فبطه آدك الأيام فتا أحاط بكل شيء علمة لذا الآخر now is talking about الله تبارك والتعالى is he has knowledge of everything because the previous point the sheikh has just mentioned that Allah is above his throne and because Allah is above his throne it doesn't mean that he's not aware of what's below he's not aware of everything and he has knowledge of it and Allah is never heedless of his creation راذا الله تبارك وتعالى بل هو مطلع عليه الله is one that's looking over them علمهم بهم he is aware of them يطلع على حركاتهم الله looks and observes their movements and their silences and matters which are very hidden about the affairs of the creation الله knows it سبحانه وتعالى as he said in the Quran الله تبارك وتعالى he says that we are the ones who created the humans and we know that which their soul whisper to them we know it and الله تبارك وتعالى he then says we are more closer to them than the worries we are more closer to them so this being what the closeness here means knowledge علم it doesn't mean الله تبارك وتعالى it doesn't mean it means it means it means it means it means الله تبارك وتعالى because Allah is a buffy throne so Allah is close to His creation by knowledge He knows everything ابن جرير الطبري رحمه الله in the ayah of Surah Al-Talaq وأن الله قد أحاط بكل شيء علمه the last ayah of Surah Al-Talaq ابن جرير الطبري رحمه الله in that ayah when he was commenting on it he said he said يقولوا جل ثناؤه الله تبارك وتعالى he says ولي تعلموا أيها الناس oh people you need to know أن الله بكل شيء that Allah تبارك وتعالى everything he has encompassed from his creation knowledge سبحانه وتعالى لا يعزوب عنه مدقال ذرط في الأرض ولا في السماء and nothing is hidden from him there is nothing that he is not aware of سبحانه وتعالى he knows everything from the affairs of his his creation سبحانه وتعالى ولا أصغر من ذلك ولا أكبر nothing little and nothing big is hidden from him يقولوا جل ثناؤه الله says in that vest and what he means by it is what فاقافوا أيها الناس oh people be scared of him because he knows everything be scared of him by opposing his command because he is going to punish you and he is aware of it فإنه لا يمنع من عقوبتك المانع and if Allah wants to punish you there is nothing out there that is able to stop him from punishing you there isn't and he is able to do so ومحيط أيضا and Allah has also encompassed سبحانه وتعالى your actions in time he knows every action which you do فلا يخفى عليه ما منها خافن وهو محصيها عليكم ليجازيكم بها يوم تجزى كل نفس ما كسبت لأن الله تبارك وتعالى he is one that's going to reward you based on the actions and the things that you put forward so Allah knows everything there is nothing hidden from him سبحانه وتعالى even though he is what he is above his throne نعم ليس كمثله شيء وهو السميع البصير the author here now he is talking about الله's characteristics the way that أهل السنة والجماعة أفهم الله's characteristics is between what بين نفين وإثبات negation and affirmation أهل السنة they negate from Allah that which he negated from himself and that which his prophet negated from him and أهل السنة أفهم for Allah that which he affirmed for himself and that which his messenger affirmed for him that is من هجو أهل السنة and that is طريقة their path in which they deal with آيات الصفات حادث الصفات what do they negate from Allah that which he negated from himself which is what كيفية والمماثلة الله تبارك وتعالى we negate from it anyhow number one we negate from it what how number one and the second one is what المماثلة we negate from it that Allah's resemblance of his characteristics towards the creation we negate that from Allah سبحانه وتعالى أهل السنة also do what they affirm they affirm for Allah characteristics what characteristics do they affirm for Allah that which Allah has affirmed for him أهل السنة they affirm it for him they affirm it for him good and wherever his prophet has affirmed for him they also affirm for him so their affirmation comes to them through what it comes through to them by the كتاب and the سنة and the negation of theirs comes through to them by what way by the كتاب and the سنة they don't say from Allah wherever our logic doesn't accept and they don't say that we affirm for Allah everything that our logic accepts they don't say that أهل السنة their نفي and their إثبات is from the كتاب and the سنة that is very important that a student of knowledge he learns that قاعدة also what we need to know is that the methodology and the path in which the Quran is up on when it comes to Allah's characteristics is that is the قاعدة you need to memorize that the affirmation the إثبات is what مفصل the إثبات is what detailed Allah's the characteristics that we affirm for Allah we go detailed when we are affirming these characteristics for Allah the negation how do we do it مجمل we make it general we don't go detailed in the negation so for example when we negate from Allah we say ليس كمثله شيء there is nothing like Allah we also say لم يلد ولم يولد ولم يكله كوفو one أحد we also say رب السموات والأرض وما بينهما فعبد هو استبر العبادة هل تعلم له سمية all of those verses do you know anyone who is equal to Allah all of those are negation which are general never do you find in the Quran will you find that the negation is detailed the nephew is never مفصل in the Quran when it comes to Allah's characteristics إلا except if there is an إثبات an affirmation that was before it in which it is then coming detailed such as آية الكرسي الله لا إله إلا هو الحي القيوم لا تأخذه سنة ولا نوم لا تأخذه سنة ولا نوم is a nephew مفصل it's a negation it's detailed yes but before it what was it حي انقيوم حي انقيوم which are if Allah is حي انقيوم لا تأخذه سنة ولا نوم so when you want to do a detailed negation you have to have before it what an affirmation which is detailed but to just say like the أهل البدع do Allah is not above Allah is not below Allah is not on the right Allah is not on the left Allah is not in where did you get it from you see also negation brothers is not praiseworthy if you had that yes it's not praiseworthy negation in and within itself is not what it's not praiseworthy because negation it shows عدم القدرة and I give you an example this wall right now does it oppress does this wall oppress right now the answer is no why doesn't this wall oppress right now the reason is because it doesn't have the ability to oppress it doesn't it can't do anything so is it praiseworthy for him that he can't oppress no it's not rather it's weakness that you are ascribing to him does that make sense when he does what do they do when they say they say Allah is able to oppress if he wants to he has the ability because stripping from Allah is not praiseworthy in and within itself does that make sense because there are those who don't oppress because they can't they are not able to oppress نعم and insha'Allah تعالى we're going to speak about these قواعد pertaining to اللسماء والصفات more insha'Allah تعالى in كتاب الطحاوي when we explain it insha'Allah تعالى so the author he got here now the ayah ليس كمثله شي وهو السميع البصير it's ayah from the Quran صورة الشورة ayah 11 صورة الشورة ayah 11 that's where he got it from the first part of the ayah is رد على المشبهة it's a refutation on who the مشبهة those who make Allah's characteristics the same as the characteristics of His creation Allah has refuted them what did He say to them ليس كمثله شيء there is nothing like Allah the next part of the ayah is what وهو السميع البصير is a refutation on who على المعطلة those who have what who have negated Allah's characteristics who have chosen not to affirm a characteristic for Allah that verse is refuting them and then الله تبارك وتعالى characteristics what are the pillars for it إثباتون ونفيون إثباتون ونفيون affirmation and negation ليس كمثله شيء there is nothing like Allah and then Allah affirmed something for Him said and what did He say وهو السميع البصير وهو السميع البصير نعم وأنه تبارك وتعالى يراه في الآخرة يراه أهل الجنة بأبصارهم ويسمعون كلامه كيف شاء وكما شاء الله سبحانه وتعالى هو بيسين نديف ججمد عكيدة أهل سنة أن الشيخ ساد وأنه وتبارك وتعالى هي الله تبارك وتعالى يراه هو بيسين في الآخرة إن يديف ججمد هو بيسين يراه أهل الجنة يراه أهل الجنة أهل الجنة ويشاهدهم كيف يشاهدهم بأبصارهم بأبصارهم بأبصارهم لا one has ever seen الله with their eyes لا one has ever seen الله with their eyes الله has not been seen with eyes no one has ever نعم but can Allah be seen in this world نعم in a dream and the Sunnah believe Allah can be seen in the dream نعم الله can be seen but Allah cannot be seen with the apsar the eyesight and insha'Allah we are going to elaborate on this part that Allah can be seen in the dream when we go to أصول السنة by Imam Muhammad Imam Muhammad speaks about it there and when we explain it they will speak about it in more details نعم so Allah is about the author the reason why he added that part to it which is what يراه أهل الجنة the people of Jannah are going to see Allah بأبصارهم with their eyesight with their eyesight he is trying to get out of what that this is something that happened before no one has seen Allah with their eyesight good and also those who believe Allah won't be seen سبحانه وتعالى like innovating and those who are corrupted but Allah the sheikh said who are the ones who are going to see Allah the people of Jannah the people of Jannah أهلو نار won't see Allah because Allah said صورة المطففين آية 15 كلا إنهم عال ربيهم يومئذي لما حجوبون آية 15 صورة المطففين Allah says about the disbelievers that they are محجوبون they are veiled from seeing الله تبارك وتعالى they won't be given that opportunity to see who to see Allah سبحانه وتعالى whereas the believers will be permitted to see Allah because Allah said السورة القيامة وجوه يومئذي ناظرة that day faces will glow the