), \frac D{\partial t}\frac{\partial \alpha}{\partial s}(s,0))
= (J(0),J'(0)).
$$
In the same way one sees that $(J(t),J'(t))$ corresponds to the tangent vector at $s = 0$ to the curve
$s \mapsto \frac{\partial \alpha}{\partial t}(s,t) = \varphi_t \circ V(s)$, which is $D\varphi_t(V'(0))$.
To summarize the preceding discussion, there is a
one-one correspondence between elements of $T_vTM$ and Jacobi fields along the geodesic $\gamma$ with $\dot\gamma(0) = v$. Note that the pair $(J(t), J'(t))$ defines a section of $TTM$ over $\gamma(t)$. We have the following key proposition:
\begin{proposition} \label{prop:key}
The image of the tangent vector $(v_1,v_2)\in T_vTM$ under the derivative of the
geodesic flow $D_v\varphi_t$ is the tangent vector $(J(t),J'(t))\in T_{\varphi_t(v)}TM$, where $J$ is the unique
Jacobi field along $\gamma$ satisfying $J(0) = v_1$ and $J'(0) = v_2$.
\end{proposition}
Any vector field of the form $J(t) = (a + bt)\dot\gamma(t)$ is a Jacobi field, since in that case $R(J,\dot\gamma) = 0$ and the Jacobi equation reduces to $J'' = 0$, which holds since $\dot\gamma' = 0$. Conversely, any Jacobi field that is always tangent to $\gamma$ must have this form. Computing the Wronskian of the Jacobi field $\dot\gamma$ and an arbitrary Jacobi field $J$ shows that $\langle J', \dot\gamma \rangle$ is constant.
It follows that if $J'(t_0) \perp \dot\gamma(t_0)$ for some $t_0$, then $J'(t) \perp \dot\gamma(t)$ for all $t$. Similarly if $J(t_0) \perp \dot\gamma(t_0)$ and $J'(t_0) \perp \dot\gamma(t_0)$ for some $t_0$, then $J(t) \perp \dot\gamma(t)$ and $J'(t) \perp \dot\gamma(t)$ for all $t$; in this case we call $J$ a {\em perpendicular Jacobi field}.
An easy consequence of the above discussion is that any Jacobi field $J$ along a geodesic $\gamma$ can be expressed uniquely as $J = J_\parallel + J_\perp$, where $J_\parallel$ is a Jacobi field tangent to
$\gamma$ and $J_\perp$ is a perpendicular Jacobi field.
\subsection{Matrix Jacobi and Riccati equations}\label{SSMJRE}
Choose an orthonormal basis $e_1=\dot\gamma(0),e_2, \ldots, e_n$ at $0$ for the tangent space at $\gamma(0)$ and parallel transport the basis along $\gamma(t)$. Let $\cR(t)$ be the matrix whose entries are
$$
\cR_{jk}(t)=\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle.
$$
Any Jacobi field can be written in terms of the basis as $J(t)=\sum_{k=1}^n y^k e_k(t)$ and the Jacobi equation can be written as
$$
\frac{d^2 y^k}{dt^2}(t) + \sum_j y^j(t)\cR_{jk}(t ) =0.
$$
A solution is determined by values and derivatives at $0$ of the $y^k$.
Let $\cJ(t)$ denote any matrix of solutions to the Jacobi equation.
When the matrix $\cJ$ is nonsingular, we can define
$$
U=\cJ'\cJ^{-1}.
$$
Then $U$ satisfies the
{\em matrix Riccati equation}:
\begin{equation} \label{MRE}
U' + U^2 + \cR = 0,
\end{equation}
where $\cR$ is the matrix above. A standard calculation using the Wronskian shows that the operator $U=\cJ'\cJ^{-1}$ is symmetric if and only if for any two columns $J_i,J_j$ of $\cJ$, we have $$\omega_{\RR^{2n}}((J_i,J_i'), (J_j,J_j'))=0,$$ where $\omega_{\RR^{2n}}$ is the standard symplectic form on $\RR^{n}$.
\subsection{Perpendicular Jacobi fields and invariant subbundles}
There are two natural subbundles of $TTM$ that are invariant under the derivative $D\varphi_t$ of the geodesic flow, the first containing the second. The first is the tangent bundle $TT^1M$ to the unit tangent bundle of $M$. Under the natural identification $T_vTM \cong T_xM\times T_xM$, for $v\in T_x^1M$,
the subspace $T_vT^1M$ is the set of all pairs $(w_0,w_1)$ such that $\la v, w_1\ra = 0$.
To see this, note that if $\alpha(s,t)$ is a variation of geodesics
generating the Jacobi field $J$ along the geodesic $\gamma$, with $\dot\gamma(0)=v$ and $\|{\partial\alpha}/{\partial t}(s,t)\| = 1$ for all $s,t$, then
$$
0 = \left. \frac{D}{\partial s}\left\|\frac{\partial \alpha}{\partial t} \right\|^2\right\vert_{(0,0)} =
\left. 2\left\la \frac{D^2}{\partial s\partial t} \alpha, \frac{\partial \alpha}{\partial t}\right\ra\right\vert_{(0,0)} =
\left. 2\left\la \frac{D^2}{\partial t\partial s} \alpha, \frac{\partial \alpha}{\partial t}\right\ra\right\vert_{(0,0)} = 2\la J'(0), \dot\gamma(0) \ra.
$$
The $D\varphi_t$-invariance of $TT^1M$ follows from the $\varphi_t$-invariance of $T^1M$.
It is reflected in the fact, noted at the end of Section~\ref{SSJfields}, that $\la J'(t), \dot \gamma\ra$ is constant for any Jacobi field $J$ along a geodesic
$\gamma$.
The second natural invariant subbundle is the orthogonal complement $\dot\varphi^\perp$ in
$TT^1M$ to the vector field $\dot\varphi$ generating the geodesic flow. Under the natural identification $T_vTM \cong T_xM\times T_xM$, for $v\in T_x^1M$, the vector $\dot\varphi(v)$ is $(v,0)$, and
the subspace $\dot\varphi^\perp(v)$ is the set of all pairs $(w_0,w_1)$ such that $\la v, w_0\ra = \la v, w_1\ra = 0$. The $D\varphi_t$-invariance
of $\dot\varphi^\perp$ follows from the observation, made at the end of section ~\ref{SSJfields}, that a Jacobi field $J$ with $J(t_0) \perp \dot\gamma(t_0)$ and $J'(t_0) \perp \dot\gamma(t_0)$ for some $t_0$ is perpendicular to $\gamma$ for all $t$.
To summarize, the space of all perpendicular Jacobi fields along $\gamma$
corresponds to the orthogonal complement to the direction of the geodesic flow $\dot\varphi(v)$ at the point $v = \dot\gamma(0) \in T^1M$. To estimate the norm of the derivative $D\varphi_t$ on $TT^1M$, it suffices to restrict attention to vectors in the invariant subspace $\dot\varphi^\perp$; that is, it suffices to estimate the growth of perpendicular Jacobi fields along geodesics.
\subsection{Consequences of negative curvature and unstable Jacobi fields}
If the sectional curvatures of the Riemannian metric are negative along $\gamma$, then it follows from the Jacobi equation that $\langle J'', J \rangle > 0$, for any Jacobi field with the property that $J(t)$ and $\dot\gamma(t)$ are linearly independent. This has the following consequence; for a proof, see \cite{Eb2}.
\begin{lemma}
\label{lem:negcon}
If the sectional curvatures are negative along $\gamma$, then
the functions $\|J(t)\|$ and $\|J(t)\|^2$ are strictly convex, for any nontrivial perpendicular Jacobi field $J$ along $\gamma$.
\end{lemma}
We also have the following results from \cite[Section 1.10]{Eb}.
Let $\gamma: (-\infty, a] \to M$ be a geodesic ray along which the sectional curvatures of the Riemannian metric are always negative. Then,
for each $w \in \dot\gamma(a)^\perp$, there is a unique perpendicular Jacobi field $J_{+,w}$ along $\gamma$ such that $J_{+,w}(a) = w$ and
$$
\|J_{+,w}(t)\| \leq \|w\| \quad\text{for all $t \leq a$.}
$$
Since $\|J_{+,w}(t)\|$ is a strictly convex function of $t$ by
Lemma~\ref{lem:negcon}, $\|J_{+,w}(t)\|$ must be strictly increasing for $t \leq a$.
In fact $J_{+,w} = \lim_{\tau \to -\infty} J_{+,w,\tau}$, where $J_{+,w,\tau}$ is the Jacobi field such that $J_{+,w,\tau}(a) = v$ and $J_{+,w,\tau}(\tau) = 0$.
We call $J_{+,w}$ an {\em unstable} Jacobi field.
For each $t \leq a$, there is a linear map $U_+(t): \dot\gamma(t)^\perp \to \dot\gamma(t)^\perp$ such that
$$
J_+'(t) = U_+(t)(J_+(t))
$$
for every unstable Jacobi field $J_+$. A Jacobi field along $\gamma$ is unstable if and only if it satisfies $J' = U_+ J$.
\begin{proposition} \label{prop:unstable}
The operators $U_+(t)$ are symmetric and positive definite. They satisfy the matrix Riccati equation (\ref{MRE}). Thus
$$
U_+' + U_+^2 + \mathcal R = 0 .
$$
In other words, for any vector $w \in \dot\gamma(t)^\perp$, we have:
$$\langle w,U_+'(w)\rangle=-\langle R(w,\dot\gamma)\dot\gamma,w\rangle-\langle w,U_+ ^2(w)\rangle.
$$
\end{proposition}
We call $U_+$ the {\em unstable solution} of the Riccati equation along the ray $\gamma$. If $v \in T^1M$ is a vector such that $\gamma_v(t)$ is defined for all $t < 0$, then we define $U_+(v)$ to be the operator $U_+(0)$ associated to the ray $\gamma_v: (-\infty,0] \to M$.
If $\gamma$ is a geodesic in a complete Riemannian manifold with negative curvature, the unstable Jacobi fields
along $\gamma$ are obtained by varying $\gamma$ through geodesics $\beta$ such that $d(\beta(t), \gamma(t)) \leq d(\beta(0), \gamma(0))$ for $t < 0$. These geodesics are orthogonal to a family of immersed hypersurfaces whose lifts to the universal cover of $M$ are called horospheres. The operators $U_+(t)$ are the second fundamental forms of horospheres.
There is an analogous definition of {\em stable} Jacobi fields and the {\em stable solution} of the Riccati equation along a ray $\gamma: [a,\infty)$.
If $\gamma : (-\infty, \infty) \to M$ is a complete geodesic, the unstable Jacobi fields along $\gamma$ are the stable Jacobi fields along the geodesic $t \mapsto \gamma(-t)$. We define $U_-(v)$ analogously to $U_+(v)$; it is symmetric and negative definite. The norm of a stable Jacobi field $J(t)$ defined on a ray $\gamma:[a,\infty) \to M$ is strictly decreasing for $t \geq a$.
Let
$$
\text{ $\mathcal D = \{ v \in T^1M: \gamma_v(t)$ is defined for all
$t\}$. }
$$
If $v \in \mathcal D$, both $U_+(v)$ and $U_-(v)$ exist. This allows us to define a splitting of the $2n -1$ dimensional space $T_vT^1M$ as the direct sum of a one dimensional space $E^0(v)$ and two spaces $E^u(v)$ and $E^s(v)$ each of dimension $n-1$. The space $E^0(v)$ is $\RR\dot\varphi(v)$, and we will have $E^u(v) \oplus E^s(v) = \dot\varphi(v)^\perp$. In our usual coordinates, $E^0(v)$ is spanned by $(v,0)$ while
$$
E^u(v) = \{ (w,U_+(v)w) : w \in v^\perp \} \quad\text{and}\quad
E^s(v) = \{ (w,U_-(v)w) : w \in v^\perp \}.
$$
The splitting at $v$ is mapped to the splitting at $\varphi_t(v)$ by $D\varphi_t$.
The next proposition shows that while the splitting $T_{\mathcal D}T^1M = E^u\oplus E^0\oplus E^s$ is defined only over the set ${\mathcal D}$, the geometry of this splitting is locally uniformly controlled.
\begin{proposition} \label{prop:ctssplit} There exists a continuous function $\delta\colon T^1M\to \RR_{>0}$ such that for all $v\in {\mathcal D}$, if $(w,w') \in E^u(v)$, then
$$
\la w, w' \ra \geq \delta(v) \|(w,w')\|_{Sas}^2,
$$
and if $(w,w') \in E^s(v)$, then
$$
\la w, w' \ra \leq - \delta(v) \|(w,w')\|_{Sas}^2.
$$
\end{proposition}
\begin{proof} It will suffice to show that the functions
$$
\delta^u(v) = \inf_{(w,w')\in E^u(v)\setminus \{0\}} \frac{\la w, w'\ra}{ \|(w,w')\|_{Sas}^2}\quad\hbox{ and }\,
\delta^s(v) = \inf_{(w,w')\in E^s(v)\setminus \{0\}} -\frac{\la w, w'\ra}{ \|(w,w')\|_{Sas}^2}
$$
are locally uniformly bounded away from $0$ for $v\in {\mathcal D}$. We prove the statement for $\delta^s$.
Suppose that $\delta^s$ is not locally bounded away from $0$. Then there would be $v \in {\mathcal D}$, a sequence of vectors $v_n$ in $\mathcal D$ with $\lim_{n \to \infty} v_n =v $ and a sequence $\xi_n \in E^s(v_n)$ such that
$\xi_n$ converges to a vector $\xi = (w,w')$ with $\la w, w'\ra = 0$.
By renormalizing we may assume that
$\|\xi_n\|_{Sas} = \|\xi \|_{Sas} = 1$ for each $n$.
Since $v\in {\mathcal D}$, there exists $\delta > 0$ such that $\gamma_v(t)$ is defined for $|t| < \delta$.
Let $J$ be the Jacobi field along the geodesic $\gamma_v$ determined by $\xi$, and let $J_n$
be the (stable) Jacobi field along $\gamma_{v_n}$ defined by $\xi_n$.
Then $(\|J\|^2)'(0) = 2 \la w, w'\ra = 0$. On the other hand,
since $\xi_n \to \xi$ and $\|J_n(t)\|$ is a decreasing function of $t$ for each $n$, we see that $\|J\|$ is nonincreasing on $(-\delta,\delta)$. It follows from this and the
strict convexity of $\|J\|^2$ given by Lemma~\ref{lem:negcon} that the function $\|J\|^2$ cannot have a critical point in the interval $(-\delta,\delta)$.
\end{proof}
This proposition has the following corollary, which will be used for the Hopf argument in Section~\ref{s=generalsetup}.
\begin{corollary}\label{c=cones} Let $\delta\colon T^1M\to \RR_{>0}$ be the function given by Proposition~\ref{prop:ctssplit}. The continuous conefields
$$
\cC^u(v) = \{(w,w')\in \dot\varphi^\perp(v) : \la w, w' \ra \geq \delta(v) \|(w,w')\|_{Sas}^2\}
$$
and
$$
\cC^s(v) = \{(w,w')\in \dot\varphi^\perp(v) : \la w, w' \ra \leq - \delta(v) \|(w,w')\|_{Sas}^2\}
$$
defined for $v\in T^1M$ intersect only at the origin, and satisfy
$$
E^u(v) \subset \cC^u(v)\quad\hbox{and}\quad E^s(v) \subset \cC^s(v),
$$
for all $v\in \cD$.
\end{corollary}
| 100,072
|
TITLE: Symmetric Group and Alternating Subgroup
QUESTION [2 upvotes]: Prove that $A_{4}$ is the only subgroup of $S_{4}$ of order $12$.
-So far I know that the best way to prove this is by contradiction, assuming that there exists another subgroup in $S_{4}$ of order $12$:
Suppose $H$ is an subgroup of $S_{4}$ of order $12$. Since the order of $S_{4}$ is $24$, the index of $H$ is $24/12=2$.
What next?
REPLY [2 votes]: Since a subgroup of $H$ (of order 12) in $S_4$ is of index 2, it must be normal.
This means it must be a union of conjugacy classes: if an element belongs to $H$ so does all of its conjugates.
In $S_4$ all of the transpositions are conjugate: $\{ (12),(13),(14),(23),(24),(34)\}$ (6 of them). The 3 cycles are all conjugate: $\{ (123),(132),(124),(142),(134),(143),(234),(243)\}$ (8 of them). The 4 cycles are all conjugate: $\{ (1234),...\}$ ($(4-1)!=6$ of them). The disjoint pairs of transpositions are conjugate: $\{(12)(34),(13)(24),(14)(23)\}$ (3 of them). Finally, the identity is just conjugate to itself.
We get: 6+8+6+3+1=24.
Now if some element is in a normal subgroup, everyone of its conjugates must be in there too. So a normal subgroup can be decomposed into the disjoint union of conjugacy classes. Keeping in mind, the identity must belong to any subgroup, how can we get 12 out of these numbers: 1,3,6,6,8? Just one way: 1+3+8. This exactly corresponds to the even permutations. :)
Another Approach Show that any subgroup of $S_n$ is either all-even or half-even/half-odd (hint: if not all-even, pick an odd member and use this to get a bijection between the even and odd elements in the subgroup).
Next, assume that we have a subgroup of order $n!/2$ which isn't $A_n$, call this $H$. Then $H$ is half-even/half-odd. Therefore, $A_n \cap H$ is a subgroup of $A_n$ of index 2.
But $A_n$ doesn't have subgroups of index 2. $A_2$ and $A_3$ are of orders 1 and 3 so they can't have a subgroups of orders 0.5 and 1.5 (silly). $A_n$, $n \geq 5$, doesn't have a subgroup of index 2, since such a subgroup would be normal and $A_n$ is simple (no non-trivial normal subgroups) for $n \geq 5$.
This just leaves $n=4$, the case you're actually interested in. In this case $A_4$ would have to have a subgroup of order 6. It doesn't. Why not? Brute force check. :(
So in the end, the only subgroup of order $n!/2$ in $S_n$ is $A_n$ (for any $n$).
REPLY [0 votes]: HINT: $A_4$ is the group of even permutations in $S_4$. For any $x\in S_4$, $x^{-1}A_4x$ is a conjugate subgroup of $A_4$ and is also order 12. Can this subgroup have odd permutations? Can there be any subgroup of $S_4$ of order 12 with odd permutations? The minimal generating set for $A_4$ is 2 transpositions (odd $\times$ odd = even) and an even 3-cycle. Could an order 12 subgroup with an odd 3-cycle contain the identity (even permutation)?
Change in approach based upon whacka's comments. Thanks!
| 28,242
|
TITLE: Kahler differentials on a smooth projective plane curve
QUESTION [4 upvotes]: Let $C = \{f=0\} \subset \mathbb{P}_k^2$ be a smooth plane curve of degree $d$.
I'm trying to find an explicit basis for $H^0(C,\Omega^1_{C/k})$. I know it should be $\frac{(d-1)(d-2)}{2}$ - dimensional. What I'm trying to find is an explicit collection of rational 1-forms on $C$ (basically elements in $k(C) \large{\frac{dx}{\partial_y f}}$) which when restricted to $C$ give a basis for all 1-forms. Sadly most of what I tried didn't get me anywhere and so I have no interesting attempts to share. Help would be really appreciated.
REPLY [4 votes]: To find the basis explicitly you can use the Poincare residue map
$$
\text{res}: H^0(\mathbb{P}^2, \Omega^2_{\mathbb{P}^2}(C)) \to H^0(C, \Omega^1_C),
$$
which in this case is an isomorphism.
Let $g(x_1, x_2) = f(1,x_1,x_2)$, any 2-form $\omega$ on $\mathbb{P}^2$ with a single pole along $C$ can be written locally in coordinates $x_1$, $x_2$ as
$$
\omega = t(x_1, t_2) \frac{dx_1 \wedge dx_2}{g(x_1,x_2)},
$$
then the residue of $\omega$ is
$$
\text{res}(\omega) = t(x_1, x_2) \frac{dx_2}{g_{x_1}(x_1,x_2)}.
$$
Section $dx_1 \wedge dx_2$ has a pole of order 3 on the hyperplane $H$ at infinity ($K_{\mathbb{P}^2} = -3H$), and $f$ has a pole of order $d$ along $H$, so $\omega$ is holomorphic when $t$ is a rational function with a possible pole of order $\leq d-3$ along $H$, such functions are polynomials in $x_1$ and $x_2$ of order $\leq d-3$. There are $l+1$ monomials of degree $l$ in two variables, so there are $\sum_{l=0}^{d-3} (l+1)=\frac{(d-1)(d-2)}{2}$ monomials of degree $\leq d-3$. The explicit basis of $H^0(C, \Omega^1_C)$ is given by residues of $\omega$'s with all monomials of such degrees.
| 108,446
|
TITLE: Renormalization of mass of an electron inside the crystal
QUESTION [4 upvotes]: In Cheng and Li's book, Gauge theory of Elementary Particle Physics, he essentially says that renormalization has nothing to do with infinities. Even in a totally finite theory, we would still have to renormalize physical quantities. For example, the mass $m^*$ of an electron inside the crystal is renormalized from the mass $m$ it has outside the crystal (due to the interaction inside the crystal). However, unlike relativistic QFT, both $m$ and $m^*$ are measurable and finite. Therefore, the correction $\delta m=m-m^*$ should also be finite.
How does one calculate this correction $\delta m$? If one uses quantum field theory, he finds that the correction to electron mass is logarithmically divergent.
REPLY [1 votes]: The theory of a solid body is the typical example of an effective field theory, typically the finite-temperature. Therefore, all integrals in momentum space are bounded from above by the physical cut-off, while loops, of course, remain. In order to compute the correction to the electron mass in a crystal You need then just to compute the self-energy of the electron by using the finite-temperature Green functions formalism. Of course, there can be a correction proportional to the mass term.
There are also typical examples from the high-energy effective field theories, like chiral perturbation theory, which describes the pseudo-scalar mesons octet interactions below the scale of the QCD chiral symmetry spontaneous breaking. In this theory the natural cut-off is the proton mass. Without EM interactions the masses $\pi^{0}, \pi^{\pm}$ are equal. However, once the EM interactions are turned on, loop corrections to the charged mesons masses appear. With finite cut-off, these corrections are, of course, finite.
| 140,988
|
\newpage
\tableofcontents
\newpage
\section*{Appendix}
\addcontentsline{toc}{part}{Appendix}
\section{Notation and organization of the Appendix}
\paragraph{Structure.} In the first Sections~\ref{APP_Section_Lin_Sys},~\ref{APP_Section_Stat} we review results from system theory and statistics. These include the main tools with which we will prove Theorems~\ref{OUT_THM_Main_Bound},~\ref{STABLE_THM_Main_Bound}. In Section~\ref{APP_Section_Fixed}, we provide PAC bounds and persistency of excitation (PE) results for a fixed time $k$ (pointwise) and fixed past horizon $p$. In Section~\ref{APP_Section_Uniform}, we generalize those PAC bounds and PE results from pointwise to uniform over all times $k$ in one epoch. In Section~\ref{APP_Section_Normalized} we prove Lemma~\ref{OUT_LEM_self_normalization}. By combining the uniform bounds and Lemma~\ref{OUT_LEM_self_normalization}, we prove in Section~\ref{APP_Section_Epoch} that the square loss suffered within one epoch is logarithmic with respect the length of the epoch. Hence, we can now prove Theorem~\ref{OUT_THM_Main_Bound}--see Section~\ref{APP_Section_TH1}. In Section~\ref{APP_Section_Stable}, we analyze the case of stable systems and prove Theorem~\ref{STABLE_THM_Main_Bound}. Finally, in Section~\ref{APP_Section_ALT}, we show how the alternative regret definition~\eqref{EXT_Online_Regret} is equivalent to ours~\eqref{FOR_EQN_Regret_Output} up to logarithmic terms.
Section~\ref{APP_Section_Log} includes some technical results about logarithmic inequalities, which are used to show that the burn-in time $N_0$ depends polynomially on the various system parameters.
\paragraph{Notation.}
Before we proceed with the regret analysis, let us introduce some notation. A summary can be found in Table~\ref{FIXED_Table_notation}
We will analyze the performance of Algorithm~\ref{OUT_ALG_Output} based mainly on a fixed epoch $i$. Since the past horizon $p$ is kept constant during an epoch, we will drop the index $p$ from $Z_{k,p}$, $G_{p}$, $\tilde{G}_{k,p}$, $\bar{V}_{k,p}$ and write $Z_{k}$, $G$, $\tilde{G}_{k}$, $\bar{V}_{k}$ instead.
Similar to the past outputs $Z_k$, we also define the past noises:
\begin{equation}\label{DEF_EQN_Past_Noises}
E_k\triangleq\matr{{ccc}e^*_{t-p}&\cdots&e^{*}_{t-1}}^*
\end{equation}
The batch past outputs, batch past noises, and batch past Kalman filter states are defined as:
\begin{equation}\label{DEF_EQN_Batch_vectors}
\bar{Z}_k\triangleq\matr{{ccc}Z_{p}&\cdots&Z_{k}},\, \bar{E}_k\triangleq\matr{{ccc}E_{p}&\cdots&E_{k}},\,
\bar{X}_k\triangleq\matr{{ccc}\hat{x}_{0}&\cdots&\hat{x}_{k-p}}
\end{equation}
This notation will simplify the sums $\sum_{t=p}^{k} Z_tZ^*_t=\bar{Z}_k\bar{Z}^*_k$, $\sum_{t=p}^{k} E_tZ^*_t=\bar{E}_k\bar{Z}_k$ etc.
Recall the definition of the correlations between the current innovation and the past outputs $S_k\triangleq\sum_{t=p}^{k}e_{t}Z^*_t$ and the regularized autocorrelations of past outputs $\bar{V}_{k}\triangleq \lambda I+\bar{Z}_k\bar{Z}^*_k$.
The innovation sequence $e_k$ is i.i.d. zero mean Gaussian. Its covariance has a closed-form expression and is defined as:
\begin{equation}\label{DEF_EQN_Innovation_Covariance}
\bar{R}\triangleq \E e_ke^*_k=CPC^*+R,
\end{equation}
where $P$ is the solution to the Riccati equation~\eqref{FOR_EQN_Riccati}.
Next we define the Toeplitz matrix $\T_k$, for some $k\ge 1$:
\begin{equation}\label{DEF_EQN_Toeplitz}
\T_k\triangleq \mathrm{Toep}(I,CK,\dots,CA^{k-2}K)=\matr{{cccc}I_m&0& &0\\CK&I_m&\cdots&0\\ \vdots&\vdots& &\vdots \\CA^{k-2}K&CA^{k-3}K&\cdots&I_m}.
\end{equation}
A useful property of system~\eqref{FOR_EQN_System_Innovation} is that the past outputs can be written as:
\begin{equation}\label{DEF_EQN_Past_Outputs_Toeplitz}
Z_t=\O_p\hat{x}_{t-p}+\T_pE_t
\end{equation}
The covariance of $\T_pE_t$ is denoted by:
\begin{equation}\label{DEF_EQN_Sigma_E}
\Sigma_E\triangleq \E \T_pE_tE^*_t\T^*_p=\T_p\diag{\bar{R},\dots,\bar{R}}\T^*_p.
\end{equation}
We define the covariance of the state predictions:
\begin{equation}\label{DEF_EQN_State_Covariance}
\Gamma_{k}\triangleq \E \hat{x}\hat{x}^*_k
\end{equation}
and the covariance of the past outputs:
\begin{equation}\label{DEF_EQN_Past_Outputs_Covariance}
\Gamma_{Z,k}\triangleq \E Z_kZ^*_k=\O_p\Gamma_{k-p}\O^*_p+\Sigma_E.
\end{equation}
Finally, let $A=SJS^{-1}$ be the Jordan form of $A$.
With the big O notation we also hide parameters like $\norm{C}_2,\norm{K}_2$, $\norm{R}_2,\norm{S}_2,\norm{S^{-1}}_2$ etc.
\begin{table}[h]\caption{Notation table for fixed past horizon $p$}\label{FIXED_Table_notation}
\begin{center}
\begin{tabular}{r l c}
\toprule
$Z_{t}$&$\triangleq \matr{{ccc}y^*_{t-p}&\cdots&y^{*}_{t-1}}^*$& Past outputs at time $t$\\
$E_{t}$&$\triangleq\matr{{ccc}e^*_{t-p}&\cdots&e^{*}_{t-1}}^*$& Past noises at time $t$\\
$\bar{Z}_k$&$\triangleq\matr{{ccc}Z_{p}&\cdots&Z_{k}}$& Batch past outputs up to time $k$\\
$\bar{E}_k$&$\triangleq\matr{{ccc}E_{p}&\cdots&E_{k}}$ &Batch past noises up to time $k$\\
$\bar{X}_k$&$\triangleq\matr{{ccc}\hat{x}_{0}&\cdots&\hat{x}_{k-p}}$ &Batch past states up to time $k$\\
$S_k$&$\triangleq \sum_{t=p}^{k}e_{t}Z^*_t$& Correlation of current noise with past outputs\\
$V_k$&$\triangleq \bar{Z}_k\bar{Z}^*_k=\sum_{t=p}^{k}Z_{t}Z^*_t$& Gram matrix of past outputs\\
$\bar{V}_k$&$\triangleq \lambda I+V_k$& Regularized Gram matrix of past outputs\\
$\bar{R}$&$\triangleq\E e_ke_k^*$& Covariance of innovations\\
$\T_k$&$\triangleq\mathrm{Toep}\paren{I,CK,\dots,CA^{k-2}K}$& See~\eqref{DEF_EQN_Toeplitz}, Toeplitz matrix of $I_m$ and system responses $CA^tK$\\
$\Sigma_{E}$&$\triangleq\E \T_pE_t E_t^*T^*_p$ & Covariance of weighted past noises\\
$\sigma_R$&$\triangleq\sigma_{\min}(R)$& Smallest singular value of $R$\\
$\Gamma_{t}$&$\triangleq\E \hat{x}_t\hat{x}^*_t$ & Covariance of Kalman filter state prediction\\
$\Gamma_{Z,t}$&$\triangleq\E Z_t Z_t^*$ & Covariance of past outputs\\
$\tilde{G}_k$&$\triangleq (\sum_{t=p}^{k}y_tZ^*_t) \bar{V}^{-1}_k$& Estimated responses\\
$G$&$\triangleq \matr{{ccc}C(A-KC)^{p-1}K&\dots&CK}$& Kalman filter responses\\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\section{Linear Systems Theory}\label{APP_Section_Lin_Sys}
\subsection{Bounds for system matrices}
Next, we provide a bound for the powers of $A$.
\begin{lemma}
Consider matrix $A$ with Jordan form $SJS^{-1}$. Let $\kappa$ be the largest Jordan block of $A$ corresponding to a unit circle eigenvalue $\abs{\lambda}=1$. Let $\kappa_{\max}$ be the largest Jordan block among all eigenvalues.
Then:
\begin{enumerate}
\item If the system is asymptotically stable $\rho(A)<1$, then $\norm{A^{i}}_2=O\paren{1}$,
$S_t=\sum_{i=0}^{t}\norm{A^i}_2=O\paren{1}$, $\snorm{\T_t}_2=O(1)$, $\snorm{\O_t}_2=O(1)$, $\snorm{\Gamma_{t}}_2=O(1)$.
\item If the system is non-explosive then $\norm{A^{i}}_2=O\paren{i^{\kappa-1}}$,
$S_t=\sum_{i=0}^{t}\norm{A^i}_2=O\paren{t^{\kappa}}$, $\snorm{\O_t}_2=O\paren{t^{\kappa}}$, $\snorm{\T_t}_2=O\paren{t^{\kappa}}$ and $\snorm{\Gamma_{t}}_2=O\paren{t^{2\kappa-1}}$.
\item For both cases
\[
\max_{0\le i\le d}\norm{A^i}_2\le O(d^{\kappa_{\max}-1})
\]
\end{enumerate}
\end{lemma}
\begin{proof}
\textbf{Proof of first part.}
By Gelfand's formula~\cite{horn2012matrix}, for every $\epsilon>0$, there exists a $i_0=i_0(\epsilon)$ such that $\norm{A^i}\le (\rho(A)+\epsilon)^i$, for all $i\ge i_0$.
Just pick $\epsilon$ such that $\rho\paren{A}+\epsilon<1$. Then,
\[
S_t\le \sum_{i=0}^{i_0} \norm{A^i}_2+\frac{1}{1-\rho(A)-\epsilon}=O\paren{1}.
\]
The proof for the other system quantities is similar.
\textbf{Proof of second part.}
Assume that $A$ is equal to a $n\times n$ Jordan block corresponding to $\lambda=1$. The proof for the other cases is similar.
Then we have that:
\[
A^{i}=\matr{{cccc}1& {i }\choose {1}&\dots& {i} \choose{ n-1}\\0&1&\dots&{i} \choose{ n-2}\\ & &\ddots &\\0&0&\dots&1}
\]
By Lemma~\ref{ALG_LEM_TOEPLITZ}, we obtain:
\[
\norm{A^{i}}_2 \le \sum_{k=0}^{n-1} {{i} \choose{ k}}\le \paren{\frac{e i}{n-1}}^{n-1}
\]
where the second inequality is classical, see Exercise~0.0.5 in~\cite{vershynin2018high}.
Hence, we have:
\[
S_t\le t \paren{\frac{e t}{n-1}}^{n-1}=O\paren{t^n}
\]
\[
\norm{\O_t}_2\le \norm{C}_2 S_{t-1}=O\paren{t^n}.
\]
By Lemma~\ref{ALG_LEM_TOEPLITZ}
\[
\norm{\T_t}_2\le 1+\norm{C}_2\norm{K}_2 S_{t-2}=O\paren{t^n}.
\]
Finally
\[
\norm{\Gamma_t}_2\le \norm{K\bar{R}K^*}_2\sum_{i=0}^{t-1}\norm{A^i}^2=tO\paren{t^{2n-2}}.
\]
\textbf{Proof of third part.} In the general case of pairs $(\lambda_j,\kappa_j)$ of eigenvalues and Jordan block sizes, similar to the previous proof:
\[
\max_{0\le i\le d}\norm{A^i}_2\le \norm{S}_2\norm{S^{-1}}_2 \max_{j} \abs{\lambda_j^{d-\kappa_j+1}}\paren{\frac{e d}{\kappa_j-1}}^{\kappa_j-1}= O(d^{\kappa_{\max}-1})
\]
\end{proof}
In the above proofs we used a standard result for the norm of (block) Toeplitz matrices. A proof can be found in~\cite{tsiamis2019finite}.
\begin{lemma}[Toeplitz norm]\label{ALG_LEM_TOEPLITZ}
Let $M\in \R^{m_1n\times m_2n}$, for some integers $n,m_1,m_2$ be an (upper) block triangular Toeplitz matrix:
\[
M=\matr{{cccccc}M_1&M_2&M_3&\cdots&\cdots&M_n\\0&M_1&M_2& & &M_{n-1}\\\vdots & & \ddots & \ddots& &\vdots\\ \\ \\ & & & &M_1&M_2\\0&0&\cdots& &0&M_1},
\]
where $M_{i}\in \R^{m_1\times m_2}$, $i=1,\dots,n$.
Then:
\[
\norm{M}_2\le \sum_{i=1}^{n}\norm{M_i}_2
\]
\end{lemma}
\subsection{Properties of covariance matrix $\Gamma_{k}$}
The following result is standard, for a proof see~\cite{tsiamis2019finite}.
\begin{lemma}[monotonicity]
Consider system~\eqref{FOR_EQN_System_Innovation}, with $\Gamma_k\triangleq \E \hat{x}_k\hat{x}^*_k$. The sequence $\Gamma_k$ is increasing in the positive semi-definite cone.
\end{lemma}
The following results is also standard, but we include a proof for completeness.
\begin{lemma}[Lyapunov difference equation]\label{SYS_LEM_Covariance}
Consider system~\eqref{FOR_EQN_System_Innovation}, with $\Gamma_k\triangleq \E \hat{x}_k\hat{x}^*_k$. Assume that the system is stable with $\rho(A)<1$. Then, the sequence $\Gamma_k$ satisfies:
\[
\Gamma_k=A\Gamma_{k-1}A^*+K\bar{R}K^*
\]
and converges to $\Gamma_{\infty}\succ 0$, the unique positive definite solution of the Lyapunov equation:
\[
\Gamma_{\infty}=A\Gamma_{\infty}A^*+K\bar{R}K^*.
\]
Moreover, there exists a $\tau=\frac{1}{\log 1/\rho(A)}\tilde{O}(\max\set{\log\mathrm{cond}(\Gamma_{\infty}),\kappa_{\max}})$ such that
\[
\Gamma_{k}\succeq \frac{1}{2}\Gamma_{\infty}, \text{ for all }k\ge \tau
\]
where $\mathrm{cond}(\Gamma_{\infty})=\frac{\sigma_{\min}(\Gamma_{\infty})}{\sigma_{\max}(\Gamma_{\infty})}$.
\end{lemma}
\begin{proof}
Since $A$ is stable $\Gamma_{\infty}=\sum_{k=0}^{\infty} A^{k}K\bar{R}K^*(A^*)^k$ is well defined and solves the Lyapunov equation.
Since $(A,K)$ is controllable $\Gamma_{\infty}$ is strictly positive definite:
\[
\Gamma_{\infty}\succeq \matr{{cccc}KR^{1/2}&AKR^{1/2}&\dots A^{n-1}KR^{1/2}}\matr{{cccc}KR^{1/2}&AKR^{1/2}&\dots A^{n-1}KR^{1/2}}^*\succ 0,
\]
where the controllability matrix $\matr{{cccc}KR^{1/2}&AKR^{1/2}&\dots A^{n-1}KR^{1/2}}$ has full rank. It is unique since the operator $\mathcal{L}(M)=M-AMA^*$ is invertible; it has eigenvalues bounded below by $1-\rho^2(A)$.
Notice that $\Gamma_0=0\preceq \Gamma_{\infty}$ and by induction, we can show that $\Gamma_k\preceq \Gamma_{\infty}$. Since $\Gamma_k$ is also monotone, it converges to the unique $\Gamma_{\infty}$.
Now form the difference
\[
\Gamma_k-\Gamma_{\infty}=-A^{k}\Gamma_{\infty}(A^*)^k.
\]
It is sufficient to find a $\tau$ such that:
\[
\norm{A^{\tau}}^2 \sigma_{\max}(\Gamma_{\infty}) \le \frac{\sigma_{\min}(\Gamma_{\infty})}{2}
\]
Since the norm of grows as fast as $ \norm{A^{\tau}}_2=O(\rho(A)^{\tau-\kappa_{\max}+1}\tau^{\kappa_{\max}})$, it is sufficient to pick:
\[
\tau\ge \frac{\kappa_{\max}\log \tau}{\log\frac{1}{\rho(A)}}-\frac{\log \mathrm{cond}(\Gamma_{\infty})/2}{\log\frac{1}{\rho(A)}}+\kappa_{\max}-1.
\]
By Lemma~\ref{LogIn_Lem_Log}, the order of $\tau$ is
\[
\tau=\frac{1}{\log 1/\rho(A)}\tilde{O}(\max\set{\log\mathrm{cond}(\Gamma_{\infty}),\kappa_{\max}})
\]
\end{proof}
\begin{lemma}[Convergence rate: Lyapunov difference equation]
Consider system~\eqref{FOR_EQN_System_Innovation}, with $\Gamma_k\triangleq \E \hat{x}_k\hat{x}^*_k$. Assume that the system is stable with $\rho(A)<1$.
\end{lemma}
\begin{proof}
\end{proof}
\subsection{Proof of Lemma~\ref{OUT_LEM_ARMA_representation}}
We first prove a result for just the observations $y_k$.
\begin{proposition}[ARMA-like representation]\label{ARMA_PROP_ARMA_form}
Consider system~\eqref{FOR_EQN_System_Innovation} with $a(s)=s^{d}-a_{d-1}s^{d-1}-\dots-a_0$ the minimal polynomial with degree $d$.
Then the outputs can be represented by the ARMA recursion:
\begin{equation}\label{ARMA_EQN_System}
y_{k}=a_{d-1}y_{k-1}+\dots+a_0 y_{k-d}+\sum_{s=0}^{d}M_{s}e_{k-s},
\end{equation}
where $M_s$ are matrices of appropriate dimensions such that:
\[
\norm{M_s}_2\le \norm{a}_1\max\set{\norm{C}_2\norm{K}_2\max_{0\le i\le d}\norm{A^{i-1}}_2,1},
\]
where $\norm{a}_1$ denotes the $\ell_1$ norm of the polynomial coefficients $1+\sum_{i=0}^{d-1} \abs{a_i}$.
\end{proposition}
\begin{proof}
We start from the fact that:
\[
y_{k-t}=CA^{d-t}\hat{x}_{k-d}+\underbrace{\sum_{s=t+1}^{d}CA^{s-t-1}Ke_{k-s}+e_{k-t}}_{\tilde{e}_{k,t}},
\]
for $t=0,\dots,d$.
From the properties of the minimal polynomial we obtain that:
\[
CA^d\hat{x}_k=a_{d-1}CA^{d-1}\hat{x}_{k-1}+\dots+a_0C\hat{x}_{k-d}
\]
which leads to:
\begin{align}
y_{k}&=a_{d-1}y_{k-1}+\dots+a_0y_{k-d}+\tilde{e}_{k,0}-\sum_{t=1}^{d}a_{d-t}\tilde{e}_{k,t}\\
&a_{d-1}y_{k-1}+\dots+a_0y_{k-d}+\sum_{s=0}^{d}M_se_{k-s},
\end{align}
with $M_0=I$ and
\[
M_s=-a_{d-s}-\sum_{t=1}^{s} a_{d-t} CA^{t-1}K+CA^{s-1}K
\]
\end{proof}
The same will now hold for the past outputs:
\[
Z_{k}=a_{d-1}Z_{k-1}+\dots+a_0Z_{k-p}+\underbrace{\sum_{s=0}^{d}\diag{M_s,\dots,M_s}E_{k-s}
}_{\delta_k}\]
where
\[
E_k\triangleq\matr{{ccc}e_{k-p}^*&\cdots&e_{k-1}^{*}}^*.
\]
We can bound $\delta_k$ by:
\[
\norm{\delta_k}_2\le (d+1)\max_{0\le s\le d}{\norm{M_s}_2}\max_{s\le k}\norm{E_{s}}_2\le (d+1)\max_{0\le s\le d}{\norm{M_s}_2}\sqrt{p}\max_{s\le k}\norm{e_{s}}_{2}.
\]
Define:
\begin{equation}\label{ARMA_EQN_Delta}
\Delta\triangleq (d+1)\norm{a}_1\max\set{\norm{C}_2\norm{K}_2\max_{0\le i\le d}\norm{A^{i-1}}_2,1}\sqrt{p}
\end{equation}
The fact that $\Delta=O(d^{\kappa_{\max}-1}\norm{a}_1\sqrt{p})$ follows from the fact that $\norm{A^{i-1}}_2=O(i^{\kappa_{\max}-2})$.\hfill$\square$
\section{Statistical Toolbox}\label{APP_Section_Stat}
\subsection{Least singular value of Toeplitz matrix}
Let $u_t\in\R^{m}$, $t=0,\dots$ be an i.i.d. sequence, where $u_k\sim\mathcal{N}(0,I)$ are isotropic Gaussians. The following results shows that the Toeplitz matrix is well conditioned with high probability.
Similar results have been reported in~\cite{sarkar2019finite,oymak2018non}.
Compared to~\cite{oymak2018non} we have better dependence between the number of samples $k$ and $\log 1/\delta$, which will allow us to prove uniform persistency of excitation.
Compared to~\cite{sarkar2019finite}, we have similar terms, but we also include universal constants.
\begin{lemma}\label{STAT_LEM_Toeplitz}
Let $u_t\in \R^{m}$, $t=0,\dots,$ be an i.i.d. sequence of Gaussian variables with unit covariance matrix.
Consider the Toeplitz matrix \[U=\matr{{cccc}u_{k-p} & u_{k-p-1}&\dots&u_{0}\\u_{k-p+1}& u_{k-p}&\dots&u_{1}\\\vdots\\u_{k-1}&u_{k-2}&\dots&u_{p-1}}.\] If
\begin{equation*}
k\ge f_1\paren{p,\delta}\triangleq p+128\paren{mp^2\log 9+p\log2+p\log\tfrac{1}{\delta}}
\end{equation*}
then with probability at least $1-\delta$:
\begin{equation}\label{NoisePE_Basic_Inequality}
\tfrac{1}{2}(k-p+1)I\preceq UU^*\preceq \tfrac{3}{2}(k-p+1)I.
\end{equation}
\end{lemma}
\begin{proof}
Consider a $1/4$-net $\N$ of the unit sphere $\S^{mp-1}$. Then, from~\citep[Exercise 4.4.3 b]{vershynin2018high}:
\begin{equation}\label{NoisePE_EQN_Proof_Net}
\norm{UU^*-(k-p+1)I}_2\le 2\sup_{v\in \N}\abs{v^*UU^*v-(k-p+1)}.
\end{equation}
Denote the partition of $v$ in $p$ blocks of length $m$ as:
\[
v^*=\matr{{cccc}v^*_1&v^*_2&\dots&v^*_p}
\]
Notice that $U^*v\in\R^{k-p}$ is zero-mean Gaussian with covariance matrix:
\begin{equation}\label{NoisePE_EQN_Toeplitz}
\Sigma=\matr{{ccccccc}1& r(1)&\cdots&r(p-1)&0&\cdots&0\\
r(1)&1&\cdots&r(p-2)&r(p-1)&\cdots&0\\ \vdots& &\ddots \\r(p-1)&r(p-2)&\cdots&1&r(1)&\cdots&0\\0&r(p-1)&\cdots&r(1)&1&\cdots&0\\& &\vdots\\ 0& 0&\cdots &0 & 0& \cdots&1}
\end{equation}
where we define the convolutions:
\[
r(t)=\sum_{s=1}^{p-t}v^*_sv_{t+s}, t=0,\dots,p-1
\]
with $r(0)=1$.
Due to the Toeplitz structure:
\[
\norm{\Sigma}_2\le 1+2\sum_{i=1}^{p-1}\abs{r(i)}\le 1+ \sum_{i,j=1,i\neq j}^{p}\norm{v}_i\norm{v}_j=(\sum_{i=1}^p \norm{v_i})^2,
\]
where the first inequality follows from Lemma~\ref{ALG_LEM_TOEPLITZ}, while the second follows from the triangle inequality.
By Cauchy-Schwartz:
\[
\norm{\Sigma}_2\le (\sqrt{p}\norm{v}_2)^2=p
\]
Moreover, for positive definite matrices the following bound holds
\[
\norm{\Sigma}^2_F\le \Tr{\Sigma}\norm{\Sigma}_2=(k-p+1)p.
\]
Hence, by Lemma~\ref{NoisePE_Scalar_Concentration} below, we obtain
\[
\P(\abs{v^*UU^*v-(k-p+1)}\ge \frac{k-p+1}{4})\le 2e^{-\frac{k-p+1}{128 p}}.
\]
Taking the union bound over the whole net $\N$, we obtain
\[
\P(\sup_{v\in N}\abs{v^*UU^*v-(k-p)}\ge \frac{k-p+1}{4})\le 2*\abs{\mathcal{N}}e^{-\frac{k-p+1}{128 p}}\le 2*9^{mp}e^{-\frac{k-p+1}{128 p}},
\]
where we used that $\abs{\mathcal{N}}\le 9^{mp}$~\cite{vershynin2018high}.
From equation~\eqref{NoisePE_EQN_Proof_Net}, we get that:
\[
\norm{UU^*-(k-p)I}_2\le \frac{k-p+1}{2}
\]
with probability at least $1-\delta$
if we choose
\[
k-p+1\ge 128\paren{mp^2\log 9+p\log2+p\log\tfrac{1}{\delta}},
\]
which completes the proof.
\end{proof}
The following result is standard but we include it to have a sense of the universal constants. It is a specialized version of the Hanson-Wright theorem~\cite{vershynin2018high}.
\begin{lemma}[Hanson-Wright specialization]\label{NoisePE_Scalar_Concentration}
Let $Z\in\R^N\sim\N\paren{0,\Sigma}$. Then, for every $t\ge 0$ the following inequalities hold:
\begin{align*}
& \P\paren{Z^*Z\ge\Tr\Sigma+t}\le \exp\paren{- \min\set{\frac{t^2}{8\snorm{\Sigma}^2_{F}},\frac{t}{8\snorm{\Sigma}_2}}}\\
& \P\paren{-Z^*Z\ge-\Tr\Sigma+t}\le \exp\paren{- \frac{t^2}{8\snorm{\Sigma}^2_{F}}}
\end{align*}
\end{lemma}
\begin{proof}
We only prove the first inequality, the second one is similar and easier. Following the standard exponential bound procedure:
\[
\P\paren{Z^*Z\ge\Tr\Sigma+t}\le e^{-\tfrac{1}{2}s(\Tr\Sigma+t)}\E\paren{e^{\frac{s}{2}Z^*Z}},\text{ for all }0\le s\le \tfrac{1}{2\snorm{\Sigma}_2}
\]
By Lemma~\ref{NoisePE_Lemma_Moment_Generating}, we obtain:
\[
\psi(t)\triangleq \log\P\paren{Z^*Z\ge\Tr\Sigma+t}\le -\tfrac{1}{2}\paren{s\Tr\Sigma+st+\log\det(I-s\Sigma)}
\]
Now let the eigenvalues of $\Sigma$ be $\lambda_i$, for $i=1,\dots,N$. Then:
\[
\psi(t)=-\tfrac{1}{2}\paren{st+\sum_{i=1}^N s\lambda_i+\log(1-s\lambda_i)}.
\]
Hence, by Lemma~\ref{NoisePE_logarithm_inequality}
\[
\psi(t)\le \tfrac{1}{2}\paren{-st+s^2\norm{\Sigma}^2_{F}},
\]
where all the products $0\le s\lambda_i\le 1/2$ since $\Sigma$ is positive definite and $s\le \tfrac{1}{2\snorm{\Sigma}_2}$ . The result follows by minimizing over $s$, from Lemma~\ref{NoisePE_Minimum}.
\end{proof}
\begin{lemma}\label{NoisePE_Lemma_Moment_Generating}
Let $Z\in\R^N\sim\N\paren{0,\Sigma}$ and $0\le s < 1/(2\snorm{\Sigma}_2)$. Then:
\[
\E(e^{\frac{s}{2}Z^*Z})=\frac{1}{\sqrt{\det\paren{I-s\Sigma}}}
\]
\end{lemma}
\begin{proof}
First, notice that for $0\le s < 1/(2\snorm{\Sigma}_2)$, the eigenvalues of $\paren{I-s\Sigma}$ are bounded away from $1/2$, hence:
\[
\Sigma^{-1}-sI \succeq \frac{1}{2\snorm{\Sigma}_2}I \succ 0
\] $I-s\Sigma$ is invertible.
Now let $P=(\Sigma^{-1}-sI)^{-1} $. By changing the measure:
\[
\E(e^{\frac{s}{2}Z^*Z})=\int \frac{(2\pi)^{-N/2}}{\sqrt{\det\Sigma}}e^{-\frac{1}{2}z^*P^{-1}z}dz=\frac{\sqrt{\det P}}{\sqrt{\det\Sigma}}=\frac{1}{\sqrt{\det\paren{I-s\Sigma}}}
\]
\end{proof}
\begin{lemma}\label{NoisePE_logarithm_inequality}
Let $0\le x\le 1/2$, then the following inequality holds
\[
x+\log{(1-x)}\ge -x^2,
\]
where $\log$ is the natural logarithm.
\end{lemma}
\begin{proof}
Let $g(x)=x+\log{(1-x)}+x^2$. Then:
\[
g'(x)=1-\frac{1}{1-x}+2x \ge 0,
\]
for $0\le x\le 1/2$. Hence, $g(x)\ge g(0)=0$
\end{proof}
\begin{lemma}\label{NoisePE_Minimum}
Let $c,\bar{s}>0$, then
\[
\min_{0\le s\le \bar{s}} -as+cs^2\le \max\set{-\frac{a^2}{4c},-\frac{a\bar{s}}{2}}
\]
\end{lemma}
\begin{proof}
By elementary calculus:
\[
s^*=\left\{
\begin{aligned}
&\frac{a}{2c},&&\text{ when }\frac{a}{2c}\le \bar{s}\\
&\bar{s}, &&\text{ when }\frac{a}{2c}> \bar{s}
\end{aligned}\right.
\]
As a result, the minimum is equal to
\[
\left\{
\begin{aligned}
&-\frac{a^2}{4c},&&\text{ when }\frac{a}{2c}\le \bar{s}\\
&-a\bar{s}+c\bar{s}^2\le -a\bar{s}/2, &&\text{ when }\frac{a}{2c}> \bar{s}
\end{aligned}\right.
\]
\end{proof}
\subsection{Self-normalized martingales}
The following theorem, which can be found in~\cite{tsiamis2019finite} is an extension of Theorem~1 in~\cite{abbasi2011improved} and Proposition~8.2 in~\cite{sarkar2018fast}.
\begin{theorem}[Cross terms,\cite{tsiamis2019finite}]\label{MART_THM_Vector}
Let $\set{\F_t}_{t=0}^{\infty}$ be a filtration. Let $\eta_{t}\in\R^m$, $t\ge 0$ be $\F_t$-measurable, independent of $\F_{t-1}$. Suppose also that $\eta_{t}\sim\mathcal{N}(0,I)$ is isotropic Gaussian.
Let $X_{t}\in\R^{d}$, $t\ge 0$ be $\F_{t-1}-$measurable. Assume that $V$ is a $d\times d$ positive definite matrix. For any $t\ge 0$, define:
\[
\bar{V}_t=V+\sum_{s=1}^{t}X_sX_s^*,\qquad S_t=\sum_{s=1}^{t} X_sH^*_s,
\]
where
\[
H^{*}_s=\matr{{ccc}\eta^*_s&\dots&\eta^*_{s+r-1}}\in\R^{rm},
\]
for some integer $r$.
Then, for any $\delta>0$, with probability at least $1-\delta$, for all $t\ge 0$
\begin{equation*}
\norm{\bar{V}_t^{-1/2} S_t }^2_2\le 8r\paren{\log\frac{r5^m}{\delta}+\frac{1}{2}\log\det\bar{V}_tV^{-1}}
\end{equation*}
\hfill $\diamond$
\end{theorem}
\subsection{Gaussian suprema}
\begin{lemma}\label{SUP_LEM_Supremum}
Consider $v_t\in \R^d\sim \N\paren{0,I}$ i.i.d., for $t=1,\dots,k$. Let $X_k\in \R^{q}$ be a linear combination:
\[
X_k\triangleq\sum_{t=1}^{k} M_{k,t}v_t, \text{ for }k=1,\dots,T
\]
where $M_{t,k}\in \R^{q\times d}$. For some $\mu>0$ define:
\[
\Sigma_{k}\triangleq \mu I+\mathbb{E}X_kX^*_k
\]
Fix a failure probability $\delta>0$.
Then with probability at least $1-\delta$:
\begin{equation}\label{BOUND_EQN_Supremum}
\sup_{k=1,\dots,T}{\snorm{\Sigma^{-1/2}_kX_k}_2}\le \sqrt{q}+\sqrt{2\log\frac{ T}{\delta}}
\end{equation}
If $\mathbb{E}X_kX^*_k$ is invertible for all $k=1,\dots,T$, the result holds for $\mu=0$.
\end{lemma}
\begin{proof}
Fix a $k$. By the following Lemma~\ref{SUP_LEM_Norm} we obtain that we probability at least $1-\delta/T$:
\[
\snorm{\Sigma^{-1/2}_kX_k}_2\le \sqrt{q}+\sqrt{2\log\frac{ T}{\delta}}.
\]
The result follows by a simple union bound.
\end{proof}
\begin{lemma}\label{SUP_LEM_Norm}
Consider $v_t\in \R^d\sim \N\paren{0,I}$ i.i.d., for $t=1,\dots,k$. Let $X_k\in \R^{q}$ be a linear combination:
\[
X_k\triangleq\sum_{t=1}^{k} M_tv_t,
\]
where $M_t\in \R^{q\times d}$. For some$\mu>0$ define:
\[
\Sigma_{k}\triangleq \mu I+\mathbb{E}X_kX^*_k
\]
Then:
\begin{equation}\label{BOUND_EQN_Normalized_Norm}
P(\snorm{\Sigma^{-1/2}_kX_k}_2> \sqrt{q}+t)\le e^{-t^2/2}
\end{equation}
If $\mathbb{E}X_kX^*_k$ is invertible the result holds for $\mu=0$.
\end{lemma}
\begin{proof}
An application of Jensen's inequality gives:
\[
\E\snorm{\Sigma^{-1/2}_kX_k}_2\le \sqrt{\E X^*_k\Sigma^{-1}_kX_k}=\sqrt{\Tr \Sigma^{-1}\mathbb{E}X_kX^*_k}\le \sqrt{q}
\]
Meanwhile,
\[
\norm{\Sigma^{-1/2}_{k}\matr{{ccc}M_1&\cdots&M_k}}_2\le 1
\]
since by definition $\Sigma_{k}\succeq \sum_{t=1}^{k}M_tM^*_t$.
Hence, the function $\snorm{\Sigma^{-1/2}_kX_k}_2$ is Lipschitz with respect to $v_{t,i}$, for $t=1,\dots,k$, $i=1,\dots,d$ with Lipschitz constant $1$.
By concentration of Lipschitz functions of independent Gaussian variables~\citep[Theorem 5.6]{boucheron2013concentration}:
\[
P(\snorm{\Sigma^{-1/2}_kX_k}_2> \sqrt{q}+t)\le e^{-t^2/2}
\]
\end{proof}
\section{PAC bounds and persistency of excitation for fixed-time and fixed-past}\label{APP_Section_Fixed}
In this section, we include results for persistence of excitation and for identification of the system parameters for a fixed time instance $k$, fixed confidence $\delta$ and a fixed past horizon $p$. To avoid excess notation, we drop the dependence on $p$ and $k$ in this section.
\begin{theorem}[PAC bounds for identification]\label{FIXED_THM_Identification}
Consider system~\eqref{FOR_EQN_System_Innovation} with observations $y_{0},\dots,y_k$. Fix a past horizon $p$ and consider the notation of Table~\ref{FIXED_Table_notation}.
Define
\begin{equation}\label{FIXED_EQN_Index_Functions}
\begin{aligned}
k_1(p,\delta)&\triangleq p+128\paren{mp^2\log 9+p\log2+p\log\tfrac{1}{\delta}}\\
k_2(k,p,\delta)&\triangleq p+\frac{64}{\min\set{4,\sigma_R}} \paren{4pn\log\paren{\frac{n\norm{\O_p}_2^2\norm{\Gamma_{k-p}}_2}{\delta}+1}+8p\log \frac{p5^m}{\delta}}
\end{aligned}
\end{equation}
With probability at least $1-5\delta$ the following events hold at the same time:
\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]
\item[a) Persistency of excitation]
\begin{equation}
\mathcal{E}_{PE}\triangleq\set{
\begin{aligned} \T_p\bar{E}_k\bar{E}_k^*\T_p^*&\succeq \frac{k-p+1}{2}\Sigma_{E}\succeq \frac{k-p+1}{2}\sigma_RI,\\
\bar{Z}_k\bar{Z}^*_k&\succeq \frac{1}{2}\O_p\bar{X}_k\bar{X}_k^*\O_p^* +\frac{1}{2}\T_p\bar{E}_k\bar{E}_k^*\T_p^* ,
\end{aligned}}
\end{equation}
if $k$ satisfies the following perstistency of excitation requirement
\begin{equation} \label{FIXED_EQN_Condition_OutputPE}
k\ge \max\set{k_1(p,\delta),k_2(k,p,\delta)}.
\end{equation}
\item[ b) Upper bound for outputs]
\begin{equation}
\mathcal{E}_{\bar{Z}}\triangleq\set{\bar{Z}_k\bar{Z}^*_k \preceq (k-p+1)\frac{mp}{\delta}\Gamma_{Z,k}} \label{FIXED_EQN_Output_Upper}
\end{equation}
\item[c) Cross term upper bound]
\begin{align}
\mathcal{E}_{\mathrm{cross}}&\triangleq\set{
\norm{S_k\bar{V}_k^{-1/2}}_{2}\le g_1\paren{k,p,\delta} \label{FIXED_EQN_Cross_Terms_Bound}},\text{ where }\\
g_1\paren{k,p,\delta}&\triangleq\sqrt{8}\sqrt{\norm{\bar{R}}mp} \sqrt{\log\frac{3mp}{\delta}+\frac{1}{2}\log(k-p+1)+\frac{1}{2mp}\log\det \paren{\Gamma_{Z,k}\lambda^{-1}+I}}\label{FIXED_EQN_Cross_Terms_g}
\end{align}
\end{enumerate}
\end{theorem}
\begin{proof}
Define the primary events:
\begin{subequations}
\begin{alignat}{2}
\mathcal{E}_{\bar{X}}\triangleq&\set{\bar{X}_k\bar{X}^*_k \preceq (k-p+1)\frac{n}{\delta}\Gamma_{k-p}}\label{FIXED_EQN_State_Upper}\\
\mathcal{E}_E\triangleq&\set{ \T_p\bar{E}_k\bar{E}_k^*\T_p^*\succeq \frac{k-p+1}{2}\Sigma_E}\\
\mathcal{E}_{XE}\triangleq&\set{ \norm{\bar{W}_k^{-1/2}\bar{X}_k\bar{E}^*_k\Sigma^{-1/2}_E}^2_2\le 8p\paren{\log \frac{p5^{m}}{\delta}+\frac{1}{2}\log\det\bar{W}_kW^{-1}}},\label{FIXED_EQN_Cross_State_Noise}\\
\mathcal{E}_{EZ}\triangleq&\set{ \norm{\bar{R}^{-1/2}S_k\bar{V}_k^{-1/2}}^2_2\le 8\paren{\log \frac{5^{m}}{\delta}+\frac{1}{2}\log\det\bar{V}_kV^{-1}}}\label{FIXED_EQN_Cross_Output_Noise}
\end{alignat}
\end{subequations}
where matrices $\bar{W}_t,\bar{L}_t,\bar{B}_t$ are appropriate Gram matrices that normalize the correlations:
\begin{align}
&\bar{W}_k\triangleq \bar{X}_k\bar{X}^*_k+W,&&W\triangleq \frac{k-p+1}{\norm{\O_p}^2_2}I\\
&\bar{V}_k\triangleq \bar{Z}_k\bar{Z}^*_k+V,&&V\triangleq\lambda I
\end{align}
We will show that $\mathcal{E}_{\bar{Z}}$ and all of the above events occur with probability at least $1-\delta$ each.
Moreover \[\mathcal{E}_{PE}\cap\mathcal{E}_{\mathrm{cross}}\supseteq \mathcal{E}_{\bar{X}}\cap\mathcal{E}_{\bar{Z}}\cap\mathcal{E}_{E}\cap\mathcal{E}_{XE}\cap\mathcal{E}_{ZE}\]
Hence by a union bound:
\[
\P(\mathcal{E}_{PE}\cap\mathcal{E}_{\bar{Z}}\cap\mathcal{E}_{\mathrm{cross}})\ge 1-5\delta.
