text:
Could tantric meditation on a guru permit boundary invasion?
Posted by: corboy ()
Date: September 02, 2018 08:45PM

in various tantric traditions (and yes, these include what is collectively termed
Tibetan Buddhism) there are meditations in which one visualizes the guru and internalizes the guru image, visualizes oneself as buddha for some other deity.

Could this carry risks of boundary puncture and identity disruption - and make a person vulnerable to grooming by a predator in guru robes?

Quote

Rosenman, S. (2003). Assaultive Projective Identification and the Plundering of the Victim's Identity... Journal of the American Academy of Psychoanalysis., 31:521-540.

Quote

Stanley Rosenman has written about how identity and even emotions are interfered with when someone else inserts thier own introject into targeted persons--victims.

Rosenman examined this only in the context of traumatic assault, such as rape or prolonged torture.

He didnt consider that this could be done by seduction, persuasion, by presenting oneself as a guru and securing peoples attention by promsing priviliged access to the ancient heritage of India.

Here on the CEI message board we are free to ponder this. Rosenman has suggested that a self impaired perpetrator may seek immortality by injected shards of his or her introject into others.

I took some rough notes on Rosenman's article.

Rosenman, S. (2003). Assaultive Projective Identification and the Plundering of the Victim's Identity... Journal of the American Academy of Psychoanalysis., 31:521-540.

(abstract of article)

Quote:
A destructive mode of projective identification is delineated: a predator's catastrophic attack calculated to cause the victim a stress disorder marked by a disarrayed identity. This discomposure enables the perpetrator to aggrandize a manifold inroad upon the victim's identity to imprint, intrude, mingle and/or lodge his representation into it; to ravage, steal from, impoverish, and/or corrupt it; finally to have his representation emerge as an internal regulator of the traumatized prey's functioning. The victim's debased integrity is manifest in the symptoms of his ensuing posttraumatic stress disorder (PTSD)."

Quote:
"The article was grim reading, but very, very informative.

Rosenman made use of testimony from people who had survived parenting from adults traumatized by war and persecution, and from persons who had survived prison atrocity, torture and other forms of prolonged, grievous assault. (This is not good stuff to read within an hour or two of bedtime--learned that the hard way.) (My note - any trauma the parent suffered would count).

The stuff that jumped out at me is that many times perpetrators present a very pathetic wounded child aspect of themselves while simultaneously intruding on and damaging vulnerable targets. ((Though Rosenman does not say so, this wounded pathetic child aspect of the perp is probably flip side of the vibrant charming child aspect of the perp when he or she is being sunny and seductive.))

The pathos literally disarms us--both before we are assaulted--and it disarms us after we are assaulted, by shaming us into distrusting our anger and revulsion at being trust busted.

This matches with how many abusive gurus (like my doctor, warped's doctor\bf, all our abusers) go into pathetic grand opera mode when feeling misunderstood--the pathos. It brings their defenders out of the woodwork (their relatives, friends, followers). Even those grievously victimized by Cohen still feel afraid to be disloyal and this has greatly muzzled the survivor's community and prevented it from becoming an effective agent for reform. Despite knowing how cruel and dangerous Cohen (one particular "guru" we've been following in CA) had been to them and that he was continuing to do this, most of the survivors pull their punches and feel afraid to be disloyal or hurtful-even a woman who was pressured and tormented to surrender over 2 million dollars still felt this hesitation. IT all fits with what Roseman describes--the seductive pathos of the tormentor.

In using pathos,the perp sets it up so the victim gets shamed and conned into feeling parental and loyal to the wounded child in the perp--and in feeling protective of the wounded child in the perp, the victim is simultaneously conned into disowning his or her own violated innocence. In this scam, its the powerholder/perp who gets the compassion and the victim ends up shafted with disorientation, shame and self loathing.

Two, Rosenman notes that perps feel soiled by what they've been through and some get a temporary sense of renewal and invigoration by disrupting the innocent happiness of their target and witnessing the person succumb to the shock that the perp went through. Rosenman noted that survivors of atrocity may resent and envy their children at having a chance at happiness and may force the children to become privy to the parent's grim outlook on life--a kind of misery loves company dynamic. (my note - they HATE our happiness. My doctor, and the LGAT he tried to suck me into, try to persuade clients that they don't know just how miserable their lives are. Many alternative practitioners use this underlying unhappiness as the "hidden cause" for illness).

So the identity and emotional life of the perp get transferred into the targeted victim and takes over the victims inner life. IF you're a child, you dont feel free to explore life on your own terms. You feel your own energy and happiness are taking something away from your parents. Roseman noted that some perps get a sense of immortality, or at least a sense of mastery (the vampire) by taking over the inner lives of their victims and (to quote the abstract) 'have (the perp's) representation emerge as an internal regulator of the traumatized prey's functioning. '

Now...doesnt this sound exactly like an LGAT's agenda--'emerge as an internal regulator of the traumatized prey's functioning.' The set up is a socially engineered assault/disorientation that punctures boundaries and fosters injection of the LGATs' "representation" so that it then takes over and runs the targets inner life for them--and sets up a conflict between true self and the foreign and predatory material injected by the LGAT. (my note - we both have had experience with either gurus or LGATs, but this injection applies to abusers in general - they become part of your internal workings).

But Rosenman's article describes another part of the process--the way perpetrators use thier pathos, their woundedness, to feel entitled to leech from others---and use their pathos to charm their victims into parenting them--and shame thier victims to ignore their own valid revulsion at being leeched from. When someone is pathetic, its hard to face that they are dangerous and using pathos as a cloaking device.

The tip off when perpetrators abuse others this is that:

*Only the perpetrator's childhood pathos and suffering gets respect.

*The perpetrators adult agency in continuing the legacy of abuse--is ignored, trivialied--or given a spiritual rationalization.

*The perpetrators adult agency is foisted onto the victim.

***And the shattered powerless victim (often groomed and pre-selected by the perp because he or she is vulnerable, is conned into parenting and nurturing the wounded child in the perp while the adult voice of the perp cons the victim into feeling responsible for the perps actions. The victim feels powerlesss and craves a sense of agency, so may accept the perpetrators definition of agency---"You made it happen. You made your own abuse." Its only abuse because you CALL it abuse.
And a false sense of agency in feeling parental toward the perpetrator...to the poin

***The victim is conned a sense of false empowerment by parenting the perpetrator.

The predation being done in in the present moment is denied. The experience of the victim suffering in the present moment is given no respect whatsoever."

Stamp out rape. Say yes.

As I wrote elsewhere:

Quote:
In New Age and Dharma Lite circles, there is a phobia against honest, frank speech.
And it may be that people who allow themselves to be emotionally spayed or neutered may at an unconscious level get off on following a leader or guru with anger management problems, because the leader gets to indulge the anger and other behavior forbidden to the followers.

One also sees this where a guru or leader milks his or her sufferings for pathos, and monopolizes compassion, while the followers are forbidden to see their own pain as equally valid.

The followers care for the guru while failing to give a damn about their own well being and fail to give a damn for the well being of each other.

All compassion is monopolized by the leader, while denied to the followers.

The leader becomes an insatiable black hole, absorbing all the emotional energy and vitality of the group. The leader is allowed to be angry. The leader is allowed to be cared for. The leader's sufferings (even when its just something that can be remedied by surgery) is exaggerated as drastic.

The followers are forbidden anger, forbidden care when suffering, told to 'turn it around' when they feel perturbed, forbidden to defend themselves or see thier own perspective as valid.

Only the leader's perspective is valid. Only the leader's experiences and emotions are real.

But from Janaki's account, the leader never has to turn it around.

A guru who is greedy and nasty can become a emtional surrogate of sorts, carrying and acting out impulses and emotions forbidden to the followers.

In Janaki's case, she mentions being in some Indian gurus ashram at age 14, then she follows Muktananda, then gets involved with BK.

That is a very very long time to be in guru-land. That very likely meant she was socialized to disown emotional boundaries early on and then rationalize this disowning of valid anger and emotional boundaries under the guise of spiritual rationalization.

The poison of an LGAT or toxic guru is that you think you are fully autonomous when in fact the toxic introject of the forum leader has been downloaded into a subconscious part of yourself and is working you as a puppet so as to defend the LGAT, disrupt dissent by trolling, or by earning money and running PR.

Stanely Rosenman--has spent a lot of time counseling and writing about the issues faced by survivors of the Nazi holocaust but also the impact the parental trauma has had upon children and even grandchildren of those who survived.

Dr Rosenman has not 'lost the moon'

[74.125.93.132]

Re: Could tantric meditation on a guru permit boundary invasion?
Date: January 29, 2021 04:54PM

Hello, As i was seeking for my own page -abuse by a "lama" (who) named (himself) "Ven Lama Dondrup Dorje"- i saw this question topic, and i want to respond to this question directly.

This is ABSOLUTELY TRUE!

And here what is stated here:
"Assaultive Projective Identification and the Plundering of the Victim's Identity"

EXACTLY this! I am completely experiencing this myself.
It is COMPLETE PARASITIC
I mean, maybe not by itself, but it can be USED this way!
The lama can ABUSE his "authority" -which he "gained" BY YOUR AGREEMENT- to molest the mind/Identity of the one targeted.

My "lama" got very angry with me and has been assaulting me ever since.
I totally feel how he attacks on "my spots" (some of his monks assist him in this character-assasination) which gives them TOTAL EXTACY as "they are the ones doing it" - assaulting my painspots which give THEM pleasure, and supposed to take away my suffering. Which is not really actually...

It is just a disgusting feeling of energy-invasions, done by jealous people who actually are jealous (this lama himself) of my strong capacity of self-healing and self-love and worth....

It has become appearant to me that this "lama" is totally and completely motivated by his own lack of self-worth and utter frustration about it...
When you would see his picture, if you look well, you can totally read this from his face: a deeply pessimistic anger and fatalistic view that he is grudgingly holding on to.

And heUSES others as he so-called is "healing them", for his own increase of self-worth.

And it is actually much worse than that. He even INITIATES his whole teaching on this basis. So from scratch, you'll be brainwashed so as to be self-hating and thus unloving to yourself, his focus is to DENY you to make you deny yourself. All teaching is fear-instilled, all based on generating fear in the mind of the other -which is a great, if not the greatest way to feast energy from others.

And AT SOME POINT will he play the heroe and COMPLETE sanctimonious saviour that will tell you to be a friend to yourself.... after months -if not YEARS- of STRAGETIC psycological manipulation.

Its all based on robbery of self-power. Which is a phrase i actually heard him mention in one of his... (STOLEN) "teachings"....

He said "By self-power"? ,,, cannot do it.
And what did i do? I DID IT. By self-power?, and BY SELF-POWER ALONE!...