faces of the righteous people and the أهلو جنة will glow you see and look إلى ربها they will what they will be looking towards and also the آية in سورة يونس آية 26 الله says للذين أحسنوا الحسنة the people who have done what and who have done what have they done the ones who have done good الله تبارك وتعالى he is prepared for them what جنة and also what حسنة also حسنة sorry زيادة الله is prepared for them just as a side benefit which is there is a فائد الله طيفة which is the جار and the مجرور the جار and the مجرور are مقدم they have been put forward and where are they sitting the جار and the مجرور which is للذين the لام which is حرف جر and اللذي which is اسم موصول which is اسم مجرور here and مقدم it was meant to be مبتد أن لا خبر here what happened the خبر has come forward before the مبتد قيل before the مبتد the فعل is what أحسنو so this is what is connected to أحسنو في عمالهم they have done righteousness in terms of their actions meaning they follow the كتاب they follow the سنة they done it with sincerity they follow the بروفس عسنة and everything where is the مبتد it's a مبتد which is مؤخر which is جنة and then what's the زيادة the زيادة is معطوفا على it's connected to what الحسنة it's معطوفا على الحسنة and this has to be something this has to be something other than the حسنة it has to be something other than the حسنة what is it الله has been seen سبحانه وتعالى so they're going to get an additional thing to جنة that they're already in they have to get something more which is what seeing Allah تبارك وتعالى seeing Allah تبارك وتعالى and it's like the verse of Allah لهم ما يشاءون فيها ولدينا مزيد مزيد here is what رؤية الله تبارك وتعالى ولهم ما يشاءون فيها they have whatever they want in جنة that's جنة ولدينا مزيد is what الله تبارك وتعالى is going to unveil from himself and they're going to see him سبحانه وتعالى so the seeing of Allah is an additional reward it's an additional reward it is better than the جنة that they're already in because Allah is referring to it as what زيادة زيادة بسعيد الخذري رضي الله تعالى عنه he narrated in the حديث صحيح البقاري that the prophet صلى الله عليه وسلم he said that بسعيد الخذري that في زمن النبي صلى الله عليه وسلم that a group of people at the time of the prophet they said to him يا رسول الله oh messenger of Allah هل نرى ربنا يوم القيامة are we going to see الله the day of judgment the prophet صلى الله عليه وسلم I said نعم yes you will هل تضارون في رؤية الشمس بالظهيرة is there anything the sun when it's daytime it's clear is there any clouds is there anything between you is it so clear can you see it do you doubt it is everything there they said they said they said ضوء اللي سفيها صحاب is there any clouds anything they said no يا رسول الله it's clear we can see it then the prophet صلى الله عليه وسلم he said عليه الصلاة والسلام the way that it's that clear and that you can see the day the sun like that you will see your load the day of judgment يوم القيامة the way that you see the sun or the full moon some narration say the full moon the way you see it and it's out there and there's no cloud between it it's the same way you're going to see الله تبارك وتعالى the day of judgment now some people are ignorant and are unaware of what the prophet compared here did he compare the moon to Allah or did he compare the seeing to the seeing he compared the seeing of the creation here with the comparing of the seeing of the day of judgment the way that it's clear here it's going to be clear the day of judgment that not saying that the moon is like Allah فتنبه be alert and aware of that نعم والجنة حق والنار حق ألو سنة also believe as the author mentioned that والجنة حق والنار حق ألو سنة believe what from the Eman of the things that they believe is that what that the jannah and the naar and know them موجودة تاني that they are both they are present right now as we speak and that they are both created by Allah لا تفنعني and they will never perish the jannah is what it's a place Allah created سبحانه وتعالى for his allies for his people and the naar Allah created it a punishment for the people of sinners the sinners and the wrongdoers and the criminals Allah created it for them whoever Allah wills he doesn't take him to the jannah even though he's a criminal even though he's a criminal Allah if he wants he won't take him to the hellfire he will take him to what to the jannah عدلا منه based on his justice سبحانه وتعالى إمام أحمد بن محمد إمام أهل السنة والجماعة إمام أهل السنة والجماعة إمام أحمد بن محمد as طبقات الحناب أبو يعلى narrated from him where he said وإن الله خلق الجنة Allah created the jannah قبل الخلق before the creation before Allah created the human being Allah created what جنة جنة جنة وخلق لها أهلا and Allah created for it a people ونعيمها دائم and it's what it's bless and the greatness in jannah and the blessings of Allah in jannah are forever to be ومن زعم anyone who claims أنه يبيد من الجنة شيء وكافر anyone who claims that the blessings that are in jannah are going to finish and it's going to come to an end he's a kafر وخلق النار Allah also created the fire قبل خلقه الخلق before Allah created the creations he also created the hellfire وخلق لها أهلا Allah also created for it a people وعذابها دائم and it's punishment is forever remaining it's punishment is forever remaining ألمام الطحوي also said in his عقيدة he said والجنة والنار مخلوقتان لا تفنيان أبدا ولا تبدا the jannah and the fire are both created they will never perish and they will لا تفنيان أبدا ولا تبيدان فناء تبيدان is both the same meaning it will never come to an end الإمام البربهال رحمه الله he said the same وكل شيء مما أوجب الله عليه الفناء يفناء everything Allah has made for it to perish and perish إلا الجنة والنار except the jannah and the fire والعرش الله is thrown والكرسي and the curse والصور والصور the trumpet والقلم the pen واللوح the لوح المحفوض ليس يفنى شيء من هذا أبدا none of those will perish ever الإمام الأجور he said the same وقد ذكر الله عز وجل في كتاب الله has mentioned in his book أهلا النار the people of the house fire الذين هم أهلها the people who are the people of house fire الله mentions in the Quran they are what يقولوا دون فيه أبدا that they will stay in the house fire forever وأن أهلا النار الذين هم أهلها and that the people of the house fire that are made for it they are what في العذاب الشديد أبدا that they are in severe punishment forever الله تعالى الله تعالى تعالى told us the same way الإمام الصابوني he said it إبنو حزم رحمه الله in his book الفصل في الملل والأهواء والنحل إبنو حزم he transmitted إجمع and an ittifak that the الجنة النار ever gonna remain he said الجنة والنار باقي تاني لا تفنياني he said إتفقت فرق الأمة كلها على أنه لا في نائل الجنة ولا لنعيمها ولا للنار ولا لعذابها إبنو حزم رحمه الله he said it is agreed by all of the فرق of the all of the groups all agreed that what that the جنة will not come to an end and its blessings and the fire will not come the hellfire will not come to an end and its punishment he also brought another إجمع in his marat he brought إجمع and this is عقيدة أهل السنة that they all gonna remain same حافل عبدالغاني عبدالواحد المغدسي he also said the same in his عقيدة book ابنو قدام in his لمعة العتقاد he said الحمبلي حافظ الحكم رحمه الله all of them they all spoke about the hellfire coming to a never coming to an end that the hellfire will never come to an end and that the جنة will not come to an end they all spoke about دعماته and مفصلن detail نعم وصراتوا حق the author now is talking about a سرات what is the سرات the سرات is a bridge who will just through it's a bridge that is over the hellfire it's a bridge on top of the hellfire every person will go over it even the مواحد even the مواحد the person of توحيد is going to go over that bridge نعم and everybody is going to go over that bridge على قدر أعماله everyone is going to go over it in accordance to their actions ويسقطوا فيه من سقط and whoever drops will drop into the hellfire you see anyone who drops will drop in it نعم and anyone who wants to know how he is going to make it on that bridge the day of judgment let him look at the way he treads on the road of this dunya today which road the road of the messenger صلى الله عليه وسلم look at yourself today where you are when it comes to the Prophet صلى الله عليه وسلم his path and how you are towards it that's the same way you're going to be on the path or the bridge of the the day of judgment the Prophet صلى الله عليه وسلم the ayah الله said to him say to the people say to them وأن هذا صلاطي مستقيما فالتبيعو ولا تتبعوا السبول فتفرق بكم عن سبيل كل هذه سبيل ادعوا إلى الله على بصيرات أنا وما نتبعني so this is like that all of them are what that this path that the messenger أن هذا صلاطي مستقيما فالتبيعو ولا تتبعوا السبول فتفرق بكم عن سبيل ذلكم وصاكم بهلعلكم تتقون the path of the messenger that he was upon was what the path of ترحيد عليه الصلاة والسلام and the sunnah in which you call to you call to the sunnah as well you don't call to innovation you don't call to sinning and you also stay away from a shirk and anything related to it so wherever you are and however you are on that path is the way you're going to be on the path of the Salat of the day of judgment the people who are going to make it or make it through the Salat the Salat الله told us the only people who gathered one characteristics الله said اي منكم إلا واريدها كان على ربك حتما مقضية ثم نجل الذين تقوا ونضروا ظاليمنا فيها جيثية in that verse Allah says اي منكم إلا واريدها there is not a person except he's going to go on that bridge he's going to go on the Salat everybody and then Allah told us ثم نجل الذين تقوا the people were going to give safety and prosperity and who are going to make it through اهو الذين تقوا the people of Taqwa اي من who are the people of Taqwa