\]
\textbf{Part a: All primary events occur with probability at least $1-\delta$.}
\noindent The fact that $\P\paren{\mathcal{E}_{\bar{X}}}\ge 1-\delta$, $\P\paren{\mathcal{E}_{\bar{Z}}}\ge 1-\delta$ follows by a Markov inequality argument--see~\cite{simchowitz2018learning}. The fact that $\P(\mathcal{E}_{E})\ge 1-\delta$ follows from Lemma~\ref{NoisePE_LEM_Noise_PE}. For the remaining events, we apply Theorem~\ref{MART_THM_Vector}. Notice that $\bar{R}^{-1/2}e_k$ and $\Sigma_E^{-1/2}\E_k$ are isotropic so that the conditions of Theorem~\ref{MART_THM_Vector} hold.
\textbf{Part b: Event $\mathcal{E}_{PE}$}
\noindent From Lemma~\ref{OutputPE_LEM_PE} below, we have that $\mathcal{E}_{PE}\supseteq \mathcal{E}_{\bar{X}}\cap\mathcal{E}_E\cap\mathcal{E}_{XE}$ if $k$ satisfies~\eqref{FIXED_EQN_Condition_OutputPE}.
\textbf{Part c: Event $\mathcal{E}_{\mathrm{cross}}$}
\noindent We show that $\mathcal{E}_{\mathrm{cross}}\supseteq \mathcal{E}_{\mathrm{\bar{Z}}}\cap\mathcal{E}_{\mathrm{ZE}}$.
Conditioned on $\mathcal{E}_{\mathrm{\bar{Z}}}\cap\mathcal{E}_{\mathrm{ZE}}$, we have
\begin{align*}
\norm{S_kV^{-1/2}_k}^2_2&\le 8\norm{\bar{R}}_2\paren{\log \frac{5^{m}}{\delta}+\frac{1}{2}\log\det\bar{V}_kV^{-1}} \\
&\le 8\norm{\bar{R}}_2\paren{\log \frac{5^{m}}{\delta}+\frac{1}{2}\log\det\paren{({k-p+1)\frac{mp}{\delta}\Gamma_{Z,k}\lambda^{-1}+I}}} \\
&\le 8\norm{\bar{R}}_2\paren{\log \frac{5^{m}}{\delta}+\frac{mp}{2}\log\frac{mp}{\delta}+\frac{mp}{2}\log(k-p+1)+\frac{1}{2}\log\det\paren{\Gamma_{Z,k}\lambda^{-1}+\frac{\delta}{mp(k-p+1)}I}}\\
&\le g_1^2(k,p,\delta)
\end{align*}
where we used the simplification $\frac{\delta}{mp(k-p+1)}<1$ and
$\log\frac{5^m}{\delta}\le \frac{mp}{2}\log\frac{3mp}{\delta}$, for $p\ge 2$.
\end{proof}
\subsection{Persistency of excitation proofs}
First, we prove persistence of excitation of the past noises in finite time.
\begin{lemma}[Noise PE]\label{NoisePE_LEM_Noise_PE}
Consider the conditions of Theorem~\ref{FIXED_THM_Identification}. If
\begin{equation}\label{NoisePE_EQN_Basic_Condition}
k\ge p+128\paren{mp^2\log 9+p\log2+p\log\tfrac{1}{\delta}}
\end{equation}
then with probability at least $1-\delta$
\begin{equation*}
\frac{k-p+1}{2} \sigma_{R}I \preceq \frac{k-p+1}{2} \Sigma_{E} \preceq \T_p \bar{E}_k \bar{E}_k^* \T^*_p\preceq \frac{3(k-p+1)}{2} \Sigma_{E}.
\end{equation*}
\hfill $\diamond$
\end{lemma}
\begin{proof}
Notice that $U_k\triangleq\Sigma^{-1/2}_E\T_pE_k$ satisfy the conditions of Lemma~\ref{STAT_LEM_Toeplitz}.
Hence, under condition~\eqref{NoisePE_EQN_Basic_Condition}, with probability at least $1-\delta$:
\[
\frac{k-p+1}{2} I \preceq \sum_{t=p}^{k}U_t U_t^* \preceq \frac{3(k-p+1)}{2} I.
\]
Multiplying from both sides with $\Sigma_E^{1/2}$ gives \[
\frac{k-p+1}{2} \Sigma_{E} \preceq \T_p \bar{E}_k \bar{E}_k^* \T^*_p\preceq \frac{3(k-p+1)}{2} \Sigma_{E}
\]
Finally, from~\cite{tsiamis2019finite}, we have $\Sigma_{E}\succeq \sigma_{R}I$.
\end{proof}
Next, we prove persistency of excitation for the past outputs.
\begin{lemma}[Output PE]\label{OutputPE_LEM_PE}
Consider the conditions of Theorem~\ref{FIXED_THM_Identification} and the definition of $\mathcal{E}_{E}$, $\mathcal{E}_{\bar{X}}$, $\mathcal{E}_{XE}$. If:
\begin{align}\label{OutputPE_EQN_OutputPE_Cond}
k\ge p-1+\frac{64}{\min\set{4,\sigma_R}} \paren{4pn\log\paren{\frac{n\norm{\O_p}_2^2\norm{\Gamma_{k-p}}_2}{\delta}+1}+8p\log \frac{p5^m}{\delta}}
\end{align}
then
\[
\set{\bar{Z}_k\bar{Z}_k^*\succeq \frac{1}{2}\O_p\bar{X}_k\bar{X}_k^*\O_p+\frac{1}{2}\T_p\bar{E}_k\bar{E}_k\T_p}\supseteq \mathcal{E}_{E}\cap\mathcal{E}_{\bar{X}}\cap\mathcal{E}_{XE}
\]
\hfill $\diamond$
\end{lemma}
\begin{proof}
The past outputs can be written as:
\[
\bar{Z}_k=\O_p\bar{X}_k+\T_p\bar{E}_k.
\]
For simplicity, we rewrite $\Sigma^{-1/2}_E\T_p\bar{E}_k=\bar{U}_k$, where $\bar{U}_k$ is defined similarly to $\bar{E}_k$ but has unit variance components.
As a result the sample-covariance matrix will be:
\[
\frac{1}{k-p+1}\bar{Z}_k\bar{Z}^*_k=\frac{1}{k-p+1}\paren{O_p\bar{X}_k\bar{X}^*_k\O^*_p+\Sigma^{1/2}_E\bar{U}_k\bar{U}^*_k\Sigma^{1/2}_E+\O_p\bar{X}_k\bar{U}^*_k\Sigma^{1/2}_E+\Sigma^{1/2}_E\bar{U}_k\bar{X}^*_k\O^*_p}
\]
The proof proceeds in two steps. First, we bound the cross-terms based on the events $\mathcal{E}_{\bar{X}},\mathcal{E}_{XE}$. Second, we show that if $k-p+1$ is large enough, then the cross-terms are small enough so that
\[
\frac{1}{k-p}\paren{\O_p\bar{X}_k\bar{U}^*_k\Sigma^{1/2}_E+\Sigma^{1/2}_E\bar{U}_k\bar{X}^*_k\O^*_p}\preceq \frac{1}{2}\frac{1}{k-p}\paren{O_p\bar{X}_k\bar{X}^*_k\O^*_p+\Sigma^{1/2}_E\bar{U}_k\bar{U}^*_k\Sigma^{1/2}_E}
\]
\noindent \textbf{Cross-term bounds}
\noindent
Conditioned on $\mathcal{E}_{\bar{X}}$
\begin{align*}
\log\det\bar{W}_kW^{-1} &\le \log\det \paren{\frac{n\norm{\O_p}^2_2 }{\delta}\Gamma_{k-p}+I}\\
&=\log\paren{\frac{n\norm{\O_p}^2_2\norm{\Gamma_{k-p}}_2}{\delta}+1}^{n}=n\log\paren{\frac{n\norm{\O_p}^2_2\norm{\Gamma_{k-p}}_2}{\delta}+1}.
\end{align*}
Conditioned also on $\mathcal{E}_{XE}$:
\[
\norm{\bar{W}_k^{-1/2}\bar{X}_k\bar{U}^*_k}^2_2\le 8p\paren{\log \frac{p5^{m}}{\delta}+\frac{1}{2}n\log\paren{\frac{n\norm{\O_p}^2_2\norm{\Gamma_{k-p}}_2}{\delta}+1}}
\]
For simplicity denote:
\[
\C_{XE}\triangleq \sqrt{8p\paren{\log \frac{p5^{m}}{\delta}+\frac{1}{2}n\log\paren{\frac{n\norm{\O_p}^2_2\norm{\Gamma_{k-p}}_2}{\delta}+1}}}.
\]
Let now $u\in\R^{mp}$, $\norm{u}_2=1$ be an arbitrary unit vector. Then, consider the quadratic form \begin{align*}
\frac{1}{k-p}\paren{u^*\O_p\bar{X}_k\bar{U}^*_k\Sigma_E^{1/2}u+u^*\Sigma^{1/2}_E\bar{U}_k\bar{X}^*_k\O^*_pu}.
\end{align*}
Conditioned on $\set{\norm{\bar{W}_k^{-1/2} \bar{X}_k\bar{U}^*_k }^2_2 \le \C_{XE}}\cap \mathcal{E}_E\cap \mathcal{E}_X$, we can bound the cross terms by:
\begin{align*}
\frac{1}{k-p+1}\norm{u^*\O_p\bar{X}_k\bar{U}^*_k\Sigma_E^{1/2}u+u^*\Sigma^{1/2}_E\bar{U}_k\bar{X}^*_k\O^*_pu}_2&\le \frac{2}{k-p}\norm{u^*\O_p \bar{W}^{1/2}_k\bar{W}_k^{-1/2}\bar{X}_k\bar{U}^*_k}_2\norm{\Sigma^{1/2}_Eu}_2\\
&\le \frac{2}{k-p+1}\norm{u^*\O_p \bar{W}^{1/2}_k}_2\norm{\bar{W}_k^{-1/2}\bar{X}_k\bar{U}^*_k}_2\norm{\Sigma^{1/2}_Eu}_2\\
&\le 2\sqrt{\frac{1}{k-p+1}u^*\O_p\bar{X}_k\bar{X}^*_k\O^*_pu+1}\frac{\C_{XE}}{\sqrt{k-p+1}}\norm{\Sigma^{1/2}_Eu}_2
\end{align*}
\noindent\textbf{Cross-terms are dominated}
\noindent To complete the proof, it is sufficient to show that if~\eqref{OutputPE_EQN_OutputPE_Cond} holds then also
\begin{align*}
2\sqrt{\frac{1}{k-p+1}u^*\O_p\bar{X}_k\bar{X}^*_k\O^*_pu+1}\frac{\C_{XE}}{\sqrt{k-p+1}}\norm{\Sigma^{1/2}_Eu}_2\le \frac{1}{2}\frac{1}{k-p+1}\paren{u^*\O_p\bar{X}_k\bar{X}^*_k\O^*_pu+u^*\Sigma^{1/2}_E\bar{U}_k\bar{U}^*_k\Sigma^{1/2}_Eu},
\end{align*}
for any unit vector $u$.
Define:
\begin{align*}
a&=\frac{1}{k-p+1}u^*\O_p\bar{X}_k\bar{X}^*_k\O^*_pu\\
b&=\frac{1}{k-p+1}u^*\Sigma_E u
\end{align*}
Notice that on $\mathcal{E}_E$ we have $\Sigma^{1/2}_E\bar{U}_k\bar{U}^*_k\Sigma^{1/2}_E\succeq \frac{k-p+1}{2}\Sigma_E$. Thus, it is sufficient to show
\begin{align*}
2\sqrt{a+1}\frac{\C_{XE}}{\sqrt{k-p+1}}\sqrt{b}\le \frac{a}{2}+\frac{b}{4}.
\end{align*}
To complete the proof, we apply the following Lemma~\ref{OutputPE_LEM_Elementary_Minimum}, where we exploit the fact that $b\ge \sigma_{\min}\paren{\Sigma_E}\ge \sigma_R$. It follows that it is sufficient to have:
\[
\C_{XE}/\sqrt{k-p+1}\le \frac{2,\sqrt{\sigma_R}}{8}
\]
\end{proof}
\begin{lemma}[Elementary Minimum]\label{OutputPE_LEM_Elementary_Minimum}
Let $a \ge 0$ and $b\ge \sigma_R>0$. Then if
\[
\gamma\le \frac{\min\set{2,\sqrt{\sigma_R}}}{8}
\]
the following inequality is true:
\[
\frac{a}{2}+\frac{b}{4}-2\sqrt{a+1}\sqrt{b}\gamma \ge 0
\]
\end{lemma}
\begin{proof}
Define the function $f(a,b)=\frac{a}{2}+\frac{b}{4}-2\sqrt{a+1}\sqrt{b}\gamma$. By optimizing over $a$, we obtain that the minimum over $a$ is:
\[
\min_{0\le a}f(a,b)=\left. \begin{aligned}
&\frac{b}{4}-2\sqrt{b}\gamma,&&\text{ if }2\gamma\sqrt{b}\le 1\\
&b\paren{\frac{1}{4}-2\gamma^2}-\frac{1}{2},&&\text{ if }2\gamma\sqrt{b}> 1
\end{aligned}\right\}.
\]
The condition $\gamma\le \frac{\min\set{2,\sqrt{\sigma_R}}}{8}$ guarantees that \[\min_{0\le a,\sigma_R\le b\le 4}f(a,b)=\frac{b}{4}-2\sqrt{b}\gamma\ge \frac{b-\sqrt{b}\sigma_R}{4}\ge 0 .\]
For $b>4$:
\[
\min_{0\le a,4< b}f(a,b)=b\paren{\frac{1}{4}-2\gamma^2}-\frac{1}{2}\ge b\paren{\frac{1}{4}-\frac{1}{8}}-\frac{1}{2}=\frac{b-4}{8} \ge 0
\]
\end{proof}
\section{Proof of Lemma~\ref{Lemma_PE}}\label{APP_Section_Uniform}
It follows from the lemma below, which is more general.
\begin{lemma}[Uniform PAC bounds]\label{UniformPE_LEM}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound}. Select a failure probability $\delta>0$. Let $T=2^{i-1}T_{\text{init}}$ for some fixed epoch $i$ with $p=\beta \log T$ the corresponding past horizon. Consider the definition of $g_1(k,p,\delta)$ in~\eqref{FIXED_EQN_Cross_Terms_g}. There exists a $N_0=\mathrm{poly}(n,\beta,\kappa,\log 1/\delta)$ such that with probability at least $1-5\sum_{k=T}^{2T-1}\frac{1}{k^2}\delta$ the following events hold:
\begin{equation}\label{UniformPE_EQN_Bounds}
\mathcal{E}_{\mathrm{unif}}\triangleq \set{ \begin{aligned}
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\preceq (k-p+1)\frac{ k^2 mp}{\delta}\Gamma_{Z,k}\\
\norm{S_k\bar{V}_k^{-1/2}}_2&\le g_1(k,p,\delta/k^2)
\end{aligned}, \text{ for all }T\le k\le 2T-1}
\end{equation}
\begin{equation}\label{UniformPE_EQN_PE}
\mathcal{E}^{\mathrm{PE}}_{\mathrm{unif}}\triangleq \set{ \sum_{j=p}^{k}Z_{j}Z^*_{j}\succeq \frac{k-p+1}{4}\sigma_{R}I,\text{ for all }\max\set{N_0,T} \le k\le 2T-1 }
\end{equation}
\end{lemma}
\begin{proof}
Consider $k_1(p,\delta)$, $k_2(k,p,\delta)$ defined in~\eqref{FIXED_EQN_Index_Functions} and
define:
\begin{equation}\label{UniformPE_EQN_N0}
N_0\triangleq \min\set{t:\: k\ge k_1(\beta\log k,\delta/k^2), k\ge k_2(k,\beta \log k, \delta/k^2),\text{ for all }k\ge t}.
\end{equation}
The dominant terms in $k_1,k_2$ increase with at most the order of $\log^2 k$:
\begin{align*}
& \max\set{k_{1}(\beta\log k,\delta/k^2),k_2(k,\beta \log k,\delta/k^2)}\\
&=O(m\beta^2 \log^2 k+n\beta\log k\log \norm{\Gamma_{k-p}}_2+\beta n \log^2 k+\beta n\log k\log \frac{1}{\delta}),
\end{align*}
while $\norm{\Gamma_{k-p}}_2=O(k^{2\kappa-1})$. By the technical Lemmas~\ref{LogIn_Lem_Log},~\ref{LogIn_Lem_LogSq} it follows that $N_0$ depends polynomially on the arguments $\beta,n,m,\log 1/\delta,\kappa$.
By the definition of $N_0$, if $k\ge \max\set{N_0,T}$ then:
\[
k\ge \max\set{k_{1}(\beta\log k,\delta/k^2),k_2(k,\beta \log k,\delta/k^2)}\ge\max\set{k_{1}(p,\delta/k^2),k_2(k,p,\delta/k^2)}
\]
Now, fix a $k$ such that $T\le k\le 2T-1$. By Theorem~\ref{FIXED_THM_Identification}, with probability at least $1-5\delta/k^2$ we have:
\begin{align*}
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\preceq (k-p+1)\frac{ k^2 mp}{\delta}\Gamma_{Z,k}\\
\norm{S_k\bar{V}_k^{-1/2}}_2&\le g_1(k,p,\delta/k^2)\\
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\succeq \frac{k-p+1}{4}\sigma_{R}I,\text{ if }k\ge N_0
\end{align*}
The uniform result follows by a union bound over all $k$ in $T,\dots,2T-1$.
\end{proof}
\section{Proof of Lemma~\ref{OUT_LEM_self_normalization}}\label{APP_Section_Normalized}
We prove a slightly more general version.
\begin{lemma}
Fix a $p$ and consider the notation of Table~\ref{FIXED_Table_notation}. Consider an $i\ge 0$. The following inequality is true:
\begin{equation}\label{EPOCH_EQN_prediction_term}
\sum_{k=T}^{2T-1}Z^*_{k-i} \bar{V}^{-1}_k Z_{k-i}\le \log\det (\bar{V}_{2T-i-1}\bar{V}^{-1}_{T-i-1})
\end{equation}
\end{lemma}
\begin{proof}
Since $\bar{V}_k$ is increasing in the positive semidefinite cone:
\[
\sum_{k=T}^{2T-1}Z^*_{k-i} \bar{V}^{-1}_k Z_{k-i}\le \sum_{k=T}^{2T-1}Z^*_{k-i} \bar{V}^{-1}_{k-i} Z_{k-i}
\]
Hence, it is sufficient to prove the inequality for $i=0$.
Recall that $\bar{V}_{k}=\bar{V}_{k-1}+Z_{k}Z^*_{k}$.
Consider the identity:
\[\det{\bar{V}_{k}}=\det\paren{\bar{V}_{k-1}+Z_kZ^{*}_k}=\det{\bar{V}_{k-1}}\det\paren{I+\bar{V}^{-1/2}_{k-1}Z_kZ^*_k\bar{V}^{-1/2}_{k-1}}=\det{\bar{V}_{k-1}}\paren{1+Z^*_{k}\bar{V}^{-1}_kZ_k}\]
where the last equality follows from the identity $\det(I+FB)=\det(I+BF)$. Rearranging the terms gives:
\[
Z^*_{k}\bar{V}^{-1}_kZ_k=1-\frac{\det \bar{V}_{k-1}}{\det \bar{V}_{k}}\le \log\det \bar{V}_{k}-\log\det \bar{V}_{k-1},
\]
where the inequality follows from the fact that
the sequence $\bar{V}_k$ is increasing in the positive semidefinite cone and the elementary inequality:
\[
1-x\le \log 1/x,\text{ for }x\le1.
\]
Since the upper bound telescopes, we finally get
\[
\sum_{k=T}^{2T-1} Z^*_k \bar{V}^{-1}_k Z_k \le \sum_{T}^{2T-1} \log\det \bar{V}_{k}-\log\det \bar{V}_{k-1}=\log\det \bar{V}_{2T-1}-\log\det \bar{V}_{T-1}
\]
\end{proof}
\section{Analysis within one epoch}\label{APP_Section_Epoch}
We will analyze the $\ell_2$ square loss for the duration of one epoch, from time $T$ up to time $2T-1$ with fixed past horizon $p=\beta \log T$. We have three cases: i) persistency of excitation is established $T\ge N_0$, where $N_0$ is defined in~\eqref{UniformPE_EQN_N0}; ii) persistency of excitation is not established $T< N_0$; iii) warm-up epoch from $0$ to $T_{\text{init}}-1$.
For now, we concentrate on the first two cases.
Consider the $\ell_2$ loss within the epoch:
\begin{equation}\label{EPOCH_EQN_Loss_Partial}
\LL_{T}^{2T-1}\triangleq\sum_{k=T}^{2T-1}\norm{\hat{y}_k-\tilde{y}_k}^2_2.
\end{equation}
Based on the notation of Table~\ref{FIXED_Table_notation}, the error between the Kalman filter prediction and our online algorithm is:
\begin{align*}
\tilde{y}_{k}-\hat{y}_{k}=&
\underbrace{S_{k-1}\bar{V}^{-1}_{k-1}Z_{k}}_{\text{regression}}+\underbrace{\lambda G\bar{V}^{-1}_{k-1}Z_{k}}_{\text{regularization}}
+\underbrace{C(A-KC)^p\paren{\sum_{i=T}^{k-1}\hat{x}_{i-p}Z^*_i\bar{V}^{-1}_{k-1}Z_k-\hat{x}_{k-p}}}_{\text{truncation bias}}\\
&=S_{k-1}\bar{V}^{-1}_{k-1}Z_{k}+\lambda G\bar{V}^{-1}_{k-1}Z_{k}
+C(A-KC)^p\paren{\bar{X}_{k-1}\bar{Z}_{k-1}\bar{V}^{-1}_{k-1} Z_k-\hat{x}_{k-p}},
\end{align*}
with the notation of Table~\ref{FIXED_Table_notation}.
By Cauchy-Schwarz,
the submultiplicative property of norm and by the fact that $\norm{\bar{Z}_{k-1}\bar{V}^{-1/2}_{k-1}}_2\le 1$:
\begin{align}
\norm{\tilde{y}_{k}-\hat{y}_{k}}^2_2&\le 4\sup_{T\le t\le 2T-1}\paren{\norm{S_{t-1}\bar{V}^{-1/2}_{t-1}}^2_2+\norm{\lambda G\bar{V}^{-1/2}_t}_2^2+\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2}\norm{\bar{V}^{-1/2}_{k-1}Z_k}^2_2\nonumber\\\label{EPOCH_EQN_Square_Loss_Terms}
&+ 4\norm{C(A-KC)^p}^2_{2}\norm{\hat{x}_{k-p}}^2_2.
\end{align}
Hence bounding the $\LL^{2T-1}_T$ consists of three steps, bounding the supremum
\[
\sup_{T\le t\le 2T-1}\paren{\norm{S_{t-1}\bar{V}^{-1/2}_{t-1}}^2_2+\norm{\lambda G\bar{V}^{-1/2}_t}_2^2+\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2}
\]
the sum
\[
\sum_{k=T}^{2T-1} \norm{\bar{V}^{-1/2}_{k-1}Z_k}^2_2,
\]
and the sum
\[
4\norm{C(A-KC)^p}^2_{2}\sum_{k=T}^{2T-1} \norm{\hat{x}_{k-p}}^2_2,
\]
This is what we do in the following theorem.
\begin{theorem}[Square loss within epoch]\label{EPOCH_THM_main}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound}. Let $a$ be the minimal polynomial of $A$ with degree $d$, $\Delta$ defined as in~\eqref{ARMA_EQN_Delta}. Fix two failure probabilities $\delta,\delta_1>0$ and consider $N_0$ defined as in~\eqref{UniformPE_EQN_N0} based on $\delta$. Let $T=2^{i-1}T_{\text{init}}$ for some fixed epoch $i\ge 1$ with $p=\beta \log T$ the corresponding past horizon. Then, with probability at least $1-5\sum_{k=T}^{2T-1}\frac{1}{k^2} \delta-2\delta_1$:
\begin{equation}\label{EPOCH_EQN_Loss_noPE}
\LL^{2T-1}_T\le \mathrm{poly}(\Delta,\norm{a}^2_2,n,\beta,\kappa,\log 1/\delta,\log 1/\delta_1)\paren{\tilde{O}(T)+\tilde{O}(\rho(A-KC)^pT^{2\kappa+1})}.
\end{equation}
If moreover $T\ge N_0$ then also:
\begin{equation}\label{EPOCH_EQN_Loss_PE}
\LL^{2T-1}_T\le \mathrm{poly}(\Delta,\norm{a}^2_2,n,\beta,\kappa,\log 1/\delta,\log 1/\delta_1)\paren{\tilde{O}(1)+\tilde{O}(\rho(A-KC)^pT^{2\kappa})}.
\end{equation}
\end{theorem}
\begin{proof}
\textbf{Uniform events occur with high probability. } Consider the primary events $\mathcal{E}_{\mathrm{unif}}$ and $\mathcal{E}^{\mathrm{PE}}_{\mathrm{unif}}$ defined in~\eqref{UniformPE_EQN_Bounds},~\eqref{UniformPE_EQN_PE} and:
\begin{align}
\mathcal{E}_x&=\set{\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{k}\hat{x}_k}^2_2\le \paren{\sqrt{n}+\sqrt{2\log \frac{4T}{\delta_1}}}^2}\\
\mathcal{E}_e&=\set{\sup_{k\le 2T-1}\norm{\bar{R}^{-1/2}e_k}^2_2\le \paren{\sqrt{m}+\sqrt{2\log\frac{2T}{\delta_1}}}^2}
\end{align}
Based on Lemma~\ref{UniformPE_LEM}, Lemma~\ref{SUP_LEM_Supremum}, and a union bound the all events $\mathcal{E}_{\mathrm{unif}}\cap \mathcal{E}^{\mathrm{PE}}_{\mathrm{unif}}\cap\mathcal{E}_x\cap\mathcal{E}_e$ occur with probability at least $1-5\sum_{k=T}^{2T-1}\frac{1}{k^2}\delta-\delta_1$. Now, we can bound all terms of the square loss based on the above events.
\noindent \textbf{Bound on $\norm{S_{k-1}\bar{V}^{-1/2}_{k-1}}^2_2$.}
From the definition of $\mathcal{E}_{\mathrm{unif}}$:
\[
\sup_{T\le k\le 2T-1} \norm{S_{k-1}\bar{V}^{-1/2}_{k-1}}^2_2\le g^2(k,p,\delta/k^2)=\mathrm{poly}(n,\beta,\kappa,\log 1/\delta)\tilde{O}(1)
\]
\noindent \textbf{Bound on $\norm{\lambda G\bar{V}^{-1/2}_t}^2_2$.}
We simply have: $\norm{\lambda G\bar{V}^{-1/2}_t}^2_2\le \lambda \norm{G}^2_2$
\noindent \textbf{Bound on $\sup_{k\le 2T}\norm{\hat{x}_k}^2_2$}
Since the covariances $\Gamma_{k}$ are increasing:
\[
\sup_{k\le 2T-1}\norm{\hat{x}_k}^2_2\le \norm{\Gamma_{2T-1}}_2\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{k}\hat{x}_k}^2_2.
\]
\noindent \textbf{Bound on $\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2$.}
Notice that $\norm{\bar{X}_{t-1}}^2_2\le 2T\sup_{k\le 2T-1}\norm{\hat{x}_k}^2_2$. From the bound above:
\[
\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2\le \mathrm{poly}(n,\log 1/\delta_1)\tilde{O}(\rho(A-KC)^{p}T^{2\kappa})
\]
since $\norm{\Gamma_{2T-1}}_2=O(T^{2\kappa-1})$.
\noindent \textbf{Bound on $\norm{C(A-KC)^p}^2_{2}\sum_{k=T}^{2T-1} \norm{\hat{x}_{k-p}}^2_2$.}
It is similar to the previous step since:
\[
\sum_{k=T}^{2T-1} \norm{\hat{x}_{k-p}}^2_2\le T \sup_{k\le 2T-1}\norm{\hat{x}_k}^2_2
\]
\noindent \textbf{Bound on the sum of $\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2$.}
By Lemma~\ref{EPOCH_LEM_Normalized}:
\[
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2\le 2d\norm{a}^2_2 \log\det(\bar{V}_{2T-1}\lambda^{-1})+2\Delta\sup_{k\le 2T}\norm{e_k}^2_2 \sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}}^2_2
\]
He have two cases:
\begin{align*}
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}}^2_2&\le \frac{1}{\sigma_R} \sum_{T}^{2T-1}\frac{1}{k-p}\le \frac{1}{\sigma_R} \log\frac{2T-p-1}{T-p-1},\text{ if }T\ge N_0\\
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}}^2_2&\le \frac{T}{\lambda},\text{ if }T< N_0.
\end{align*}
The terms $\log\det(\bar{V}_{2T-1}\lambda^{-1})$, $\sup_{k\le 2T}\norm{e_k}^2_2$ can be bounded based on $\mathcal{E}_{\mathrm{unif}},\mathcal{E}_{e}$.
\end{proof}
\subsection{Bounding the sum of $\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2$}
In the following lemma, we bound term $\norm{\bar{V}^{-1/2}_{k-1} Z_k}_2$, which the key to obtaining bounds in the non-explosive regime. We will apply this in both cases i), ii).
\begin{lemma}[Normalized matrix $\bar{V}^{-1/2}_{k-1} Z_k$]\label{EPOCH_LEM_Normalized}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound}. Let $a$ be the minimal polynomial of $A$ with degree $d$ and $\Delta$ defined as in~\eqref{ARMA_EQN_Delta}. Let $T=2^{i-1}T_{\text{init}}$ for some fixed epoch $i$ with $p=\beta \log T$ the corresponding past horizon. Then,
\begin{equation}
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2\le 2d\norm{a}^2_2 \log\det(\bar{V}_{2T}\lambda^{-1})+2\Delta\sup_{k\le 2T}\norm{e_k}^2_2 \sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}}^2_2.
\end{equation}
\end{lemma}
\begin{proof}
We replace $Z_k=a_{d-1}Z_{k-1}+\dots+a_0 Z_{k-d}+\delta_k$. Then by two applications of Cauchy-Schwarz:
\[
\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2\le 2(a^2_{d-1}+\dots+a^2_0)\sum_{i=0}^{d-1} Z_{k-i}\bar{V}^{-1}_{k-1}Z_{k-i}+2 \norm{\bar{V}^{-1/2}_{k-1}\delta_k}^2_2.
\]
Now by Lemma~\ref{OUT_LEM_self_normalization} and Lemma~\ref{OUT_LEM_ARMA_representation} it follows that:
\begin{equation*}
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2\le 2\norm{a}^2_2 \sum_{i=0}^{d-1}\log\det(\bar{V}_{2T-i}\lambda^{-1})+2\Delta \sup_{k\le 2T}\norm{e_k}_2 \sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}}^2_2.
\end{equation*}
The result follows from the fact that the sequence $\bar{V}_{2T-i}$ is monotone.
\end{proof}
\subsection{Warm-up epoch}
\begin{lemma}[Warm-up epoch]\label{EPOCH_LEM_Warm_up}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound} and the $\ell_2$ loss $\LL_{T_{\text{init}}}=\sum_{t=0}^{T_{\text{init}}}\norm{\hat{y}_t-\tilde{y}_t}^2_2$. Then with probability at least $1-\delta$:
\[
\LL_{T_{\text{init}}}\le \mathrm{poly}(m,\log1/\delta)T_{\text{init}}\norm{C\Gamma_{T_{\text{init}}}C^*+\bar{R}}_2\sqrt{\log T_{\text{init}}}=\left.\begin{aligned}&\tilde{O}(T^{2\kappa}_{\text{init}}), \text{ if }\kappa\ge 1
\\ &\tilde{O}(T_{\text{init}}), \text{ if }\kappa=0
\end{aligned}\right\}\]
\end{lemma}
\begin{proof}
During this time, we have $\tilde{y}_k=0$. Let $\Gamma_{y,k}=C\Gamma_{k}C^*+\bar{R}$. Then,
\[
\LL_{T_{\text{init}}}\le T_{\text{init}} \sup_{k\le T_{\text{init}}}\norm{\hat{y}_k}^2_2 \le T_{\text{init}}\norm{\Gamma_{y,T_{\text{init}}}}_2 \sup_{k\le T_{\text{init}}}\norm{\Gamma^{-1/2}_{y,T_{\text{init}}}\hat{y}_k}^2_2.
\]
By Lemma~\ref{SUP_LEM_Supremum} and by monotonicity of $\Gamma_{k}$, with probability at least $1-\delta$:
\[
\LL_{T_{\text{init}}}\le T_{\text{init}}\norm{\Gamma_{y,T_{\text{init}}}}_2 \paren{\sqrt{m}+\sqrt{2\log\frac{T_{\text{init}}}{\delta}}}=\left.\begin{aligned}&\tilde{O}(T^{2\kappa}_{\text{init}}), \text{ if }\kappa\ge 1
\\ &\tilde{O}(T_{\text{init}}), \text{ if }\kappa=0 \end{aligned}\right\}
\]
\end{proof}
\section{Proof of Theorem~\ref{OUT_THM_Main_Bound}}\label{APP_Section_TH1}
Recall that the regret can be decomposed in two terms:
\[
\Reg_N= \LL_N+2\sum_{k=0}^{N}{e^*_k\paren{\hat{y}_k-\tilde{y}_k}}
\]
where $\LL_N$ is the square loss and the other term is a martingale.
\textbf{Square loss bound.}
Without loss of generality assume that $N=2T_i-1=T_{\text{init}}2^{i}$ is the end of an epoch, where $i$ is the total number of epochs. The number of epochs $i$ depends logarithmically on $N$. Then the square loss $\LL_N$ can be written as:
\[
\LL_N=\LL_{T_{\text{init}}}+\sum_{j=1}^{i-1}\LL^{2T_j-1}_{T_{j}}.
\]
Let $N_0$ be defined as in~\eqref{UniformPE_EQN_N0}. Select
\begin{equation}\label{OUT_EQN_beta_choice}
\beta\ge \frac{\kappa}{\log 1/\rho(A-KC)}
\end{equation}
Then by Theorem~\ref{EPOCH_THM_main}, Lemma~\ref{EPOCH_LEM_Warm_up}, and a union bound, with probability at least $1-(5\frac{\pi^2}{6}+1)\delta-i\delta_1$:
\[
\LL_N=\tilde{O}(T^{2\kappa}_{\text{init}})+\mathrm{poly}(\Delta,\norm{a}^2_2,n,\beta,\kappa,\log 1/\delta,\log 1/\delta_1)\paren{\tilde{O}(N_0)+\tilde{O}(1)}
\]
To complete the bound on $\LL_N$, replace $\delta_1$ with $\delta/{i}$. Since $i$ depends logarithmically on $N$ we finally obtain that with probability at least $1-(5\frac{\pi^2}{6}+2)\delta$:
\[
\LL_N=\tilde{O}(T^{2\kappa}_{\text{init}})+\mathrm{poly}(\Delta,\norm{a}^2_2,n,\beta,\kappa,\log 1/\delta)\paren{\tilde{O}(N_0)+\tilde{O}(1)}
\]
\textbf{Martingale term bound.}
Denote $u_k\triangleq \bar{R}^{-1/2}e_k$ and $z_k\triangleq R^{1/2}\paren{\hat{y}_k-\tilde{y}_k}$. Then
$\sum_{t=1}^{N}e^*_t\paren{\hat{y}_t-\tilde{y}_t}=\sum_{t=1}^{N}u^*_tz_t=\sum_{t=1}^{N}\sum_{i=1}^{m}u_{t,i}z_{t,i}$.
To apply Theorem~\ref{MART_THM_Vector} we need to slightly modify the definition of the filtration. Let $\F_{t,i}\triangleq \sigma(\F_{t}\cup\set{u_{t+1,1},\dots,u_{t+1,i}})$, with $\F_{t+1}\equiv \F_{t,m}$ and define:
\begin{align}
\tilde{\F}_{0}&=\F_0\\
\tilde{\F}_{s}&=\F_{t,s-tm},\text{ if }tm+1 \le s \le (t+1)m
\end{align}
By applying Theorem~\ref{MART_THM_Vector} with $\tilde{\F}_s$ we can bound the sum in terms of the square loss $\LL_N$.
With probability at least $1-\delta$:
\begin{align}
\paren{\sum_{t=1}^{N}z^*_tz_t+1}^{-1/2}\sum_{t=1}^{N}u^*_tz_t &\le 8\log \frac{5}{\delta}+4\log \paren{\sum_{t=1}^{N}z^*_tz_t+1}
\end{align}
or
\begin{align}
\sum_{t=1}^{N}u^*_tz_t \le \paren{\norm{\bar{R}}_2\LL_N+1}^{1/2}\paren{8\log \frac{5}{\delta}+4\log \paren{\norm{\bar{R}}_2\LL_N+1}}
\end{align}
where we used the fact that $z_k^*z_k=(\hat{y}_k-\tilde{y}_k)^*R(\hat{y}_k-\tilde{y}_k)\le \norm{\bar{R}}_2\norm{\hat{y}_k-\tilde{y}_k}^2_2$.
\textbf{Final step}
Finally, by a union bound, with probability at least $1-(5\frac{\pi^2}{6}+3)\delta$:
\[
\Reg_N=\tilde{O}(T^{2\kappa}_{\text{init}})+\mathrm{poly}(\Delta,\norm{a}^2_2,n,\beta,\kappa,\log 1/\delta)\paren{\tilde{O}(N_0)+\tilde{O}(1)}
\]
\hfill $\square$
\section{Stable case}\label{APP_Section_Stable}
In the case of stable systems, we can exploit the fact that the covariance matrix $\Gamma_k$ converges exponentially fast to a stead-state covariance $\Gamma_{\infty}$--see Lemma~\ref{SYS_LEM_Covariance}.
We have the following pointwise persistency of excitation result.
Define the controllability matrix
\begin{equation}\label{STABLE_EQN_Controllability}
\C_t\triangleq\matr{{cccc}A^{t}K\bar{R}^{1/2}&\dots&AK\bar{R}^{1/2}&K\bar{R}^{1/2}}
\end{equation}
Since $(A,K)$ is controllable we also have that $\mathrm{rank}(\C_t)=n$, for $t\ge n$. As a result, the covariance matrix, is always strictly positive definite $\sigma_{\min}(\Gamma_t)\ge \sigma_{\min}(\C_n\C_n^*)>0$, for $t\ge n$--see also Lemma~\ref{SYS_LEM_Covariance}.
\begin{lemma}[Stable: pointwise persistency of excitation]\label{STABLE_LEM_PE_Pontwise}
Consider system~\eqref{FOR_EQN_System_Innovation} with observations $y_{0},\dots,y_k$. Pick a $\tau\ge n$.
Define
\begin{equation}\label{STABLE_EQN_Index_Functions}
\begin{aligned}
k_3(\tau,\delta)&\triangleq \tau+128\paren{m\tau^2\log 9+\tau\log2+\tau\log\tfrac{1}{\delta}}\\
k_4(k,\tau,\delta)&\triangleq \tau+\frac{64}{\min\set{8,\sigma_{\min}(\Gamma_{\tau})}} \paren{4\tau n\log\paren{\frac{n\norm{A^{\tau}}_2^2\norm{\Gamma_{k-\tau}}_2}{\delta}+1}+8\tau \log \frac{\tau 5^m}{\delta}}.
\end{aligned}
\end{equation}
With probability at least $1-3\delta$, if \[k\ge k_3(\tau,\delta),k_4(k,\tau,\delta)\]
then
\begin{equation}\label{STABLE_EQN_PE_Pointwise}
\sum_{t=0}^k \hat{x}_t\hat{x}^*_t \succeq \frac{k-\tau+1}{4}\Gamma_{\tau}
\end{equation}
\end{lemma}
\begin{proof}
Define $u_t\triangleq \bar{R}^{-1/2}e_k$:
\[
U_t\triangleq \matr{{c}u_{t-\tau}\\\vdots\\u_{t-1}}
\]
Observe that:
\begin{align*}
\hat{x}_t&=A^{\tau}\hat{x}_{t-\tau}+\C_{\tau}U_t,\\
\Gamma_{\tau}&=\C_{\tau}\C_{\tau}^*.
\end{align*}
Expanding the correlations gives:
\begin{align}
\sum_{t=0}^k \hat{x}_t\hat{x}^*_t &\succeq \sum_{t=\tau}^k \hat{x}_t\hat{x}^*_t \\
&= A^{\tau}\sum_{t=\tau}^{k}\hat{x}_{t-\tau}\hat{x}^*_{t-\tau}(A^*)^{\tau}+A^{\tau}\sum_{t=\tau}^{k}\hat{x}_{t-\tau}U^*_{t}\C_\tau^*+\C_{\tau}\sum_{t=\tau}^{k}U_t\hat{x}^*_{t-\tau}(A^*)^{\tau}+\C_{\tau}\sum_{t=\tau}^{k}U_tU^*_{t}\C_\tau^*
\end{align}
The proof now is similar to Theorem~\ref{FIXED_THM_Identification} and Lemma~\ref{OutputPE_LEM_PE}. We will show that the cross terms are dominated.
Define the primary events:
\begin{subequations}
\begin{alignat}{2}
\mathcal{E}_{\bar{X}}\triangleq&\set{\sum_{t=0}^{k-\tau}\hat{x}_t\hat{x}^*_t \preceq (k-\tau+1)\frac{n}{\delta}\Gamma_{k-\tau}}\label{STABLE_EQN_State_Upper}\\
\mathcal{E}_E\triangleq&\set{ \sum_{t=\tau}^{k}U_kU_k^*\succeq \frac{k-\tau+1}{2}I}\\
\mathcal{E}_{XE}\triangleq&\set{ \norm{\bar{W}_k^{-1/2}A^{\tau}\sum_{t=\tau}^{k}\hat{x}_{t}U^*_t}^2_2\le 8\tau\paren{\log \frac{\tau5^{m}}{\delta}+\frac{1}{2}\log\det\bar{W}_kW^{-1}}},\label{STABLE_EQN_Cross_State_Noise}
\end{alignat}
\end{subequations}
where matrices $\bar{W}_t, W$ are:
\begin{align}
&\bar{W}_k\triangleq A^{\tau}\sum_{t=0}^{k-\tau}\hat{x}_t\hat{x}^*_t (A^*)^{\tau}+W,&&W\triangleq (k-p+1)I.
\end{align}
The events $\mathcal{E}_{\bar{X}}, \mathcal{E}_{XE}$ occur with probability at least $1-\delta$ each--see proof of Theorem~\ref{FIXED_THM_Identification} and Theorem~\ref{MART_THM_Vector}.
By Lemma~\ref{STAT_LEM_Toeplitz}, if $k\ge k_3(\tau,\delta)$, then also $\mathcal{E}_E$ occurs with probability at least $1-\delta$.
By a union bound $\mathcal{E}_{\bar{X}},\mathcal{E}_{E},\mathcal{E}_{XE}$ occur with probability at least $1-3\delta$.
What remains to show is that these three events imply~\eqref{STABLE_EQN_PE_Pointwise}.
The remaining proof is omitted since it is identical with the one of Lemma~\ref{OutputPE_LEM_PE}.
\end{proof}
\subsection{Proof of Lemma~\ref{Lemma_PE_stable}}
It follows from the lemma below, which is more general.
\begin{lemma}[Stable case: Uniform PAC bounds]\label{STABLE_UniformPE_LEM}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound} with $\rho(A)<1$. Select a failure probability $\delta>0$. Let $T=2^{i-1}T_{\text{init}}$ for some fixed epoch $i$ with $p=\beta \log T$ the corresponding past horizon. Consider also the definition of $g_1(k,p,\delta)$ in~\eqref{FIXED_EQN_Cross_Terms_g}. There exists a $N_0=\mathrm{poly}(n,\beta,\kappa,\log 1/\delta,\log 1/\rho(A))$ such that with probability at least $1-8\sum_{k=T}^{2T-1}\frac{1}{k^2}\delta$ the following events hold:
\begin{equation}\label{STABLE_UniformPE_EQN_Bounds}
\mathcal{E}_{\mathrm{unif}}\triangleq \set{ \begin{aligned}
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\preceq (k-p+1)\frac{ k^2 mp}{\delta}\Gamma_{Z,k}\\
\norm{S_k\bar{V}_k^{-1/2}}_2&\le g_1(k,p,\delta/k^2)
\end{aligned}, \text{ for all }T\le k\le 2T-1}
\end{equation}
\begin{equation}\label{STABLE_UniformPE_EQN_PE}
\mathcal{E}^{\mathrm{PE}}_{\mathrm{st,unif}}\triangleq \set{ \sum_{j=p}^{k}Z_{j}Z^*_{j}\succeq \frac{k-p+1}{32}\Gamma_{Z,k+1},\text{ for all }\max\set{N_0,T}\le k\le 2T-1}
\end{equation}
\end{lemma}
\begin{proof}
Pick $\tau$ such that $\Gamma_{\tau}\succeq \frac{1}{2}\Gamma_{\infty}$. By Lemma~\ref{SYS_LEM_Covariance},
\[\tau=\tilde{O}(\frac{1}{\log 1/\rho(A)}\max\set{\log\mathrm{cond}(\Gamma_{\infty}),\kappa_{\max}}).\]
Next, consider also definitions of $k_1(p,\delta)$, $k_2(k,p,\delta)$ in~\eqref{FIXED_EQN_Index_Functions}, and the definitions of $k_3(\tau,\delta)$, $k_4(k,\tau,\delta)$ in~\eqref{STABLE_EQN_Index_Functions}. Define:
\begin{equation}\label{STABLE_UniformPE_EQN_N0}
N_0\triangleq \min\set{t:\: \begin{aligned} k\ge k_1(\beta\log k,\delta/k^2)&, k\ge k_2(k,\beta \log k, \delta/k^2)\\k-\beta\log k \ge k_3(\tau,\delta/k^2)&, k-\beta\log k\ge k_4(k,\tau, \delta/k^2)\end{aligned},\text{ for all }k\ge t}.
\end{equation}
Similar to the proof of Lemma~\ref{UniformPE_LEM}, by the technical Lemmas~\ref{LogIn_Lem_Log},~\ref{LogIn_Lem_LogSq} it follows that $N_0$ depends polynomially on the arguments $\beta,n,m,\log 1/\delta,\tau$.
By the definition of $N_0$, if $k\ge N_0,T$, then:
\[
k\ge \max\set{k_{1}(\beta\log k,\delta/k^2),k_2(k,\beta \log k,\delta/k^2)}\ge\max\set{k_{1}(p,\delta/k^2),k_2(k,p,\delta/k^2)}.
\]
Moreover,
\[
k-p\ge \max\set{k_{3}(\tau,\delta/k^2),k_4(k,\tau,\delta/k^2)}.
\]
Now, fix a $k$ such that $T\le k\le 2T-1$. By Theorem~\ref{FIXED_THM_Identification}, with probability at least $1-5\delta/k^2$ we have:
\begin{align*}
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\preceq (k-p+1)\frac{ k^2 mp}{\delta}\Gamma_{Z,k}\\
\norm{S_k\bar{V}_k^{-1/2}}_2&\le g_1(k,p,\delta/k^2)\\
\sum_{j=p}^{k}Z_{j}Z^*_{j}&\succeq \frac{1}{2}\O_p\bar{X}_k\bar{X}_k\O^*_p+\frac{k-p+1}{4}\Sigma_E,\text{ if }k\ge N_0
\end{align*}
Meanwhile by Lemma~\ref{STABLE_LEM_PE_Pontwise}, with probability at least $1-3\delta/k^2$:
\[
\bar{X}_k\bar{X}_k\succeq \frac{k-p-\tau+1}{4}\Gamma_{\tau}\succeq \frac{k-p-\tau+1}{8}\Gamma_{\infty}\succeq \frac{k-p+1}{16}\Gamma_{\infty},\text{ if }k\ge N_0
\]
where in the last inequality we used the fact that $k-p\ge k_3\ge 2\tau.$
Combining both we have:
\[
\sum_{j=p}^{k}Z_{j}Z^*_{j}\succeq \frac{k-p+1}{32}\paren{\O_p\Gamma_{\infty}\O^*_p+\Sigma_E}\succeq\frac{k-p+1}{32}\Gamma_{Z,k+1},\text{ if }k\ge N_0
\]
The uniform result follows by a union bound over all $T\le k\le 2T-1$.
\end{proof}
\subsection{Proof of Theorem~\ref{STABLE_THM_Main_Bound}}
Similar to the non-explosive case, we analyze the square loss for a single epoch.
\begin{theorem}[Square loss within epoch]\label{STABLE_EPOCH_THM_main}
Consider the conditions of Theorem~\ref{OUT_THM_Main_Bound}. Fix two failure probabilities $\delta,\delta_1>0$ and consider $N_0$ defined as in~\eqref{STABLE_UniformPE_EQN_N0} based on $\delta$. Let $T=2^{i-1}T_{\text{init}}$ for some fixed epoch $i\ge 1$ with $p=\beta \log T$ the corresponding past horizon. Then, with probability at least $1-8\sum_{k=T}^{2T-1}\frac{1}{k^2} \delta-2\delta_1$:
\begin{equation}\label{STABLE_EPOCH_EQN_Loss_noPE}
\LL^{2T-1}_T\le \mathrm{poly}(n,\beta,\log 1/\delta,\log 1/\delta_1)\paren{\tilde{O}(T)+\tilde{O}(\rho(A-KC)^{2p}T^2)}.
\end{equation}
If moreover $T\ge N_0$ then also:
\begin{equation}\label{STABLE_EPOCH_EQN_Loss_PE}
\LL^{2T-1}_T\le \mathrm{poly}(n,\beta,\log 1/\delta,\log 1/\delta_1)\paren{\tilde{O}(1)+\tilde{O}(\rho(A-KC)^{2p}T)}.
\end{equation}
\end{theorem}
\begin{proof}
\textbf{Uniform events occur with high probability. } Consider the primary events $\mathcal{E}_{\mathrm{unif}}$ and $\mathcal{E}^{\mathrm{PE}}_{\mathrm{st,unif}}$ defined in~\eqref{STABLE_UniformPE_EQN_Bounds},~\eqref{STABLE_UniformPE_EQN_PE} and:
\begin{align}
\mathcal{E}_x&=\set{\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{k}\hat{x}_k}^2_2\le \paren{\sqrt{n}+\sqrt{2\log \frac{4T}{\delta_1}}}^2}\\
\mathcal{E}_z&=\set{\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{Z,k}Z_k}^2_2\le \paren{\sqrt{pm}+\sqrt{2\log\frac{2T}{\delta_1}}}^2}
\end{align}
Based on Lemma~\ref{STABLE_UniformPE_LEM}, Lemma~\ref{SUP_LEM_Supremum}, and a union bound the all events $\mathcal{E}_{\mathrm{unif}}\cap \mathcal{E}^{\mathrm{PE}}_{\mathrm{st,unif}}\cap\mathcal{E}_x\cap\mathcal{E}_z$ occur with probability at least $1-8\sum_{k=T}^{2T-1}\frac{1}{k^2}\delta-\delta_1$. Now we proceed as in the proof of Theorem~\ref{EPOCH_THM_main}. We bound the square loss based on the above events.
\noindent\textbf{Bound on $\norm{S_{k-1}\bar{V}^{-1/2}_{k-1}}^2_2$. }
From the definition of $\mathcal{E}_{\mathrm{unif}}:$
\[
\sup_{T\le k\le 2T-1} \norm{S_{k-1}\bar{V}^{-1/2}_{k-1}}^2_2\le g^2(k,p,\delta/k^2)=\mathrm{poly}(n,\beta,\log 1/\delta)\tilde{O}(1)
\]
\noindent \textbf{Bound on $\norm{\lambda G\bar{V}^{-1/2}_t}^2_2$.}
We simply have: $\norm{\lambda G\bar{V}^{-1/2}_t}^2_2\le \lambda \norm{G}^2_2$
\noindent\textbf{Bound on $\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2$.}
Notice that
\[\norm{\bar{X}_{t-1}}^2_2\le 2T\norm{\Gamma_{2T-1}}_2\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{k}\hat{x}_k}^2_2.\]
Based on $\mathcal{E}_x:$
\[
\norm{C(A-KC)^p}^2_2\norm{\bar{X}_{t-1}}^2_2\le \mathrm{poly}(n,\log 1/\delta_1)\tilde{O}(\rho(A-KC)^{2p}T)
\]
\noindent\textbf{Bound on $\norm{C(A-KC)^p}^2_{2}\sum_{k=T}^{2T-1} \norm{\hat{x}_{k-p}}^2_2$.}
It is similar to the previous step:
\[
\sum_{k=T}^{2T-1} \norm{\hat{x}_{k-p}}^2_2\le T \norm{\Gamma_{2T-1}}_2\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{k}\hat{x}_k}^2_2
\]
\noindent\textbf{Bound on the sum of $\norm{\bar{V}^{-1/2}_{k-1} Z_k}^2_2$.}
We have:
\begin{align*}
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}Z_k}^2_2&\le \paren{\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}\Gamma^{1/2}_{Z,k}}^2_2}\sup_{k\le 2T-1}\norm{\Gamma^{-1/2}_{Z,k}Z_k}^2_2
\end{align*}
There are two cases:
\begin{align*}
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}\Gamma^{1/2}_{Z,k}}^2_2&\le 32 \frac{2T-p-1}{T-p-1},\text{ if }T\ge N_0\\
\sum_{k=T}^{2T-1}\norm{\bar{V}^{-1/2}_{k-1}\Gamma^{1/2}_{Z,k}}^2_2&\le \frac{T}{\lambda} \norm{\Gamma_{Z,2T-1}}_2,\text{ if }T< N_0.
\end{align*}
Meanwhile, we upper-bound $\norm{\Gamma^{-1/2}_{Z,k}Z_k}^2_2$ based on $\mathcal{E}_z$.
\noindent\textbf{Final bound.} It follows from~\eqref{EPOCH_EQN_Square_Loss_Terms} and the above bounds.
\end{proof}
To prove Theorem~\ref{STABLE_THM_Main_Bound}, we now follow the same steps as in the proof of Theorem~\ref{OUT_THM_Main_Bound}, which are omitted here. It is sufficient to select
\begin{equation}\label{STABLE_EQN_beta_choice}
\beta\ge \frac{1}{\log 1/\rho(A-KC)}
\end{equation}
The final result is the following: with probability at least $1-(8\frac{\pi^2}{6}+3)\delta$:
\[
\Reg_N=\tilde{O}(T_{\text{init}})+\mathrm{poly}(n,\beta,\log 1/\delta)\paren{\tilde{O}(N_0)+\tilde{O}(1)}
\]
\hfill $\square$
\section{Alternative regret definition}\label{APP_Section_ALT}
In this section, we sketch how the online learning definition~\eqref{EXT_Online_Regret} of regret, i.e. the best linear predictor if we knew all $N$ data beforehand, is equivalent to our definition~\eqref{FOR_EQN_Regret_Output}.
\begin{lemma}
Consider system~\eqref{FOR_EQN_System_Innovation} with $\rho(A)\leq 1$.
Let $y_0,\dots,y_N$ be sequence of system observations with $\hat{y}_0,\dots,\hat{y}_N$ being the respective Kalman filter predictions. Fix a failure probability $\delta>0$. There exists a $N_0=\mathrm{poly}\paren{\log1/\delta}$ such that with probability at least $1-\delta$, if $N> N_0$ then:
\begin{equation}\label{ALT_Result}
\sum_{k=0}^{N}\norm{y_k-\hat{y}_k}^2_2-\inf_{\mathcal{G}}\sum^{N}_{k=0}\norm{y_k-\sum^{k}_{t=1}g_{t}y_{k-t}}^2\le \mathrm{poly}(\log 1/\delta)\tilde{O}(1).
\end{equation}
\end{lemma}
\begin{proof}
We only sketch the proof here.
We have
\begin{align}
&\inf_{\mathcal{G}}\sum^{N}_{k=0}\norm{y_k-\sum^{k}_{t=1}g_{t}y_{k-t}}^2\\
&=\inf_{\mathcal{G}}\sum_{k=0}^{N} \paren{\norm{y_k-\sum_{t=1}^{p}g_ty_{k-t}}^2+\norm{\sum_{t=p+1}^{k}g_ty_{k-t}}^2-2(y_k-\sum_{t=1}^{p}g_ty_{k-t})^*\sum_{t=p+1}^{k}g_ty_{k-t}}.
\end{align}
If we bound the magnitudes of $y_{k}$ based on Lemma~\eqref{SUP_LEM_Supremum}, then with probability at least $1-\delta$
\begin{align}
\inf_{\mathcal{G}}\sum^{N}_{k=0}\norm{y_k-\sum^{k}_{t=1}g_{t}y_{k-t}}^2&\ge \min_{g_1,\dots,g_p}\sum_{k=0}^{N} \paren{\norm{y_k-\sum_{t=1}^{p}g_ty_{k-t}}^2}-\tilde{O}(\rho^{p}\mathrm{poly}(N,\log 1/\delta)),
\end{align}
where the minimum is over all possible values for $g_i$.
Let
\[
Z_k=\matr{{ccc}y^*_{k-p}&\cdots&y^*_{k-1}}, \text{ for }k=0,\dots, N
\]
with $y_{t}=0$ if $t<0$.
Define
\begin{align*}
E^{+}&\triangleq \matr{{ccc}e_0&\cdots&e_{N}},\\
Y^{+}&\triangleq \matr{{ccc}y_0&\cdots&y_{N}},\\
Z&\triangleq \matr{{ccc}Z_0&\cdots&Z_{N}},\\
\end{align*}
Let $\tilde{G}=\matr{{ccc}\tilde{g}_1&\cdots&\tilde{g}_p}$ be the solution of $\min_{g_1,\dots,g_p}\sum_{k=0}^{N} \paren{\norm{y_k-\sum_{t=1}^{p}g_ty_{k-t}}^2}$. We can show that:
\[
\tilde{G}=YZ^*(ZZ^*)^{-1}.
\]
Hence,
\begin{equation}\label{ALT_EQN_AUX_1}
\min_{g_1,\dots,g_p}\sum_{k=0}^{N} \paren{\norm{y_k-\sum_{t=1}^{p}g_ty_{k-t}}^2}=\norm{Y-\tilde{G}Z}^2_F=\norm{Y-YZ^*(ZZ^*)^{-1}Z}^2_F,
\end{equation}
where $\norm{\cdot}_F$ denotes the Frobenius norm.
But we have:
\[
Y=GZ+E^++O(\rho^p\mathrm{poly}(N))
\]
Replacing $Y$ in~\eqref{ALT_EQN_AUX_1}, we obtain:
\[
\min_{g_1,\dots,g_p}\sum_{k=0}^{N} \paren{\norm{y_k-\sum_{t=1}^{p}g_ty_{k-t}}^2}=\norm{E}^2_F-\norm{EZ^*(ZZ^*)^{-1/2}}^2_F-\tilde{O}(\rho^{p}\mathrm{poly}(N,\log 1/\delta))
\]
But the term $\norm{E}^2_F=\sum_{k=0}^{N}\norm{y_k-\hat{y}_k}^2_2$ is the Kalman filter prediction error. Using Theorem~\ref{FIXED_THM_Identification}, we can show that there exists a $N_0$ such that $ZZ^*\succeq (k-p+1)\sigma_R I$ with high probability if $N\ge N_0$. Meanwhile, using Theorem~\ref{MART_THM_Vector} combined with persistency of excitation, we can show that with probability at least $1-\delta$:
\[
\norm{EZ^*(ZZ^*)^{-1/2}}^2_F=\mathrm{poly}(\log 1/\delta)\tilde{O}(1)
\]
Combining all results:
\[
\inf_{\mathcal{G}}\sum^{N}_{k=0}\norm{y_k-\sum^{k}_{t=1}g_{t}y_{k-t}}^2\ge \sum_{k=0}^{N}\norm{y_k-\hat{y}_k}^2_2-\mathrm{poly}(\log 1/\delta)\tilde{O}(\rho^{p}\mathrm{poly}(N))
\]
Choosing $p=c\log N$ for sufficiently large $c$ gives the result.
\end{proof}
\section{Technical lemmas}\label{APP_Section_Log}
\begin{lemma}\label{LogIn_Lem_Log}
Let $c>0$ be a positive constant. Consider the inequality:
\[
k\ge c \log k
\]
Then, a sufficient condition for the above inequality to hold is:
\[
k\ge \max\set{2c\log 2c,1}
\]
\end{lemma}
\begin{proof}
If $c\le e$, then the inequality is satisfied for all $k>0$. To see why this holds consider $f(k)=k-e\log k$. The minimum is attained at $f(e)=e-e\log e= 0$. Hence, $k\ge e \log k\ge c\log k$.
Next, we analyze the case $c>e$.
We have that the function $k-c\log k$ is increasing for $k\ge c$. Moreover, $2c\log 2c \ge c$. As a result if $k\ge 2c\log 2c$ then also:
\[
k-c\log k\ge 2c\log 2c -c\log(2c\log 2c)=c\log 2c-c\log\log 2c\ge c\log 2c-\frac{c}{e}\log 2c\ge 0
\]
where we used Lemma~\ref{LogIn_Lem_LogLog}.
\end{proof}
\begin{lemma}\label{LogIn_Lem_LogSq}
Let $c$ be a positive constant. Consider the inequality:
\[
k\ge c \log^2 k
\]
Then, a sufficient condition for the above inequality to hold is:
\[
k\ge \max\set{4c\log^2 4c,4c\log 4c,1}
\]
\end{lemma}
\begin{proof}
If $c\le 1$, then the inequality is satisfied for $k\ge 1$. To see why this holds define $f(k)=k-\log^2{k}$. Its derivative $f'(k)=1-2\frac{\log k}{k}$ is always positive for $k\ge 1$ since from the proof of Lemma~\ref{LogIn_Lem_Log} $k\ge 2\log k$. Hence $f(k)\ge f(1)=1$.
Consider now the case $c>1$ and define $g(k)=k-c\log^2 k$. Its derivative is $g'(k)=1-2c\frac{\log k}{k}$. From Lemma~\ref{LogIn_Lem_Log} $g'(k)\ge 0$, for $k\ge \max\set{4c\log 4c,1}$.
Now, pick $k_1=4c\log^2 4c$ and observe that $k_1\ge 4c\log 4c$ since $4c>e$ and $\log 4c>1$. Since $g$ is increasing for $k\ge k_1$, it is sufficient to prove that $g(k_1)>0$.
We compute:
\[
c\log^2(k_1)=c\paren{\log 4c+\log\log 4c}^2\stackrel{(i)}{\le}c\paren{\log 4c+\frac{1}{e}\log 4c}^2\le 4c\log^2 4c=k_1,
\]
where $(i)$ follows from Lemma~\ref{LogIn_Lem_LogLog} below.
\end{proof}
\begin{lemma}\label{LogIn_Lem_LogLog}
Let $c\ge e$, then the following inequality holds:
\[
\log\log c \le \frac{1}{e}\log c\]
\end{lemma}
\begin{proof}
Consider function $f(c)=\frac{1}{e}\log c-\log\log c$ and compute the derivative:
\[
f'(c)=\frac{1}{ec}-\frac{1}{c\log c}
\]
The minimum is attained at $e^e$. Hence
\[
f(c)\ge f(e^e)=0
\]
for all $c\ge e$.
\end{proof}
| 47,714
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BRCA-MoNet. Each node represents a drug. A group of nodes linked by edges of the same color represent a MoA. The black edge linked two MoAs that show correlated effects.
| 164,953
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\begin{document}
\begin{abstract}
According to a well-known theorem of Cram\'er and Wold,
if $P$ and $Q$ are two Borel probability measures on $\RR^d$
whose projections $P_L,Q_L$ onto each line $L$ in $\RR^d$ satisfy $P_L=Q_L$, then $P=Q$.