This than, is than "skillfully" identified as ego.

COMPLETE BRAINWASHING!!!

I would like to suggest to make an Investigation form for this kind of abuse.
Almost by dreaming could i set up a list of questions which WILL identify this kind of abuse!

Let me know if there is an interest into this, than it can slowly evolve...
For it literally comes bit by bit. The informations just pop up from the subconscious memories.... I have loads and loads of information about this, and i actually cannot manage it properly. Because IM NOT OUT OF TRAUMA!!! Yet..

But i know this will serve MANY MANY people, and leave a lot of people disillusionalised of outer heroes....

Which pretend that you should and have to be, like sticky glue, be allways dependant on the guru (and thus allways be the little kid while you are full-grown........)

This ofcourse will be identified as Western ego and what not... Ingratitude?
And so forth... For which you will burn in hell for many eons ofcourse...
I hope they did not forgot to mention this to you (lol)

Anyway, i hope this is a bit of help and answer to the question.

I say YES!!!... This is totally my experience.
And i TOTALLY aware of how my lama" try to completely reverse, my victimhood into his power. How he completely try to reverse, my self-power into so-called being helped and aided by them, so as i have to be forever grateful and what not... ==> And THUS be ACTUALLY DIS-EMPOWERED, whjile pretending themselves to be empowering me...

Its quite a funny thing but actually not funny....
It is ONLY funny to the PERPATRATORS of this rediculous practices....

YES there is true and real properly intended practices that Buddhists can do to help others.... If they cannot help themselves.
But there are BOUNDARIES, exactly as stated in this topic
Which some just tend to completely disrespect.

And parasitically "offer themselves" for your healing, but in fact they completely robb you of identity and self-power, thus make you LESS powerful, which NEVER can be the way as it was INTENDED by the Buddha's.

IF ever they gave this practices.... Buddha is symbol for complete Integrity.
Anyone who acts in opposition to that, or not in accordance to it, is by definition a complete ripoff and shame to it.

And as you may understand already, is motivated NOT by Buddha's Ideals!
Which is total Integrity.

If you ask me, this ugly kind of people should be WHIPPED! For shaming the Buddha and perverting the TRUE teachings and intentions of what is Identified as completely pure beings of diamond (perfect-ed) Integrity.

It should not be tolerated. However it is not the way to set up policegroups to put punishment on this kind of people. All in perspective.

There SHOULD BE groups that investigate the integrity and true intention of this kind of individuals and cults...

As Buddha would say "use your own judgement"...

Greatings, and wish you quick realisation and if needed recovery. I havent read any further than the 2 phrases which i mentioned at the start of my response. For i completely understand the meaning of it.

There are MANY MANY ways of realising that which is to be realised....
And what i have actually concluded, a multiple of times already, is that i am actually my own guru...

The way i came to "higer///deeper" realisations of the Buddhist thing, -which in my case was a complete abuse of the real Buddhist teaching-, was to recognise whom i have allways been....

Which is crazy if you think of it. From realising whom i have allways been, and how i have allways conducted my life, i understood this that i... had all the answers already... And have been doing the Dharma all my life.. Naturally.

This angry jealous lama decided to destroy me. And more cunningly, to "help me" destroy myself...

It is very hard for me to understand (and accept), that this kind of individuals truely exist... but as you see they do.

Pathetic individuals truely exist. Those who CLAIM harmony but are much disharmonious with others themselves in their aggressive endeavor they seek to stress on others, which they call their students and children.... from some kind of distorted attachment.

Wishing you fast recovery.
(all im saying is from my own experience, i have no clue about whom this is, so i have no single accusation against anyone. But this is my experience, and what I have come to understand - which have MASSIVELY discouraged my inclination to Buddhism. In spite of having met true authentic pure loving teachers whic htruely give freely, and allow you to keep your own freedom. For YOUR life is meant to be YOURS, and they understand that and respect it.

Those who do not respect this are nothing like teachers. They got it all wrong.

Re: Could tantric meditation on a guru permit boundary invasion?
Posted by: facet ()
Date: January 29, 2021 10:01PM

Quote
Could this carry risks of boundary puncture and identity disruption - and make a person vulnerable to grooming by a predator in guru robes?

Big resounding Yes !

It may not be well proven in researched yet but will share from my own personal research observations, is that the internalisation is directly related to the neuropathology of the human brain.. if one lets ones self go enough identity wise and as the above poster states, this in many cases (and unless usably aparent already) is initiated by using existing hatred and disgust towards ones self , and initiating a new one (sex or certain sex act is bad, food is bad, choices bad, having children bad, thoughts bad, unclean, etc) to be repulsed by ones own nature enough that the throwing away occurs.. and the neuropathology of the leading person is then gradually mimicked in the brain.

So, brain wiring following brain wiring where our own brain wiring is not currently being decided by ourselves.

It’s not magic it’s just nature.

Controlling people manage much of what they do by these means, it is akin to parasitic nature as the above poster states.

Luckily, neuroplasticity can be changed for ourselves it would just take a lot of will on the part of the controlled to get out, or the grace that reality changes in some way to allow the space.

Brain chemistry deep into the situation will not easily allow exit.

Edited 1 time(s). Last edit at 01/29/2021 10:03PM by facet.

Re: Could tantric meditation on a guru permit boundary invasion?
Date: January 31, 2021 10:47PM

What Facet is saying makes total sense.
As this is what i have been experiencing.

I feel how i have been robbed from my own identity, sweet self-nature.
And became more like a monstrous type of person :S

ITs very strange and hard to explain, but what i am totally sure of, is that many areas of hurt within me have been assaulted. This was a revenge assault by the collective of my past teacher and his most close group.. Witchcraft i would say.

For safety purposes, it may be better not to speak of the experience and "how its done", for its not easy to understand.

One should SEE what is really happening, that would most likely give a more appropriate discription.

But what i can tell, is that they want me NOT to be peaceful, they want me NOT to be/reside in my inner peace, and SEEK to TRAPP me in opposite(s)...

For example, winning and losing (in competition), and the tension and stress that comes with it... (over a long period of time without resolving the mental activity). There IS most often a fear involved in competition.
From having sense of reality and possibility. This is where the TRICK lies for causing stress to someone who engages in competition.

This teacher i had is VERY VERY much punishing. Rediculously repetitve in it and actually secretly it seems to me that he is very jealous as well....

For he engaged in long-term character assasination and afterwards trapped me by 4 year long consistent image-manipulation to do with that and other stuff...
Why also other stuff?AT THE SAME TIME! ... Because they know i'm supposed to be " very" skillful... And want to punish me as much as they can, as long as they can. Why multiple attack at once? Because of group-effort...
The teacher was angry about my brightness (smart) and i think very jealous at previous accomplishment.

Thus far have i experienced lot of " soul-loss", AUTHENTIC SOUL-loss.
Which hadnt much to do with (what i learned from) him actually.
BUT due to some of his practices it allowed me to blossom these qualities within me..

Anyway, i'd like to mention as well that what you say about using self-hatred and self-disgust, TOTALLY CORRECT!

This guy even talked to me in a blaming way, that i had " no self-love" ....
Which was a result of his manipulation though. But i actually learned very deep self-love. This was ACUTALLY what i was practicing while i was in his Centre and he got me all depressed with his consistent targeting manipulation...

MANY MANY SPELLS been hexed upon me with his evil intended, choice and use of words.... I felt this very deeply. And im sure this gives him a little boner..
It give him a little boner everytime every day he succeeds in keeping me trapped and unrecovered...

It seems to be totally HIS MAIN GOAL IN LIFE nowadays!! Its utter rediculous.

HEy, i gtg fast now, i have an appointment.
bb

Re: Could tantric meditation on a guru permit boundary invasion?
Posted by: facet ()
Date: February 01, 2021 03:10AM

You describe the vast majority of people who are into these things.

The majority have a desperate need for control, and are high manipulators, it is just that it is often not realised until we’re in trouble.

Remember, it’s not magic, it’s just nature. They behave badly to create the circumstances and call it magical power, power over others, but it is false, abuse is not power. It’s just abuse.

Spreading false rumours about a person is one of the no.1 ways people try to puff themselves up with having some powerful magic ability to harm others, it is even hidden agenda in those who make it a big deal to talk about concept of karma. It works to keep people trapped in fear.

Our own neuropathology is always there, it is the strongest over anything, so luckily no “soul” or person is ever lost. It feels this way only due to the neuro mimicry issue mentioned before. The mimicry is not true, so it cannot be permanent.

Focusing on ones own self habits before knowing the controller and instilling those whilst creating your own new ones (healthy ones) is a way out.

Anonymous study -

A : Woman in her mid 60’s, grey hair, short stature, unfriendly in her physical approach with a special effort toward false friendliness via words and feigned use of emotion. Is quick to tell others upon meeting them that she is a “witch”. This serves her highly controlling nature where she prefers people do her will as she believes that only her way to do things is the right way. Everyone else is at a fault and imperfect. Others who do not do her bidding (not able to be controlled) are “bullies” in her mind, and become objects of hatred where the projection of her “bullies” turns into vile, wild, and untrue rumours spread in order to cause them social issues, eventual distress, and that often leading to illness. The “witch” classifies this behaviour as her “magic” power, and uses it to back up her claims with others not yet targeted, when in fact it is only the poor behaviour involved that is the catalyst.

It is later found that in reality, the lady presenting has a diagnosis of Pure OCD and a long history of antisocial behaviour, greatly struggling with forming stable abuse free relationships in her life, and greatly struggling to settle in any home as she has been frequently moved on for her own safety, as officials who do not know of her behaviour and disbelieve victims, take her claims of “bullies” and the stories that accompany the claim quite seriously.

It’s not magic :-).

Edited 3 time(s). Last edit at 02/01/2021 03:22AM by facet.

Re: Could tantric meditation on a guru permit boundary invasion?
Posted by: facet ()
Date: February 01, 2021 03:45AM

Here’s another example:

Man, in his late 50’s, split from his wife and estranged from her and their two children. Presents as friendly though has an exceptionally aggressive undertone, inability to accept and therefore process anger. Is now running a “spiritual” workshop where he presents many YouTube conspiracy theories to attendees and urges them to repeat the spiritual workshop as it is a form of protection and renders the attendee indestructible. One of the attendees is always picked out as a “nuisance” of the group, even though they are not, and attempts are made to bring the person down with belittling comments in front of all of the group, whilst not fully going there, but displaying his edge of anger toward them.

The man charges $100 for the first attendance, but because attendees are returning they then receive a very special price of$80 instead. Attendees are repeatedly told that they are maintaining their protection and imperviousness to the topic by repeat attendance.