go back to the definition of طلق من حبيب that's the best definition جامع المانع it's a very comprehensive definition a definition that gives a whole meaning the word it's correct meaning طلق من حبيب you will find it in ابن راجب حملي he's كتاب جامع علوم الحكم I think he brings it in the Hadith of إرباظ المنسارية or he brings it in the Hadith of أبيضر which is إتق اللي حيث ما كنت or in the Hadith of إرباظ المنسارية وعدنا رسول الله صلى الله عليه وسلم المعظه he brings the أقوال there and the definitions are regarding the word تقوى الإمام الأصبهاني رحمه الله there are two ways to say الأصبهاني الأصبهاني is قوام الحجة أبو قاسم تيمي the author of the كتاب الحجة في بيان المحجة you can say أصبهاني with a back and you can also say أصفهاني back or if is the same both are said الإمام أبو قاسم التيمي قوام السنة قوام السنة he has a book called الحجة في بيان المحجة دار الراية published it دار الراية he said والصرات حق the the is what يجوز عليه من شاء الله he would go over it the one who he would go through it makes it ويصقط في جهنما من شاء الله and he would drop in the hellfire whoever Allah wills for him ولهم أنوار على قدر عمالهم and the people have light and they glow that day based on their actions you see how seriously it is also إمام محمد said the same thing in his in the there's a a narration of خلال أبو بكر الخلال he has a narration where he narrates from who إمام محمد محمد he said إمام محمد said والصراط حق وطع على سواء جهنما that the الصراط is حق and it's been placed on top of جهنما ويمر الناس عليها and the people will go over it ويمر الناس علي والجنة من وراء ذلك and جنة is after after the after the after the after all of that نسأل الله السلام إمام محمد said we ask that he makes us safe it is something we ask الله in our five daily prayers every time إهدن الصراط المستقين Oh Allah guide us on a straight path why are we saying Oh Allah guide us on a straight path the reason why we do is because we want to be those who do not drop on the صراط of the day of judgment الله also said in صورة الفتح إنا فتحنا لك فتح مبينة ليغفر لك الله ما تقدم و يتم نعمته عليك و يهديك صراط المستقيمة و ينصورك الله نصلا عزيزة و يهديك ف الله تجايد أم محمد صراط اي باث that is صورة الفتح I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to I want to وَالمِزَانُوا حَقٌّ لَهُ كِفَّةَانِي المزان هو حق المزان هو ماذا؟ إنه مخصوص في which Allah scales things أهلُ سُنَّة have transmitted that there are things Allah scales there are what? there are things from them are actions are scale are weighed on it on the day of judgment what is weighed on it is عمال people's actions the person's action is taken and it's weighed also the individual is weighed narrations that say أمان would be brought the day of judgment يُتَابِ رَجُنِ يَوْمَ الْقِيَامَةِ السَّمِينُ الْبَدِينِ the person who is chubby and big فَا يُضَعُ فِي الْمِزَانِ he is placed on the scale فَلَ يَزِنُ عِنْدَ اللَّهِ جَنَا حَبَعُوضًا and he doesn't weigh in the eye of Allah a wing of a mosquito so he's weighed this guy is big he's chubby he's placed on the scale and he doesn't weigh in the eyes of Allah a wing of a mosquito also what he's weighed is the scrolls in which the actions were written on and the evidence for that is the حديث of بطاقة the حديث of the man who لا إله إلا الله saved him the card that had said it said لا إله إلا الله in it then the hadith says he has on the scale تسعة و تسعين سجلة 99 scrolls are on the scale and then a card is brought out and it said to him this card and then he said يا رب what will this card do in comparison to the scale what's already there the 99 scrolls of sins and then Allah said no one is oppressed today everyone will be fairly judged and so what is taken then the card is then placed on the scale and it outweighs all his sins so what's weighed now the place that the actions were written on all of those are what all of those are the weighing that are done on the day of judgment so the sheikh said والميزان the ميزان حق له كفة تاني كفة ميزوات it has two sides it's got two sides in which the scale it would be weighed and evidence for that is صورة القارع on ayah 6 to ayah 9 ونضع الموازين القصة اليوم القيامة فلا تظلم نفس شيئا وإن كان مثقال حبة من خردل اتينا بها وكفى بنا حاسبين الله also says فأما من فأما من تقولت موازينه فهو في عيشة الراضية وأما من خفة موازينه فأمه هاوية وما أدرى كما هي نور حامية الإمام الإمام القيرواني رحمة الله إذا مقدمة of his رسالة عقيدة بوك which is the introduction it was a matter of عقيدة الإمام القيرواني he's from the Ayma of Malikia he's from the Ayma of Malikia he was عقيدة وازوات it was عقيدة أهل سنة والجماعة he said وطوبع الموازين اللي وزني أعمال العباد he said that the scale will be placed and what will be put on it the day of judgment what will be put on it and the actions of the people the actions of that people will be put on it and as I said that's one of the things that are weighed the day of judgment that is what's weighed the day of judgment and also the body of the person I told you the hadith of the man who is going to be brought also the hadith of عبدالله عبدالله من المسعود when the Sahabas they laughed at him how he is what and they laughed at him إلى دقة ساقيه إلى دقة ساقيه how small and little his shin was when they looked they laughed at him and then the Prophet ﷺ he said أتعجبون من دقة ساقيه are you guys laughing and how small his shins are أما إنهما في الميزان الأفقال من جاب الإحدين the Prophet ﷺ he said for verily he's sacks which are very little in size are heavy in the in the scale of Allah ﷺ the day of judgment then the mountain of then the mountain of أحد so that shows you that people are going to be weighed the day of the day of judgment نعم also the houd the fountain والحوض المكرم the honorable houd the honorable fountain which our Prophet ﷺ is حوض it's for him عليه الصلاة والسلام in which ثورة الكوثر speaks about الله تبارك وتعالى يسأل إنه أعطينا كالكوثر فصل لي لي ربك وانحر إن شانئك هو الأبتر الله تبارك وتعالى he told us إنه عباس that he has given نبي الله محمد الحوض the kowther is his the Prophet ﷺ also told us himself إنه حديث صحيح البقاري من حديث عبدالله من عمر العاص that he الله gave him a houd and the Prophet described it for us صلى الله عليه وسلم he told us that that طوله the width and the length are both the same طوله كعارضه it's length and it's width are the same كل منها كل منهم each and every one of them is مصافة شهر it's a traveling of a month the Prophet told us أباريقه أعدد نجوم السماء the utensils are like the sky


Suppose p is a permutation of the rows of the identity matrix where pn vp is equal to a. We say that a and a' are orthogonally equivalent. Earlier we proved that isomorphic graphs have orthogonally equivalent adjacency matrices and conversely if the adjacency matrices are orthogonally equivalent then the graphs are isomorphic. And this gives us a way to easily determine if two graphs are isomorphic, provided we can easily find the right permutation matrix. But this is a hard problem. However, orthogonal equivalence is a special case of similar matrices. Suppose pn vp is equal to a, then we say that a and a' are similar matrices. The main difference is that p might not be a permutation matrix. But here's a key fact about similar matrices. Similar matrices have the same eigenvalues with the same multiplicities. Could this be useful? Let's find out. So before we start, we note that the adjacency matrix for a graph is square, symmetric, with real entries. And these are the required conditions for a powerful theorem about the eigenvalues of a. If a is a real symmetric matrix, all of the eigenvalues are real, eigenvectors for distinct eigenvalues are orthogonal, and a has an eigen decomposition that looks suspiciously like our formula for the orthogonal equivalence. So we might find the adjacency matrix for the graph and then find its eigenvalues. So our adjacency matrix will be, and we can find its eigenvalues using a complicated method of going online and using Wolfram Alpha. And this gives us our eigenvalues. What's useful here is that because all of the eigenvalues are real and there's a complete set of eigenvectors, then there are easy numerical methods for finding the eigenvalues. What if we switch the vertices around? So let's try a relabeling like, how about this one? Our new adjacency matrix is, and we find the eigenvalues to be, which are the same as they were before. Since isomorphic graphs have similar adjacency matrices, this gives us the following result. If two graphs are isomorphic, their adjacency matrices have the same eigenvalues with the same multiplicities. The contra positive then tells us that if there is a difference in the eigenvalues or the multiplicities, then the graphs cannot be isomorphic. For example, we'll find the adjacency matrices and eigenvalues for the two graphs shown. The graph on the left has adjacency matrix, and the eigenvalues are. The graph on the right has adjacency matrix, and the eigenvalues are. And since the eigenvalues are different, the graphs are not isomorphic. Or we can do the same thing for these graphs. The graph on the left has adjacency matrix, and eigenvalues. The graph on the right has adjacency matrix, and eigenvalues. And since the graphs have the same eigenvalues with the same multiplicities, they are isomorphic. Or are they? The graphs can't be isomorphic since one has two components while the other is connected. And so this leads to two problems. Suppose two graphs have the same number of vertices, the same number of edges, the same degree sequence, and the same eigenvalues. These are necessary conditions for the two graphs to be isomorphic. But even then they might not be isomorphic. We can't prove they're isomorphic until we find an isomorphism. So let's take a look at that next.