Our main result is that, if $P$ and $Q$ are both elliptical distributions,
then, to show that $P=Q$, it suffices merely to check that $P_L=Q_L$ for a certain set of $(d^2+d)/2$ lines $L$.
Moreover $(d^2+d)/2$ is optimal. The class of elliptical distributions contains the Gaussian
distributions as well as many other multivariate distributions of interest.
We use our results to derive a statistical test for equality of elliptical distributions,
and we carry out a small simulation study of the test.
\end{abstract}
\keywords{Cram\'er--Wold; projection; elliptical distribution; Kolmogorov--Smirnov test}
\subjclass[2010]{Primary 60B11, Secondary 60E10, 62H15}
\maketitle
\section{Introduction and statement of main results}
Given a Borel probability measure $P$ on $\RR^d$ and a vector subspace $H$ of $\RR^d$,
we write $P_H$ for the projection of $P$ onto $H$, namely the Borel probability measure on $H$ given by
\[
P_H(B):=P(\pi_H^{-1}(B)),
\]
where $\pi_H:\RR^d\to H$ is the orthogonal projection of $\RR^d$ onto $H$.
We shall be particularly interested in the case when $H$ is a line $L$.
One can view $P_L$ as the marginal distribution of $P$ along $L$.
According to a well-known theorem of Cram\'er and Wold \cite{CW36},
if $P,Q$ are two Borel probability measures on $\RR^d$,
and if $P_L=Q_L$ for all lines $L$, then $P=Q$.
In other words, a probability measure on $\RR^d$ is determined by its complete set of one-dimensional marginal
distributions.
There are several extensions of this theorem,
in which one assumes more about the nature of the measures $P,Q$
and less about the set of lines $L$ for which $P_L=Q_L$.
For example, if $P$ and $Q$ have moment generating functions that are finite in a neighbourhood of the origin,
and if $P_L=Q_L$ for all lines $L$ in a set of positive measure (in a natural sense),
then $P=Q$.
Articles on this subject include those of R\'enyi \cite{Re52},
Gilbert \cite{Gi55}, B\'elisle--Mass\'e--Ransford \cite{BMR97}
and Cuesta-Albertos--Fraiman--Ransford \cite{CFR07}.
If one assumes yet more about $P$ and $Q$, then it is even possible
to differentiate between them using only a finite set of projections.
Heppes \cite{He56} showed that, if $P$ and $Q$ are supported on a finite set of cardinality $k$,
and if $H_1,\dots,H_{k+1}$ are vector subspaces such that $H_i^\perp\cap H_j^\perp=\{0\}$ whenever $i\ne j$,
then $P_{H_j}=Q_{H_j}$ for all~$j$ implies that $P=Q$.
Our goal in this note is to establish an analogue of Heppes' result for a certain family of continuous distributions,
namely the so-called elliptical distributions. Here is the definition.
\begin{definition}\label{D:elliptical}
A Borel measure $P$ on $\RR^d$ is an \emph{elliptical distribution}
on $\RR^d$ if its characteristic function has the form
\begin{equation}\label{E:elliptical}
\phi_P(\xi)=e^{i\mu\cdot\xi}\psi(\xi^T \Sigma \xi)
\quad(\xi\in\RR^d),
\end{equation}
where $\Sigma$ is a real positive semi-definite $d\times d$ matrix,
$\mu$ is a vector in $\RR^d$, and $\psi:[0,\infty)\to\CC$ is a continuous function.
The measure $P$ is said to be \emph{centred} if $\mu=0$.
\end{definition}
If $P$ is an elliptic distribution with finite second moments, then $\mu$ represents the mean and $\Sigma$ the covariance matrix. However, there are elliptic distributions even whose first moments are infinite.
The most important elliptical distributions are surely the Gaussian distributions. They correspond to taking $\psi(t)=e^{-t/2}$ in \eqref{E:elliptical}. However, there are numerous other examples of interest,
including multivariate Student distributions, Cauchy distributions, Bessel distributions, logistic distributions, stable laws, Kotz-type distributions, and multi-uniform distributions.
For background on the general theory of elliptical distributions, we refer to \cite{CHS81} and \cite{FKN90}. The latter reference also contains a more complete list of examples (see \cite[Table~3.1]{FKN90}).
To state our results, we need a further definition.
\begin{definition}
A set $S$ of vectors in $\RR^d$ is a
\emph{symmetric-matrix uniqueness set} (or \emph{sm-uniqueness set} for short)
if the only real symmetric $d\times d$ matrix $A$ satisfying
$x^T Ax=0$ for all $x\in S$ is the zero matrix.
\end{definition}
For example, if $v_1,\dots,v_d$ are linearly independent vectors in $\RR^d$
and if we define $S:=\{v_i+v_j:1\le i\le j\le d\}$, then $S$ is an sm-uniqueness set.
Also, every sm-uniqueness set must contain at least $(d^2+d)/2$ elements,
so the example $S$ above is minimal. We shall justify these statements in
\S\ref{S:uniqueness} below, where sm-uniqueness sets will be discussed in more detail.
We can now state our first main theorem.
Given $x\in\RR^d\setminus\{0\}$, we denote by $\langle x\rangle$
the one-dimensional subspace spanned by $x$.
\begin{theorem}\label{T:CWelliptic}
Let $S\subset\RR^d$ be a symmetric-matrix uniqueness set.
If $P,Q$ are elliptical measures on $\RR^d$ such that $P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x\in S$, then $P=Q$.
\end{theorem}
The following corollary is worthy of note. It follows from the fact that, in $\RR^2$, any set of three vectors, none of which is a multiple of the others, forms an sm-uniqueness set.
\begin{corollary}\label{C:CWelliptic}
An elliptical distribution $\RR^2$ is determined by its marginals along any three distinct lines.
\end{corollary}
\begin{remarks}
(1) Perhaps the main interest of the theorem lies the fact that,
to differentiate between two elliptical distributions,
only finitely many projections are needed.
In that sense, the theorem is an analogue of Heppes' theorem, mentioned earlier. However, there are also important differences between the two results. Obviously one result treats discrete measures and the other treats continuous ones. Also, in Heppes' theorem, the projections are onto general subspaces $H$, whereas in Theorem~\ref{T:CWelliptic} the projections are onto lines $L$.
Notice that, by the original Cram\'er--Wold theorem, $P_H=Q_H$ if and only if $P_L=Q_L$ for all lines $L\subset H$. Thus Theorem~\ref{T:CWelliptic} is `better' in some sense.
(2) Among the numerous variants of the Cram\'er--Wold theorem, Theorem~\ref{T:CWelliptic} is unusual (maybe even unique) in that no assumption is made about the finiteness of moments of $P$ and $Q$. As mentioned earlier,
there are elliptical distributions whose first moments are infinite, for example the multivariate Cauchy distributions.
(3) In Theorem~\ref{T:CWelliptic}, it is important to assume that \emph{both} $P$ and $Q$ are elliptical distributions. If we suppose merely that one of them is elliptical, then the result no longer holds.
(In the terminology of \cite{BMR97}, Theorem~\ref{T:CWelliptic} is not a strong-determination result.)
Indeed, given any finite set $\cH$
of $(d-1)$-dimensional subspaces of $\RR^d$, there exist probability measures $P,Q$ on $\RR^d$ with $P$ Gaussian and $P_H=Q_H$ for all $H\in\cH$, but $P\ne Q$. Such examples were constructed by
Hamedani and Tata \cite{HT75} in the case $d=2$, and by
Manjunath and Parthasarathy \cite{MP12} for general~$d$.
\end{remarks}
We now turn to the question of sharpness.
The following result shows that Theorem~\ref{T:CWelliptic} is optimal in a certain sense.
In particular, it explains why symmetric-matrix uniqueness sets enter the picture.
We say that a Borel probability measure $P$ on $\RR^d$ is \emph{non-degenerate}
if it is not supported in any hyperplane in $\RR^d$.
\begin{theorem}\label{T:CWellipticsharp}
Let $P$ be a non-degenerate elliptical distribution on $\RR^d$.
Let $S\subset\RR^d$, and suppose that $S$ is not a symmetric-matrix uniqueness set.
Then there exists a non-degenerate elliptical distribution $Q$ on $\RR^d$
such that $P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x\in S$, but $P\ne Q$.
\end{theorem}
As remarked earlier, a symmetric-matrix uniqueness set in $\RR^d$ must contain at least $(d^2+d)/2$ elements.
We thus obtain the following corollary.
\begin{corollary}\label{C:CWellipticsharp}
Let $P$ be a non-degenerate elliptical distribution on $\RR^d$.
Let $\cL$ be a set of lines in $\RR^d$ containing strictly fewer than $(d^2+d)/2$ lines.
Then there exists a non-degenerate elliptical distribution $Q$ on $\RR^d$
such that $P_{L}=Q_{L}$ for all $L\in\cL$, but $P\ne Q$.
\end{corollary}
The rest of the paper is organized as follows.
In \S\ref{S:uniqueness} we discuss in more detail the notion of symmetric-matrix uniqueness sets.
The proofs of our two main results, Theorems~\ref{T:CWelliptic} and \ref{T:CWellipticsharp}
are presented in \S\ref{S:proof1} and \S\ref{S:proof2} respectively.
In \S\ref{S:statistics} we use Theorem~\ref{T:CWelliptic} to derive a statistical test for equality of
elliptical distributions, and we carry out a small simulation study of the test.
Finally, in \S\ref{S:conclusion} we make some concluding remarks.
\section{Symmetric-matrix uniqueness sets}\label{S:uniqueness}
Recall that a set $S$ of vectors in $\RR^d$ is a
\emph{symmetric-matrix uniqueness set} or \emph{sm-uniqueness set}
if the only real symmetric $d\times d$ matrix $A$ satisfying
$x^T Ax=0$ for all $x\in S$ is the zero matrix.
We now examine these sets in more detail,
beginning with the following result, which furnishes some
simple examples.
\begin{proposition}\label{P:uniquenessexample}
Let $v_1,\dots,v_d$ be linearly independent vectors in $\RR^d$, and let
\[
S:=\{v_j+v_k: 1\le j\le k\le d\}.
\]
Then $S$ is an sm-uniqueness set.
\end{proposition}
\begin{proof}
If $A$ is any symmetric matrix, then, for all $j,k$, we have
\[
2v_j^T A v_k=(v_j+v_k)^TA(v_j+v_k)-\frac{1}{4}(v_j+v_j)^TA(v_j+v_j)-
\frac{1}{4}(v_k+v_k)^TA(v_k+v_k).
\]
Hence, if $x^TAx=0$ for all $x\in S$, then $v_j^TAv_k=0$ for all $j,k$,
and so $A=0$.
\end{proof}
\begin{corollary}
If $v_1,v_2,v_3\in\RR^2$ and no $v_j$ is a multiple of any other, then $\{v_1,v_2,v_3\}$ is an
sm-uniqueness set.
\end{corollary}
\begin{proof}
Clearly $v_1,v_2$ are linearly independent. Also, replacing them by suitable non-zero multiples of themselves,
we can suppose that $v_3=v_1+v_2$. The result therefore follows from Proposition~\ref{P:uniquenessexample}.
\end{proof}
The set $S$ in Proposition~\ref{P:uniquenessexample}
contains $(d^2+d)/2$ elements.
The next result shows that this number is minimal.
\begin{proposition}
An sm-uniqueness set in $\RR^d$ must contain at least $(d^2+d)/2$ elements.
\end{proposition}
\begin{proof}
Let $v_1,\dots,v_m\in\RR^d$
with $m<(d^2+d)/2$. The vector space $V$ of all $d\times d$ symmetric matrices has dimension $(d^2+d)/2$, so the linear map $A\mapsto (v_1^TAv_1,\dots,v_m^TAv_m):V\to\RR^m$
must have a non-zero kernel. Thus there exists $A\ne0$
such that $v_j^TAv_j=0$ for $j=1,\dots,m$. Hence $\{v_1,\dots,v_m\}$ is not an sm-uniqueness set.
\end{proof}
\begin{proposition}\label{P:span}
If $S$ is an sm-uniqueness set in $\RR^d$, then $S$ spans $\RR^d$.
\end{proposition}
\begin{proof}
Suppose, on the contrary, that $S$ does not span $\RR^d$.
Then we can find a unit vector $v\in S^\perp$.
Let $R$ be a $d\times d$ rotation matrix such that $Rv=e_1$,
and let $A$ be the $d\times d$ symmetric matrix given by
$A:=R^TDR$, where $D:=\diag(1,0,\dots,0)$.
If $x\in S$, then $x\perp v$, so $Rx\perp Rv=e_1$, so
$Rx\in\spn\{e_2,\dots,e_d\}$ and hence $(Rx)^TD(Rx)=0$.
In other words, $x^TAx=0$ for all $x\in S$.
On the other hand, $v^TAv=(Rv)^TDRv=e_1^TDe_1=1$, so $A\ne0$.
We conclude that $S$ is not an sm-uniqueness set.
\end{proof}
The next result is not really needed in what follows,
but it provides a necessary and sufficient condition for a
set of vectors in $\RR^d$ to form an sm-uniqueness set.
\begin{proposition}\label{P:criterion}
Given a vector $x\in\RR^d$, say $x=(x_1,\dots,x_d)$,
let $\hat{x}$ be the upper-triangular $d\times d$ matrix with entries
$\hat{x}_{ij}:=x_ix_j~( 1\le i\le j\le d)$.
A set $S\subset\RR^d$ is an sm-uniqueness
set if and only if
$\{\hat{x}:x\in S\}$
spans the space of upper-triangular $d\times d$ matrices.
\end{proposition}
\begin{proof}[Sketch of Proof]
Denote by $\sym^2(\RR^d)$ the symmetric tensor product of $\RR^d$ with itself.
For each symmetric $d\times d$ matrix $A$,
there is a unique linear functional $\tilde{A}:\sym^2(\RR^d)\to\RR$ such that $\tilde{A}(x\otimes x)=x^TAx$ for all $x\in\RR^d$.
Thus a set $S\subset \RR^d$ is an sm-uniqueness
set iff $\{x\otimes x:x\in S\}$
spans $\sym^2(\RR^d)$.
Let $e_1,\dots,e_d$ be the standard unit basis of $\RR^d$.
Then $\sym^2(\RR^d)$ has a basis $\{E_{ij}:1\le i\le j\le d\}$,
where $E_{ij}:=(e_i\otimes e_j+e_j\otimes e_i)/2$.
Expressing $x\otimes x$ in terms of this basis, we have
\[
x\otimes x=\sum_{1\le i\le j\le d}x_ix_j E_{ij}.
\]
The result follows easily from this.
\end{proof}
Since the space of upper-triangular $d\times d$ matrices has dimension $(d^2+d)/2$,
we deduce the following corollary.
\begin{corollary}\label{C:criterion}
Let $S$ be a set of $(d^2+d)/2$ vectors in $\RR^d$.
For each $x\in S$, define $\hat{x}$ as in
Proposition~\ref{P:criterion}.
Then $S$ is an sm-uniqueness set if and only if
the matrices $\{\hat{x}:x\in S\}$ are linearly independent.
\end{corollary}
If we view an upper-triangular $d\times d$ matrix as a column vector of length $D:=(d^2+d)/2$,
then Corollary~\ref{C:criterion} becomes a criterion expressed in terms of the linear independence of $D$ vectors in $\RR^D$, which can in turn be reformulated as the non-vanishing of a $D\times D$ determinant. This furnishes a
systematic method of determining whether a given set of $(d^2+d)/2$ vectors in $\RR^d$ is an sm-uniqueness set.
\section{Proof of Theorem~\ref{T:CWelliptic}}\label{S:proof1}
We break the proof into a series of lemmas.
The first of these is fairly standard.
\begin{lemma}\label{L:cf}
Let $P,Q$ be Borel probability measures on $\RR^d$
and let $L$ be a line in $\RR^d$. Then $P_{L}=Q_{L}$
if and only if the characteristic functions of $P,Q$ satisfy
$\phi_P(\xi)=\phi_Q(\xi)$ for all $\xi\in L$.
\end{lemma}
\begin{proof}
Let $\xi\in L$. Then $x\cdot\xi=\pi_L(x)\cdot\xi$ for all $x\in\RR^d$.
Consequently,
\[
\phi_{P}(\xi)=\int_{\RR^d}e^{ix\cdot\xi}\,dP(x)
=\int_{\RR^d}e^{i\pi_L(x)\cdot\xi}\,dP(x)
=\int_{L}e^{i y\cdot\xi}\,dP_L(y),
\]
and similarly for $Q$. Hence, if $P_L=Q_L$,
then $\phi_{P}(\xi)=\phi_{Q}(\xi)$.
Conversely, if $\phi_P(\xi)=\phi_Q(\xi)$ for all $\xi\in L$,
then the calculation above shows that $P_L$ and $Q_L$ have the
same characteristic function, and consequently $P_L=Q_L$.
\end{proof}
\begin{lemma}\label{L:mu1=mu2}
Let $P,Q$ be elliptical distributions, with characteristic functions
\[
\phi_P(\xi)=e^{i\mu_1\cdot\xi}\psi_1(\xi^T\Sigma_1\xi)
\quad\text{and}\quad
\phi_Q(\xi)=e^{i\mu_2\cdot\xi}\psi_2(\xi^T\Sigma_2\xi)
\quad(\xi\in\RR^d).
\]
If $P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x$ in some spanning set of $\RR^d$,
then $\mu_1=\mu_2$.
\end{lemma}
\begin{proof}
By Lemma~\ref{L:cf}, if $P_{\langle x\rangle}=Q_{\langle x\rangle}$,
then $\phi_P=\phi_Q$ on $\langle x\rangle$,
in other words $\phi_P(tx)=\phi_Q(tx)$ for all $t\in\RR$.
Recalling the form of $\phi_P,\phi_Q$, we obtain
\begin{equation}\label{E:basiceqn}
e^{it\mu_1\cdot x}\psi_1(t^2x^T\Sigma_1 x)=
e^{it\mu_2\cdot x}\psi_2(t^2 x^T\Sigma_2x)
\qquad(t\in\RR).
\end{equation}
Since $\phi_P,\phi_Q$ are characteristic functions,
they are both equal to $1$ at the origin.
It follows that $\psi_1(0)=\psi_2(0)=1$.
As $\psi_1,\psi_2$ are continuous, there exists $\delta>0$
such that both $\psi_1(t^2x^T\Sigma_1x)\ne0$ and $\psi_2(t^2x^T\Sigma_2x)\ne0$ for all $t\in[-\delta,\delta]$.
For each such $t$, we may divide equation \eqref{E:basiceqn}
by the same equation in which $t$ is replaced by $-t$. This gives
\[
e^{2it\mu_1\cdot x}=e^{2it\mu_2\cdot x} \quad(t\in[-\delta,\delta]).
\]
Differentiating with respect to $t$ and setting $t=0$,
we obtain $\mu_1\cdot x=\mu_2\cdot x$.
Finally, if this holds for all $x$ in a spanning set of $\RR^d$,
then $\mu_1=\mu_2$.
\end{proof}
\begin{lemma}\label{L:samepsi}
Let $P,Q$ be centred elliptical distributions on $\RR^d$
such that $P_{\langle x\rangle}=Q_{\langle x\rangle}$
for all $x$ in some spanning set of $\RR^d$.
Then there exists a continuous function $\psi:[0,\infty)\to\CC$
and positive semi-definite $d\times d$ matrices
$\Sigma_1,\Sigma_2$ such that
\begin{equation}\label{E:samepsi}
\phi_P(\xi)=\psi(\xi^T\Sigma_1\xi)
\quad\text{and}\quad
\phi_Q(\xi)=\psi(\xi^T\Sigma_2\xi)
\quad(\xi\in\RR^d).
\end{equation}
\end{lemma}
\begin{proof}
By assumption, there exist continuous functions
$\psi_1,\psi_2:[0,\infty)\to\CC$ and positive semi-definite
$d\times d$ matrices $\Sigma_1',\Sigma_2'$ such that
\[
\phi_P(\xi)=\psi_1(\xi^T\Sigma_1'\xi)
\quad\text{and}\quad
\phi_Q(\xi)=\psi_2(\xi^T\Sigma_2'\xi)
\quad(\xi\in\RR^d).
\]
The point of the lemma is to show that, adjusting $\Sigma_1',\Sigma_2'$ if necessary, we may take $\psi_1=\psi_2$.
If $\Sigma_1'=\Sigma_2'=0$, then we may as well take $\psi_1=\psi_2\equiv1$.
Otherwise, at least one of $\Sigma_1',
\Sigma_2'$ is non-zero, say $\Sigma_1'\ne0$.
Since $\Sigma_1'$ is positive semi-definite, any spanning
set of $\RR^d$ contains a vector $x$ such that $x^T\Sigma_1'x>0$.
Hence, there exists $x_0\in\RR^d$ such that both $P_{\langle x_0\rangle}=Q_{\langle x_0\rangle}$ and $x_0^T\Sigma_1'x_0>0$.
By Lemma~\ref{L:cf},
\[
\psi_1(t^2x_0^T\Sigma_1' x_0)= \psi_2(t^2 x_0^T\Sigma_2'x_0)
\qquad(t\in\RR).
\]
It follows that
\[
\psi_1(s)=\psi_2(as) \quad(s\ge0),
\]
where $a:=(x_0^T\Sigma_2'x_0)/(x_0^T\Sigma_1'x_0)\ge0$.
Feeding this information into the formula for $\phi_P$, we obtain
\[
\phi_P(\xi)=\psi_2(\xi^T(a\Sigma_1')\xi)
\quad(\xi\in\RR^d).
\]
Thus \eqref{E:samepsi} holds with $\psi:=\psi_2$ and $\Sigma_1:=a\Sigma_1'$ and $\Sigma_2:=\Sigma_2'$.
\end{proof}
\begin{lemma}\label{L:end}
Let $P,Q$ be probability measures on $\RR^d$ with characteristic functions given by \eqref{E:samepsi}.
Let $S$ be an sm-uniqueness set,
and suppose that $P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x\in S$. Then $P=Q$.
\end{lemma}
\begin{proof}
We show that either $\Sigma_1=\Sigma_2$ or $\psi$ is constant.
Either way, this gives that $\phi_P=\phi_Q$ and hence that $P=Q$.
Suppose then that $\Sigma_1\ne\Sigma_2$.
As $S$ is an sm-uniqueness set,
there exists $x_0\in S$ such that
$x_0^T(\Sigma_1-\Sigma_2)x_0\ne0$.
Exchanging the roles of $P,Q$ if necessary, we can suppose that
\[
x_0^T\Sigma_1x_0>x_0^T\Sigma_2 x_0\ge0.
\]
Since $P_{\langle x_0\rangle}=Q_{\langle x_0\rangle}$, Lemma~\ref{L:cf} gives
\[
\psi(t^2x_0^T\Sigma_1 x_0)=\psi(t^2x_0^T\Sigma_2 x_0)
\quad(t\in\RR).
\]
It follows that
\[
\psi(s)=\psi(as) \quad(s\ge0),
\]
where $a:=(x_0^T\Sigma_2x_0)/(x_0^T\Sigma_1 x_0)\in[0,1)$.
Iterating this equation, and using the continuity of $\psi$, we obtain
\[
\psi(s)=\psi(as)=\psi(a^2s)=\cdots=\psi(a^ns)\underset{n\to\infty}\longrightarrow \psi(0) \quad(s\ge0).
\]
Therefore $\psi$ is constant, as claimed.
\end{proof}
\begin{proof}[Completion of proof of Theorem~\ref{T:CWelliptic}]
Let $P,Q$ be elliptical distributions on $\RR^d$, with characteristic
functions
\[
\phi_P(\xi)=e^{i\mu_1\cdot\xi}\psi_1(\xi^T\Sigma_1\xi)
\quad\text{and}\quad
\phi_Q(\xi)=e^{i\mu_2\cdot\xi}\psi_2(\xi^T\Sigma_2\xi).
\]
Suppose that
$P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x\in S$,
where $S\subset\RR^d$ is an sm-uniqueness set.
Our goal is to prove that $P=Q$.
By Proposition~\ref{P:span}, $S$ is a spanning set for $\RR^d$.
By Lemma~\ref{L:mu1=mu2}, it follows that $\mu_1=\mu_2$. Translating both $P,Q$ by $-\mu_1$,
we may suppose that both measures are centred.
Lemma~\ref{L:samepsi} then shows that $P,Q$ have characteristic functions given by \eqref{E:samepsi},
and finally Lemma~\ref{L:end} gives $P=Q$.
\end{proof}
\section{Proof of Theorem~\ref{T:CWellipticsharp}}\label{S:proof2}
Once again, we break the proof up into lemmas.
The first lemma characterizes those elliptical distributions on $\RR^d$
that are non-degenerate
(i.e., not supported on any hyperplane in $\RR^d$).
\begin{lemma}\label{L:nondegen}
Let $P$ be an elliptical distribution on $\RR^d$ with characteristic
function
\[
\phi_P(\xi)=e^{i\mu\cdot\xi}\psi(\xi^T\Sigma\xi)
\quad(\xi\in\RR^d).
\]
Then $P$ is non-degenerate if and only if $\psi$ is non-constant and $\Sigma$ is strictly positive definite.
\end{lemma}
\begin{proof}
Suppose that $P$ is supported on the hyperplane
$H:=\{x\in\RR^d: y\cdot x=c\}$,
where $y\in\RR^d\setminus\{0\}$ and $c\in\RR$.
Then $P_{\langle y\rangle} =\delta_{x_0}$, where $x_0$ is the unique point in $\langle y\rangle \cap H$. By Lemma~\ref{L:cf},
\[
\phi_P(ty)=e^{ity\cdot x_0} \quad(t\in\RR).
\]
Recalling the form of $\phi_P$, we deduce that
\[
e^{it\mu\cdot y}\psi(t^2 y^T\Sigma y)=e^{ity\cdot x_0} \quad(t\in\RR).
\]
The argument used to prove Lemma~\ref{L:mu1=mu2} shows that
$\mu\cdot y=x_0\cdot y$, and hence that
\[
\psi(t^2 y^T\Sigma y)=1 \quad(t\in\RR).
\]
This in turn implies that either $\psi\equiv 1$ or $y^T\Sigma y=0$.
Thus either $\psi$ is constant or $\Sigma$ is not strictly positive definite.
The converse is proved by running the same argument backwards.
\end{proof}
\begin{lemma}\label{L:Qexists}
Let $P$ be a non-degenerate elliptical distribution on $\RR^d$,
with characteristic function
\[
\phi_P(\xi)=e^{i\mu_1.\xi}\psi(\xi^T\Sigma_1 \xi)
\quad(\xi\in\RR^d).
\]
Then, for each vector $\mu_2\in\RR^d$ and each positive semi-definite $d\times d$ matrix $\Sigma_2$, there exists an elliptical distribution $Q$ on $\RR^d$ with characteristic function
\[
\phi_Q(\xi)=e^{i\mu_2.\xi}\psi(\xi^T\Sigma_2 \xi)
\quad(\xi\in\RR^d).
\]
\end{lemma}
\begin{proof}
Since $\Sigma_1,\Sigma_2$ are positive semi-definite, we can write them as $\Sigma_1=S_1^TS_1$ and $\Sigma_2=S_2^TS_2$,
where $S_1,S_2$ are $d\times d$ matrices. Further, as $P$ is non-degenerate, Lemma~\ref{L:nondegen} implies that $\Sigma_1$ is strictly positive definite, and so $S_1$ is invertible. Let $A:\RR^d\to\RR^d$ be the affine map defined by
\[
Ax:=S_1^{-1}S_2(x-\mu_1)+\mu_2 \quad(x\in\RR^d),
\]
and set $Q:=PA^{-1}$. We shall show that $\phi_Q$ has the required form.
First of all, we remark that, if $\tilde{P}$ and $\tilde{Q}$ denote the translates of $P,Q$ by $-\mu_1$ and $-\mu_2$ respectively, then
\[
\phi_{\tilde{P}}(\xi)=e^{-i\mu_1\cdot\xi}\phi_P(\xi)
\quad\text{and}\quad
\phi_{\tilde{Q}}(\xi)=e^{-i\mu_2\cdot\xi}\phi_Q(\xi)
\quad(\xi\in\RR^d).
\]
Further, $\tilde{Q}=\tilde{P}T^{-1}$, where $T:\RR^d\to\RR^d$ is the linear map given by
\[
Tx:=S_1^{-1}S_2x \quad(x\in\RR^d).
\]
By a calculation similar to that in the proof of Lemma~\ref{L:cf}, we have
\[
\phi_{\tilde{P}T^{-1}}(\xi)=\phi_{\tilde{P}}(T\xi) \quad(\xi\in\RR^d).
\]
Putting all of this together, we get
\[
e^{-i\mu_2\cdot\xi}\phi_Q(\xi)
=\phi_{\tilde{Q}}(\xi)
=\phi_{\tilde{P}T^{-1}}(\xi)
=\phi_{\tilde{P}}(T\xi)
=e^{-i\mu_1\cdot\xi}\phi_P(T\xi)
=e^{-i\mu_1\cdot\xi}\phi_P(S_1^{-1}S_2\xi).
\]
Recalling the form of $\phi_P$ and the fact that $\Sigma_j=S_j^TS_j$ for $j=1,2$, we deduce that
\[
e^{-i\mu_2\cdot\xi}\phi_Q(\xi)=e^{-i\mu_1\cdot\xi}e^{i\mu_1\cdot\xi}\psi(\xi^TS_2^TS_1^{-T}\Sigma_1S_1^{-1}S_2\xi)
=\psi(\xi^TS_2^TS_2\xi)=\psi(\xi^T\Sigma_2\xi).
\]
Thus $\phi_Q$ does indeed have the required form.
\end{proof}
\begin{proof}[Completion of the proof of Theorem~\ref{T:CWellipticsharp}]
By Lemma~\ref{L:nondegen}, since $P$ is a non-degenerate elliptical distribution, it characteristic function is given by
\[
\phi_P(\xi)=e^{i\mu.\xi}\psi(\xi^T\Sigma_1\xi)\quad(\xi\in\RR^d),
\]
where $\psi$ is non-constant and $\Sigma_1$ is strictly positive definite.
As $S$ is not an sm-uniqueness set, there exists a
symmetric $d\times d$ matrix $A$ such that $x^TAx=0$ for all $x\in S$ but $A\ne0$. If $\epsilon>0$ is small enough, then $\Sigma_1+\epsilon A$ is also strictly positive definite. Fix such an $\epsilon$, and set $\Sigma_2:=\Sigma_1+\epsilon A$. By Lemma~\ref{L:Qexists},
there exists an elliptical distribution $Q$ on $\RR^d$ such that
\[
\phi_Q(\xi)=e^{i\mu.\xi}\psi(\xi^T\Sigma_2\xi)\quad(\xi\in\RR^d).
\]
By Lemma~\ref{L:nondegen} again, $Q$ is non-degenerate, since $\psi$ is non-constant $\Sigma_2$ is positive definite. Also, if $x\in S$, then
\[
x^T\Sigma_2x=x^T\Sigma_1x+x^TAx=x^T\Sigma_1x,
\]
so $\phi_Q(tx)=\phi_P(tx)$ for all $t$.
Hence $P_{\langle x\rangle}=Q_{\langle x\rangle}$ for all $x\in S$.
On the other hand, since $\Sigma_1\ne\Sigma_2$ and $\psi$ is non-constant, the argument of the proof of Lemma~\ref{L:end} shows that $P\ne Q$.
\end{proof}
\section{A statistical application}\label{S:statistics}
\subsection{A Kolmogorov--Smirnov test for elliptical distributions}
Several of the variants of the Cram\'er--Wold theorem mentioned in the introduction are useful in deriving
statistical tests for the equality of multivariate distributions (see e.g.\ \cite{CFR06, CF09, FMR22a, FMR22b}).
Theorem~\ref{T:CWelliptic} is no exception.
In this section, we propose a test for the one- and two-sample problems for elliptical distributions,
based on Theorem~\ref{T:CWelliptic}.
The problem of goodness-of-fit testing for elliptic distributions is discussed in several recent articles (see e.g.\ \cite{CJMZ22, DL20, HMS21}), but always with more restrictive assumptions than ours (supposing, for example, that the generator function $\psi$ is known). We also remark that the idea of using projections of elliptical distributions is mentioned in \cite{No13}, though that article addresses a different problem.
Since the one- and two-sample problems are very similar, we describe just the two-sample problem.
Given two samples $\aleph_n:=\{X_1, \ldots, X_n\}\subset \mathbb R^d$
and $\gimel_m:=\{Y_1, \ldots, Y_m\}\subset \mathbb R^d$ of multivariate elliptical distributions $P$ and $Q$,
we consider the testing problem
\[
\mathbf {H0}: P = Q \quad\text{vs}\quad P \neq Q,
\]
based on the samples $\aleph_n, \gimel_m$.
Let $\{v_1, \dots v_D\}\subset\RR^d$ be a fixed symmetric-matrix uniqueness set, where $D=(d^2 +d)/2$.
Let $F_{n,\langle v_j\rangle}$ be the empirical distribution of $P_{\langle v_j\rangle}$ and
$G_{m,\langle v_j\rangle}$ be the empirical distribution of $Q_{\langle v_j\rangle}$, for $j=1, \ldots, D$.
We consider the test statistic
\[
\sqrt{\frac{nm}{n+m}} KS_{n,m,D}
:= \sqrt{\frac{nm}{n+m}} \max_{j=1, \ldots, D} \| F_{n,\langle v_j\rangle}-G_{m,\langle v_j\rangle}\|_{\infty}.
\]
Since the statistic is not distribution-free,
in order to obtain the critical value for a level-$\alpha$ test,
we approximate the distribution using bootstrap on the original samples $\aleph_n, \gimel_m$
by generating a large enough number $B$ of values of $KS_{n,m}$,
for each bootstrap sample
choosing $n,m$ vectors from $\aleph_n \cup \gimel_m$ with replacement.
See for instance \cite{HM88} for the two-sample bootstrap.
We then take as critical value $c^*_{\alpha}$,
the $(1-\alpha)$-quantile of the empirical bootstrap sample, i.e., we reject the null hypothesis when
\[
\sqrt{\frac{nm}{n+m}} KS_{n,m,D} > c^*_{\alpha}.
\]
The validity of the bootstrap in this case follows from \cite[Theorems~3 and~4]{Pr95}.
Therefore the proposed test has asymptotic level $\alpha$.
Also it is consistent since,
under the alternative hypothesis, $P\ne Q$, so, by Theorem~\ref{T:CWelliptic},
$\max_{j=1, \ldots, D} \|F_{n,\langle v_j\rangle}-G_{m,\langle v_j\rangle }\|_{\infty} >0$,
and thus
$\sqrt{\frac{nm}{n+m} } KS_{n,m,D} \to \infty$.
We summarize our conclusions in the following proposition.
\begin{proposition}
\begin{enumerate}
\item The bootstrap version of the test has asymptotic level $\alpha$.
\item Under the alternative,
\[
\sqrt{\frac{nm}{n+m} } KS_{n,m,D} \to \infty,
\]
i.e., the test is consistent.
\end{enumerate}
\end{proposition}
\subsection{A small simulation study}
In this subsection, we study the power of the proposed test for two elliptical distributions
$F_1$ and $F_2$ belonging to the multivariate Student distribution in $\mathbb{R}^d$.
More precisely, we suppose that
\[
F_i \sim t_{\nu_i} (\mu_i, \Sigma_i) \quad (i=1,2),
\]
where $\nu_i \in \mathbb{N}$ are the numbers of degrees of freedom,
$\mu_i \in \mathbb{R}^d$ are the means,
and $\Sigma_i \in \mathbb R^{d \times d}$ are the covariance matrices.
We consider three different scenarios.
In the first scenario, we fix $\mu_i=0$ and $\Sigma_i = \mathbf{I}_{d\times d}$ (the $d\times d$ identity matrix), for $i=1,2$ and we vary only the degrees of freedom:
$\nu_1=1$ and $\nu_2 \in \{1,2,3,4\}$. The power functions are plotted in Figure \ref{Fig:esc1} for different sample sizes.
In the second scenario, we vary only the mean value of one of the distributions,
taking $\nu_i=2$ and $\Sigma_i= \mathbf{I}_{d\times d}$, for $i=1,2$,
while $\mu_1=0$ and $\mu_2 \in [0,1/2]$. See Figure \ref{Fig:esc2}.
Lastly, in the third scenario ,we vary only the covariance matrices. We fix $\nu_i=2$ and $\mu_i=0$ for $i=1,2$,
while $\Sigma_1= \mathbf{I}_{d \times d}$ and $\Sigma_2= \mathbf{I}_{d \times d} + \theta \mathbf{1}_{d \times d}$, where $\mathbf{1}_{d \times d}$ denotes the matrix with value $1$ in all its entries, and $\theta \in [0,1]$.
See Figure \ref{Fig:esc3} .
In all cases we generate iid samples of equal sizes $n= n_1=n_2$, where $n \in \{100,500,1000\}$ and $d \in \{5,10,50\}$ for each scenario. The null distribution is approximated by bootstrap.
For the power function, we report the proportion of rejections in 10000 replicates.
The results are quite encouraging in all the three scenarios considered.
\begin{figure}[htb]
\centering
\subfloat{\includegraphics[width=138mm]{Figures/Power_Gr.pdf}}
\caption{Power function in scenario 1, changing the degrees of freedom and the sample sizes.}
\label{Fig:esc1}
\end{figure}
\begin{figure}[htb]
\centering
\subfloat{\includegraphics[width=138mm]{Figures/Power_mu.pdf}}
\caption{Power function in scenario 2, changing the mean value and the sample sizes.}
\label{Fig:esc2}
\end{figure}
\begin{figure}[htb]
\centering
\subfloat{\includegraphics[width=138mm]{Figures/Power_sd.pdf}}
\caption{Power function in scenario 3, changing the covariance matrix as a function of $\theta$ and the sample sizes.}
\label{Fig:esc3}
\end{figure}
\section{Conclusion}\label{S:conclusion}
In this article we have studied the problem of which sets $\cL$ of lines in $\RR^d$ determine elliptical distributions
in the sense that, if $P,Q$ are elliptical distributions whose projections satisfy $P_L=Q_L$ for all $L\in\cL$,
then $P=Q$.
Combining our two main results, Theorems~\ref{T:CWelliptic} and \ref{T:CWellipticsharp},
we have a precise characterization of such $\cL$.
In particular, there exist such sets $\cL$ of cardinality $(d^2+d)/2$, and this number is best possible.
We have also applied our results to derive a Kolmogorov--Smirnov type test for equality of elliptical distributions,
and we have carried out a small simulation study of this test in the case of multivariate Student distributions,
the results of which are quite encouraging.
Let us conclude with a question. Our main result, Theorem~\ref{T:CWelliptic}, is, in a certain sense, a
continuous analogue of Heppes' result on finitely supported distributions. Recently, the authors
obtained the following quantitative form of Heppes' theorem, expressed in terms of the total variation metric
$d_{TV}(P,Q):=\sup\{|P(B)-Q(B)|: B \text{~Borel}\}$.
\begin{theorem}[\protect{\cite[Theorem~2.1]{FMR22b}}]
Let $Q$ be a probability measure on $\RR^d$
whose support contains at most $k$ points.
Let $H_1,\dots,H_{k+1}$ be subspaces of $\RR^d$ such that
$H_i^\perp\cap H_j^\perp=\{0\}$ whenever $i\ne j$.
Then, for every Borel probability measure $P$ on $\RR^d$, we have
$d_{TV}(P,Q)\le \sum_{j=1}^{k+1} d_{TV}(P_{H_j},Q_{H_j})$.
\end{theorem}
Is there likewise a quantitative version of Theorem~\ref{T:CWelliptic}?
\section*{Declarations}
The authors have no competing interests to declare that are relevant to the content of this article.
\bibliographystyle{amsplain}
\bibliography{biblist}
\end{document}
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\section{Introduction}
The study of patterns in a word is
related to several topics and it has
various applications in several research domains, ranging from practical applications to theoretical considerations.
Among these patterns, we can mention palindromes and repetitions such as squares, cubes, periods, overlaps, etc.
They appear in different contexts in computer science such as data compression (LZW overlap), searching algorithms, structure of indexes \cite{lothaire3}, digital geometry \cite{bgl}, in number theory about Diophantine approximation and transcendence statements~\cite{BorisA}, and in physics in connection with Schr\"odinger operators~\cite{ABCD}, among several others.
Palindromic complexity has received noticeable attention and a comprehensive survey was provided by Allouche et al.~\cite{ABCD}.
In particular, Droubay, Justin and Pirillo established that the number of distinct palindromic factors of a word is bounded by its length plus one and that finite Sturmian (and even episturmian) words realize it~\cite{DrJuPi}.
For the square factors the situation seems to be significantly harder to handle.
Indeed, Fraenkel and Simpson conjectured in~\cite{FraenkelS98} that the number of distinct non-empty squares of a finite word $w$ is bounded by its length $|w|$ and they proved that this number is bounded by $2|w|$.
After that Ilie~\cite{Ilie} strengthened this bound to $2|w|-\Theta(\log(|w|))$; Lam~\cite{lam}
improved this result to $\frac{95}{48}|w|$; Deza, Franek and Thierry~\cite{DezaFT15}
achieved a bound of $\frac{11}{6}|w|$; Thierry~\cite{thie} refined this bound to $\frac{3}{2}|w|$. We remark that the Fraenkel and Simpson's conjecture implies that the square complexity of a finite word is also bounded by its length plus one by counting the empty word. In this article, we prove the following theorem:
\begin{thm}
\label{th:sw}
For any finite word $w$, let $S(w)$ denote the number of its distinct square factors, $|w|$ denote the length of $w$ and $|\Alphabet(w)|$ denote the number of distinct letters in $w$, then we have $$S(w) \leq |w|-|\Alphabet(w)|+1.$$
\end{thm}
This result is a stronger version of the conjecture of Fraenkel and Simpson stated in~\cite{FraenkelS98}.
The strategy of the proof is as follows: for a given word $w$, we first recall the definition of Rauzy graph and define small circuits in it; we then prove that the total number of small circuits in the union of the Rauzy graphs of $w$ is bounded by $|w|-|\Alphabet(w)|$; finally we conclude by proving that there exists an injection from the set of distinct nonempty squares of $w$ to the set of small circuits in the union of the Rauzy graphs of $w$.\\
\section{Preliminaries}
Let us recall the basic terminology about words. By {\em word} we mean a finite concatenation of symbols $w = w_1 w_2 \cdots w_{n}$, with $n \in \N$. The {\em length} of $w$, denoted $|w|$, is $n$ and we say that the symbol $w_i$ is at the {\em position} $i$. The set $\Alphabet(w)=\left\{w_i| 1\leq i \leq n\right\}$ is called the {\em alphabet} of $w$ and its elements are called {\em letters}. Let $|\Alphabet(w)|$ denote the cardinality of $\Alphabet(w)$. A word of length $0$ is called the {\em empty word} and it is denoted by $\varepsilon$. For any word $u$, we have $u=\varepsilon u=u\varepsilon$.
A word $u$ is called a {\em factor} of $w$ if $w = pus$ for some words $p,s$. The set of all factors of $w$ is denoted by $\fac(w)$. For any integer $i$ satisfying $1 \leq i \leq |w|$, let $L_w(i)$ be the set of all length-$i$ factors of $w$ and let $C_w(i)$ denote the cardinality of $L_i(w)$.
Two finite words $u$ and $v$ are {\em conjugate} when there exist words $x,y$ such that $u=xy$ and $v=yx$.
The conjugacy class of a word $w$ is denoted by $[w]$. Let $w=w_1w_2...w_t$ be a finite word, for any integer $i$ satisfying $1 < i \leq t$, let us define $w_{s}(i)=w_iw_{i+1}...w_{t}$ and $w_p(i)=w_1w_2...w_{i-1}$ and let $w_s(1)=w$ and $w_p(1)=\epsilon$.
Thus, $[w]=\left\{w(i)|w(i)=w_s(i)w_p(i), i=1,2,...t\right\}$.
For any natural number $k$, we define the {\em $k$-power} of a finite word $u$ to be the concatenation of $k$ copies of $u$, and it is denoted by $u^k$. Particularly, a \emph{square} is a word $w$ of the form $w=uu$.
A word $w$ is said to be {\em primitive} if it is not a power of another word. Let $\prim(w)$ denote the set of primitive factors of $w$. For any word $u$ and any rational number $\alpha$, the $\alpha$-power of $u$ is defined to be $u^au_0$ where $u_0$ is a prefix of $u$, $a$ is the integer part of $\alpha$, and $|u^au_0|=\alpha |u|$. The $\alpha$-power of $u$ is denoted by $u^{\alpha}$. Let $\alpha$ be a positive rational number larger than $1$, the word $w$ is said to be {\em of the period $\alpha$} if there exists a word $u$ such that $w=u^{\alpha}$.\\
Here we recall some basic lemmas concerning repetitions:\\
\begin{lemma}[Fine and Wilf~\cite{fiwi}]
\label{period}
Let $w$ be a word having $k$ and $l$ for periods. If $|w| \geq k+l-\gcd (k,l)$ then $\gcd(k,l)$ is also a period of $w$.
\end{lemma}
\begin{lemma}[Lyndon and Schützenberger~\cite{lyndon}]
\label{two-words}
Let $x$ and $y$ be two words such that $xy = yx$. Then there exist a primitive $p$ and two non-negative integers $i, j$ such that $x = p^i$ and y = $p^j$.
\end{lemma}
\begin{lemma}[Lyndon and Schützenberger~\cite{lyndon}]
\label{three-words}
Let $x,y,z$ be three words such that $xy = yz$ and $|y| \neq 0$. Then there exist two words $u, v$ with $|u|\neq 0$ and some positive integer $i$ such that
$x = uv, y = (uv)^iu, z = vu$.
\end{lemma}
Here we recall some elementary definitions and proprieties concerning graphs from Berge~\cite{berge}.\\
A {\em (non-oriented) graph} consists of a nonempty set of {\em vertices} $V$ and a set of {\em edges} $E$. A vertex $a$ represents an endpoint of an edge and an edge joins two vertices $a, b$.
A {\em chain} is a sequence of edges $e_1,e_2, \cdots, e_k$, in which each edge $e_i$ ($1<i<k$) has one vertex in common with $e_{i-1}$ and the other vertex in common with $e_{i+1}$. A {\em cycle} is a finite chain which begins with a vertex $x$ and ends at the same vertex. A graph is called {\em connected} if for any couple of vertices $a,b$ in this graph, there exists a chain which begins at $a$ and ends at $b$.
A graph is called {\em oriented} if its edges are oriented from one endpoint to the other. An oriented graph is called weakly connected if it is connected as a non-oriented graph.
Let $G$ be a connected graph and let $\left\{e_1,e_2 \cdots e_l\right\}$, $\left\{v_1,v_2 \cdots v_s\right\}$ denote respectively the edge set and the vertex set of $G$. The number $\chi(G)=l-s+1$ is called the {\em cyclomatic number} of $G$.
Let $C$ be a cycle in $G$. A vector $\mu(C)=(c_1,c_2 \cdots c_l)$ in the $l$-dimensional space $\mathbb{R}^l$ is called the {\em vector-cycle corresponding to $C$} if $c_i$ is the number of visits of the edge $e_i$ in the cycle $C$ for all $i$ satisfying $1 \leq i \leq l$. The cycle $C_1,C_2, \cdots, C_k,...$ are said to be {\em independent} if their corresponding vectors are linearly independent.
\begin{lemma}[Theorem 2, Chapter 4 in~\cite{berge}]
\label{book}
the cyclomatic number of a graph is the maximum number of independent cycles in this graph.
\end{lemma}
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Dear Professionals, Please have the below requirement and send me your consultant resumes ASAP
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| 311,624
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Whipped Cream for Valentine Cake
This recipe for whipped cream is for our romantic valentine cake.
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Ingredients
- 2 cups heavy cream
- 2 tablespoons confectioners' sugar
- 1 teaspoon pure vanilla extract
Directions
Step 1
In the bowl of an electric mixer fitted with the whisk attachment, combine the cream, sugar, and vanilla. Beat on high speed until soft peaks form, about 5 minutes.
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| 406,765
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| 411,133
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Know more about this business than we do? Cool! Please submit any corrections or missing details you may have.Help us make it right
Be the first cleveland.com user to add photos or videos of Skidmore Tim Excavating
Skidmore Tim Excavating can be found at S Spruce St 889. The following is offered: B2B Contractors. The entry is present with us since Sep 10, 2010 and was last updated on Nov 14, 2013. In Jefferson there are 7 other B2B Contractors. An overview can be found here.
| 82,209
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Featured by Megan Wetherall
The Flavor of Resilience: Loló, A Beloved Restaurant in San Francisco’s Mission District, Offers Comfort During A Pandemic
The transporting power of food and memories.
Wine And Cheese
One morning, at 4 a.m., I go to Rungis with Bernard Noel, the daytime chef at Chez Denise, to shop … Continued
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At seven o’clock in the morning, eight butchers, who have been working all night preparing meat for scores of Parisian … Continued
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Enduring
If you don't boil it into oblivion or hurl it at a passing outcast, the humble, dependable turnip may surprise you with its sweet warmth and nourishing soul.
Dinner Mint
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Eating The Yule Log
It's the best kind of Christmas tree.
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Mints: Sweet Springtime for the Mouth
Our favorite mints from around the world.
| 160,960
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National Geographic (NatGeo), the international news magazine, dedicated its April 2019 issue to the challenges of urban living and the progression of big cities. When I came across the hard copy of this issue, entitled "Cities: Ideas for a Brighter Future", my heart reached out for Dhaka, the city where I was born and the city of my love. I started going through the pages of the magazine reading up on ideas offered by some of the world's best thinkers, architects, urban planners, and civic leaders on how to build or rebuild their own favourite cities for the future. As a lifelong DhakaphiIe, I assumed and believed that Dhaka, obviously, would be a prime candidate for NatGeo's attention and will benefit immensely from the contribution that could be provided by the team assembled by the magazine. Unfortunately, none of the essays in the Cities Issue of NatGeo addressed Dhaka's problems specifically. Nevertheless, what gave me reasons to rejoice was the affirmation that many of the world's largest, most crowded, and also the so-called "unliveable" cities face and manage the same set of problems that Dhaka does.
I have written in these columns in the past on some of Dhaka's pressing needs. Trash removal, ease of traffic congestion, pollution-free air, affordable housing, playgrounds. You name it. Yes, Dhaka has to solve these problems and many more. But reading the NatGeo articles gave me hope that none of them is insurmountable. If other developing countries and cash-strapped municipalities can do it, how can Dhaka be an exception?
Why did NatGeo zero in on the cities? Because, more than two-thirds of the world's population will be living in cities in 2050 compared to about half of the population in 2010, according to the UN. It is forecasted that the world's population will reach 8.6 billion in 2030 and 9.8 billion in 2050 from 7.0 billion in 2010. Dhaka's 2019 population is now estimated at 20,283,552 and will reach 31.23 million in 2035. In 1950, the population of Dhaka was 335,760. Dhaka has grown by 2,686,375 since 2015, which represents a 3.62 percent annual change. And the trend will continue. Let us hope that the growth rate slows down slightly between 2020 and 2035. According to one source, the growth rate of the city is projected to slow down from 3.56 percent annually in 2025 to 2.15 percent in 2035.
But, what do we expect the future city to look like? NatGeo tasked the internationally renowned architectural and urban planning firm Skidmore, Owings and Merrill (SOM) to explore and come up with a guideline to answer the 64 billion question, "How to design a city of the future?" My jaws dropped when I read the guidelines provided by NatGeo's editors.
The city of the future must meet 8 targets: i) It must have a plan to allow "ecology" to guide development; ii) The water resources are protected and systems are designed to capture, treat, and reuse it; iii) Energy is renewable and the city becomes more liveable even as it becomes more densely populated; iv) All waste becomes a resource; v) Food is grown locally and sustainably; vi) High-speed rail to improve mobility; vii) The culture and heritage of the increasingly diverse population are publicly supported and; viii) The infrastructure is carbon-neutral, and the economy is largely automated and online.
NatGeo's consultants paid glowing tribute to the city planners of Tokyo, the world's most populous metropolitan area. "Tokyo is one of the safest, cleanest, most dynamic, and most innovative cities." I was not surprised, but I was looking for clues that might help us understand why Dhaka had become so unliveable.
When I talk to someone who lives in Dhaka, or returned from a short visit there, we always return to our favourite topic, "Life in Dhaka". We discuss the new shopping malls, coffee shops sprouting up in every corner of the city, the flyovers and mega-projects, and the enhanced quality of life thanks to modern technology. The conversation then inevitably turns to tales of hours stuck in traffic, cost of decent healthcare in Dhaka, and the state of public transportation projects in the city. But, what cheered me up even more was the news that things are getting better in some, albeit small, pockets of the city.
On Facebook, I am a member of a group that calls itself "Dhaka- 400 years. History in Photographs". On this page, members post facsimiles from various sources, both public and private. On a lazy day, I spend hours going over the black and white, sometimes colour, images of the city and livelihood, stretching from 100 years ago to more recent decades. The frames from the '70s and '80s hark back to the days when Dhaka roads were less congested and the neighbourhoods in Old Dhaka, Motijheel, Dhanmondi, and Gulshan were pristine. But, I ask: Was Dhaka more liveable then as compared with the megacity it is today? I also wonder if this is a legitimate question since we are comparing apples and oranges. Dhaka of yesteryears was not the same metropolis we see today. Yes, we have more crowd and congestion today, but we also have better jobs, modern hospitals, and improved quality of life.
The most important conversation for all stakeholders ought to be: how do we see the Dhaka of the future? What are we willing to give up for the convenience of living in Dhaka? Urban life involves trade-offs, and we may gain big benefits only in return for suffering big disadvantages.
Turning to the NatGeo spotlight on cities, the takeaways are plenty.
First of, each city needs to manage three key areas: traffic gridlock, affordability, and the demand for civic amenities.
Secondly, the urban planners of Dhaka today face the same issues that their compatriots in Kazakhstan or Uganda confront. However, the solutions in each case is unique. Jared Diamond, Professor of Geography at UCLA, wrote, "It all comes down to compromises. As the world becomes increasingly urban, will all of us be forced to adopt more of Singapore's solutions?" pointing to the model adopted by Singapore, a tiny urban city with six million people packed into about 722 square kms. "Singapore's government monitors its citizens closely, to make sure that individuals don't harm the community." As an example, he cites the following: "Smart-technology sensors measure (or will measure) the traffic on every street, the movements of every car…They will track the water and electricity consumption of every household and will note the time whenever a household toilet is flushed." Are Dhakaites willing to accept such intrusions?
Thirdly, Jan Gehl, an urban designer in Copenhagen, Denmark who is revered for his insights, discounts the notion that one size fits all. Nonetheless, all city planners share a common vision, he says. "Waking up every morning and knowing that the city is a little.
| 286,526
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Source: Gilbert Carrasquillo / Getty
Vivica A. Fox has sparked quite a fire on social media, after she suggested that her ex-boyfriend, 50 Cent might be gay.
The actress went on Watch What Happens Live this weekend, where she claimed that her famous ex was a “booty snatcher,” and as well as questioned his sexuality while on a previous cover of XXL with Soulja Boy.
Since then, 50 has been flaming her on social media for her comments, while Soulja has also come at Vivica for bringing him into the mix.
She made sure to clear up the mess, by retracting that Soulja Boy could possibly be gay for being on the cover with 50.
She tweeted an apology, writing:
This doesn’t, however, clear up her accusations towards 50 Cent, whom she previously claimed was the love of her life last year.
According to TMZ, they revealed that Soulja has accepted Vivica’s apology this time around, and will be letting the situation go.
Hopefully this feud can be settled between Vivica and 50, as he already has some other beefs on his shoulders.
SOURCE: TMZ | PHOTO CREDIT: Getty
Vivica A. Fox Apologizes To Soulja Boy After 50 Cent Comments & He Responds was originally published on globalgrind.com
| 254,369
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TITLE: Orthogonality of summed wave functions
QUESTION [4 upvotes]: Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to $\Psi_4=\Psi_1-\Psi_2$.
My naive solution. From the premises, we know that
$$\int_{-\infty}^\infty \Psi_1^*\Psi_1 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_2 dx=1$$
and
$$\int_{-\infty}^\infty \Psi_1^*\Psi_2 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_1 dx=0$$
We also have $(z_1+z_2)^*=z_1^*+z_2^*$
$$\int_{-\infty}^\infty \Psi_3^*\Psi_4 dx = \int_{-\infty}^\infty (\Psi_1+\Psi_2)^*(\Psi_1-\Psi_2)dx \\
=\int_{-\infty}^\infty(\Psi_1^*+\Psi_2^*)(\Psi_1-\Psi_2)dx\\
=\int_{-\infty}^\infty(\Psi_1^*\Psi_1-\Psi_1^*\Psi_2+\Psi_2^*\Psi_1-\Psi_2^*\Psi_2)dx\\
=1-0+0-1=0\,,$$
which is equivalent with what we wanted to prove. Is this a legitimate proof? Is there any simpler way to do this? I am afraid I still haven't grasped how wave functions behave mathematically, so I may have missed somethings very obvious here.
Edit: The solution manual somehow uses normalization factors for $\Psi_3$ and $\Psi_4$. How are these factors when you don't actually know the exact functions? And how does this relate to the concept of orthogonality?
REPLY [3 votes]: This problem could be done more simply through the application of linear algebra. You want to prove that
$$\langle \psi_1 - \psi_2 | \psi_1 + \psi_2 \rangle = 0$$
The inner product is analogous to the dot product of linear algebra, and it is distributive. Distributing, we find that
$$\begin{aligned}
\langle \psi_1 - \psi_2 | \psi_1 + \psi_2 \rangle &= \langle \psi_1 - \psi_2 | \psi_1 \rangle + \langle \psi_1 - \psi_2 | \psi_2 \rangle \\
&= \langle \psi_1 | \psi_1 \rangle - \langle \psi_2 | \psi_1 \rangle + \langle \psi_1 | \psi_2 \rangle - \langle \psi_2 | \psi_2 \rangle
\end{aligned}
$$
Because $\psi_1$ and $\psi_2$ are orthogonal and normalized, you know $\langle \psi_i | \psi_j \rangle = \delta_{i j}$. Substituting, the above expression evaluates to $1 - 0 + 0 - 1 = 0$, demonstrating that the two vectors are indeed orthogonal.
Your approach - using the integrals - was also valid, and fundamentally similar to mine here. However, by noting that the relation you used ($\langle \psi_1 | \psi_2 \rangle = \int_{-\infty}^{\infty} \! \psi_1^* \psi_2 \, \mathrm{d}x$) satisfied the definition of an inner product, the integrals can be omitted.
| 153,732
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And still, each day begins with a sense of removal and distance. As if I put myself in exile, broken only if the will is there to do it.
Again I am up at dawn. You’d think I’ve seen all there is to be seen at sunrise, but that’s not so. Today, for instance, there are wisps of cloud that filter light in new ways. When the sun does break through, the early morning rays, paired with the lightly green of the now fully clothed tree limbs push aside the tedious stuff of the past month – the move, the mice, the twisting roots of weeds in the flowerbeds – all made small, because in the morning, I can take a solitary walk along the road and admire this:
I walk through the fields that are now farmed by the truck farmers -- all Laotian, bringing to Madison's farmers markets, as the the famously wonderful R.W. Apple of the NY Times once said -- a suggestion of bigger and better flavors than those we're so used to.
I nudge my girl to go walking with me. Her heart says yes, but her tired urban soul keeps her in the farmhouse room, the lemon room, where lilacs bend toward her window and the smell of May and the smell of pine doors and floorboards makes her sigh and drift further into sleep, unbothered by the sound of birds and sparrows, robins and somewhere, not too far, mating cranes. Apple blossoms pave the path outside, bumble bees hover around the blossoms that remain, and if this all doesn’t feel remote and far from downtown Chicago, or downtown Madison then I don’t know what does.
She is to return to the city this evening, but for now we do country things – she wants the rhubarb compote my reader noted in one of the Ocean comments and so we pick many many stalks of this pleasantly tart vegetable, chop them up and set it all on the stove to simmer.
We’re to eat a meal in town later in the morning, with others joining us, but still, this is the kind of warm day where a bowl of fruit with honey and kefir, a chunk of a doughnut, a cup of coffee, are a good excuse to stay on the porch for a while and talk about cats, old barns and jobs, and all things in between.
| 128,483
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TITLE: $P(X=c)=0$ for normally distributed $X$?
QUESTION [1 upvotes]: Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
REPLY [2 votes]: If $X$ is any continuous random variable (e.g. normally distributed), then $P(X=c) = 0$ for all $c$ in $\mathbb{R}$, no matter what distribution $X$ has or what point $c$ is. Points are too small to have non-zero probability of being hit by $X$.
If by the quantity $b$ you mean the variance -- normally, we write $X \sim N(\mu,\sigma^2)$ -- then $b$ is zero if and only if $X$ is constant. No normally distributed r.v. can have zero variance. If $X$ is constant with constant value $c$, then $P(X=c) = 1$ and $P(X=d) = 0$ for all other $d \ne c$.
| 152,864
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- New
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6028 Toffee / Black
UPC Code:680887069767
Leather crossbody with adjustable shoulder strap. Two top zip and one open compartments, back zip pocket. Interior- zip and slide pockets. Dimension: 9.75 x 7 x 1.5 in.
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TITLE: Is $\mathbb{F}_{2011^2}[x]/(x^4-6x-12)$ a field?
QUESTION [5 upvotes]: I'm studying for my qualifying exam and this was one of the questions in the question bank under Field and Galois theory section. I'm currently stuck on this question.
Is $\mathbb{F}_{2011^2}[x]/(x^4-6x-12)$ a field?
I'm guessing the answer is no but I don't know how do I prove that $x^4-6x-12$ is an irreducible polynomial over $\mathbb{F}_{2011^2}$. I'm trying to use the following theorem: "let $p$ be a prime. Over $\mathbb{F}_p, x^{p^n}−x$ factors as the product of all monic irreducible polynomials of degrees $d\mid n$."
REPLY [7 votes]: Extending the comment.
If $f(x)=x^4-6x-12$ is reducible over $\Bbb{F}_{2011}$ then it is also reducible over the extension field. It follows that the quotient ring is not a field.
But all the extensions of finite fields are normal. So if $f(x)$ is irreducible over $\Bbb{F}_{2011}$ then it splits over the field $K=\Bbb{F}_{2011^4}$. By the basic properties of finite fields $K$ contains a copy of $F=\Bbb{F}_{2011^2}$. So the zeros of $f(x)$ have quadratic minimal polynomials over $F$. Therefore $f(x)$ is reducible over $F$, and we can conclude that this quotient ring is not a field irrespective of whether $f(x)$ is irreducible over the prime field or not.
This was a trick question in the sense that the same argument works for any quartic in place of $f(x)$. All because $\gcd(4,2)>1$.
A general related result is that a degree $m$ polynomial $g(x)$, irreducible in $\Bbb{F}_q[x]$, remains irreducible over $K=\Bbb{F}_{q^n}$ if and only if $\gcd(m,n)=1$. The proof is similar. The roots of $g(x)$ reside in $\Bbb{F}_{q^m}$ which is a subfield of $L=\Bbb{F}_{q^\ell}$, where $\ell=\operatorname{lcm}(m,n)$. Therefore the minimal polynomials of those roots over $K$ have degree $\ell/n=m/\gcd(m,n)$. Those minimal polynomials are the factors of $g(x)$ in $K[x]$.