In reality, the man is officially diagnosed with depressive disorder after his wife left, both she and the children becoming estranged due to his ongoing engagement with and undertaking of conspiracy theory beliefs which was freaking them out and taking up 99% of his time alone. The mother wishing to protect her children as well as reengage in a healthier outlook. The man had lost his job due to his lack of interest in it over the conspiracies, and now lives in his vehicle, his only income is that of the workshop he urged others to frequently attend after preying on their fears connected to the conspiracy materials that he had shown them on YouTube.

The attendee picked out as a “nuisance”
and as receiver of the edge of anger is done intentionally in order to socially control the rest of the group into complying with what he is saying, as it would be highly uncomfortable to be picked out as the “nuisance”, all others largely comply to avoid.

Again, not magic.. just piss poor behaviour and treatment of others :-)

Edited 1 time(s). Last edit at 02/01/2021 03:47AM by facet.


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# Improper integrals - Integration Techniques

### Improper integrals

First, we will learn about Type 1 improper integrals. These types of improper integrals have bounds which have positive or negative infinity. Then we will look at Type 2 improper integrals. These improper integrals happen when the function is undefined at a specific place or area within the region of integration. For these integrals, we will have to use limits. If the limit exists and is finite, then the integral can be solved. Otherwise, the integral will be unsolvable.

format: plain


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# Basic Statistical Analysis of LWR Pin Power Data

This article details the basic mathematical formulas used to statistically compare pin power data sets from a reference and an alternate LWR.

Due to the migration of our articles from MediaWiki to Markdown, the formulas are not showing up properly. Unfortunately, Markdown does not have a lot of support for mathematical characters. We will soon be migrating back to MediaWiki pages, at which point this article will be back to normal.

## Pin Power Dataset Representation

First, let's view the pin power dataset of each LWR as a 4D matrix P(i, j, k, l) where

Variable Represents
i pin row
j pin column
k axial level
l assembly number

In order to compare two of these pin power datasets, we shall define the following 4D pin power datasets:

Variable Represents
A reference power data set
B alternate power data set
C power difference data set

where

_C = B - A _ (a basic power difference)

or

_C = (B - A) / A _ (a relative power difference)

In addition, each element in C can be weighted by a 4D matrix W.

## Derived Quantities

There are four general derived quantities used in pin power analysis. The first is the simple difference which is simply the dataset C. The second is the absolute difference which is the absolute value of C or |C|. The third is the average difference is a weighted average calculated with the following formula.

$\frac{\sum(C{\cdot}W)}{{\sum}W}$

Finally, the root-mean-square difference is a weighted RMS difference determined by the formula below.

$\sqrt[2]{\frac{\sum(C^{2}{\cdot}W)}{{\sum}W}}$

The following sections detail how to calculate the above quantities over a variety of dimensions to develop statistical results from pin power differences.

## 1D Axial Power Results

The axial power difference is defined by the formula below.

$C(k) = \sqrt[2]{\frac{\sum_{ijl}(C{\cdot}W)}{{\sum_{ijl}}W}}$

The average axial power difference is defined as

$\frac{\sum_k[C(k){\cdot}dz]}{{\sum_k}dz}$

where the dz, the axial weight, is $dz = mesh(k+1) - mesh(k)$. Here mesh is a 1D array containing the actual physical distance between each axial level k.

The RMS axial power difference is defined by the formula below.

$\sqrt[2]{\frac{\sum_k[C(k)^{2}{\cdot}dz]}{{\sum_k}dz}}$

## 2D Radial Power Results

The radial power difference is defined as

$C(i,j,l) = \sqrt[2]{\frac{\sum_{k}(C{\cdot}W)}{{\sum_{k}}W}}$

The average radial power difference is defined as

$\frac{\sum_{ijl}[C(i,j,l){\cdot}dr]}{{\sum_{ijl}}dr}$

where the dr, the radial weight, is $dr(i,j,l) = \sum_kW$

The RMS radial power difference is defined by the formula below.

$\sqrt[2]{\frac{\sum_k[C(i,j,l)^{2}{\cdot}dr]}{{\sum_{ijl}}dr}}$

## 3D Assembly Power Results

The assembly power difference is defined as

$C(k,l) = \sqrt[2]{\frac{\sum_{ij}(C{\cdot}W)}{{\sum_{ij}}W}}$

The average assembly power difference is defined as

$\frac{\sum_{kl}[C(k,l){\cdot}da]}{{\sum_{kl}}da}$

where the da, the assembly weight, is $da(k,l) = \sum_{ij}W$

The RMS assembly power difference is defined by the formula below.

$\sqrt[2]{\frac{\sum_{kl}[C(k,l)^{2}{\cdot}da]}{{\sum_{kl}}da}}$

In addition to the above quantities, the form factor found at each pin for the reference dataset is defined as

$ff_{A}(i,j,k,l) = A(i,j,k,l) / A(k,l)$

and for the alternate dataset

$ff_{B}(i,j,k,l) = B(i,j,k,l) / B(k,l)$

where the form factor difference is

$ff_{C} = ff_{B} - ff_{A}$


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# Finding two languages satisfying conditions

Let $$E_{TM} = \left \{ \langle M\rangle \mid L(M) = \emptyset \right\}$$

Prove that there are two languages $$L_1, L_2$$ such that

1. $$L_1, L_2$$ are infinite.
2. $$L_1 \cup L_2 = E_{TM}$$
3. $$L_1 \cap L_2 = \emptyset$$
4. $$L_1$$ is decidable, $$L_2$$ is not recognizable.

I'm finding it really hard to find two languages that satisfy these conditions.

Especially the second condition, which two language unify to $$E_{TM}$$?

• Hint 1, can you construct one $M$ such that $L(M)=\emptyset$? Hint 2, can you construct infinitely many such $M$? Hint 3, let $L_2=E_{TM}\setminus L_1$. – John L. Mar 27 '19 at 20:04
• Thank you, I'll try again! – Alan Mar 28 '19 at 8:30
• If you has figured out this problem, please write an answer (yes, you can answer your own question). – xskxzr Mar 28 '19 at 8:53
• Certainly, I will – Alan Mar 28 '19 at 8:55
• The padding lemma states that there is a total computable function that given any (encoding of) TM $M$, it returns a TM $N$ having a "longer" encoding than the one of $M$, where $L(M)=L(N)$. Essentially, the function adds a few redundant states to $M$ to make its encoding "longer". You could try to apply this function many times, iterating it. – chi Mar 28 '19 at 12:30

A slight improvement:

$$L_2 = E_{TM} \backslash L_1$$ (since we require that $$L_1 \cap L_2 = \emptyset$$).

$$L_1$$ must be a language of turing machines, i.e. $$L_1 = \{\langle M\rangle \mid \dots\}$$, so that $$L_1 \cup L_2 = E_{TM}$$.

So let's set $$L_1 =$$ $$\{\langle M\rangle\mid C_0\text{ is a reject state}\}$$.

The following holds:

• $$L_1$$ is infinite and decidable (easy to decide given $$M$$, what the configuration of the first state is).

• $$L_2$$ is infinite and unrecognizable.

• thank you for your answer! – Alan Apr 3 '19 at 18:32


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# Tau

Not to be a party-pooper on π day, but the excellent Tau Manifesto seems to have gotten an update in the time since it was first published, now with rebuttals to common arguments against the π-loyalists.

My favorite new section, towards the end:

We come now to the final objection. I know, I know, "π in the sky" is so very clever. And yet, τ itself is pregnant with possibilities. τism tells us: it is not τ that is a piece of π, but π that is a piece of τ—-one-half τ, to be exact. The identity e=1 says: "Be 1 with the τ." And though the observation that "A rotation by one turn is 1" may sound like a τ-tology, it is the true nature of the τ. As we contemplate this nature to seek the way of the τ, we must remember that τism is based on reason, not on faith: τists are never πous.

Remember, τ day is June 28.


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# What is the physical origin of acoustic modes in a duct?

Lets deal with wave propagation in a cylindrical duct.

We ask the question: "what is the general form of a pressure wave which can propagate through the duct?" In answering this, we assume that the pressure wave in question has harmonic time dependence, i.e. $\exp(i\omega t)$ dependence.

The critical thing about solutions of the wave equation is that they must obey the boundary conditions of the duct. Without going into detail, for an infinite length duct of radius $a$ the solution is:

$p(r,\theta,x,t) = \exp(i(\omega t-n\theta) J_n(z_{mn}r/a) \left( A_{mn}\exp(-ik_{mn}x) + B_{mn}\exp(ik_{mn}x) \right)$

Where $J$ is the bessel function with corresponding zeros $z_{mn}$. Mathematically the significance of $m,n$ is simply that we can assign any positive integer value to these as a feasible solution. This determines the mode mathematically. As an aside it should be pointed out that for a particular value of $\omega = \omega_c$ only certain modes with low enough $\max(m.n)$ can actually propagate- any "higher order" modes will decay rapidly along $x$. $\omega_c$ is therefore known as the cut-off frequency.

My question now is how are modes created in the first place by an acoustic source? Does the source select a particular mode or are many present at the same time? Please explain.

To start with an answer I thought of a vibrating disc or membrane in the duct which vibrates with velocity $v = \hat{V}\exp(i\omega t)$ in the $x$ (axial) direction. Assuming linearity, the pressure wave form should have same harmonic time dependence. Now it is important for the sake of this argument that $v$ is exactly sinusoidal, i.e. I am not assuming there would be any higher order coefficients in the Fourier decomposition of $v$. So what mode or modes would initially (even if they are cut-off) be set up by this scenario?


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## How is multiplying by a fraction the same as dividing by the inverse of that fraction

Supposedly this proof answers my question.

8 / 16 = .5
8 / 8 = 1
8 / 4 = 2
8 / 2 = 4
8 / 1 = 8
8 / .5 = 16
8 * 2/1 = 8 / .5

I'm just wondering how this proof answers my question?
 PhysOrg.com mathematics news on PhysOrg.com >> Pendulum swings back on 350-year-old mathematical mystery>> Bayesian statistics theorem holds its own - but use with caution>> Math technique de-clutters cancer-cell data, revealing tumor evolution, treatment leads
 It doesn't ... and that's not a proof o__O It follows kind of from "definition" of division: $x \over y$ actually means $x \cdot y^{-1}$ where $y^{-1}$ is the number such that $y \cdot y^{-1} = 1$, called the "multiplicative inverse" or "reciprocal" of y. This number is unique. Obviously $({a \over b})^{-1} = {b \over a}$ (since ${a \over b} \cdot {b \over a} = 1$) as long as we have $a, b \ne 0$. (Also of note: $0^{-1}$ does not exist! So it turns out that since ${x \over y} = x \cdot y^{-1}$, setting $y = {a \over b}$ we get "${{x} \over {a \over b}}$" $= {x} \cdot ({a \over b})^{-1} = {x} \cdot {b \over a}$ (again provided $a, b \ne 0$). I hope this explanation helps.
 ooooooooooooooooooooooooooh awesome! :D Very much appreciated!