This happens when you go no contact with the narcissist. When you go no contact with the narcissist, you may feel guilty. You may regret your decision. You may miss the narcissist. And you may wonder if they miss you. But the narcissist does not miss you. They don't miss anyone. You may assume that they're missing you. Because you miss them. But they do not think or feel the way we do. Instead, they get very angry. Because they're so used to having you under their control. So they see no contact as unfair treatment. They see it as though you're doing something wrong. As though you're intending to hurt them. When you're involved with a narcissist. You become an extension of them. But when you start to think for yourself. You begin to separate yourself from them. Which will cause a narcissistic injury and narcissistic rage. But some narcissists will hide their rage. And instead, they will give you the silent treatment. Because they still see you as their property. And they believe it's only a matter of time until you come back to them. Because many victims do go back to them. They begin to think that it was their fault. But even if you don't go back to the narcissist. Eventually they will hoove you. They will send you a text message. They will come back to you.  They will tell you they understand. How much they've hurt you. And they will promise to change. And if you fall for it. From that point on they will only be looking for ways to punish you. They may already have a new supply. They may have someone else who they're involved with. But the reason why they're coming back to you. Is because their new supply is not as good as they expected them to be. Maybe their new supply doesn't have as much money as they thought. Or maybe they're not willing to put up with the narcissist. So they will declare their love for you. But it's just an act. They're playing you. Once you let them back in. You will begin to see who they really are. They will turn on you. They will punish you for going no contact. Because they don't really care about you. They're entitled. And they want to be in control. When you go no contact. It causes a narcissistic injury. It causes them to lose control.  To regain that control. But once they've done that. They will only devalue you and discard you. Once you've gone no contact. You must stay no contact. Don't give them an open door. Don't give them freedom of access. You will only regret it in the end. Thank you for watching. I hope this video resonates with you. Please like, comment, share and subscribe. If you would like to donate. My PayPal link is in the video description. Thank you for watching. I'll talk to you soon.


So, now I will go and start take you through the ok. So, we now calculate the partition function of the system and you can kind of you just look at it that we wrote down this ideal gas. So, ideal gas is the total Hamiltonian is the just kinetic energy ideal gas means a particle does not interact with each other. So, we do not have any potential energy if you do not have any potential energy then it is just kinetic energy and kinetic energy means it is just sum over the here as we shown in this equation here is just that now I want to get the so why that means that I want to get the partition function and as I have shown the partition function here is 1 over n factorial and then 1 over n factorial and h to the power 3 n then d r 1 d r n then d p 1 d p n e to the power minus beta h. So, this would be the partition function now look at this partition function the there is no interaction so position of the position of the atom. So, I have a monatomic like I have taken a monatomic gas of spheres and they do not interact with each other like an argon or your noble gas in load density and then interestingly if I do that then I can this particle they do not interact with each other.  So, I have essentially n n integrations identical n integrations where the product like these are just product. So, then I have to the power n and then I have h to the power 3 n here and so this is then my partition function where q is now given by this integration of d p x d p y d p z d x d y d z e to the power minus beta h this is the partition function of a single particle and then that I can now do because d x d y d z there is no in beta h there is no position.  So, they are again uncoupled. So, a 3 Gaussian integrals and that integral I can do. So, then I know root over this. So, this is 3 of them and each comes with a root then and there are 2 m. So, 2 m goes over 2 pi m k b t and k b t is there beta is 1 over k b t. So, when I do that I get this. So, this is the partition function of a single molecule in ideal gas this is the single particle partition function this quantity single particle partition function. So, that thing is done. So, then we can now go back and construct the next part next. So, now we got the partition function and the partition function q n is 1 over n factorial and then 2 pi m k b t by h square 3 n by 2 this is the partition function. So, now we know that from canonical ensemble we have done that in lecture I think 4th or 5th that free edge equal to minus k b t l n q this quantity. So, now we can put the q that we this q here. So, then we get minus l n this become minus l n n factorial and this become n this is q to the power n this is q to the power n. So, n l n q. So, and this is q n equal to q to the power n by n factorial that thing is used here. Now, we put the value of q there which we derived before. So, we get the l n n factorial starlings approximation is made. So, l n n factorial equal to n l n n minus n if we do that this minus n actually shows up here as f a and we already have v. So, q is already have this n 3 n by 2. So, I bring out that. So, n a I have n here. So, this only 3 by 2. So, so n comes out. So, n k b t here this k b t and q to the power n capital q is q to the power n that brings it n. So, n k b t l n then 2 pi m k b t h square 3 by 2 v this is multiplied by whole thing here v by n because I they separated out l n v and the o 1 is this comes out. So, this comes from these things you can just work it out the basic things which are you will find out what is fine. So, important thing to understand now that everything is in logarithmic. So, I have a logarithmic here.  So, that comes from the volume integral and so in l n that part is l n v to the power n. So, that becomes in l n v. So, now I can calculate from here the pressure I can calculate the pressure from here and that is by thermodynamic relation pressure minus d a d v k b t and then you can do that look at the let us do these things here. So, this is l n.    So, n k b t in front. So, n k b t by v by doing the derivative then p v by n k b t. So, this is the ideal gas law. So, now I have a microscopic derivation of p v equal to n k b t which was surmised by doing experimental enormous amount of experimental studies and also what comes out from Maxwell Boltzmann kinetic theory of gases, but here these no kinetic theory of gases no collisions with the wall no cross section that we have done you have to do the collisions of the wall with the kinematic description. We did not take do any kinematic description anywhere we did have a kinetic energy of course, but we do not have a kinematic description, but this is very important to understand that this is the pressure of the system of molecules which is contained in a volume v this is really magical this derivation. Next what goes on something even more magical and that is the yeah. Now, we have been something really very very important and this is something one of the major reason of reviewing this class once more and also we will do that in the in the in the in the in the in the diatomic gas also there are certain very far reaching consequences and we do not actually this kind of jumps that it comes from almost nowhere because we did not expect such a beautiful relation. So, early in our derivation or study of a such a major area of research sort of formidable area of research like in like in statistical mechanics which is actually as I told you again is one of the two major disciplines of our thinking our theory or understanding of huge number of phenomena very very first class I spent fully on describing that thing ok. So, now we have the partition function and now we have the free energy because we have the free energy a equal to we have this beautiful relation in now I know entropy because I know entropy by this I know entropy as equal to minus dA dB t I have a qn I know the qn because I know qn is 1 over n factorial h to the power 3n small q to the power n and small q I know is 2 pi m kB t 3 by 2 and volume v I know everything. So, I know qn and small q is the single particle and then I know free free energy exactly and then I know that I can now calculate entropy I can calculate absolute value of the entropy it is very important it could summarize cannot give me. Now, yes you can say I can get it from ideal crystal all these things but it is ok but you know everything have to be done numerically. Now I can do that and the beauty of this thing is now that if I do I take now look at this I show I take qn I take ln of this thing now look at that. So, if free energy has temperature in two places one is in temperature t one is temperature t this is one another place it has is in temperature t here in small q here. So, when I have to take these derivative I have to take care of this temperature and I have to take care of this temperature ok. Now, I do that first one is easy d a d t just this goes off I have kB t ln qn t v and this my qn t v great that is why 2 pi m kB t by h square all these things come in wonderful now I have to do the second derivative that now kB I do not hit kB with d t the kB t I come here and do the derivative here ok. Now, as I told you it is not q to the power n. So, n comes out I have a small q and I have to take time a temperature derivative the small q ok then ln q. So, then I do ln q then ln 2 pi m kB plus ln t I have a kB t sitting in front when I do that ok. So, now I can take that derivative I can do this derivative on this quantity and I can then I combine the terms this you can do yourself and it is left as an exercise you can do it yourself and you will get that entropy just do it yourself it just takes a couple of minutes then you get that this quantity this entropy. So, this part temperature part is ln q the ln is here and q is here 2 pi m kB t and that is a volume v this volume v is here v by n and so everything remains the same except e becomes 5 by 2 because one extra term comes in it 3 by 2 n comes because this derivative this derivative this 3 by 2 that comes out. So, then that t 3 by 2 here. So, when you take the derivative of that then 3 by 2 comes out and t becomes half the t becomes half then you have a t to the bar t it is become t to the power half you have the t here. So, you combine that t with the t that becomes half here after taking derivative. So, that become t to the power 3 by 2 again.  So, the main thing is to combine and you can get that term. So, this is the then this is an exact expression for an entropy a high degree non-trivial equation which is this thing that entropy of n number of ideal gas law is n k b l n 2 by m k b t 3 by 2 k b t by d.  