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TITLE: If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
QUESTION [2 upvotes]: I've been thinking how to prove that an analytic function $f$ in the domain $D$ is a constant if $f'(z)=0$ in every point in $D$, but I haven't figured it out yet.
What I was thinking is to use Cauchy-Riemann equations, but it didn't work well...
If this is not true, I would like to know the counterexample...
Here is what I tried:
Let $f(z)=u(x,y)+iv(x,y)$
$\color{blue}{(1)}\text{(Cauchy-Riemann})\begin{cases}
u_y=-v_x\\
u_x=v_y
\end{cases}$
$\color{blue}{(2)}\lim\limits_{\Delta z \to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z }=0$ for all $z_0$ in $D$
I'm stuck here...
REPLY [1 votes]: With Cauchy-Riemann: Note that $f' = u_x +iv_x$ at all points. Therefore $u_x, v_x \equiv 0.$ Use C-R to conclude $u_y, v_y \equiv 0.$ Thus $\nabla u, \nabla v \equiv 0,$ so $u,v$ are both constant, which implies $f$ is constant. (The connectness of $D$ was used here.)
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\chapter{Simplicial techniques}\label{Chapt:simp}
The theory of simplicial sets and simplicial spaces is classical.
Simplicial sets were first defined in~\cite{EilenbersZilber:50}. The geometric realization
was defined in~\cite{Milnor:57}. Other references to simplicial techniques
are~\cite{May:67},~\cite{GoerssJardine:99} and~\cite{DwyerHenn:01}.
For the theory of simplicial spaces see~\cite{May:72},~\cite{Segal:74}
and~\cite{Madsen:94}.
This chapter will recall from the literature the simplicial techniques which
are relevant to this thesis. One reason for including this material is for completeness,
but also important is the viewpoint and the notation.
\section{The category $\catDelta$ and its relatives}
\begin{Def}
Let $[n]$ be the ordered set $\{0<1<\ldots<n\}$. The category $\catDelta$ has one object $[n]$ for each
non-negative integer $n$, and the morphisms are ordering preserving functions $\phi:[m]\rightarrow[n]$.
\end{Def}
It is customary to let $\delta_i:[n-1]\rightarrow[n]$ be the order preserving function that misses $i$,
and $\sigma_i:[n+1]\rightarrow[n]$ be the order preserving function that hits $i$ twice. The $\delta$'s and $\sigma$'s
generate all morphisms in $\catDelta$.
\begin{Def}
A \textit{simplicial set} is a functor $X_{\bullet}:\catDelta^{\op}\rightarrow\Ens$.
A \textit{simplicial space} is a functor $X_{\bullet}:\catDelta^{\op}\rightarrow\Top$.
More generally, one can define simplicial objects in any category.
\end{Def}
Observe that simplicial sets can be considered as simplicial spaces by giving each $X_n$ the
discrete topology. Hence, in most cases we can do our constructions
for simplicial spaces, the corresponding results for simplicial sets follow implicitly.
Given a simplicial space $X_{\bullet}$, the following notation and terminology is standard:
The space $X_n$ is called the \textit{$n$-simplices} of $X_{\bullet}$. $\delta_i:[n-1]\rightarrow[n]$
induces a map $d_i:X_n\rightarrow X_{n-1}$ called the \textit{$i$'th face map}, and
$\sigma_i:[n+1]\rightarrow[n]$ induces the \textit{$i$'th degeneracy map}, $s_i:X_n\rightarrow X_{n+1}$.
A simplex $x$ in $X_n$ is said to be \textit{degenerate} if $x=s_ix'$ for some $i$ and $x'\in X_{n-1}$.
We denote by $sX_{n-1}$ the subspace of $X_n$ consisting of the degenerate simplices.
\subsection{Geometric realization of simplicial spaces}
Simplicial spaces are combinatorial models for topological spaces, and
geometric realization is the functor which turns a simplicial space
into the topological space for which it is a model.
The geometric realization, due to Milnor~\cite{Milnor:57}, has several good properties,
it commutes with products and it commutes with all colimits.
See~\cite{May:67} or~\cite{DwyerHenn:01}.
Furthermore, every point in the geometric realization is uniquely determined
as the interior point of a non-degenerate simplex. We give a modern
formulation of this result; giving a filtration for the geometric realization.
Geometric realization of simplicial spaces is defined
using a functor $\Delta:\catDelta\rightarrow\Top$. We send $[n]$ to the space
$\Delta^n=\{(t_0,\ldots,t_n)\in\R^{n+1}\;|\; \sigma_i t_i=1, t_i\geq 0\}$.
And we call $\Delta^n$ the \textit{topological $n$-simplex}.
On morphisms the functor is defined by sending $\delta_i:[n-1]\rightarrow [n]$ to the map
\[
\delta_i(t_0,\ldots,t_{n-1})=(t_0,\ldots,t_{i-1},0,t_i,\ldots,t_{n-1})\quad,
\]
and $\sigma_i:[n+1]\rightarrow [n]$ to
\[
\sigma_i(t_0,\ldots,t_{n+1})=(t_0,\ldots,t_{i-1},t_i+t_{i+1},t_{i+2},\ldots,t_{n+1})\quad.
\]
Now we define the \textit{geometric realization} of a simplicial space $X_{\bullet}$ as the coend
\[
|X_{\bullet}|=\int^{[n]\in\catDelta}X_n\times \Delta^n\quad.
\]
Coends are defined in section~IX.5 in~\cite{MacLane:98}. The space
$|X_{\bullet}|$ is isomorphic to the quotient of $\coprod X_n\times \Delta^n$
where we identify $(x,\phi(\mathbf{t}))$ with $(\phi^*x,\mathbf{t})$
for all morphisms $\phi$ in $\catDelta$.
\begin{Rem}\label{Rem:prodformulaforgeomreal}
In order to have convenient technical properties, one should form
the geometric realization in the category of compactly generated spaces (=weak Hausdorff $k$-spaces),
see~\cite{McCord:69}. This ensures, for example, that the product theorem holds.
\end{Rem}
There is also a presimplicial realization, defined using only the injective morphisms in $\catDelta$.
The injective morphisms are those generated by the $\delta$'s. Let $i\catDelta$ denote this subcategory.
We define the \textit{presimplicial realization} as the coend
\[
\|X_{\bullet}\|=\int^{[n]\in i\catDelta}X_n\times \Delta^n\quad.
\]
This space is the quotient of $\coprod X_n\times \Delta^n$
where we identify $(x,\delta_i(\mathbf{t}))$ with $(d_i(x),\mathbf{t})$
for all $\delta_i$'s.
Whereas the geometrical realization, $|X_{\bullet}|$, has better formal properties, it is often
easier to prove results about the homotopy of the presimplicial realization, $\|X_{\bullet}\|$.
And we can compare the two realizations via a natural map
\[
\|X_{\bullet}\|\rightarrow |X_{\bullet}|\quad.
\]
It is natural to ask when this is a weak homotopy equivalence. This question is answered by
Segal in~\cite{Segal:74} and by May in~\cite{May:72}. We follow Segal and define:
\begin{Def}
A simplicial space $X_{\bullet}$ is \textit{good} if for all $n$ and $i$, the map
$s_i:X_n\rightarrow X_{n+1}$ is a closed cofibration.
\end{Def}
We refer to our remark~\ref{Rem:cofibdef} or one of the articles~\cite{Steenrod:67} or~\cite{Strom:66}
for the definition of a closed cofibration. Observe that any simplicial set automatically is good, since
an injective map between discrete spaces is a closed cofibration. Now, Segal shows in his proposition~A.1(iv) that:
\begin{Prop}\label{Prop:geomandpresimplrealization}
If $X_{\bullet}$ is a good simplicial space, then the natural map
$\|X_{\bullet}\|\rightarrow |X_{\bullet}|$ is a weak equivalence.
\end{Prop}
Let us now describe the realizations more carefully. We have already mentioned Milnor's result,
that a point in $|X_{\bullet}|$ is uniquely given as the interior point of a non-degenerate simplex.
There is a similar description for the presimplicial realization. We now give a
modern formulation of these statements:
\begin{Constr}\label{Constr:presimplfilt}
First we consider the case of the presimplicial realization.
Recall that $\|X_{\bullet}\|$ is the quotient space formed from $\coprod X_n\times \Delta^n$
by identifying $(x,\delta^i(\mathbf{t}))$ with $(d_i x,\mathbf{t})$ for all morphisms $d_i$.
We define a filtration by letting the $q$'th space, $F_q\|X_{\bullet}\|$, $q\geq0$,
be the image of $\coprod_{n\leq q} X_n\times \Delta^n$ in $\|X_{\bullet}\|$.
Notice that $F_q\|X_{\bullet}\|$ is the pushout of
\[
X_q\times\Delta^q\leftarrow X_q\times\partial\Delta^q\rightarrow F_{q-1}\|X_{\bullet}\|\quad.
\]
Now observe that
\begin{itemize}
\item[] $\colim F_q\| X_{\bullet}\|$ is equal to $\| X_{\bullet}\|$, and
\item[] each $X_q\times\partial\Delta^q\rightarrow X_q\times\Delta^q$ is a closed cofibration. It follows that also
$F_{q-1}\| X_{\bullet}\|\rightarrow F_q\| X_{\bullet}\|$ is a closed cofibration.
\end{itemize}
The last observation explains why the presimplicial realization behaves so well homotopically;
it is easy to give inductive arguments using the pushout diagram relating $F_{q-1}\| X_{\bullet}\|$
to $F_q\| X_{\bullet}\|$.
\end{Constr}
\begin{Constr}\label{Constr:filtergeomrealization}
Next consider the geometric realization.
Recall that $|X_{\bullet}|$ is the quotient of $\coprod X_n\times \Delta^n$
where we identify $(x,\phi(\mathbf{t}))$ with $(\phi^* x,\mathbf{t})$ for all morphisms $\phi$ in $\catDelta$.
We define $F_q|X_{\bullet}|$, $q\geq 0$, to be the image of $\coprod_{n\leq q} X_n\times \Delta^n$.
Recall that the degenerate simplices, $sX_{q-1}$, are the
points in $X_q$ which are in the image of some map $s_i:X_{q-1}\rightarrow X_q$.
Notice that the following diagram is pushout:
\[
\begin{CD}
X_q\times\partial\Delta^q\cup sX_{q-1}\times\Delta^q @>>> F_{q-1}|X_{\bullet}|\\
@V{j}VV @VVV\\
X_q\times \Delta^q@>>> F_q|X_{\bullet}|
\end{CD}\quad.
\]
Here the map $j$ comes from the square
\[
\begin{CD}
sX_{q-1}\times\partial\Delta^q @>>> sX_{q-1}\times\Delta^q\\
@VVV @VVV\\
X_{q}\times\partial\Delta^q @>>> X_{q}\times\Delta^q
\end{CD}\quad.
\]
By Lillig's union theorem, see~\cite{Lillig:73}, we have that
$j$ is a closed cofibration whenever $sX_{q-1}\subset X_q$ is a closed cofibration.
Now observe that
\begin{itemize}
\item[] $\colim F_q| X_{\bullet}|$ is equal to $| X_{\bullet}|$, and
\item[] if each $sX_{q-1}\subset X_q$ is a closed cofibration, then
$X_q\times\partial\Delta^q\cup sX_{q-1}\times\Delta^q\rightarrow X_q\times\Delta^q$ and
$F_{q-1}| X_{\bullet}|\rightarrow F_q| X_{\bullet}|$ are closed cofibrations.
\end{itemize}
The last observation explains why good simplicial spaces behave well with respect to homotopy.
\end{Constr}
These filtrations are extremely useful when proving results about realizations.
We illustrate this by proving a few well known facts:
\begin{Prop}
Let $f_{\bullet}:X_{\bullet}\rightarrow Y_{\bullet}$ be a map of simplicial spaces
such that each $f_q$ is a weak homotopy equivalence. Then the induced map
\[
\|f_{\bullet}\|:\|X_{\bullet}\|\rightarrow \|Y_{\bullet}\|
\]
also is a weak homotopy equivalence.
\end{Prop}
\begin{proof}
We use the filtration from construction~\ref{Constr:presimplfilt}, and prove inductively
that $F_q\|X_{\bullet}\|\rightarrow F_q\|Y_{\bullet}\|$ is a weak homotopy equivalence.
It will follow that $\|f\|$ is a weak homotopy equivalence.
$F_0\|X_{\bullet}\|=X_0\rightarrow Y_0=F_0\|Y_{\bullet}\|$ is a weak homotopy equivalence by assumption.
Now consider the inductive step. We have the diagram
\[
\begin{CD}
X_q\times\Delta^q @<{j}<< X_q\times\partial\Delta^q @>>> F_{q-1}\|X_{\bullet}\|\\
@V{f_q\times\id}VV @V{f_q\times\id}VV @VVV\\
Y_q\times\Delta^q @<{j'}<< Y_q\times\partial\Delta^q @>>> F_{q-1}\|Y_{\bullet}\|
\end{CD}\quad.
\]
Here $j$ and $j'$ are closed cofibrations, while all vertical maps are weak homotopy equivalences.
By proposition~\ref{Prop:gluewe}, the map of the row-wise pushouts,
\[
F_q\|X_{\bullet}\|\rightarrow F_q\|Y_{\bullet}\|
\]
also is a weak homotopy equivalence. This finishes the proof.
\end{proof}
A straight forward corollary of this proposition together with
proposition~\ref{Prop:geomandpresimplrealization} is:
\begin{Cor}
If $f_{\bullet}:X_{\bullet}\rightarrow Y_{\bullet}$ is a map between good
simplicial spaces and each $f_q$ is a weak homotopy equivalence, then
\[
|f_{\bullet}|:|X_{\bullet}|\rightarrow |Y_{\bullet}|
\]
also is a weak homotopy equivalence.
\end{Cor}
Let us also prove the product theorem.
Given two simplicial spaces, $X_{\bullet}$ and $Y_{\bullet}$, we define their product
$X_{\bullet}\times Y_{\bullet}$ to be the simplicial space with $n$-simplices
$X_n\times Y_n$. We have a natural map
$\eta:|X_{\bullet}\times Y_{\bullet}|\rightarrow|X_{\bullet}|\times |Y_{\bullet}|$
defined using the natural projections from $X_{\bullet}\times Y_{\bullet}$
into $X_{\bullet}$ and $Y_{\bullet}$. The product theorem states that $\eta$ is a
homeomorphism. It is hard to find a proof in the literature,
which is of the generality suggested in remark~\ref{Rem:prodformulaforgeomreal}.
This is the reason for including a proof here:
\begin{Prop}
Since the geometric realization is formed in the category of compactly generated spaces,
the natural map
\[
\eta:|X_{\bullet}\times Y_{\bullet}|\rightarrow|X_{\bullet}|\times |Y_{\bullet}|
\]
is a homeomorphism.
\end{Prop}
\begin{proof}
It is well known that $\eta$ is a continuous bijection, see theorem~2 in~\cite{Milnor:57} or
theorem~11.5 in~\cite{May:72}. The hard part is to check that $\eta^{-1}$ is continuous.
May proves this when ``spaces'' is the category of compactly generated Hausdorff spaces,
although his proof for continuity of $\eta^{-1}$ is not particularly clear. The author
hopes that the argument below will be more understandable, but in essence the proofs are the same.
We use the filtration of the geometrical realization given in construction~\ref{Constr:filtergeomrealization}.
The product $|X_{\bullet}|\times|Y_{\bullet}|$ inherits a filtration given by
\[
F_q(|X_{\bullet}|\times|Y_{\bullet}|)=\bigcup_{m+n=q}F_m|X_{\bullet}|\times F_n|Y_{\bullet}|\quad.
\]
And by the constructions one can see that $\eta$ restricts to a continuous bijection
$F_q|X_{\bullet}\times Y_{\bullet}|\cong F_q(|X_{\bullet}|\times|Y_{\bullet}|)$.
A continuous bijection $\eta$ between compactly generated spaces is a homeomorphism,
if $\eta^{-1}(K)$ is compact whenever $K$ is compact. This follows from the definition
of compactly generated by lemma~2.1 in~\cite{McCord:69}.
We will now try to apply lemma~2.8 in~\cite{McCord:69}.
Suppose that $K\subset |X_{\bullet}|\times|Y_{\bullet}|$ is a compact subset. Then $K$
is contained in $F_q(|X_{\bullet}|\times|Y_{\bullet}|)$ for some $q$.
$q$ is now fixed.
Below we will specify $Z_{\alpha}$'s such that for all $\alpha$ we have commutative diagrams
\[
\begin{CD}
\coprod_{p\leq q} X_p\times Y_p\times \Delta^p @<<<
Z_{\alpha}\subset \coprod_{n,m}X_n\times \Delta^n\times Y_m\times \Delta^m\\
@VVV @VV{\pi}V\\
F_q|X_{\bullet}\times Y_{\bullet}| @>{\eta_q}>> |X_{\bullet}|\times |Y_{\bullet}|
\end{CD}\quad.
\]
Here the vertical maps are surjective
and $\eta_q$ is injective. Furthermore, the target of the map $\pi$
has the quotient topology.
The $Z_{\alpha}$'s depend on the standard triangulation of $\Delta^n\times\Delta^m$.
For given $m$ and $n$ let $i_{\alpha}:\Delta^{n+m}\rightarrow \Delta^n\times\Delta^m$
be the inclusion of a maximal topological simplex in this triangulation. We set
$Z_{\alpha}=X_n\times Y_m\times \Delta^{n+m}$, included in
$\coprod_{n,m}X_n\times \Delta^n\times Y_m\times \Delta^m$
via $i_{\alpha}$. Moreover, via the appropriate degeneracy maps, there are
maps $Z_{\alpha}\rightarrow X_p\times Y_p\times \Delta^p$, $p=n+m$, such that the diagram
above commutes. These maps are explicitly constructed in May's proof.
Now observe that the collection of $Z_{\alpha}$'s with $n+m\leq q$ covers the image
of $F_q|X_{\bullet}\times Y_{\bullet}|$. This collection is finite. Thus all conditions
of McCord's lemma~2.8 are satisfied. This implies that $\eta_q$ is an embedding, and
consequently we have that $\eta^{-1}(K)=\eta_q^{-1}(K)$ is compact. And we are done.
\end{proof}
\subsection{Crossed simplicial categories}
Techniques involving simplicial sets or simplicial spaces are extremely useful when working with
topological spaces. However, if we want to consider involutions, $S^1$- or $O(2)$-actions on our spaces,
it is handy to replace $\catDelta$ by other categories; $\catDeltaT$, $\catDeltaC$ and $\catDeltaD$.
We will recall the notion of a crossed simplicial group from~\cite{FiedorowiczLoday:91}.
The categories mentioned above are examples of such, they will be defined below
and we will introduce notation for their morphisms.
\begin{Def}
A sequence of groups $\{G_n\}$, $n\geq0$, is a \textit{crossed simplicial group}
if it is equipped with the following structure. There is a small category $\catDeltaG$,
which is part of the structure, such that
\begin{itemize}
\item[] the objects of $\catDeltaG$ are $[n]$, $n\geq0$,
\item[] $\catDeltaG$ contains $\catDelta$ as a subcategory,
\item[] the automorphisms of $[n]$ in $\catDeltaG$ is the opposite group of $G_n$, and
\item[] any morphism in $\catDeltaG([m],[n])$ can be uniquely written as a composite $\phi\circ g$,
where $\phi\in\catDelta([m],[n])$ and $g\in G^{\op}_m$.
\end{itemize}
\end{Def}
\begin{Rem}\label{Rem:Gissimplset}
The last axiom implies that for any $g\in G_n$ and $\phi\in\catDelta([m],[n])$ there exist unique
$\phi^*(g)\in G_m$ and $g^*(\phi)\in \catDelta([m],[n])$ such that
\[
g\circ \phi= g^*(\phi)\circ \phi^*(g)\quad.
\]
The functor that sends $[n]$ to $G_n$ and $\phi$ to $\phi^*:G_n\rightarrow G_m$ gives $G_{\bullet}$ the
structure of a simplicial set.
\end{Rem}
Unlike~\cite{FiedorowiczLoday:91}, our focus will not be these simplicial sets, but
rather the categories $\catDeltaG$ and their analogue of simplicial sets and spaces, i.e.
functors from $\catDeltaG^{\op}$ into sets and spaces. We will therefore refer to $\catDeltaG$
as a \textit{crossed simplicial category}.
Here are the crossed simplicial categories relevant for this thesis,
they are taken from the examples~2, 4, 5 and~7 in~\cite{FiedorowiczLoday:91}:
\begin{Def}
Define $\catDeltaT$ to be the crossed simplicial category with
the automorphism group of $[n]$ cyclic of order $2$.
Let $\rho_n$ be the generator of the automorphism group and put
$\rho_{n}\delta_i=\delta_{n-i}\rho_{n-1}$ and
$\rho_{n}\sigma_i=\sigma_{n-i}\rho_{n+1}$.
\end{Def}
\begin{Def}\label{Def:catDeltaC}
Let $\catDeltaC$ be the crossed simplicial category
where the automorphism group of $[n]$ is cyclic of order $(n+1)$.
We name the preferred generator $\tau_n$, and introduce the relations:
\begin{align*}
\tau_{n}\delta_0=\delta_{n}\quad&\text{and}\quad \tau_n\delta_i=\delta_{i-1}\tau_{n-1}\text{, for $1\leq i\leq n$, and}\\
\tau_{n}\sigma_0=\sigma_{n}\tau_{n+1}^2\quad&\text{and}\quad \tau_{n}\sigma_i=\sigma_{i-1}\tau_{n+1}\text{, for $1\leq i\leq n$.}
\end{align*}
\end{Def}
\begin{Def}\label{Def:catDeltaD}
Let $\catDeltaD$ be crossed simplicial category
where the automorphism group of $[n]$ is the dihedral group of order $2(n+1)$.
We name the preferred generators $\rho_n$ and $\tau_n$, where $\rho_n^2=\tau_n^{n+1}=\id$ and $\rho_n\tau_n=\tau_n^{-1}\rho_n$,
and introduce the relations:
\begin{align*}
\rho_{n}\delta_i&=\delta_{n-i}\rho_{n-1},\\
\rho_{n}\sigma_i&=\sigma_{n-i}\rho_{n+1},\\
\tau_{n}\delta_0=\delta_{n}\quad&\text{and}\quad \tau_n\delta_i=\delta_{i-1}\tau_{n-1}\text{, for $1\leq i\leq n$, and}\\
\tau_{n}\sigma_0=\sigma_{n}\tau_{n+1}^2\quad&\text{and}\quad \tau_{n}\sigma_i=\sigma_{i-1}\tau_{n+1}\text{, for $1\leq i\leq n$.}
\end{align*}
\end{Def}
\begin{Def}
Let $\catDeltaC_r$, $r\geq 1$, be the crossed simplicial category
where the automorphism group of $[n]$ is cyclic of order $r(n+1)$.
We name the preferred generator $\tau_n$, where $\tau_n^{r(n+1)}=\id$, and introduce the same relations
as in definition~\ref{Def:catDeltaC}.
\end{Def}
\begin{Def}
Let $\catDeltaD_r$, $r\geq 1$, be crossed simplicial category
where the automorphism group of $[n]$ is the dihedral group of order $2r(n+1)$.
We name the preferred generators $\rho_n$ and $\tau_n$, where $\rho_n^2=\tau_n^{r(n+1)}=\id$ and $\rho_n\tau_n=\tau_n^{-1}\rho_n$,
and introduce the same relations as in definition~\ref{Def:catDeltaD}.
\end{Def}
We now give names to these crossed simplicial categories, and call $\catDeltaT$, $\catDeltaC$, $\catDeltaD$, $\catDeltaC_r$ and $\catDeltaD_r$
the \textit{involutive simplicial category}, the \textit{cyclic category}, the \textit{dihedral category},
the \textit{$r$-cyclic category} and the \textit{$r$-dihedral category} respectively.
Notice that $\catDeltaC_1=\catDeltaC$ and $\catDeltaD_1=\catDeltaD$. We see that $\catDeltaC_r$ is a subcategory of $\catDeltaD_r$,
and that $\catDeltaT$ is a subcategory of $\catDeltaD_r$, for any $r\geq1$.
Our reason for introducing crossed simplicial categories is to
study $G_{\bullet}$-objects in some category $\mathscr{C}$:
\begin{Def}
Let $\catDeltaG$ be a crossed simplicial category
and $\mathscr{C}$ any category.
A \textit{$G_{\bullet}$-object} in $\mathscr{C}$
is a functor $\catDeltaG^{\op}\rightarrow\mathscr{C}$. A \textit{$G_{\bullet}$-map}
between $G_{\bullet}$-objects is a natural transformation of functors.
\end{Def}
If $\catDeltaG$ is one of the crossed simplicial categories above and $\mathscr{C}$ is $\Top$, the category of
(compactly generated) spaces,
then we call $G_{\bullet}$-objects for \textit{involutive simplicial spaces},
\textit{cyclic spaces}, \textit{dihedral spaces}, \textit{$r$-cyclic spaces} and \textit{$r$-dihedral spaces} accordingly,
and similarly we replace the word ``spaces'' by ``sets'' when $\mathscr{C}=\Ens$, the category of sets.
Given an $r$-dihedral space $X_{\bullet}$ we have the following notation: The map induced by
$\delta_i$ is denoted by $d_i:X_{n}\rightarrow X_{n-1}$ and called the \textit{$i$'th face map}.
The map induced by $\sigma_i$ is denoted $s_i:X_{n}\rightarrow X_{n+1}$ and called the
\textit{$i$'th degeneracy map}. The map induced by $\rho_n$ is denoted by $r_n:X_n\rightarrow X_n$
and called the \textit{involutive operator}. And the map induced by $\tau_n$ is denoted by
$t_n:X_n\rightarrow X_n$ and called the \textit{cyclic operator}.
For an $r$-cyclic space, we use the same notation and terminology, but in this case there are
no involutive operators. Analogously, there are no cyclic operators for involutive simplicial spaces.
\subsection{Geometric realization of $G_{\bullet}$-spaces}
We now turn toward the geometric realization of
$G_{\bullet}$-spaces. Via the inclusion $j:\catDelta\rightarrow\catDeltaG$ we associate to
any $G_{\bullet}$-space $X_{\bullet}$ its underlying simplicial space $j^*X_{\bullet}$, which is given as
the composition
$\catDelta^{\op}\xrightarrow{j}\catDeltaG^{\op}\xrightarrow{X_{\bullet}}\Top$.
And we define:
\begin{Def}
The geometric realization of a $G_{\bullet}$-space $X_{\bullet}$ is the geometric realization
of its underlying simplicial space $j^*X_{\bullet}$.
\end{Def}
From the article~\cite{FiedorowiczLoday:91} we now summarize
results about the geometric realization of a $G_{\bullet}$-space.
\begin{Thm}\label{Thm:MainThmOfCrossedSimplCat}
Let $\catDeltaG$ be a crossed simplicial category, and $X_{\bullet}$ a simplicial space.
We have:
\begin{itemize}
\item[] The functor $j^*$ from $G_{\bullet}$-spaces to simplicial spaces
has a left adjoint, denoted by $F_G$, and there are projection maps
$p_1:|F_G(X_{\bullet})|\rightarrow |G_{\bullet}|$ and $p_2:|F_G(X_{\bullet})|\rightarrow |X_{\bullet}|$.
\item[] The map $(p_1,p_2):|F_G(X_{\bullet})|\rightarrow |G_{\bullet}|\times |X_{\bullet}|$
is a homeomorphism.
\item[] For any simplicial map $f_{\bullet}:X_{\bullet}\rightarrow Y_{\bullet}$ the following diagrams commute:
\[
\begin{CD}
|F_G X_{\bullet}| @>{|F_G f_{\bullet}|}>> |F_G Y_{\bullet}|\\
@V{p_2}VV @VV{p_2}V\\
|X_{\bullet}| @>{|f_{\bullet}|}>> |Y_{\bullet}|
\end{CD}\quad\text{and}\quad
\begin{CD}
|F_G X_{\bullet}| @>{|F_G f_{\bullet}|}>> |F_G Y_{\bullet}|\\
@V{p_1}VV @VV{p_1}V\\
|G_{\bullet}| @= |G_{\bullet}|
\end{CD}\quad.
\]
\item[] Since $F_G$ is a left adjoint, there are canonical natural transformations
$\mu_{\bullet}:F_G(F_G(X_{\bullet}))\rightarrow F_G(X_{\bullet})$ and $\iota_{\bullet}:X_{\bullet}\rightarrow F_G(X_{\bullet})$.
And the following diagrams commute:
\[
\begin{CD}
|F_G(F_G( X_{\bullet}))| @>{|\mu_{\bullet}|}>> |F_G (X_{\bullet})|\\
@V{p_2}VV @VV{p_2}V\\
|F_G(X_{\bullet})| @>{p_2}>> |X_{\bullet}|
\end{CD}\quad\text{and}\quad
\begin{CD}
|X_{\bullet}| @>{|\iota_{\bullet}|}>> |F_G X_{\bullet}|\\
@V{=}VV @VV{p_2}V\\
|X_{\bullet}| @= |X_{\bullet}|
\end{CD}\quad.
\]
\item[] There is a canonical isomorphism $G_{\bullet}\cong F_G(*)$ and the composition
$|G_{\bullet}|\cong |F_G(*)|\xrightarrow{p_1}|G_{\bullet}|$ is the identity.
\item[] Let $1$ denote the point in $|G_{\bullet}|$ determined by the unit in $G_0$. The following diagram commutes:
\[
\begin{CD}
|X_{\bullet}| @>{|\iota_{\bullet}|}>> |F_G(X_{\bullet})|\\
@VVV @VV{p_1}V\\
\{1\} @>>> |G_{\bullet}|\\
\end{CD}\quad.
\]
\item[] $|G_{\bullet}|$ is a topological group.
\item[] If $X_{\bullet}$ is a $G_{\bullet}$-space, then there is an
induced action $|G_{\bullet}|\times|X_{\bullet}|\rightarrow|X_{\bullet}|$.
\item[] $(p_1,p_2):|F_G(X_{\bullet})|\rightarrow |G_{\bullet}|\times |X_{\bullet}|$ is an equivariant homeomorphism.
\item[] For every $n$ there is an inclusion of $G_n$ in $|G_{\bullet}|$ as a discrete subgroup.
\end{itemize}
\end{Thm}
For a proof see propositions~4.4,~5.1,~5.3 and~5.13 in~\cite{FiedorowiczLoday:91}.
\begin{Rem}\label{Rem:sescsg}
Chasing Fiedorowicz and Loday's proof of theorem~\ref{Thm:MainThmOfCrossedSimplCat} above,
it is not hard to see that
all results are natural with respect to a morphism $\catDeltaG\rightarrow\catDeltaG'$
of crossed simplicial categories. In particular we get an induced homomorphism of topological groups
$|G_{\bullet}|\rightarrow|G_{\bullet}'|$. Furthermore, it is possible to consider short exact sequences
of crossed simplicial categories. It is more convenient to write such a sequence in terms of
the corresponding crossed simplicial groups. The sequence
\[
0\rightarrow G''_{\bullet}\rightarrow G_{\bullet}\rightarrow G'_{\bullet}\rightarrow 0
\]
is short exact if the evaluation at each $[n]$ is. Taking the geometric realization
one gets a sequence
\[
|G''_{\bullet}|\rightarrow |G_{\bullet}|\xrightarrow{f} |G'_{\bullet}|\quad,
\]
which an extension of topological groups.
\end{Rem}
Let us now determine what the group $|G_{\bullet}|$ is for our crossed simplicial categories.
\begin{Exa}
Consider the involutive simplicial category, $\catDeltaT$. The automorphism group, $G_n^{\op}$,
of $[n]$ in $\catDeltaT$ is isomorphic to $\Z/2$.
Recall that $G_{\bullet}$ is a
simplicial set, the face and degeneracy maps are given by the formula in remark~\ref{Rem:Gissimplset}.
The degeneracy map $s_0$ is always injective. By counting the order of $G_n$, we immediately see
that the only non-degenerate simplices lie in degree $0$. Hence, we have that
$|G_{\bullet}|$, in this case, is the group $\Z/2$. This means that the geometric realization of an involutive
simplicial space is a topological space with involution.
\end{Exa}
\begin{Exa}
Next consider the cyclic category, $\catDeltaC$.
Using the formula from remark~\ref{Rem:Gissimplset}, we find that the non-degenerate
simplices are $\tau_0\in G_0$ and $\tau_1\in G_1$. Hence, $|G_{\bullet}|\cong S^1$.
We now determine the group structure.
A theorem by von Neumann says that any compact, locally Euclidean topological group
is a Lie group, see theorem~57 in~\cite{Pontrjagin:39}.
The theory of Lie groups now tells us that the only topological group structure on $S^1$
is the ordinary group structure.
\end{Exa}
\begin{Exa}
Now look at the $r$-cyclic category, $\catDeltaC_r$. Let $G_{\bullet}$
be the associated crossed simplicial group.
To determine $|G_{\bullet}|$ as a topological space, we find the non-degenerate simplices.
The $0$-simplices, $G_0=C_r$, are non-degenerate. Recall from remark~\ref{Rem:Gissimplset}
the formula defining the simplicial structure on $G_{\bullet}$. The relation
\[
\tau_0\sigma_0=\sigma_0\tau_1^2
\]
implies that $s_0(\tau_0^i)=\tau_1^{2i}$. Hence, $\tau_1$, $\tau_1^3$,$\ldots$, $\tau_1^{2r-1}$
are the non-degenerate simplices in $G_1$. Playing with the relations in $\catDeltaC_r$,
we see that there are no more non-degenerate simplices. Furthermore, we have that
$d_0(\tau_1^{2i-1})=\tau_0^{i-1}$ and $d_1(\tau_1^{2i-1})=\tau_0^{i}$. Hence,
$|G_{\bullet}|\cong S^1$. And $S^1$ has a unique structure as a topological group.
\end{Exa}
\begin{Exa}
Let us now study the $r$-dihedral category, $\catDeltaD_r$.
We can use the definition of the category and the formula from remark~\ref{Rem:Gissimplset}
to determine the simplicial structure on the associated simplicial group $G_{\bullet}$.
Finding non-degenerate simplices and calculating the face maps, we see that
\[
|G_{\bullet}|\cong S^1\times \Z/2
\]
as topological spaces.
Hence there are two possibilities for the group structure on $|G_{\bullet}|$:
it is isomorphic either to $S^1\times\Z/2$ or $O(2)$.
By the last statement of theorem~\ref{Thm:MainThmOfCrossedSimplCat},
$|G_{\bullet}|$ contains dihedral subgroups.
This excludes $S^1\times\Z/2$, so $|G_{\bullet}|=O(2)$.
\end{Exa}
The theorem~\ref{Thm:MainThmOfCrossedSimplCat} above tells us that the geometric
realization of a $G_{\bullet}$-space has a $|G_{\bullet}|$ action.
However, it is usually the case that the action takes one out of the topological simplex one starts in.
In particular, the $q$'th space of the filtration $F_q|X_{\bullet}|$ is seldom
$|G_{\bullet}|$-equivariant.
In many situations it would be easier if the action stayed inside the topological simplices and
the filtration had $|G_{\bullet}|$-action.
We can achieve this by defining the topological $|G_{\bullet}|$-simplices according to
the crossed simplicial category under consideration.
Let $\catDeltaG$ be a crossed simplicial category. Consider the representable functors
\[
\catDeltaG(-,[n]):\catDeltaG^{\op}\rightarrow \Ens\quad.
\]
\begin{Def}
Let $\Delta G:\catDeltaG\rightarrow \Top$ be the functor with $\Delta G^n=|\catDeltaG(-,[n])|$.
The \textit{topological $|G_{\bullet}|$-simplices} are the spaces $\Delta G^n$, $n\geq0$.
\end{Def}
Observe that the representable functor $\catDeltaG(-,[n])$ is $F_G(\catDelta(-,[n]))$, hence we have
homeomorphisms $\Delta G^n=|\catDeltaG(-,[n])|\cong|G_{\bullet}|\times \Delta^n$.
So the $|G_{\bullet}|$-action does not take points outside $\Delta G^n$.
Using the functor $\Delta G^{\bullet}$ we can now define a geometric realization of
$G_{\bullet}$-spaces $X$ given by:
\[
|X|_{\Delta G}=\int^{[n]\in\catDeltaG}X_n\times \Delta G^n\quad.
\]
This space is isomorphic to the quotient of $\coprod X_n\times \Delta G^n$
where we identify $(x,\phi(\mathbf{t}))$ with $(\phi^*x,\mathbf{t})$
for all morphisms $\phi$ in $\catDeltaG$.
\begin{Lem}\label{Lem:equalityofrealizations}
There is a natural homeomorphism $|X|_{\Delta G}\cong|X|$ for $\catDeltaG^{\op}$-spaces $X$.
\end{Lem}
\begin{proof}
Consider the functor
$F:(\catDelta\times\catDeltaG)^{\op}\times (\catDelta\times\catDeltaG)\rightarrow \Top$
given by
\[
F([n_o],[m_o],[n],[m])=X_{m_o}\times\catDeltaG(j(n_o),m)\times\Delta^n\quad.
\]
We have that
\[
\int^{[n]\in\catDelta}F([n],[m_o],[n],[m])\cong X_{m_o}\times\Delta G^{m}
\]
and
\[
\int^{[m]\in\catDeltaG}F([n_o],[m],[n],[m])\cong X(j(n_o))\times\Delta^{n}\quad.
\]
The result now follows from the Fubini theorem for coends, see \S{}IX.8 in~\cite{MacLane:98}:
coends can be interchanged.
\end{proof}
To achieve full control of the $|G_{\bullet}|$-action on $|X_{\bullet}|$, it suffices to have
an explicit description of the functor $\Delta G^{\bullet}$. This description should
specify the map $\Delta G^n\rightarrow\Delta G^m$ induced by a morphism $\phi:[n]\rightarrow[m]$ in $\catDeltaG$.
In the case $\catDeltaC$, this description is given implicitly in proposition~2.7 in~\cite{DwyerHopkinsKan:85},
and more explicitly in theorem~3.4 in~\cite{Jones:87}. For the $r$-cyclic case a formula is given by
lemma~1.6 in~\cite{BokstedtHsiangMadsen:93}, and by formula~(2.1.3) in~\cite{Madsen:94}. In general it is just
a question about writing out the equivariant homeomorphism
$(p_1,p_2):|F_G(\Delta^n_{\bullet})|\rightarrow |G_{\bullet}|\times |\Delta^n_{\bullet}|$
from theorem~\ref{Thm:MainThmOfCrossedSimplCat}. Here $\Delta^n_{\bullet}$ is the simplicial $n$-simplex
$\catDelta(-,[n])$.
Explicitly we have in our cases:
\begin{Exa}
For the involutive simplicial category $\catDeltaT$ we define the functor $\Delta T^{\bullet}$ by
sending $[n]$ to $\Z/2\times\Delta^n$. We write $\Z/2$ multiplicatively. The generators of $\catDeltaT$
induce the following maps:
\begin{align*}
\delta_i(\epsilon;t_0,\ldots,t_n)&= (\epsilon; t_0,\ldots, t_{i-1},0,t_i,\ldots,t_n)\quad,\\
\sigma_i(\epsilon;t_0,\ldots,t_n)&= (\epsilon; t_0,\ldots, t_{i-1},t_i+t_{i+1},t_{i+2},\ldots,t_n)\quad\text{, and}\\
\rho_n(\epsilon;t_0,\ldots,t_n)&= (-\epsilon; t_n,t_{n-1},\ldots, t_1,t_0)\quad.
\end{align*}
\end{Exa}
\begin{Exa}
For the $r$-cyclic category $\catDeltaC_r$ we define the functor $\Delta C_r^{\bullet}$ by
sending $[n]$ to $S^1\times\Delta^n$. We identify $S^1$ with the quotient $\R/\Z$.
The generators of $\catDeltaC_r$
induce the following maps:
\begin{align*}
\delta_i(\theta;t_0,\ldots,t_n)&= (\theta; t_0,\ldots, t_{i-1},0,t_i,\ldots,t_n)\quad,\\
\sigma_i(\theta;t_0,\ldots,t_n)&= (\theta; t_0,\ldots, t_{i-1},t_i+t_{i+1},t_{i+2},\ldots,t_n)\quad\text{, and}\\
\tau_n(\theta;t_0,\ldots,t_n)&= (\theta-\frac{1}{r}t_0; t_1,t_{2},\ldots, t_n,t_0)\quad.
\end{align*}
\end{Exa}
\begin{Exa}\label{Exa:dihedraltopsimpl}
For the $r$-dihedral category $\catDeltaD_r$ we define the functor $\Delta D_r^{\bullet}$ by
sending $[n]$ to $O(2)\times\Delta^n$. $O(2)$ is the space of orthogonal $2\times 2$-matrices.
For $t\in\R/\Z$ let $R(t)$ denote the rotation matrix
$\begin{pmatrix}\cos(2\pi t)&\sin (2\pi t)\\ -\sin(2\pi t)& \cos(2\pi t)\end{pmatrix}$, and let
$T$ be the matrix $\begin{pmatrix} 0 & 1\\ 1& 0\end{pmatrix}$.
The generators of $\catDeltaD_r$
induce the following maps:
\begin{align*}
\delta_i(M;t_0,\ldots,t_n)&= (M; t_0,\ldots, t_{i-1},0,t_i,\ldots,t_n)\quad,\\
\sigma_i(M;t_0,\ldots,t_n)&= (M; t_0,\ldots, t_{i-1},t_i+t_{i+1},t_{i+2},\ldots,t_n)\quad,\\
\tau_n(M;t_0,\ldots,t_n)&= (M R(-\frac{1}{r}t_0); t_1,t_{2},\ldots, t_n,t_0)\quad\text{, and}\\
\rho_n(M;t_0,\ldots,t_n)&= (M T;t_n,t_{n-1},\ldots, t_1,t_0)\quad.
\end{align*}
\end{Exa}
\subsection{Filtering the geometric realization}
Similar to the constructions~\ref{Constr:presimplfilt} and~\ref{Constr:filtergeomrealization},
we now design a filtration of $|X_{\bullet}|$, when $X_{\bullet}$ is a $G_{\bullet}$-space.
This filtration is $|G_{\bullet}|$-equivariant.
\begin{Constr}\label{Constr:filtercrossedsimplreal}
Let $\catDeltaG$ be a crossed simplicial category and $X_{\bullet}$ a $G_{\bullet}$-space.
The drawback of using the filtration above to study $|X_{\bullet}|$ is that
$F_q|X_{\bullet}|$ has no $|G_{\bullet}|$ action. Therefore we define another
filtration $F_q^G|X_{\bullet}|$.
Recall that $|X_{\bullet}|$ can be described as the quotient of
$\coprod X_n\times\Delta G^n$ where we identify $(x,\phi(\mathbf{t}))$
with $(\phi^* x,\mathbf{t})$ for all morphisms $\phi$ in $\catDeltaG$.
Define $F^G_q|X_{\bullet}|$ to be the image of $\coprod_{n\leq q} X_n\times \Delta G^n$.
We define the \textit{$G_{\bullet}$-degenerate simplices} of
$X_{q}$ to be the subspace $s^GX_{q-1}$ consisting of all points which lie in the image of
some map $\phi^*:X_{q-1}\rightarrow X_q$, $\phi\in\catDeltaG([q],[q-1])$.
Recall that the opposite group of $G_q$ is the automorphisms of $[q]$ in $\catDeltaG$.
Hence $X_q$ and $sX_{q-1}$ have $G_q$ actions, while
$\Delta G^q$ and $\partial \Delta G^q$ have $G_q^{\op}$ actions.
Let $X_q\times_{G_q} \Delta G^q$ denote the quotient of the product
where we have identified $(gx,\mb{t})$ with $(x,g^*\mb{t})$ for every $g$ in $G_q$.
We now have a pushout diagram
\[
\begin{CD}
X_q\times_{G_q}\partial\Delta G^q\cup sX_{q-1}\times_{G_q}\Delta G^q @>>> F_{q-1}^G|X_{\bullet}|\\
@V{i}VV @VVV\\
X_q\times_{G_q} \Delta G^q @>>> F_q^G|X_{\bullet}|
\end{CD}\quad.
\]
\end{Constr}
\begin{Rem}
Here is a warning: In general it is not true that natural map
$X_0\times_{G_0}|G_{\bullet}|\rightarrow F_0^G|X_{\bullet}|$ is an homeomorphism,
but it is always an equivariant quotient map.
\end{Rem}
\subsection{Edgewise subdivision}
Above we have seen that both cyclic and $r$-cyclic spaces yield $S^1$-spaces
after geometric realization. Similarly both dihedral and $r$-dihedral spaces
realize to $O(2)$-spaces. So why do we bother with the $r$-cyclic and $r$-dihedral
categories? Observe that neither the $S^1$- nor the $O(2)$-action is simplicial.
Let $C$ be a finite cyclic group. Notice that $C$ embeds as
a normal subgroup of both $S^1$ and $O(2)$. The answer to the question is that
$C$-fixed points can be studied simplicially whenever the order of $C$ divides $r$.
After making precise the observations above, we shall define the $c$'th
edgewise subdivision, $c\geq 1$.
This is a functor $\sd_c$ from $r$-cyclic spaces to $rc$-cyclic spaces,
and similarly from $r$-dihedral spaces to $rc$-dihedral spaces.
The edgewise subdivisions come with natural equivariant homeomorphisms
$D_c:|\sd_c X_{\bullet}|\rightarrow |X_{\bullet}|$.
In particular we can replace a cyclic space
with an $r$-cyclic space for the purpose of studying its restricted $C_r$-action.
Let $C$ be a finite cyclic group $C$ of order $c$.
Recall from example~\ref{Exa:dihedraltopsimpl} that $R(t)\in O(2)$ denotes a rotation by $2\pi t$,
while $T\in O(2)$ is a reflection. We identify $C$ as the normal subgroup of $O(2)$
generated by $R(\frac{1}{c})$.
Now we construct homomorphisms
\[
\rho_C: O(2)\rightarrow O(2)/C
\]
by letting $\rho_C(R(t))=R(\frac{t}{c})$ and $\rho_C(T)=T$. Observe that $\rho_C$
is an isomorphism. The restriction of $\rho_C$ to $S^1$ is the ``$c$-th root map''
$S^1\xrightarrow{\cong}S^1/C$.
Two basic facts are: The $C$-fixed point space of an $O(2)$-space $Y$ is an $O(2)/C$-space $Y^C$,
and an $O(2)/C$-space $Z$ can be viewed as an $O(2)$-space $\rho_C^*Z$ via the isomorphism
$\rho_C$.
After these preliminaries we show:
\begin{Prop}
Assume that $X_{\bullet}$ is an $r$-dihedral space and $C$ a finite cyclic group
of order $c$. Assume that $c$ divides $r$ and let $cs=r$.
Each $X_n$ has a $C$-action and $X_{\bullet}^C$
is an $s$-dihedral space. Furthermore, there is a natural $O(2)$-equivariant homeomorphism
\[
\rho_C^*|X_{\bullet}|^C\cong |X_{\bullet}^C|\quad.
\]
A similar result holds for $r$-cyclic spaces.
\end{Prop}
\begin{proof}
The $C$ action on $X_n$ is given by the map $t_n^{s(n+1)}$.
Observe that all the operators $d_i$, $s_i$, $t_n$ and $r_n$
preserve $C$-fixed points. Hence, $X_{\bullet}^C$ is an
$r$-dihedral space. But since $t_n^{s(n+1)}$
is the identity when restricted to $X_n^C$, we see that
$X_{\bullet}^C$ satisfies the identities for an $s$-dihedral space.
To define the natural $O(2)$-homeomorphism we use the filtration
from construction~\ref{Constr:filtercrossedsimplreal}. Assume inductively
that we have an $O(2)$-homeomorphism
\[
\rho_C^*F_{n-1}^{\catDeltaD_r}|X_{\bullet}|^C\cong F_{n-1}^{\catDeltaD_s}|X_{\bullet}^C|\quad.
\]
Recall that the automorphism group
of $[n]$ in $\catDeltaD_r$ is the dihedral group $D_{2r(n+1)}$ of order $2r(n+1)$.
If $Y$ is a $D_{2r(n+1)}$-space, then we may form the induced $O(2)$-space
$Y\times_{D_{2r(n+1)}} O(2)$. It is a basic fact about induced $O(2)$-spaces
and $C$-fixed points, compare lemma~\ref{Lem:fixedpointsofinducedspaces},
that
\[
\left(Y\times_{D_{2r(n+1)}} O(2)\right)^C\cong Y^C\times_{D_{2r(n+1)}/C} O(2)/C\quad.
\]
For the induction step we inspect the $n$-simplices, and calculate:
\begin{align*}
\rho_C^*\left(X_n\times_{D_{2r(n+1)}}\Delta D_r^n\right)^C
&\cong \rho_C^*\left( (X_n\times\Delta^n)\times_{D_{2r(n+1)}} O(2)\right)^C\\
&\cong \rho_C^*\left( (X_n\times\Delta^n)^C\times_{D_{2r(n+1)}/C} O(2)/C\right)\\
&\cong (X_n\times\Delta^n)^C\times_{D_{2s(n+1)}}\rho_C^{-1}(O(2)/C)\\
&\cong (X_n^C\times\Delta^n)\times_{D_{2s(n+1)}}O(2)\\
&\cong X_n\times_{D_{2s(n+1)}}\Delta D_s^n\quad.
\end{align*}
Similarly, we have an $O(2)$-equivariant homeomorphism for the degenerate points. And these
$O(2)$-homeomorphisms fit into a diagram
\scriptsize
\[
\begin{CD}
\rho_C^*\left(X_n\times_{D_{2r(n+1)}}\Delta D_r^n\right)^C
&\leftarrow& \rho_C^*\left(X_n\times_{D_{2r(n+1)}}\partial\Delta D_r^n\cup sX_{n-1}\times_{D_{2r(n+1)}}\Delta D_r^n\right)^C
&\rightarrow& \rho_C^*F_{n-1}^{\catDeltaD_r}|X_{\bullet}|^C\\
@V{\cong}VV @V{\cong}VV @VV{\cong}V\\
X_n\times_{D_{2s(n+1)}}\Delta D_s^n
&\leftarrow& X_n^C\times_{D_{2s(n+1)}}\partial\Delta D_s^n\cup sX_{n-1}^C\times_{D_{2s(n+1)}}\Delta D_s^n
&\rightarrow& F_{n-1}^{\catDeltaD_s}|X_{\bullet}^C|
\end{CD}\quad.
\]
\normalsize
By construction~\ref{Constr:filtercrossedsimplreal} we see that the map
of the row-wise pushouts is $F_{n}^{\catDeltaD_s}|X_{\bullet}^C|\cong \rho_C^*F_{n}^{\catDeltaD_r}|X_{\bullet}|^C$.
The statement for $r$-cyclic spaces is proved similarly.
\end{proof}
We now define the \textit{edgewise subdivision functor} $\sd_c:\catDeltaD_{rc}\rightarrow \catDeltaD_r$.
The idea behind $\sd_c$ is to send the ordered set $[q]$ to the disjoint union of $c$ copies of $[q]$:
\[
\sd_c[q]=[q]\amalg\cdots\amalg[q]=[c(q+1)-1]\quad.
\]
This yields the following formulas for $\sd_c$ of the generators in the dihedral case:
\begin{align*}
\sd_c(\delta_i)&=\delta_{i+(c-1)(q+1)}\cdots\delta_{i+(q+1)}\delta_i\quad,\\
\sd_c(\sigma_i)&=\sigma_i\sigma_{i+(q+2)}\cdots\sigma_{i+(c-1)(q+2)}\quad,\\
\sd_c(\tau_q)&=\tau_{c(q+1)-1}\quad\text{, and}\\
\sd_c(\rho_q)&=\rho_{c(q+1)-1}\quad.
\end{align*}
Observe that $\sd_c$ restricts to functors $\catDeltaC_{rc}\rightarrow \catDeltaC_r$,
$\catDeltaT\rightarrow \catDeltaT$ and $\catDelta\rightarrow \catDelta$.
\begin{Def}
Let $X_{\bullet}$ be an $r$-dihedral space. Its \textit{$c$'th edgewise subdivision}, $\sd_cX_{\bullet}$ is the
composition $\catDeltaD_{rc}^{\op}\xrightarrow{\sd_c}\catDeltaD_r^{\op}\xrightarrow{X_{\bullet}}\Top$.
Similarly, we also define the $c$'th edgewise subdivision of $r$-cyclic, involutive simplicial and
simplicial spaces.
\end{Def}
To compare the geometric realization of $\sd_cX_{\bullet}$ and $X_{\bullet}$, we first
define a diagonal map from the topological $rc$-dihedral $q$-simplex $\Delta D_{rc}^q$
to the topological $r$-dihedral $(c(q+1)-1)$-simplex $\Delta D_r^q$. This map is given by
\[
(M;t_0,\ldots,t_q)\mapsto
(M;\frac{1}{c}t_0,\ldots,\frac{1}{c}t_q,\frac{1}{c}t_0,\ldots,\frac{1}{c}t_q,\ldots,\frac{1}{c}t_0,\ldots,\frac{1}{c}t_q)
\quad.
\]
This map is $O(2)$-equivariant.
Varying $q$, we get a natural transformation $\Delta D_{rc}^{\bullet}\rightarrow\Delta D_{r}^{\bullet}\circ\sd_c$.
Using a trick with coends we define a natural $O(2)$-map $D_c:|\sd_cX_{\bullet}|\rightarrow|X_{\bullet}|$.
Consider
\[
\int^{[p]\in\catDeltaD_{rc}}\int^{[q]\in\catDeltaD_r} X_q\times\catDeltaD_{r}(\sd_c[p],[q])\times\Delta D_{rc}^p\quad.
\]
Observe that the evaluation
\[
\int^{[q]\in\catDeltaD_r} X_q\times\catDeltaD_{r}(\sd_c[p],[q])\rightarrow (\sd_c X)_p
\]
is a homeomorphism. (The identity in $\catDeltaD_{r}(\sd_c[p],\sd_c[p])$ gives an inverse map.)
It follows that the double coend above equals
\[
\int^{[p]\in\catDeltaD_{rc}} (\sd_cX)_p\times\Delta D_{rc}^p=|\sd_cX_{\bullet}|\quad.
\]
On the other hand, by the Fubini theorem for coends, we can consider the coend over $[p]\in\catDeltaD_{rc}$ first.
Via the diagonal map given above, we get a natural $O(2)$-map
\begin{align*}
\int^{[q]\in\catDeltaD_r}&\int^{[p]\in\catDeltaD_{rc}} X_q\times\catDeltaD_{r}(\sd_c[p],[q])\times\Delta D_{rc}^p\\
&\cong \int^{[q]\in\catDeltaD_r}X_q\times
\left(\int^{[p]\in\catDeltaD_{rc}}\catDeltaD_{r}(\sd_c[p],[q])\times\Delta D_{rc}^p\right)\\
&\xrightarrow{\text{diagonal}} \int^{[q]\in\catDeltaD_r}X_q\times
\left(\int^{[p]\in\catDeltaD_{rc}}\catDeltaD_{r}(\sd_c[p],[q])\times\Delta D_{r}^{\sd_c[p]}\right)\\
&\xrightarrow{\text{evaluate}} \int^{[q]\in\catDeltaD_r}X_q\times\Delta D_r^q\\
&=|X_{\bullet}|\quad.
\end{align*}
Putting this together we see that the diagonal map on topological simplices gives a natural $O(2)$-map
\[
D_c:|\sd_cX_{\bullet}|\rightarrow|X_{\bullet}|\quad.
\]
Similarly, for the cyclic, the involutive simplicial and the simplicial categories we have a
natural $S^1$-map, $\Z/2$-map and map respectively.
\begin{Prop}
Let $X_{\bullet}$ be an $r$-dihedral space, an $r$-cyclic space, an involutive simplicial space
or a simplicial space.
In all cases, the (equivariant) map $D_c:|\sd_cX_{\bullet}|\rightarrow|X_{\bullet}|$
is a homeomorphism.
\end{Prop}
\begin{proof}
Recall that we can compute the geometric realization either over the crossed simplicial category or
over $\catDelta$. Because both methods yield the same space, lemma~\ref{Lem:equalityofrealizations},
it is enough to inspect the map in the simplicial case.
The proof for simplicial sets, lemma~1.1 in~\cite{BokstedtHsiangMadsen:93} applies also to
the case of simplicial spaces: One first checks by explicit computation that
$D_c$ is a homeomorphism when $X_{\bullet}$ is the simplicial $1$-simplex $\catDelta(-,[1])$.
It follows that $D_c$ also is a homeomorphism for products $\catDelta(-,[1])^{\times q}$.
Then it holds for the simplicial $q$-simplex because of the retraction
$\catDelta(-,[q])\xrightarrow{i}\catDelta(-,[1])^{\times q}\xrightarrow{r}\catDelta(-,[q])$.
Let $\eta_q$ denote the inverse of $D_c:|\sd_c\catDelta(-,[q])|\rightarrow|\catDelta(-,[q])|$.
For general simplicial spaces $X_{\bullet}$ we now define the inverse as follows:
\begin{align*}
|X_{\bullet}| &= \int^{[q]\in\catDelta}X_q\times\Delta^q\\
&= \int^{[q]\in\catDelta}X_q\times|\catDelta(-,[q])|\\
&\xrightarrow{\id\times\eta_q}\int^{[q]\in\catDelta}X_q\times|\sd_c\catDelta(-,[q])|\\
&=\int^{[q]\in\catDelta}X_q\times\left(\int^{[p]\in\catDelta}\catDelta(\sd_c[p],[q])\times\Delta^p\right)\\
&=\int^{[q]\in\catDelta}\int^{[p]\in\catDelta} X_q\times\catDelta(\sd_c[p],[q])\times\Delta^p\\
&=|\sd_cX_{\bullet}|\quad.
\end{align*}
\end{proof}
\section{Homotopy colimits over topological categories}
In this short section we will define the homotopy colimit of a continuous functor
over a topological category. Also, we give a condition
on $F$ such that $\hocolim_{\mathscr{C}}F\rightarrow B\mathscr{C}$ is a $\lambda$-quasi cofibration.
Assume that $\mathscr{C}$ is a small topological category; we have a discrete set of objects,
while for each pair of objects, $a,b\in\mathscr{C}$, we have a topological space $\mathscr{C}(a,b)$
of morphisms from $a$ to $b$.
For continuous functors $F:\mathscr{C}\rightarrow\Top$ we would like to define a homotopy colimit.
\begin{Def}
We define $\hocolim_{\mathscr{C}}F$ as the realization of a simplicial space. Its $q$-simplices are
\[
X_q=\coprod_{a_0,\ldots,a_q\in\mathscr{C}} \mathscr{C}(a_{q-1},a_q)
\times\cdots\times \mathscr{C}(a_{0},a_1)\times F(a_0)\quad.
\]
Face and degeneracy maps are given by
\begin{align*}
d_i(f_{q-1},\ldots,f_0;x)&=\begin{cases}
(f_{q-1},\ldots,f_1;f_0(x))&\text{for $i=0$,}\\
(f_{q-1},\ldots,f_{i+1},f_{i}\circ f_{i-1},f_{i-2},\ldots,f_0;x)&\text{for $0<i<q$,}\\
(f_{q-2},\ldots,f_0;x)&\text{for $i=q$, and}
\end{cases}\\
s_i(f_{q-1},\ldots,f_0;x)&=(f_{q-1},\ldots,f_i,\id_{a_i},f_{i-1},\ldots,f_0;x)\quad.
\end{align*}
\end{Def}
$\hocolim$ is functorial. If $\tau:F\rightarrow F'$ is a natural transformation, then there is an induced map
\[
\hocolim_{\mathscr{C}}F\rightarrow \hocolim_{\mathscr{C}}F'\quad.
\]
Furthermore, if $j:\mathscr{D}\rightarrow\mathscr{C}$ is a functor, then there is an induced map
\[
\hocolim_{\mathscr{D}}j^*F\rightarrow \hocolim_{\mathscr{C}}F\quad,
\]
where $j^*f$ is the composite $f\circ F:\mathscr{D}\rightarrow\Top$.
\begin{Prop}\label{prop:hocolimhomotopy}
If $\tau:j_0\rightarrow j_1$ is a natural transformation between continuous
functors $\mathscr{D}\rightarrow\mathscr{C}$,
then there is a simplicial homotopy between
\[
\hocolim_{\mathscr{D}}j_0^*F\xrightarrow{(j_0)_*} \hocolim_{\mathscr{C}}F
\]
and
\[
\hocolim_{\mathscr{D}}j_0^*F\xrightarrow{\tau_*}\hocolim_{\mathscr{D}}j_1^*F\xrightarrow{(j_1)_*}\hocolim_{\mathscr{C}}F
\]
for any continuous functor $F:\mathscr{C}\rightarrow\Top$.
\end{Prop}
\begin{proof}
We define a simplicial homotopy. It is given by maps
\begin{align*}
h_i:\coprod &\mathscr{D}(b_{q-1},b_q)\times\cdots\times \mathscr{D}(b_{0},b_1)\times F(j_0(b_0))\\
&\rightarrow \coprod \mathscr{C}(a_{q},a_{q+1})\times\mathscr{C}(a_{q-1},a_q)\times\cdots\times \mathscr{C}(a_{0},a_1)\times F(a_0)
\end{align*}
for $0\leq i\leq q$.
To define the $h_i$'s we consider the diagram in $\mathscr{C}$:
\[
\begin{CD}
j_0(b_0)@>{j_0(f_0)}>> j_0(b_1) @>{j_0(f_1)}>> \cdots @>{j_0(f_{q-1})}>> j_0(b_q)\\
@V{\tau_{b_0}}VV @V{\tau_{b_1}}VV &\cdots& @VV{\tau_{b_q}}V\\
j_1(b_0)@>{j_1(f_0)}>> j_1(b_1) @>{j_1(f_1)}>> \cdots @>{j_1(f_{q-1})}>> j_1(b_q)
\end{CD}\quad.
\]
$h_i$ is now given by the formula:
\[
h_i(f_{q-1},\ldots,f_0;x)=(j_1(f_{q-1}),\ldots,j_1(f_i),\tau_{b_i},j_0(f_{i-1}),\ldots,j_0(f_0);x)\quad.
\]
It is easily checked that this is the required simplicial homotopy.
\end{proof}
We now define $\lambda$-quasi fibrations:
\begin{Def}
A map $p:E\rightarrow B$ is a \textit{$\lambda$-quasi fibration} if for any $b\in B$ the induced map
$\pi_i(E,p^{-1}(b))\rightarrow \pi_i (B,b)$ is an isomorphism for $0\leq i< \lambda$ and a surjection
for $i=\lambda$.
\end{Def}
\begin{Prop}\label{prop:gluelqf}
Consider the diagram:
\[
\begin{CD}
E_2@<{F}<< E_0 @>>> E_1\\
@V{p_2}VV @VV{p_0}V @VV{p_1}V\\
B_2@<{f}<< B_0 @>{i}>> B_1
\end{CD}\quad.
\]
Assume that the $p_i$'s are $\lambda$-quasi fibrations with $p_i^{-1}(b)$ path-connected for all $i$ and $b\in B_i$.