 Similar discussions for: How is multiplying by a fraction the same as dividing by the inverse of that fraction Thread Forum Replies Calculus & Beyond Homework 22 Precalculus Mathematics Homework 3 General Math 6 Precalculus Mathematics Homework 4 Precalculus Mathematics Homework 6


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Armstrong's axioms more precisely 'inference rules': what's the distinction?

"more precisely 'inference rules'" is what wikipedia says twice.

Other references I've looked at (the Alice book, Ullman lecture notes cited, Maier textbook) name them as Armstrong's Axioms, use them for inference. talk about axiomatisation and decidability, etc.

Wikipedia's links to axiom vs inference rule isn't helping me. "a rule of inference's action is purely syntactic, and does not need to preserve any semantic property:" Ok I can see that for (for example) modus ponens: it wants a premise like $p \supset q$ and a premise looking like $p$.

Hmm take the Armstrong Axiom of reflexivity

If $X$ is a set of attributes and $Y$ is a subset of $X$, then $X$ holds $Y$.

If $Y \subseteq X$ then $X \rightarrow Y$.

How to determine syntactically that $Y \subseteq X$?

So what is wikipedia trying to say? What distinguishes an axiom (which might be of the form $p \supset q$) vs an inference rule?

• – D.W. Mar 27 '18 at 15:21
• Please state the question in the title. – reinierpost Mar 27 '18 at 22:24

First, terminologically, "axiom" and "inference rule" are often used as roughly interchangeable as they tend to serve similar purposes. There are technical distinctions, which themselves can vary slightly, but outside the study of formal logic or related systems, these distinctions aren't that important.

In the context of formal logic, an axiom is a formula of the logic that is a theorem by definition. A rule of inference is a way of deducing new theorems from old theorems. Rules are always part of the logic, while axioms are separated into logical axioms, which are viewed as part of the logic, and non-logical axioms, which are viewed as part of the particular theory you are studying within the logic. Many logical axioms can be presented as rules and vice versa. For example, the axiom $P\supset(Q\supset P)$ roughly corresponds to the rule of weakening $\cfrac{\Gamma\vdash P}{\Gamma,Q\vdash P}$.

From a meta-logical perspective, rules of inference can be viewed as (definitional) axioms about the logic being studied (not to be confused with the "formal" axioms within the logic. Philosophically, "believing" in the conclusions of a logical theory requires "believing" the rules of inference just as much as "believing" the axioms.

So are Armstrong's "Axioms" axioms or rules of inference? It depends on how you formalize them. The "axioms" strongly suggest a deductive system, i.e. a system of rules, and that deductive system is quite useful in actually calculating functional dependencies. But, we could just as well take $\to$ and $\subseteq$ as relation symbols in a first-order theory and then the "axioms" really would be non-logical axioms of that theory. This could be viewed as meta-logical axioms about the deductive system, but it can just as reasonably be viewed as axiomatizing what "functional dependency" means without any deductive system in mind.

As for how to determine if $Y\subseteq X$ if we do indeed take a deductive system view of Armstrong's Axioms, in this context $X$ and $Y$ are meta-variables that stand for sets of attributes. They would be instantiated with expressions like $\{\mathsf{name},\mathsf{address}\}$. So for the rule you referenced, you'd simply be able to calculate that $\{\mathsf{name}\}\subseteq\{\mathsf{name},\mathsf{address}\}$ and $\{\mathsf{age}\}\not\subseteq\{\mathsf{name},\mathsf{address}\}$. An algorithm to calculate this could be formalized as a collection of inference rules itself, but there is no reason to do this. We can just assume that it can, in fact, be calculated. Obviously it can be in the intended use-case where the "sets" are explicitly presented, finite sets of elements with decidable equality.

Even for modus ponens, we have $P$ and $Q\supset R$ and we can apply modus ponens if $P=Q$, but checking the (syntactic) equality of two formulas while not hard is not completely trivial. Indeed, weakening is often presented with a rule like $\cfrac{\Gamma\vdash P\qquad \Gamma\subseteq\Delta}{\Delta\vdash P}$ and it is just understood that you can, in fact, calculate whether one set of formulas is a subset of another. It may also be presented as $\cfrac{\Gamma\vdash P}{\Gamma\cup\Delta\vdash P}$ where, again, it is just assumed that you have some way of calculating $\Gamma\cup\Delta$.

• Thank you @Derek, that makes perfect sense. So the reason for calling them 'inference rules' might be that X, Y are strictly speaking meta-variables. – AntC Mar 28 '18 at 1:19
• This explains neither what an axiom is nor what a rule of inference is so it's not clear why you think it is going to clarify things. – philipxy Mar 28 '18 at 13:25
• @philipxy I explicitly disclaim doing so in the first paragraph. Whether you call Armstrong's Axioms "axioms" or "rules" doesn't really matter as long as you understand what they are saying and how to calculate with them. In the context of database theory, the distinction between "axioms" and "rules" is rarely, if ever, going to matter. Nevertheless, you are right, it is a major focus of the question that I just brushed aside, so I will add some more text about that. – Derek Elkins left SE Mar 28 '18 at 18:43
• Thank you again for the additions. I came at this from considering the FOL corresponding to A's axioms given some relation predicate P(a, b, c, d) and a FOL expression for the FD {a, b} → {c, d}. Then A's axioms are just logical consequences of the FD. Interpreting the meta-variables into FOL got me stumped. – AntC Mar 28 '18 at 23:29
• @AntC A's axioms are consequences of FDs holding, but only trivially, because they are true regardless of any FD holding or not holding. Because each of A's axioms states that certain things follow from FDs that are things that follow from them. You don't need to assume A's axioms, because you can prove A's axioms from nothing (like A did) & because assuming them plus given FDs doesn't entail any more than what the FDs already entail. But A's axioms have been chosen so that certain repeated plugging in of given & derived FDs gives all & only the derivable FDs that hold when the given ones do. – philipxy Mar 29 '18 at 3:58


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The parallelogram identity on groups and deformations of the trivial character in SL_2(C). (arXiv:1805.06293v1 [math.GR])

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth point if dim H_1(G,C)>2.查看全文

Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth point if dim H_1(G,C)>2.


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# This equation has multiple correct options;If AB = A and BA = B, then1. ${A^2}B = {A^2}$ 2. ${B^2}A = {B^2}$ 3. $ABA = A$ 4. $BAB = B$

Last updated date: 23rd Mar 2023
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Hint: We are going to solve every option from the question and apply the required conditions and operation.

Given, ${AB} = {A}$
Multiply by A
$\Rightarrow A \cdot AB = A \cdot A \Rightarrow {A^2}B = {A^2}$
So (1) option is correct

Given, ${BA} = {B}$
Multiply by B
$\Rightarrow B \cdot BA = B \cdot B \Rightarrow {B^2}A = {B^2}$
So (2) option is also correct

Given, BA = B
Multiply by A
$ABA = AB$
We know that AB = A
$\Rightarrow ABA = A$
So (3) option is correct

Take AB = A
Multiply by B
$\Rightarrow BAB = BA$
We know that BA = B
$\Rightarrow BAB = B$
So, (4) option is correct
Hence all options are correct.
Note: Such problems you have to use the conditions given and try to apply some operations in given conditions. So you can easily get an answer.


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# How to determine if an IIR filter is causal or non-causal?

It may sound impractical, as only causal system are passively realizable, but what I am facing is a general statement made regarding recursive digital filters: for them to be stable, the poles should be inside the unit circle. I am not able to see that how without knowing type of system that the system is causal or non causal one can comment on location of poles for system to be stable.

I also know that for system to be stable its output should only be dependent on values of input at or before observation time for instant below is causal

• I'm assuming in your third equation you meant to write $y[n+K]$ instead of $y[n-K]$? – Florian Jul 1 at 15:21
• "... only causal system are passively realizable ..." I wouldn't know how to realize non-causal systems with active electronics either. – Matt L. Jul 1 at 15:45
• @MattL. Aren't you assuming that $n$ is temporal rather than spatial? – Rodrigo de Azevedo Jul 6 at 17:36
• @RodrigodeAzevedo: Sure I am, causality is always about time. – Matt L. Jul 6 at 18:21

First of all, it's not correct to say "poles should (always) be inside the unit circle for an LTI system to be stable" ; unless it's implied that system is also causal. Otherwise, if the system is noncausal, then its poles should be outside of unit circle for the system being stable.

For IIR systems that are described by LCCDEs causality must be externally imposed on the system (consistent wit the initial conditions), and cannot be derived from the equation alone. For example, the system

$$y[n] + a y[n-1] = b x[n]$$ cannot be concluded to be causal, as the equation can also be written as

$$y[n] = (-1/a) y[n+1] -(b/a) x[n+1]$$ which is (apparently) indicating a noncausal dependence on the current output on both future input and future output.

Therefore this difference equation can signify both a causal and a noncausal system. Its solution should be derived by assuming a causal system or noncausal system.

This is also understood by the fact that a given transfer function $$H(z)$$ will have a corresponding LCCDE representation, but will have multiple ROCs. For each ROC the LCCDE should be solved accordingly, yielding a different solution for each assumption. But the LCCDE is the same.

FIR systems do not have poles (except at origin or infinity) and have no issues with stability and they don't posses regions of convergences on the z-plane. Hence their equation will be signifying the causality such as: $$y[n] = a x[n+1] + b x[n] + c x[n-1]$$ is a noncausal FIR system and you cannot manipulate the given equation into a causal form.

Or the following system $$y[n] = a x[n] + b x[n-1] + c x[n-2]$$ is a causal FIR system and you cannot manipulate the given equation into a noncausal form...

So causality shall be an assumed property for IIR systems.

Substitute $$m = n+K$$, i.e., $$n=m-K$$. This gives $$y[m-K] = -a_1 y[m-K+1] - \ldots - a_{K-1} y[m-1] - a_K y[m] + b_0 x[m-K] + b_1 x[m-K-1] + \ldots+ b_L x[m-K-L].$$ Rearranging gives $$y[m] = -\frac{a_{K-1}}{a_K} y[m-1] - \frac{a_{K-2}}{a_K} y[m-2]-\ldots-\frac{a_1}{a_K}y[m-K+1] - \frac{1}{a_K}y[m-K] + \frac{b_0}{a_K}x[m-K] + \ldots + \frac{b_L}{a_K}x[m-K-L].$$

That's a causal filter to me.