So, it becomes k b t by p this is more useful because most of the time we want at a constant pressure not at constant volume and we know the number of particles and if we know and at temperature constant temperature constant pressure constant n. So, this more of an experimental thing then this is an exact expression of the entropy of the system beauty of it is the following the v by n that comes here when you do the calculations is a very important thing is very important to understand from that that what makes everything an extensive quantity. So, the entropy that I have here is just entropy is proportional to n as it should be everything else inside is an intensive property the temperature is intensive property pressure is an intensive property. So, the entropy is extensive because it is proportional to n, but this is a beauty of an equation it is called circuit tetrodéquation circuit tetrodéquation called a it is often called S t equation this is equation which is one of the. So, you are now the basic idea of my telling all these things again in great detail is to give you an idea how the statistical mechanics flows just like in quantum mechanics you do particle in a box you learn that how you solve Schrodinger time independence Schrodinger equation by putting boundary values and when you do that low and behold this sketch energy levels appear and then immediately find an use of that particle in a box explaining the spectra of butadiene or conjugate polymers. Then you go harmonic oscillator and low and behold you can read some idea of the vibrational spectroscopy rigid rotator you get rotational spectroscopy those are very preliminary very simple things that we solve time independence Schrodinger equation, but we get back huge amount of information as if without doing any work that you are getting such formidable and very fundamental things this is essentially analogous to particle in a box or harmonic oscillator we will use harmonic oscillator immediately now, but important thing to know that we are getting we are we are getting p v equal to n k b t we are getting an expression a beautiful expression of entropy and absolute value of entropy although ideal gas, but it will find lot of use in even today is research this equation finds a lot of use. So, next what we will go we will do the so one can now calculate the translational entropy this is called translational we have taken on the translation account so entropy the values of the absolute value of the entropy see in thermodynamics cannot give you absolute value of the entropy absolute value of the entropy this is the neon argon and krypton in atmospheric pressure, but then atmosphere pressure of course is probably not a but whatever condition is not written here I do not remember, but the experimental condition that we can calculate this translation into the particle then we can we can an experimental way of finding the entropy and it agrees agrees with the experimental values of this entropy of these three gases they are very similar to each other. So, now I just want to tell you why these things are so important and so what are the circuit retro equation one of the important applications of circuit retro equation of modern times is this quantity which is the very recently this has been used by several people that you want to calculate the you want to calculate that how a drug goes into mind order major groups of DNA and that is very important to know because of many chemo therapeutic drugs they go and bind to many other drugs in mind to DNA is the target of many therapeutic studies and so we need to know what is the binding energy of this molecule drug molecule this is now done in the following way that you consider your drug molecule is a drug molecule is going to go and bind to mind order major group of the DNA. So, now this has a lot of entropy in the ideal when it is in solution now in the solution we can treat it as an ideal gas if it is a single DNA molecule of a low density sorry single drug molecule or very low density drug molecule that goes and combines to these DNA. So, that we need to know the entropy loss of entropy of the drug when it binds there so we need to know the what is the entropy of this molecule in solution that is why circuit retro equation is used then you want to do for example gas liquid nutrition in a low temperature gas is going to go to bubble to form a bubble like in atmosphere to form a nucleation then one need to calculate how much entropy how much entropy the gas particles are losing in order to form the bubble because you know what is from the bubble it moves very slowly and it moves as a one particle. So, n number of gas molecules like water forms and goes from the water droplet then it is a huge amount of loss of entropy so that actually resist formation of the droplet formation of the cloud bubbles and in particular that is why high temperature when the water molecules are still there but they have lot of entropy and they will not form the bubble they will not nucleate they will not nucleate to form a water droplet which can form as a ring. So, entropy of these water molecules in the gas phase it is not just translation entropy this is a rotation entropy there is a vibration entropy we will talk of that so all those entropies this is a very important role so this is done by this gas liquid nucleation is a huge area of research and in the study of this nucleation against this entropy this in the nucleation theory again this entropy that we derived today finds out you. Now, finally to end this part that specific heat we know 3 by 2 R and that is very trivially derived now if you can go T it is C v T ds dT you go to circuitry equation do the ds dT then you will find that it just becomes 1 over N k B T by T that T cancels and you get C v equal 3 by 2 R and that is one of the P v equal to RT along with the ideal gas law C v equal to 3 by 2 R and C p is 5 by 2 R and that is because another way to derive is E equal to 3 by 2 RT that you know then dE dT is C v this is C v and that is 3 by 2 R that is the derivation that is the you know from ideal from the theory of gas but this is the derivation here coming from entropy directly with the entropy. So, it is the alternate derivation same result and is since equipartition theory is involved I cannot say it is more fundamental because equipartition theory is as fundamental as the gate but this is a statistical mechanical derivation with starting from partition function. So, this is about the ideal gas law what monotonic gas and ideal gas law that I wanted to do again in order to tell you the take you through a second time the things.


Welcome to Toffy TV, it is the match preview, Brighton Hovalbyn vs Everton at the Amex. Last year, somehow spectacular these days. We won 5-1 and yet, even on that day, I was not fully convinced we were going to win till the fifth one went in. Na, mi'n ifa. That was exactly the same. It was one of the highlights, well it was the highlight of last season in regards to football games that we watched and it would be great if we could do it again. You say highlight, it was highlight obviously when the game was finished and we won 5-1, but I didn't feel like it was a highlight when I was watching it. I don't think I've ever been so nervous. When they got one back, it was kind of like, oh here we go, good draw. But it was a great win last season and it's kind of hopefully this weekend we can pull, obviously we don't feel like we're going to be pulling that type of result off again but even if we get something, that may be a great result because they're a very good team aren't they? Yeah, it's a tough fiction as we saw the goodness in this season, 80% of the possession. I mean we were very unlucky, let's be honest, the goodness and it was a, even though they had a lot of the ball and they weren't really doing anything and obviously it was a wicked deflection that took her over Jordan Pickford with about 6 or 7 minutes left of the game. In a game that, well, it felt like we were just going to get over the line and sometimes they're the biggest disappointments, aren't they, when you don't? We just need to get back to the way we were doing things away from home, not that we've spectacularly gone away from them of late. But we could just do it one way or the other and go, oh I wasn't quite expecting that because that's the general feeling around I think at the moment is obviously a lot of worry, there's a lot of frustration with the point situation and the ownership situation. We could just do it, just go on. I was great that, really surprised by that and you know, just, it's that getting away from it. Yeah, exactly, and I think the fans need a bit of a lift and also the players need to, obviously we need to start picking up points and we've stopped picking up points. Even obviously the home game recently, Ain't Crystal Palace, which was really disappointing, but if you look at your own dashes, it's evidently away from home, it has performed really well. Like I was talking before about the Brentford game, I feel like we need that type of performance tomorrow, where we need to start, I think one thing I've noticed about having away from home is when they start quick, they get an early goal, Brentford away was like that. Obviously West Ham places like that, we've gone this season and picked up points, but obviously we're a really tough game. What is it, the very last one game at home this season? So is it West Ham and Newcastle, did they lose to Newcastle? No, they batted Newcastle, I think it's just one win, I think they lost to Newcastle since August. They've come into a bit of form, obviously they've got a bit of a break at the moment with the Europa League as well, they haven't got many games with that because they qualified straight through to the last 32. But I think we've just got to go there and make it difficult, but just try and stay in the game if it is, like if we are 1-0 down or it is 0-0. I think it's just important that we just don't lose our shape and try and nick something, but again it's going to be an extremely difficult game and it'd be one of those games where if you could walk away with a point you'd be delighted. I mean their home record is good but it's not amazing, they've drawn quite a lot of games and a lot of the time it seems to be that maybe when they've got a couple of games so obviously they're playing in Europe, but obviously they're playing the FA Cup next week as well.  I just wonder if he'll take one eye off and maybe just rotate it a little bit just because there's a couple of games this week. No he could because I've mentioned before on the daily it's like he's changed his goalkeeper 15 times this season obviously for Bergen and Steele and he hasn't really had a set of team all season, even if you look at the full back areas they haven't really kept a solid full solid team all season, they've been a bit all over the place at times and like you mentioned there he does like to tinker with his team so it's one of them where they could be one of them. They could take their eye off the ball, obviously with Wolves coming up, I think it was in the FA Cup next week. Because the reality is we need to, I think if they play their best and they probably nine times out of time they probably will beat us but we didn't need to have a bit of an off day hopefully and we just need to take our chances when we get them because the reality is we haven't been taking our chances. No that's the top and bottom of it isn't it? We don't have a goal scorer because I mean the court has been off for two months and it's just, we all knew the night he went out against Bay and Lee. We all worried and then to see him missing and it's not a coincidence we know our style of play is very reliant on him as a player playing in just behind Dom and away games has been huge and it's where we've picked up the most points of it. The thing about the situation for me is it can't become or I think it has become maybe for some people a crutch and going don't worry about the on-phone because we're good away from home but the reality is we play some tough sides now. So we've beaten teams that you'd go actually we should be beating those teams away from home but we're going up against tough sides now away and that is my worry but at least we have D'Corey Bach who's been a main part of it and can take the pressure of Dom on a Carvelling and add something that has been desperately missing in the two months since he's been out and he's only played two home games in that. Obviously when he came back in January against Vera and then obviously played on Monday night against against Crystal Palace so we haven't seen him since the Bay and I play away and it'll be interesting to see if that affects our changes things because we've been great at walls and obviously we've played like city. But the teams