If $i$ is a cofibration, the right square pullback, and $p_0^{-1}(b)\rightarrow p_2^{-1}(f(b))$ $\lambda$-connected
for all $b\in B_0$, then the induced map of pushouts $p:E\rightarrow B$ is a $\lambda$-quasi fibration.
\end{Prop}
\begin{proof}
We can assume that $f$ is a cofibration, if not one can replace $B_2$ by the mapping cylinder $M_f$, and
$E_2$ by the pullback $r^*E_2$ over the retraction $r:M_f\rightarrow B_2$.
Moreover, we can assume that $F$ is a cofibration, if not we can replace $E_2$ by $M_F$. Using that $f$ is injective
it follows that $M_F\rightarrow B_2$ is a $\lambda$-quasi fibration.
Now compare the long exact sequences of homotopy groups
for the triples $(E_1,E_0,p_0^{-1}(b))$ and $(B_1,B_0,b)$, where $b\in B_0$.
Since $p_0^{-1}(b)=p_1^{-1}(b)$, remember that the right square is pullback, and using that $p_0$ and $p_1$ are
$\lambda$-quasi fibrations, we get that $\pi_i(E_1,E_0)\rightarrow \pi_i(B_1,B_0)$ is an isomorphism
for $0\leq i<\lambda$ and surjective for $i=\lambda$.
Regarding the connectedness of $\pi_i(E_2,p_0^{-1}(b))\rightarrow \pi_i(B_2,b)$, we reason as follows:
Since $p_0^{-1}(b)\rightarrow p_2^{-1}(f(b))$ is $\lambda$-connected, we get that $\pi_i(p_2^{-1}(f(b)),p_0^{-1}(b))$
is the trivial group when $i\leq\lambda$. Now consider the long
exact sequence of homotopy groups for $(E_2,p_2^{-1}(f(b)),p_0^{-1}(b))$. The homomorphism
$\pi_i(E_2,p_0^{-1}(b))\rightarrow \pi_i(E_2,p_2^{-1}(f(b)))$ is an isomorphism for $i<\lambda$ and
surjective for $i=\lambda$.
Using that $p_2$ is a $\lambda$-quasi fibration, the composed map
\[
\pi_i(E_2,p_0^{-1}(b))\rightarrow \pi_i(E_2,p_2^{-1}(f(b)))\rightarrow \pi_i(B_2,b)
\]
is also an isomorphism for $i<\lambda$ and surjective for $i=\lambda$.
Comparing the long exact sequences of homotopy groups
for $(E_2,E_0,p_0^{-1}(b))$ and $(B_2,B_0,b)$, we see that
$\pi_i(E_2,E_0)\rightarrow \pi_i(B_2,B_0)$ is an isomorphism for $0\leq i<\lambda$ and
surjective for $i=\lambda$.
Since the maps under consideration are cofibrations, the Mayer-Vietoris property for homotopy groups holds as stated
in~\cite{Hatcher:02} proposition~4K.1. Therefore, we have that $\pi_i(E,E_1)\rightarrow \pi_i(B,B_1)$
is an isomorphism for $i\leq \lambda$ and surjective for $i=\lambda$. The same is also true for
$\pi_i(E,E_2)\rightarrow \pi_i(B,B_2)$.
At last we can check whether $p:E\rightarrow B$ is a $\lambda$-quasi fibration. If $b\in B_2$ we
compare the long exact sequences of homotopy groups for $(E,E_2,p^{-1}(b))$ and $(B,B_2,b)$. By the five lemma
we see that $\pi_i(E,p^{-1}(b))\rightarrow \pi_i(B,b)$ is an isomorphism
for $i\leq \lambda$ and surjective for $i=\lambda$.
When $b\in B_1\smallsetminus B_0$, we compare long exact sequences of homotopy
groups for $(E,E_1,p^{-1}(b))$ and $(B,B_1,b)$.
The same conclusion holds.
\end{proof}
Observe that for any functor $F:\mathscr{C}\rightarrow\Top$ there is a natural map
\[
\hocolim_{\mathscr{C}}F\rightarrow B\mathscr{C}\quad.
\]
Here $B\mathscr{C}$ is the bar construction (=geometric realization of the nerve).
In some cases, this map is a $\lambda$-quasi fibration:
\begin{Prop}\label{prop:hocolimqf}
If the induced map $F(a)\rightarrow F(b)$ is $\lambda$-connected for all morphisms of $\mathscr{C}$,
$\mathscr{C}$ is well-pointed
and all $F(a)$'s are path-connected, then
\[
\hocolim_{\mathscr{C}}F\rightarrow B\mathscr{C}
\]
is a $\lambda$-quasi fibration.
\end{Prop}
\begin{proof}
As above let $X_{\bullet}$ denote the simplicial space whose realization is $\hocolim_{\mathscr{C}}F$.
Now compare the presimplicial realization with the geometric realization:
\[
\begin{CD}
F(a_0)@>>> \| X_{\bullet} \| @>>> \|B_{\bullet}\mathscr{C} \|\\
@V{=}VV @VV{\simeq}V @VV{\simeq}V\\
F(a_0)@>>> | X_{\bullet} | @>>> |B_{\bullet}\mathscr{C} |\\
\end{CD}\quad.
\]
Here $F(a_0)$ is the fiber over some point $b$ in $\|B_{\bullet}\mathscr{C}\|$.
The fiber over $b$'s image in $|B_{\bullet}\mathscr{C}|$ is identical. This can be seen by inspecting the
definition of the degeneracy maps.
Since $\mathscr{C}$ is well-pointed, it follows that $X_{\bullet}$ and $B_{\bullet}\mathscr{C}$ are
good simplicial spaces. Hence, the vertical maps are weak equivalences. Therefore it is enough to
show that $\| X_{\bullet} \| \rightarrow \|B_{\bullet}\mathscr{C} \|$ is a $\lambda$-quasi fibration.
Following Quillen, we now consider the skeletal filtration of the presimplicial realization.
\[
\begin{CD}
F_{q-1}\|X_{\bullet}\|@<<< X_q\times \partial\Delta^q @>>> X_q\times \Delta^q\\
@VVV @VVV @VVV\\
F_{q-1}\|B_{\bullet}\mathscr{C}\|@<<< B_{q}\mathscr{C}\times \partial\Delta^q @>>> B_{q}\mathscr{C}\times \Delta^q
\end{CD}\quad.
\]
This diagram satisfies the conditions of proposition~\ref{prop:gluelqf}, so the map of pushouts
$F_{q}\|X_{\bullet}\|\rightarrow F_{q}\|B_{\bullet}\mathscr{C}\|$ is a $\lambda$-quasi fibration.
Now the result follows since the direct limit of $\lambda$-quasi fibrations is a $\lambda$-quasi fibration.
\end{proof}
| 4,925
|
TITLE: Divergence of inverse of metric tensor
QUESTION [2 upvotes]: I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?!
I'm not so familiar with the divergence of the second ranked tensor. However, I think one can write
$$\nabla_j g^{ij}=\partial_jg^{ij}+\Gamma^i_{kj}g^{kj}+\Gamma^j_{kj}g^{ki}$$
using the identity $g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})$ and $\Gamma^j_{kj}=\partial_k~log~\sqrt g$, so
$$\nabla_j g^{ij}=\partial_jg^{ij}-\frac{1}{\sqrt g }\partial_j(\sqrt g g^{ij})+\partial^i~log~\sqrt g$$
therefore
$$\nabla_j g^{ij}=\partial_jg^{ij}-\partial_jg^{ij}-g^{ij}\partial_j~log~\sqrt g+\partial^i~log~\sqrt g=0$$
which is weird to me. Where am I doing anything wrong?
REPLY [8 votes]: We know that the Levi-Civita connection satisfies $\nabla_a g_{bc} = 0$ and the product rule. The definition of the inverse metric $g^{ab}$ is $g^{ab}g_{bc} = \delta^a_c$. Therefore, we have:
$$\begin{align}
0 &= \nabla_a \delta^b_c \\
&= \nabla_a (g^{bd}g_{dc}) \\
&= (\nabla_a g^{bd}) g_{dc} + g^{bd} \nabla_a g_{dc} \\
&= (\nabla_a g^{bd}) g_{dc} \end{align}$$
Upon multiplying both sides by $g^{ce}$ and renaming indices we get that indeed $\nabla_a g^{bc} =0$. If you want the divergence $\nabla_a g^{ac}$ then it's just a matter of setting $b=a$; we still get zero.
| 67,695
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analytic philosophers list
Services
But neither the high average intelligence of the analytic philosophers nor the occasional emergence of an analytic philosopher energetic and creative enough to overcome a disadvantaged early environment (e.g., a Donald Davidson, a Bernard Williams, a Daniel Dennett) may be enough to save the discipline from decadent scholasticism. Bertrand Russell - Can't think of the specific paper, but something about his work on language This helps keep discussion in the comments on topic and relevant to the linked material. use the following search parameters to narrow your results: To learn more about what is and is not considered philosophy for the purposes of this subreddit, see our FAQ. If you have unrelated thoughts or don't wish to read the content, please post your own thread or simply refrain from commenting. Hahahaha "living off it." Analytic philosophy (sometimes analytical philosophy) is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century. [37] Alston, grappling with the consequences of analytic philosophy of language, worked on the nature of religious language. [â]WittgensteinBerySwne 0 points1 point2 points 7 years ago (1 child). For example, in this view, saying, "Killing is wrong", is equivalent to saying, "Boo to murder", or saying the word "murder" with a particular tone of disapproval. Kripke's Naming and Necessity, already mentioned, for philsophy of language and logic. Owing largely to Gettier's 1963 paper "Is Justified True Belief Knowledge? The other, known as "Oxford philosophy", involved J. L. It evolved into more sophisticated non-cognitivist theories such as the expressivism of Charles Stevenson, and the universal prescriptivism of R.M. Analytic philosophy of religion has also been preoccupied with Wittgenstein, as well as his interpretation of Søren Kierkegaard's philosophy of religion. Although many discussions are continuations of old ones from previous decades and centuries, the debate remains active. Anscombe's 1958 "Modern Moral Philosophy" sparked a revival of Aristotle's virtue ethical approach and John Rawls's 1971 A Theory of Justice restored interest in Kantian ethical philosophy. (2000). Therefore if there is a purpose or meaning to life, it lies in maximizing, perfecting, and improving states of consciousness.. An alternative higher-order theory, the higher-order global states (HOGS) model, is offered by Robert van Gulick.[25]. Users must follow all reddit-wide spam guidelines, and in addition must not submit more than one post per day on /r/philosophy. [15]. Users are also strongly encouraged to post abstracts for other linked material. Many traditional philosophical problems are dismissed because their terms are too vague, while those that remain are subjected to a rigorous logical analysis. Those definitions often include an emphasis on conceptual analysis: A.P. Phillips in Philosophy's Cool Place, which rests on an interpretation of a passage from Wittgenstein's "Culture and Value. The logical positivists opined that statements about value—including all ethical and aesthetic judgments—are non-cognitive; that is, they cannot be objectively verified or falsified. Posts must not be behind any sort of paywall or registration wall. His conception of behaviorism became much more inclusive over his career. I recommend this anthology for a good grounding in analytic philosophy. I believe he's on record as saying that sometimes he says things not because he believes them to be true or even likely, but because he considers them interesting ideas. But because value judgments are of significant importance in human life, it became incumbent on logical positivism to develop an explanation of the nature and meaning of value judgments. Examples include Ryle, who tried to dispose of "Descartes' myth", and Wittgenstein. The history of analytic philosophy (taken in the narrower sense of "20th-/21st-century analytic philosophy") is usually thought to begin with the rejection of British idealism, a neo-Hegelian movement. In reaction to what he considered excesses of logical positivism, Karl Popper insisted on the role of falsification in the philosophy of science—although his general method was also part of the analytic tradition. Adams worked on the relationship of faith and morality. But although Iâve learned how to do philosophy, nobody ever told me how do it, and, so far as I would guess, nobody will have told you how to do it, or is likely to tell you how ⦠While there is a general consensus for the global neuronal workspace model of consciousness,[24] there are many opinions as to the specifics. Another development of political philosophy was the emergence of the school of analytical Marxism.. To see him in a top five list is somewhat bizarre. Behaviorism later became much less popular, in favor of type physicalism or functionalism, theories that identified mental states with brain states. The second is in logical positivism and its attitude that unverifiable statements are meaningless. Thus the fifth phase, beginning in the mid 1960s and continuing beyond the end of the twentieth century, is characterized by eclecticism or pluralism. them. For example, he refutes Skinner's classical version of behaviorism while offering an alternative that is nonetheless behaviorist in nature. [â]tablefor1 0 points1 point2 points 7 years ago (0 children). In Saul Kripke's publication Naming and Necessity, he argued influentially that flaws in common theories of proper names are indicative of larger misunderstandings of the metaphysics of necessity and possibility. The work of these later philosophers have furthered Cohen's work by bringing to bear modern social science methods, such as rational choice theory, to supplement Cohen's use of analytic philosophical techniques in the interpretation of Marxian theory. [54] The philosophy of biology has also undergone considerable growth, particularly due to the considerable debate in recent years over the nature of evolution, particularly natural selection. Additionally, Russell adopted Frege's predicate logic as his primary philosophical method, a method Russell thought could expose the underlying structure of philosophical problems. [30][31][32] In education, applied ethics addressed themes such as punishment in schools, equality of educational opportunity, and education for democracy. [â][deleted] 0 points1 point2 points 7 years ago (2 children). The debate moves on. G.E.M. Exceptions are made only for posts about philosophers with substantive content, e.g. /r/philosophy is intended for philosophical material and discussion. More important, the development of modern symbolic logic seemed to promise help in solving philosophical problemsâand logic is as a ⦠Karl Popper - Two Meanings of Falsifiability, or any of his lectures on falsifiability. This is a typical problem: Someone comes up with a good idea and gets so excited that they run around acting as if their shiny new hammer turns everything into nails. Please contact the moderators for pre-approval. Popper's The Logic of Scientific Discovery or his Postscript are, I think, better than any of his lectures simply because they're far more detailed expressing his turns to methodology and metaphysics. A significant feature of analytic philosophy since approximately 1970 has been the emergence of applied ethics—an interest in the application of moral principles to specific practical issues. (I understand that the divide is not always clear-cut - and whether the division is a healthy one is a discussion worth having, but a separate one - though I think most of us would be able to confidentially call most philosophers one or the other. Verificationism is now considered completely discredited by the people who are right about things. The claims of ethics, aesthetics, and theology were consequently reduced to pseudo-statements, neither empirically true nor false and therefore meaningless. Analytic philosophy is one of two major branches of philosophy defined by its emphasis on formal logic, philosophy of language and scientism. [34]:3, As with the study of ethics, early analytic philosophy tended to avoid the study of philosophy of religion, largely dismissing (as per the logical positivists) the subject as part of metaphysics and therefore meaningless. [â]chzchbo 0 points1 point2 points 7 years ago (0 children), Good luck! Thank you all - I'm keeping a running list and will, in the coming years, try to tackle as many of these works as I can. Central figures in this historical development of analytic philosophy are Gottlob Frege, Bertrand Russell, G. E. Moore, and Ludwig Wittgenstein. Everett, Anthony and Thomas Hofweber (eds.) J.O. He thereby argued that the universe is the totality of actual states of affairs and that these states of affairs can be expressed by the language of first-order predicate logic. The following is a list of the most cited articles based on citations published in the last three years, according to CrossRef. In his important book The Principles of Mathematics, he ⦠Indeed, while the debate remains fierce, it is still strongly influenced by those authors from the first half of the century: Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, J.L. Hello Gents and Ladies! The best-known member of this school is G. A. Cohen, whose 1978 work, Karl Marx's Theory of History: A Defence, is generally considered to represent the genesis of this school. [â]Sloph 3 points4 points5 points 7 years ago (9 children). Other prominent analytical Marxists include the economist John Roemer, the social scientist Jon Elster, and the sociologist Erik Olin Wright. Spring 2018 AMA Series - Announcement & Hub Post! Indeed, the first analytic philosophers were German or Austrian (Frege, Wittgenstein). The first is G.E. [9] An important aspect of British idealism was logical holism—the opinion that there are aspects of the world that can be known only by knowing the whole world. Lennox, James G., "Darwinism and Neo-Darwinism" in Sakar and Plutynski (eds.). Skinner's problem wasn't that he was completely wrong, but that he took it too far. [16]. Originating. Urmson's article "On Grading" called the is/ought distinction into question. [â]NeoPlatonist 0 points1 point2 points 7 years ago (0 children), [â]ConclusivePostscript 2 points3 points4 points 7 years ago (1 child), Wittgensteinâs Philosophical Investigations. ), [â]wza 17 points18 points19 points 7 years ago (3 children). [â]arkasia 5 points6 points7 points 7 years ago (1 child). During this time, utilitarianism was the only non-skeptical type of ethics to remain popular. (see linked article states for the problem statement; thesis; alternatives; and objections, and analysis), Frances Power Cobbe and Nineteenth-Century Moral Philosophy. (1998). To look up a philosopher you know the name of, click on the first letter of their last name. This is closely related to the opinion that relations between items are internal relations, that is, properties of the nature of those items. Start by looking at Quine's school. After World War II, during the late 1940s and 1950s, analytic philosophy became involved with ordinary-language analysis. These works are essential reading for anyone who wishes to understand the contemporary philosophical climate. Interest in its historical development is increasing, but there has hitherto been no sustained attempt to elucidate what it currently amounts to, and how it differs from so-called 'continental' philosophy. The Argument from Reason: A Case Against Metaphysical Naturalism. At the end of the 1960s some philosophers began to try constructing a systematic theory of meaning for natural languages, and on the basis of such a theory to formulate specific metaphysical statements. This explains the readiness with which typically analytical philosophers often get fasci-nated with typical speculative philosophers. It's not because he's some strange holdover from the bad old days, but rather that he's taken these two stances and evolved them forward. Links to Google Translated versions of posts are not allowed. Susanne Langer[59] and Nelson Goodman[60] addressed these problems in an analytic style during the 1950s and 1960s. Other important figures in its history include the logical positivists (particularly Rudolf Carnap), W. V. O. Quine, Saul Kripke, and Karl Popper. Google Analytics lets you measure your advertising ROI as well as track your Flash, video, and social networking sites and applications. All links to either audio or video content require abstracts of the posted material, posted as a comment in the thread. Analytic philosophy - Analytic philosophy - History of analytic philosophy: During the last decades of the 19th century, English philosophy was dominated by an absolute idealism derived from the German philosopher G.W.F. See here for an example of a suitable abstract. I don't understand your problem with the question- obviously some of the same names will pop up, but I've noticed a suprising amount of variation in the comments. Today, contemporary normative ethics is dominated by three schools: consequentialism, virtue ethics, and deontology. Order now. [51], Metaphysics remains a fertile topic of research, having recovered from the attacks of A.J. [29], Topics of special interest for applied ethics include environmental issues, animal rights, and the many challenges created by advancing medical science. Encouraging other users to commit suicide, even in the abstract, is strictly forbidden. Bas van Fraassen - Scientific Image (I may show my bias by including van Fraassen) [26] It was only with the emergence of ordinary language philosophers that ethics started to become an acceptable area of inquiry for analytic philosophers. These theories were not without their critics. As a result, analytic philosophers avoided normative ethics and instead began meta-ethical investigations into the nature of moral terms, statements, and judgments. Plantinga, Mackie and Flew debated the logical validity of the free will defense as a way to solve the problem of evil. Austin's philosophy of speech acts. Van Inwagen, Peter, and Dean Zimmerman (eds.) For them, philosophy concerned the clarification of thoughts, rather than having a distinct subject matter of its own. Philosophy of language is a topic that has decreased in activity during the last four decades, as evidenced by the fact that few major philosophers today treat it as a primary research topic. Instead, the logical positivists adopted an emotivist theory, which was that value judgments expressed the attitude of the speaker. Born in 1872 into an aristocratic family, Bertrand Russell is widely regarded as one of the founders of Analytic philosophy, which is today the dominant philosophical tradition in the English-speaking world. Go on Jstor and search for "It Ain't Necessarily So" and "Meaning and Reference.". [â][deleted] 5 points6 points7 points 7 years ago (2 children). The following (not exhaustive) list of items require moderator pre-approval: meta-posts, posts to products, services or surveys, links to other areas of reddit, AMAs. I'm an undergrad sophomore majoring in philosophy with at least April through the first part of August on my hands before I head back. Thomas Samuel Kuhn with his formulation of paradigm shifts and Paul Feyerabend with his epistemological anarchism are significant for these discussions. defe⦠By the time he's writing "On Behaviorism" the differences between his behaviorism and the cognitive paradigm that replaced it are less significant than philosophers familiar only with his early work realize. . Another factor at work here is that when people speak of Skinner, they are usually speaking of early Skinner. Twentieth-century meta-ethics has two origins. [49], Among the developments that resulted in the revival of metaphysical theorizing were Quine's attack on the analytic–synthetic distinction, which was generally considered to weaken Carnap's distinction between existence questions internal to a framework and those external to it. Russell's many appearances on the BBC also helped to promote the public understanding of ideas. People like me, who have been trying to do philosophy for more than forty years, do in due course learn, if theyâre lucky, how to do what theyâve been trying to do: that is, they do learn how to do philosophy. For summaries and some criticism of the different higher-order theories, see Van Gulick, Robert (2006) "Mirror Mirror – Is That All?" is the best example for this. [7], The history of analytic philosophy (taken in the narrower sense of "20th-/21st-century analytic philosophy") is usually thought to begin with the rejection of British idealism, a neo-Hegelian movement. Finally, analytic philosophy has featured a certain number of philosophers who were dualists, and recently forms of property dualism have had a resurgence; the most prominent representative is David Chalmers. Since Goodman, aesthetics as a discipline for analytic philosophers has flourished. A perfect example would be verificationism/behaviourism. Be careful about this list.. Users with a history of such comments may be banned. Another example, perhaps better, is logical positivism. He wrote a number of books aimed at the general public, including The History of Western Philosophy which became enormously popular, and in 1950 he was awarded the Nobel Prize in Literature. Dennett's version is more balanced; it explains rather than denying. [â]fitzgeraldthisside 4 points5 points6 points 7 years ago (1 child). One striking difference with respect to early analytic philosophy was the revival of metaphysical theorizing during the second half of the 20th century. However, verificationism is now completely discredited, and almost completely irrelevant to any current major debate in analytic philosophy. Word-for-word, I don't think there's anything as influential as the Gettier Paper. I also suggest you check out the British Idealists (ex. Phillips became two of the most prominent philosophers on Wittgenstein's philosophy of religion.[42]. ". Quine. [38] Analytic epistemology and metaphysics has formed the basis for some philosophically-sophisticated theistic arguments, like those of the reformed epistemologists like Plantinga.. ),. [2][3][4] It also takes things piecemeal, "an attempt to focus philosophical reflection on smaller problems that lead to answers to bigger questions. Essential Analytic Philosophy This is a list of some of the works which comprise the so called Analytic tradition including American Pragmatism and Ordinary Language Philosophy. He definitely deserves to be on that list. Members of this school seek to apply techniques of analytic philosophy and modern social science such as rational choice theory to clarify the theories of Karl Marx and his successors. Linguistic philosophy gave way to the philosophy of language, the philosophy of language gave way to metaphysics, and this gave way to a variety of philosophical sub-disciplines. free will) require more development. Ayer and the logical positivists. It then takes decades for someone to fish out the good parts from the corpse of the original version and rehabilitate its public image. As a historical development, analytic philosophy refers to certain developments in early 20th-centuryphilosophy that were the historical antecedents of the current practice. I would add that Searleâs work in philosophy of mind and language has been very influential, and Plantingaâs work in epistemology and philosophy of religion is also significant. This is extremely historically significant in analytic philosophy. Nagel's "What is it Like to be a Bat?" Perhaps the most influential being Elizabeth Anscombe, whose monograph Intention was called by Donald Davidson "the most important treatment of action since Aristotle". Hegel. [â]blacksheep1 0 points1 point2 points 7 years ago (0 children). Analytic philosophy is a branch and tradition of philosophy using analysis which is popular in the Western World and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era and continues today. I would've liked the "must-read" version of the question better, had it not suffered from the same content-killing generality, and were it not something you could easily answer with a quick look at Wikipedia.
2020 analytic philosophers list
| 215,019
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Pinit Scientific Anglers Mastery Series Anadro Fly Lines(0) No Reviews yet SKU: SA-10502 $79.95 In Stock Available: 8 Gift Certificate Amount Is Variable Amount Line Item Weight - WF10F WF4F WF5F WF6F WF7F WF8F WF9F Quantity -+ Add to Shopping Cart Reviews 0 BackRatings & ReviewsNo reviews availableBe the first to Write a Review
| 322,057
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NOAA Climate Change Data Manipulation Charge: Scandal or Nothing to See Here?
Settled science and confirmation bias all the way down
Did National Oceanic and Atmospheric Administration researchers rush and manipulate data back in 2015 in order to publish a high-impact study in Science disproving the notion that the rate of man-made global warming has slowed significantly after 2000? That is certainly the way that an explosive article at the Daily Mail portrayed the claims by prominent and just retired NOAA data slinger John Bates against his former (also now retired) colleague Tom Karl. Characterizing Bates as a whistleblower, the Mail reported that'.
Specifically, Karl and his colleagues in their "pausebuster" 2015 study used improperly archived and vetted data on sea surface and land temperature trends that showed considerably more warming than other datasets did at the time. "The central estimate for the rate of warming during the first 15 years of the 21st century is at least as great as the last half of the 20th century. These results do not support the notion of a 'slowdown' in the increase of global surface temperature," concluded the study.
Bates' claims have reignited the debate over just how "settled" the science of man-made climate change is. Interestingly, Energy & Environment News reports that in an interview with Bates that he expressed a "significantly more nuanced take" about what happened with the NOAA data than the one found in the Mail. According to E&E News:.
On the other hand, it is the plain fact that Bates did assert in a his February 4 post "Climate scientists versus climate data" over at the invaluable Climate Etc. website run by climate researcher Judith Curry that Karl had put his thumb on the scale by urging colleagues to make adjustments to the temperature data that maximized warming. So what claim is Bates really making? Did Karl and colleagues purposedly manipulate the data to get the result they wanted or were they just irresponsibly sloppy and less transparent than they should have been about what they had done? Or is Bates saying he thinks that the sloppiness and lack of transparency was deliberately used to hide data manipulation?
All too predictably, this contretemps has most everyone rushing to find data that confirms what they already think. "No Data Manipulation in 2015 Climate Study, Researchers Say," headlines The New York Times. "As planet warms, doubters launch a new attack on famous climate change study," reports The Washington Post. "House Committee to 'Push Ahead' With Investigation Into Alleged Climate Data Manipulation at NOAA," reports The Daily Caller, citing claims from Committee on Science, Space and Technology aides that other unnamed NOAA whistleblowers are coming forward. Fox News headlines, "Federal scientist cooked the climate change books ahead of Obama presentation, whistle blower charges."
Defenders of Karl's 2015 NOAA article rightly point to an independent Science Advances study just published in January that basically concluded that the study's temperature adjustments were properly done and that the increase in sea surface temperatures had not slowed down after 2000.
That being said, it is a bit puzzling that the Science Advances study does not cite another prominent study from Nature Climate Change published in February 2016 in which a group of researchers led by Canadian climate scientist John Fyfe concluded that global warming hiatus is real. Bates does cite the 2016 Nature Climate Change study as evidence against the findings reported by Karl and his colleagues in 2015. Clearly, the Nature Climate Change study's conclusions strongly contradicted Karl's 2015 Science article and the new results reported in Science Advances. Interestingly, neither the Times nor the Post stories mention the Nature Climate Change study, but Fox News did.
Apparently, NOAA is considering an investigation into Bates' allegations and House Committee on Science, Space and Technology Chair Lamar Smith (R-Tex.) is renewing his demand that NOAA turn over emails related to the how the 2015 study was managed.
For those who want to wade further into charges and counter-charges click on over to Climate, Etc. where Bates is responding to various critics' claims.
| 162,746
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TITLE: Proving properties of a family of measurable sets
QUESTION [0 upvotes]: Let $m$ be a lebesgue measure and $\{A_n\}_{n\in \mathbb{N}}$ a family of measurable sets on $[0,1]$,and $F=\{x:\forall n\in\mathbb{N} \exists k\geq n \text{ s.t } x\in A_k \}$
Prove the following:
$F=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$
if $m(A_n)>\delta>0$ for all $n$ then $m(F)>\delta$
if $\sum_{n=1}^{\infty}m(A_n)<\infty$ then $m(f)=0$
there is a familly $\{A_n\}_{n\in \mathbb{N}}$ s.t $\sum_{n=1}^{\infty}m(A_n)=\infty$ and $m(F)=0$
1) I started with taking and element from the RHS and LHS and show it bidirectional contained to prove the equality.
Let $x\in \bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$ that mean the for all $n\in \mathbb{N}$ the element $x\in\bigcup_{k=n}^{\infty}A_k$ which means that for all $n\in \mathbb{N}$ there are some $k\geq n$ such that $x\in A_k$ but this is the definition of $F$ so $x\in F$.
Now to take $x\in F$ and show that $x\in \bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k$ seems to be the same claims as before
2) I have thought about way to start using the definition that $E$ is lebesgue measurable then for all $A$ we have $m^*(A)=m^*(A\cap E)+^*(A\cap E^C)$ but I do not have much to say on the compliment of the set $A_k$
3) I tried using the subadditive of the measure and the fact the a bounded set a measure $0$ but did not manage
4) I do not have a clear idea how to approach this
REPLY [1 votes]: 2)Νote that $F=\limsup_nA_n$
Use the inequality $$m(\limsup_nA_n) \geq \limsup_nm(A_n)$$
3) It is the Borel-Cantelli lemma.
4)Take the sets $A_n=[0,\frac{1}{n}]$
| 207,733
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Hot Links! What Should Kim and Kris Do With Their Wedding Swag Now?
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/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Anatole Dedecker
-/
import topology.separation
/-!
# Extending a function from a subset
The main definition of this file is `extend_from A f` where `f : X → Y`
and `A : set X`. This defines a new function `g : X → Y` which maps any
`x₀ : X` to the limit of `f` as `x` tends to `x₀`, if such a limit exists.
This is analoguous to the way `dense_inducing.extend` "extends" a function
`f : X → Z` to a function `g : Y → Z` along a dense inducing `i : X → Y`.
The main theorem we prove about this definition is `continuous_on_extend_from`
which states that, for `extend_from A f` to be continuous on a set `B ⊆ closure A`,
it suffices that `f` converges within `A` at any point of `B`, provided that
`f` is a function to a T₃ space.
-/
noncomputable theory
open_locale topological_space
open filter set
variables {X Y : Type*} [topological_space X] [topological_space Y]
/-- Extend a function from a set `A`. The resulting function `g` is such that
at any `x₀`, if `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `g x₀` is defined to be one of these `y`. Else, `g x₀` could be anything. -/
def extend_from (A : set X) (f : X → Y) : X → Y :=
λ x, @@lim _ ⟨f x⟩ (𝓝[A] x) f
/-- If `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `f` tends to `extend_from A f x` as `x` tends to `x₀`. -/
lemma tendsto_extend_from {A : set X} {f : X → Y} {x : X}
(h : ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : tendsto f (𝓝[A] x) (𝓝 $ extend_from A f x) :=
tendsto_nhds_lim h
lemma extend_from_eq [t2_space Y] {A : set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A)
(hf : tendsto f (𝓝[A] x) (𝓝 y)) : extend_from A f x = y :=
begin
haveI := mem_closure_iff_nhds_within_ne_bot.mp hx,
exact tendsto_nhds_unique (tendsto_nhds_lim ⟨y, hf⟩) hf,
end
lemma extend_from_extends [t2_space Y] {f : X → Y} {A : set X} (hf : continuous_on f A) :
∀ x ∈ A, extend_from A f x = f x :=
λ x x_in, extend_from_eq (subset_closure x_in) (hf x x_in)
/-- If `f` is a function to a T₃ space `Y` which has a limit within `A` at any
point of a set `B ⊆ closure A`, then `extend_from A f` is continuous on `B`. -/
lemma continuous_on_extend_from [t3_space Y] {f : X → Y} {A B : set X} (hB : B ⊆ closure A)
(hf : ∀ x ∈ B, ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : continuous_on (extend_from A f) B :=
begin
set φ := extend_from A f,
intros x x_in,
suffices : ∀ V' ∈ 𝓝 (φ x), is_closed V' → φ ⁻¹' V' ∈ 𝓝[B] x,
by simpa [continuous_within_at, (closed_nhds_basis _).tendsto_right_iff],
intros V' V'_in V'_closed,
obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, is_open V ∧ V ∩ A ⊆ f ⁻¹' V',
{ have := tendsto_extend_from (hf x x_in),
rcases (nhds_within_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩,
use [V, is_open.mem_nhds V_op hxV, V_op, hV] },
suffices : ∀ y ∈ V ∩ B, φ y ∈ V',
from mem_of_superset (inter_mem_inf V_in $ mem_principal_self B) this,
rintros y ⟨hyV, hyB⟩,
haveI := mem_closure_iff_nhds_within_ne_bot.mp (hB hyB),
have limy : tendsto f (𝓝[A] y) (𝓝 $ φ y) := tendsto_extend_from (hf y hyB),
have hVy : V ∈ 𝓝 y := is_open.mem_nhds V_op hyV,
have : V ∩ A ∈ (𝓝[A] y),
by simpa [inter_comm] using inter_mem_nhds_within _ hVy,
exact V'_closed.mem_of_tendsto limy (mem_of_superset this hV)
end
/-- If a function `f` to a T₃ space `Y` has a limit within a
dense set `A` for any `x`, then `extend_from A f` is continuous. -/
lemma continuous_extend_from [t3_space Y] {f : X → Y} {A : set X} (hA : dense A)
(hf : ∀ x, ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : continuous (extend_from A f) :=
begin
rw continuous_iff_continuous_on_univ,
exact continuous_on_extend_from (λ x _, hA x) (by simpa using hf)
end
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The Botswana Congress Party, BCP, says it wants the Directorate of Public Prosecution (DPP) to prosecute the minister who allegedly attempted to rape a senior government official during an official trip.
“Allegations of rape concerning a minister are matters of national importance,” said a press statement, issued by BCP Executive Secretary, Mr. Kagiso Tshwene. “The public has the right to know whether it is true or not and if such allegations have been reported.”
Tshwene said it would be very disappointing if no action is taken to clear the matter.
“It is further disappointing for the victim to have been transferred as a way to cover up.”
As BCP, he said, they are not trying to politicize the whole issue but stated that this issue is very sensitive and needs urgent attention.
Speaking to Sunday Standard, the secretary general of the Botswana Democratic Party, Women’s Wing, Keneilwe Mathangwane, said, “We have not yet received any such complaint.”
Mathangwane added, “I do not want to be dragged into the issue because, as I told you before, I am not aware of such a case and the Women’s Wing has never discussed any rape issues either in our workshops or congresses.”
The Sunday Standard broke the story two weeks ago but the police have not yet given any concrete explanation why the case has not been investigated or whether they are investigating such a case at all.
| 392,408
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Kobe Bryant has two jerseys retired by the Lakers. Five NBA championship winning legend. Sadly, I only saw him play once in the Staples. 2009 against the Denver Nuggets. Carmelo Anthony was a nightmare but Kobe won out the duel. Trevor Ariza had a great game but left next season. Ron Artest replaced him and a second Championship was won.
Watching the Lakers play Golden State, the new best team at present. It's close but with Durrant it will be a hard battle.
Kobe's speech and video poem were inspiring directions to what any sports player should want to become.
Sent from my SM-G955U using Tapatalk
Results 1 to 6 of 6
Nba
Last edited by GreenLake; 19-12-2017 at 08:13 AM. Reason: Was pissed and typos were horrific
Seen the Mamba play just the once when team USA played team GB in Manchester for a 2012 warm up. Would have loved to have seen him in the purple and gold but never got it organised.
The young kids on the team now are playing well but just encountering the usual issues of being a young team. Ingram and Ball will be fun to watch over the next few years!
Well.........I've been to see the Rocks play a few times and they sucked
. You do get to have a beer at the games though
Lakers are at Rockets tomorrow night and also on New Years Eve.
I'm seeing the Celtics game in London while I'm back, should be fun21.05.2016. I Was There.
.net PM board "Biggest Slaver of the year 2014"
Less talk, more gifs.
:
Seen the Bulls play at the United centre, totally different experience from being at a football or rugby match here!
loved it though, great experience
Bookmarks
| 41,190
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TITLE: Proof by induction that $x_{n} + by_{n} = a$
QUESTION [1 upvotes]: I have an algorithm where a and b $\in$ Z+
MUL(a,b)
x = a
y = 0
WHILE x >= b DO
x = x-b
y = y+1
IF x = 0 THEN
RETURN(y)
ELSE
RETURN(-1)
I need to prove by induction that
$x_{n} + by_{n} = a$
is true if $x_{n}$ and $y_{n}$ are the values for x and y after n runs of the WHILE-loop.
My approach is to to prove that its true for the basecase where the WHILE-loop runs 0 times.
When the WHILE-loop runs 0 times (n = 0), we know that y = 0 since y is counting the amount of times the WHILE-loop is ran.
So we have:
\begin{equation}
\begin{split}
x_{0} + by_{n} =& a \\
x_{0} + b*0 =& a \\
x_{0} =& a \\
\end{split}
\end{equation}
If we input $a = 1$ og $b = 2$ into $MUL()$. The result will be be $y = 0$ and $x = 1 = a$.
I'm not sure if this basecase step is correct and I am not sure how to proceed from here. I know I need to prove if it's true for n + 1. But how I do it I don't know. I hope somebody can help me!
REPLY [0 votes]: Strictly speaking, we do not know anything like "$y$ is counting the amount of times the WHILE loop is run" beforehand; however, this (i.e., $y_n=n$) may be a corollary of an analysis of the code.
After $0$ loops, i.e., immediately before entering the while loop, only x = a and y = 0 are executed, hence indeed $x_0=a$ and $y_0=0$. Thus
$$\tag1 x_n+by_n=a\qquad\text{and}\qquad y_n=n$$
holds for the base case $n=0$
Assume $x_n+by_n=a$ and $y_n=n$ after $n$ loops. Then in the next loop, the statements x = x - b and y = y + 1 are executed. It follows that $x_{n+1}=x_n-b$ and $y_{n+1}=y_n+1$. Therefore, using the induction hypothesis,
$$x_{n+1}+by_{n+1}=x_n-b+b(y_n+1)=x_n+by_n=a $$
and
$$y_{n+1}=x_n+1=n+1,$$
i.e., $(1)$ with $n\leftarrow n+1$. Therefore we have shows that $(1)$ holds for all $n\in\Bbb N_0$.
Once this is shown, we see from $(1)$ that
$$x_n=a-by_n=a-bn. $$
So for $n$ large enough, $x_n\ge b$ will not hold, and the WHILE loop will terminate.
Let $n$ be minimal with $x_n<b$. Then the return value of MUL will be $y_n$ if $x_n=0$, and $-1$ otherwise. As $x_n=0$ implies $by_n=a$, we conclude that for $a,b\in \Bbb Z^+$,
$$\text{MUL}(a,b)=\begin{cases}\frac ab&\text{if }b\text{ divides }a\\-1&\text{otherwise}\end{cases} $$
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121 Comments
- dheller, on 10/12/2007, -2/+42Wouldn't it be great if we could do this? Take off at 5p EST and land at 4p PST.
Too bad they are only marketing to the exec class to start.
- atlantisceo, on 10/12/2007, -2/+21Not the newest news if you watch the aviation industry, but definitely worth a digg.
Boeing is actually using some of the technology developed for their version of the high-speed jet on their new 787 (which is supposed to be available in late 2007 I believe). The 787 will travel faster than just about every other jetliner out there today. Normally when an aircraft is traveling near the speed of sound the plane is producing tons of drag - Boeing has figured a way to push this point to a higher speed. Very cool technology.
@stealthboy - don't mistake Boeing/Lockheed with the makers of the Concorde. Also, there has been decades between that very old tech and this very new tech. Apples and Oranges...
- Vlatro, on 10/12/2007, -2/+21"Time to make some huge leaps again. I want NY to Melbourne in 2 hours. Hell, I want NY to Melbourne in 2 minutes"
2 Min. Let's see... That's about 10,355 Miles (estimate compliments of indo.com)
That's 86.3 Miles per second
The speed of sound is 0.211446403 miles per second
Your speed divided by the speed of sound = Mach 408.1
Air friction would basically incinerate even the best insulations at that speed. You'd be riding a wave of superheated plasma, even your ashes would be burned up. Of course accelerating that fast to begin with would force all the blood in your body through the tissue and skin right out of your back, and you'd be unconscious before the pain even registered. Of course the amount of fuel required to maintain that speed for 2 minuets would be substantial, but I'm betting it would most likely ignite under the extreme temperatures, turning the vehicle into a giant flying Molotov cocktail.
But on the bright side, they would be awfully hard to Hijack. Maybe that would shorten the lines at the security checks.
- jav1231, on 10/12/2007, -2/+21"Even at say 35,000 feet you can't break the speed of sound over a populated area without angering some group"
So what's the downside? :)
- Chewie67, on 10/12/2007, -5/+24Yeah, it will be for private jets to start with, but if it works, and there is demand, it will move over to the mainstream.
If you think about it, advances in travel really have been at a stand-still for about 60 years now. From 1900 to 1950 we went from steam trains and horse drawn carriages to jet airliners and personal automobiles. Since then, we've just been refining, not innovating.
Time to make some huge leaps again. I want NY to Melbourne in 2 hours. Hell, I want NY to Melbourne in 2 minutes.
- saleens281, on 10/12/2007, -0/+19@all of you talking about supersonic over land mass:
You can't do it because of the sonic boom it creates. This jet has no sonic boom, therefore there's absolutely no reason not to allow it to fly over land mass at supersonic speeds. Critical thinking skills help every now and again, try em ;)
- NSMike, on 10/12/2007, -1/+18Okay, to every idiot out there whose attitude is, "He he! NARF! Concorde!" Shut up. Seriously, shut up. I've already seen about 7 or 8 Concorde comments.
Last time I checked, there aren't many aeronautical engineers on digg. So unless you actually know what you're talking about, shut your hole. Do you honestly believe that someone else didn't look at the designs for a new civilian supersonic aircraft and NOT take into account the Concorde? Did you honestly think someone would decide, "Hey, let's re-brand that old SST as a new, hip business jet, and hope they conveniently forget what a failure the other one was! Ignore the technical problems and improvements made over the past several decades since the first Concorde! We want the SAME dud product!"
Shut up.
- sinfree, on 10/12/2007, -3/+18You die.
- 0crabby0, on 10/12/2007, -0/+11I remember something in the 1970's about using evacuated tunnels to travel fast below ground(+400mph?).
Plus I just think terminals that look like this are just cool
- MOJIRA, on 05/17/2008, -5/+16EIGHTY-EIGHT MILES PER HOUR!!!!
- Terc, on 10/12/2007, -2/+13@DBCubix,
Did you even read the Digg headline? Supersonic travel over land is EXACTLY what we're talking about here.
- ajchavar, on 10/12/2007, -7/+17why would you even think that that was an acceptable thing to say?
- mrbambastik, on 10/12/2007, -7/+17Wired's article states something untrue: "Concorde was barred from flying at supersonic speeds over the United States [...] because excessive noise was produced by pressure waves colliding in the plane's wake.".
The thing that killed the Concorde was purely and simply American sour grapes when Boeing finally admitted that their own late entry into supersonic air travel was over budget, overdue and over weight and would never fly. There were plenty of American airline with options to buy, but they all pulled out when the American government then decided to ban overland commercial supersonic flight, making the aircraft practically useless to American airlines. Of course, many military aircraft continue to fly supersonic over the American mainland, and cows still give uncurdled milk, children are not thrown from their beds by the sonic shock-wave, and there are not hoards of angry sleep-deprived and shell-shocked American citizens beating at the doors of congress to limit this evil.
- atlantisceo, on 10/12/2007, -1/+10I think you missed the sarcasm...
- tofuoni, on 10/12/2007, -1/+9"I'm sure at that speed it can't be good."
You're sure? Really? What makes you sure? What is your educational background? The truth is, you can't possibly be sure. What you are is afraid. You have fear that the fuel effeciency isn't good. You're uncertain because you have no knowledge of jet engines. And you doubt because you're typically cynical.
So that's what your comment amounts to, fear, uncertainty, and doubt. Good job.
- stealthboy, on 10/12/2007, -0/+7Uhh... the Concorde failed because it was way to expensive and most of the initial buyers backed out, leaving an expensive niche market. The failure of the one Concorde in 2000 was because of a shredded tire after it ran over a strip of metal on the runway; it did not "fall apart in mid flight". Do some research before you spout off.
- thefirelane, on 10/12/2007, -0/+7>Why haven't we made any progress since then? Air flight from NY to LA is virtually identical today as it was in 40 years ago.
We have made progress, the flights are much more numerous and less expensive.
- merreborn, on 10/12/2007, -0/+6The concorde failed for many reasons, not the least of which was a Concorde bursting into flame on take off in 2000.
- atlantisceo, on 10/12/2007, -2/+8Not at all - they are using engines vastly different from today's turbofans. Some are using scram jets, others are using variations. These aircraft won't be worse for the environment than what we have now.
What we need to get rid of is the straight turbojets from the 1960's and 70's that can still be found on tons of business jets (and even some really old jetliners). Those things emit black clouds when running.
- pcheaven2k, on 10/12/2007, -1/+7I also want to point out that the sonic boom is caused by the air that rushes over the surface of the plane and collides behind the plane. These new jet designs disperse that air as it travels across the fuselage so that the wake behind the plane is much much larger and therefore there is very little collision of air.
- FanofFilm, on 10/12/2007, -0/+5Actually, I am an aerospace engineer. And I am on digg. And you're right about all the other stuff.
- pcheaven2k, on 10/12/2007, -1/+6Alright let me point out a few things.
1.) The Concord was designed and they were all built like 30 years ago (some maybe a few years less than that)
2.) The Concord was NOT DESIGNED worth a damn, it was basically a quickly slapped together airplane with oversized engines and extra fuel capacity. It was never designed to be quiet or to fly Mach+ over land
3.) The new corporate jets being built by BOEING/LOCKHEED are being designed from the ground up to be super fast, light weight, extremely quiet and fuel efficient.
Additionally this new bread of Mach+ planes are all being designed with special wing configurations/angles/surfaces that disipate the air more widely and therefore minimize or maybe even elimenate the sonic boom effect. Also, from what I have read these planes will fly at Mach 1.8 and burn less fuel than a freaking Cesna flying at 300 mph.
- SlowOnTheUptake, on 10/12/2007, -0/+5I'd be content if they could arrange to get me from the main concourse to the gate in two hours.
- matts0344, on 10/12/2007, -2/+7I want NY to London in 1 hour!
Transatlantic train ftw, too bad it will cost trillions and a century to build.
Guess the next best thing is supersonic jets.
- Xalorous, on 10/12/2007, -0/+4@smackfu - sonic boom is a legal issue, the reason it was not a problem is that they only flew the thing > M when over the ocean. They only flew it over the ocean because the only flights long enough to be practical were transoceanic flights. The reason for the downfall of the Concorde is that it was extremely expensive to operate, and the fleet was aging to the point where it needed to be replaced.
And if you'll notice, the jet in question will accomodate 12 passengers. Not exactly a jetliner. This is just a souped up corporate jet.
- Leathersoup, on 10/12/2007, -2/+6My big issue with SUV's isn't the fuel efficiency. As far as I'm concerned if you can afford to waste fuel for a status symbol all the power to them. My problem is people who own SUVs but can't drive.
If people are unable to drive properly, they should not be allowed to own such a large vehicle. SUVs seem to be compensation for a lack of ability to drive. I think of them as the bullies of the road.
- JAKN, on 10/12/2007, -0/+4@0crabby0
there is a guy who is really pushing for "Evacuated Tube Transport" - but he says we can do it above ground, and even through water (have my doubts). He quotes a study for some science tunnel (could be a particle accelerator) that is as close to a perfect vaccum as you could expect - and it held it's pressure for 1 year with no discernable changes.
He lectures at the Engineering school I graduated from, about once a year. Apparently he's got a business deal in China right now. Based on this comment I clearly wasn't paying enough attention.
- wistar, on 10/12/2007, -0/+4As if Europe allowed supersonic overflight.
- JasonPrini, on 10/12/2007, -0/+4Concorde : Maiden flight 2 March 1969
They were designed almost 50 years ago. The first prototypes were being built in the mid 60s.
From the Wikipedia entry "...although the outside air temperature was typically -60 °C, air friction would heat the external skin at the front of the plane to around +120 °C making the windows warm to the touch..."
They flew for over 30 years.... badly designed?
- rnelsonee, on 10/12/2007, -0/+4RTFA.... they've reduced the sonic boom to 1% of the levels that are usually associated with sonic booms.
- wistar, on 10/12/2007, -0/+4This is not what killed Concorde. Concorde wasn't killed. It fThere was no supersonic overflight of Europe allowed, either. Extraordinary cost of operation throughout the entire SST realm was the dagger to the heart.
- dougmc, on 10/12/2007, -0/+3`Of course, many military aircraft continue to fly supersonic over the American mainland'
Sure, if they're actually in a combat or emergency situation, but if not, only in certain areas.
For more details, read
- HRF1, on 10/12/2007, -0/+3It's not the 4 to 6 hour flights that I want shortened, it's the 12 to 13 hour flights from Japan to Chicago or New York I want shortened with improved aero transportation! Flights from LAX to LGA seems like nothing if you've ever flown cross continent.
- beand1p, on 10/12/2007, -1/+4@ 0crabby0
I don't think that is a terminal, I think that looks like a leaked Halo 3 map.
- NSMike, on 10/12/2007, -1/+4Supposed to. Then again, by now, we were supposed to be commuting to Mars to work, if you follow past predictions.
- ImmortalLobster, on 10/12/2007, -0/+3Someone here mentioned a lack of Aero Engineers here, well hell, ill just register then ;)
The idea is actually rather cool looking, seems that the point of the inverse V tail is to funnel the shockwave through the tail creating a very focused beam of noise, rather then dispersing the wave all over the place, which is whats heard on the ground. rather the concept I see here is the shocks would be focused in such a way that they all ricohette in the tail, and either disperse, or as I said, create a focused wave which would not be audible outside of the chamber, kind of like a hose, the sound cannot escape this channel much like water cant escape a hose. eventually the waves would disapate and unless your standing directly behind it...unlikely, your not going hear it.
with regards to the FAA, the FAA blocked supersonic flight over the US becuase of the noise, certain planes can, and most likely will be able get a specal permit to breach that rule provided they can prove the plane will not produce an 'audible' supersonic boom.
as to the concorde....someone mentioned ***** design, he/she is right. the planes could go fast, yes. but they were indeed literally falling apart in mid air, also they were inefficient as hell with fuel consumption.
Prior to 9/11 and the 787 Dreamliner, Boeing was developing the Boeing Sonic Cruiser, a plane desighned to cruise at .99 mach, but after the airline economic collapse of 9/11 they dropped the project and produced the 787 which sacrifices speed for fuel efficiency.
- ZachPruckowski, on 10/12/2007, -0/+3Well, the execs would be the early adopters. Then once it gets cheap enough (component prices coming down, more fab plants come online, better designs, whatever) and gets big enough (100 seat jets or so), we can talk about using it for vacations or something. I'm sure it'd be great for a East Coast to Las Vegas flight Friday afternoon/evening and back on Sunday afternoon. Save 3 hours of flight time, the casinos have you gambling for 3 more hours.
- ImmortalLobster, on 10/12/2007, -0/+3Remember the concept of the sound barrier, after breaching it flight was easier on the other side? the Lift to drag curve drops off drastically behind Mach1
see this image, which is accurate to most any application
- sporkman, on 10/12/2007, -3/+6did you read the ***** article? I didn't think so.
it states why this will be better than the concorde, it's quieter, and faster.
and the reason the concorde failed is because they were ***** quality, and were falling apart in mid flight.
- rnelsonee, on 10/12/2007, -0/+3The 'whisper' comment in the artice was talking about the sonic boom, not engine noise. The article is devoid of any real detail, but this probalby involves breaking up the waves at the front on the sonic wave, so they hit the ground at different times. How much of a delay will determine how much quieter their sonic boom will be from a traditional one - although it is hard to imagine something sounding like a 'whisper'. Hell, even if it is a whisper sound, you're still subjecting millions of Americans to the sound whenever you fly from NY to LA.
- jzulli, on 10/12/2007, -0/+3That's a funny thought.. traveling back in time!
- iamhrh, on 10/12/2007, -1/+4Good god wired's website blows. It is always so slow!
- ImmortalLobster, on 10/12/2007, -0/+2Agreed, the wing failed at I believe 147% instead of the mandatory 150%, am I correct?
what I was saying just the political jumble a cross country deal brings to the table as t is, eitherway, Airbuses current slump is its own doing
- NospmisRemoh, on 10/12/2007, -0/+2"...he sonic boom is caused by the air that rushes over the surface of the plane and collides behind the plane..."
That is not quite what really what happens. There is a cone (or wedge) shaped shock wave that forms from the leading edge of any surface on the plane. The shape of the shock wave depends on the speed and the shape of the leading edge. This shock wave the first (an louder) sonic boom at ground level. The expansion fan at the trailing edge (the opposite of a shock wave) is the second part of the sonic boom. It is not really caused by air colliding but rather a sudden change in pressure across shock waves and expansion fans.
- FanofFilm, on 10/12/2007, -0/+2i took it he, like some other comments here, was refering to some kind of political jealousy on the part of the american government after the boeing and lockheed SST projects failed before they could even get to production, where the concorde project was able to succeed.
airbus problem with the a380 is that they just can't meet ultimate load requirements. it's not american political pressure holding it back. it's poor engineering.
- ImmortalLobster, on 10/12/2007, -0/+2Actually, a plane flying in a supersonic state gets better fuel efficiency then a plane flying in subsonic, this aircraft and related designs could pull the ticket prices back down some.
- ImmortalLobster, on 10/12/2007, -0/+2Buelldozer: Dropp -off trend continues, the faster the plane goes the less drag, its inverse in comparison to normal subsonic flight
dougmc: the image was from google image search, needed a fast example,but every drag-to speed curve will sho that, once Mach oneis breached, drag goes down. Yes, planes designed for supersonic flight exhibit less drag, but they also exhibit less lift, which means, *gasp* more fuel needed to keep it afloat. shockwaves produce more lift then normal air, and the waves also distribute skin friction further over the aircraft thereby decreasing overall drag. there is no simple way to put all this into writing. but once a plane is cruising in supersonic speeds, it is indeed more fuel efficient
- atlantisceo, on 10/12/2007, -0/+2If "slightly quieter" is 1% the volume of the original, sure. If I were 1% my height I wouldn't be slightly shorter.
- atlantisceo, on 10/12/2007, -0/+2Because all of the shuttle crashes happened within 10 years of the vehicle's introduction...
What bull - the new plane wouldn't be certified by the FAA (and used by airliners) unless it was safe. The shuttle was always known to be risky (I believe NASA expected an 1 accident for every 200 launches). The new SST is completely different - a plane fling Mach 1.8 isn't inherently more dangerous than one going Mach 0.8.
It is this type of thinking that slows down development of new technologies. Just stop.
- ImmortalLobster, on 10/12/2007, -0/+2No, from the description, the sonic boom wouldnt be audible unless you elect to stand right behind it, and the future is smaller aircraft, since less people are flying and securiy is getting tighter, dollars and sense it makes perfect sense ^_^
- Show 51 - 100 of 119 discussions
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Vanuatu Accommodation
Vanuatu is a "Y" shaped chain of 83 islands lying 2,500km northeast of Sydney Australia. The main accommodation is located on the island of Efate, the two other popular island destinations are Tanna Island which is famous for its volcano and Espiritu Santo for its diving.
Efate
Benjor Beach Club luxury villas (suitable for up to 10 people).
Breakas Beach Resort
Enjoy your holiday at this unique boutique resort on the magnificent surf beach at Pango. Nestled amongst beautifully landscaped grounds, the exclusive Breaka's Beach Resort guarantees to rejuvenate, pamper and indulge every guest. Relax at the Infinity Pool Bar while sipping a South Pacific Cocktail. An elegant and cosmopolitan resort, Breaka's offer 15 Bungalows situated on a sweeping, white sandy beach, fringed with palms.
Chantillys on the Bay
Chantilly's On The Bay offers self contained accommodation just 5km right from Chantilly's own jetty and the shops are just minutes away.
Erakor Island Resort and Spa
Eratap Beach Resort is a small, exclusive resort located less than 20 minutes drive south east of Port Vila. Swim off the jetty in the turquoise lagoon, snorkel for giant star fish or organize to be dropped off on a deserted island, the choices of activities are endless. This boutique resort caters for guests seeking a private, high quality resort with personal service in a beautifully preserved natural environment making this the perfect location for your next holiday to Vanuatu.
Hideaway Island Resort
Hideaway Island Resort and Marine Sanctuary is a picture perfect tropical island with sand and coral beaches surrounded by turquoise blue water. It is only a short 15 minute bus and ferry trip form the airport and Port Vila. Hideaway Island's Marine Sanctuary is one of the few places on earth where you can hand feed hundreds of 'tame' brilliantly coloured fish right from the beach. As a marine sanctuary there are many wonderful dive sites near by and their excellent facilities and dive instructors will ensure you have memorable diving experiences.
Iririki Island Resort
The pearl of the South Pacific - Iririki is Vanuatu's premier boutique Resort Hotel. Iririki Island Resort is situated on the site of the former British High Commissioner's residence on a private 69 acre island. It's only three minutes across the bay from Port Vila by complimentary ferry and ferries run 24 hours a day. Iririki has all the comforts of an international resort with privacy in a prime position. Iririki's no children policy enhances the romantic ambience.
Iririki Snorkelers Cove
Iririki Snorkelers Cove is an exciting new precinct of Iririki Island Resorts and Spa. This additional resort complex includes contemporary style Deluxe Rooms and Penthouses fitted out to five star standards. The resort also includes facilities in the Lagoon Pool Village including one of the largest saltwater swimming pools in the Pacific. Children are welcome at Iririki Snorkelers Cove making the resort an inviting escape for families, couples and groups.
Kaiviti Village Motel
Less than 1 km from the town centre of Port Vila and located in the middle of the restaurant belt, the Kaiviti Village Motel offers a range of self-catering units that are ideal for families or those combining business with pleasure. This Vanuatu accommodation offers an unobtrusive home-away-from-home atmosphere.
Le Lagon Resort Vanuatu
Le Lagon Resort Vanuatu is nestled in 75 acres of exotic tropical gardens fronting the shores of the beautiful crystal waters of Erakor Lagoon. There are 141 rooms of various categories ranging from standard rooms and deluxe rooms to bungalows, all with a garden or lagoon views. In addition there are four luxurious over-water suites, all with lagoon views. All offer tranquil garden or lagoon views, each one traditionally finished with a thatched roof. Each room opens out to a furnished patio or balcony, perfect for enjoying one of the South Pacific's finest tropical settings.
Le Meridien Port Vila Resort, Spa & Casino
Le Meridien Port Vila Resort, Spa and Casino is Vanuatu's premier international deluxe resort. This low rise resort is set on 50 lush landscaped acres with a private beach frontage overlooking the tranquil waters of Erakor Lagoon. Features include a nine hole golf course, floodlit tennis courts and a children's "Penguin Village". The resort has 150 guest rooms featuring local arts and materials with a relaxed and friendly Melanesian atmosphere. There are also 10 Lagoon Bungalows 'fales' offering unrivalled seclusion and luxury, and linked to the main resort by a 160 metre suspension bridge.
Mangoes Resort & Restaurant
Mangoes Resort & Restaurant is situated on a ridge close to the centre of the Port Vila township, with truly magnificent views across the tranquil waters of the Erakor lagoon. This “exclusive adults” resort is a peaceful tropical escape for couples and individuals to relax and enjoy their Vanuatu experience, away from children.
Moorings Hotel
The Moorings Hotel offers modern and stylish rooms overlooking beautiful Port Vila Harbour, capturing the tropical breezes and spectacular sunsets or opening onto the “infinity” pool that runs to the waters edge. Located directly on the harbour in Port Vila on the main Island of Efate, just 10 minutes from the airport and only a short stroll to the town centre, this resort offers that special “island feel” making your stay truly memorable.
Poppys on the Lagoon
Poppy's on the Lagoon is set on the shores of Erakor Lagoon and purpose designed to take advantage of the cooling breeze of the S charming town centre, where there are plenty of restaurants and shops.
Sunset Bungalows Resort
Sunset Bungalows Resort is a stylish boutique resort, located just 5 minutes from Port Vila. This Vanuatu Resort has a maximum occupancy of 38 guests with a restricted 18 years and over policy. Sunset Bungalows Resort is a secret place to explore, sensational sunsets and balmy moonlit nights under the Vanuatu stars, enhance the romance for any couple, especially honeymooners. You will leave Sunset Bungalows Resort with many unforgettable memories.
The Melanesian Port Vila Orchid Rooms. There's a gaming lounge, 24 hour Mediterranean style restaurant, two bars, pool and tennis courts and a collection of crafts and artifacts throughout the hotel that rivals the Museum!
The Sebel Vanuatu
Overlooking Vila Harbour, The Sebel Vanuatu boasts a stunning waterfront position and extensive water views from every guestroom. Ideal for the business or leisure traveler, this contemporary hotel is conveniently located close to the shops, restaurants and local markets of Port Vila.
Vila Chaumieres
'Vila Chaumieres' is French for 'charming cottages'. It's an ideal location for lovers of fine food and wines looking for an island holiday away from the larger resorts. There are four comfortably furnished garden bungalows nestled in a lush, tropical peaceful garden setting and two rooms with spacious balconies overlooking the lagoon. All are air-conditioned with overhead fans, ensuites and a refrigerator. Vila Chaumieres caters only for adults and makes for the perfect tranquil, relaxed and romantic escape.
Espiritu Santo and Surrounding Islands
Aore Resort
Aore Resort is located on 10 acres and is an absolute beachfront resort that backs onto a cattle/coconut Plantation. The accommodation consists of 10 Double/Twin Bungalows and 8 Family bungalows. The bungalows have private bathrooms, ceiling fans, refrigerators, tea and coffee-making facilities and large balconies.
Bokissa Private Island Resort
Out on the frontier of the Pacific exists the original coral eco island “Bokissa” in an atmosphere that's a little wild… and seriously private. Presently there is only a maximum 36 guests on the island at any one time.
White Grass Ocean Resort
White Grass Ocean Resort is an ideal starting point from which to explore Tanna Island. The Resort is situated only minutes from the Tanna Island airport, with 12_12<<
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What do you get when you have a popular fastfood restaurant make a game console that could have competed against the likes of PlayStation 5 and Xbox Series X?
A prank, of course.
The notion comes from a recently launched ad by no other than KFC itself via Twitter which came with the #PowerYourHunger that depicts an almost convincing presentation suggesting that the food company is venturing to gaming.
Related stories
- KFC Tray Typer: use your phone while eating without greasing up the display
- KFC’s smart restaurant will suggest what you should order based on your face
It’s good marketing, for sure. Especially as far as enticing audiences attention to the brand.
One can also tell that the creative persons behind the advertisement have a very good taste for humor to even come up with it, particularly in light of the recently held big reveal amongst bona fide game console makers, like Sony.
But there is indeed a sense of mystery to the ad when it teased the “11.12.20” date as part of the video campaign. What that entails is still up for interpretation and future revelation—but it is likely not going to be a traditional console we have been seeing.
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TITLE: Help proof by induction
QUESTION [1 upvotes]: I just got my first induction assignment in a new course. They want me to prove by induction that:
$$\sum_{s=1}^k s*s! = (K+1)!-1.$$
The way I understand induction is that I test for the first value. Then for n and then n+1, to show the given expression is true.
I've done it for 1 and n, however I'm stuck at n+1.
$$\sum_{s=1}^{n+1} s*s!=(n+1)!-1+(n+1)*(n+1)!$$
Using maple I can see the expression is $-1+(n+2)!$ (which is true) however i dont know how to reduce/rewrite $(n+1)!-1+(n+1)(n+1)!$ to $-1+(n+2)!.$$
I've asked my friends and a older student, but to no avail. I'm hoping you guys can help.
REPLY [0 votes]: First proof the statement for some value, which you did (induction basis).
Then induction step: Let's assume that the statement you want to proof holds for K = n. Then you have to establish that it also holds for n+1.
So we know that $\sum_{s=1}^{n} s \cdot s! = (n+1)! - 1$ (induction hypothesis)
$\sum_{s=1}^{n+1} s \cdot s! = \sum_{s=1}^{n} s \cdot s! + (n+1) \cdot (n+1)! = (n+1)! - 1 + (n+1) \cdot (n+1)! = (n+1)! + (n+1) \cdot (n+1)! - 1 = (1 + (n+1)) \cdot (n+1)! - 1 = (n+2) \cdot (n+1)! - 1 = (n+2)! - 1 = ((n+1)+1)! - 1$
So then we know that the statement also holds for n+1. This concludes the proof.
So in proofs by induction you explicitly use the induction hypothesis for the induction step.
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Early ride on the Zamboni – Sunday, January 9th
Jan 9, 2011, 8:59 PM EST
It seems like we’re in the time of year in which teams play games with some pretty odd start times, so let’s take a look at some of the contests that finished a little earlier this Sunday.
Carolina 4, Atlanta 3 (OT)
All of a sudden, it’s looking like the Hurricanes might have a legitimate shot at a playoff spot … and that might come at the expense of the upstart Thrashers. At one point it seemed like Erik Cole scored a goal that made it 4-1, but it was disallowed. Atlanta tied it up to send it into overtime, but Cole ended up getting rewarded for his hard work by scoring the game-winner.
Seventh-ranked Atlanta has a five point lead over Carolina, but the Thrashers played in five more games so that lead is misleadingly small if the Hurricanes can put together some victories.
New Jersey 6, Tampa Bay 3
For only the second time this season, the Devils won a game in which they were trailing heading into the third period. It was 2-1 Lightning but the Devils produced a stunning five goal final frame to – at least briefly – break out of a considerable funk. Martin Brodeur is showing signs of life as he stopped all 19 shots doing relief work in Saturday’s 2-1 loss against Philly and made 33 stops in this one.
These are the kind of games the Lightning need to win if they hope to maintain their grasp on the top spot in the Southeast.
Andrew Raycroft win a match between two of the league’s most efficient backups, earning a 26-save shutout as the Stars beat Jose Theodore and the Wild 4-0. Dallas has now won seven games on the road, showing that they can beat just about any team if they’re on – even when Kari Lehtonen gets the night off.
Jamie Benn scored two twice in this one, including a highlight reel tally that is a likely candidate for Sunday’s goal of the)
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DEVELOPMENT OF FLUID-SEDIMENT-SEABED INTERACTION MODEL AND ITS APPLICATION
Abstract
Keywords
References
Bear, J. 1972. Dynamics of Fluids in Porous Media, American Elsevier Pub. Co., New York, 764 p.
Jiang, Q., S. Takahashi, Y. Muranishi, and M. Isobe. 2000. A VOF-FEM model for the interaction among waves, soils and structures, Proceedings of Coastal Engineering, JSCE, 47, 51-55 (in Japanese).
Kunugi, T. 2000. MARS for multiphase calculation, Computational Fluid Dynamics Journal, 9(1), 1-10.
Lee, K. H., and N. Mizutani. 2006. Local scour near a vertical submerged breakwater and development of its time domain analysis, Annual Journal of Coastal Engineering, JSCE, 53, 501-505 (in Japanese).
Liu, X., and M. H. García. 2008. Three-dimensional numerical model with free water surface and mesh deformation for local sediment scour, Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE, 134(4), 203-217.
Mizutani, N., W. G. McDougal, and A. M. Mostafa. 1996. BEM-FEM combined analysis of nonlinear interaction between wave and submerged breakwater, Proceedings of 25th International Conference on Coastal Engineering, ASCE, Orlando, FL, 2377-2390.
Nakamura, T., Y. Kuramitsu, and N. Mizutani. 2008. Tsunami scour around a square structure, Coastal Engineering Journal, JSCE, 50(2), 209-246.
Nakamura, T., and S. C. Yim. 2011. A nonlinear three-dimensional coupled fluid-sediment interaction model for large seabed deformation, Journal of Offshore Mechanics and Arctic Engineering, ASME, 133(3), 031103-1-031103-14.
Nakamura, T., and N. Mizutani. 2012. Sediment transport model considering pore-water pressure in surface layer of seabed and its application to local scouring due to tsunami, Journal of Japan Society of Civil Engineers, Series B2 (Coastal Engineering), JSCE, 68(2), I_216-I_220 (in Japanese).
Sandhu, R. S., and E. L. Wilson. 1969. Finite element analysis of seepage in elastic media, Journal of the Engineering Mechanics Division, ASCE, 95, EM3, 641-652.
Takahashi, S., K. Suzuki, Y. Muranishi, and M. Isobe. 2002. U-π form VOF-FEM program simulating wave-soil interaction: CADMAS-GEO-SURF, Proceedings of Coastal Engineering, JSCE, 49, 881-885 (in Japanese).
Tonkin, S., H. Yeh, F. Kato, and S. Sato. 2003. Tsunami scour around a cylinder, Journal of Fluid Mechanics, 496, 165-192.
Yamamoto, T. 1977. Wave induced instability in seabeds, Proceeding of Coastal Sediments ’77, ASCE, 898-913.
DOI:
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TITLE: Sharkovskii's theorem
QUESTION [0 upvotes]: the Sharkovskii's theorem states that for any continuous function $f:I\rightarrow I$, where $I = [0,1]$, if $\exists x$ such that $f\circ f\circ f (x) = x$, then for all $n\in \mathbb{N},\; \exists x,\; \underbrace{f\circ f\circ \dots \circ f}_{n \;\text{times}}(x) = x$. We note it $f^n(x) = x$.
Let $f$ be such a function. Then let $P$ be :
$$P = \{ x\in I\,|\,\exists n \in\mathbb{N},\, f^n (x) = x\}$$
How to prove that $P$ is dense in $I$ ?
REPLY [1 votes]: Why you expect that $P$ should be dense in $I$? Let $f \colon I \rightarrow I$ be the constant function $f(x)=0$. Then $f(0)=0$ and $0$ is a periodic point of any period, i.e. $f^n(0)=0$ for any $n \in \mathbb{N}$. In this case, we see that $P = \{0\}$, i.e. $P$ is not dense in $I$.
Another example is $f(x) = \exp(1-x)$, satisfying $f(1) =1$. But we have $f(x) > 1 > x$ for any $x \neq 1$ and therefore $f^n(x) > x$ for any $x\neq 1$.
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Whether you are feeling happy, angry, silly or just bored, Netflix has a show that will match your every mood. In fact, to save you time, OK! Magazine has picked the top Netflix shows to match your every mood. As the cherry on top, you can watch just one show or binge the entire series in the comfort of your own home.
More: Magnificent Movie Mania! Classic Movies That You Can Stream Today
Feeling bored? Tiger King and its quirky star Joe Exotic will not only pull you out of the doldrums, but it will have you seeing the world of big cat breeding in an entirely different light. Feeling anxious? The Great British Baking Show is sure to soothe you as you listen to talk of pies and cookies.
More: A Quick Pick-Me-Up: 24 Inspirational Movies You Can Watch Online
Whether you crave an apocalyptic escape series like The Walking Dead or something on the opposite end of the spectrum like Wet Hot American Summer: First Day of Camp, we’ve got you covered. Even though the story lines and topics are all over the map, one thing all of these shows have in common is that they have a stellar cast that is sure to keep you watching.
Ready to binge? Check out our top 10 Netflix shows for your top ten moods!
For access to all our exclusive celebrity videos and interviews – Subscribe on YouTube!
Sound off in the comments below!
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Where Do You Stand: Raising Children in a Two-Religion Household
In today's world, it's not uncommon to marry outside of your race, culture, or religion. That's all fine and well when you don't have kids. But when children are brought into the mix, deciding how to raise them, religiously speaking, could be problematic. What if both parents have strong beliefs in their faith? Some families have no problem teaching kids both (Chanukah Bush anyone?) but is that confusing for the kids? Should one parent convert to the other's religion? Of course, this matter should be worked out before getting married and starting a family, but tell me, where do you stand on raising kids in a duel-religion household?
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MINATEC construction projects right on track
Categorie(s) : Life @ MINATEC, MINATEC, News
Published : 3 February 2014
The many construction projects underway at MINATEC are all advancing according to plan. The second floor of the Skills Center Building is set to be completed by end-2015, and will accommodate over 500 Leti and CNRS researchers. The additional floors being added to the B2I building will be ready in May to house Leti’s bridging technology research team. And in February work will begin on plans to expand the Phelma 2 building and build a 500-seat auditorium in front of Maison des Micro et Nanotechnologies (MMNT).
But that’s not all. This summer will see construction work begin on the Photonics Platform—a modern complex comprising laboratories, clean rooms, and offices—with delivery scheduled in 2016. At the end of this year, work should begin on the Software Center addition to MMNT; bids to select the prime contractor are currently under review.
Contact: stephane.siebert@cea.fr
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Inspections, Compliance, Enforcement, and Criminal Investigations
Electronic Data Systems 07-Feb-05
FEI: 3000992933
VL #: 05200033
February 7, 2005
WARNING LETTER
CERTIFIED MAIL
RETURN RECEIPT REQEUSTED
Fielding “Butch” Cochran, Vice President
Military Health Programs Account
Electronic Data Systems
13600 EDS Drive
Herndon, VA 20171
Dear Mr. Cochran:
During an inspection of your establishment located at 5113 Leesburg Pike, Skyline 4, Suite 300, Falls Church, VA on September 13 through October 5, 2004, United States Food and Drug Administration (FDA) investigators, [redacted] and [redacted] determined that your establishment manufactures the Defense Blood Standard System (DBSS) software, version 3.04. This product is a device as defined by Section 201(h) of the Federal Food, Drug, and Cosmetic Act (the FD&C Act).
The above stated inspection revealed that this device is adulterated within the meaning of Section 501 (h) of the FD&C Act, in that the methods used in, or the facilities or controls used for, the manufacture, packing, storage, or installation of the DBSS are not in conformance with Current Good manufacturing Practice (CGMP) requirements of the Quality System (QS) regulation for medical devices, as specified in Title 21, Code of Federal Regulations (CFR), Part 820.
Quality System Regulation
At the close of the inspection, your firm was issued a list of inspectional observations, Form FDA-483, which identified a number of significant QS regulation violations including, but not limited to, the following:
1. Your firm failed to ensure that the procedures for implementing corrective and preventive actions were complete, as required by 21 CFR 820.100(a), [FDA-483, Item 1]. Specifically,
a. Your firm’s complaint handling procedures do not address how to enter data into the [redacted] electronic system to provide accurate figures to management on the number of trouble tickets received for similar issues or the number of complaints that are actually closed and resolved versus closed but unresolved.
b. Terms and categories entered into the various fields of the [redacted] have not been defined in a procedure to assure that they are used consistently to accurately determine the status of all software requests. The [redacted] is the software application used for tracking software requests, software testing problems, and software corrections.
c. Procedure [redacted] does not address the appropriate documentation for problems for which no workaround is possible nor does it address how the documentation should be entered into the [redacted]. While workaround forms for at least [redacted] system incident reports state "no workaround: or "no workaround possible," [redacted] records note that workarounds exist for these problems.
2. The software used by your firm as part of the quality system has not been fully validated for its intended use according to an established protocol, as required by 21 CFR 820.70(i), [FDA-483, Item 2]. Specifically,
a. The [redacted] software application, used for complaint handling, has not been validated to assure that the queries initiated to track and trend complaints yield accurate results. For example, the actual status of trouble tickets cannot be determined since they are categorized [redacted] whether or not the underlying issue has been corrected or otherwise resolved.
b. Information generated from data in [redacted] cannot be relied upon as accurate. For example: (1) [redacted] system report records indicate that workarounds exist when in fact no workarounds exists. (2) At least [redacted] system reports corrected in DBSS Version [redacted] are linked to DBSS Version [redacted] in [redacted]. (3) Individual software requests that are closed when linked to an open master software request are not always shown on the master request record.
3. Your firm failed to address design input requirements that are incomplete, as required by 21 CFR 820.30(c) [FDA-483, Item 31. Specifically,
a. Functional requirements for the DBSS are in some cases very high level. For example, the requirements for Donor Reporting merely state that the application must enable the user to identify, from a list of persons, the donor on whom the report should be based, and to print an Autologous Unit Status Report, a Donor History Report, a Person Audit Trail Report, a Deferral Audit Trail Report, and an Individual Donor Orders Report for a specific donor. The requirements do not address the files to be accessed, the fields to be printed, or the format of the ave been written for problems with these reports. [redacted] have been written for problems with these reports. For example, [redacted] concerns the military personnel’s SSN appearing where the donor’s SSN should appear on the Donor Audit Trail Report.
b. Detailed design specifications for DBSS version 3.04 could not be located Design specifications for DBSS Version [redacted], which is [redacted] and DBSS Version [redacted] which is currently being tested, only address changes to the current 3.04 version.
4. Your firm’s acceptance criteria were not complete prior to the performance of verification activities, as required by 21 CFR 820.30(f) [FDA-483, Item 4]. Specifically,
a. Unit testing of the [redacted] functionality in the DBSS Version [redacted] was completed and accept as passing although the output of the [redacted] report was incorrect. Unit testing was documented as passing on September 2, 2004. System integration testing on September 3, 2004 found that the merge report generated after the merge is incorrect in that the person shown as deleted and the person shown as kept are reversed on the report.
b. [redacted] initiated b a user site in 1999, documented this particular problem with the report. The [redacted] as invalidated by the SRRB (System Request Review Board) on August 12, 2004, with the rationale, “works as intended.”
5. Your firm fails to maintain complete complaint files, as required by 21 CFR 820.198(a) [FDA-483, Item 51. Specifically, investigation files are not complete. Facsimiles received from users depicting the software problem and screen-prints of the Customer Support personnel's re-creation of the problem are not retained. For example, files for trouble ticket [redacted] and related [redacted] concerning the incorrect SSN on the Donor Audit TRail REport, did not contain hard copy documentation from the user or the Customer Support personnel's re-creation of the problem.
6. Your firm’s device master record does not include or refer to the location of all software specifications, as required by 21 CFR 820.181(a) [FDA-483, Item 6]. Specifically, the device master record for DBSS Version [redacted] does not reference at least [redacted] system requests that were included in that version.
FDA evaluated the corrective actions taken to correct the objectionable conditions noted during the previous February-March 2003 inspection at your establishment. While EDS has taken steps to correct the previously cited deficiencies, several deficiencies from the previous inspection are identical or similar to those observed during the current inspection at your establishment.
EDS Response
FDA has received your letter dated November 2, 2004, responding to the Form FDA-483, (Inspectional Observations), issued and discussed with you at the conclusion of the inspection. We would like to comment on the following statement in your letter, “We talked to [redacted] the FDA Inspection Team lead, and she agreed that EDS should not address corrective actions en u-r response to each observation because EDS will no longer be in a position to take such actions.” At the conclusion of the inspection, EDS management stated that EDS would not respond to the FDA-483 observations due to the loss of the DBSS sustainment contract. Investigator [redacted] responded that she understood. Investigator [redacted] did not agree that EDS should not respond to the FDA-483 observations, nor did she instruct EDS not to respond. This letter is not intended to be an all inclusive list of the deficiencies at your facility. It is your responsibility to assure adherence to all applicable FDA regulations and the FD&C Act. The specific violations noted in this. letter and on the Form FDA-483 issued at the conclusion of the inspection may be symptomatic of serious underlying problems with your firm’s manufacturing and quality assurance systems. Federal agencies are advised of the issuance of Warning Letters regarding medical devices so that they may take this information into consideration when awarding contracts.
You should take prompt action to correct these violations. Failure to promptly correct these violations may result in regulatory action being initiated by.FDA without further notice. These actions include, but are not limited to, seizure, injunction, and/or civil penalties.
Please notify this office in writing, within fifteen (15) working days of receipt of this letter, of
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\subsubsection*{Discrete polygons}
A \textit{discrete polygon} is a simply connected subgraph $\Omega$ of $\Z^2$,
{or $\delta \Z^2$,}
with $2N$ marked boundary points $x_1, x_2, \ldots , x_{2N}$ in counterclockwise order, whose precise definition is given below.
Firstly, we define the \textit{medial polygon}. We give orientation to edges of the medial graph $(\Z^2)^\diamond$ in the following way: edges of each face containing a vertex of $\Z^2$ are oriented clockwise, and edges of each face containing a vertex of $(\Z^2)^{\bullet}$ are oriented counterclockwise.
Let $x_1^\diamond,\ldots, x_{2N}^\diamond$ be $2N$ distinct medial vertices. Let $(x_1^\diamond \, x_2^\diamond), (x_2^\diamond \, x_3^\diamond), \ldots , (x_{2N}^\diamond \, x_{1}^\diamond)$ be $2N$ oriented paths on $(\Z^2)^\diamond$ satisfying the following conditions\footnote{We use the convention that $x_{2N+1}^\diamond := x_{1}^\diamond$.}:
\begin{itemize}[leftmargin=2em]
\item
each path $(x_{2r-1}^\diamond \, x_{2r}^\diamond)$ has counterclockwise oriented edges for $1\leq r \leq N$;
\item
each path $(x_{2r}^\diamond \, x_{2r+1}^\diamond)$ has clockwise oriented edges for $1\leq r \leq N$;
\item
all of the paths are edge-avoiding and satisfy $(x_{i-1}^\diamond \, x_i^\diamond) \cap (x_i^\diamond \, x_{i+1}^\diamond) = \{x_i^\diamond\}$ for $1\leq i \leq 2N$;
\item
if $j\notin \{i+1,i-1\}$, then $(x_{i-1}^\diamond \, x_{i}^\diamond) \cap (x_{j-1}^\diamond \, x_j^\diamond) = \emptyset$;
\item
the infinite connected component of $(\Z^2)^\diamond\setminus \smash{\overset{2N}{\underset{i=1}{\bigcup}}}
(x_i^\diamond \, x_{i+1}^\diamond)$ lies on the right of the oriented path $(x_1^\diamond \, x_2^\diamond)$.
\end{itemize}
Given $\{(x_i^\diamond \, x_{i+1}^\diamond) \colon 1\leq i\leq 2N\}$, the medial polygon $(\Omega^\diamond; x_1^\diamond,\ldots, x_{2N}^\diamond)$ is defined
as the subgraph of $(\Z^2)^\diamond$ induced by the vertices lying on or enclosed by the non-oriented loop obtained by concatenating all of $(x_i^\diamond \, x_{i+1}^\diamond)$.
For each $i \in \{1,2,\ldots,2N\}$, the \textit{outer corner} $y_{i}^{\diamond}\in (\mathbb{Z}^2)^\diamond\setminus\Omega^\diamond$ is defined to be a medial vertex adjacent to $x_i^\diamond$, and the \textit{outer corner edge} $e_i^\diamond$ is defined to be the medial edge connecting them.
\smallbreak
Secondly, we define the \textit{primal polygon}
$(\Omega;x_1,\ldots,x_{2N})$ induced by $(\Omega^\diamond;x_1^\diamond,\ldots,x_{2N}^\diamond)$ as follows:
\begin{itemize}[leftmargin=2em]
\item
the edge set $E(\Omega)$ consists of edges passing through endpoints of medial edges in
$E(\Omega^\diamond)\setminus \smash{\overset{N}{\underset{r=1}{\bigcup}}}
(x_{2r}^\diamond \, x_{2r+1}^\diamond)$;
\item
the vertex set $V(\Omega)$ consists of endpoints of edges in $E(G)$;
\item
the marked boundary vertex $x_i$ is defined to be the vertex in $\Omega$ nearest to $x_i^\diamond$ for each $1\leq i\leq 2N$;
\item
the arc $(x_{2r-1} \, x_{2r})$ is the set of edges whose midpoints are vertices in $(x_{2r-1}^\diamond \, x_{2r}^\diamond)\cap \partial \Omega^\delta$ for $1\leq r \leq N$.
\end{itemize}
Lastly, we define the \textit{dual polygon} $(\Omega^{\bullet};x_1^{\bullet},\ldots,x_{2N}^{\bullet})$ induced by $(\Omega^\diamond; x_1^\diamond,\ldots,x_{2N}^\diamond)$ in a similar way.
More precisely, $\Omega^{\bullet}$ is the subgraph of $(\Z^2)^{\bullet}$ with edge set consisting of edges passing through endpoints of medial edges in $E(\Omega^\diamond)\setminus \smash{\bigcup_{r=1}^{N}} (x_{2r-1}^\diamond \, x_{2r}^\diamond)$ and vertex set consisting of the endpoints of these edges.
The marked boundary vertex $x_i^{\bullet}$ is defined to be the vertex in $\Omega^{\bullet}$ nearest to $x_i^\diamond$ for $1\leq i\leq 2N$.
The boundary arc $(x_{2r}^{\bullet} \, x_{2r+1}^{\bullet})$ is the set of edges whose midpoints are vertices in $(x_{2r}^\diamond \, x_{2r+1}^\diamond)\cap \Omega^\diamond$ for $1\leq r \leq N$.
\subsubsection*{Boundary conditions}
We shall focus on the critical FK-Ising model on the primal polygon $(\Omega^\delta; x_1^\delta,\ldots,x_{2N}^\delta)$, with the following boundary conditions:
first, every other boundary arc is wired,
\begin{align*}
(x_{2r-1}^{\delta} \, x_{2r}^{\delta}) \text{ is wired,} \quad \text{ for all } r \in\{1,2,\ldots, N\} ,
\end{align*}
and second, these $N$ wired arcs are further wired together
outside of $\Omega^{\delta}$ according to a planar link pattern $\beta\in\LP_N$ as in~\eqref{eqn::linkpatterns_ordering}
--- see Figure~\ref{fig::6points}.
In this setup, the model is said to have \textit{boundary condition} $\beta$. We denote by $\PP_{\beta}^{\delta}$ the law, and by $\mathbb{E}_{\beta}^{\delta}$ the expectation, of the critical model on $(\Omega^{\delta}; x_{1}^{\delta},\ldots,x_{2N}^{\delta})$ with boundary condition $\beta$.
\subsection*{Loop representation and interfaces}
Let $\omega\in \{0,1\}^{E(\Omega^\delta)}$ be a configuration of the FK-Ising model with boundary condition $\beta\in \LP_N$ on the primal polygon $(\Omega^\delta; x_1^\delta,\ldots,x_{2N}^\delta)$. Given $\omega$, we can draw self-avoiding loops on $\Omega^{\delta, \diamond}$ as follows: a loop arriving at a vertex of $\Omega^{\delta,\diamond }$ always makes a turn of $\pm\pi/2$, so as not to cross the open or dual open edges through this vertex. Given $\omega$, the \textit{loop representation} contains loops and $N$ pairwise-disjoint and self-avoiding \textit{interfaces} connecting the $2N$ outer corners $y_{1}^{\delta,\diamond}, \ldots,y_{2N}^{\delta,\diamond}$ of the medial polygon $(\Omega^{\delta,\diamond};x_1^{\delta,\diamond},\ldots,x_{2N}^{\delta,\diamond})$. For each $i\in \{1,2,\ldots,2N\}$, we denote by $\eta_i^\delta$ the interface starting from the medial vertex $y_{i}^{\delta,\diamond}$.
{See Figure~\ref{fig::loop_representation} for an illustration of the interfaces. }
\subsubsection*{Convergence of polygons}
Let $\{\Omega^{\delta}\}_{\delta>0}$ and $\Omega$ be simply-connected open sets $\Omega^{\delta}, \Omega\subsetneq\C$, all containing a common point $u$. We say that $\Omega^{\delta}$ converges to $\Omega$ in the sense of \textit{kernel convergence} with respect to $u$,
denoted $\Omega^{\delta}\to\Omega$, if
\begin{enumerate}[label=\textnormal{(\arabic*):}, ref=\arabic*]
\item
every $z\in\Omega$ has some neighborhood $U_z$ such that $U_z\subset\Omega^{\delta}$, for all small enough $\delta > 0$; and
\item
for every point $p\in\partial\Omega$, there exists a sequence $p^{\delta}\in\partial\Omega^{\delta}$ such that $p^{\delta}\to p$.
\end{enumerate}
If $\Omega^{\delta}\to\Omega$ in the sense of kernel convergence with respect to $u$, the same convergence holds with respect to any $\tilde{u}\in\Omega$.
We say that $\Omega^{\delta}\to\Omega$ in the Carath\'{e}odory sense as $\delta\to 0$.
Note that $\Omega^{\delta}\to \Omega$ in the Carath\'{e}odory sense if and only if there exist conformal maps $\varphi_{\delta}$ from $\Omega^{\delta}$ onto $\U$, and a conformal map $\varphi$ from $\Omega$ onto $\U$, such that $\varphi_{\delta}^{-1}\to\varphi^{-1}$ locally uniformly on $\U$ as $\delta\to 0$, see~\cite[Theorem~1.8]{Pommerenke}.
For polygons, we say that a sequence of discrete polygons $(\Omega^{\delta}; x_1^{\delta}, \ldots, x_{2N}^{\delta})$
converges as $\delta \to 0$ to a polygon $(\Omega; x_1, \ldots, x_{2N})$ in the \textit{Carath\'{e}odory sense}
if there exist conformal maps $\varphi_{\delta}$ from $\Omega^{\delta}$ onto $\U$,
and a conformal map $\varphi$ from $\Omega$ onto $\U$,
such that $\varphi_{\delta}^{-1} \to \varphi^{-1}$ locally uniformly on $\U$,
and $\varphi_{\delta}(x_j^{\delta}) \to \varphi(x_j)$ for all $1\le j\le 2N$.
Note that Carath\'{e}odory convergence allows wild behavior of the boundaries around the marked points.
In order to ensure
precompactness of the interfaces in Theorem~\ref{thm::FKIsing_Loewner}, we need a convergence on polygons stronger than the above Carath\'{e}odory convergence.
The following notion was introduced by Karrila in~\cite{KarrilaConformalImage}, see in particular~\cite[Theorem~4.2]{KarrilaConformalImage}. (See also~\cite{KarrilaMultipleSLELocalGlobal} and~\cite{ChelkakWanMassiveLERW}.)
\begin{definition} \label{def:closeCara}
We say that a sequence of discrete polygons $(\Omega^{\delta}; x_1^{\delta}, \ldots, x_{2N}^{\delta})$
converges as $\delta \to 0$ to a polygon $(\Omega; x_1, \ldots, x_{2N})$ in the \textit{close-Carath\'{e}odory sense} if it converges in the Carath\'{e}odory sense, and in addition, for each $j\in\{1, \ldots, 2N\}$, we have $x_j^{\delta}\to x_j$ as $\delta\to 0$ and the following is fulfilled:
Given a reference point $u\in\Omega$ and
$r>0$ small enough, let $S_r$ be the arc of $\partial B(x_j,r)\cap\Omega$ disconnecting (in $\Omega$) $x_j$ from $u$ and from all other arcs of this set. We require that, for each $r$ small enough and for all sufficiently small $\delta$ (depending on $r$), the boundary point $x_j^{\delta}$ is connected to the midpoint of $S_r$ inside $\Omega^{\delta}\cap B(x_j,r)$.
\end{definition}
In this setup, the FK-Ising interfaces, and more generally, the random-cluster interfaces for any parameter $q\in [1,4)$, always have a convergent subsequence in the curve space $(X,\metric)$.
\begin{lemma}\label{lem::FKIsing_tightness}
Assume the same setup as in
Conjecture~\ref{conj::rcm_Loewner}.
Fix $i\in\{1, \ldots, 2N\}$. The family of laws of $\{\eta_i^{\delta}\}_{\delta>0}$ is
precompact in the space $(X,\metric)$ of curves with the metric~\eqref{eq::curve_metric}.
Furthermore, any subsequential limit $\eta_i$ does not hit any other point in $\{x_1, x_2, \ldots, x_{2N}\}$ except its two endpoints, almost surely.
\end{lemma}
\begin{proof}
The proof is standard nowadays.
For instance, $q=2$ is treated in~\cite[Lemmas~4.1 and~5.4]{IzyurovMultipleFKIsing}.
The main tools are the so-called RSW bounds from~\cite{DuminilCopinHonglerNolinRSWFKIsing, KemppainenSmirnovRandomCurves}
--- see also~\cite{KarrilaConformalImage, KarrilaMultipleSLELocalGlobal}.
The case of general $q\in [1,4)$ follows from~\cite[Theorem~6]{DuminilSidoraviciusTassionContinuityPhaseTransition} and~\cite[Section~1.4]{DCMTRCMFractalProperties}.
\end{proof}
In this rest of this section, we fix $q=2$ and thus focus on the critical FK-Ising model.
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Certainly when I got to medical school, I had role models of the kind of physicians I wanted to be. I had an uncle who, looking back, was probably not the most-educated physician around, but he carried it off so well..'
The flip side of suicide is that it leaves a lingering question in the minds of the people who survived. It's like a cancer that's metastasized. The suicide is the cancer and the metastasis is all these people saying, Why? Why? Why?.
I still find the best way to understand a hospitalized patient whose care I am taking over is not by staring at the computer screen but by going to see the patient; it's only at the bedside that I can figure out what is important.
I've never bought this idea of taking a therapeutic distance. If I see a student or house staff cry, I take great faith in that. That's a great person; they're going to be a great doctor. it..
I'm a great believer in geography being destiny.
I joke, but only half joke, that if you show up in an American hospital missing a finger, no one will believe you until they get a CAT scan, MRI and orthopedic consult..
Modern society has evolved to the point where we counter the old-fashioned fatalism surrounding the word 'cancer' by embracing the idea of the Uber-mind - that our will possesses nearly supernatural powers.
For un-subscribe please check the mail footer.
| 172,921
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This is a large format Vimeo video of a selection of Helmi Flick’s cat photographs. As she has taken thousands upon thousands of outstanding photographs the selection was made randomly. It was a case of whatever came to hand. That may sound careless but it is not. Almost any set of Helmi photos would be equally suitable. Although for some inexplicable reason I chose lots of Ragdolls. Perhaps, some Freudian thing happened. Ragdolls do suit the music, though.
Why did I make it? These are reasons:
- The first reason is poor. I embedded this slideshow somewhere on the site before but can’t find it. If I can’t find it, who can 😉 So, I thought I’d embed it again.
- To show off Helmi’s cat photographs. She is possible the best cat photographer in the world; certainly in the very top bracket and there are quite a few cat photographers. Channan, who is also American, and whose photos are used by the CFA is not as good. His lighting is nowhere near as good; much less vibrant. Helmi’s photos are used by TICA the other large American cat association. Although TICA is an international cat association.
- Vimeo videos are very high quality. They are known for that. This makes them particularly suitable for Helmi’s pictures in a video.
- I discovered that Apple’s well known photographer’s software called “Aperture” has the facility to do these fancy slideshows. I like the breaking up and reforming of the image. The music is bundled-in to the program so I used it. Although I don’t know whether Apple purchased the rights for its customers to use the music. I would hope and expect that to be the case. I have assumed it to be the case!
The cats in order of appearance are (excluding the cover photograph):
- Maine Coon
- Maine Coon
- Bengal
- Ragdoll
- Ragdoll
- Ragdoll
- Maine Coon kittens
- Ragdoll
- American Curl (dilute peachy coat)
- Moggie – red tabby
- Ragdoll
- Bengal
- Maine Coon – red tabby
- Norwegian Forest Kittens – I think ; )
- Norwegian Forest Kittens – I think ; )
- Persian – bicolor
- Ragdoll
- Ragdoll
- Russian Blue
You can see 104 cat breeds in a slide show on this page – no music and highspeed.
Facebook Discussion
Those effects are fun aren’t they.. I played around with them at one time. Pretty impressive how Apple makes high quality stuff so accessible.
I’m learning to play with GIMP. It’s made hundreds of my old photos useable on my website. Us old grouches gotta do something to occupy us.
Helmi is possibly the best in the world at what she does.
I forget what GIMP is!
Fun video, Michael. FWIW, I also think the 4 kittens are Norwegian Forest cats. Seems they have a classic NFC head profile. Took me a while to get to grips with seeing the differences between NFCs and Maine coons. From my POC hosted site a while back (thanks Michael)
Thanks VG. You know your cats 😉
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3 April 2019
Japan Credits Overseas TV Content Sales With Triggering Tourism Boom
- Photo: Mount Yotei: The setting for a recent edition of the popular Love Hokkaidō travelogue.
- Photo: Anime: The most popular Japanese cultural export since the kimono.
- Photo: The Pearl: Being fictitious has been no impediment to this Hong Kong skyscraper attracting tourists.
Given the apparent link between the surge in the number of overseas tourists arriving in Japan and escalating foreign sales of the country's domestically-produced TV content, is there a formula here that other destinations could possibly follow?
With the number of overseas tourists visiting Japan having soared over recent years, government and tourism officials are crediting at least part of this upturn to the success the country's TV industry has had in selling content abroad. It is believed that the positive depiction of Japan in such programming – now a staple of many overseas conventional broadcast and streaming services – has inspired many viewers to move the country to the top of their must-visit list.
In total, Japan welcomed more than 28 million foreign tourists in 2017 – a huge increase on the eight million arrivals recorded for 2010. For many, there is a telling correlation between this tourism uptick and sales of Japan-made television content.
Back in 2010, the total overseas sales value of Japanese broadcast content was logged at 6.6 billion yen (US$60 million). By the end of 2016, however, the corresponding figure was 39.4 billion yen, with the increase due to both the rapidly rising number of content-hungry platforms and a more aggressive export policy by Japanese programme makers. A combination of these two elements had led to far higher numbers of non-domestic viewers being exposed to positive depictions of Japanese culture, lifestyle and tourism attractions than ever before.
The volume of cultural and lifestyle programming exported, however, is dwarfed by the overseas sales of Japan-sourced anime video content. Anime – the country's proprietary and highly distinctive graphic style – has been the backbone of the country's cartoon industry since the 1960s and has, more recently, also conquered the computer-gaming sector. Although originally a strictly Japanese phenomenon, it is now globally popular among younger viewers and gamers, with its ubiquity almost certainly having also helped to drive visitor numbers.
According to the Ministry of Internal Affairs and Communications, anime-related sales accounted for 80% of all revenue from exported Japanese broadcast content in 2016. The remaining 20% largely stemmed from drama and variety show programming.
Apart from its undoubted cross-border popularity, another reason why the figures are skewed in favour of anime is that it is far easier to dub into the language of any local market than big budget dramas or fast-moving, supposedly spontaneous, variety shows. With the primary overseas markets for Japanese broadcast content seen as North America and other countries across Asia, overcoming this language barrier is regarded as a major challenge for non-anime programme makers if the export momentum is to be maintained.
Anime aside, programme makers in some of Japan's more tourist-friendly regions are seen as having notched up considerable success in promoting their appeal to overseas tourists. One of the clear leaders had been Hokkaidō, the northernmost of the country's primary islands.
Programme makers there were somewhat ahead of the curve, having produced content celebrating the local landscape and way of life since the 1990s. Indeed, depictions of the island's snowy winter climate – which has made it one of Japan's leading skiing centres – and legendary seafood / dairy dishes are seen as having played key roles in luring visitors from Europe, the US and across Asia.
Among the areas that have followed suit is Kyūshū, with the southwestern island finding a ready overseas audience for the locally-made travelogues that highlighted its active volcanoes and natural hot springs. Of late, other popular visitor destinations – including Tōkai and Hokuriku – have also got in on the act, with their efforts credited with producing a substantial boost for their respective local tourist economies. In addition to locally-themed strands, programmes dedicated to Japanese cuisine (particularly ramen) and the more idiosyncratic elements of the country's national culture are also seen as having helped pique the interest of would-be visitors.
Given Japan's success in scaling the league of the world's most visited countries and regions, other popular tourist destinations have been wondering if they, too, could benefit from such a television-led windfall. Asked whether adopting a similar policy might make sense for Hong Kong, Liz Shackleton, Asia Editor for Screen International, the leading publication for the global film business, said: "Locally-produced films and TV shows that showcase the city's food and culture could undoubtedly have a positive impact on tourism numbers, especially as many of the streaming services are now well-placed to distribute this kind of content globally.
"In fact, something very similar is already happening on the mainland. This has seen local content creators producing documentaries that are being shown on a global basis. A particular success here has been Tencent's Flavorful Origins, which is now introducing a wide variety of Chinese cuisine to viewers in more than 200 countries and regions via Netflix."
Adding a warning that Hong Kong could be at risk of missing out on any tourism-TV-related bonanza, she said: "When international productions make use of any location, it nearly always contributes to an increase in local tourism, with New Zealand – the setting for The Lord of The Rings trilogy – being, perhaps, the best-known example. To date, Hong Kong has been proactive in this area and has already provided the location for several major international movies, notably Tomb Raider, Transformers: Age of Extinction, Ghost in The Shell and Skyscraper.
"In recent years, however, there has been a huge global shift towards episodic TV production. This has seen many countries in Asia introduce production incentives and funding schemes to encourage the major streaming companies, the likes of Netflix and Amazon, to use them for location shooting. Hong Kong, though, doesn't currently offer any such incentives but, with the global landscape becoming increasingly competitive, it may have to reconsider its stance."
As a sign of the close links between on-screen appearances and tourism, a number of overseas visitors to Hong Kong have, apparently, been determined to tour the smoking ruins of The Pearl, a 240-storey building in the city's central business district that Dwayne "The Rock" Johnson fails to save from destruction in Skyscraper (2018). Clearly, the fact it was wholly fictitious was deemed no good reason to omit it from their tour schedule.
Shota Maruko, Tokyo Office
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\begin{document}
\title{Some abstract versions of G\"odel's second incompleteness theorem based on non-classical logics}
\author{Lev Beklemishev and Daniyar Shamkanov \\
Steklov Mathematical Institute, Moscow\\
Moscow M.V.~Lomonosov State University \\
National Research University Higher School of Economics, Moscow\\
}
\maketitle
{\em To Albert Visser, a remarkable logician and a dear friend, whose papers and conversations are a source of constant inspiration}
\begin{abstract} We study abstract versions of G\"odel's second incompleteness theorem and formulate generalizations of L\"ob's derivability conditions that work for logics weaker than the classical one. We isolate the role of contraction rule in G\"odel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating G\"odel's argument.
\end{abstract}
\section{Introduction}
One of the topics that have been fascinating logicians over the years is G\"odel's second incompleteness theorem (G2).
Both mathematically and philosophically G2 is well known to be more problematic than his first incompleteness theorem (G1).
G1 and Rosser's Theorem are well understood in the context of recursion theory. Abstract logic-free formulations have been given by Kleene \cite{Kle50} (`symmetric form'), Smullyan \cite{Smu94} (`representation systems') and others. Sometimes G2 is considered as a minor addition to G1, whose role is to exhibit a specific form of the sentence independent from a given theory, namely its consistency assertion. However, starting with the work of Kreisel, Orey, Feferman, and others, who provided various nontrivial uses of G2, it has been gradually understood that the two results are of a rather different nature and scope. G2 has more to do with the (modal-logical) properties of the provability predicate and the phenomenon of self-reference in sufficiently expressive systems. A satisfactory general mathematical context for G2, however, still seems to be lacking.
The main difficulties in G2 are due to the fact that we cannot easily delineate a class of formulas that `mean' consistency. Thus, the most intuitively appealing formulation of G2 --- \emph{sufficiently strong consistent theories cannot prove their own consistency} --- remains non-mathematical.
For a concrete formal system, such as Peano arithmetic $\PA$, one can usually write out a specific `natural' formula $\Con_\PA$ and declare it to be the expression of consistency. This approach is rather common in mathematics but has several deficiencies: Firstly, it ties the statement to a very particular formula, coding mechanism etc., and provides no clue why this choice is better than the other ones. Secondly, instead of a general theorem working uniformly for a wide class of theories, we only obtain a specific statement for an individual theory such as $\PA$. We do not know what is the natural consistency assertion for an arbitrary extension of $\PA$. Thus, we have a problem with translating our informal intuition into strict mathematical terms.
The way to better understand G2 is through investigating its range and generalizations. A lucky circumstance is that G2 also holds for larger syntactically defined classes of consistency formulas, some of which are apparently intensionally correct (adequately express consistency), but some are not. Thus, it is still possible to formulate mathematical results in certain important aspects
\emph{more} (rather than less) general than the broad intuitive formulation of G2 above.
A universally accepted approach to general formulations of G2 appeared in the fundamental paper by Feferman~\cite{Fef60} who showed, among other things, that G2 holds for all consistency assertions defined by $\Sigma_1$-numerations.
Feferman deals with first-order theories $T$ in the language containing that of $\PA$ and specified by recursively enumerable (r.e.)\ sets of axioms. Feferman assumes fixed some natural G\"odel numbering of the syntax of $T$ as well as some specific axiomatization of first order logic. A $\Sigma_1$-formula $\alpha(x)$ defining the set of G\"odel numbers of axioms of $T$ in the standard model of $\PA$ is called a \emph{$\Sigma_1$-numeration} of $T$.\footnote{Feferman deals with the notion of r.e.\ formula rather than with the equivalent notion of $\Sigma_1$-formula more common today.} It determines the
provability formula $\Prs{\ga}(x)$ and the corresponding consistency assertion $\Con_\ga$. Feferman's statement of G2 is that for all consistent theories $T$ given by $\Sigma_1$-numerations $\ga$ and containing a sufficiently strong fragment of $\PA$, the formula $\Con_\ga$ is unprovable in $T$.
This theorem is considerably more general than any specific instance of G2 for an individual theory $T$. However, it also presupposes quite a lot: first order logic and its axiomatization, G\"odel numbering, the way formula $\Prs{\ga}$ is built from $\ga$.
Exploring bounds to G2 leads to relaxing various assumptions involved in Feferman's statement:
\bi
\item One can weaken the axioms of arithmetic (for a representative selection see Bezboruah--Shepherdson~\cite{BezShe}, Pudl\'ak~\cite{Pud85}, Wilkie--Paris~\cite{WP}, Adamowicz--Zdanowski~\cite{AZ11}, Willard~\cite{Wil01,Wil06}).
\item One can consider theories modulo interpretability. This approach started with the work of Feferman~\cite{Fef60}. In the recent years it lead to particularly attractive coding-free formulations of generalizations of G2 due to Harvey Friedman and Albert Visser~(see \cite{Vis93,Vis11,Vis12}).
\item One can weaken the requirements on the proof predicate aka derivability conditions (see Feferman~\cite{Fef60}, L\"ob~\cite{Lob55}, Jeroslow~\cite{Jer70,Jer73}).
\item One can weaken the logic.
\ei
It is the latter two aspects, less studied in the literature, that we are going to comment on in this note. Firstly, let us briefly recall the history of derivability conditions.
G\"odel \cite{God31} gave a sketch of a proof of G2 and a promise to provide full details in a subsequent publication. This promise has not been fulfilled, and a detailed proof of this theorem --- for a system $Z$ related to first-order arithmetic $\PA$ --- only appeared in a monograph by Hilbert and Bernays \cite{HB}. In order to structure a rather lengthy proof Hilbert and Bernays formulated certain conditions on the proof predicate in $Z$, sufficient for the proof of G2. Later Martin L\"ob~\cite{Lob55} gave an elegant form to these conditions by stating them fully in terms of the provability predicate $\Pr(x)$ and obtained an important strengthening of G2 known as L\"ob's Theorem. Essentially the same properties of the provability predicate were earlier noted by G\"odel in his note~\cite{God33}, where he proposed to treat the provability predicate as a connective $\Box$ in modal logic, though the idea that these conditions constitute necessary requirements on a provability predicate most likely only appeared later. For the sake of brevity we call the G\"odel--Hilbert--Bernays--L\"ob conditions simply \emph{L\"ob's conditions} below.
A traditional proof of G2 (for arithmetical theories) consists of a derivation of G2 from the fixed point lemma using L\"ob's conditions (see e.g.~\cite{Smo77}). An accurate justification of these conditions is technically not so easy, and a rare textbook provides enough details here, however see Smory\'nski~\cite{Smo85} and Rautenberg~\cite{Rau06} for readable expositions.
L\"ob's conditions are applicable to formal theories at least containing the connective of implication and closed under the \emph{modus ponens} rule. Here we give more general abstract formulations of G2 which presuppose very little about logic. They are rather close in the spirit and the level of generality to the recursion-theoretic formulations of G1 due to Smullyan. When a good implication is added to the language one essentially obtains the familiar L\"ob's conditions. However, we show that G\"odel's argument presupposes admissibility of the contraction rule restricted to $\Box$-formulas in the logic under consideration. Moreover, the uniqueness of G\"odelian fixed point is based on the similarly restricted form of weakening.
In the last part of the paper we present a system invalidating a formalized version of G2. We consider a version of propositional modal logic K4 based on the contraction-free fragment of classical logic extended by fixed point operators (defined for any formulas modalized in the fixed point variables). By means of a cut-elimination theorem for this system we establish the failure of G2 and some other properties such as the infinity of the G\"odelian and Henkinian fixed points.
\section{Abstract provability structures}
\bd \label{cns}
Let us call \emph{an abstract consequence relation} a structure $S=(L_S,\leq_S,\top,\bot)$, where $L_S$ is a set of \emph{sentences} of $S$, $\leq_S$ is a transitive reflexive relation on $L_S$, $\top$ and $\bot$ are distinguished elements of $L_S$ (`axiom' and `contradiction'). A sentence $x\in L_S$ is called \emph{provable in $S$}, if $\top\leq_S x$, and \emph{refutable in $S$}, if $x\leq_S \bot$. Sentences $x,y$ are called \emph{equivalent in $S$}, if $x\leq_S y$ and $y\leq_S x$. The equivalence of $x$ and $y$ will be denoted $x=_S y$.
\ed
The structure $S$ represents syntactical (rather than semantical) data about the theory in question.
In a typical case, for example, for arithmetical theories $S$, the relation $x\leq_S y$ denotes the provability of $y$ from hypothesis $x$, whereas $\top$ and $\bot$ are some standard provable and refutable formulas, respectively, e.g., $0=0$ and $0\neq 0$.
In concrete situations we can enrich this structure by additional data, for example, by the conjunction and the implication connectives. Notice that we do not assume either $\bot\leq_S x$ or $x\leq_S \top$, nor do we assume the existence of any logical connectives (such as negation) in $S$.
$S$ is called \emph{inconsistent} if $\top\leq_S \bot$, otherwise it is called \emph{consistent}. By transitivity, if $S$ is consistent then no sentence is both provable and refutable. $S$ is called \emph{complete} if every $x\in L_S$ is either provable or refutable. $S$ is called \emph{r.e.}, if $L_S$ is recursive and $\leq_S$ is r.e.\ (as a binary relation). $T$ is called an \emph{extension} of $S$ if $L_T=L_S$ and $\leq_S$ is contained in $\leq_T$.
Let $P_S$ and $R_S$ denote the sets of provable and of refutable sentences of $S$, respectively. If $S$ is consistent and r.e., then $P_S$ and $R_S$ is a pair of disjoint r.e.\ sets. We say that $S$ \emph{separates pairs of disjoint r.e.\ sets} if for each such pair $(A,B)$ there is a total computable function $f$ such that
$$\al{n\in A} f(n)\in P_S \text{ and } \al{n\in B} f(n)\in R_S.$$
The following statement is a natural version of G1 and Rosser's theorem for abstract consequence relations (\'a la Kleene and Smullyan); we omit the standard proof.
\bpr
\begin{enumr}
\item If $S$ is r.e., consistent and complete, then both $P_S$ and $R_S$ are decidable.
\item If $S$ is r.e.\ and separates disjoint pairs of r.e.\ sets, then every consistent extension of $S$ is incomplete and undecidable.
\end{enumr}
\epr
Next we introduce two operators $\Box,\boxtimes:L_S\to L_S$ representing provability and refutability predicates in $S$.
\bd
\emph{Provability} and \emph{refutability operators} for an abstract consequence relation $S$ are functions $\Box,\boxtimes:L_S\to L_S$ satisfying the following conditions, for all $x,y\in L_S$:
\label{prop-def}
\renewcommand{\labelenumi}{C\arabic{enumi}.}
\ben
\item $x\leq_S y \ \Imp \ \Box x\leq_S \Box y$, $\boxtimes y\leq \Boxt x$. \label{g1}
\item $\top\leq_S \Boxt \bot$; \label{g2}
\item $x\leq_S\Box y,\ x\leq_S\Boxt y\ \Imp x\leq_S \Boxt \top$; \label{g3}
\item $\Boxt x\leq_S\Box\Boxt x$. \label{g4}
\een
The algebra $(L_S,\leq_S,\top,\bot,\square,\Boxt)$ is called an \emph{abstract provability structure} (APS).
\ed
Intuitively, $\Box x$ is the sentence expressing the provability of a sentence $x$, whereas $\Boxt x$ expresses its refutability in $S$. Condition C\ref{g1} means that provability of $y$ follows from provability of $x$ whenever $y$ is derivable from $x$; similarly, refutability of $y$ implies refutability of $x$. Conditions C\ref{g2} and C\ref{g3} are axioms for contradiction: according to C\ref{g2}, refutability of $\bot$ is provable in $S$; according to
C\ref{g3}, $\top$ is refutable if some sentence $y$ is both provable and refutable.
Finally, Condition \ref{g4} means that the refutability of $x$ can be formally checked in $S$. It is an analogue of L\"ob's condition L2 (see below).
Note that we consider the refutability operator on a par with the provability operator, since we do not assume that the logic of $S$ necessarily has a well-defined operation of negation, that is, we cannot always define $\Boxt x$ as $\Box\neg x$.
\brem It is rather natural to additionally require that $\Box\bot=_S\Boxt\top$: refutability of $\top$ and provability of $\bot$ are expressed by the same statement $\top\leq_S\bot$. Yet, it is not, strictly speaking, needed in this very abstract context, and we take $\Boxt\top$ as our default expression of inconsistency.
\erem
\bd We say that an abstract provability structure $S$ \emph{has a G\"odelian fixed point} if there is a sentence $p\in L_S$ such that $p=_S\Boxt p$.
\ed
Notice that G\"odel considered a dual sentence $q$ expressing its own unprovability in $S$. R. Jeroslow \cite{Jer73} noticed that the sentence stating its own refutability allows to prove G2 under somewhat more general conditions than those of L\"ob. In our formalism the sentence $q$ is not expressible, therefore we are using Jeroslow's idea.
A very abstract version of G2 can now be stated as follows.
\bt \label{god2-abs}
Suppose an APS $S$ has a G\"odelian fixed point.
\begin{enumr}
\item If $S$ is consistent, then $\Boxt \top$ is irrefutable in $S$.
\item $\Boxt\Boxt \top \leq_S \Boxt \top$, that is, Statement \emph{(i)} is formalizable in $S$.
\end{enumr}
\et
\bp\ Let $p=_S\Boxt p$. First we prove Statement (ii) omitting the subscript $_S$ everywhere:
\ben
\item $\Boxt p\leq \Box\Boxt p\leq \Box p$ by C\ref{g4} and C\ref{g1};
\item $p=\Boxt p\leq \Boxt\top$ by C\ref{g3} (since $\Boxt p\leq \Boxt p$);
\item $\Boxt\Boxt\top\leq \Boxt p=p\leq\Boxt \top$ by C\ref{g1}.
\een
Proof of Statement (i):
Assume $\Boxt \top\leq \bot$. By the previous argument $p\leq \Boxt \top$, hence $p\leq\bot$. By C\ref{g1}, $\Boxt \bot\leq \Boxt p = p \leq \bot$.
Therefore, by C\ref{g2}, $\top\leq \Boxt \bot\leq \bot$. \ep
The following statement shows that under some additional condition the G\"odelian--Jeroslowian fixed point is unique modulo equivalence in $S$ and coincides with the inconsistency assertion for $S$. Therefore, the existence of such a fixed point is not only sufficient but also necessary for the validity of (a formalized version of) G2. The additional condition is
\bi
\item[C5.] $x\leq_S \top$, for all $x\in L_S$.
\ei
\bt \label{fixp} Assume C5 holds for $S$. Then $p=_S\Boxt \top$ for all G\"odelian fixed points $p$ and (if such a sentence exists) $$\Boxt\Boxt \top =_S \Boxt \top.$$
\et
\bp\ We know that $p\leq \Boxt \top$. Since $p=\Boxt p\leq\top$ we obtain $\Boxt\top\leq \Boxt p = p$. Hence $p=\Boxt\top$ and therefore $\Boxt\Boxt \top = \Boxt \top.$
\ep
\section{Consequence relations with implication}
Classical L\"ob's conditions emerge for APS with an implication. A decent implication can be defined for consequence relations representing derivability of a sentence from a (multi)set of assumptions. In other words, we now go to a more general but less symmetric format $\Gamma\vdash\phi$, where $\Gamma$ is a finite multiset and $\phi$ an element of a given set $L$. In order to avoid confusion we use the more standard notation $\vdash$ instead of $\leq$ and will follow the standard conventions of sequential proof format. In particular, $\Gamma,\phi$ denotes the result of adjoining $\phi\in L_S$ to a multiset of sentences $\Gamma$, and $\Gamma,\Delta$ denotes the multiset union of $\Gamma$ and $\Delta$.\footnote{Our strive for generality does not go as far as to consider lists of formulas rather than multisets.}
\bd \emph{A consequence relation with an implication on $L$} is a structure $S=(L_S,\vdash,\to,\top,\bot)$ where
$\vdash$ is a binary relation between finite multisets of elements of $L_S$ and elements of $L_S$; $\to$ is a binary operation on $L$; $\top$ and $\bot$ are distinguished elements of $L$ such that the following conditions hold:
\ben
\renewcommand{\labelenumi}{I\arabic{enumi}.}
\item $\phi\vdash \phi$;
\item if $\Gamma,\psi\vdash \phi$ and $\Delta\vdash\psi$ then $\Gamma,\Delta\vdash\phi$;
\item $\Gamma,\phi\vdash\psi \iff \Gamma\vdash \phi\to\psi$;
\item $\Gamma,\top\vdash\phi\iff \Gamma\vdash\phi.$
\een
\ed
Notice that Conditions I1 and I2 generalize reflexivity and transitivity of $\leq$. Setting $\phi\leq_S\psi$ as $\phi\vdash\psi$ yields an abstract consequence relation in the sense of Definition~\ref{cns}. Condition I3 speaks for itself. Condition I4 conveniently stipulates that provability from the empty multiset of assumptions is the same as provability from $\top$. It also implies $\top\to\bot=_S\bot$.
Similarly to the implication one can consider consequence relations with other additional connectives of which we are mostly interested in conjunction.
\bd \emph{Conjunction} is a binary operator $\otimes:L_S^2\to L_S$ satisfying
$$\Gamma,\phi,\psi\vdash \theta \iff \Gamma, \phi\otimes\psi\vdash\theta.$$
\ed
If conjunction is available, then $\phi_1,\dots,\phi_n\vdash\psi$ holds in $S$ if and only if $\phi_1\otimes\cdots\otimes\phi_n\vdash\psi$. Hence, in the presence of conjunction in $S$ the relation $\leq_S$ uniquely determines the corresponding multiset consequence relation $\Gamma\vdash\phi$.
For consequence relations with an implication we can define negation $\neg\phi$ by $\phi\to\bot$. The following simple lemma shows that the implication respects the deductive equivalence relation in $S$ and the negation satisfies the contraposition principle.
\bl \label{imp-prop}
\begin{enumr}
\item If $\Gamma\vdash \phi\to\psi$ and $\Delta\vdash\phi$, then $\Gamma,\Delta\vdash\psi$;
\item $\phi_1=_S\phi_2$ and $\psi_1=_S\psi_2$ implies $ (\phi_1\to\psi_1)=_S (\phi_2\to\psi_2);$
\item $\Gamma,\phi\vdash\psi$ implies $\Gamma,\neg\psi\vdash\neg\phi$.
\end{enumr}
\el
Next we turn to the derivability conditions. Assume $S$ is a consequence relation with an implication.
\bd $\Box:L_S\to L_S$ satisfies L\"ob's derivability conditions for $S$ if
\ben
\renewcommand{\labelenumi}{L\arabic{enumi}.}
\item $\Box(\phi\to\psi)\vdash \Box\phi\to\Box\psi$;
\item $\Box\phi\vdash\Box\Box\phi$;
\item $\vdash\phi$ implies $\vdash\Box\phi$.
\een
\ed
\bl For any consequence relation with an implication the following statements are equivalent:
\begin{enumr}
\item $\Box$ satisfies L\"ob's conditions for $S$;
\item $\Box$ satisfies L2 and $S$ is closed under the rule
$$\frac{\Gamma\vdash\phi}{\Box\Gamma\vdash\Box\phi};$$
\item $S$ is closed under the rule
$$\frac{\Gamma,\Box\Delta\vdash\phi}{\Box\Gamma,\Box\Delta\vdash\Box\phi}.$$
\end{enumr}
\el
\brem Notice that the last rule is formulated slightly differently from the more standard rule for modal logic K4:
$$\frac{\Gamma,\Box\Gamma\vdash\phi}{\Box\Gamma\vdash\Box\phi}.$$
The latter has a form of built-in contraction that we are not assuming here.
\erem
It is natural to define refutability $\boxtimes\phi$ as provability of negation $\Box\neg\phi$. Notice that since $\bot=_S\top\to\bot$ we have $\Boxt\top=_S \Box\bot$, whenever L1 holds for $\Box$. However, as the example in Section \ref{sec-ex1} shows, this translation does not always yield an APS in the sense of Definition \ref{prop-def}. To sort things out we need to consider two additional conditions on the consequence relation.
\bd A consequence relation with an implication
\bi
\item[-] \emph{satisfies contraction} if
$\Gamma,\phi,\phi\vdash\psi$ implies $\Gamma,\phi\vdash\psi$;
\item[-] \emph{satisfies weakening} if
$\Gamma\vdash\psi$ implies $\Gamma,\phi\vdash\psi$, for any $\phi$.
\ei
\ed
The first condition intuitively means that any hypothesis can be used several times in a derivation. Recall that for Girard's linear logic this condition is not met, however it is postulated, for example, for relevant logics. It turns out that a certain amount of contraction is essential for the proof of G2.
The second condition corresponds to the requirement $x\leq_S \top$ that was needed to guarantee that $\Boxt \top$ is a G\"odelian fixed point and that such a fixed point is unique.
For consequence relations with an implication we have the following proposition.
\bpr \label{s-lob}
Suppose $S$ satisfies contraction, $\Box:L_S\to L_S$ satisfies L\"ob's conditions for $S$ and $\Boxt\phi:=\Box(\phi\to\bot)$. Then $(L_S,\leq_S,\Box,\boxtimes,\top,\bot)$ is an APS.
\epr
\bp By Lemma \ref{imp-prop} $\phi\vdash\psi$ implies $\neg\psi\vdash\neg\phi$. This yields Conditions C\ref{g1} and C\ref{g4}. Condition C\ref{g2} obviously follows from Condition 1 for a good consequence relation. Let us prove C\ref{g3}.
By Lemma \ref{imp-prop}(i) we have: $\phi,\neg\phi,\top\vdash \bot$. Hence, $\phi,\neg\phi\vdash\top\to\bot$, therefore $\Box\phi,\Box\neg\phi\vdash\Box\neg\top$ by Condition 1. The rules of transitivity and contraction imply that, if $\Gamma\vdash\Box\phi$ and $\Gamma\vdash\Box\neg\phi$, then $\Gamma\vdash\Box\neg\top$. \ep
Thus, from Proposition \ref{s-lob} we obtain the following expected corollary, parallel to Theorem \ref{god2-abs}, for consequence relations satisfying contraction.
\bt \label{g2contr} Suppose $S$ satisfies contraction and $\Box$ satisfies L\"ob's conditions for $S$.
Then Theorem \ref{god2-abs} holds for $S$.
\et
For an analogue of Theorem \ref{fixp} on the uniqueness of a G\"odelian fixed point we also need a weakening property.
\bt \label{g2weak} Suppose $S$ satisfies contraction and weakening and $\Box$ satisfies L\"ob's conditions for $S$. Then all G\"odelian fixed points in $S$ (if exist) are equivalent to $\Boxt\top=_S\Box\bot$. \et
\brem
As it turns out, contraction and weakening for $S$, though natural, are somewhat excessive requirements for the validity of Theorems \ref{g2contr} and \ref{g2weak}. A consequence relation with an implication
\bi
\item[-] \emph{satisfies $\Box$-contraction} if
$\Gamma,\Box\phi,\Box\phi\vdash\psi$ implies $\Gamma,\Box\phi\vdash\psi$;
\item[-] \emph{satisfies $\Box$-weakening} if
$\Gamma\vdash\phi$ implies $\Gamma,\Box\psi\vdash\phi$, for any $\psi$.
\ei
Conditions C3 and C5 of APS can also be weakened to
\bi
\item[C3$'$.] $\Boxt x\leq_S\Box y,\ \Boxt x\leq_S\Boxt y\ \Imp \Boxt x\leq_S \Boxt \top$;
\item[C5$'$.] $\Boxt x\leq_S \top$.
\ei
With these modifications, the proofs of Theorems \ref{god2-abs} and \ref{fixp} stay the same, which in turn yields more general versions of Theorems \ref{g2contr} and \ref{g2weak} for consequence relations satisfying only $\Box$-contraction and $\Box$-weakening.
The property of $\Box$-contraction actually holds for some meaningful arithmetical systems lacking general contraction rule, for example, for a version of Peano arithmetic based on affine predicate logic considered by the second author of this paper (as yet, unpublished).
\erem
\section{A non-G\"{o}delian theory with fixed points} \label{sec-ex1}
In view of Theorems \ref{g2contr} and \ref{g2weak} it is natural to ask whether the assumptions of $\Box$-contraction and $\Box$-weakening are substantial for these results. More specifically, two questions immediately present themselves:
\ben
\item Does there exist a consequence relation with an implication satisfying L\"ob's conditions for $\Box$ in which a G\"odelian fixed point exists, but G2 fails? (The failure of G2 can be understood in two different senses --- as a failure of its formalized version, and as a failure of its non-formalized version. Our example will show the failure of the formalized version.)
\item Do G\"odelian fixed points in such a system $S$ have to be unique, even if $S$ satisfies weakening?
\een
In this section we provide an example showing that the answer to the first question is positive and to the second one negative. Moreover, we formulate a system in which there are many more fixed points than are officially required for a proof of G2. Our system $\mathsf{S}$ is a version of modal logic K4 based on the multiplicative $\{\to,\otimes,\bot\}$ fragment of a classical logic without contraction. It also has a built-in fixed point operator where the expression $\mathsf{fp}\, x. A(x)$ denotes some fixed point of $A(x)$ for formulas $A$ modalized in the variable $x$. Thus, one will be able to derive $$\mathsf{fp}\, x. A(x)=_S A(\mathsf{fp}\, x. A(x)),$$
for each formula $A(x)$ modalized in $x$.
Let us now turn to the exact definitions.
\medskip
Consider the set of formulas $\mathsf{Fm}_0$ given by the grammar:
$$ A ::= p \,\,|\,\, x \,\,|\,\, \bot \,\,|\,\, (A \rightarrow A) \,\,|\,\, \Box A \;, $$
where $p$ stands for \emph{atomic propositions} and $x$ stands for \emph{variables} (the alphabets of atomic propositions and variables are disjoint). We define the \emph{set of formulas of $\mathsf{S}$} by extending the set $\mathsf{Fm}_0$ by a new constructor: if $A$ is a formula and all free occurrences of $x$ in $A$ are within the scope of modal operators, then $\mathsf{fp}\, x. A$ is a formula, and $\mathsf{fp}\, x$ binds all free occurrences of $x$. A formula $B$ is \emph{closed} if it does not contain any free occurrences of variables. For a closed formula $B$, we denote by $A[B/\!/x]$ the result of replacing all free occurrences of $x$ in $A$ by $B$.
We also put $\neg A := A\rightarrow \bot$, $\top := \neg \bot$ and $A \otimes B := \neg (A \rightarrow \neg B)$.
A \textit{sequent} is an expression of the form $\Gamma \Rightarrow \Delta$, where $\Gamma$ and $\Delta$ are finite multisets of closed formulas.
The sequent calculus $\mathsf{S}$ is defined in the standard way by the following initial sequents and inference rules:
\begin{gather*}
\AXC{ $\Gamma, A \Rightarrow A, \Delta $}
\DisplayProof \qquad
\AXC{ $\Gamma , \bot \Rightarrow \Delta$}
\DisplayProof \\\\
\AXC{$\Gamma, A[\mathsf{fp}\, x.\, A/\!/x] \Rightarrow \Delta$}
\LeftLabel{$(\mathsf{fix_L})$}
\UIC{$\Gamma, \mathsf{fp}\, x.\, A \Rightarrow \Delta$}
\DisplayProof \qquad
\AXC{$\Gamma \Rightarrow A[\mathsf{fp}\, x.\, A/\!/x],\Delta$}
\LeftLabel{$(\mathsf{fix_R})$}
\UIC{$\Gamma \Rightarrow \mathsf{fp}\, x.\, A ,\Delta$}
\DisplayProof
\end{gather*}
\begin{gather*}
\AXC{$\Gamma , B \Rightarrow \Delta$}
\AXC{$\Sigma \Rightarrow A, \Pi$}
\LeftLabel{$(\mathsf{\rightarrow_L})$}
\BIC{$\Gamma , \Sigma, A \rightarrow B \Rightarrow \Pi, \Delta$}
\DisplayProof \qquad
\AXC{$\Gamma, A \Rightarrow B ,\Delta$}
\LeftLabel{$(\mathsf{\rightarrow_R})$}
\UIC{$\Gamma \Rightarrow A \rightarrow B ,\Delta$}
\DisplayProof \\\\
\AXC{$\Sigma, \Box \Pi \Rightarrow A$}
\LeftLabel{$(\mathsf{\Box})$}
\UIC{$\Gamma, \Box \Sigma, \Box \Pi \Rightarrow \Box A , \Delta$}
\DisplayProof \;.
\end{gather*}
Explicitly displayed formulas in the conclusions of the rules are called \emph{principal formulas} of the corresponding inferences.
In the rules $(\mathsf{fix_L})$, $(\mathsf{fix_R})$, $(\mathsf{\rightarrow_L})$ and $(\mathsf{\rightarrow_R})$, the elements of
$\Gamma$, $\Delta$, $\Sigma$ and $\Pi$ are called \emph{side formulas}.
In initial sequents and in applications of the rule $(\mathsf{\Box})$, the elements of
$\Gamma$ and $\Delta$ are \emph{weakening formulas}. We call the elements of $\Box \Sigma$ and $\Box \Pi$ in the corresponding applications of $(\mathsf{\Box})$ \emph{active formulas}. In addition, explicitly displayed formulas in initial sequents are called \emph{axiomatic formulas}.
A \emph{proof in $\mathsf{S}$} is a finite tree whose nodes are marked by sequents and leaves are marked by initial sequents that is constructed according to the rules of the sequent calculus. A sequent $\Gamma \Rightarrow \Delta$ \emph{is provable in $\mathsf{S}$} if
there is a proof with the root marked by $\Gamma \Rightarrow \Delta$.
We associate with $\mathsf{S}$ a consequence relation with an implication and conjunction in the usual way by letting $\Gamma\vdash_\mathsf{S}\phi$ iff $\Gamma\Imp\phi$ is provable in $\mathsf{S}$. The main thing we need to prove about $\mathsf{S}$ is the closure of
$\mathsf{S}$ under the cut rule, which would show that $\Gamma\vdash_\mathsf{S}\phi$ is indeed a well-defined consequence relation (see Theorem~\ref{cutelim} below).
Since $\mathsf{S}$ is cut-free, the following propositions are easy to establish. Firstly, we obtain the failure of formalized G2.
\bpr
The sequent $\Box (\Box \bot \rightarrow \bot) \Rightarrow \Box \bot$ is not provable in $\mathsf{S}$.
\epr
Recall that an inference rule is called admissible (for a given proof system) if, for every instance of the rule, the conclusion is provable whenever all premises are provable.
\bpr
The L\"{o}b rule and the Henkin rule
\begin{gather*}
\AXC{$\Box A \Rightarrow A$}
\LeftLabel{$(\mathsf{L\ddot{o}b})$}
\RightLabel{ }
\UIC{$ \quad \Rightarrow A$}
\DisplayProof
\qquad
\AXC{$\Box A \Rightarrow A$}
\AXC{$A\Rightarrow \Box A$}
\LeftLabel{$(\mathsf{Hen})$}
\RightLabel{ }
\BIC{$ \quad \Rightarrow A$}
\DisplayProof
\end{gather*}
are not admissible in $\mathsf{S}$.
\epr
\begin{proof}
Consider the Henkin fixed point $\mathsf{fp}\, x.\, \Box x$. The sequent $\Rightarrow \mathsf{fp}\, x.\, \Box x$ is not provable in $\mathsf{S}$. Hence, the Henkin rule is not admissible and so is the stronger L\"ob rule.
\end{proof}
\bpr
There are infinitely many Henkinian and G\"odelian fixed points in $\mathsf S$.
\epr
\bp The routine of bound variables in $\mathsf S$ is such that the formulas
$\mathsf{fp}\, x_i.\Box x_i$ for graphically distinct variables $x_i$ are all inequivalent. (There is no rule of bound variables renaming and, in fact, it is easy to convince oneself that there are no cut-free proofs in $\mathsf S$ of the sequents $\mathsf{fp}\, x_i.\Box x_i\Imp \mathsf{fp}\, x_j.\Box x_j$, for $i\neq j$.) The same holds for the G\"odelian fixed points of $\mathsf S$.
\ep
\section{Cut-admissibility for $\mathsf S$}
For a proof of the cut-admissibility theorem for $\mathsf S$ we need the following standard lemma. Let the \emph{size $\lVert \pi \rVert$ of a proof $\pi$} be the number of nodes in $\pi$.
\bl
The weakening rule
\begin{gather*}
\AXC{$\Gamma \Rightarrow \Delta$}
\LeftLabel{$(\mathsf{weak})$}
\RightLabel{ }
\UIC{$ \Sigma , \Gamma \Rightarrow \Delta, \Pi$}
\DisplayProof
\end{gather*}
is admissible for $\mathsf{S}$, and its conclusion has a proof of at most the same size as the premise.
\el
\bt \label{cutelim}
The cut rule
\begin{gather*}
\AXC{$\Gamma \Rightarrow \Delta, A$}
\AXC{$A, \Sigma \Rightarrow \Pi$}
\LeftLabel{$(\mathsf{cut})$}
\RightLabel{ ,}
\BIC{$\Gamma, \Sigma \Rightarrow \Pi, \Delta$}
\DisplayProof
\end{gather*}
is admissible for $\mathsf{S}$. Moreover, if $\pi_1$ and $\pi_2$ are proofs of the premises of $(\mathsf{cut})$, then the conclusion of $(\mathsf{cut})$ has a proof with the size being less than $ \lVert \pi_1 \rVert + \lVert \pi_2 \rVert$.
\et
\begin{proof}
Assume we have an inference
\begin{gather*}
\AXC{$\pi_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma \Rightarrow \Delta, A$}
\AXC{$\pi_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$A, \Sigma \Rightarrow \Pi$}
\LeftLabel{$(\mathsf{cut})$}
\RightLabel{ ,}
\BIC{$\Gamma, \Sigma \Rightarrow \Pi, \Delta$}
\DisplayProof
\end{gather*}
where $\pi_1$ and $\pi_2$ are proofs in $\mathsf{S}$. We proof by induction on $\lVert \pi_1 \rVert + \lVert \pi_2 \rVert$ that for any formula $A$ there exists a proof $\mathcal{E}_A(\pi_1,\pi_2)$ of $\Gamma, \Sigma \Rightarrow \Pi, \Delta$ with the size being less than $\lVert \pi_1 \rVert + \lVert \pi_2 \rVert $.
Consider the final inference in $\pi_1$. If the formula $A$ is in a position of a weakening formula in it, then we erase $A$ in $\pi_1$ and extend the sequent $\Gamma \Rightarrow \Delta$ to $\Gamma, \Sigma \Rightarrow \Pi, \Delta$ by adding new weakening formulas. This transformation of $\pi_1$ defines $\mathcal{E}_A(\pi_1,\pi_2)$. Moreover, we have $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert = \lVert \pi_1 \rVert < \lVert \pi_1 \rVert + \lVert \pi_2 \rVert$.
Suppose the formula $A$ is an axiomatic formula in the final inference of $\pi_1$. Then the proof $\pi_1$ consists of an initial sequent and the multiset $\Gamma$ has the form $ \Gamma_0, A$. We obtain $\mathcal{E}_A(\pi_1,\pi_2)$ by applying the admissible rule $(\mathsf{weak})$:
\begin{gather*}
\AXC{$\pi_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$A, \Sigma \Rightarrow \Pi$}
\LeftLabel{$(\mathsf{weak})$}
\RightLabel{ .}
\UIC{$\Gamma_0, A, \Sigma \Rightarrow \Pi, \Delta$}
\DisplayProof
\end{gather*}
We have $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert \leqslant \lVert \pi_2 \rVert <\lVert \pi_1 \rVert + \lVert \pi_2 \rVert$.
Now suppose the formula $A$ is a side formula. Then the final inference in $\pi_1$ can be $(\mathsf{fix_L})$, $(\mathsf{fix_R})$, $(\mathsf{\rightarrow_L})$ or $(\mathsf{\rightarrow_R})$.
In the case of $(\mathsf{\rightarrow_R})$, the proof $\pi_1$ has the form
\begin{gather*}
\AXC{$\pi^\prime_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma, B \Rightarrow C, \Delta_0, A$}
\LeftLabel{$(\mathsf{\rightarrow_R})$}
\RightLabel{ ,}
\UIC{$\Gamma \Rightarrow B\rightarrow C, \Delta_0, A$}
\DisplayProof
\end{gather*}
where $\Delta = B\rightarrow C,\Delta_0$. We define $\mathcal{E}_A(\pi_1,\pi_2)$ as
\begin{gather*}
\AXC{$\mathcal{E}_A(\pi^\prime_1,\pi_2)$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma, B, \Sigma \Rightarrow \Pi,C, \Delta_0$}
\LeftLabel{$(\mathsf{\rightarrow_R})$}
\RightLabel{ .}
\UIC{$\Gamma, \Sigma \Rightarrow \Pi, B\rightarrow C, \Delta_0$}
\DisplayProof
\end{gather*}
The proof $\mathcal{E}_A(\pi^\prime_1,\pi_2)$ is defined by the induction hypothesis for $\pi^\prime_1$ and $\pi_2$. We also have $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert = \lVert \mathcal{E}_A(\pi^\prime_1,\pi_2) \rVert +1 < \lVert \pi^\prime_1 \rVert+ \lVert \pi_2 \rVert + 1= \lVert \pi_1 \rVert+ \lVert \pi_2 \rVert$.
In the case of $(\mathsf{fix_R})$, the proof $\pi_1$ has the form
\begin{gather*}
\AXC{$\pi^\prime_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma \Rightarrow B[\mathsf{fp} \, x.\, B/\!/x] \Delta_0, A$}
\LeftLabel{$(\mathsf{fix_R})$}
\RightLabel{ ,}
\UIC{$\Gamma \Rightarrow \mathsf{fp} \, x.\, B, \Delta_0, A$}
\DisplayProof
\end{gather*}
where $\Delta = \mathsf{fp} \, x.\, B,\Delta_0$. We define $\mathcal{E}_A(\pi_1,\pi_2)$ as
\begin{gather*}
\AXC{$\mathcal{E}_A(\pi^\prime_1,\pi_2)$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma, \Sigma \Rightarrow \Pi,B[\mathsf{fp} \, x.\, B/\!/p], \Delta_0$}
\LeftLabel{$(\mathsf{fix_R})$}
\RightLabel{ .}
\UIC{$\Gamma, \Sigma \Rightarrow \Pi, \mathsf{fp} \, x.\, B, \Delta_0$}
\DisplayProof
\end{gather*}
The proof $\mathcal{E}_A(\pi^\prime_1,\pi_2)$ is defined by the induction hypothesis, and $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert = \lVert \mathcal{E}_A(\pi^\prime_1,\pi_2) \rVert +1 < \lVert \pi^\prime_1 \rVert+ \lVert \pi_2 \rVert +1= \lVert \pi_1 \rVert+ \lVert \pi_2 \rVert$.
The remaining cases of $(\mathsf{\rightarrow_L})$ and $(\mathsf{fix_L})$ can be analyzed analogously, so we omit them.
Now consider the final inference in $\pi_2$. If the formula $A$ is a weakening, an axiomatic or a side formula in it, then we can define $\mathcal{E}_A(\pi_1,\pi_2)$ in a similar way to the previous cases.
Suppose that the formula $A$ is a principal or an active formula in the final inferences of $\pi_1$ and $\pi_2$.
Then $A$ has the form $\mathsf{fp} \, x. \,A_0$, $A_0\rightarrow A_1$ or $\Box A_0$.
If $A=\Box A_0$, then
$\pi_2$ has one of the two forms
\begin{gather*}
\AXC{$\pi^\prime_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$ A_0, \Sigma_1, \Box \Sigma_2\Rightarrow D$}
\LeftLabel{$(\mathsf{\Box})$}
\RightLabel{ }
\UIC{$ \Sigma_0, \Box A_0 ,\Box \Sigma_1, \Box \Sigma_2 \Rightarrow \Box D, \Pi_0$}
\DisplayProof \qquad
\AXC{$\pi^\prime_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$ \Sigma_1, \Box A_0,\Box \Sigma_2\Rightarrow D$}
\LeftLabel{$(\mathsf{\Box})$}
\RightLabel{ ,}
\UIC{$ \Sigma_0, \Box \Sigma_1, \Box A_0 , \Box \Sigma_2 \Rightarrow \Box D, \Pi_0$}
\DisplayProof
\end{gather*}
where $\Sigma = \Sigma_0, \Box \Sigma_1, \Box \Sigma_2$ and $\Pi = \Box D, \Pi_0$.
In addition, the proof $\pi_1$ has the form
\begin{gather*}
\AXC{$\pi^\prime_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma_1, \Box \Gamma_2 \Rightarrow A_0$}
\LeftLabel{$(\mathsf{\Box})$}
\RightLabel{ ,}
\UIC{$\Gamma_0, \Box \Gamma_1, \Box \Gamma_2 \Rightarrow \Box A_0, \Delta$}
\DisplayProof
\end{gather*}
where $\Gamma = \Gamma_0, \Box \Gamma_1, \Box \Gamma_2$. If $\pi_2$ has the first form, then we define $\mathcal{E}_A(\pi_1,\pi_2)$ as
\begin{gather*}
\AXC{$\mathcal{E}_{A_0}(\pi^\prime_1,\pi^\prime_2)$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma_1, \Box \Gamma_2, \Sigma_1, \Box \Sigma_2 \Rightarrow D$}
\LeftLabel{$(\mathsf{\Box})$}
\RightLabel{ .}
\UIC{$\Gamma_0, \Box \Gamma_1, \Box \Gamma_2,\Sigma_0, \Box \Sigma_1, \Box \Sigma_2 \Rightarrow \Box D, \Pi_0, \Delta$}
\DisplayProof
\end{gather*}
We have $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert = \lVert \mathcal{E}_{A_0}(\pi^\prime_1,\pi^\prime_2)\rVert +1 < \lVert \pi^\prime_1\rVert + \lVert \pi^\prime_2\rVert +1 < \lVert \pi_1\rVert + \lVert \pi_2\rVert$.
If $\pi_2$ has the second form, then we define $\mathcal{E}_A(\pi_1,\pi_2)$ as
\begin{gather*}
\AXC{$\mathcal{E}_A(f(\pi_1),\pi^\prime_2)$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Box \Gamma_1, \Box \Gamma_2, \Sigma_1, \Box \Sigma_2 \Rightarrow D$}
\LeftLabel{$(\mathsf{\Box})$}
\RightLabel{ ,}
\UIC{$\Gamma_0, \Box \Gamma_1, \Box \Gamma_2,\Sigma_0, \Box \Sigma_1, \Box \Sigma_2 \Rightarrow \Box D, \Pi_0, \Delta$}
\DisplayProof
\end{gather*}
where $f(\pi_1)$ is the proof obtained by erasing multisets $\Gamma_0$ and $\Delta$ from the conclusion of $\pi_1$.
We have $\lVert \mathcal{E}_A(\pi_1,\pi_2) \rVert = \lVert \mathcal{E}_A(f(\pi_1),\pi^\prime_2)\rVert +1 < \lVert f(\pi_1)\rVert + \lVert \pi^\prime_2\rVert +1 = \lVert \pi_1\rVert + \lVert \pi_2\rVert$.
In the case of $A = \mathsf{fp} \, x. \,A_0 $, the proofs $\pi_1$ and $\pi_2$ have the form
\begin{gather*}
\AXC{$\pi^\prime_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$\Gamma \Rightarrow \Delta, A_0[ \mathsf{fp} \, x. \,A_0 /\!/x]$}
\LeftLabel{$(\mathsf{fix_R})$}
\UIC{$\Gamma\Rightarrow \Delta, \mathsf{fp} \, x. \,A_0 $}
\DisplayProof \qquad
\AXC{$\pi^\prime_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$A_0[ \mathsf{fp} \, x. \,A_0 /\!/x], \Sigma \Rightarrow \Pi$}
\LeftLabel{$(\mathsf{fix_L})$}
\RightLabel{ .}
\UIC{$\mathsf{fp} \, x. \,A_0 ,\Sigma \Rightarrow \Pi$}
\DisplayProof
\end{gather*}
We put $\mathcal{E}_A(\pi_1,\pi_2) = \mathcal{E}_{A_0[ \mathsf{fp} \, x. \,A_0 /\!/x]}(\pi^\prime_1,\pi^\prime_2)$ and see that
$\lVert \mathcal{E}_A(\pi_1,\pi_2)\rVert
= \lVert \mathcal{E}_{A_0[ \mathsf{fp} \, x. \,A_0 /\!/x]}(\pi^\prime_1,\pi^\prime_2)\rVert
< \lVert \pi^\prime_1 \rVert + \lVert \pi^\prime_2 \rVert < \lVert \pi_1
\rVert + \lVert\pi_2 \rVert$.
If $A=A_0 \rightarrow A_1$, then the proofs $\pi_1$ and $\pi_2$ have the form
\begin{gather*}
\AXC{$\pi^\prime_1$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$A_0, \Gamma \Rightarrow \Delta, A_1$}
\LeftLabel{$(\mathsf{\rightarrow_R})$}
\UIC{$\Gamma\Rightarrow \Delta, A_0 \rightarrow A_1$}
\DisplayProof \qquad
\AXC{$\pi^\prime_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$ A_1, \Sigma_1 \Rightarrow \Pi_1$}
\AXC{$\pi^{\prime \prime}_2$}
\noLine
\UIC{\vdots}
\noLine
\UIC{$ \Sigma_0 \Rightarrow \Pi_0, A_0$}
\LeftLabel{$(\mathsf{\rightarrow_L})$}
\RightLabel{ ,}
\BIC{$\Sigma_0, A_0 \rightarrow A_1 ,\Sigma_1 \Rightarrow \Pi_0, \Pi_1$}
\DisplayProof
\end{gather*}
where $\Sigma =\Sigma_0, \Sigma_1$ and $\Pi= \Pi_0, \Pi_1 $. By the induction hypothesis, $\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1)$ is defined and $ \lVert \mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1)\rVert < \lVert\pi^{\prime\prime}_2 \rVert +\lVert\pi^\prime_1 \rVert$.
Since $\lVert\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1) \rVert+ \lVert\pi^\prime_2 \rVert<\lVert\pi^{\prime\prime}_2 \rVert +\lVert\pi^\prime_1 \rVert+ \lVert\pi^\prime_2 \rVert<\lVert\pi_1 \rVert+ \lVert\pi_2 \rVert $, then $ \mathcal{E}_{A_1} (\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1), \pi^\prime_2)$ is defined by the induction hypothesis. We put $\mathcal{E}_A(\pi_1,\pi_2) = \mathcal{E}_{A_1} (\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1), \pi^\prime_2)$. In addition, we have
$\lVert \mathcal{E}_A(\pi_1,\pi_2)\rVert = \lVert \mathcal{E}_{A_1} (\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1), \pi^\prime_2)\rVert < \lVert\mathcal{E}_{A_0}(\pi^{\prime\prime}_2,\pi^\prime_1) \rVert+ \lVert\pi^\prime_2 \rVert<\lVert\pi_1 \rVert+ \lVert\pi_2 \rVert$.
\end{proof}
\section{Conclusions and future work}
The preliminary results presented in this paper indicate the following conclusions:
\bi
\item Derivability conditions can be stated in a way not assuming much about logic. However,
\item G\"odel's argument presupposes a certain amount of contraction for the logic under consideration.
\ei
The role of contraction rule here is somewhat similar to its role in Liar-type paradoxes including Russell's paradox in set theory. Thus, Vyacheslav Grishin (see~\cite{Gri74,Gri82}) pioneered the study of set theory with full comprehension based on a logic without contraction. He demonstrated that the pure comprehension scheme is consistent in this logic. He also showed, however, that the extensionality principle allows for this system to actually \emph{prove} contraction even if there is no postulated contraction in the logic.
One can also consider systems of arithmetic based on contraction-free logic, see e.g.~Restall~\cite[Chapter 11]{Res94}. For one such system, considered by the second author of this paper, the rule of $\Box$-contraction is admissible, which according to our results still yields G2. Thus, we are still missing convincing examples of mathematical theories based on weak logics for which G2 would fail.
\bi
\item For consequence relations with an implication and with $\Box$ satisfying L\"ob's conditions, the existence of appropriately many fixed points does not imply their uniqueness. Nor does it imply formalized versions of G2 and L\"ob's theorem $\Box(\Box\phi\to\phi)\vdash \Box\phi$.
\ei
This shows that the move from \emph{diagonalized algebras} in the sense of R.~Magari, i.e., Boolean algebras with $\Box$ satisfying L\"ob's conditions and having fixed points, to \emph{diagonalizable algebras} (modal algebras satisfying L\"ob's identity) is, in general, not possible for logics without contraction and weakening. See Smory\'nski~\cite{Smo82a,Smo85} for a nice exposition of the original setup.
\bi
\item One can also show that the admissibility of L\"ob's rule does not, in general, imply a formalized version of G2.
\ei
A system $\mathsf S^*$ witnessing this property can be obtained by extending the notion of proof in the system $\mathsf S$ to possibly non-well-founded proof trees. Infinite proofs may arise because of the presence of the fixed point rules. For $\mathsf S^*$, unlike $\mathsf S$, one can show that L\"ob's rule is admissible. Yet, formalized G2 is still underivable.
The analysis of $\mathsf S^*$ is based on another cut-admissibility theorem, which we postpone to a later publication.
We remark that the system $\mathsf S$ does not provide a counterexample to the non-formalized version of G2, since $\Imp \neg \Box\bot$ is not provable. We believe that such a counterexample can be constructed by extending the language of $\mathsf S$ by an operator similar to $!$ from linear logic and adding to $S$ a fixed point of the form $a=\Diamond!a$. However, a confirmation of this hypothesis is left for future work.
\section{Acknowledgements}
The authors would like to thank Johan van Benthem for useful comments and questions. This work is supported by the Russian Foundation for Basic Research, grant 15-01-09218a, and by the Presidential council for support of leading scientific schools.
\input{gen_goedel.bbl}
\end{document}
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En español | WASHINGTON—Today, residents of all ages.
the projects, which must be completed by November 4, 2019,.
View the full list of grantees: Jessica Winn, jwinn@aarp.org, 202-434-2506, @AAARPMedia
| 338,479
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TITLE: Can We Write the Differential in Terms of Covectors?
QUESTION [3 upvotes]: Let $f:\mathbf R^n\to \mathbf R$ be a smooth map.
We can write $df:T\mathbf R^n\to \mathbf R$ neatly as
$$
df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i
$$
For a function $f:M\to \mathbf R$ defined on a smooth manifold $M$, given local coordinates $(x_1, \ldots, x_n)$ about a point $p$, we can write
$$
df = \sum_{i=1}^n(\partial /\partial x_i)f\ dx_i
$$
Is there a similar neat way of writing the differential of the map $f:\mathbf R^n\to \mathbf R^m$, and more generally of $f:M\to N$, where $M$ and $N$ are smooth manifolds?
REPLY [0 votes]: You've given a coordinate representation of the linear map $df_p:T_pM\to T_{f(p)}R$. One can work with it as a row-vector or introduce the "gradient vector" or indeed write it in the basis $\{dx_i$}.
Generally, for any $df_p:T_pM\to T_{f(p)}N$, its coordinate representation is simply the Jacobian matrix which could too be written in the corresponding basis matrices but usually its action on a tangent $v\in T_pM$ is simply written as a matrix-vector product $Jv.$
| 63,454
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\section{Introduction}
\label{sec:introduction}
In digital communication systems, data detection and decoding is a fundamental task at the receiver. The maximum a-posteriori (MAP) decoding approach is the optimal one to minimize the symbol or sequence error probability. If the channel model is known, the decoder can be realized according to known algorithms. For instance, for a linear time invariant (LTI) channel with additive Gaussian noise, the decoder comprises a first stage where the channel impulse response is estimated. Then, a max-log-MAP sequence estimator algorithm is implemented. The decoding metric is the Euclidean distance between the received and hypothesized transmitted message filtered by the channel impulse response. The task becomes more intriguing for channels that cannot be easily modelled or are unknown. In such a case, a learning strategy turns out to be an attractive solution.
Recent advancements in machine learning for communications have fostered the design of systems with a data-driven paradigm. As a consequence, physical layer design has been reinterpreted using machine learning techniques to improve the coding and decoding scheme performance \cite{Oshea2017, Nachmani2018, Dorner2018, Alberge2019, Stark2019}. In particular, the work in \cite{Oshea2017} introduced the concept of autoencoder-based communication systems. In contrast to traditional bottom-up approaches, the encoder and decoder blocks can be jointly learned during the end-to-end autoencoder training phase \cite{Oshea2017}. The autoencoder is a deep neural network that maps a sequence of input bits $\mathbf{s}$ into a sequence of output bits $\hat{\mathbf{s}}$. The input sequence is generally transformed into data symbols $x$ that are then fed into an intermediate channel layer (or network), which introduces constraints, distortions and uncertainties. Given the intermediate channel model, the autoencoder is typically trained by minimizing the cross-entropy loss function, so that it essentially performs a classification task. However, it is known that training a classifier via cross-entropy often suffers from overfitting issues, especially for large networks \cite{Zhang2017}. Moreover, autoencoders for communications shall consider the channel capacity during the learning process in order to produce optimal channel input samples (latent codes). In this direction, the work in \cite{Letizia2021} proposed to use a mutual information regularizer to control and estimate the amount of information stored in the latent representation. In addition, it proposed to use label smoothing to improve the network decoding ability. In \cite{Letizia2021}, the mutual information was computed with the neural estimator MINE \cite{Mine2018} by inserting an appropriate block in the autoencoder architecture. MINE was also exploited in \cite{Wunder2019} to study optimal coding schemes when no channel model is attainable. Nevertheless, MINE suffers from high bias and variance. Thus, new mutual information estimators have been recently proposed to subvert such limitations \cite{Song2020, Poole2019a, LetiziaNIPS}. Among them, a promising approach is given by the discriminative mutual information estimator (DIME) \cite{LetiziaNIPS}, a neural network that directly estimates the density ratio
\begin{equation}
\label{eq:density_ratio}
R(\mathbf{x},\mathbf{y}) = \frac{p_{XY}(\mathbf{x},\mathbf{y})}{p_X(\mathbf{x})\cdot p_Y(\mathbf{y})}
\end{equation}
instead of the individual densities in \eqref{eq:density_ratio}. Concurrently, it is expected that novel and optimal channel coding techniques based on mutual information learning, estimation and maximization can significantly impact the development of beyond 5G communication technologies.
In this paper, inspired by the DIME estimator and the $f$-GAN training objectives \cite{Nowozin2016}, we firstly propose a new family of estimators referred to as $f$-DIME. We secondly propose $\gamma$-DIME, a family of estimators that can be used in the end-to-end autoencoder training process to target the channel capacity. We then include the developed estimators in the capacity-driven autoencoder proposed in \cite{Letizia2021} and evaluate their performance in terms of block-error-rate (BLER) and accuracy of the mutual information estimation w.r.t. to MINE.
The rest of the paper is organized as follows. Section \ref{sec:autoencoders} revisits the autoencoder-based communication systems and discusses the major advantages of using a mutual information regularization term in the loss function. Section \ref{sec:theory} reviews some variational lower bounds on the mutual information and the related estimators. Section \ref{sec:f-DIME} presents a new set of discriminative estimators. Section \ref{sec:results} compares the estimators and illustrates the results. Finally, conclusions are reported.
| 75,618
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TITLE: Prove that a given CFG grammer $G$ is equivalent to language $L$
QUESTION [1 upvotes]: I need help to prove that the given CFG grammar $G$ is equivalent to language $L$:
as
$S\to 0S1 \mid SS \mid \varepsilon$
and
$L=\{w\in\{0,1\}^* \mid \#_0(w)=\#_1(w)\text{ and for any prefix } u \text{ of } w:\ \#_0(u)\ge\#_1(u)\}$
I need help only in proving: $L\subseteq L(G)$.
Thanks!
REPLY [0 votes]: The inclusion $L \subseteq L(G)$ can be shown by (strong) induction on the length of $w \in L$.
Clearly any $0$-length string in $L$ can be generated by the given CFG.
Suppose for some $n > 0$ that every string in $L$ of length $<n$ can be generated by the CFG. Let $w$ be a string in $L$ of length $n$. Consider the least $k > 1$ such that the prefix of $w$ of length $k$ has as many $\mathtt{0}$s as $\mathtt{1}$s. (Note that $k \leq n$.)
If $k = n$, then $w = \mathtt{0} v \mathtt{1}$, where $v$ is a string of length $n-2$. By definition of $k$ one can show that $v \in L$.
As there are as many $\mathtt{0}$s as $\mathtt{1}$s in $w = \mathtt{0} v \mathtt{1}$, there are as many $\mathtt{0}$s as $\mathtt{1}$s in $v$.
If some prefix of $v$ has more $\mathtt{1}$s than $\mathtt{0}$s, then the shortest such prefix has exactly one more $\mathtt{1}$ than $\mathtt{0}$; call this prefix $v^\prime$. But then $\mathtt{0} v^\prime$ is a prefix of $w$ with as many $\mathtt{0}$s as $\mathtt{1}$s, contradicting our choice of $k$. Therefore every prefix of $v$ must has at least as many $\mathtt{0}$s as $\mathtt{1}$s.
By the induction hypothesis $v$ can be generated by the CFG, and by first invoking the rule $S \to \mathtt{0} S \mathtt{1}$ and then generating $v$ we can then generate $w$ from the CFG.
If $k < n$, then $w = \mathtt{0}v_0\mathtt{1}v_1$, where $v_0$ has length $k-2 < n$ and $v_1$ has length $<n$. By definition of $k$ one can show that $v_0, v_1 \in L$.
As there are as many $\mathtt{0}$s as $\mathtt{1}$s in $\mathtt{0}v_0\mathtt{1}$, there must be as many $\mathtt{0}$s as $\mathtt{1}$s in $v_0$.
(Using the same argument as above, every prefix of $v_0$ has at least as many $\mathtt{0}$s as $\mathtt{1}$s.)
As there are as many $\mathtt{0}$s as $\mathtt{1}$s in both $w = \mathtt{0}v_0\mathtt{1}v_1$ and $\mathtt{0}v_0\mathtt{1}$, there are as many $\mathtt{0}$s as $\mathtt{1}$s in $v_1$.
If $v_1^\prime$ is a prefix of $v_1$, then $\mathtt{0} v_0 \mathtt{1} v_1^\prime$ is a prefix of $w$, as so has at least as many $\mathtt{0}$s as $\mathtt{1}$s. As there are an equal number of $\mathtt{0}$s and $\mathtt{1}$s in $\mathtt{0} v_0 \mathtt{1}$, there must be at least as many $\mathtt{0}$s as $\mathtt{1}$s in $v_1^\prime$.
By the induction hypothesis both $v_0$ and $v_1$ can be generated by the CFG. We may then generate $w$ from the CFG by $$S \to SS \to \mathtt{0} S \mathtt{1} S \overbrace{\to \cdots \to \mathtt{0} v_0 \mathtt{1} S}^{\text{steps to generate }v_0} \; \overbrace{\to \cdots \to \mathtt{0} v_0 \mathtt{1} v_1}^{\text{steps to generate }v_1}$$
| 116,770
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TITLE: How to Treat Coefficient as Constant To Approximately Solve PDE?
QUESTION [0 upvotes]: I am interested in analytically deriving an approximate solution to $\frac{\partial}{\partial t}f(x,t)=c(x,t) \frac{\partial}{\partial x^2}f(x,t)$ subject to given initial and boundary conditions. If the coefficient $c(x,t)$ was constant, then I knew how to obtain the solution of the equation exactly. Now that $c(x,t)$ varies in both $x$ and $t$, I do not have an analytical form, but still probably my best tool is to treat $c(x,t)$ as a constant $c$. I can think of two strategies for doing that, and I want to know which one makes more sense to you.
Literally replacing $c(x,t)$ by a constant, e.g. by its average value across $x \in \mathbb{R}$ and $t \geq 0$.
Treating $c(x,t)$ as a constant $c$ when solving the PDE. Once the solution is obtained, the we replace $c$ with $c(x,t)$ in the solution form.
Idea #2 is a bit crazy, but the actual $c(x,t)$ is not even nearly a constant in my problem, so I am worried that idea #1 completely misrepresents the problem. I appreciate opinions about which one of the two ideas may lead to a higher quality approximate solution and what is the justification for that.
Thanks
Golabi
REPLY [1 votes]: I am no expert, and perhaps there is an easy way to obtain the analytical solution... if one makes the variable change $w(\eta(x,t)) = u(x,t)$ with $\eta(x,t) = \frac{x}{\sqrt{t}},$ the equation becomes
$$w''(\eta) - \frac{\eta}{C(\eta)} w'(\eta) = 0,$$ which can then be directly integrated using an integrating factor, and then integrated directly again. Note that $C(\eta) = c(x,t)$ and I have assumed that $c \neq 0$ in order to divide by it. If $C(\eta)$ is a "nice" function, you'll be able to get the explicit form of the solution.
As for just solving the heat equation and plugging in $c(x,t)$ at the end... I'd say keep dreaming. Replacing $c(x,t)$ by some sort of average... I'd say its highly unlikely that doing either of these will capture the approximate nature of the actual solution.
One way, which is likely more difficult than just solving the equation numerically or analytically, could be to specify a grid of squares (boxes) over your domain, then treat $c$ as a constant (say, the average) over each of those boxes. As you refine the grid size, I suppose you may get a reasonable approximation. But even this is likely to result in discontinuities between boxes, which is qualitatively not what the heat equation does.
Edit: Consider the problem where $c = 1$ and $u_t = u_{xx}$ on $|x| < \infty, t > 0$ with the initial condition $u(x,0) = H(x),$ the heaviside function. If one does the transform above, one ultimately concludes that the solution to this problem is $$u(x,t) = \frac{1}{2}\bigg(1 + erf\big(\frac{x}{2\sqrt{t}}\big)\bigg).$$
Upon differentiating, one sees that $u_x(x,t) = \frac{1}{\sqrt{4\pi t}} e^{-x^2/4t}.$ One may recognize that as $t \to 0,$ this function becomes a better and better approximation to the delta function. This motivates saying that perhaps the solution to the problem
$$u_t = u_{xx}; \; u(x,0) = \delta(x); \; |x| < \infty, t>0$$ is in fact $u(x,t) = \frac{1}{\sqrt{4\pi t}} e^{-x^2/4t}.$ Recall that this indeed is the fundamental solution to the heat equation, or the heat kernel, infinite domain Green's function, what have you.
This is a lot to do but for what purpose? Well, now consider solving
$$u_t = c(x,t)u_{xx}; \; u(x,0) = H(x); \; |x| < \infty, t>0.$$ Following along the steps outlined originally, one might be able to get a nice analytic solution to this problem for nice $c(x,t),$ and then maybe by differentiating this analytic solution, one will obtain the solution to $$u_t = c(x,t)u_{xx}; \; u(x,0) = \delta(x); \; |x| < \infty, t>0.$$
There are some serious details being left out in that final step, but hey, its a possibility.
| 109,221
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\begin{document}
\theoremstyle{plain}
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\newtheorem*{conjecture*}{Conjecture}
\numberwithin{equation}{thm}
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\newcommand{\Modulione}{\mathfrak{M}_{1,n+3}}
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\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\Spec}{\mathrm{Spec}}
\newcommand{\Jac}{\mathrm{Jac}}
\newcommand{\modulinm}{\mathfrak{M}_{AR}}
\newcommand{\mc}{\mathfrak{M}_{1,6}}
\newcommand{\mx}{\mathfrak{M}_{3,6}}
\newcommand{\tmc}{\widetilde{\mathfrak{M}}_{1,6}}
\newcommand{\tmx}{\widetilde{\mathfrak{M}}_{3,6}}
\newcommand{\smc}{\mathfrak{M}^s_{1,6}}
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\newcommand{\proofend}{\hspace*{13cm} $\square$ \\}
\thanks{This work is supported by Chinese Universities Scientific Fund (CUSF), Anhui Initiative in Quantum Information Technologies (AHY150200) and National Natural Science Foundation of China (Grant No. 11622109, No. 11721101).}
\maketitle
\begin{abstract}
In the infinite series of complete families of Calabi-Yau manifolds $\tilde{f}_n: \tilde{\mathcal{X}}_n\rightarrow \mathfrak{M}_{n, n+3}$, where $n$ is an odd number, arising from cyclic covers of $\P^n$ branching along hyperplane arrangements (\cite{SXZ13}), the set of CM points is dense for $n=1, 3$ and finite for $n\geq 5$.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
Recall that to a $\Q$-Hodge structure ($\Q$-HS) $h: \C^*\to \mathrm{GL}_{\R}(V\otimes_{\Q}\R)$, one attaches a $\Q$-algebraic group $MT(V)$, the so-called Mumford-Tate group, which is defined to be the smallest $\Q$-algebraic subgroup of $\mathrm{GL}_{\Q}(V)$ whose real points contain $h(\C^*)$. A basic result about Mumford-Tate group is that it is equal to the maximal $\Q$-algebraic subgroup of $\mathrm{GL}_{\Q}(V)$ which fixes all Hodge cycles occurring in $\bigoplus_{m,n}V^{\otimes m}\otimes V^{*\otimes n}$. A $\Q$-HS is said to be \emph{CM} if its attached Mumford-Tate group is commutative. Thus, the set of CM $\Q$-HSs makes a special kind of Hodge structures.
Indeed, a weight one $\Q$-polarized Hodge structure is of CM if and only if its corresponding Abelian variety, defined up to isogeny, has complex multiplication. It is well known that in the moduli space $\sA_g$ of $g$-dimensional (principally) polarized Abelian varieties, the set of CM points (a CM point is such a moduli point that represents an Abelian variety having complex multiplication) is Zariski dense. The same property holds for any Shimura subvariety in $\sA_g$. The celebrated Andr\'{e}-Oort conjecture for the $\sA_g$ case, which was recently solved by Tsimermann in \cite{Tsimerman}, asserts the converse statement: any irreducible component of the Zariski closure of a subset of CM points in $\sA_g$ is a Shimura subvariety. In general, we say a complex smooth projective variety $X$ has CM if
the $\Q$-HS on the middle cohomology $H^n(\mathcal{X}_s, \Q)$ is CM. Let $S$ be a smooth connected quasi-projective variety over $\C$ and $f: \mathcal{X}\rightarrow S$ be a smooth projective family,
we say a point $s\in S$ is a CM point if the fiber $\mathcal{X}_s$ has CM. In view of the Andr\'{e}-Oort conjecture,
we are naturally led to studying the distribution of CM points in a smooth projective family.
As a vast generalization of the Andr\'{e}-Oort conjecture, Bruno Klingler proposed
a general conjecture about the distribution of CM points in a $\Z$-variation of mixed
Hodge structures (Conjecture 5.3 in \cite{Klingler}). To our purpose, we state
Klingler's conjecture in the following form.
\begin{conjecture*}[Klingler]\label{conj:Klingler}
Let $S$ be a smooth connected quasi-projective variety over $\C$ and $f: \mathcal{X}\rightarrow S$ be a smooth projective family of relative dimension $n$. If $S$ contains a Zariski dense set of CM points then the period map
of the $\Z$-variation of
Hodge structures on the
middle cohomology group $R^nf_*\Z_{\mathcal{X}}$ factors through
a Shimura variety.
\end{conjecture*}
Our aim is to examine Conjecture for an infinite series of families of Calabi-Yau (CY) manifolds, initially studied in our previous work \cite{SXZ13}. For each odd number $n$, let $r=\frac{n+3}{2}$, and let $\mathfrak{M}_{n, n+3}$ be the moduli space of $n+3$ hyperplanes arrangements of $\P^n$ in general position. In \cite{SXZ13}, we investigated several basic properties of universal families $f_n: \mathcal{X}_n\rightarrow \mathfrak{M}_{n, n+3}$ of $r$-fold cyclic covers of $\P^n$ branched along $n+3$ hyperplanes in general position. Note that the $n=1$ case is nothing but the classical universal family of elliptic curves. Among other results, we showed in loc. cit. that the family $f_n$ admits a simultaneous crepant resolution $\tilde{\mathcal{X}}_n\rightarrow \mathcal{X}_n$; the resulting CY family $\tilde{f}_n:\tilde{\mathcal{X}}_n\rightarrow \mathfrak{M}_{n, n+3}$ is actually complete and versal at each point; they all have Yukawa coupling length one. In this note, we are going to show the following
\begin{theorem*}\label{thm introduction: finite CM}
The set of CM points of the family $\tilde{f}_n$ is Zariski dense for $n=1,3$ and finite for $n\geq 5$.
\end{theorem*}
It was early shown in our work \cite{SXZ15} that the period map of $\tilde f_n$ does not factor through a Shimura variety if $n\geq 5$ (see Theorem 6.7 \cite{SXZ15}). Therefore, the above theorem provides an evidence to Conjecture. Note that it remains an interesting open problem whether the one-dimensional mirror quintic CY family contains only finitely many CM points or not. By a result of Deligne \cite{Deligne} on the monodromy group of the mirror quintic family, it should be the case after Conjecture. As far as we know, our result provides the first example of complete CY families with finite CM points.
\iffalse
As for the proof, we use a correspondence between an $r$-fold cyclic covers of $\P^n$ branched along $n+3$ hyperplanes in general position and an $r$-fold cyclic covers of $\P^1$ branched at $n+3$ distinct points. Then we can deduce our main result from a finiteness result of Chen-Lu-Zuo \cite{CLZ} about
CM superelliptic curves.
\fi
\section{Families from hyperplane arrangements}\label{sec:Families from hyperplane arrangements}
Given a hyperplane arrangement $\mathfrak{A}$ in $\P^n$ in general position, the cyclic cover of $\P^n$ branched along $\mathfrak{A}$ is an interesting algebraic variety. When the hyperplane arrangement $\mathfrak{A}$ moves in the coarse moduli space of hyperplane arrangements, we get a family of projective varieties. In this section, we collect some known facts about Hodge structures of these cyclic covers. Detailed proofs of all of the statements in this section can be found in \cite{SXZ13}.
We say an ordered arrangement $\mathfrak{A}=(H_1,\cdots, H_m)$ of hyperplanes in $\P^n$ is in general position if no $n+1$ of the hyperplanes intersect in a point, or equivalently, if the divisor $\sum_{i=1}^mH_i$ has simple normal crossings.
Given an odd number $n$, and let $r=\frac{n+3}{2}$. For each ordered hyperplane arrangement $(H_1,\cdots, H_{n+3})$ in $\P^n$ in general position, we can define a (unique up to isomorphism) degree $r$ cyclic cover of $\P^n$ branched along the divisor $\sum_{i=1}^{n+3}H_i$. In this way, if we denote the coarse moduli space of ordered $n+3$ hyperplane arrangements in $\P^n$ in general position by $\mathfrak{M}_{n, n+3}$, then we obtain a universal family $f_n:\mathcal{X}_{n}\rightarrow \mathfrak{M}_{n, n+3}$ of degree $r$ cyclic covers of $\P^n$ branched along $n+3$ hyperplane arrangements in general position. In \cite{SXZ13}, we constructed a simultaneous crepant resolution $\pi: \tilde{\mathcal{X}}_{n}\rightarrow \mathcal{X}_{n}$ for the family $f$ without changing the middle cohomology of fibers. Moreover, this simultaneous crepant resolution gives an $n$-dimensional projective Calabi-Yau family which is maximal in the sense that its Kodaira-Spencer map is an isomorphism at each point of $\mathfrak{M}_{n, n+3}$. We denote this smooth projective Calabi-Yau family by $\tilde{f}_n:\tilde{\mathcal{X}}_{n}\rightarrow \mathfrak{M}_{n, n+3}$.
Now we recall the relation between a cyclic cover of $\P^1$ branched along points and that of $\P^n$ branched along hyperplane arrangements.
Suppose $(p_1,\cdots, p_{n+3})$ is a collection of $n+3$ distinct points on $\P^1$, and put $H_i=\{p_i\}\times \P^1\times \cdots \times \P^1$. By the natural identification between $\P^n$ and the symmetric power $Sym^n(\P^1)$ of $\P^1$, we can view each $H_i$ as a hyperplane in $\P^n$. Then it can be shown that $(H_1,\cdots, H_{n+3})$ is a hyperplane arrangement in $\P^n$ in general position. A direct computation shows that this construction gives an isomorphism between the moduli spaces $\mathfrak{M}_{1, n+3}$ and $\mathfrak{M}_{n, n+3}$. Moreover, for $r=\frac{n+3}{2}$, if we denote $C$ as the $r$-fold cyclic cover of $\P^1$ branched along the $n+3$ points $(p_1,\cdots, p_{n+3})$, and $X$ as the $r$-fold cyclic cover of $\P^n$ branched along the corresponding hyperplane arrangement $(H_1,\cdots, H_{n+3})$, then we have an isomorphism
\begin{equation}\label{equation:relation between X and C}
X \simeq C^n/S_n\ltimes N.
\end{equation}
Here $N$ is the kernel of the summation homomorphism $(\Z/r\Z)^n \rightarrow \Z/r\Z$. The action of $\Z/r\Z$ on $C$ is induced from the cyclic cover structure, and $S_n$ acts on $C^n$ by permutating the $n$ factors.
From the isomorphism (\ref{equation:relation between X and C}),
we see the natural $\Q$-mixed Hodge structure on the middle cohomology group $H^n(X, \Q)$ is pure. Moreover, since the simultaneous crepant resolution $\tilde{\mathcal{X}}_{n}\rightarrow \mathcal{X}_{n}$ of the universal family $\mathcal{X}_{n}\xrightarrow{f_n} \mathfrak{M}_{n,n+3}$ does not change the middle cohomologies of the fibers, the two $\Q$-VHS (rational variation of Hodge structures) $R^n\tilde{f}_{n*}\Q_{\tilde{\mathcal{X}}_{n}}$ and $R^n f_{n*}\Q_{\mathcal{X}_{n}}$ are isomorphic.
\section{Proof of the main result}
For the smooth projective family of Calabi-Yau $n$-folds $\tilde{f}_n:\tilde{\mathcal{X}}_{AR}\rightarrow \mathfrak{M}_{n, n+3}$, we have the following proposition:
\begin{proposition}\label{prop:finite CM}
The family $\tilde{f}_3$ has a dense set of fibers that has CM. If $n\geq 5$ is an odd number, then up to isomorphism, there exist at most finitely many fibers of $\tilde{f}_n$ that has CM.
\end{proposition}
\begin{proof}
First note that since the two $\Q$-VHS $R^n\tilde{f}_{n*}\Q_{\tilde{\mathcal{X}}_{n}}$ and $R^n f_{n*}\Q_{\mathcal{X}_{n}}$ are isomorphic, we only need to prove this theorem for the family $f_n$.
By the discussions in Section \ref{sec:Families from hyperplane arrangements}, for each fiber $X$ of $f_n$, we can find $n+3$ distinct point $p_1, \cdots, p_{n+3}$ on $\P^1$, such that
\begin{equation}\label{equ:cor between X and C in general}
X\simeq C^n/S_n\ltimes N
\end{equation}
Here $C$ is the $r$-fold cyclic cover of $\P^1$ branched along $p_1, \cdots, p_{n+3}$, the group $N$ is the kernel of the summation homomorphism $\oplus_{i=1}^n\Z/ r\Z\xrightarrow{\sum}\Z/r\Z$, and $S_n$ is the permutation group of $n$ elements. The action of $S_n$ on the product $C^n$ is by permutating the factors, and the action of $N$ is induced by the cyclic cover action of $\Z/r\Z$ on $C$.
By the isomorphism \eqref{equ:cor between X and C in general}, we get an isomorphism of cohomology groups:
\begin{equation}\label{equ:cor between cohomology groups}
H^n(X, \ \Q)\simeq H^n(C^n, \ \Q)^{S_n\ltimes N}
\end{equation}
where $H^n(C^n, \ \Q)^{S_n\ltimes N}$ means the $S_n\ltimes N$-fixed part of $H^n(C^n, \ \Q)$. Let $W=H^1(C, \ \Q)$, and $V=H^n(X, \ \Q)$. As a subgroup of $\oplus_{i=1}^{n} \Z/ r\Z$, the group $N$ acts naturally on the tensor product $W^{\otimes n}$. We can also describe the action of $S_n$ on $W^{\otimes n}$:
$$
\sigma \cdot \alpha_1\otimes \alpha_2\otimes \cdots \otimes \alpha_n=(-1)^{sgn (\sigma)}\cdot \alpha_{\sigma^{-1}(1)}\otimes \alpha_{\sigma^{-1}(2)}\otimes \cdots \otimes \alpha_{\sigma^{-1}(n)}
$$
for $\sigma\in S_n$ and $\alpha_i\in W$ , $1\leq i\leq n$. Here $sgn(\sigma)=\pm 1$ means the signal of the permutation $\sigma$. By K\"unnenth formula, the isomorphism \eqref{equ:cor between cohomology groups} implies the following isomorphism:
\begin{equation}\label{equ:cor between W and V}
V\simeq (W^{\otimes n})^{S_n\ltimes N}
\end{equation}
Note $V$ admits a $\Q$-Hodge structure of weight $n$, and $W$ admits a weight one $\Q$-Hodge structure. Moreover, the isomorphism \eqref{equ:cor between W and V} is an isomorphism between $\Q$-Hodge structures of weight $n$, with the Hodge structure on $(W^{\otimes n})^{S_n\ltimes N}$ induced from the Hodge structure on $W$. A direct computation shows that the genus $g$ of $C$ satisfies the following relation:
$$
2g=(n+1)(r-1)
$$
Recall $r=\frac{n+3}{2}$. So if $n\geq 5$, then $g\geq 9$. Note the main result in \cite{CLZ} asserts that if $g\geq 8$, then up to isomorphism, there exist at most finitely many superelliptic curves of genus $g$ with CM Jacobian. Then the $n\geq 5$ cases follows from this result and the following claim. For the $n=3$ case, the
corresponding family of curves is the universal family of cyclic triple
covers of $\P^1$ branched along six distinct points, and by \cite{DM} this
curve family is a Shimura family. It is well-known that the CM points
are dense in a Shimura family. Then the $n=3$ case also follows from the following claim.
\textbf{Claim}: $V$ \textit{ has CM if and only if } $W$ \textit{ has CM}.
\end{proof}
\textbf{Proof of Claim}:
We first define the $\Q$-subgroup $G$ of $GL(W)$ consisting of elements commutating with the action of $\Z/r\Z$ on $W$. i.e.,
$$
G:=\{g\in GL(V)| g\cdot \alpha=\alpha \cdot g, \ \forall \alpha \in \Z/r\Z\}
$$
Then $G$ acts on $V=(W^{\otimes n})^{S_n\ltimes N}$ by:
$$
g\cdot \alpha_1\otimes \alpha_2\otimes \cdots \otimes \alpha_n= g\cdot \alpha_1\otimes g\cdot \alpha_2\otimes \cdots \otimes g\cdot \alpha_n
$$
for $g\in G$ and $\alpha_1\otimes \alpha_2\otimes \cdots \otimes \alpha_n\in (W^{\otimes n})^{S_n\ltimes N}$.
In this way, we get a homomorphism between $\Q$-algebraic groups:
$$
\varphi: G\rightarrow GL(V).
$$
By considering the complex points, we can verify directly that the kernel $K$ of $\varphi$ is a diagonalizable group lying in the center of $G$. We also have the following commutative diagram:
\begin{diagram}\label{diag:MT group}
\C^*& \rTo^{h_W} & G(\R)\\
& \rdTo_{h_V} &\dTo_{\varphi} \\
& &GL(V_{\R})
\end{diagram}
Here $h_V$ and $h_W$ are the morphisms associated with the Hodge structures on $V$ and $W$ respectively.
Note the Mumford-Tate group $MT(W)$ is contained in $G$, and by the commutative diagram above, we get $MT(V)\subset \varphi(MT(W))$. Hence $W$ has CM implies $V$ has CM.
Note also the following exact sequence:
$$
1\rightarrow K \rightarrow \varphi^{-1}(MT(V))\rightarrow MT(V)
$$
By this exact sequence, if $V$ has CM, then $\varphi^{-1}(MT(V))$ is a diagonalizable group whose real points contain $h_W(\C^*)$, and hence $MT(W)\subset \varphi^{-1}(MT(V))$ is a torus, and $W$ has CM. This finishes the proof of Claim. \hfill $\square$
As a corollary, we can deduce our main result.
\begin{theorem}
If $n=3$, then the set of CM points of the family $\tilde{f}_n$ is Zariski dense in $\mathfrak{M}_{n, n+3}$. If $n\geq 5$ is an odd number, then the set of CM points of the family $\tilde{f}_n$ is finite.
\end{theorem}
\begin{proof}
We only need to prove the $n\geq 5$ cases. Let $\mathcal{M}_n$ be the coarse moduli space of the fibers of $\tilde{f}_n$,
then the family $\tilde{f}_n$ induces a morphism of quasi-projective varieties
$\varphi: \mathfrak{M}_{n, n+3}\rightarrow \mathcal{M}_n$. Let
$S$ be the set of CM points in $\mathfrak{M}_{n, n+3}$. Then Proposition
\ref{prop:finite CM} implies that the image $\varphi(S)$ is a finite set in
$\mathcal{M}_n$. On the other hand, since the Kodaira-Spencer map of $\tilde{f}_n$
is an isomorphism at each point of $\mathfrak{M}_{n, n+3}$, we know each fiber of
$\varphi$ is a zero-dimensional close subvariety of $\mathfrak{M}_{n, n+3}$.
Then $S=\varphi^{-1}(\varphi(S))$ is a zero-dimensional close subvariety of $\mathfrak{M}_{n, n+3}$.
So $S$ is a finite set.
\end{proof}
\textbf{Acknowledgements}
We thank Professor Kang Zuo for several enlightening conversations related to this work. A primitive form of Conjecture \ref{conj:Klingler} for Calabi-Yau varieties has already appeared in the work \cite{Zhang} of Yi Zhang, who left us on April 2nd, 2019. The first named author is very grateful to Yi Zhang for his generous help and warm encouragement during different periods of the career. The sudden pass-away of Yi Zhang is a great loss to his family, teachers and friends.
\iffalse This work is partially supported by Chinese Universities Scientific Fund (CUSF) and Anhui Initiative in Quantum Information Technologies (AHY150200). The first named author is supported by National Natural Science Foundation of China (Grant No. 11622109, No. 11721101).\fi
| 1,877
|
TITLE: Construction of new ellipse
QUESTION [6 upvotes]: Using a pencil, the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that the new geometric figure which the pencil drew is also an ellipse (with the same foci as the first ellipse)?
REPLY [2 votes]: Partial answer
Given the two ellipses, we'll check if the thread around the inner ellipse is constant from any point on the outer ellipse.
We can describe the inner and outer ellipses as follows:
$$
\begin{align}
x &= a \cos(\alpha) && 0<a<e,\ 0<b, \label{1}\tag{1}\\
y &= b \sin(\alpha) && \alpha \in (-\pi, \pi],\ \beta \in (-\pi, \pi]\\
x' &= e \cos(\beta)\\
y' &= \sqrt{e^2-a^2+b^2}\ \sin(\beta)
\end{align}
$$
The $\sqrt{e^2-a^2+b^2}$ is to make the ellipses have the same foci.
First we want to know where (which two values of $\alpha$) the thread leaves the inner ellipse.
We do this by looking which tangent lines of the inner ellipse go through $(x', y')$.
The tangent lines can be found by adding a multiple of the derivative:
$$
\begin{align}
x' &= a \cos(\alpha) + t \cdot (-a\sin(\alpha)) \qquad t \in \mathbb{R} \\
y' &= b \sin(\alpha) + t \cdot (b \cos(\alpha))
\end{align}
$$
Let $c = \frac{x'}{a}$ and $d = \frac{y'}{b}$, we have to solve:
$$
\begin{align}
c &= \cos(\alpha) - t \sin(\alpha)\\
d &= \sin(\alpha) + t \cos(\alpha)
\end{align}
$$
This seems pretty hard, but using Mathematica we get:
$$
\begin{align}
m &= d \pm \sqrt{c^2+d^2-1} \label{2}\tag{2} \\
\alpha &= 2 \arctan \left( \displaystyle \frac{m}{c+1} \right) \\
t &= \frac{1-c^2-m^2}{2 m}
\end{align}
$$
The formula for $\alpha$ is undefined for $c = -1$, but that's no problem, because
we'll only check the right part ($\beta \in [-\pi/2, \pi/2]$) of the outer ellipse,
which is sufficient because of symmetry. This also prevents us to run into a potential
problem with the discontinuity of $\alpha$ and $\beta$ at $\pi$.
We can plug $\alpha$ back in $\eqref{1}$ to find the points on the ellipse:
$$
\begin{align}
x &= a \cos \left(2 \arctan \left( \frac{m}{c+1} \right) \right) = a \frac{-d m+c+1}{d m+c^2+c} \\
y &= b \sin \left(2 \arctan \left( \frac{m}{c+1} \right) \right) = b \frac{m(c+1)}{d m+c^2+c}
\end{align}
$$
Now we can make a formula for the length of the thread $L$. In $L_3$ the $\alpha_1$ refers to the lower of
the two values of $\alpha$ (choosing $-$ in $\eqref{2}$), and $\alpha_2$ to the higher value (choosing $+$ in $\eqref{2}$).
The integrand is the Pythagorean theorem applied to the derivative of $\eqref{1}$.
$$
\begin{align}
L_1 &= \sqrt{(x' - x_1)^2 + (y' - y_1)^2} \\
L_2 &= \sqrt{(x' - x_2)^2 + (y' - y_2)^2} \\
L_3 &= \int_{\alpha_2}^{\alpha_1+2\pi} \sqrt{a^2 \sin(\alpha)^2 + b^2 \cos(\alpha)^2} \ \mathrm{d}\alpha \\
L &= L_1 + L_2 + L_3
\end{align}
$$
The only thing left to do is proving $L$ is constant. We can do this by proving that its derivative to $\beta$ is equal to 0.
When writing $L$ as function of $a$, $b$, $e$ and $\beta$, the formula gets pretty big. So you'll want to use a computer for this.
We can get rid of the integral in $L_3$ because:
$$
\begin{align}
\frac{\mathrm{d}L_3}{\mathrm{d}\beta} &= \frac{\mathrm{d}\alpha_1}{\mathrm{d}\beta} \sqrt{a^2 \sin(\alpha_1)^2 + b^2 \cos(\alpha_1)^2} -
\frac{\mathrm{d}\alpha_2}{\mathrm{d}\beta} \sqrt{a^2 \sin(\alpha_2)^2 + b^2 \cos(\alpha_2)^2}
\end{align}
$$
With help of Mathematica we see that $\frac{\mathrm{d}L}{\mathrm{d}\beta}$ is equal to: (Mathematica code)
1/2 ((2 (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) (Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] - (b^2 e sinβ (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) + (b^2 (a + e Sqrt[1 - sinβ^2]) Sqrt[(-1 + sinβ^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2)))] (a^4 sinβ + b^2 e^2 sinβ - a^2 (b^2 + e^2) sinβ + a Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (a^2 b^2 (a + e Sqrt[1 - sinβ^2]) (-Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/ a) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^( 3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) - b sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[-1 + ((-a^2 + b^2 + e^2) sinβ^2)/b^2 - (e^2 (-1 + sinβ^2))/ a^2]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)) + (2 (a + e Sqrt[1 - sinβ^2]) (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) + a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2))) (-e sinβ + (a^3 (a b^2 - a (-a^2 + b^2 + e^2) sinβ^2 + b^2 e Sqrt[1 - sinβ^2] + sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) - ( sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])/ a))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2) - (a^2 b (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + sinβ (b^2 e + 2 a (-a^2 + b^2 + e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (-b^3 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a b sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])))))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))/(\[Sqrt](((a + e Sqrt[1 - sinβ^2])^2 (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) + a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2)))^2)/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))^2 + (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))^2)) + (-((2 (a + e Sqrt[1 - sinβ^2]) (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) - a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2))) (-e sinβ - (a^3 (-(( e sinβ)/a) - ( Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/a))/b^2 - ( sinβ Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (b Sqrt[-a^2 + b^2 + e^2] + ( b (-a^2 + b^2) (a^2 - e^2) sinβ)/( a Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])))/b^3))/(a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2) + ( a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/ b^2) - (a^3 (a b^2 - a (-a^2 + b^2 + e^2) sinβ^2 + b^2 e Sqrt[1 - sinβ^2] - sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + ( sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])/ a))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)))/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) + 2 (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) (Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] - (b^2 (a + e Sqrt[1 - sinβ^2]) Sqrt[(-1 + sinβ^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2)))] (-a^4 sinβ - b^2 e^2 sinβ + a^2 (b^2 + e^2) sinβ + a Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (b^2 e sinβ (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (a^2 b^2 (a + e Sqrt[1 - sinβ^2]) (Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/ a) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^( 3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + b sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[-1 + ((-a^2 + b^2 + e^2) sinβ^2)/b^2 - (e^2 (-1 + sinβ^2))/ a^2]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)))/(\[Sqrt](((a + e Sqrt[1 - sinβ^2])^2 (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) - a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2)))^2)/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))^2 + (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))^2)) - (2 a b (a^3 e sinβ + a^4 sinβ Sqrt[1 - sinβ^2] - a^2 (b^2 + e^2) sinβ Sqrt[1 - sinβ^2] + a (-e^3 sinβ + Sqrt[-(a^2 - e^2) (a^2 - b^2 - e^2) (-1 + sinβ^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) + e (b^2 e sinβ Sqrt[1 - sinβ^2] + Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])) Sqrt[-((( 4 a^2 (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2)/( b^2 (a + e Sqrt[1 - sinβ^2])^2) + b^2 (-1 + (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2))^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))))])/((-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) Abs[ 1 + (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2)]) - (2 a b (-a^3 e sinβ - a^4 sinβ Sqrt[1 - sinβ^2] + a^2 (b^2 + e^2) sinβ Sqrt[1 - sinβ^2] + a (e^3 sinβ + Sqrt[-(a^2 - e^2) (a^2 - b^2 - e^2) (-1 + sinβ^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) + e (-b^2 e sinβ Sqrt[1 - sinβ^2] + Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])) Sqrt[-((( 4 (a^2 Sqrt[-a^2 + b^2 + e^2] sinβ + a Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])^2)/( b^2 (a + e Sqrt[1 - sinβ^2])^2) + b^2 (-1 + (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2))^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))))])/((b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) Abs[ 1 + (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2)]))
Inserting arbitrary values for $a$, $b$, $e$ and $sin\beta$ and calculating it numerically seems to always yield 0.
It is left as an exercise for the reader to prove this is true for all $0<a<e, 0<b, -1 \le sin\beta \le 1$.
| 132,210
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Tesla Is Bringing Autonomous Cars to Dubai
This move could ultimately lead to safer roadways everywhere.
Tesla is now in the United Arab Emirates (UAE). The electric vehicle manufacturer and sustainable technology innovator has brought its charging, service, and support infrastructures, as well as autonomous models to Dubai. The country’s Roads and Transport Authority (RTA) ordered 200 Model S sedans and Model X SUVs with ‘fully-self-driving-capability’ from Tesla.
The deal was formalized Monday between Tesla CEO Elon Musk and Mattar Al Tayer, the UAE’s Director General and Chairman of the Board of Executive Directors of the RTA.
In a statement, Al Tayer said:. It is also part of the Dubai Smart Autonomous Mobility Strategy aimed at transforming 25 [percent] of total journeys in Dubai into autonomous journeys by 2030. The agreement also reflects RTA’s efforts towards providing driverless transportation solutions through undertaking technological tests of autonomous transit means.
Tesla expects to begin delivery of the vehicles by July of this year.
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The RTA will be the one oversee regulation of the vehicles in Dubai. The self-driving capable vehicles will be added to the Dubai Taxi Corporation’s limousine fleet. Studies have shown that autonomous vehicles can actually save lives, so far reducing Tesla crashes by up to 40 percent. Some suggest that driverless cars could save an estimated 32,000 lives per year. And as soon as Tesla rolls out its improved autopilot self-driving software, Dubai expects the usual regulatory approvals to come with ease. Perhaps this wave of autonomous vehicles in Dubai will lead to safer future roadways.
Futurism Readers: Find out how much you could save by switching to solar power at UnderstandSolar.com. By signing up through this link, Futurism.com may receive a small commission.
| 102,647
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Wimpy gamma-ray burst delights astronomers
A new type of cosmic explosion that occurred late last year could shed light on the death of massive stars, astronomers say.
The explosion was more powerful than supernovae, explosions marking the death of a huge star, but weaker than gamma-ray bursts (GRB), the mysterious and most brilliant blasts in the universe.
"I was stunned that my observations ... showed that this event confirmed the existence of a new class of bursts," said graduate student Alicia Soderberg, of the California Institute of Technology in Pasadena, who reported the finding today in the journal Nature. "It was like hitting the jackpot."
Scientists had thought all GRBs had a standard energy and the same intrinsic brightness until the discovery of the cosmic blast that occurred on 3 December 2003 known by its date of birth, GRB 031203.
"With this new event we realise it is not true. There are sub-energetic bursts that are less luminous with fainter emission, which means there is not a standard energy," Soderberg said.
"Perhaps there is some sort of continuum between the two explosions that we didn't realise before," she added.
The new blast occurred about 1.6 billion light-years away. A light-year is about 10 trillion kilometres, the distance light travels in a year.
It was also much closer than other GRBs and about 1000 times weaker.
Astronomers do not know what causes GRBs. They are thought to occur when stars collapse possibly to become a black hole, creating a huge gravitational pull from which nothing can escape.
But last month, cosmologist Professor Stephen Hawking said he believed some material oozes out of black holes over billions of years through irregularities on their surface.
Have we been missing faint gamma-ray bursts?
Scientists from the Space Research Institute of the Russian Academy of Sciences, in Moscow, who also observed the cosmic blast and reported it in Nature, believed other similar blasts have occurred but have not been detected. And in 1998 astronomers reported the extremely faint GRB 980425.
NASA's forthcoming Swift mission, which will study GRBs, could provide more information about the explosions.
"This is an intriguing discovery," said Shrinivas Kulkarni, a professor of astronomy and planetary science at the California Institute of Technology and a co-author of one of the reports.
"I expect a treasure trove of such events to be identified by NASA's Swift mission," he said.
"I am convinced that further discoveries and studies of this new class of hybrid events will forward our understanding of the death of massive stars."
| 319,629
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<< < alp off > >>
A short distance from the target, in right ascension, where
the telescope is pointed for various purposes.
→ alpha; → offset.
In → CCD detectors, same as
→ bias and → offset.
→ bias; → offset.
A.
A short distance from the target, in → declination,
where the → telescope is pointed for various purposes.
→ delta; → offset..
Same as → Greisen-Zatsepin-Kuzmin cutoff.
→ Greisen-Zatsepin-Kuzmin cutoff.
The radiation law which states that at thermal equilibrium the ratio of
the energy emitted by a body to the energy absorbed by it depends only
on the temperature of the body.
Gustav Robert Kirchhoff (1824-1887), a German physicist who made
major contributions to the understanding of electric circuits, spectroscopy,
and the emission of black-body radiation from heated objects; → law.-2022 by M. Heydari-Malayeri
| 173,940
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\begin{document}
\title[A simple algorithm to compute link polynomials]{A simple algorithm to compute link polynomials defined by using skein relations}
\author{Xuezhi Zhao}
\address{Department of Mathematics \& Institute of mathematics and interdisciplinary
science, Capital Normal University, Beijing 100048, CHINA}
\email{zhaoxve@mail.cnu.edu.cn}
\thanks{The authors are supported by National
Natural Science Foundation of China (Grant No. 10931005)}
\keywords{Link; polynomial; braid representative}
\subjclass[2010]{57M25}
\maketitle
\begin{abstract}
We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete link invariant are obtained.
\end{abstract}
\newcommand{\brp}[3]{\sigma_{#1_{#2}}^{\epsilon'_{#2}}\cdots \sigma_{#1_{#3}}^{\epsilon'_{#3}}}
\newcommand{\br}[3]{\sigma_{{#1}_{#2}}^{\epsilon_{#2}}\cdots \sigma_{{#1}_{#3}}^{\epsilon_{#3}}}
\newcommand{\brord}{<_{br}}
\section{Introduction}
Link polynomials are important topological invariants to distinguish links and knots. Many efforts were made to give more effective methods to calculate them (see\cite{Kauffman, book}). It is known that computing the Jones polynomial is generally $\sharp P$- hard \cite{W}, and hence it is expected to
require exponential time in the worst case.
As we know, many link polynomials can be defined by using the so-called skein relation. For instance, HOMFLY polynomial $P(\cdot)$ (see \cite{HOMFLY}), which contains the information of Alexander polynomial, Conway polynomial, Jones polynomial, and etc.,
could be obtained inductively as follows:
\begin{equation}\label{skein-relation}
\begin{array}{l}
P(\mbox{unknot}) =1, \\
\ell\, P(L_+) +\ell^{-1}P(L_-) +m P(L_0) =0,\ \ \ \ \mbox{(skein relation)}
\end{array}
\end{equation}
where $L_+$, $L_-$ and $L_0$ are three link diagrams which are different only on a local region, as indicated in the following figures.
\begin{equation}\label{3L}
\setlength{\unitlength}{0.7mm}
\mbox{
\ \ \ \ $L_+$:\ \
\begin{picture}(14,14)(0,-2)
\qbezier(6.79,5.89)(10,7.5)(10,10)
\qbezier(3.21,5.89)(0,7.5)(0,10)
\qbezier(3.21,4.11)(0,2.5)(0,0)
\qbezier(6.79,4.11)(10,2.5)(10,0)
\qbezier(3.21,5.89)(5,5)(6.79,4.11)
\put(0,0){\vector(0,-1){3}}
\put(10,0){\vector(0,-1){3}}
\end{picture}
\ \ \ \ \ $L_-$:\ \
\begin{picture}(14,14)(0,-2)
\qbezier(6.79,5.89)(10,7.5)(10,10)
\qbezier(3.21,5.89)(0,7.5)(0,10)
\qbezier(3.21,4.11)(0,2.5)(0,0)
\qbezier(6.79,4.11)(10,2.5)(10,0)
\qbezier(3.21,4.11)(5,5)(6.79,5.89)
\put(0,0){\vector(0,-1){3}}
\put(10,0){\vector(0,-1){3}}
\end{picture}
\ \ \ \ $L_0$:\ \
\begin{picture}(14,14)(0,-2)
\qbezier(10,0)(4,5)(10,10)
\qbezier(0,0)(6,5)(0,10)
\put(3,3){\vector(-1,-1){3}}
\put(7,3){\vector(1,-1){3}}
\end{picture}
}
\end{equation}
In this paper, we shall provide a simple algorithm to calculate link polynomials, if these polynomials are defined by using skein relations. Links are considered as closed braids, and hence are oriented by from top to bottom orientation on braids. Our reduction is based on a new total order of the set of all braid representatives.
\section{Braid group and an order of braid representatives}
The Artin $n$-strands braid group $B_n$ has classical generators $\sigma_1, \sigma_2, \ldots \sigma_{n-1}$,
and two types of relations:
\begin{equation}\label{relation1}
\sigma_i\sigma_j = \sigma_j\sigma_i \ \ \ \ \mbox{for }\ |i-j|>1,
\end{equation}
\begin{equation}\label{relation2}
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} \ \ \ \ \mbox{for }\ j=1,2,\ldots, n-2.
\end{equation}
Geometrically, elements in $B_n$ can be regarded as $n$ strings, the product of two braids is a joining from top to bottom.
Each generator $\sigma_j$ is given as follow.
\begin{center}
\setlength{\unitlength}{1.3mm}
\begin{picture}(55,16)(-20,0)
\put(-5,12){\makebox(0,0)[cb]{$j\!\!-\!\!1$}}
\put(5,12){\makebox(0,0)[cb]{$j\!\!+\!\!1$}}
\put(0,12){\makebox(0,0)[cb]{$j$}}
\put(-20,12){\makebox(0,0)[cb]{$1$}}
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\end{center}
We shall write arrays of the form $[n; b_1, \ldots, b_k]$ for the braid representatives. Here, each $b_j$ is a non-zero integer with $|b_j|<n$. The array $[n; b_1, \ldots, b_k]$ indicates the element $\sigma_{|b_1|}^{\sgn(b_1)}\cdots\sigma_{|b_k|}^{\sgn(b_k)}$ in $B_n$.
\begin{Definition}
Given a braid representative $\beta=[n; b_1,\ldots, b_k]$, the $m$-th weight $w_m(\beta)$ of $\beta$ is defined to the number $\sharp \{b_j \mid |b_j| = m \}$ of indices $b_j$ in the representative $\beta$ having absolute value $m$.
\end{Definition}
All weights of a given braid representative are zero but finitely many ones. Explicitly, $w_i([n; b_1, \ldots, b_k])=0$ for a braid representative of elements in $B_n$ if $i\ge n$.
By using these weights, we can define an total order on all braid representatives as follows.
\begin{Definition}\label{new-order}
Let $\beta=[n; b_1,\ldots, b_k]$ and $\gamma=[m; c_1,\ldots, c_l]$ be two braid representatives. We say that $\beta$ is smaller that $\gamma$, denoted $\beta \brord \gamma$, if one of the following conditions is satisfied:
(1) $k<l$;
(2) $k=l$, $n<m$;
(3) $k=l$, $n=m$, and there is an integer $p$ such that $w_i(\beta)=w_i(\gamma)$ for $i=1,2,\ldots, p-1$ and $w_p(\beta)<w_p(\gamma)$;
(4) $k=l$, $n=m$, $w_i(\beta)=w_i(\gamma)$ for all $i$, and there is an integer $q$ such that $|b_j|=|c_j|$ for $j=1,2,\ldots, q-1$ and $|b_q|<|c_q|$;
(5) $k=l$, $n=m$, $w_i(\beta)=w_i(\gamma)$ for all $i$, $|b_j|=|c_j|$ for $j=1,2,\ldots, k$, and there is an integer $q$ such that $b_j=c_j$ for $j=1,2,\ldots, q-1$ and $b_q<c_q$ (i.e. $b_q=-c_q<0$).
\end{Definition}
It is well-known that each link can be considered as a closed braid (see \cite{BirmanBook}). Clearly, with respect to the order $\brord$, the set of all braid representatives turns out to be a total order set. For any given braid representative $\beta$, there are finitely many braid representatives which are smaller than $\beta$.
Hence, we have
\begin{Theorem}
Each link has a unique minimal braid representative according to the order $\brord$. Hence the minimal braid representative is a complete link invariant.
\end{Theorem}
Looking for orders on set of all braids is also an interesting topic (see \cite{De}). Our order ``$\brord$'' gives naturally a total order on set of all braids. If we disregard the difference of braids at their weights in above definition, our definition coincides with the order introduced in \cite{Minimum Braids}. Our main improvement makes it possible to compute inductively link polynomial according to this. It seems that the order in \cite{Minimum Braids} does not work.
\section{An algorithm}
\def\ltl{{\rm ol}}
In this section, we shall give the key algorithm, showing the way to use the skein relation to make braid representatives smaller.
\begin{Definition}\label{eq}
An equivalence relation $\sim$ on the set of all braid representatives is defined to be one generated by following elementary relations:
(1) $[n; b_1,\ldots, b_k] \sim [n; b_2, b_1, b_3, \ldots, b_k ]$ if $||b_1|-|b_2||>1$;
(2) $[n; b_1,\ldots, b_k] \sim [n; \sgn(b_3)|b_2|, \sgn(b_2)|b_1|,\sgn(b_1)|b_2|, b_4, \ldots, b_k ]$ if $||b_1|-|b_2||=1$, $|b_1|=|b_3|$, but $\sgn(b_1)=-\sgn(b_2)=\sgn(b_3)$ does not hold;
(3) $[n; b_1,\ldots, b_k] \sim [n; b_3, \ldots, b_k ]$ if $b_1=-b_2$;
(4) $[n; b_1,\ldots, b_k] \sim [n-1; b_2-\sgn(b_2), \dots, b_k-\sgn(b_k)]$ if $|b_j|>|b_1|$ for $j=2,\ldots, k$;
(5) $[n; b_1,\ldots, b_k] \sim [n; b_2, \ldots, b_k, b_1]$.
\end{Definition}
From \cite[Corollary 2.3.1]{BirmanBook}, we obtain immediately that
\begin{Lemma}
Two braid representatives $\beta$ and $\gamma$ are equivalent if and only if corresponding closed braids $\hat\beta$ and $\hat\gamma$ are the same link.
\end{Lemma}
To calculate link polynomial by using skein relation, we shall convert a calculation of polynomial of a link given by a braid representative into those given by two simple braid representatives. Here, we need to find a ``good'' braid representative so that the two reduced braid representatives are both smaller than given one with respect to the order ``$\brord$''.
To this end, we introduce a technical concept for braid representatives.
\begin{Definition}\label{leading-tag}
Given a braid representative $\beta=[n; b_1,\ldots, b_k]$, the {\em ordered leading tag length} $\ltl(\beta)$ of $\beta$ is a non-negative integer defined as follows:
(1) If $|b_1| \ne \min_{1\le j\le k}\{|b_j|\}$, then $\ltl(\beta)$ is $0$.
(2) If $|b_1| = \min_{1\le j\le k}\{|b_j|\}$, then $\ltl(\beta)$ is the maximal subscript $q$ such that $|b_j| = |b_1|+j-1$ for $j=1,2,\ldots, q$.
\end{Definition}
Now, we provide our key algorithm. In each step braid representatives decrease according to our order ``$\brord$'' in given equivalence classes. Meanwhile, ordered leading tag length is becoming longer.
\begin{Algorithm}\label{alg} (Simplify a braid representative of a link)
{\bf Input}: a braid representative $\beta$ of given link $L$.
{\bf Output}: a braid representative $\gamma$ of link $L$ with $\gamma\sim\beta$ such that either $\gamma\brord \beta$ or $\gamma= \beta$.
In each of following steps, assume that we start with a renewed braid representative $\beta=[n; b_1,\ldots, b_k]$.
Step 1: If $k=0$, then stop. Otherwise, find the $b_m$ such that $|b_m|=\min_{1\le j\le k}\{|b_j|\}$, and $|b_j|>|b_m|$ for $j=1,2,\ldots, m-1$. Replace $\beta$ with $[n; b_m, b_{m+1}, \ldots, b_k, b_{1}, \ldots, b_{k-1}]$ \emph{(Elementary relation (5))}.
Step 2: If $|b_j|>|b_1|$ for $j=2,\ldots, k$, i.e. $w(\beta_{|b_1|})=1$, replace $\beta$ with $\beta'=[n-1; b_2-\sgn(b_2), \dots, b_k-\sgn(b_k)]$ \emph{(Elementary relation (4))}, and then go to step 1. Otherwise, go to next step.
Step 3: If there is an index $b_j$ such that $b_j=-b_{j+1}$, then replace $\beta$ with the representation $[n; b_1,\ldots, b_{j-1}, b_{j+2} \ldots, b_{k}]$. Repeat this step until $\beta$ can not be renewed. If length reduction happens in this step, then go to step 1, otherwise go to next step.
Step 4: Having $\ltl(\beta)=q>0$, there are three cases:
Case 4.1 $|b_{q+1}| > |b_q|+1$: Let $p$ be the maximal subscript such that $|b_j|> |b_q|+1$ for $j=q+1, \ldots, p$. Replace $\beta$ with $[n; b_1,\ldots, b_q, b_{p+1}, \ldots, b_k, b_{q+1}, \ldots, b_p]$ \emph{(Repeating of several elementary relations (1) and (5))}, and then go to step 3;
Case 4.2 $|b_{q+1}| = |b_q|$: Stop;
Case 4.3 $|b_{q+1}| < |b_q|$: There must be an integer $m$ with $(1\le m\le q-1)$ such that $|b_m|=|b_{q+1}|$. Replace $\beta$ with $$\beta'=[n; b_1, \ldots, b_m, b_{m+1}, b_{q+1}, b_{m+2}, \ldots, b_q, b_{q+2}, \ldots, b_k]$$ \emph{(Elementary relation (1) and (5))}.
For the sake of simplification, the new $\beta$ is still written as $[n; b_1, \ldots, b_k]$. Now, we have $|b_j| = |b_1|+j-1$ for $j=1,2,\ldots, m+1$, and $|b_{m+2}| = |b_m|$. There are two subcases:
Subcase 4.3.1 $\sgn(b_m) = -\sgn(b_{m+1}) = \sgn(b_{m+2})$: Stop.
Subcase 4.3.2 $\sgn(b_m) = -\sgn(b_{m+1}) = \sgn(b_{m+2})$ does not hold: Replace $\beta$ with $$[n; b_1, \ldots, b_{m-1}, \sgn(b_{m+1})|b_{m}|, \sgn(b_m)|b_{m+1}|, b_{m+3}, \ldots, b_k,
\sgn(b_{m+2})|b_{m+1}|].$$ \emph{(In fact, $\beta$ is equivalent to $$[n; b_1, \ldots, b_{m-1}, \sgn(b_{m+2})|b_{m+1}|, \sgn(b_{m+1})|b_{m}|, \sgn(b_m)|b_{m+1}|, b_{m+3}, \ldots, b_k]$$ by elementary relation (2), and hence to $$[n; \sgn(b_{m+2})|b_{m+1}|, b_1, \ldots, b_{m-1}, \sgn(b_{m+1})|b_{m}|, \sgn(b_m)|b_{m+1}|, b_{m+3}, \ldots, b_k]$$ by elementary relation (1) and (5). The latter is equivalent to our new $\beta$ by elementary relation (5))}. And then go to step 1.
\end{Algorithm}
Main features of above algorithm is summarized as follows.
\begin{Lemma}\label{output-brp}
Given any braid representative $\beta$, Algorithm~\ref{alg} terminates at a braid representative $\gamma=[m, c_1, \ldots, c_l]\sim\beta$ satisfying one of the following conditions:
(1) $\gamma=[m; - ]$, i.e. $l=0$;
(2) $\ltl(\gamma)=q>0$ and $c_{q+1}=c_q$;
(3) $\ltl(\gamma)=q>0$, $c_{q+1}=c_{q-1}$, and $\sgn(c_{q+1})=-\sgn(c_q)$.
\end{Lemma}
\begin{proof}
Equivalency of all steps are explained in the brackets in algorithm description. Three cases of terminated braid representatives are respectively those terminated at step 1, case 4.2 of step 4 and case 4.3.1 of step 4.
\end{proof}
We are ready to show that our algorithm really works in calculating HOMFLY polynomial.
\begin{Theorem}
If a braid representative of a link $L$ is given, the calculation of HOMFLY polynomial of $L$ can be fulfilled inductively by using skein relation and Algorithm~\ref{alg}.
\end{Theorem}
\begin{proof}
Given a braid representative $\beta$ of link $L$, Algorithm~\ref{alg} leads to a new braid representative $\gamma=[m; c_1, \ldots, c_l]$ for $L$, as indicated in Lemma~\ref{output-brp}.
If the first case of Lemma~\ref{output-brp} happens, we are done because the link $L$ is a trivial circle.
In the other two cases, let $\gamma'=[m; c_1, \ldots, c_{q-1}, -c_{q}, c_{q+1}, \ldots, c_l]$ be the braid representative obtained from $\gamma$ by changing the sign of $q$-th index, and let $\gamma''=[m; c_1, \ldots, c_{q-1}, c_{q+1}, \ldots, c_l]$ be the braid representative obtained from $\gamma$ by dropping the $q$-th index. Consider the region around the crossing indicated by $c_q$ (i.e. $\sigma_{c_q}$), the corresponding three closed braids $\hat\gamma$, $\hat\gamma'$ and $\hat\gamma''$ have the relation: either $L_+=\hat\gamma$, $L_-=\hat\gamma'$ and $L_0=\hat\gamma''$ (when $c_q>0$); or $L_+=\hat\gamma'$, $L_-=\hat\gamma$ and $L_0=\hat\gamma''$ (when $c_q<0$), where $L_+$, $L_-$ and $L_0$ are those as illustrated in (\ref{3L}). From the skein relation, the calculation of the HOMFLY polynomial of $\hat\gamma$ is reduced down to those of $\hat\gamma'$ and $\hat\gamma''$. Thus, it is sufficient to show that as links, $\hat\gamma'$ and $\hat\gamma''$ have braid representatives which are smaller than $\gamma$ with respect to the order $\brord$.
Clearly, we have that $\gamma''\brord\gamma$ because $\gamma''$ has less indices. If $\gamma$ is in the case (2) of Lemma~\ref{output-brp}, then $c_{q+1}=c_q$. As elements in braid group $B_m$, $\gamma'$ is the same as $[m; c_1, \ldots, c_{q-1}, c_{q+2}, \ldots, c_l]$, which is smaller than $\gamma$. If $\gamma$ is in the case (3) of Lemma~\ref{output-brp}, then $c_{q+1}=c_{q-1}$, and $\sgn(c_{q+1})=-\sgn(c_q)$. The elementary relations (5) and (2) in Definition~\ref{eq} imply that $\gamma'$ is equivalent to $$\delta=[m; c_1, \ldots, c_{q-2}, -c_q, c_{q-1}, -c_q, c_{q+2}, \ldots, c_l]$$ (cf. relation~(\ref{relation2})), which is smaller than $\gamma$ because $\gamma$ and $\delta$ have the same number of indices, $w_i(\delta)=w_i(\gamma)$ for $i=1, 2, \ldots, |c_{q-1}|-1$ but $w_{|c_{q-1}|}(\delta)=w_{|c_{q-1}|}(\gamma)-1$.
\end{proof}
Let us illustrate our method by using a concrete example. Consider the knot with braid representative $[4; -1,2,3,-1,3,2,-3]$. (The first knot having crossing number $6$.)
$$\begin{array}{cl}
& P([4; -1,2,3,-1,3,2,-3]) \\
= & P([4; -1,2,-1, 3,3,2,-3]) \\
= & -\ell^{-2}P([4; -1,-2,-1, 3,3,2,-3])-\ell^{-1}mP([4; -1,-1, 3,3,2,-3]) \\
= & -\ell^{-2}P([4; -2,-1, -2, 3,3,2,-3])-\ell^{-1}mP([4; -1,-1, 3,3,2,-3]) \\
= & -\ell^{-2}P([4; -1, -2, 3,3,2,-3,-2])-\ell^{-1}mP([4; -1,-1, 3,3,2,-3]) \\
= & -\ell^{-2}P([3; -1, 2,2,1,-2,-1])-\ell^{-1}mP([4; -1,-1, 3,3,2,-3]) \\
= & \ell^{-2}(\ell^{-2}P([3; -1, -2,2,1,-2,-1])+\ell^{-1}mP([3; -1,2,1,-2,-1]))\\ & \ \ +\ell^{-1}m(\ell^2 P([4; 1,-1, 3,3,2,-3]) + \ell mP([4; -1, 3,3,2,-3])) \\
= & \ell^{-4}P([3; -2,-1])+\ell^{-3}mP([3; 2,1, -2, -2,-1])\\
& \ \ +\ell mP([4; 3,3,2,-3]) + m^2P([4; -1, 2,-3, 3,3]) \\
= & \ell^{-4}P([3; -1,-2])+\ell^{-3}mP([3; 1, -2, -2,-1, 2])\\
& \ \ +\ell mP([4; 2,3]) + m^2P([4; -1, 2, 3]) \\
= & \ell^{-4}P([1; -])+\ell^{-3}mP([3; 1, -2, -2,-1, 2]) +\ell mP([2; -]) + m^2P([1; -]) \\
= & (\ell^{-4}+m^2)P([1; -])+\ell mP([2; -]) \\
& \ \ +\ell^{-3}m(-\ell^2 P([3; 1, 2, -2,-1, 2])-\ell m P([3; 1,-2,-1, 2])) \\
= & (\ell^{-4}+m^2)P([1; -])+\ell mP([2; -]) -\ell^{-1}m P([3; 2])-\ell^{-2} m^2 P([3; 1,-2,-1, 2])) \\
= & (\ell^{-4}+m^2)P([1; -])+\ell mP([2; -]) -\ell^{-1}m P([3; 2])-\ell^{-2} m^2 P([3; 2,-1,-2, 2])) \\
= & (\ell^{-4}+m^2)P([1; -])+\ell mP([2; -]) -\ell^{-1}m P([2; -])-\ell^{-2} m^2 P([1; -]))
\end{array}
$$
Since $P([1; -])=1$ and $P([2; -])= -\ell m^{-1}- \ell^{-1} m^{-1}$, we obtain that
$$ P([4; -1,2,3,-1,3,2,-3]) = \ell^4 +m^2 -\ell^{-2}m^2 -\ell^2+\ell^{-2}.
$$
\section{Computing remarks}
In order to verify our algorithm, we make a programm by using Mathematica. Thank to the listing of knots in terms of braid representatives, we calculate the HOMFLY polynomials of knots up to cross number $12$. For these $2977$ knots, the total running time is 430 second. Meanwhile, we record, for each knot $K$, the maximal number $ND(K)$ of link diagrams during calculation.
We obtain that
\begin{equation}\label{exp}
\exp(\sum_{K}\frac{\ln(ND(K))}{bc(K)})=1.42,
\end{equation}
where $bc(K)$ is the braid crossing of knot $K$. The number $ND(K)$ indicates how many nodes we need to store the temporary braid representatives in calculating the HOMFLY polynomial of the knot $K$. The equation~(\ref{exp}) gives us a geometric average of growth rate of number of nodes according braid crossing if it is considered as to be exponential. The complicities is about $1.42^c$, where $c$ is the crossing number. Comparing traditional method (with complexity $2^c$), our algorithm is reasonable.
Of course, there are many methods to compute link polynomial, such as \cite{Murakami, eg, SBY}, which may have less complexities in some restricted cases. Our algorithm can be applied to arbitrary link and arbitrary link polynomial as long as skein relation is satisfied.
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Special topics
Upstream Petroleum Operations in Ghana: An Overview of Activities and Environmental Impacts.
For an undeniable fact, exploitation of petroleum resources has sumptuous economic implications for a country, as well as in most countries has also reduced enormously natural oil pollution from oil seeps. However, activities of oil companies as perceived by the environmentalist have serious, unplayful impacts on the environment. Research done by a group of scientists from the Ambrose Alli University of Nigeria on “The effects of petroleum exploration and production operations on the heavy metals contents of soil and groundwater in the Niger Delta” shows a high pollution of the groundwater systems and soils. The heavy metals investigated including, Iron, Copper, Cadmium, Chromium, Lead and Nickel and their existence in extreme concentration in water and food and subsequent consumption poses serious health problems and damage to internal organs. For example, excess consumption of lead can lead to plubism, and damage the brain, Central Nervous System, kidney, liver and the reproductive system. In this article, cardinal activities in upstream operations including seismic, drilling, development and production, and transportation shall be scrutinized in the light of their environmental impacts.
Seismic Acquisition
Although there has been a lack of scientific evidence on the detrimental effect of acoustic waves on marine life, the geophysical industry has not been able to prove this. There are serious concerns on the effects of seismic waves on whales, sea turtles, prawns, lobsters, and other commercial fish species. In the Gulf of Mexico, United Kingdom, Canada, Australia and most part of the world it is a standard procedure that a seismic contractor performs ramp-up procedure to warn sea animals and make provision for 24 hour observer crews onboard seismic vessel to observes the presence of sea animals that likely to be harmed by acoustic waves. Ramp-up procedure drives sea animal at least about 500 meters from a seismic array. Research conducted in the Norwegian seas has proven that seismic shooting could cause fish to travel tens of kilometers, and some may not return unless after a few weeks. In onshore data acquisition, vast area of vegetation needs to be cleared to improve accessibility to Vibroesis and other seismic acquisition equipment. The destruction to vegetation is made more pronounced in mangroves and forests.
Drilling
The predominant discharges during drilling are cuttings; small chips of formation rock cut by the drilling bit and the mud used in cooling and lubricating the drilling bit, carry the cuttings out of the hole and counter-balance the pressure of gas when it is reached. It is generally accepted that drill mud cause greatest harm in exploration activities. In an experiment to monitor the damage caused by drill mud in an exploration drilling in 1981-82 on Georges Bank, barium, a tracer of drilling mud doubled in concentration 35 km eastward and increased six fold 65 km in the western direction. Oil based muds have been known to have detrimental effects on benthic organisms because of it redox potential.
Development and Production
The development stage is characterized by further drilling and installation of pumps, and separation equipment together with their peripherals. Directional drilling where several wells can be drilled from a single platform has reduced the amount of “footprint” and subsequent contamination of soil and destruction of vegetation in onshore and offshore expeditions.
In countries where there is no commercial market for associated gas, the gas is flared. The World Resources Institute Report World Resources 1994-95 indicates that total gas flaring in 1991 produced a contribution of 256 million tones of Carbon Dioxide emissions which represent some 1 per cent of global carbon dioxide emissions (22, 672 million tones).
Particulates which are generated from other burning sources such as well testing contribute enormously to atmospheric pollution. Apart from the emission of carbon dioxide and carbon monoxide, nitrogen oxides and hydrogen sulphide gases are introduced into the atmosphere; the quantity of which depends on the content of nitrogen and sulphur in the oil.
In production, the major waste is produced water containing inorganic salts, heavy metals, solids, production chemicals, hydrocarbons and occasionally Naturally Occurring Radioactive Material (NORM); these have minimal effect on the environment. Nonetheless the release of produced water into freshwater bodies requires special care. Transportation Oil transportation has been a major source of pollution through oil spills and leakages. Oil spills occur as a result of unguaranteed mechanical processes that are involved in oil transportation. For a period of fourteen years, from 1986 to 2000, the Nigerian petroleum industry suffered a total of 3,854 oil spills squirting a total of 437,810 barrels of oil into the Nigerian environment. Oil spills have been known since antiquity and they appear to be inevitable in oil production.
Pipeline installation requires massive dredging of seabed and excavations of soil, which poses serious threat to sensitive environments. In onshore areas, estuaries, wet land and mangroves are disturbed massively and may result in flooding. Offshore activities are associated with dredging of seabed and destruction marine habitat.
Human, Socio-economic and Cultural Impacts
There are serious concerns on land use and fishing in the Western shores of the country. Until recently, exploration activities and fishing have not existed peacefully in this part of the country. With the influx of skilled and unskilled labor into the Western Region, it is highly anticipated that local population levels are going to surge together with diverse unfamiliar cultures. Sociologist would be interested on the impact of diverse cultures on the fragile Ghanaian culture.
Socio-economic imbalance would be created due to new employment opportunities, income differentials, inflation, differences in per capita income when different members of local groups benefit unevenly from induced economic changes. Autochthonous people often react when they are persuaded the activities of oil companies are detrimental to their social economic and physical wellbeing, as it has been the modus operandi in the oil prolific Niger Delta region. In some cases oil pipelines are vandalized by local people as in their opinion, they want to take their share of the national oil cake. The end results of which have been mostly fatal.
By June, 10th 2006, a total of $ 400 billion has accrued from Nigeria’s 50 years of oil and gas exploration and production, yet there are millions of Nigerians wallowing in extreme poverty. This has a major incentive to the concerns being raised about the possible repetition of the “paradox of plenty” on the Ghanaian side of the oil industry.
Conclusions
A barrel of oil is pumped along several barrels of environmental issues. Would we as a nation forfeit exploiting our natural resources with the primary intention of developing the nation because of the associated environmental issues? Certainly no; hence environmental pollution is inevitable. There are several international conventions on the conduct of Exploration and Production (E&P) companies. However, depending on the political environment, and as a means of maximizing profit, oil companies would relinquish their environmental responsibilities. It therefore becomes imperative on governmental and non-governmental watchdogs to monitor the activities of E&P companies. At this juncture, the Ghana National Petroleum Corporation (GNPC)—the Lord of petroleum exploration and production activities in Ghana, the Ministry of Environment and Science, the Environmental Protection Agency and the Ministry of Energy should be commended for the indomitable, synergetic work they have done supervising the exploration and development stage of the Jubilee Field. We trust them that during the production stage, the environmental laws would be strictly enforced to ensure a safer environment both in the short and long terms.
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LHOTEL
262 rue St-Jacques Ouest, Montréal, Québec H2Y 1N1, Canada
Andy Warhol, Roy Lichtenstein, Robert Rauschenburg—you’ll find their art here, among other pop treasures. LHOTEL brings the glamor of the international art scene to Old Montreal.
YOU SHOULD KNOW While the location is great for exploring the Old Port, rue St-Jacques is deadly dull after 6 p.m.
room
The rooms have second empire molding and impressive 14-foot ceilings. Rooms that aren’t painted bright hues of yellow, blue and green have polished wood-paneling. Vintage photos of movie stars, Warhols, and Persian rugs give everything a homey feel.
bathroom
Airy and bright, the bathrooms have body products by Gilchrist & Soames.
lobby
Again, chalk full of art, model boats, cube seats, sofas, and sculpture. The Botera Wine Bar is lit with electric blue neon tubing that works well with the contemporary art.
dining
A hot continental breakfast buffet is the only meal available ($19), and worth every penny to eat under the curved glass and wrought iron ceiling in the dining room.
LOCATION
Getting Around
Business guests appreciate the location in the middle of the financial district and a block from the city's convention center. Just a 10-minute walk from the Square Victoria metro in one direction and Place d’arm metor in the other direction, there’s no reason to rent a car. There’s very little parking and the streets are narrow, so this is no place for a Hummer. Take your Hummer elsewhere, thank you.
Restaurants
European foodies should head to Toqué! (20-minute walk), for the famous fois gras. For something more intimate, there’s Barroco (10-minute walk). Order the bone-in rib eye and truffle butter.
Bars
There are quirky microbrewery labels at Joverse (10-minute walk), or choose from the hot, cold, nigori or sparkling sake at Kyo Bar Japonaise (10-minute walk).
WHY WE LIKE IT
It's a slice of MOMA in Montreal. This hotel feels like a (wealthy) curator's private home, with lots of fancy guest bedrooms. We like the stately lobby, with a bar and boutique, but we really love wandering the hallways and poking our noses into the meeting rooms, looking at pieces of original modern art. The owner, an international fashion mogul, lives on the top floor, keeping standards up to snuff.
QUICK FACTS
HOTEL INFO
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.
The first blog of this TfL series showed that a key difference between the public transport funding models in London and Paris is that dependency on fare revenues is particularly low in the latter. Of the five cities analysed, the French capital is the only one where fare revenues accounted for less than 30 per cent of total revenue. So how does Paris fund its public transport system?
Paris has a funding tool called “Versement Transport” (VT). This gives local government the powers to create a local payroll tax for firms with more than 11 employees. It’s a system that has been in place in Greater Paris since 1973 and today it varies between 1.4 per cent and 2.6 per cent of gross wages, depending on their place of residency.
Figure 1 shows the importance of VT to Paris’s public transport finances. It accounts for 52 per cent of all revenues and around three quarters of funding from taxes (2018). Central government subsidies are a significant component of Paris’ transport – 18 per cent of total revenue – which is higher than what TfL’s previous received from the Government (around 10 per cent).
Source: Île-de-France Mobilités (2018)
The Parisian funding model, by raising revenue with VT, stems from a belief that the transport network is for the city’s good and so should be funded by most of its population. By collecting revenues from most of its workers, regardless of whether they use the network regularly, this approach charges working residents who indirectly benefit from the public transport system (i.e. better air quality and less congested roads). Raising revenue through VT also ensure fares are relatively affordable (around £65 a month). In addition, this system makes local authorities less vulnerable to both central policy (i.e. cutting existing subsidies) and ridership, which is a significant factor in a post-pandemic world.
Unlike Hong Kong’s model, which relies on land development that could take several years to replicate, a VT-style contribution could allow TfL to raise further revenue in the short to medium term. Centre for Cities estimates that £1 billion could be raised (around the projected funding gap) with a 0.6 per cent contribution from London’s gross wages (on average £20.40 a month per worker).
Raising taxes of course would be politically difficult but a VT-style contribution would at least not be as regressive as raising council taxes. It would in principle free up public resources to invest in either other policy areas or elsewhere in the country.
In practical terms, London cannot introduce such a scheme without primary legislation because the local government does not have the powers to enact its own taxes. London’s congestion charge and the Workplace Parking Levy had to pass the same process. If the Mayor of London wants to move towards a Parisian-style model in the years ahead – to reduce TfL’s dependency on fares – fiscal devolution should be a core priority in negotiations with Government.
This blog is part of the TfL series, where the Centre for Cities explores different Mass Transit System funding models around the globe. The final piece of research of the series will set a range of policy recommendations, based on the models analysed.
A short blog series examining different Mass Transit System funding models from around the globe.
The second blog in Centre for Cities' TfL series shows that while Hong Kong’s mass transit operator may not be the answer to TfL’s short-term funding issues, it needs to be considered in the long-term.
The first blog in Centre for Cities' TfL series looks at how the pandemic has affected London’s transport network, and why TfL should move away from a fare-driven funding model.
The fourth blog in Centre for Cities' TfL series looks at Singapore's urban mobility model which shows that congestion charging and ULEZ are not the only policies available to simultaneously raise revenue and reduce car use.
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| 407,280
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$ 108.99 SALE: $ 98.99.
customer shoe reviews
| 313,055
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TITLE: Non-associative commutative "group"
QUESTION [4 upvotes]: When dealing with some hash functions that I was trying to speed up, I toyed with a binary operation with the goal to "approximate" the addition on $\{0,1\}^*$ when seen as binary representation of the positive integers:
$$(a,b) \in (\{0,1\}^*)^2 \mapsto a \oplus b \oplus ((a \land b) \ll 1)$$ where $\oplus$ is bitwise XOR and $\land$ is bitwise AND, and $\ll$ is left-shift by 1 position. (The purpose of $((a \land b) \ll 1)$ is to simulate the "carry-bit" operation.)
Formal definition. Let $\{0,1\}^\mathbb{N}$ denote the collection of functions $f:\mathbb{N}\to \{0,1\}$ and let $$\{0,1\}^* = \{x \in \{0,1\}^\mathbb{N}: \exists N\in\mathbb{N}(\forall k\in\mathbb{N}(k\geq N\implies x(k)=0))\}.$$
Denote by $\ll 1$ the shift by one position, i.e. $\ll 1 : x \in \{0,1\}^* \to x'\in \{0,1\}^*$ where $x'(0) = 0$ and $x'(n+1) = x(n)$ for all $n\in \mathbb{N}$. We usually write $x \ll 1$ instead of $\ll 1(x)$.
For any $a,b\in\{0,1\}^*$ let us write $$a +_2 b := a \oplus b \oplus ((a\land b) \ll 1).$$
It is easily seen that $+_2$ is not associative, and that $0$ is a neutral element for every $x\in \{0,1\}^*$. Moreover, $+_2$ is clearly commutative
Question. Given $a,b\in \{0,1\}^*$, is there $x\in \{0,1\}^*$ such that $a +_2 x = b$? Is $x$ necessarily unique?
Further question. (Need not be answered for acceptance.) If the answer to the above question is positive, we would have a kind of "non-associative group". Do these have a proper name? Do they occur "naturally" somewhere in mathematics?
REPLY [5 votes]: The answer to the first question is yes:
Claim. Let $a, b \in \{0, 1 \}^{\ast}$ and let $n \ge 0$ be the least integer such that $a(i) = b(i) = 0$ for every $i > n$. Then the equation $$a +_2 x = b$$ has a unique solution $x \in \{0, 1 \}^{\ast}$ which is recursively defined by $x(0) = a(0) + b(0)$, $x(i) = a(i - 1)x(i - 1) + a(i) + b(i)$ for $0 < i \le n$ and
$x(n + 1) = a(n)x(n)$, $x(i) = 0$ for $i > n + 1$.
Equivalently, the solution $x$ is given by
$$
\begin{array}{lll}
x(0) &=& a(0) + b(0), \\
x(1) &=& a(0)(a(0) + b(0)) + a(1) + b(1),\\
... & = & ..., \\
x(n) & = & a(0) \cdots a(n - 1)(a(0) + b(0)) + a(1) \cdots a(n - 1)(a(1) + b(1)) + \cdots + a(n - 1)(a(n - 1) + b(n - 1)) + a(n) + b(n),
\end{array}$$
$x(n + 1) = a(n)x(n)$
and $x(i) = 0$ for $n > n + 1$ where the operations $+$ and $\cdot$ refer to addition and multiplication in $\mathbb{Z} / 2 \mathbb{Z}$ which we identified with $\{0, 1 \}$.
Proof. The equation is equivalent to $x \oplus ((a \wedge x) \ll 1) = a \oplus b$ which reads as a linear system over $\mathbb{Z} / 2 \mathbb{Z}$.
| 211,788
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The Day Thou Gavest : Lyrics.
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Printable PDF
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\begin{document}
\date{June 3, 2018}
\title{Erratum to ``Deformed Calabi--Yau Completions''}
\author{Bernhard Keller}
\address{Universit\'e Paris Diderot -- Paris 7\\
UFR de Math\'ematiques\\
Institut de Math\'ematiques de Jussieu--PRG, UMR 7586 du CNRS \\
Case 7012\\
B\^{a}timent Chevaleret\\
75205 Paris Cedex 13\\
France
}
\email{bernhard.keller@imj-prg.fr}
\urladdr{https://webusers.imj-prg.fr/~bernhard.keller/}
\begin{abstract}
We correct an error and some inaccuracies
that occurred in ``Deformed Calabi--Yau completions''. The most important point is that,
as pointed out by W.~K.~Yeung,
to show that the deformed Calabi-Yau completion has
the Calabi--Yau property, one needs to assume that the deformation parameter
comes from negative cyclic homology. Notice that this does hold in the case
of Ginzburg dg algebras.
\end{abstract}
\keywords{Calabi--Yau completion, Ginzburg dg algebra}
\subjclass[2010]{18E30 (primary); 13F60, 16E35, 16E45, 18E35, 18G60 (secondary)}
\maketitle
\section{Calabi--Yau completions}
\label{s:CY-completions}
We refer to \cite{Keller11b} for unexplained notation and terminology.
Let $k$ be a commutative ring, $n$ an integer and $\ca$ a dg $k$-category.
We may and will assume that $\ca$ is cofibrant
over $k$, \ie each morphism complex $\ca(X,Y)$ is cofibrant as a complex
of $k$-modules. Moreover, we assume that $\ca$ is homologically smooth
\cite{KontsevichSoibelman06},
\ie $\ca$ is perfect in the derived category $\cd(\ca^e)$ of $\ca$-bimodules. For a
dg $\ca$-bimodule $M$, put
\[
M^\vee=\RHom_{\ca^e}(M,\ca^e).
\]
Recall \cite{Ginzburg06} that $\ca$ is {\em (bimodule) $n$-Calabi--Yau} if there is an isomorphism
\[
\phi: \Sigma^n \ca^\vee \iso \ca
\]
in the derived category of $\ca$-bimodules.
The symmetry property originally imposed on $\phi$ is automatic as shown in
Appendix~C of \cite{VandenBergh15}. Following
\cite{ThanhofferVandenBergh12, KontsevichVlassopoulos13, Yeung18},
we define an {\em $n$-Calabi-Yau structure on $\ca$}
as the datum of a class $\eta$ in negative cyclic homology $HN_n(\ca)$
which is {\em non degenerate}, \ie whose
image under the canonical maps
\[
HN_n(\ca) \to HH_n(\ca,\ca) \iso \Hom_{\cd(\ca^e)}(\Sigma^n \ca^\wedge, \ca)
\]
is an isomorphism. Following \cite{VandenBergh15}, such a structure
is called {\em exact} if $\eta$ is an image under Connes' map
$B: HC_{n-1} (\ca) \to HN_n(\ca)$.
Let $\Theta_\ca$ be the inverse dualizing complex $\ca^\vee$ and
$\theta$ a cofibrant replacement of $\Sigma^{n-1}\Theta_\ca$.
The {\em $n$-Calabi--Yau completion $\Pi_n(\ca)$} was defined in \cite{Keller11b}
as the tensor category $T_\ca(\theta)$. The following theorem
is a more precise version of Theorem~4.8 of \cite{Keller11b}.
We include a proof since the statement is slightly
stronger and the new proof more transparent.
\begin{theorem} \label{thm:cy-completion}
The Calabi-Yau completion $\Pi_n(\ca)$ is homologically
smooth and carries a canonical exact $n$-Calabi-Yau structure.
\end{theorem}
\begin{remark} For finitely cellular dg categories $\ca$,
W.~K.~Yeung gives two proofs of this theorem: one in section~3.3 of \cite{Yeung16} and a
more geometric one in section~2.3 of \cite{Yeung18}. The theorem generalizes his result
to arbitrary homologically smooth dg categories $\ca$.
\end{remark}
\begin{remark} It is well-known that the Calabi-Yau completion,
being a generalization of the $2$-Calabi--Yau preprojective algebra \cite{CrawleyBoevey00}, should be viewed as a (shifted) non commutative
cotangent bundle. This is very nicely explained in section~2.3 of \cite{Yeung18}.
Alternatively, one may view it as a (shifted) non commutative
total space of the canonical bundle.
This is made rigorous in section~3.5 of \cite{IkedaQiu18}.
\end{remark}
\begin{proof}[Proof of the Theorem] Put $\cb=\Pi_n(\ca)$. Notice that $\cb$ is
augmented over $\ca$ in the sense that we have canonical dg functors
$\ca\to\cb\to\ca$ whose composition is the identity. Thus, Hochschild and cyclic
homology of $\ca$ are canonically direct summands of those of $\cb$.
We call the supplementary summands the {\em reduced} Hochschild respectively
cyclic homology of $\cb$. We would like to compute them.
Since $\cb$ is a tensor category (though over the non commutative ground
category $\ca$), it suffices to adapt the results of section~3.1 of \cite{Loday98}.
The analogue of the
small resolution $C^{\mathrm{sm}}(T(V))$ of remark 3.1.3 of \cite{Loday98}
is the exact bimodule sequence
\begin{equation} \label{eq:small-resolution}
\xymatrix{0 \ar[r] & \cb\ten_\ca\theta\ten_\ca\cb \ar[r]^-{b'} & \cb\ten_\ca\cb \ar[r] &
\cb\ar[r] & 0}\ko
\end{equation}
where the map $b'$ takes $x\ten t\ten y$ to $xt\ten y -x\ten t y$ and the second
map is composition.
By taking the derived tensor product over $\cb^e$ with $\cb$, we find that
Hochschild homology of $\cb$ is computed by the cone over the induced morphism
\[
\xymatrix{ (\cb\ten_\ca\theta\ten_\ca\cb)\lten_{\cb^e} \cb \ar[r] &
(\cb\ten_\ca\cb)\lten_{\cb^e}\cb.}
\]
Notice that $\theta$ is cofibrant over $\ca^e$ so that
$\cb\ten_\ca\theta\ten_\ca\cb=\theta\ten_{\ca^e}\cb^e$ is cofibrant over
$\cb^e$ and for the first term, we have
\[
(\cb\ten_\ca\theta\ten_\ca\cb)\lten_{\cb^e} \cb =(\theta\ten_{\ca^e}\cb^e)\lten_{\cb^e}\cb
=\theta\ten_{\ca^e}\cb.
\]
As for the second term, notice that $\cb$ is cofibrant over $\ca$ so that
we have
\[
\cb\ten_\ca\cb=\cb\lten_\ca\cb=\ca\lten_{\ca^e}\cb^e
\]
and therefore
\[
(\cb\ten_\ca\cb)\lten_{\cb^e}\cb = (\ca\lten_{\ca^e}\cb^e)\lten_{\cb^e}\cb =
\ca\lten_{\ca^e}\cb.
\]
Now notice that in $\cb=\ca\oplus\theta\oplus(\theta\ten_\ca\theta)\oplus \cdots$,
all the summands are cofibrant over $\ca^e$ except the first one. Since we
are interested in {\em reduced} Hochschild homology, the first term does
not matter: Let us put $\ol{\cb}=\cb/\ca$. Then $\ol{\cb}$ is cofibrant over
$\ca^e$ and we have $\ca\lten_{\ca^e}\ol{\cb} = \ca\ten_{\ca^e}\ol{\cb}$.
So the reduced Hochschild homology of $\cb$ is computed by the cone over
the morphism
\[
\xymatrix{\theta\ten_{\ca^e} \cb \ar[r]^-b & \ca\ten_{\ca^e}\ol{\cb}.}
\]
It is not hard to check that the map $b$ is given by
\[
t\ten u \mapsto 1_y \ten tu - (-1)^{|t||u|} 1_x \ten ut \ko
\]
where $t\in \theta(x,y)$ and $u\in\cb(y,x)$. Notice that its kernel has the summand
$\theta \ten_{\ca^e}\ca$. The construction shows that the cone over $b$
is the analogue of the complex $C^{\mathrm{small}}(T(V))=(V\ten A \to A)$ of section~3.1.1 of \cite{Loday98}, where $A=T(V)$. Now proceding further along this line, it is not
hard to check that the analogue of the complex
\[
\xymatrix{\ldots \ar[r]^-\gamma & V\ten A \ar[r]^-b & A \ar[r]^-\gamma & V\ten A \ar[r]^-b &
A \ar[r] & 0}
\]
computing the cyclic homology of $A=T(V)$ in Prop.~3.1.5 of \cite{Loday98}
is the sum total dg module of the periodic complex
\[
\xymatrix{\ldots \ar[r]^-\gamma & \theta \ten_{\ca^e}\cb \ar[r]^-b &
\ca\ten_{\ca^e}\ol{\cb} \ar[r]^-\gamma & \theta\ten_{\ca^e}\cb \ar[r]^-b &
\ca\ten_{\ca^e}\ol{\cb} \ar[r] & 0}\ko
\]
which computes the reduced cyclic homology of $\cb$ and where
$\gamma$ is given by
\[
a \ten (t_1 \ldots t_n) \mapsto
\sum_{i=1}^n \pm t_i \ten (t_{i+1} \ldots t_n a t_1 \ldots t_{i-1})\ko
\]
where the sign is given by the Koszul sign rule.
Let us now exhibit a canonical element in the reduced $(n-1)$th cyclic homology
of $\cb$. We have canonical quasi-isomorphisms
\[
\xymatrix{\theta \ten_{\ca^e} \ca & \theta\ten_{\ca^e} \bp\ca \ar[l] \ar[r] & \Sigma^{n-1}\Hom_{\ca^e}(\bp \ca,\bp\ca)} \ko
\]
where $\bp\ca\to\ca$ is an $\ca^e$-cofibrant resolution of $\ca$.
The identity on the right hand side corresponds to a Casimir element $c$ on the
left hand side. This element yields a canonical class of homological
degree $n-1$ in the summand
$\ca\ten_{\ca^e}\theta\cong\theta\ten_{\ca^e}\ca$ of the last term of the complex\[
\xymatrix{\ldots \ar[r]^-\gamma & \theta \ten_{\ca^e}\cb \ar[r]^-b &
\ca\ten_{\ca^e}\ol{\cb} \ar[r]^-\gamma & \theta\ten_{\ca^e}\cb \ar[r]^-b &
\ca\ten_{\ca^e}\ol{\cb} \ar[r] & 0}
\]
and thus a canonical class in $HC_{n-1}^{\mathrm{red}}(\cb)$. Under Connes'
map $HC_{n-1}^{\mathrm{red}}(\cb) \to HH_n^{\mathrm{red}}(\cb)$, which
is induced by $\gamma$, this class clearly corresponds to the class of the
Casimir element $c$ in the summand $\theta \ten_{\ca^e} \ca$ of
$\theta\ten_{\ca^e}\cb$. It remains to be checked that this class
is non degenerate,\ie that it is taken to an isomorphism by the canonical maps
\[
\cb\lten_{\cb^e}\cb \iso \cb\lten_{\cb^e}\Theta^\vee \iso
\RHom_{\cb^e}(\Theta, \cb) \ko
\]
where $\Theta$ is the inverse dualizing complex $\cb^\vee$.
By the exact sequence (\ref{eq:small-resolution}), the $\cb$-bimodule $\cb$
is quasi-isomorphic to the cone over the map
\[
\xymatrix{\cb\ten_\ca\theta\ten_\ca\cb \ar[r]^-{b'} & \cb\ten_\ca\cb.}
\]
As we have already noted, the left hand term is $\cb^e$-cofibrant but the right
hand term is not. We choose a surjective $\ca^e$-cofibrant resolution $\bp\ca\to\ca$.
It yields a surjective quasi-isomorphism
\[
\xymatrix{\cb\ten_\ca\bp\ca\ten_\ca\cb \ar[r] & \cb\ten_\ca\cb}\ko
\]
where now the left hand term is $\cb^e$-cofibrant. We choose a lift
$\tilde{b}'$ of $b'$ along this quasi-isomorphism. We find that $\cb$ has as
$\cb^e$-cofibrant resolution the cone over the map
\[
\xymatrix{\cb\ten_\ca\theta\ten_\ca\cb \ar[r]^-{\tilde{b}'} &
\cb\ten_\ca\bp\ca\ten_\ca\cb}\ko
\]
which we can rewrite as
\[
\xymatrix{\theta\ten_{\ca^e}\cb^e \ar[r] & \bp\ca\ten_{\ca^e}\cb^e}.
\]
By applying $\Hom_{\cb^e}(?,\cb^e)$ to this cone and using the adjunction,
we find that $\Theta$ is the cylinder over
\[
\xymatrix{\Hom_{\ca^e}(\bp\ca,\cb^e) \ar[r] & \Hom_{\ca^e}(\theta,\cb^e).}
\]
We know that the class of $c$ in $\theta\ten_{\ca^e}\ca\subset \theta\ten_{\ca^e}\cb$ yields a morphism from this cylinder shifted by $n$ degrees to the
$\cb^e$-cofibrant resolution of $\cb$, \ie the cone over
\[
\xymatrix{\cb^e\ten_{\ca^e}\theta \ar[r] & \cb^e\ten_{\ca^e} \bp \ca.}
\]
This morphism is invertible in $\cd(\cb^e)$.
Indeed, it is not hard to see that in the left respectively right hand term,
the class $c$ induces the canonical isomorphisms of $\cd(\cb^e)$
\[
\Hom_{\ca^e}(\bp\ca,\cb^e) \iso\cb^e\ten_{\ca^e}\Sigma^{1-n}\theta
\quad\mbox{resp.}\quad
\Hom_{\ca^e}(\theta,\cb^e) \iso \cb^e\ten_{\ca^e}\Sigma^{1-n}\bp\ca.
\]
\end{proof}
\section{Deformed Calabi--Yau completions}
We keep the notations and assumptions of section~\ref{s:CY-completions}.
In particular, $\ca$ is assumed to be homologically smooth.
Let $c$ be a class in the Hochschild homology $HH_{n-2}(\ca)$.
Via the canonical isomorphism
\[
HH_{n-2}(\ca) \iso \Hom_{\cd(\ca^e)}(\theta,\Sigma\ca)\ko
\]
the class $c$ may be lifted to a closed degree~$1$ bimodule morphism
$\delta: \theta\to\ca$. The {\em deformed Calabi--Yau completion $\Pi_n(\ca,c)$}
was defined in section~5 of \cite{Keller11b} as obtained from
$\Pi_n(\ca)=T_\ca(\theta)$ by replacing the differential $d$ with the unique
derivation of the tensor category extending
\[
\left[\begin{array}{cc} d & 0 \\ \delta & d \end{array} \right] :
\theta\oplus \ca \to \theta\oplus\ca.
\]
Up to isomorphism, it only depends on $\ca$ and the class $c$. It was claimed in
Theorem~5.2 of \cite{Keller11b}, that the deformed Calabi--Yau
completion is always $n$-Calabi--Yau. A counter-example to this
claim is given by W.~K.~Yeung at the end of section~3.3 in \cite{Yeung16}.
As pointed out by Yeung, a sufficient condition for the deformed
$n$-Calabi--Yau completion to be $n$-Calabi--Yau is suggested by
the work of Thanhoffer de V\"olcsey--Van den Bergh \cite{ThanhofferVandenBergh12}:
It suffices that the class $c$ lifts to the negative cyclic homology $HN_{n-2}(\ca)$.
The following is Theorem~3.17 of \cite{Yeung16}.
\begin{theorem}[Yeung] Suppose that $\ca$ is finitely cellular and $c$ is a class in
$HH_{n-2}(\ca)$. Any lift $\tilde{c}$ of $c$ to the negative cyclic homology
$HN_{n-2}(\ca)$ determines an $n$-Calabi--Yau structure on the
deformed Calabi--Yau completion $\Pi_n(\ca,c)$.
\end{theorem}
Notice that the assumptions do hold for Ginzburg dg categories.
One would expect that, like theorem~\ref{thm:cy-completion}, this theorem
should generalize to arbitrary homologically smooth dg categories.
\section{Correction of some inaccuracies in ``Deformed Calabi--Yau completions"}
\label{s:inaccuracies}
In section~3.6, one should point out in addition
that the assumption that $Q$ is projective over $k$ implies that it is
projective as an $\cR$-bimodule (since $\cR$ is discrete, so
is $\cR^e$), which in turn implies that
the filtration (3.6.1) is split in the category of $\cR$-bimodules.
At the end of the proof of Theorem~4.8, it is claimed that the transpose
conjugate of $\tilde{\lambda}$ maps to $\rho$. This is not true. However,
the image of $\tilde{\lambda}$ and $\rho$ define the same class in homology.
The statement of Theorem~6.1 is incorrect. One has to assume in addition
that the set of `minimal relations' $R$ generates the
ideal $I$ (from the definition of $R$, it only follows that
$R$ topologically generates the $J$-adic completion of $I$).
Moreover, the condition 3) in the proof of the theorem should be
replaced with
\begin{quote}
for all $n\geq 1$, the differential $d$ maps $V^{-n-1}$ to $T_n$
and induces an isomorphism from $V^{-n-1}$ onto the head
of the $H^0(T_n)$-bimodule $H^{-n}(T_n)$, where $T_n$
denotes the dg category $T_{\mathcal{R}}(V^0\oplus\cdots \oplus V^{-n})$.
\end{quote}
\section*{Acknowledgments}
I am very grateful to W.~K.~Yeung for providing a counter-example to Theorem~5.2
of \cite{Keller11b} and indicating a way to correct the error. I thank
Xiaofa Chen, Martin Kalck, Kai Wang and Dong Yang for pointing out
the inaccuracies corrected in section~\ref{s:inaccuracies}.
\def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
| 158,617
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It’s unlike us to post an entire album, but then again it’s unlike albums to be as good as this. Even at almost a year old Submotion Orchestra‘s Finest Hour still rolls and breaks along, perfectly full of energy and yet still almost absurdly understated. The sound reminds me of Belleruche in that it combines the traditionally electronic genres with an acoustic, organic vibe which is hard to beat.
Choosing the right music to soundtrack these next few weeks is pretty important for me. Job applications have come good and the blog is flexing the occasional muscle in good directions which brings in a huge amount of satisfaction. But at the same time exams are exams and it’s good to do good in exams. So rather than trawl through 8tracks for study mixes which just end up leading me on a day long tangent of musical discovery I’m sticking to a diet made up purely of Submotion Orchestra, Belleruche and 2 playlists in my iTunes (Claudia Mix and Blaz Mix). I might dabble in a bit of arrange, some Griggleschpot and, if things get really bad, some Ludovicio Einaudi.
What will probably end up happening is I will spend days typing up a Malcolm Middleton interview we did yesterday followed by endless tweaks to the blog design, then sorting out the multi-blog mixtape we are hosting on Monday, then getting my festival diary together, then doing some promo for the gig we are curating on the 10th May in Exeter with Said The Whale…
“What you do instead of your work is your real work.“
-Roger Ebert
marcus
| 272,203
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