From the difference equation alone you cannot tell if a system is causal or not. For example, the difference equation

$$y[n]=-a_1y[n-1]-a_2y[n-2]+b_0x[n]\tag{1}$$

can be used to describe three different systems. The first one is a causal system, as suggested by the form given in $$(1)$$.

However, rewriting $$(1)$$ as

$$y[n-2]=-\frac{a_1}{a_2}y[n-1]-\frac{1}{a_2}y[n]+\frac{b_0}{a_2}x[n]\tag{2}$$

suggests an anti-causal system, where all output values depend on future values of the input and output.

Yet a third way of rewriting $$(1)$$ suggests a non-causal (two-sided) system, where the current output value depends on past and on future values of the input and output:

$$y[n-1]=-\frac{a_2}{a_1}y[n-2]-\frac{1}{a_1}y[n]+\frac{b_0}{a_1}x[n]\tag{3}$$

It's the initial conditions that can define such a system to be causal. In the $$\mathcal{Z}$$-transform domain, the three different system interpretations $$(1)-(3)$$ correspond to three different regions of convergence.


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## Introduction

The prospect of generating topologically protected entangled states of several photons is a highly intriguing proposition1,2,3. Specifically, topological protection can enable robust transport of quantum information across disordered photonic structures without degradation4,5, just as efficiently as for single-particle wavepackets6,7,8,9,10.

In recent years, we have witnessed several experimental demonstrations of topological protection at the single-photon level in integrated one-dimensional lattice systems. Notably, Wang and co-workers11 showed that the fundamental quantum features of spatially entangled biphoton-states can be protected against disorder in the so-called Su-Schrieffer-Heeger (SSH) topological lattice. Interestingly, SSH lattices turned out to be equally effective in protecting polarization-entangled photon pairs12. Another important ingredient was provided by Tambasco et al.13 showing that Hong-Ou-Mandel two-photon interference of topological edge–modes is feasible, by implementing a topological beamsplitter in a judiciously engineered time-dependent Harper-model.

Concurrently, on the theory front several ideas have been suggested to investigate topological two-photon effects in linear14,15 and nonlinear16 lattice systems. In this regard, an intriguing proposition was recently put forward17, where the Bose-Hubbard model, which is topologically trivial for single particles, becomes topologically nontrivial for two interacting photons. That is, particle interactions have a dramatic impact on topological properties, not only modifying the topology of the spectra but also creating a topological order in otherwise topologically trivial systems.

In order to maximize the potential of topological photonic networks for transferring quantum information, it is indispensable to have a considerable number of edge modes at our disposal. One possibility is to use two-dimensional topological systems, which intrinsically support a multitude of topological edge-states18,19,20.

In two-dimensional photonic topologial insulators, single-particle edge-states reside in the gap existing between the energy bands supporting the bulk states21,22,23. Thus, breaking the topological protection requires disorder with sufficient strength to close the bandgap. For states describing two indistinguishable photons, the same bandgap is fundamentally lacking. The reason is because the propagation eigenvalues $${\lambda }_{12}^{(2)}$$ for two-photon eigenstates in a photonic system are given by the sum of the eigenvalues λ1, λ2 corresponding to the constituent individual photons, $${\lambda }_{12}^{(2)}={\lambda }_{1}+{\lambda }_{2}$$. This implies that we can keep $${\lambda }_{12}^{(2)}$$ constant while increasing λ1 and simultaneously decreasing λ2, or vice versa. In this way, we can combine two single-photon bulk states, one from the lower and one from the upper band, to create a biphoton bulk–bulk state whose energy lies inside the single-particle bandgap. This fundamental additive property of the single-particle eigenvalues removes the bandgap and leads to massive degeneracies of the edge–edge, edge–bulk, and bulk–bulk two-photon states. Hence, considering the lack of the topological bandgap for two-photon systems, it is not clear whether topological protection will be automatically granted to two-particle states provided single particles are topologically protected in the same system.

In solids, the degeneracies described above lead to the decay of two-electron edge states when electron–electron correlations are substantial24,25,26. This decay mechanism is reminiscent of auto-ionization, where electron–electron correlations lead to energy exchange between the two particles, coupling two bound electronic excitations to an energy-degenerate bound-continuum two-electron state27,28.

Still, photonic systems are fundamentally different from solids, as the two photons do not readily interact with each other29. Consequently, the evolution operator for two-photon states, U(2)(z), breaks down into the product of two propagators for individual single-photon states, U(2)(z) = U(z) U(z)30. Thus, a natural question to ask is whether such a factorization and the absence of bangap will prevent decoherence and dissipation of non-factorizable two-photon edge-states into the bulk?

In this work, we analyze possible mechanisms of dissipation of two-photon edge states into the bulk of two different topological insulator system, the Haldane lattice model and an aperiodic lattice corresponding to the quantum Hall effect. Our results show that the key to topological protection is to minimize the disorder-induced overlap of the initial two-photon (joint) spectrum with the edge–bulk and bulk–bulk spectral regions.

## Results

### Theoretical approach

In lattice systems, static disorder can be introduced in either the site energies—termed diagonal disorder31—or in the coupling coefficients—so-called off-diagonal disorder32. In either case, static disorder is represented by a single-particle operator $${\hat{V}}^{(1)}$$. Since such perturbation is time-independent, energy conserving resonant coupling into the bulk is absent within first-order perturbation theory—the single-particle transition induced by $${\hat{V}}^{(1)}$$ does not preserve energy. The process that can resonantly couple a two-photon edge–edge state to a bulk–bulk, or to a bulk–edge, state would require a correlated change of states for both photons and it might arise within the second-order corrections in $${\hat{V}}^{(1)}$$.

To see this, we examine the second-order transition matrix elements between an initial two-photon edge–edge state $$\left|{\rm{i}}\right\rangle =\left|{n}_{{\rm{i}}},{m}_{{\rm{i}}}\right\rangle$$ and a final edge–bulk, or bulk–bulk, state $$\left|{\rm{f}}\right\rangle =\left|{n}_{{\rm{f}}},{m}_{{\rm{f}}}\right\rangle$$

$${V}_{{\rm{i}},{\rm{f}}}^{(2)}=\mathop{\sum} \limits_{j^{\prime} }\frac{{V}_{{\rm{f}}j^{\prime} }^{(1)}{V}_{j^{\prime} {\rm{i}}}^{(1)}}{{\lambda }_{j^{\prime} }^{(2)}-{\lambda }_{{\rm{i}}}^{(2)}},$$
(1)

where $$\left|j^{\prime} \right\rangle =\left|n^{\prime} ,m^{\prime} \right\rangle$$ labels intermediate virtual states and $${\lambda }_{j^{\prime} }^{(2)}={\lambda }_{n^{\prime} }+{\lambda }_{m^{\prime} }$$. The single-particle nature of $${\hat{V}}^{(1)}$$ ensures that only two terms corresponding to the two possible time-orderings of the two single-particle transitions are left in the sum

$${V}_{{\rm{i}},{\rm{f}}}^{(2)}={V}_{{n}_{{\rm{f}}},{n}_{{\rm{i}}}}^{(1)}{V}_{{m}_{{\rm{f}}},{m}_{{\rm{i}}}}^{(1)}\left[\frac{1}{{\lambda }_{{n}_{{\rm{f}}}}-{\lambda }_{{n}_{{\rm{i}}}}}+\frac{1}{{\lambda }_{{m}_{{\rm{f}}}}-{\lambda }_{{m}_{{\rm{i}}}}}\right].$$
(2)

The stationary nature of the disorder dictates that real transitions from the edge–edge states can occur only if the initial eigenvalue $${\lambda }_{{\rm{i}}}^{(2)}={\lambda }_{{n}_{{\rm{i}}}}+{\lambda }_{{m}_{{\rm{i}}}}$$ is equal to the final one $${\lambda }_{{\rm{f}}}^{(2)}={\lambda }_{{n}_{{\rm{f}}}}+{\lambda }_{{m}_{{\rm{f}}}}$$. Therefore, $${\lambda }_{{n}_{{\rm{f}}}}-{\lambda }_{{n}_{{\rm{i}}}}=-({\lambda }_{{m}_{{\rm{f}}}}-{\lambda }_{{m}_{{\rm{i}}}})$$, and the two terms in $${V}_{{\rm{i}},{\rm{f}}}^{(2)}$$ exactly cancel each other, $${V}_{{\rm{i}},{\rm{f}}}^{(2)}=0$$. That is, two-particle dissipation from each product state is zero, and the same is true for any entangled two-particle state ψ(2) represented as a quantum superposition of two possible distinguishable configurations $${\psi }_{{\rm{i}}}^{(2)}$$, e.g., $${\psi }^{(2)}={\psi }_{1}^{(2)}+{\psi }_{2}^{(2)}$$. Physically, this destructive interference is a direct outcome of the indistinguishability of the photons, which ensures that the two-particle eigenvalue is a sum of the two single-particle eigenvalues, and that the two-particle propagator is a product of the single-particle counterparts.

In contrast to dissipation, the situation with dephasing can be different: while each constituent state in an entangled superposition can be protected against disorder1, the overall superposition is, in general, not. To be precise, motion through different disordered regions may lead to disorder-induced random phase shifts between the states destroying the entanglement. To avoid this fate, all states in the superposition must travel across the same spatial region of the photonic structure, such that they are affected by disorder in the exact same manner33. These effects have been explored for spatial path-entangled states1, and for states built from an entangled superposition of an initial non-stationary state $${\psi }_{1}^{(2)}$$ with its time-delayed replica $${\psi }_{1}^{(2)}(\tau )$$, that is, entangled states in the time-domain2. In these two cases, however, the entanglement of the states can be related to the entanglement of symmetrized wavefunction of identical particles34. Consequently, the states exhibit the lowest possible amount of entanglement, as indicated by the corresponding Schmidt numbers SN = 2. Throughout this work we use the Schmidt number to quantify the amount of entanglement: SN = 1 denotes complete separability while SN 1 corresponds to high entanglement35.

A more appealing type of highly entangled two-photon states are multimode optical Gaussian states in which both photons are most likely to be found inhabiting any waveguide, within an excitation window, simultaneously36. The importance of such states is based on the fact that any phase difference arising among the paths becomes enhanced by a factor of two in comparison with single photon states37. Naturally, the enhanced phase sensitivity of such highly entangled two-photon states manifests as faster diffraction of the associated wavepackets propagating in any photonic system, periodic and disordered38. Therefore, it is not clear to what extent topological protection will persist for these types of highly entangled states.

### Propagation of entangled two-photon states in disordered topological lattices

In what follows we analyze the impact of disorder onto a continuum of two-photon states that extend from the correlated to the anti-correlated limits, passing through a completely separable state. For our analysis we consider two topological lattices, one periodic and one aperiodic. In the periodic case we consider the Haldane model39, and for the aperiodic we use a square lattice whose single-particle dynamics corresponds to the quantum Hall effect6,40 (information, S. Supporting material). The results for the Haldane model are presented here, while the quantum Hall effect lattice is discussed in the Supplementary Note 5.