that were supposed to beat you know this is I mean I'm not saying we're supposed to be bright but people have got it in their head because they were there. Because we won there last season. Yeah because we won there and because he was a big part of us winning there last season so I think that is key and it's key because he's come back against Palace he's had that so I looked at Monday night and looked at his game to get back into it. He's come through that the manager's confirmed he's 30 he hasn't suffered any. So this is the game now he looked on he could make a big difference to us. Now he can he's I've heard you say many times he is key to Everton and he's key to all show and dice he wants to play and yeah and if you look at the stats the stats don't lie away from home he's our most important player and at home he's probably not as he probably doesn't do it as much but he's still very important. But I think for Everton to if you look at Everton's away form it's been basically reliant on the call. But from my point of view I think the money from here I think the team point of view they've got to start getting wins back on the board because what is it I think it's two months now isn't it without a win and you don't want to get into that habit. And this is the third season in a row that's happened as well. You know what happened under Benitez it happened under Lampard. You know we going through long stretches without a win and I face you so. That's under the start of the season. Yeah, yeah, yeah, but it's that middle bit isn't it that. It's it's like you think about it where in those in all three seasons you look at it and go well if we'd won a couple of games in that run the difference that would have made to us you know. Under Melisa's that's where we were dragged into the relegation fight. We were bobbing around 12 for ages and they just weren't winning. And suddenly we just like it looked like we just dropped like a stone you know we beat Arsenal under Lampard last year it was the same. It was just the same. It was just we'd be Crystal Palace away at home sorry and then we just didn't win a game and it was just piles on and piles on and piles on. You know last season with Dice he came in he obviously came in when that spell was happening for Lampard. And what he did at least was he was winning a game maybe every every three games or something. That's what we need at the moment you know every three or four games of where winning and we're getting a draw because we know we're good at the back. We just can't put the we just can't put the ball in the back of the net and it's just the difference. That's where that's where a Brighton they had a good start of the season. But now since that you know they just seem to be winning every three games or wherever it is. And that's why they are in that round the top seven and that's why they are still looking to get into Europe via the league. And they are in Europe because they just managed to keep ticking over and ticking over and that's where someone like us has got to get back to that. Just winning those because the difference it makes is huge. Now and especially with obviously what's going on off the pitch as well in regards to potential points coming back. The next couple of days potentially but we've been told 48 hours and then that's still not sitting here. But it could be a good lift for the players as well going to see if they could get something back in the next day or two. But yeah Brighton I think they've won the last four out of six I think it is. So yeah I think their aim now will be looking at having they got maniated in the four points off maniated or five points off maniated. So they're in a great spot of still getting having a really good season again. I think that was the worry we spoke a few times at the end of the season. The thing for them is can they back it up. But it's interesting Brighton isn't it because we have suddenly got this idea of what they are and who they are.  This is a team that have only had one good season in the Premier League. Don't be wrong they've built to it and I'm not saying there's like the one season wonders. They've built to where they are last season brilliant for them and then they sold like you know two of their most important players. And they've had injuries and if you know things have slightly changed for them and it's more games and you like the expectations gone off. And you think actually this is a really solid season for them and the idea that they're not doing as well as people thought they were is a great credit to them and what they've done. And it's weird how sometimes we flick that switch and they're not having a great run or this or that. Well they're coming off the back of a 5-0 win against Sheffield United you know where they obviously, making all gates obviously got into Sheffield United but he took the time and then got the goals in the second half and exploded. But let's just have a look at the team from last week for Brighton against Sheffield United. And as you can see the brilliant in goal Lewis Duncoff has been massively important for them on set pieces, he's hugely important. Pascal Growson there, Lam Tee. A Dengar was good. And Danny Welbech o front I think is massively underrated I think as a player massively underrated. But it's gross that I just want to look at for a minute. Let's have a little look at Pascal Growson's numbers, 23 games, 4 goals, XG of 5.3 hours, 6'9 and XA of 6.4. 8. Don't have to whisper in it. And as you can see his heat map there all over the pitch. It's funny because I've watched him a few times this season he's one of those players you can play anyway. Like he's played it left back for them this season, played it right back. Attack of midfield, right wing, left wing and this heat map obviously suggests that as well. He's just a very capable footballer, very good on the ball, very good on set pieces as well. So he's another one. He will come as well. I think he's 32 years old. He just looks dead easy for them when he plays. I think that's a good thing with Brighton. They've got a good mix of youth and experience.  What good pros haven't they? Milner and Llanar are really good pros for them. Like One Stuff at Big Clubs, James Milner, City Liverpool, England, Llanar, Liverpool. I'm sure that probably helps players settle down as well when they come into the squad. Some of the foreign lads. Perhaps the manager as well because the reality is that type of players will run the dressing room I imagine. The manager probably doesn't have to get involved as much. His job is to coach the team. Obviously he's done another good job this season. People start looking at him won't they? Now with the vacancy maybe Stan. Let's have a look at the staff pack from the game. This Saturday at the Amac Stadium, Everton played Brighton in the Premier League. The two teams have met in the Premier League a total of 13 times. Three draws between the two teams with four wins going to Brighton and six to Everton. Four of Everton wins came at Goodersham while two were at the Amac Stadium including last year's 5-1 win. In the reverse fixture this season at Goodersham Park, Everton and Brighton drew 1-1. Go far this season Brighton sits 7th. They have won 10 games, drawn 8 and lost 7. Everton sits 17th after a 10-point seduction. They have won 8 games, drawn 6 and lost 11. Everton this season have kept 8 clean sheets but Brighton have kept 3. Brighton's top goalscorer this season is Jal Pedro with 8 goals. Everton's top scorer is Adelaide de Corrie with 6. Leading the assists between the two teams is Pascal Groge with 9 assists. Dwight McNeill follows with 5. Everton's top the tackles list for all the clubs this season with 507 total tackles. Brighton has 17th with 407 although they do see a lot of the ball during the games. There you go. Good work from Ned Day getting all the stats for you. Obviously we've won the last couple of years there as well haven't we? Not just last year with a great win but the year before.  Frank never went there. Frank never went there. And he loved going back down south didn't he? So it's a bit weird. So what's your Mondays? Well, as soon as they get back home. I think our actual record down there, I think we've lost 2 since we've been in the Premier League. We had the very controversial 3-2 when there was a penalty that never was. Then there was one under silver, we got beaten. It was the game after we beat Burnley 5-1 wasn't it? Over Christmas that time. It was 2 days later wasn't it? So the record has been quite good there. But let's have a little look at the side obviously for everything I played on Monday nights against Crystal Palace. What changes do you think are needed? I mean I can see 2 immediately. Well I was saying before, I actually think he might play James Garner out wide. Come on. Give me your workings out. So I was talking about the Brentford game earlier on in the season. And he played McNeill on the left and he played Garner on the right. And he played O'nana, Decori and Garner again in the middle. And I actually think away from home that can work. Because I think that right side is going to be whoever is playing it right back is going to have a tough job with Matoma.  And I think James Garner, what I noticed as well when he does that he comes inside and makes it a bit like a compact midfield. I just got a funny feeling when I go with James Garner because again I know you mentioned that on your ratings. I don't think he wants to drop any of them. And I think that could be his best way of getting them all in rather than leaving O'nana off the team. Which again, I think O'nana has got a start on Saturday. Well the last time when we played McGullison we didn't have O'nana that day. And he played Garner and Guy in the middle of the park. And we only had 80% of the ball that day as well. He has got form for it. You know what, it wouldn't surprise me if he left O'nana out again almost looking at the situation of not having loaded the ball. But I think that would be massively counterproductive for wanting to play a certain way. I think if you're going to play on the counterattack you have to at least get on the ball and not rely on the long ball. And I think that's where O'nana, for me O'nana starts absolutely. And then for me you have to make the choices, the manager between Garner and Disagarner and Guy. And I think I said on Monday night that I thought he bottled it. I think I thought he bottled that decision. Now he'll tell me differently I imagine. But I just thought he bottled it. O'nana is the site because the easy drop because he's being out injured. But I just think certainly at home you need O'nana now. Whether you'll leave O'nana out again, I don't know how you can do that. I really don't. I just think he breaks up to play really well O'nana on his way from home. And he can get you with the picture of one pass where I don't think the other two can do that. I think there's almost this idea that he's like a luxury player and he's not. He's a great tackleer. You read the game well and as you just said there, it's passing get you on the go. And because he's such a good passer and his control is so good. It looks simple what he does, but it's not simple. Well he looked when he came on against City, he was probably everything's best player there. And he played that ball three to better and now he's offside. But he can do that, that's what I mean. He tried it a few times the other night, to be honest. O'nana obviously scored the goal. He was everything's best player in the whole game. Listen I think you've got to play your best players. It's tough on because Jack Harrison did he do enough to get his place back when he come on the other night, I don't know. I think he done enough to get his place back off Ashley Young. Now whether you're suggesting a tactical change, now maybe that's the strategy. But to me everything you've got to choose is right McNeill and Harrison. And the reality is McNeill will probably start you to set pieces because he's the only one who looks like that. No he is, he is a