In optics, a first-order approximation of the Haldane model can be implemented using a honeycomb lattice composed of helical waveguides as illustrated in Fig. 1a, see pioneering work41. In this system, every waveguide has a nearest-neighbor coupling κ1 and a complex second-nearest-neighbor coupling κ2 or $${\kappa }_{2}^{* }$$, see Fig. 1b. At the single-photon level, the Haldane lattice is governed by the Hamiltonian1

$$\hat{H}=\mathop{\sum} _{i}{\beta }_{i}{\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}+{\kappa }_{1}\mathop{\sum} _{\langle i,j\rangle }\left({\hat{a}}_{i}^{\dagger }{\hat{a}}_{j}+{\hat{a}}_{j}^{\dagger }{\hat{a}}_{i}\right)+i{\kappa }_{2}\mathop{\sum} _{\langle \langle i,j\rangle \rangle }\left({\hat{a}}_{i}^{\dagger }{\hat{a}}_{j}-{\hat{a}}_{j}^{\dagger }{\hat{a}}_{i}\right),$$
(3)

where βi represents the propagation constant of the i-th waveguide and the corresponding optical mode is represented by the creation (annihilation) operator, $${\hat{a}}_{i}^{\dagger }$$ ($${\hat{a}}_{i}$$). Notice, in a disorder-free lattice βi = β. The symbols 〈〉 and 〈〈〉〉 indicate summation over nearest and next-nearest-neighbor sites, respectively. The lattice used in our simulations is a ribbon with Ny = 90 hexagons in the y-direction and Nx = 10 hexagons in x-direction, Fig. 1c. We normalized the units in terms of κ1 throughout this work, and set κ2 = iκ1/5.

For pure states of two indistinguishable noninteracting particles the Hamiltonian is H2 = HI + IH, where H is the single-particle Hamiltonian and I is the identity operator42. The two-photon eigenstates are given by the symmetric tensor-product combinations of the single-photon eigenstates

$$|{\phi }_{m,n}^{(2)}\rangle =\left\{\begin{array}{l}\left|{\phi }_{m}\right\rangle \otimes \left|{\phi }_{n}\right\rangle \iff m=n,\hfill\\ \frac{1}{\sqrt{2}}\left(\left|{\phi }_{m}\right\rangle \otimes \left|{\phi }_{n}\right\rangle +\left|{\phi }_{n}\right\rangle \otimes \left|{\phi }_{m}\right\rangle \right)\iff m\,\ne\, n.\end{array}\right.$$
(4)

As alluded to above, the two-photon eigenvalues are the sums of the single-photon ones, $${\lambda }_{m,n}^{(2)}={\lambda }_{m}+{\lambda }_{n}$$.

In the absence of disorder, the eigenvalue spectrum for single-photon states in a finite lattice exhibits topological edge states in the bandgap43, Fig. 1d. In contrast, for two indistinguishable photons, the spectrum does not have a bandgap: the edge–edge states can have the same eigenvalues $${\lambda }_{n,m}^{(2)}={\lambda }_{n}+{\lambda }_{m}$$ as those lying in the bulk–bulk region, Fig. 1e.

To include disorder, we separate the lattice shown in Fig. 1c into three regions1. The left and right parts of the system are disorder-free, while its middle part exhibits diagonal disorder31, that is, random modifications of the on-site refractive index taken from a normal distribution with zero mean and variance σ = 1. Importantly, taking σ = 1 ensures that the disorder strength does not destroy the topological protection for single photons, since σ = 1 corresponds to half the size of the topological bandgap. The two-photon eigenspectrum in the presence of disorder is shown in Fig. 1f.

We now send trial two-photon wavepackets into the system. They are built from single-photon edge states and vary continuously from an unentangled product state, with Schmidt number SN = 1, to highly entangled two-photon states, SN 144,45, with the two photons either correlated or anti-correlated in space46.

To construct these states, we begin with protected single-photon states as a template, $$|{\tilde{\psi }}_{\sigma }^{(1)}\rangle =\mathop{\sum }\nolimits_{j = 1}^{{M}_{{\rm{e}}}}{(-1)}^{j}{e}^{-\frac{{\left({x}_{0}-j\right)}^{2}}{2{\sigma }^{2}}}\left|j\right\rangle$$, where $$\left|j\right\rangle$$ describes a photon initialized in waveguide j, Me = 20 is the selected range of waveguides in the upper-left edge of our system, and x0 = (Me + 1)/2 = 10.5 is the center of this range. These single-photon wavepackets travel through both clean and disordered lattice without scattering to the bulk or back scattering. Importantly, the alternating sign (−1)j in the amplitude ensures that the wavepacket has proper momentum and resides in the single-photon edge subspace.

Next, we construct our trial two-photon states as follows

$$|{\tilde{\psi }}_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle =\mathop{\sum }\limits_{j,k=1}^{{M}_{{\rm{e}}}}{\psi }_{j,k}\left|j,k\right\rangle =\mathop{\sum }\limits_{j,k=1}^{{M}_{{\rm{e}}}}{(-1)}^{j+k}{e}^{-\frac{{(j-k)}^{2}}{4{\sigma }_{{\rm{a}}}^{2}}-\frac{{\left({x}_{0}-(j+k)/2\right)}^{2}}{{\sigma }_{{\rm{c}}}^{2}}}\left|j,k\right\rangle .$$
(5)

Here, $$\left|j,k\right\rangle$$ represents the state where a photon starts at waveguide j and its twin at k. The spatial two-photon correlations are controlled by the parameters σc and σa. For σcσa we have a spatially correlated state, in which both photons most probably enter into the same waveguide simultaneously38. For σaσc we obtain a spatially anti-correlated state, in which the two photons enter at two waveguides symmetrically lying on opposite sides of the window covered by the wavefunction46.

Finally, we must ensure that the initial wavepackets only include edge states. To this end, we project our state onto the two-photon eigenstates $$|{\phi }_{m,n}^{(2)}\rangle$$ of the system and then remove the components belonging to the subspaces $${\mathcal{B}}\otimes {\mathcal{E}}$$ and $${\mathcal{B}}\otimes {\mathcal{B}}$$, keeping only states that belong to the edge–edge subspace

$$|{\psi }_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle =\frac{1}{A}\mathop{\sum }\limits_{m,n}^{{\mathcal{E}}\otimes {\mathcal{E}}}\mathop{\sum }\limits_{j,k=1}^{{M}_{{\rm{e}}}}{\psi }_{j,k}\langle {\phi }_{m,n}^{(2)}| j,k\rangle |{\phi }_{m,n}^{(2)}\rangle ,$$
(6)

where A is the normalization constant. It is worth noting that two-photon states described by Eq. (6) are a lattice adoption of Gaussian two-mode squeezed states47, which are a commonplace choice in quantum optical experiments. The corresponding spatial $${P}_{j,k}=| \langle j,k| {\psi }_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle {| }^{2}$$ and spectral $${S}_{m,n}=| \langle {\phi }_{m,n}^{(2)}| {\psi }_{{\sigma }_{{\rm{c}}},{\sigma }_{{\rm{a}}}}^{(2)}\rangle {| }^{2}$$ correlation maps of our initial states, Eq. (6), are shown in Fig. 2. Tuning σa and σc, one can go from the spatially correlated state $$|{\psi }_{{\rm{c}}}^{(2)}\rangle$$, Fig. 2a, to the product state $$|{\psi }_{{\rm{p}}}^{(2)}\rangle$$, Fig. 2b, and to the spatially anti-correlated state $$|{\psi }_{{\rm{a}}}^{(2)}\rangle$$, Fig. 2c. Note the relation between spatial and spectral distributions: the state $$|{\psi }_{{\rm{c}}}^{(2)}\rangle$$, which is strongly correlated in space, Fig. 2a, is strongly anti-correlated spectrally Fig. 2d, and vice versa for $$|{\psi }_{{\rm{a}}}^{(2)}\rangle$$. Irrespective of their correlation maps, all these states occupy the same spatial area on the upper-left edge of the lattice, see Supplementary Note 1. The Schmidt number for $$|{\psi }_{{\rm{c}},{\rm{a}}}^{(2)}\rangle$$ is SN = 13, while for $$|{\psi }_{{\rm{p}}}^{(2)}\rangle$$ we have SN = 1. We now explore the robustness of these two-photon states as they traverse the disordered lattice. We begin with the product state $$|{\psi }_{{\rm{p}}}^{(2)}\rangle$$. To characterize the impact of disorder, we compute the fidelity30 that is given as the overlap of the state $$|{\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}})\rangle$$ after it has traversed the lattice with the reference state $$|{\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{m}}})\rangle$$ obtained after propagating the same state $$|{\psi }_{{\rm{p}}}^{(2)}\rangle$$ in a disorder-free lattice, see Supplementary Note 2. The two wavepackets are taken at slightly different propagation distances zf and zm to account for the somewhat different travel distance in a disordered lattice. We find the fidelity $${F}_{{\rm{p}}}=| \langle {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}})| {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{m}}})\rangle {| }^{2}=0.98$$, confirming that both the single-photon states and their product are immune to disorder. The edge–mode content of the evolved state is almost 100%, $${E}_{{\rm{p}}}=\mathop{\sum }\nolimits_{n,m}^{{\mathcal{E}}\otimes {\mathcal{E}}}| \langle {\phi }_{n,m}^{(2)}| {\psi }_{{\rm{p}}}^{(2)}({z}_{{\rm{f}}})\rangle {| }^{2}=0.9934$$. The product state traverses the lattice without distortion, see Supplementary Movie M1, in spite of the degeneracy between the two-photon edge–edge and bulk–bulk states.

Figure 3a, b visualize this outcome by showing the single-photon spatial distribution and two-photon spectral correlation maps for the two-photon product state $$|{\psi }_{{\rm{p}}}^{(2)}\rangle$$ traversing the disordered lattice. The spatial single-photon probability density R(n) is given by the diagonal elements $${\rho }_{nn}^{(1)}$$ of the reduced single-photon density matrix $${\hat{\rho }}^{(1)}$$, $$R(n)\equiv \left\langle n\right|{\hat{\rho }}^{(1)}\left|n\right\rangle \equiv {\rho }_{nn}^{(1)}$$48. The reduced single-photon density matrix $${\hat{\rho }}^{(1)}$$ is obtained from the two-photon density matrix $${\hat{\rho }}^{(2)}$$ in the usual way, $${\hat{\rho }}^{(1)}=\mathop{\sum }\nolimits_{m}^{M}\left\langle m\right|{\hat{\rho }}^{(2)}\left|m\right\rangle$$30. As expected, the spectral composition of the wavepacket remains undisturbed and the wavepacket propagates through the disordered region without leaving the edge.

We now turn our attention to entangled two-photon states. Figure 3c, d depict R(n) and the spectral correlation maps for the two-photon state $$|{\psi }_{{\rm{c}}}^{(2)}\rangle$$ traversing the “clean” lattice, and Fig. 3e, f show the same for the disordered lattice. While in the absence of disorder, R(n) stays on the edge and the highly correlated two-photon spectral distribution is unchanged (panels c, d), the disorder strongly affects these states. Figure 3e shows strong dissipation into the bulk as soon as the entangled wavepacket encounters the disordered region. The spectral distribution spreads all over the system, with both bulk–bulk and bulk–edge states becoming occupied, Fig. 3f. A similar result is obtained for $$|{\psi }_{{\rm{a}}}^{(2)}\rangle$$, except that the cross-like shape observed in Fig. 3f is flipped towards the opposite diagonal, Supplementary Note 3.

To quantify the probability fraction of the states scattered into the bulk we compute the edge–mode content. For $$|{\psi }_{{\rm{c}}}^{(2)}\rangle$$ the edge–mode content after traversing the disordered lattice is Ec = 0.4524, while for $$|{\psi }_{{\rm{a}}}^{(2)}\rangle$$ it gives Ea = 0.4453. Thus, >50% of both types of states is scattered into the bulk. The part of the states that survives the disordered region and stays on the edge remains strongly correlated in the spectral domain: the edge–edge part of its spectral content preserves the initial shape, see the right column in Fig. 3f. However, the spectral phase of the state is scrambled. To illustrate this point, we have renormalized the transmitted edge part of the two-photon wavepacket to unity and computed its fidelity FN by overlapping it with the reference two-photon wavepacket from a clean system, yielding FN = 0.405.

### The topological window of protection

We find that the conduit for dissipation of the two-photon edge–edge states is always provided by the edge–bulk states, which are degenerate in energy with the edge–edge states. Once disorder induces transitions into the edge–bulk states, they further transfer the amplitudes into the energy-degenerate bulk–bulk states, see Supplementary Movies M2 and M3. Hence, the key to topological protection is to minimize the disorder-induced overlap of the initial joint spectrum with the edge–bulk and bulk–bulk spectral regions, keeping it as close to the center as possible. That is, there is a topological protection window for two-photon states that offers the key guideline for designing robust two-photon states. To infer the protection window, we sent a probe product state with σc = σa = 0.01 through an ensemble of 1000 disordered lattices. This initial state is very well localized onto the edge region in real space, ensuring that all components within the state travel along very close paths. The spectral content of the state before and after the disorder is shown in in Fig. 4a, b. The components that have survived the impact of disorder are within the marked window—the topological window of protection. The joint spectral correlation map of any entangled state with varying σa and σc must fit inside this protection window to be robust against disorder.

In practice, to increase the amount of entanglement we need to increase σa$$\left({\sigma }_{{\rm{c}}}\right)$$ while decreasing σc$$\left({\sigma }_{{\rm{a}}}\right)$$, and by doing so the joint spectrum unavoidably tends to fall outside the protection window. However, we can always find two-photon states with a considerable amount of entanglement, which are protected. To elucidate this we have scanned the edge-mode content of the two-photon states after propagation through the disordered region as a function of σa and σc. In Fig. 4c, d we show the contour maps of the edge–mode content as we vary σa and σc. Figure 4c shows the edge–mode content of the two-photon states after propagation through the disordered region, with the diagonal corresponding to the product states, that is, states with σa = σc. The states with the highest degree of entanglement correspond to very different σa and σc and, therefore, they are found in the top left and lower right corners in Fig. 4c. In general, highly entangled states lay on the top left $$\left({\sigma }_{{\rm{a}}}\ll {\sigma }_{{\rm{c}}}\right)$$ and bottom right $$\left({\sigma }_{{\rm{a}}}\gg {\sigma }_{{\rm{c}}}\right)$$ corners and the edge–mode content quickly drops below 0.5. The reason is because as one increases σa, or σc, the tails of the spectral correlation ellipse fall outside of the protection window and, as a result, the states scatter into the bulk. Similarly, uncorrelated states may experience the same fate when they are initially confined into a small spatial region, which is the case for states with $${\sigma }_{{\rm{a}}}={\sigma }_{{\rm{c}}}\in \left(0,2.5\right)$$. Figure 4d shows the key figure of merit, ESN, the product of the Schmidt number SN and the edge–mode content E. The bright yellow islands indicate the best two-photon states, which combine robustness against disorder with high degree of entanglement. Importantly, the spectral correlation ellipse of these states always fits into the protection window shown in Fig. 4b. It is worth mentioning the features exhibited by the contour maps are generic as similar structures are obtained for disordered Haldane lattices with different dimensions, see Supplementary Note 4. This demonstrates that, in principle, one can create states with high Schmidt number and edge–mode content close to unity.

As evidence that our results are generic, in the sense that they apply to other two-dimensional topological systems, in the Supplementary Note 5 we have performed a similar analysis for an aperiodic topological lattice system6,40. We have found that the contour map of the edge-mode content E is not symmetric, implying that the correlated states are slightly less protected than their anti-correlated mirror-images. Nevertheless, we obtain the same qualitative features as in the Haldane model.

## Discussion

Before concluding, we would like to outline possible ways to generate the initial states and address the potential challenges for experimental observations of these effects. The initially highly correlated states can be implemented using standard spontaneous-parametric-down-conversion nonlinear crystals to generate photon pairs that are coupled to the edge of the lattice using a positive achromatic doublet lens as demonstrated in38. Anti-correlated photon pairs can be generated by applying the fractional Fourier transform to the highly correlated states49. The Haldane lattice has been previously demonstrated using femtosecond laser written waveguides as reported in41. Hence, the challenges are reduced to optimizing the fabrication for minimal scattering, absorption and bending losses associated with the helical waveguides.

These results lead to the following conclusions. Two issues have to be considered when constructing two-photon entangled edge states in topological systems: their dissipation into the bulk and the relative dephasing between the different components comprising the entangled state. Regarding dissipation, the two-photon edge states can be protected just as well as the single-photon edge states. Further, phase scrambling can also be minimized if the different components of the entangled state travel along the same path in the edge region. Both aims are achieved by keeping the spectral correlation map of the two-photon state in the center of the window of protection. Thus, attempts to increase entanglement must be balanced against keeping the spectral correlation maps of the two-photon states within the narrow spectral region at the very center of the single-photon gap—the topological window of protection. This limits the degree of entanglement one can safely encode in practice, but presents a clear strategy for creating useful states with high degree of entanglement and robustness.

Looking forward, one could take advantage of the static nature of disorder to circumvent entanglement-induced dissipation into the bulk. While the disorder-induced relative phase between the different product-state components of the entangled wavepacket may appear random due to the random nature of disorder, for static disorder scrambling and dissipation are nevertheless fixed. This opens an opportunity to find the windows of protection as we have done in the cases considered here, and generate robust wavepackets tailored to the particular disordered system at hand. From a practical perspective, the stability of entangled states up to relatively high Schmidt numbers offers practical guidelines for generating useful entangled edge states in topological photonic systems. Finally, our work may open the door to study topological protection of highly entangled multiphoton non-Gaussian states that fulfill the protection conditions.


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# Fast integration of a posterior distribution

I wish to infer the posterior distribution on the probability of success $\theta$ in some binomial process, the twist being that I know that $\theta$ lies in the interval [0.5, 1].

The trouble is that a Beta distribution that is supported on [0.5, 1] is no longer conjugate. In particular, the prior looks like this ($p$ is the pdf):

$p(\theta) \propto {(\theta-0.5)}^{\alpha-1} {(1-\theta)}^{\beta-1}$

The likelihood, given $x$ successes in $n$ trials, is the usual binomial likelihood:

$p(n,x | \theta) \propto \theta^x {(1-\theta)}^{n-x}$

The posterior, as it stands, looks like this:

$p( \theta | n, x) \propto \theta^x {(\theta-0.5)}^{\alpha-1} {(1-\theta)}^{n-x+\beta-1}$

I would like to be able to make CDF calculations (i.e. compute integrals) of the posterior efficiently. (numerical integration, e.g., may be too slow for my purposes). I am unable to figure out how to do this.

Much appreciate any help the forum can provide.

• It sounds like the only reason you use beta is because its conjugate? uniform over the interval is fine if you have a reasonable amount of data ($\alpha=\beta=1$) - then you just have an incomplete beta function to evaluate, most software packages can do this for you. Plus numerical integration is very quick because you have a short interval $[0.5,1]$ - uniform grid sampling would work well. – probabilityislogic Apr 18 '11 at 10:26
• The indefinite integral can be computed exactly in terms of hypergeometric functions and gamma functions. With integral values of $\alpha$, $\beta$, $n$, and $x$, this reduces to a polynomial in the limit of integration. If you don't need to perform general analysis of the expressions, though, then numerical quadrature as suggested by @probability will be fast and effective. It should be adaptive sampling, not uniform, though: the singularities at 1/2 and 1 for small values of $\alpha$ and $\beta$ might otherwise cause havoc. – whuber Apr 18 '11 at 15:43
• @probabilityislogic: if I can, I'd prefer to use something other than a uniform prior, since my prior is quite informative, and for speed, I'd like to observe as few actual binomial trials as possible. Beta seemed the natural alternative. But your answer induced me to consider a simple piece-wise uniform prior, which may have enough information for my purposes. – Venu Satuluri Apr 18 '11 at 22:04

How accurate does your posterior cdf need to be? You might consider replacing the continuous prior with a discrete approximation:

$p^*(\theta) \propto p(\theta) 1(\theta\in t_1, \dots, t_k)$

where $p(\theta)$ is your original continuous prior.

Then to compute the posterior you just calculate likelihood x prior

$p(\theta|x) \propto p^*(\theta)p(x|\theta)$

over the support of the prior $t_1, \dots, t_k$ and renormalize.

This is called "griddy Gibbs" by some. It can be quite effective if you have an informative prior in which case you can choose the grid points non-uniformly (and, of course, if you can live with a discrete approximation coarse enough to be computationally feasible).