set. I mean that's the thing if you're going to play away and you're shown from set pieces, you have to look at it and go well who would replace Dwight McNeill? And what are the chances of scoring a goal from say Harrison going on the left and him crossing the ball? What are the chances of a scoring compared to the chances of Dwight McNeill putting a set piece in which he's done numerous times this season? And the answer you come to is Dwight McNeill will start the game and it's rightly so. Maybe if Dan Doom is available and you have a different way of playing you go on that maybe. But I just think for the set pieces having D'Cori in there, having Onara in there, it makes more sense. And I think that's where the manager looks at it and looks at the way it's all set up. Would you bring Harrison in for him? I'd have Harrison on the right because he's been left out of game now. He's been left out of game. I just think there has to be a way we play that isn't just long ball. You can go long ball with onara on the team, but at least with onara on the team you're not relying on long ball. I don't know if it'll be as bad as what it was on Monday, but we showed there last year what kind of attack inside we can be, a counter attack inside we can be. With the players we've got and there's not loads of changes from last year when we played there. You know it would still be Dominic Arventloon, it would still be D'Cori. You know it's still Dwight McNeill who had a brilliant game down there last season, maybe his best game in an event shape by the way. Everything just clicked that day and who knows it. I'm not expecting that to happen like it did last year. But Brighton do a form, Brighton do allow people to chances. So the manager's got choices, I just hope he doesn't take the safe option and just be like oh I'm playing too holding the fielders. No, I agree. I just think he needs a fresh in and up after Monday. Also it would be interesting to see if he keeps playing Godfrey at right back of Coleman's fifth as well. I think he will because of the pace of his home. He can get back on it, but it'll be interesting one. Yeah, everything could do with something this weekend obviously. We aren't in the bottom three as Luton don't play this weekend. And if we can get something this weekend then just again add pressure on. Cos the pressure's been on us, that's the way it feels at the moment. It feels like the pressure's on us. I think for us there's a way to feel it as well so even if we can get something as well. I think we could bring them closer into it and like to grab food. It's about getting closer to those people. So it's not just becomes a one horse, two horse race. You want to get away from that and obviously getting some points back. You would imagine we'll find out no later than next week. You would imagine as well. Get a result here, get some points back. And before you know it, the table might look a little bit better. Everyone feels a little bit better about it. So let us know your thoughts in the comments. Tell us how you're feeling ahead of this one. It's going to be tough, but we've proved that the last couple of years we can go there and we can do a job. So let us know your thoughts in the comments. Make sure to give this video a like, subscribe if you haven't already. And join us over on Toffy TV premiere for daily live exclusives. And no vids, well loads of vids, but vids without ads and podcasts without ads. The link is in the description, QR codes on the screen now. See you later.


The nature of war is never going to change, but the character of war is changing before our eyes. Modernization strategies focused on one goal, makes soldiers and units far more lethal and effective than any other adversary.


Red Lake Police and emergency personnel are still searching for two missing people after their boat capsized on lower Red Lake. According to the Red Lake Police Department, a commercial Red Lake Fisheries boat capsized along with three people who were on board. Law enforcement discovered the capsized boat shortly before 4 o'clock this afternoon. One person has been found and was sent to the hospital. A search and dive team continues to search the area where the boat was discovered. It is unknown at this time what caused the boat to capsize. As more information becomes available, we will bring it to you online on our website at lptv.org slash news and tomorrow night right here on Lakeland news. If you've enjoyed this segment of Lakeland news, please consider making a tax deductible contribution to Lakeland Public Television.


There are certain other factors that a financial analyst needs to consider during the process of capital budgeting. These factors include NPV profile, ranking conflicts between NPV and IRR and multiple or no IRR. So far as the NPV profile is concerned, it is a graph that shows relationship between multiple NPVs against the corresponding discount rates. In NPV profile, we see that the relationship between NPV and the discount rate is a downward sloping and the curve is convex from the origin. Dear students, we can see an NPV profile on the screen. This NPV profile shows that NPV against the discount rate is showing a downward trend or we can see that it is a convex from the below. This line has three distinctive features at 0% discount rate. We see that NPV is the maximum that is $34,000 whereas at a point where the discount rate is 19.52%, NPV is equal to 0 and hence 19.52% is the discount rate. Till this discount rate, the NPV is positive but decreasing and at the discount rate above than this IRR, the NPV is start becoming negative. So far as ranking conflicts between NPV and IRR is concerned, we see that for a single and conventional project, there exists no conflict between IRR and NPV as both the criteria is in agreement with each other. But when we talk about mutually exclusive projects, then these two criterion may not be in agreement and this disagreement may be due to two different factors like differing cash flow patterns of the mutually exclusive projects and the project skills. Now there is a golden rule that in such cases NPV rules the IRR. On the screen, dear friends, you can see a graph that is showing ranking conflicts due to differing projects cash flows. We have two projects, project A and B. We have an initial cash outflow which is equal followed by a series of cash inflows having different patterns. NPV of project A is 53.59 where is project B has NPV of 73.21 and project A has IRR of 21.86% and project B has an IRR of 18.92%. Now if we see these two projects independently, then both of the projects can be accepted as project A has higher IRR but project B has higher NPV. Now if we see the NPV profile of these two projects, we see that in the left panel we see dotted line that shows project B's NPV profile and a thick dark line that shows NPV profile of project A. There is a point at which these two lines cross each other. Now the point where NPV's of different projects cross each other, this point is called as a crossover rate. This crossover rate is 15.09% where both the projects have equal NPV of 27.98%. If we go deeply into this graph, we see that from 0% to the crossover rate, project B has higher NPV over project A but beyond this crossover rate, project A has higher NPV than the project B. Now on the screen, we see cash flows of two different projects having different skills. Project A has initial outflow of $100, project B has initial outflow of $400. So we can say that project A is smaller than project B. These projects have future cash inflows. Now project A has an NPV of $58.49 whereas project B has NPV of $138.88. Project A has IRR of 34.9% whereas project B has IRR of 25.21%. If we see these two projects as independent one, we can choose both of these two projects as project A has higher IRR over project B and project B has higher NPV of project A. When we see these NPV profiles of these two projects at the left below panel, we see that project B's NPV profile is shown through thick dotted line whereas project A has an NPV profile depicted through thick flat line. The point where these two projects have their NPVs crossed each other, the point is termed as crossover rate. This crossover rate is 21.86%. At this rate, both of the projects have equal NPV. Now before these discount rates, project B has higher NPV over project A and beyond these discount rates, project A has higher NPV than project B. NPV assumes reinvestment of the future cash flows at the discount rate and not at the IRR. In fact, IRR is always unknown. IRR assumes no investment of future cash flows at the same rate throughout the life of the project. If the future cash flows can be reinvested at the IRR, then there is no need to use required rate of return as the discount rate. In this case, we see that NPV criteria uses the most realistic discount rate which is the opportunity cost of capital. Another benefit to using NPV as the project DCN criteria is that the NPV shows the gain earned by a project in terms of currency units and not in terms of percentage relative to the invested capital. The IRR gives a rate that might not last for the whole life of the project. Therefore we see that NPV is theoretically a sound DCN criteria. There appears another problem with reference to the IRR and that is we might have multiple IRRs or we have no IRR. On the screen, we see that we have a multiple IRR problem in which we have a project that has two different signs, one at the time of zero. The project has negative cash outflow as an initial cash outflow. Then at time two, the project has another cash outflow that is showing negative sign. In this case, we might have multiple IRRs. We see the NPV profile, we see that we have an IRR at first that is 100% and then we have another IRR that is 200%. If we observe this NPV profile, we see that from 0% to 100%, the NPV is negative. At 100%, it is 0 from 102 to 100% between the two IRRs, the NPV is positive. And from the second IRR onward, the NPV is showing a negative value. What is the solution of multiple IRR problems? We have a solution to this problem and the solution is the modified IRR. Modified IRR is the rate that equates the present value of terminal value of the all cash flows of a project to the present value of cash outflows of the project. In other words, we can say that we compute a present value of all terminal values of the project and we equalize this value with the initial cash outflow. Now how we compute the present value of terminal values of the project cash inflows? We use a VEC that is weighted average cost of capital or cost of financing. We use this VEC to discount the future cash flows of the project. And then we use another rate and that is the modified IRR rate. We use this rate to equate the discounted terminal value with the equality of the initial cash outflow. In this way, the modified IRR avoids the problem of multiple IRR. We see that an example where $100 is the initial outflow then we have future cash inflows of $10, $60 and $80. Now we grow, we discount in fact the future cash flows at 10% which is the VEC and in this way we have a terminal value of $158.1. We have now and cash inflow cash outflow of $100 and when we equalize these two values using a certain discount rate, this discount rate is termed as modified IRR. We may have situation where we may find no IRR for the project on the screen. Here students we see that we have a cash outflow of $100 then we have cash inflow of $250 at time 2 but $300 as a negative cash inflow or outflow for the period 1. If we see the NPV profile of this project, we see that we have a 0% to 400% and we have nowhere a 0 NPV during this discount range. In other words we find there is no IRR for this project but if an NPV profile has no IRR is there a problem or this NPV is of no use. We note an NPV profile with no IRR means that there is no discount rate that can transform this NPV into a 0 NPV. So this another mean that project has all positive cash inflows for the days to come. In this case there exist no IRR for this project.


This is the interface to the carbon cycle model that you'll be working with in this module.  It's a complicated looking thing down here, but all these boxes are just places where carbon can reside, including the atmosphere here. And the amount of carbon in the atmosphere determines the CO2 concentration in the atmosphere, which then affects other parts of this. Let me point out a couple of things about this diagram. One is that there's a fossil fuels reservoir here that includes a flow called fossil fuel burning, FFB, that adds carbon to the atmosphere. And this is controlled by different kinds of emissions scenarios that, you know, dictate the amount of carbon released at each year through this process of burning fossil fuels. It also includes a reservoir for permafrost carbon. So this carbon is stored in permafrost, but if the permafrost melts, then that will be added to the atmosphere here. And that's controlled by a switch that we can either turn on or off to consider the effect of that. The CO2 concentration in this carbon cycle model then feeds into a climate model. You can see the edge of it over here. It's the same climate model that we worked with before, but so there's a connection between these two. And then this determines the global temperature change that then in turn affects the carbon cycle. So these two are linked very closely to each other. Now let's see how you operate this. This carbon cycle model will allow you to impose three different emissions scenarios on it. One is the business as usual scenario. If you click on this, you see what that scenario is, just an increasing emission of carbon over time. It begins in 1880 and then ends in 2100. So it goes into the future a bit. Here's another emission scenario, one that involves moderate reductions after some period of time here. And then finally there's a third emission scenario here. It's shown like this, more drastic reductions through time. So you can activate those three different scenarios by using these two switches here. So the way it comes right now, it will impose the business as usual BAU scenario. That's A2 here. So if I run it, it's going to impose that emission scenario on from 1880 up to 2100. You see the temperature change. And you can see the CO2 concentration and atmosphere rising up here at the very alarming levels. If you click again, it shows the pH in the oceans going down, down, down, getting more acidic. And it shows how much of the carbon stays in the atmosphere, the airborne fraction, how much goes into the oceans, and how much goes into the terrestrial reservoirs. Click again, it shows the history of fossil fuel burning in detail from that emission scenario. And then through here, this shows the permafrost melting that's turned off now so that it didn't do anything. And then this graph and the next one show just the amount of carbon in terms of gigatons in those different reservoirs over time. And this shows the total anthropogenic change, which is the cumulative effect of burning fossil fuels and having land use changes. So if we wanted to impose a different scenario, we would turn this switch off. And now we're looking at one of these reduction scenarios. If I have this switch down like that, it's going to implement the A1B scenario. And if I click this, there it goes. And then if I click this switch, it's going to impose the B1 scenario here. So if I run it, you'll see that take effect. All right, so that's how you activate those different scenarios. If I want to turn on the permafrost melting, I would hit this switch here. Now, as the temperature rises up above a certain level, once it gets above one degree, then the permafrost melting begins. So I'm going to turn that off here.  In this case, that's the standard scenario. So that's the essence of how this carbon cycle model works.


Så det turns out that those peaks in the energy landscape are important because they will teach us about how fast things happen. Let's look at little on that from another point of view. And I'm going to be looking at processes where I specifically consider how long it takes to get over the barrier, not just whether we get over them. So let's assume that I have a process that happens along some sort of reaction coordinate going from an initial state zero to an end state one. And here we have some sort of free energy, we can be physicists here, F. There is a starting free energy here, there is a barrier and there is an end free energy that's lower, so I would like to be in the lower one. But for this to happen I first need to get to cross the barrier here. And that barrier relative to the starting state has a free energy barrier that says delta F hash. You can call it anything you want. Now, it's not entirely trivial to calculate how fast that process will happen, but I can reason about it if we think about orders of magnitude and don't worry about the details. If I have a total of capital N molecules in my system, at any point in time the fraction of molecules that are going to be up here relative to the total amount of molecules in the system. Well, let's call that N hash divided by N. We already know that relative populations don't require a constant and this corresponds to e raised to minus delta F hash divided by kT, right? Plain and simple. Once we are at this peak, half the molecules are going to fall down to the left and half to the right, so some of them will now be at the end state, that's great. In principle I should include this factor too, but there will be worse approximations I make, so I will just say that if we are up here, they are all going to fall down in a tau, time tau. Time, tau reaction, time for a single molecule. How long that is depends. If it's diffusion it might be nanoseconds or so. If it's galaxy formation, well, then you should ask somebody else. And I would now like to know that how long does it take before this N hash might be tiny here, this might be one in a billion of molecules. So how long does it take before roughly all the molecules or at least half of them or something have reacted? Well, I can calculate that. If I have N molecules in total and each time this happens N hash of them have gone up. The time it will roughly take for all of them to go through that change would be say time t or t0. So that would then be roughly N divided by N hash because N hash would go through in each batch and each batch would take in the ballpark of this time tau. This is then simply the time for the elementary step multiplied by e raised to in this case it's going to be plus delta F hash divided by kT. This makes a lot of sense and this is also the reason why I don't worry too much about tau. If I look at the same process but this barrier is suddenly a factor too higher, I know here roughly how that is expected to change the time. Or if I make tau significantly lower, sorry delta F significantly lower. I can also reason about this relative to the temperature scale I have. If the free energy barrier here delta F is significantly smaller than kT. Remember what was that at room temperature? 0.6 kK per mole. So if delta F is significantly smaller than 0.6 kK per mole that means that these barriers are just going to be like gravel in the road. I can just bike over them, I won't notice it. But on the other hand if the energy barriers are significantly larger than kT then it's going to be like brick walls. They will really stop me and it will take a very long time to go over them and they're going to dominate my process. And armed with this I can actually at least estimate roughly how fast reactions will happen and I can also in particular compare different reactions when the tau is roughly the same. The books goes through some details to argue what tau should be for chemical reactions based on diffusion and it can be instructive to follow that. But I'm going to do something else. I will directly move to slightly more complicated processes. Because let's face it, there are very few processes in life that are so simple that you just have one step. What if I have a process that first is one step and then a second step we need to go through. Well, in this case first I have to go across the first barrier here. And then you might think that the second step can start from this intermediate point. But no it can't. It has to start from the starting point. The reason for that is that once I'm sitting here and you keep adding energy, what's first going to happen is that I will fall down here again. But if I just use those as delta F1 and delta F2. Then I can just adapt the reaction times. Because first my molecules will have to go through F1 and then they will have to go through F2. Technically you can make this more complicated by considering backflow and everything. But trust me it really becomes that simple. This will of course get super complicated if you have 55 reactions here. But look at the exponential dependence here. If I have two barriers and one is say a factor of three higher. It turns out that the higher barrier is completely going to dominate that. That's going to be the rate limiting factor, the time limiting factor here. So compared to the highest barrier, if the other ones are significantly smaller and I need to go through them sequentially. I can forget about the other ones. Only the highest barrier will matter. So I can approximate a multistate sequential process by only the highest barrier. And then I can use that very simple expression. Which is of course the reason why we are introducing it. Now all processes are in sequential. What if somebody tells you that there is free candy on the other side of the building or so. So half the students might take the path to the left. Unfortunately you will meet the dean there. So there is going to be a high free energy barrier there. While the other half of the students they will take the path to the right. You are lucky. You will only meet the vice dean. So it's not quite as much as an obstacle. But in this case both processes can happen in parallel. And then we are going to need a slightly different way to treat that. And what we do then is that we introduce something called reaction rates. Which is just one over the reaction time. Literally, it's very simple. So the reaction time describes that it would take this molecule one millisecond to go through a barrier. The reaction rate would then say well that means that there are 1000 molecules per unit of time that go through the reaction. And as for a serial process sequential you add up reaction times. For a parallel process you add up rates. If 50 students per unit of time go to the left. And another 100 students per unit of time go on the right. There will be 150 students in total getting to the candy. And this too becomes simpler. The lower the barrier is the higher the rate. And that will go as e to the minus exponent right. So there will be a very strong dependence that almost all the reactions will happen across the lowest barrier. Can you imagine any way of simplifying that? Yes, we simply take the lowest barrier and ignore the other ones. And we can simplify that too with one of these so-called first order processes. So to sum that up, for a sequential serial process we can simplify it by just considering the highest barrier. For a parallel process with many paths that are independent of each other we can simplify it and only study the lowest free energy barrier.