• Do you have any suggestions for how to choose the grid points (non-uniformly)? If I need to choose k grid points, and say I restrict the grid points to only n possibilities, then the brute-force approach is to evaluate the likelihood of each (n choose k) combination, and choose the max. likelihood one. Are there alternatives? (Sorry for the late comment, I just saw your answer today.) – Venu Satuluri Sep 15 '11 at 18:24

There may be a simpler approach, simply by applying the usual Beta conjugate to the binomial, and then requiring $\theta \in [\frac12,1]$. You can do this with an indicator function, for example as in

$$p(\theta) \propto \theta^{\alpha-1} {(1-\theta)}^{\beta-1} \mathbb{1}[{\tfrac12 \le \theta \le 1}]$$

Now apply your $p(n,x | \theta) \propto \theta^x {(1-\theta)}^{n-x}$ to get the posterior density

$$p(\theta|x) \propto \theta^{\alpha+x-1} {(1-\theta)}^{\beta+n-x-1} \mathbb{1}[{\tfrac12 \le \theta \le 1}].$$

The posterior cumulative distribution function for $\theta \in [\frac12,1]$ is then $\dfrac{I_\theta(\alpha+x, \beta+n-x)-I_{\frac12}(\alpha+x, \beta+n-x)}{1-I_{\frac12}(\alpha+x, \beta+n-x)}$ with $I$ representing a regularised incomplete beta function or the cumulative distribution function of a Beta distribution, which any decent statistical program will calculate quickly, such as R's pbeta function.

You can always use Monte Carlo Integration or the Midpoint method. With Monte Carlo, you simply generate a bunch of points in your parameter space and see if they are in the area or volume or hyper-dimensional space you are trying to integrate.

From: http://farside.ph.utexas.edu/teaching/329/lectures/node109.html "Let us now consider the so-called Monte-Carlo method for evaluating multi-dimensional integrals. Consider, for example, the evaluation of the area, , enclosed by a curve, . Suppose that the curve lies wholly within some simple domain of area , as illustrated in Fig. 97. Let us generate points which are randomly distributed throughout . Suppose that of these points lie within curve . Our estimate for the area enclosed by the curve is simply" the ratio of random points in the space times the size of the space. The link has a nice picture and a description of the inferior midpoint method that I would suggest skipping.

• Because this is a 1D, not multidimensional, integral, numerical methods (as suggested by @probabilityislogic) tend to be much faster and more accurate than Monte Carlo methods. – whuber Apr 18 '11 at 15:37


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1.(iii)    Show that the statement             p: “If $x$ is a real number such that $x^3 + 4x = 0$, then $x$ is 0” is true by             (iii) method of contrapositive

If $x$ is a real number such that $x^3 + 4x = 0$, then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$.

q: x is 0.

In order to prove the statement “if p then q”

Contrapositive Method:  By assuming that q is false, prove that p must be false.

So,

q is false: $\dpi{100} x \neq 0$

$\dpi{100} \implies$ x.(Positive number) $\dpi{100} \neq$ 0.(Positive number)

$\dpi{100} \implies x(x^2 + 4) \neq 0(x^2 + 4)$

$\dpi{100} \implies x(x^2 + 4) \neq 0 \implies x^3 + 4x \neq 0$

Therefore p is false.

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## 07 August 2014

### Coursera Courses

In this post I review some of the Coursera courses that I completed.

Algorithms: Design and Analysis, Part 2. The course covers greedy, dynamic programming, and NP-completeness. The lectures were very clear, not too fast and not too slow. Caveat: I watched lectures only when I needed. My strategy was to try to solve the problems. If I had trouble, I looked at the slides. If the slides were cryptic, then I looked at the lectures. So, I ended up seeing very few lectures. I even learned an algorithm that I knew I should know but didn't: Karger's. I think it's my favorite example of a randomized algorithm.

Introduction to Genetics and Evolution. The course covers the basic math of genetics. I'll give two examples of things I remember now, more than one year after.

One idea is the Hardy-Weinberg equilibrium. Suppose each individual has one of the three genes $aa$, $aA$, and $AA$. I'll use the fancy word allele: it means one of $a$ and $A$. Reproduction works as follows: Randomly choose an allele from each of the two parents, and put them together. Hardy-Weinberg says that in an infinite population, the percent of each allele is constant. So, if you start with half $a$s and half $A$s, you'll remain with half $a$s and half $A$s.

The other idea is genetic drift. It says what happens when the population is not infinite. What happens is that you'll have random variations. Eventually, these lead to the extinction of one of the alleles.

Programming Languages. This course uses ML, Racket, and Ruby to illustrate basic principles of programming languages. As usual in this kind of courses, you write a few interpreters. The best time to take this course is after you've learned your second language. (I learned BASIC when I was 6; Pascal and C++ came when I was 14. I think I would've benefited most from this course when I was 15. So, if you are 15, then you should try it!)

The Modern World: Global History since 1760. What I remember most vividly about this course is rather strange. It is the soothing voice of the lecturer. Even when I was disagreeing with what he said, I felt compelled to keep listening.

From what I recall, the quizzes were boring: They simply tested a bunch of facts that happened to be mentioned in lectures. The videos, on the other hand, always told a nice story. These stories were clearly addressed to someone living in 2013. For example, the lectures not only said that the West was weak in 1760. They stressed this fact repeatedly, because a world in which the West is weak and insignificant is really difficult to imagine for the typical user of Coursera.

Natural Language Processing. For this course, like for the Algorithms one, I did problems first. I looked at the slides and watched the lectures only when I needed. I ended up looking at most slides and almost no lecture video. Which, I guess, means that the slides are self-sufficient. From what I recall, doing the homeworks involved a substantial effort.

The topics covered are things like probabilistic grammars, tagging problems (e.g., noun, verb, …), log-linear models, learning.

Discrete Optimization. Like for the other computer-related courses, I went directly for the homeworks. In addition, I delayed doing most homeworks for the last week. That turned out to be a huge mistake. I simply couldn't do the homeworks at the rate of one per day, as I planned. In fact, I thought at some point that I won't even get a passing grade. I did, but I also learned that Coursera sometimes has tough courses. So, what were the homeworks like? I recall: knapsack, graph coloring, euclidean traveling salesman. There was also some combination of bin packing and euclidean traveling salesman. More importantly, I recall some tricks that I learned while doing the homeworks. For example, for knapsack I used two tricks to get a good solution. The standard dynamic programming runs in time proportional to the total weight. This is troublesome because weight can be very large. In that case, use a trick from physics: pick your units of measurement. Strictly speaking, the unit of measure doesn't matter. But, when one works in kilometers, one usually doesn't keep track of millimeters. Thus, by ‘picking the unit’ I actually mean ‘ignore the less important bits’. When ignoring bits, you'll want to round up. This way, you never try to fit more than possible in the knapsack. The other trick is to repeat this process using the items that remained out, with better precision.

For travelling salesmen, I wrote some complicated solution that barely got medium points. It did two main things, as I recall. First, it used dynamic programming to do local optimizations. Second, it greedily improved the tour by using a set of moves that I noticed empirically. For an example of move, think of a crossing of edges. These two edges are far apart in the current best tour, so local optimizations won't notice the crossing. But, of course, you can look at all pairs of edges and check for intersections. Once you notice the interesection, it's clear what to do to shorten the tour.

A Brief History of Humankind. This course was like the other history course: trivial (useless?) quizzes, but fascinating lectures. Where the Modern History course would talk about personalities and their decisions, the Humankind History course would talk about the way of life of a typical person. In this sense, the two courses complement each-other nicely.

The most vivid image that this course left in my mind is that of ‘shared imaginary worlds’. Why do money have value? Because we believe that others believe that money have value. This kind of circular reasoning has a certain appeal for a computer scientist.

Linear and Integer Programming. This course was too easy for my taste. They do recommend a resource that looks intriguing: the convex optimization book by Stephen Boyd.

Automata. This course had two kinds of homework: multiple-choice quizzes and programming assignments. Unexpectedly, I enjoyed the quizzes more. Each quiz question tries to test whether you understood some concept. You get treated as a computer program: The quiz presents you with some (smallish) input data, you have to provide the right output. If you repeat the quiz, you get some other input data that you have to process. For one of the first quizzes, I managed to get a question wrong three times in a row. I was trying to do it quickly, and I ended up being very slow. The fourth time I moved slowly and checked my work. I think this phenomenon appears often in real life: People make fun of those that appear slow, but the slow ones often finish first.

Back to the course. Why did I not like the programming assignments? Because I was supposed to modify some Python scripts that were kinda crap.

To answer some questions, I had to look at the slides. These were extremely clear — I never needed to look at the video lectures.

Social and Economic Networks: Models and Analysis. I learned something new from almost every video lecture. The quizzes were rather easy, but only after I learned the vocabulary. It wasn't possible to grasp the definitions from the slides, but the videos were very clear. In other words, if you take this course and you don't already work in social networks, you'll have to watch the videos. Which is a good thing.

Here's an example of the kind of stuff I remember. Social networks are graphs. In various circumstances it makes sense to talk about the utility of a social network. A simple way to define this is as the sum of utilities of each vertex. For an example, think of the co-author graph. The vertices are authors, the edges represent an ongoing collaboration. One particular author benefits from the time given by its co-authors. In turn, this particular author offers his time to their co-authors. In the simplest model, the time is divided equally. Thus, the utility of a vertex $x$ is $1+\sum_{y \in N(x)} 1/|N(y)|$. (The $1$ comes from the time spent by $x$ on their own stuff, and $1/|N(y)|$ appears because $y$ divides its time equally between its neighbors $N(y)$, one of those neighbours being $x$.) Once you have such a simple model, you can ask questions like: Which configurations are pairwise stable (meaning that there is no local incentive to create or delete edges)? Which configurations correspond to global optima? For the answers, you'll have to take the course. (Well, you could find out the answers yourself, of course.)

Statistical Mechanics: Algorithms and Computations. This course is mostly about Markov chains. You can think of Markov chains as a way to sample complicated probability distributions, or as a way to estimate integrals. Come to think of it, the course is more about sampling and integrating than about Markov chains. For example, when you get to the quantum stuff, you see that computing path integrals is better done using Levy paths. These are a direct sampling technique, which works better than the Markov chain approach, in this case.

Also, these techniques can be used for optimization problems. One example is the traveling salesman problem, which makes an appearance in one of the reviews above. I was conceited, and expected that the program I wrote with more than one day of work would do better. After all, I looked at the Python program provided by the instructors and it seemed very similar, except: (1) it did no local optimization using dynamic programming, (2) the vocabulary of moves was a small subset of the one I accumulated. And yet, their implementation worked much better. Wat?? I had to take a look. It turns out there was only one difference: They sometimes allowed a move that made the tour worse. (They used simulated annealing, rather than greedy.) In other words, my solution was getting stuck in local optima. I'm not sure why I didn't think of this when I was doing the discrete optimization course. (It seems obvious with hindsight.) A good rule of thumb: Whenever you use greedy for an optimization problem, at least consider trying simulated annealing.

Starting Soon. Out of these courses, the ones that start soon are the following: