الحمد لله رب العالمين له الحمد الحسن للسناء الجميل وشدوى الله إلا الله وحده لا شريك له يقول الحق وهو يهدي السبيل وشدوى نسيتنا ونبينا محمد صلى الله عليه وعلى آله وأصحابه والتابعين لهم بحسان إلى يوم الدين أما بعد. last lesson in our شرحة of the science of Quran I mentioned the last point the objectives that I want to achieve from this module or this course what did I say where is the objectives and the aims that I had I mentioned three does anyone have their notes beautiful did anyone hear can somebody say it loudly as well when I was talking about the aims and the objectives that I want to achieve from this course إن شاء الله تعالى المدخل تعلموا القرآن what did I mention yeah my aim and objective is three things remember I said three things number one was what التعرف على العلم to define terms that we're going to give a lot of importance to definitions and knowing the meanings of words. هل يمكنك سيرات؟ so إن شاء الله تعالى today we're going to define revelation so today we're going to define the following today's class is going to be on these four points إن شاء الله تعالى we're going to do تعريفه تعريف الواحي إن شاء الله تعالى we're going to define what revelation is we're going to speak about its types إن شاء الله تعالى we're going to speak about the way that the revelation used to come down and إن شاء الله تعالى we're going to we're also going to refute the doubts that have been brought regarding the revelation of the Qur'an today's class is going to be these four points إن شاء الله تعالى when we leave you should know the answer and you should have an understanding to these four points number one is تعريفه الواحي what does وحي mean? what does وحي mean? what's the revelation? second is the أقفان the revelation how much is it categorized into? what are its types? the third one is كيف ياته? its way how do the revelations used to come down on the Prophet ﷺ? and the fourth one is ورد الشرح التي أفورت حاولة we're going to refute the doubts that have been brought regarding the وحي the revelation number one we're going to define the revelation تعريفه الواحي we're going to define the revelation we're going to speak about the types of revelation there are now are we all together? number three is how the revelation used to come down we'll speak about it إن شاء الله تعالى and the fourth one is if we get time this is نافلة القول it's not the جبدة and it's not the صوب المدوح but we'll try to respond to some of the doubts that have been brought regarding the revelation of the Qur'an okay وثيرات حاوله that's been spread regarding it what does the word وحي mean? again we're going to define it in the لغة and we're going to define it أصطلحن again we'll do that what does the revelation لغة أمي and what does it mean? أصطلحن the word وحي the revelation in the Arabic language in which the Qur'an used it in the Qur'an used it sometimes the word وحي in the linguistic meaning and it is four or three شاء الله تعالى three the three uses does that language have the Qur'an uses it and this is the linguistic one not the technical one the Qur'an uses the linguistic one and he also uses the technical one the first one is الإلحام الفطري الإلحام so I'm just going to write this one up here الإلحام الفطري we'll explain that in the Qur'an الإلحام الفطري the second one is الإلحام الغريزي the third one is الإشارة السريعة not three of those Qur'an used it you guys are going to have to help me here with the translation in English and الإلحام means to be given what the understanding of something what would be the best definition for the word إلحام in the English language inspiration any other word any other word inspiration any other word okay let's use that last one word I could also think of الإلحام الفطري means innate inspiration لكن شكل سامطه بإنكتر النبوات he mentions that الإلحام الفطري if they want to give it to the person الإلحام الفطري is Allah سبحانه وتعالى inspired a person innate he to do something it's an innate thing it's something that's natural إذا الله سبحانه وتعالى قرآن وإلحينا إلى أم موسى أم أرضعي وإلحينا أعوش سمطه أم مضى أو ماذا موسى يعني الله يقول إليه موسى مضى بإنكتر النبوات that is in every mother to breastfeed her child is it something that's in every mother huh does that make sense look what Allah said وإلحينا وإلحينا is وحي إلا أم موسى أم أرضعي الله says we gave إلحام الفطري to the mother of mousa to do what that doesn't mean revelation is coming down on the mother of mousa because revelation does not come on any woman صحيح شخص سمطه is this إجمع even if it's not against it that it doesn't come down on a woman so وإلحينا إلى أم موسى huh intuition now صحيح الله gave mousa's mother the understanding to breastfeed her child she did that so this is إلحام فطري the difference between إلحام فطري and إلحام غريضي إلحام غريضي is for the animal is for the animal if the innate ability has given to the animal this is given to the human and this is given to the animal and is the one Allah says وعوحة ربكة إلى النحلي الله has sent down he gave إلحام غريضي to the bee الله يزوى الوحي for the bee yeah the second language the third one is الإشارة السريعة not specifically but is done on the hand ونصمري بس جشة on the hand as Allah has sent in the ayah فخرج Allah has sent خرج على قوني من المحراب فإلحى إليهم وانسبحوه فإلحى هي ميت أي وأشارة سريعا to use your hand and gesture to somebody افكري الله used it as the word وحي أولي تجدر to use a which one the number one and number two are the same the distance between the two is the first one is for the humans and the second is what for the animal the bee is the animal right does that make sense the third one is الإشارة إشارة is to what is to gesture right الله has sent فخرج على قونه he came out on his people ونبلأ يذكرية came out on his people and what did he do فأوحى إليهم اي فأشارة إليهم he gesture to them وانسبحوه to exalt the second one is إشارة إشارة is to what is to gesture تغيرتم سبحانه وتعالى إذن صورة مريم صح إذن آية صورة صورة مريم these are all the linguistic usage تستثمين وحيلة وطوكن about الان وحيلة وطوكن about نمذذة وحيلة the one that comes down on a bee and the one that comes down on the mother of the mother of the mother of the mother of the mother of the mother لكن إستلاحه one is what there's so many definitions each definition there's a so I'm just going to say the easiest one for you now are we all together brothers even if it's not a comprehensive definition don't worry about it as time goes on we will start working on finding the best definition إن شاء الله تعالى تجرى لكن the easiest one for you now is the one that إبن القي منشز رحمه الله which is هو كلام الله استلاحاً أمرب this one now استلاحاً رب الوحي يميز يميز كلام الله باستيشة الله number one المنزل that was sent down على نبي من أنبيائه it's sent on a prophet from the prophet of Allah so it's three things it's first of all كلام الله كلام الله it's speak of Allah second is what المنزل it's sent down three who is it sent down نبي من to a prophet from the prophets this I make sense that's هو حي استلاحاً when we say كلام الله when we say the speech of Allah what enter there yeah the Qur'an and the sunnah when we take the sunnah in there is no is the sunnah not the speech of Allah it is sunnah is the speech of Allah من حيث المعنى لا لا من حيث الله the meaning is Allah's كلام سبحانه وتعالى the meaning is speech of Allah not in terms of وادي and we don't go too much into that now i'm confused you're wrong there is the speech of Allah المنزل that was sent down on a prophet from the prophets so everything that was sent down to نبي الله آدم to نبي الله محمد all of it it's called the way it's called the way it's called the way so it's كلام الله it's speech of Allah المنزل that was sent down and it was sent down to a prophet from the prophets so الحمد لله we've defined يحي مغتن واستلاحاً to technically we defined it and technically we have defined it true or false we did right good now i'm going to move on to its types have i defined it its types is what it's two the أقصامه is what it's two and i really gave it to you الحمد لله it's the لغة and the thing that came out of it and the third one which is استلاحاً فيوانا كولئ اما شرعاً فيوانا كولئ the types are what لغة and الشرع that's number one and that's number two these are the two types of revelations the linguistic type of the linguistic one and the what and the technical one those are the two types of revelation are you all together does everyone understand that okay now we're going to go into the third one which is كيف يأتره how the revelation used to come down the revelation the way that it used to come down is three ways as الامام البغاوية he mentioned رحمه الله when it came to the ayah ما كان لبشر ان يكلمه الله when it came to the ayah الامام البغاوية we're going to mention them he mentions that the revelation the way that he used to come down is four how did the revelation come down three ways the first one is ما يقومه منامم that which used to be a sleep was a dream the second one is ما يكونه الامام this is the one which is confused with the illham of the first one or the illham of the second one the illham of the fourth one no this is the illham of the third one okay it's not the illham of the fourth one the third one is the way we touch on each one ما يكونه ما يكونه مكالمه بين الربي والعبض ما يكونه the fourth one is ما يكونه بواسطة أمين السماعة pay attention here إن شاء الله تعالى then you can write it the way that he used to come down is four ما يكونه منامم it would be a dream it's perfect it will be sleeping and it will come down to him as a dream that's one ما يكونه الامام that which was an illham الله تباركه تعالى will guide the prophet to do this will inspire the prophet to do this this is the شرعي one it's not fitri to everybody it's not ghezi which is lowery this is شرعي meaning the illham that the prophet get it's something that the people have to do does that make sense number four ما يكونه مكالمه the prophet will actually be speaking to بين الربي والعبض الله will be talking to the prophet and the prophet will be talking to Allah and they will be taking directly from what الله عزي جله the fourth one is ما يكونه بواسطة أمين السماعة جبري will come down on what huh جبري will come down to the prophet and will give him the message and as ibn tanya mentioned and ibn jama'a and suwati and others that the one that the Qur'an had is this one the Qur'an is only this one the fifth one are we together about this the Qur'an ما يكون بواسطة أمين السماعة the Qur'an is the fifth one our Qur'an is one this one the other religions they had the other wasa'ita the other means are we together the Qur'an came down to the prophet صلى الله عليه وسلم the revelation of the Qur'an came down بواسطة أمين السماعة وستأذن السفرات الله سد وإنه يلتنزير رب العالمين نازل بجره الأمين على قلبك لتكون من المنجلين برسان العربين مبين that the Qur'an was down three degrees كريستوكلي are we together there are disputes amongst the scholars each point can be responded to but we're never going to that the second one is ما يقوله الهامن the prophet will be inspired to do something he will be what he will be inspired to what to do something this is a revelation ما يقول من مكالومة بين العبد العرب the slave talking to Allah dialogue between prophets the prophet saying أم الله عزيجل لكن يقوله النوثة الله سر يدي ودي الله سلم القرآن وكلّم الله النوثة تكلمة أقول لبس ثور أقول ثاني الصورة صورة الشيرع اي فور عاية 51 عاية 51 صورة الشيرع الله منشل these four types let's go over it المميذ بغوي المميذ بغيز كرام I'm going to tell you he builds all four out of this he says the first one is what where's the first one I mentioned yeah where's the first one from the fourth المميذ بغوي فرد the third part of the ayah the ayah is long we're going to read it all the first part is وما كان لبشر أن يكلمه الله إلا وحيا he said it's manam and it's also ilham these two are the first part of the ayah these two are what ما كان لبشر أن يكلمه الله إلا وحيا that's the first part read it together ما يقود مكلمه بين العبدي وربدي he says الله سبحانه وتعالى he says it's a passage to me that's the second one it is قبي ورائح جاب that's it it's the third one right and the fourth one is قومور سيلة رسولة أو أو يور سيلة رسولة هل ترى كيف يقومون بشكل جيد في one ayah so the whole ayah that one ayah so i51 is where it's a shura it tells me the fourth part in there who extracted it right the name that's important الإمام and إمامة بغري extract it but all those four are in that one ayah does that make sense brothers am i making sense yeah so we mentioned which one is the Quran yeah the Quran is جيبريل we mentioned that الله mentioned in my place إمامة وتنزينه رب العالم ونزل بهروف للأمن على قلبك إمامة لدى أعلى الله ستقول مفزله ويوف القدر القرآن came from جيبريل beautiful now i want to talk about something important which is who came who gave it to النبي الله محمد the Quran who put it in i just mentioned who جيبريل where did he take it from this is also spoken about when revelation is spoken about جيبريل gave it to نبي الله محمد صح لكن where did you get it from yeah there are two مدهب in this issue i want you to take this down can i rub this off there is two مدهب in this issue okay there are what two مدهب the first مدهب they say جيبريل took it from what لوح المحفوظ and that is the كلام of أهل البداح the speech of the innovators not the speech of أهل السنة that he took it from لوح المحفوظ meaning جيبريل read لوح المحفوظ and then he came to نبي الله محمد and he gave it to him this is not the معتقة of أهل السنة are you with me does that make sense they say that جيبريل went to لوح المحفوظ he read لوح المحفوظ and then he came to نبي الله محمد and he gave the قرآن to him this is not what أهل السنة والجماعة believe so what is it that they believe they believe تلقاه جيبريل أما جيبرائيل those ways it can be said تلقاه جيبريل جيبريل he took it السماعة من الله هارف من الله جيبريل هارف من الله جيبريل هاد الله صوتاً the sound and he had the hearth the wordings أما ده ألفاظه are you with me about this أهل السنة والجماعة believe he had it جيبريل from الله عز و جل and he gave it نبي الله محمد as well and محمد صلى الله عليه وسلم نبي الله محمد صلى الله عليه وسلم he took it from who جيبريل does that make sense so if you say reason why because they don't want to accept that Allah speaks are we all together so this مدهب is not a مدهب which is مدهب أهل السنة this is the مدهب we believe which is that جيبريل took it from who question جيبريل when he came to and how did do i want to speak about that now okay جيبريل used to come in two forms take this note note this down as well جيبريل used to come into what two forms what are the two forms that he used to come in he used to come in the form of himself and the second one is that he used to come in the form of a human the form of an angel he will come and he brings the Quran and he will never come in the form of a human note this down it's important information جيبريل when he would come to the messengers صلى الله عليه وسلم to bring the Quran he was in his form are we all together brothers if جيبريل came he would come in what in his form and he wouldn't come what and he wouldn't come as a he wouldn't come as a بشر because if he did Allah said in the Quran that the the Quraysh and the disbelievers what did they used to say إنما يعلمه بشر human is teaching him this they used to accuse the Prophet ﷺ of taking this knowledge from a human and Allah said a human did not teach him this so جيبريل did not come in the form of a human are we all together brothers does that make sense لكن he would if he was bringing knowledge of the حديث and the سلح yes because of the famous حديث بينما نحن جلوسنا عند رسول الله عليه وسلم حتى جلس إلى النبي صلى الله عليه وسلم فأسنا درقبتي إلى رقبتي ووضع كفية على فخيدي وقال يا محمد أخبيرني عليه الإسلام وثم with him this when he came and when he would come in the form of the human he would come in one of two forms one of two forms either a bedouin or he would come in the form of the companion ديحة الكلبي are we all together so جيبريل would come in his actual form when he was being in the Quran as an angel pay attention to that because he can't come in a human form if he did then he would go against the آي إنما يعلومه بشر as humans teaching him this and the human was not teaching him this does that make sense and that is the جيبريل come in his actual form the valuation and the glorification of the Quran but if he used to come with other valuation other than the Quran which is the sunnah of the valuation right yeah if the sunnah of the valuation وما ينتقع عن الهواء إن هو إله يحييها حافظ الحكم يدزيسي فالسنة النبي is another valuation عليه ما قد أطلق اليحياني وإنما طريقه الرواية ففتقر الراوي إلى الدراية the sunnah is a what فالسنة النبي is another valuation but when he came with the sunnah جيبريل he would come in the form of a human okay he would come in the form of a human sometimes he would come as a Bedouin عرابي or he would come in the form of the companion ديشة الكلبي ديشية and if you go to سير عالم النبلة بايمان الزهبي he mentions that ديشية the reason why جيبريل used to come into his form because he was a very handsome man two companions were the two most handsome companions of the prophet ﷺ the first one is ديحية and the second is جريرينا عبدالله البجري جهور جريرينا عبدالله البجري جريرينا عبدالله دهدر عمال سير عالم النبلة he composed between who is handsome not handsome he talks about the narration of the lakwal لها بي تكسبها ده read it إن شاء الله تعالى okay how was the revelation when it came down in the message of ﷺ how is his situation read it together this is the last part that we want to speak about إن شاء الله تعالى بإذن الله الكريم which is how was the message of ﷺ when the revelation will come down on him because a hadith that summarizes how the message of ﷺ was when the revelation will come down on him عيش رضي الله تعالى عنها she said أن الحارث إبراهشام حارث إبراهشام he came to the messenger حارث إبراهشام he came to the messenger ﷺ and then he said to him كيف يأتيك الوحيو so this is how how did the revelation used to come to you how so he asked the messenger ﷺ and the messenger ﷺ he said أحيان يأتيلي في مثل صلصلة الجرسي the messenger ﷺ he said that the revelation will come down to me sometimes as the reading of the bell so the first one the person I mentioned was it will come to me مثل صلصلة الجرسي sometimes it comes to me like the reading of the bell the person who I said أشدبه عليها the one that was the hardest on me was this one so let's stand over this one that's the first one حالة الوحيوة the first situation in which the revelation used to come down and the messenger ﷺ is what yeah the first situation حالة الوحيوة it was what the first situation مثل صلصلة الجرس it was like a ringing and the prophet ﷺ he said this one is the hardest one for me it's like a bell noise big noise would be made and Imam Surah I mentioned the reason this noise would be made so the prophet ﷺ be awake fully focused this one was the one that the messenger ﷺ if he would be on his riding beast as Imam Ahmed said in his narration that if the prophet ﷺ was on his riding beast even the riding beast would be moved by the when it came down on him like this the neck of the animal would be pulled they would be moving from the bottom the animal the riding beast camel the horse whenever he was on the animal would shake from beneath its float they didn't react but he said رضي الله تعالى عنه that mercy if this type of revelation came down on the prophet and he was sleeping on my thigh his head ﷺ will become so heavy that it felt like my legs was breaking that I could never look at my legs ever again عايش رب الله تعالى عنه and she said that if it was a cold day he would sweat from this one excessively ﷺ so the first one is مثل سلسلة الجرسي the very high noise in another riding and another riding it mentions the حديث of أم رضي الله تعالى عنه it was دهي كدهي النحل it wasn't a bell it was like the noise that the bees make what's that called the buzzing هل this is humming we use buzzing less buzzing sounds better so the buzzing of the bee it was that noise but it was very strong this was the harshest one for the Prophet ﷺ are we here together the second situation was in which the حديث mentions is sometimes the angel will come to me in the form of a what he will come to me in the form of a female bee he will come to me in the form of a female bee so he will speak to me and I will understand what he was saying to me and this we mentioned already it's not referring to the Qur'an this is not referring to the this revelation is not referring to to the Qur'an because the Qur'an he will come to him على ملكيته as an angel not to the human being I think we'll stop there Inshallah I'll take questions and answers بإذن الله الكريم we'll leave the shepherds for the next lesson بإذن الله الكريم any questions that anyone may have فضل so the Prophet ﷺ what did I say the Qur'an came down from him جيبيل but did every single time جيبيل come to the Prophet ﷺ do you think the Prophet ﷺ I think with the Qur'anic revelation and the Prophet ﷺ on جيبيل was it every single time جيبيل came to him the scholars they said لا جيبيل will come once a year to the Prophet ﷺ when he came to preparing the Qur'an with him are we together brothers are we together once which was in a band and he said the last year the Prophet ﷺ was passing away how many times did he come to him هه twice he came to him إبراقيم said that when the Prophet ﷺ said رأيه to جيبيل I saw جيبيل وعليه 600 جراح and he had 600 wings each wing one of his wings it can block of the whole horizon إبراقيم said that he did see جيبيل after that that doesn't eliminate that he saw جيبيل after it are we together that was when he saw it he saw it جيبيل came to him after that because revelation was still coming down on him he mentioned that in his كتابة البداع الفرائد رحمه الله تعالى which one and the rest is إصلاع المعارج was it they come down on إصلاع المعارج the two lost eyes no that's what in the nation this state he took it from Allah directly we yeah so we know the Prophet ﷺ the night of إصلاع المعارج he took rulings from Allah عز وجل and Allah told him صلاة and حكام other rulings but did he take Qur'an of the night when إصلاع المعارج was a revelation that came down on him none of the narrations mentioned that none of the narrations mentioned that was to just speak the ayah Qur'an yeah the thing is that I haven't looked I can't remember the narrations authenticity so that's why I did not say that's why I don't remember if it's authentic or not like إن شاء الله تعالى look into the last آيات of صورة البقرة لا شك إبنو كثير will be it in his تفسير so we'll verify that إن شاء الله so the issue of قراءات were coming to it إن شاء الله تعالى it's one of the things that we're going to speak about in great details especially when we talk about the قراءات and the أحروف السبع what is the difference and how is it and how does it work because it's slightly confusing if we talk about the قراءات and the أحروف السبع but all of this came down from الله سبحانه وتعالى every single one of it came down from الله عز وجل فضل see the Qur'an the Prophet ﷺ was told to read the Mu'awwa the tale when he had the Ain done to him but he came down before that so the آيات some of the قصور they didn't just come down once they came twice so he came down twice the second thing is sometime the Prophet ﷺ will receive a revelation sent to him and he'll be told to use that revelation for something like the Mu'awwa that came the Mu'awwa that came they came down before and when the Prophet ﷺ was the magic was done to him ﷺ الله instructed him to read it but it wasn't that it came down at that particular moment now the Qur'an جيبري الهالف from الله عز وجل الله says in the Qur'an إنه أنزلناه في ليلة القادر إنه لد أعياء الله says إنه أنزلناه في ليلة مباركة إنه كنا منزلين القرآن نعلم أنه all came down once the whole Qur'an but where did it come down on سماء الدنيا and from there it came down منجمن منجمن means what bit by bit لكن جيبري الهالف it all from Allah عز وجل and then he was then instructed each verse because Allah knows the incident and the things that are going to take place each part الله was commanding him that part take it down now are you with me brothers and so each verse and each was coming down on that particular instant instant but prior to that جيبري الهالف all from Allah سبحانه و تعالى to convey it to the messenger صلى الله عليه وسلم but it was all placed in سماء الدنيا and it was coming down from there منجمن bit by bit that's the only way to reconcile between the أقوال of the هل العلم and on my channel I spoke about this in more details when I spoke about the تفسير والصورة ليلة القدر أقوال of him العباس and others who differed with him فضل رمضان the prophet صلى الله عليه وسلم he used to finish the Qur'an with جيبري they used to revise and that was once a year the revision and the last year that the messenger صلى الله عليه وسلم was dying جيبري الهالف came and he finished the Qur'an this time with him twice not once how many times did he finish it this time they finished it twice and scholars they took from this if the person grows old in age and they realize that they're getting closer to their grave they should multiply their righteous deeds are you with me if you feel that you've grown old and now you've aged and you're more closer to your date of leaving this world you should multiply your righteous deeds because the prophet صلى الله عليه وسلم but that year he was gonna depart from the world صلى الله عليه وسلم so he increased in his recitation of the Qur'an in that Ramadan some people when they become old in age they become more businessmen and they start liking dunya more and start becoming investing in this and that the older you get the more you should connect yourself to the best treat the more you should start getting staying in the masjid and knowing that the time is going to finish there are scholars who hold that opinion some scholars who hold that the revelation is not restricted to jibreel but each point that they brought and the discussion pertaining it that we've left it for some other time because we just wanted your overview but that's why I said the other day if you guys can read the Kitab and it's called فِي علوم القرآن it's very important something that I just mentioned I go over it quickly you can go and research even more and you might even differ with me after your research you might think that that view that I took is not strong this is the beauty but you have to do your research and read the Kitab اليت قال فِي علوم القرآن by suyoti and other books that are written in the science of the Qur'an are you with me brothers now the Qur'an the sequence the ayat are what the Prophet ﷺ said it as like in the surah the companions did it that's what they said like the surah of the Qur'an the companions are the ones who ordered it like this and this is the call of the Jumhoor like in the ayat and the surah that it's in the Prophet ﷺ did that he commanded it he would say to the companion take this ayat and place it in surah كذا وكذا and it will be placed nope the Qur'an صورة اقرأ بسم ربك اللي came down the first six came down and then صورة المدثل came down so it would mean an ayah from this would come an ayah from the Prophet ﷺ would command the companions he would say take this verse and place it in that surah take that verse and place it into the surah he read it on the Prophet ﷺ like in the surah the scholars they these are مسائل which are the khilaf is too much I'm just trying to be in the but that's why it's good to read the Kitab صيوطر brings all the views he brings the evidences and it was the Kitab was sent out right to everybody was it sent out via email huh it was sent by email to everyone huh people didn't receive via email so if it could be sent to them via email it will help them a lot shall we conclude there إن شاء الله تعالى فاط الله ها so the can that be made into a legislation no how about then it can't be it's not a it's not a legislation الهام what takes more presidents over that is what is أقل logic and the person's mindset all of those they don't have a place in legislation the path to Allah and what pleases Allah and what Allah loves that cannot be known by a person's innate ability or even dreams somebody might say I saw the Prophet in my dream and the Prophet صلى الله عليه وسلم told me that you don't have to pray صلى الله عليه وسلم anymore he said it to me والله are you with me what you saw in your dream whatever command that he gave you or they gave you will not stop on will not stop the حكم of Allah that's upon you whatever it may be we'll conclude there إن شاء الله تعالى anything which I have said that was wrong or incorrect is from me إن شاء الله that Allah is messing to our free from it سبحانك اللهم بحمدك أشهد ولا إله إلا الله أستغفرك أتوب إلي


So, welcome to all of you. As promised today, we will take up one topic in point set topology and then we will take one more topic which we will discuss very thoroughly, point set topology because these are very much used in algebraic topology. The first one is our function spaces. Recall that when you have a homotopy from some x cross i to y, then you can think of just a family of functions from x to y indexed by the interval i. Each function is definitely continuous. Therefore, it belongs to the set of continuous functions from x to y. That can be thought of as a subspace of the product of y taken x times and then you have the product topology. But the product topology is not very good because it corresponds to point wise convergence. What we need is some kind of uniform convergence and the correct thing one has to cook up and that is why different topologies on these function spaces is needed depending upon your requirement. That is why the study of function spaces. The one which is most useful in algebraic topology is called compact open topology. So, today I would like to introduce you to that aspect. Later on, we will talk about quotient spaces also. Given two topological spaces, let us denote by y power x, the set of all continuous functions from x to y. In point set topology, this notation is used to set of all functions from x to y. Namely, the product of y taken x times, x many copies of y. Since we are not interested in all functions, we are only interested in continuous functions because x is not an arbitrary indexing set here, but a topological space itself and we are interested in only continuous functions. We will use this notation, the simplest notation for most use sort of concept. So, there should not be any confusion with this one. So, y power x will denote all continuous functions from x to y for us. So, now topology on this one is called compact open topology. So, the topology is described by describing a sub-base and sub-base consists of these like inner product notation, bracket KU. All those functions which are continuous from x to y such that they take the compact set KF of K inside U. So, this should remind you that we are more or less using some kind of continuity and epsilon delta condition. U playing the role of epsilon and K playing the role of this delta neighborhood or whatever, but there is no point reference. This should be for all. So, that is why this will become uniform continuity. This with respect to this kind of things, this condition automatically. Uniform continuity on compact substance. So, this is the catch here. F of K is contained inside U captures the uniform continuity aspect. So, these K comma U will form sub-base just means that I have to take finitely many such sets, intersect them and look at all such sets. They will form a base. Thus, the topology on this y power x is now such that any open set is arbitrary union of finite intersections of members K1, U1, K2, U2 and KN, UN. Ki's are compact and Ui's are open. So, this is the topology. So, just for a while, whenever it is unclear what topology we are using, we will use this notation compact open or compact open topology. Once you are familiar with this, we will not even mention that. Like we are all the time writing just R with its usual topology. So, let us look at the first position here. Its y power x with compact open topology is finer than the product topology. Therefore, it is Hausdorff respectively. It is regular. If y is Hausdorff, y power x is Hausdorff. If anything which is finer than the product topology, then that will be also Hausdorff. When I say just y power x, it is the product topology, right? y power x is finer, I am claiming. So, it will be Hausdorff. Similarly, product of regular space is regular. Then you take the finer topology, CO, it will be also regular. So, I have to see why this is finer than this one. To see that, you see the single point is always compact. Therefore, in this collection here, single point comma u, these are already there. And what are those? They are the sub-basic open sets for the product topology. Therefore, this collection has already more elements in it than the standard sub-based topology of space for the product topology. Oh, singleton x is compact. That is all I am using. So, the first position is very nice. So, you also know that there are more open sets in CO than in product topology. Therefore, convergence here will be stronger. Point wise convergence may not be a sequence which is point wise convergence function, may not be uniformly convergent. You know that. Whereas, if you have this uniformly convergent, then it will be point wise convergence also. So, if some sequence of functions is convergent in this topology, in the ordinary product topology, also it will be convergent because this is finer than the product topology. The central result that we want is summed up in one single theorem, three parts of that. The second part is the most important one for us which is of immediate use. So, let us be done with it and that is all about function spaces that we are going to discuss. So, this is called exponential correspondence. So, here is an important extra hypothesis here, restriction here namely x is locally compact hostile. y and z could be any topology spaces. For simplicity, you can assume y and z are also locally compact if you do not remember which one to be locally compact. But assuming y and z are locally compact does not help you at all, x must be locally compact and hostile. So, what is it involved here is evaluation map e y power x cross x to y, what is this evaluation? Evaluation of f at x, f is a function from x to y, the second slot is x, evaluate it at f of x. So, this is what evaluation map, this will be automatically a continuous function. In fact, this is continuous even if you take even if you take here the product topology, this should be nothing but the projection map to the one of the factors. This x is an element of x, so it is x projection of this map. Since I have put a stronger topology here, automatically this should be also continuous. So, this is the way you have to see, but you can directly prove that also. So, evaluation map is continuous. Now, look at the function g from y to z to y power x, some function is continuous if and only if you take g cross identity from z cross x to y cross x y power x cross x. So, z cross x you land here with g cross identity and then follow it by identity, follow it by evaluation map. If that composite is continuous, g will be continuous. If g is continuous, the product is continuous, composite is continuous is easy. So, what is important here is if the composite is continuous, then g is continuous. So, this is the key thing which allowed the classical approach of saying parameterized family and so on. This was when formalized here. So, let us see what is the proof of this one. Before that, let us look at the third condition here. This is actually tells you the law of exponentials is valid inside these continuous functions just like the it is valid for sets. You must be knowing that cardinality of y power x power z is the same thing as cardinality of y power cardinality of z into cardinality of x product. This is same thing as a raise to b raise to c is a raise to b c for numbers. Same thing is true for cardinality of when you take these things are all functions, but this is also true now for continuous functions is the beauty here. Well, namely if z is also locally compact hostile, so x i position then the function psi from y power x raise to z. What is this one? This is set of all continuous functions from z to y power x which occurs in statement b here, b here. That corresponds to under b, it corresponds to a function from z cross x to y. If that is continuous, it will be continuous and vice versa under these exponential components psi of j is e composite, g composite, g cross identity is just like here. So, c is actually a consequence of b, but we will have to work out. What happens if this bijection is fine, but this is actually homeomorphism is what we have. So, let us go through this proof one by one. So, I am writing a full detail of why the function evaluation map is continuous. I have given you the idea why it is continuous. Let us write down some full detail here. Given an open set u inside y, we have to show that e inverse of u is open in y power x cross x. So, take a point f naught x naught belonging to e inverse of u. I must produce a basic open set which contains a point f naught x naught and contained inside e power u. Then I am done. What is the meaning of this belongs to e inverse of u? It implies that f naught is first of all continuous function because it is inside y power x. And f naught of x naught can evaluate it is inside u, e of that is inside u. Now, you see now because we have assumed that x is locally compact of x. There exists a compact neighborhood k of x naught such that f naught of k is contained inside u. This means that f naught is inside k u.   So, you have to find a x naught f naught, you must find a neighborhood cross a neighborhood here. That is the product topology from y power x cross x. That is why. So, this is clear cut proof, no hand waving. Now, that is about to be. From a, we need to prove only one part because we have already proved e is continuous, e is continuous. If g is continuous, g cross identity will be continuous, composing with e will be continuous. So, only the converse part we have to prove. Suppose this is continuous, then we have to show that g is continuous. For this, it is enough to show that g inverse of a basic, sub basic open set is open whenever what is a basic open, k is compact and u is open. By sure inverse image of every sub basic open set is open then the function is continuous. So, how to show g inverse of k is continuous. So, take a point here, say z naught belonging to z. Let us say g of z naught is inside k u, that is the meaning of g inverse of k u. z naught is in g inverse of k u means g z naught is in k u. Then for every point inside k, what we get is e cross g cross, e composite g cross identity of z naught comma k by definition of what? g z naught comma k, e of that which is g z naught operating upon k. Remember, g z naught, g is a map from where x to y, right. So, g z naught, sorry, g is a map from z to y raise to x. So, g z naught itself is a map from x to y. So, g z naught of k belongs to this open set. So, there exist open sets vk, wk of z and x because this is a product, right. So, a neighborhood here and neighborhood here respectively such that z naught is in vk, k is in vk, wk and e of this product is contained inside u. If a point goes inside this one, the composite is continuous. So, entire function operating on a neighborhood contains this continuity of e composite g cross identity. Now, we use the fact that k is compact. We can pass on to a finite cover k contained is a i range of 1 to n vk i. What I have done? For each point k inside k, we have formed a wk. So, all these wks is a cover for k. Therefore, I say there is a finite cover wk i. k is contained inside wk i. So, let us call this union as w, okay. On the other hand, let us take the neighborhood for this z naught to be intersection of vk i. Then e of g cross identity of this intersection here and union there that will be in consent inside it, okay. Because once it is intersection, this will be true for every element no matter where you take the second one to be in one of them. So, it is contained inside, okay.  The whole w goes inside u, okay. So, g of v is contained inside k u, okay. What do you mean by k u? Every element of g v takes k inside u. But they are taking whole w inside u. Therefore, they are taking k also inside. Since v is a neighborhood of z naught, we have done because we have formed a neighborhood here. g of that is contained inside k. g started with g of z naught contained inside k u and formed a neighborhood now. So, this shows that g is spontaneous. Is that clear? Yes, sir. The compactness, local compactness of x is used here. Now, you see in c, in statement c, there is local compactness assumption on z also. You know why? Because in this statement z is also going up exponential. It is also superscript. So, it is easy to remember whenever the space goes up here on the superscript, those things must be locally compact. The things at the bottom could be anything. This is y power x. So, if something should work properly, x must be locally compact. So, thing is raised to z. So, z must be locally compact. So, z cross x, both of them are locally compact. So, product is also locally compact. So, that is why z is also locally compact necessary. Why could still be anything? So, I am telling you a trick to remember. Things which go up must be locally compact. All right? So, let us prove c. The idea of giving all these proofs is so that you will get familiar with compact open topology. Otherwise, one can just take the statement and go away. Finally, we are going to use only the statements. From b, it follows that the map whatever you are writing here, psi, its map, how we define psi of g is e composite g cross identity. Whenever g is continuous, you have defined this continuous. So, you do not need the b. But even the inverse is continuous whenever this is continuous, this is continuous, that will need the part b. It is well defined and its inverse is also well defined. That is what it is. It is well defined and it is a bijection. So, for that bijective part, you need part b there. Applying b with y power x raised to z in place of z and z cross x in place of, this is in its not is, in place of x, continuity of psi is the same as continuity of, see I am applying b here, e composite psi cross instead of identity of x and identity z cross x. So, y raised to x cross z to cross, z cross x I take to y. If I show that this is continuous, then it will imply that psi is continuous. So, this is where I am using b very strongly. What is this map? You see some lambda here from z to y power x. Z is a point of z, x is a point of x. So, lambda z and x, you take lambda z, it is a function from x to y, you can evaluate it on x. This is lambda z x. By b, it will follow that, you know, if this whole thing is continuous, then this is continuous. But why this is continuous? Continuity of this one is obvious because it is again from b. So, this latter map is nothing but, you can think of this as y power x power z cross z bracket, I am putting the bracket other way around cross x. So, from here y, where are you coming, y power x cross x, this is continuous. From here to here again I will evaluate, it is continuous. So, this is given by lambda z x, lambda z you are evaluating cross x. So, it is like evaluation carried twice. That is why again from b, again from a it is continuous. From b this will be continuous. So, that proves that the evaluation map is, you know, evaluation map is continuous. That was psi is continuous. The proof of psi inverse is continuous exactly similar to actual tautology. Everything is if and only if we can go ahead. So, I will leave it to you as an exercise. You have to spend some time to get familiar with what is going on, playing with these exponentials and so on. So, you can check this that psi inverse is also continuous. What that in b? We do not need the local compactness effects to prove that the map g from z to n is written from continuity of the corresponding map. The local compactness effects is needed in the other way implications because it is needed in part a. One important special case of statement b is the homotopy version which we will be using all the time. This is the special case when x itself is i, z cross i to y, in cutoff z cross x to y. z cross i to y is a homotopy. This is continuous if and only if it corresponds to the function a little h. I have put little h here. Remember I have put little h t. Little h t is where all functions from z to y. So, now that family is whole family is a function from i to y. So, h itself is a function from z to y raised to y. h t of z is capital Z t. So, h is a function. h of z is a function from i to y operating upon t is h z. So, that is a meaning of it. This is a very special case and this goes in the in the suffix up here. So, this must be locally compact. But you have actually it is compact hostile. So, of course, it is locally compact hostile. Therefore, the joint continuity of this one ensures you that a family like this is a just one single function continuous function from z into the function space y raised to y. Set of all map all path is from path is in y with compact open topology. I think we will stop here because the next thing is a different topic namely quotient spaces which we will discuss in the next module.


Okay, so let's do this. Let's play with fractions. We'll start simply by adding fractions so going over the notes and when we have one third plus two-thirds If the denominator is the same It stays the same It's already done it and then we can just add the two numerators together one plus two is three and If we have three out of three it is also known as a whole number one So that's when the denominators are the same What if they're different? What do we do if they're different? So let's say it's one over three and Two over four and we needed to add them together. What's the strategy there? We can't add them unless these two denominators are the same So we need to convert them and we do this by what I call ratio so we need to find the lowest common Multiple LCM of both of these fractions or both of these denominators So the lowest common multiples go like this What do both of these numbers multiply? Together with that is the same number. All right, so let's go through our multiples of three one times three Three. Yeah, two times three six three times three equals nine four times three Equals twelve. Let's leave it at twelve and here's why Well, let's do number let's do the four so one times four equals four two times four equals eight three times four equals twelve now Can you see that here and here? We have both twelve that is the lowest common multiple So we can convert one over three plus two over four two Over twelve here and over twelve here and we're adding them together We know that it's going to equal twelve as the denominator but We can't just add one plus two because we've scaled up both of these fractions So we need to learn now how to scale the numerators up and here's how you do it, okay? Firstly, and this is will help you with your ratios two three Times something equals twelve now when we were doing it back here We figured out that three times something or something times three, which is the same equals twelve We go down. Oh, here it is the magic number is four So three times four equals twelve now with these fractions and ratios We have to do the same thing to the top as we did to the bottom. So it's going to be one multiplied by four one multiplied by four is four So four over twelve is the same as saying one over three. It's the same ratio. Yeah So if you were to ask if you were to get asked to simplify four over twelve to a simpler fraction The answer would actually be one over three But this time we're making a bigger so that we can Add these two fractions together, right? Let's do the next one What is four? multiplied by to get twelve Did you have three as the answer you're correct? So four times three equals twelve and again Whatever you do to the bottom you have to do to the top so two times by three will equal Six so we've got six Right, we're done. Where are we done? Let's add these two together to get our final answer four plus six equals ten and There we have it Ten over twelve, but let's simplify it. How do we simplify ten over twelve? We need to make it into a smaller number the way that I go about it is work through my Singular numbers. We don't use one because one divided into ten is ten. Nothing's happened One divided to twelve is twelve, okay What about two two can two divide into ten? The answer is yes because ten is an even number. How many times does two go into ten? Let's go old-school. We can even use fingers here ten two four six eight ten Five is your answer. So we have five What about twelve? Let's use our fingers again two four six eight ten twelve six So the answer is six and that's how you add Numbers or fractions that have a That don't have a common denominator So step one find the lowest common multiple L C and remember that and Then you can add them once you add them make sure that you scale The fractions the same way as you do with the denominator to the numerator and that's how you do it Okay next bit next bit Now again, if that's going too quick, you can always pause it rewind the video and watch it again Okay So what about subtraction? Let's say we have four fifths Take away three fifths If the denominator is the same The denominator stays the same and we can do four take three. Yeah Taken away the numerators four take three equals one One over five is the answer. That's easy, but it's been in high school. We need to get a little bit more challenging Let's make it Four over five take away Three over Let's go with six Three over six four over five takes three over six Yeah, so first thing we need to do is we need to make these two numbers the same We need to make a lower cop find the lowest common multiple of the denominates. So let's do that one times five five two times five ten three times five 15 Four times five because 20 who's getting a bit of a recall from private school. We're doing our times tables. This is why five times five 25 And then six times five equals 30 now I could just keep going if I wanted to but let's stop at six Okay, let's do about six times tables one times six six two times six twelve three times six 18 four times six 24 five times six 30 ah There it is There's our number now. I purpose purposely stop at 30 because I know that's the lowest common multiple But I've been doing this forever. I'm old So let's have a look. Let's convert this to 30 here take away 30 here and then we'll have our answer here over 30 What are our numerators going to be? Well, let's have a look. How do we get from five to six? Sorry five to 30 we multiplied by six Whatever we do to the bottom we do to the top Four times six. Oh, we got that here. This is an easy answer 24 That was easy. Yeah What about this one? Oh six to 30. How do we get from six to 30? Oh, we multiply by five five times six equals 30 and Then we have to multiply this one by five Don't get confused because that's the six that we use and we multiply by five That's sound number so times five and whatever we do to this one. We do to the top as well. So times by five That's where we get the number from three times five equals 15 Excellent. So that is the same as that but with common denominators Let's finish the problem 24 take 15 is Well, let's have a look. Let's do it here just in case you forgot how to subtract Four take five. You can't do borrow the one carry the four fourteen take five equals nine And then one take one equals zero our answer is nine So our answer here is nine over 30 Now, can we simplify that fraction? Let's go. We're going to ignore the one. We're going to go to the two. Can two divide evenly into nine? No, it can't. Can three divide evenly into nine? Yes, it can and Before we continue, we're going to make sure that three can divide into 30 evenly as well Yes, it can. How many times? Let's use our fingers Three goes into nine three six nine three There's the first one what about 30 Three six nine twelve fifteen eight and twenty one twenty four twenty seven thirty Ten there's our answer three over ten Okay, we've done it. We've done it. We've got our answer. Yay Okay, so now we've done that We're going to look at the next stop Okay, the next problem you're going to get a problem that Looks like this now I'm gonna go a little bit quicker and skip the finding the lowest common denominator. Okay, so let's say we have five Over six plus and then in brackets. What's the number that I've set up? Negative seven over nine. Okay now What we can do here is first that we need to find the lowest common multiple I'm going to skip a step and I'm going to tell you straight away that that equals two over 18 over 18 over 18 Okay, and the top two numbers are going to be 15 and and 14 Okay, 15 and 14 So from here, we need to know what the middle bit is now Because we have a plus and a minus We need to convert that now when we're adding a negative. We're actually subtracting. Okay, so We have 15 and 14 The plus and minus if you remember your integers from year seven is actually a takeaway So we have a takeaway here. So the answer is one over 18. Okay, done done That's the first one. Now another way to do that is Let's say let's say We have a common. Let's say we have a common denominator Okay, let's make it a little bit easier. Let's say we have three over five Plus and then brackets minus one over five So it's a minus altogether. Yeah, so a lot same thing now Quick tip minus one over five equals minus one over five like this equals one over minus five They're all the same. Okay, but Let's make it the top one. Let's use this one as an example. Okay So we're going to rewrite this like this so three Over five plus negative one over five and then when it comes to our Coding let's call it coding. It actually looks like this Three plus negative one over five because When we have a common denominator, remember it stays the same and then we have three plus negative one When we're adding a negative, we're actually taking it away. This here becomes a negative Three take one is two over five And that's your final answer Okay done next bit next bit next bit All right now. This is my favorite bit. I prefer this over the addition and subtraction. All right, let's erase this Let's go for the Multiplication. I'm just gonna check if we still got time for this recording. Yes plenty of time We're 14 minutes in and if you're ready stuck feel free to go back and try again We're gonna go with multiply first Multiply okay with multiplying fractions. There's actually less work involved. Okay, so let's say we have three over five multiply by two over four okay, all you have to do is multiply both Numerators and in brackets, I'm gonna put top and then also multiply both denominators Which is the bottom numbers Now if you're still quite confused with what all those words are Let's label it a half That's the numerator numerator and that's the denominator Okay, I probably should have done that at the very beginning of the lesson But here we go. We've unlocked it now if you're less confused you can go all the way back to the start and go for it Okay, so multiply both numbers the top so we're gonna multiply these two together Okay, anytime you see this set up three times two is six. That's the top number and then four times five or five times four is 20 There's our number. Let's simplify it how many times there's two going to six three and How many times there's two going to 20? Okay, so there's our first one. Let's say multiply. That's it the end the end I'm going to do one more for negatives. I'm going to do one more for negatives. Okay, so for the negatives Let's say we have negative one over five multiply by two over three Okay, now remember one over five Negative is equal to negative one over five as well. So let's write it like that. Let's make that a little bit easier Let's get rid of this and do that we can do that We're allowed to do that now remember from the notes here if you didn't write them down Rewind this about a minute and copy those notes down. That's the video video We multiply the top two together now negative one times two. What do you think the answer is? It's actually negative two Okay, now a Here's some more notes for you if you haven't figured it out a negative Multiply by a positive Will give you a negative and also the other way around because remember three times four is also four times three So therefore positive Times a negative is also a negative There are two more. Can you figure them out? Pause the video and see if you can write them down and unpause negative times a negative Is actually a positive Okay, it's like two wrongs make a right sort of thing and then a positive times a positive positive equals a positive Remember those those will be with you throughout your maths career all throughout high school. There it is Now let's go back to this. That's correct because negative one negative times a Positive now we know that's a positive because there's no pluses or minuses in front of that We don't need to put a plus and then because we're automatically going to know that it's positive. That's the rule Okay, positive equals a negative. That's why we wrote negative two and we know our ones times tables We know our two times tables one times two is two and that's where we put the negative there Right next one five times three five as a positive Times three. That's also a positive. Oh, that's not it positive positive positive So the answer is gonna be a positive five times three is 15 We're done. That's our answer and you can do that in any way whether it's here whether it's here Anyway, just as long as you follow these laws. Okay, right last one. Let's do divide divide Isn't that much harder, but you have to remember three words Keep change and flip So let's have a look at this problem one divide one over five divided by two over three Now here's where the words come in label this keep Which means? leave it Alone Leave it alone. Don't do anything. This one is Change, what are we changing it to? To multiply Change to multiply so from this To this okay, and then this last one Flip They're the three magic words keep change and flip we can also Use the word swap Swap over swap around rotate whatever words you want to use So the two and the three become the three and the two Okay, let's write all of that Here based off of what you just learned So first step first number one over five. What does it tell me keep leave it alone? Okay, I'll leave it alone. I'll keep it like it is one over five. What's the next step? Change to multiply. Oh, we've got to change this. Oh this to this. Okay, I'm gonna write that here done Last one flip swap over two over three. Oh Three over two. See how we've just rotated them around Done. Oh, I know how to do this. This is just multiply You just multiply the tops and the bottoms one times three is three Five times two is ten done drop the mic. We're finished How are we cool? Let's move on. Yeah Okay, the last one last one and I'm going to do negatives. Okay, I'm gonna do negatives and a vision This is as hard as it gets for now. Let's use Some different numbers Let's go with over eight. Let's make it a negative eight. Let's make it. Let's make it and then we'll go negative three over four. So what do we do here? Keep leave it alone. Okay. I'll just write it again. That was easy Change to multiply. Oh, that was easy Flip swap over. Okay, negative three and four. So the four now goes on top and the negative three goes to the bottom again We've just rotated and now let's do it. Oh We have to just multiply the top and the bottom two times four. That's easy Eight What about negative eight times negative three? Oh, what happens if you don't know if you're not sure rewind it a few minutes Go back to when I had the positive and negative notes But if you've written them down well done So, you know that now a negative times a negative equals a positive so eight times three eight sixteen twenty four There's our answer Can we convert this? Yeah, we can. Two goes into eight. How many times? four Two goes into eight. Uh, two goes into 24. How many times? 12 Can we convert that? Absolutely. Two goes into four. How many times? Twice. Two goes into 12. How many times? Six. Can we convert that? Oh Yeah, why not? Why not? Why not? Two goes into twice Sorry, two goes into twice once Two goes into six times three times One third. Could we have done this quicker instead of doing these? Yes, this is how. What goes into eight and what goes into 24 evenly? The number four. How many times does four go into eight? Twice. How many times does four go into 24? Six times. So we could skip that step. Okay, and then they're both even still so two goes into two one Two goes into two three times. Ladies and gentlemen, we are done with fractions. That's how you do it Thank you for tuning in. You've been a wonderful audience. Good night


This study examined how atmospheric turbulence affects the performance of wind turbines. It found that the turbulence level of the incoming air has a significant impact on the turbine's ability to generate power. Higher turbulence levels result in faster recovery of the turbine-induced wake, as well as more intense turbulence near the turbine. Additionally, the study showed that turbulence plays a role in the generation of turbulent shear stresses, which can be used to improve the efficiency of wind turbines. This article was authored by Yuting Wu and Fernando Port Agile.


Concurrent schedules, a schedule of reinforcement in which two or more contingencies of reinforcement elements operate independently and simultaneously for two or more behaviors. Examples of concurrent schedules are numerous. They include everything from gambling to everyday life. Your behavior is constantly under multiple schedule control. Those multiple schedules, if you start to tease them out, you could start to identify what piece is reinforcing what behavior. The fact that they're happening all kind of on top of each other is when we say they are concurrent schedules. In the laboratory, we can specifically establish those schedules and tie them to a specific behavior, so it is clear to understand what pieces are reinforcing what behavior.


All right, where's, uh, where's D-Lack?  All right, go ahead. Sure. Is this maybe becoming your team's identity? Well, you hope so. That's what we said when, when Mr. Blank hired me and Terry. I mean, we wanted a tough minded football team and we've started to build that.  Yeah, sure. I mean, we, you'd love to be on four minute and went easy in the four quarter. That's a really good football team. I mean, they're five and two. This is a really tough place to play. It's a divisional arrival.  And, um, our guys don't flinch. You know, it's, uh, I was joking, Mr. Blank. I'll get a text later from my mother telling me to stop doing that. We're just trying to win. Um, you know, so we'll do it any way we can. And when you put the ball in Matt Ryan's hand, a game on the line, our money's on Matt. So that's how I feel. It's a national football league, Michael. It's not for soft souls. If you get comfortable, shame on you. He's a, he's a really good football player. I mean, obviously we have a lot of faith in CP and Matt and the line. I mean, they don't have credit, right? I mean, you, the drive before try to stay aggressive credit to the Saints. You know, they may plays. Um, you know, we weren't going to get into the three yard and cloud of dust. So we, we tried a couple of different things. We got the penalty in the first town. I told the drive before. And again, we went into a little zone read, see if we could get them. Get some yards going. They played it pretty well. And, you know, it shows the faith we have in Felipe because those are big time and situation right there. But I trusted Felipe with the football and he was smart. And I wasn't what we wanted. And they jamed us up on the next one to play pass.   They scored a go ahead, but I looked up and there's plenty of time. And we had our timeouts. I was like, good, you know, we stopped at those two point conversions. They went on both of those. You know, that's when you get into the game. They understand the consequences. They miss, you know, miss one. So they're forced to go for it again. And we stopped them both. Those are two huge plays in this game. The two point conversions for our defense. What did you say to Matt? What's going on there before that last round? Same thing as they don't want. We have a neutral mindset and we come in there. We talk about it. So, hey, look, if we get this match up over here, let's take it. And that's what I boil down to. They didn't give us that. We're going to something else. So that's pretty simple. And he's as cool as they can be. So I don't know what this is. Again, I don't want to put you on the spot basity, but his comebacks are getting up there, right? Yeah. Yeah. So that helps when he had a veteran hall of fame type quarterback. So that certainly makes it easy. Is there something that you can see from a guy like Matt Ryan that can tell you that you can't be a guy like that? Because not every guy can do that. Yeah. Just look at his, look at his whole career. That's what's a body of work. My experience with him. I mean, already what's been done this season. So, and not enough credit here too. I mean, guys that continue to operate. You know, the Josh Harris world. I mean, even the whole operation. Those are things that those guys are real pros. Him and Colquitt. And then obviously Ku's been clutched. Your first game as a head coach for the Falcons in this building. Any sense for the specialist of this rivalry? Well, it's an intense rivalry. These guys have been the standard in this division the last four years. So I know Tampa won the Super Bowl last year. It's a really tough division. But these guys have, since Sean Payton's been here, they've had a lot of sustained success. And so it's a good win for us. All right, wraps up. All right. Thanks, guys. No, no, no. I'm, you probably already covered for last. Go for it. How big was that plate? It's huge. I'm not sure. Yeah. I mean, you can get an explosive play to start a drive like that. Certainly makes it easier as a play caller.  It's an enormous play. Had you seen something that made you think that plate that was going to be there? Yeah. Just kind of kind of how they've been playing the third down in two minutes. And it's chess matches. So this is a really good defense and a defensive staff. And they made their adjustments and we made ours. And like I said, we put them out there and we got the match that we wanted. And Matt and C.P. made it work. I was going to say that you made a certain look for it. Sure. I mean, I'm not going to get into all of our schemes, but we got the look we wanted right there. And just, you know, you get the big, exact strip. Yeah. And then it gets hard. You know that we're going to come, but how did you guys just stay there and try to keep them from getting downhill? Well, I mean, there's a lot of unsung heroes on this football team. There's been a lot of change. And like I said, we've got a real football staff. It's really collaborative from our entire scouting to our, you know, coaches and everybody. And it's a real team. You know, there's no politicians involved and we've had a vision and they underrated. And there's a lot of credit it goes. You bring in James Waters, right? He made an enormous play today. Mike Bonnell, he's been pretty solid. You get Anthony Rush. I mean, gosh, mighty. He made some good plays. I mean, these are guys that said, well, you continue to turn the roster and they've stepped up. And we've got a team that believes in no egos and it's a fun group to coach. On the string of three penalties, you know, there's a person about four games. Well, you know, look, these guys, look, the officials in this league have a hard job.  I think these guys did a nice job. Certainly in the moment, you can get frustrated. But again, I could see what they saw. And like I said, I got no issues with the officiating. It's hard. That's a hard job. And the more rules they make, it's like the bills they put through Congress.  But I think our officials in our league, they got a tough job. And my experience as a head coach, they've done a nice job. And Kuhl's kick at the end in 29-yarders didn't have to take the center out there, actually nailed to get it, too. Sure. You know, you're playing the game.  You don't want to have to roll back out there and kick to Deontay Harris. He's an electric football player. So that's good situational football. Like I said, I mean, you would like to not have the whole scrum at the end.  I mean, we have a mechanism for that. And then you get there and you make sure you operate and you milk the clock down. No different than what we did in Miami. What is the sure-thank factor of Kuhl from inside 30? I think almost any NFL kicker, he would feel pretty good inside the 30. You said you've got a mechanism that they're going to try to let you score. Sure, I mean, I had every team that's a little secret.  We've seen it happen. You've seen it happen in the Super Bowl. Obviously, the Giants Patriots 1 comes to mind. I mean, it's happened. Got a situation in Miami this year or almost identical to it. Almost identical. So you just got to prepare for that. That's why you practice situational football. I know it gets Miami inside the team. You want the ball in your hands and you have to have that mindset. Sure. Can you kind of speak more? It's really our whole team. But that's our leader and, you know, Grady and those guys and, like I said, it takes an entire team and those guys have bought in a fun group of coach. That's what you did. I mean, the whole mechanism, like you get in there. Sure. Like you'd love to have a comfortable leave. Everybody wants to be comfortable. That's not the NFL. Not when you're on the road playing a good team and our mindset to be, we're going to finish games. And it's never going to be perfect when I try to be, but it shows more about their mental toughness and when you roll it out there, you got to stay in the present. And here's what the call is. All right, if you're off track, well, we're going to. And that's my job. That's why they pay me to do it. Coach. Again, I'm not going to get into our practice schedule. But I'm not going to get into it. We have a really good football staff. And I just, I just, every team does, I mean, every team works at pro football, but I'm not going to get into our specifics there, Michael. I appreciate the question, but keep that in house. With the growing team that's, you know, gone through a bunch along the way, just how much pride do you take in now having three last second field goal wins all on the road? I think a lot of pride in it. But the credit goes to those guys, the men in that locker room. They're the ones that make it happen. So I said, I really, really enjoy coaching this football team. Same thing in London, D-Led. It's pro football. Yeah, it's our job. We're in the solution business. That doesn't change. You have injuries, part of this football game. It's a long season. We're in a 17 game season. And we're right now, we're right in the middle, we're in the mix. And so we have a huge game to go down to Dallas next week. It's a really good football team. And that's what it is. And you're going to need contributions from everybody. You're never going to have that little clean sheet of depth chart. When you go in there, it's never going to stay the same. And it's our job to adapt. And our guys have. I'm not a doctor. Just, we'll update you like I do every week. And I don't, I can't speculate on that. I did. Yeah, I stole from Mike Brable. I thought it was a good thing that Raves did. It made a lot of sense. So I did it. There's a lot of things I've stolen from Mike Brable and a lot of good coaches. I've been around. Appreciate it guys. Thank you.


This is stock number 10999A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used SUV headquarters. Today we are checking out this super clean 2016 Buick enclave with the leather package all-wheel drive. This vehicle has the 3.6-liter V6 motor which pumps out 288 horsepower. This vehicle has been fully safe and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. And this enclave is 100% ready to go. Crimson red tint coat is the color. We shoot all of our videos in 1080p, 60 frames per second. So if you have HD capabilities on your computer tablet or smartphone device, turn them on right now because it is definitely your best way to check out the quality and condition of the vehicle before seeing it in person. And if you'd like to check out all the photos on this enclave and the upper right hand part of your screen is a link right to our website. Click that and check us out there. It comes with the factory alloy rims with the chrome covers. And it has Goodyear Assurance Weather Ready tires. These are pretty new tires on here and they are 255, 60, R19s. I would say they probably have 80, 90% of the tread left. Very new tires on there. Front fender is in excellent condition. I didn't see any dents or dings on that. You'll notice it has the LED running lights, the HID headlamps, and the LED fog lamps. Front bumper is in really nice shape. I didn't see any major dents, dings or cracks on it. Grille is in nice shape in the hood. I didn't see any dents or dings on there either. Passenger side front fender. Excellent condition. And the passenger side rim. No major scuffs or scrapes on that. As you go down this side of the vehicle, take note of how clean the body is, how reflective and mirror-like that paint is. We take these HD videos so if you are far away, or even if you're close by and you just cannot make the trip down but you're still interested in purchasing the vehicle, you can see the vehicle, hear the vehicle, and have confidence in what you were looking at before you even get here. So that when you do get here, there's absolutely no surprises. Has the dual glass roof here with the sunroof. And that's why we offer the Summit Auto Fastpass option in the upper right hand part of your screen. Link right to our website where you can apply for financing. Get pre-approved. See a payment quote. Even appraise a trade. All from the comfort of your own home. So when you get here all you got to do is drive the vehicle. Love it. You'll be all set to go. Back rim is in nice shape and the back tires have just as much tread as the front tires. Coming around to the back of the vehicle, rear bumper is in nice shape. Does have the back of parking sensors. Chrome looks like it's in nice condition. The rear gate looks like it's in nice condition. You can really get a good idea of that metal flake in the paint. It is a power lift gate. You can see that that works really nice. Back storage area is very clean. There is a little bit of storage underneath here. And to get these seats down you just pull that strap there and they fold down like so. So there's a manual down seat. Just grab the strap to pull it back up. And to get that rear gate down you just press this button right here. Come on down like so. And as you go down the driver's side just as clean as that passenger's side. No dents or dings on the rear quarter. This back rim. Pretty nice condition. One tiny little scuff I noticed on it but other than that every rim on this vehicle is perfect other than the little scuff right there. But it's a used vehicle and that's why we do the videos. And down the rest of the side of the vehicle looks really good. Didn't see any dents. Didn't see any dings. You got blind spot monitoring. Built in directional signals in the mirrors as well. Inside the leather package gives you the light gray leather interior. There are no rips. There are no tears on the seats. Both these seats are power. You get driver's side lumbar. Factory floor mats throughout. Auto headlamps. Tilt and telescopic steering wheel. Power windows, power locks and power mirrors. Memory driver's seat. And as we hop inside the vehicle here you can see that this one has 73,732 miles. Instrument cluster is very nice and clean. You do get a heated leather wrapped and wood-grain trimmed steering wheel. Bluetooth and audio controls on the right. Cruise controls and heated steering wheel controls on the left. Comes with the Buick IntelliLink system. AM, FM and Sirius XM radio capabilities. This is also where your backup camera is going to show up. And you got your home screen, a couple different apps and stuff such as Pandora and Stitcher Smart Radio. You got information center controls, dual climate control, the nice wood-grain trim throughout. Down here are your heated seat buttons. Your rear windshield wiper buttons. And are your rear window wipers. And then you also have the traction control and the gate. You can turn that on or off. USB jack, automatic transmission, passenger side format is in excellent shape. Passenger side seat, no rips or tears on there. Smells very clean inside this enclave. I don't think it's ever been smoked in. It has the dual glass roof. Like I said the power sunroof in front and then the fixed glass in back. You do get map lights and home link buttons for your garage door security systems and lane systems right there. And then on-star capabilities in that rear view mirror. We'll take a quick look at the back seats and then we'll check out under the hood. The back seats are the quad seats and they are in nice shape. Didn't see any rips or tears on them. Passenger side looks good too. You do get rear air controls and two USB chargers. Factory floor map back here. These seats slide forward to get to the third row which is in really nice shape. No rips or tears back there. Comes with the third row floor mat. All the seat tracks are nice and clean. And those seats go down like so. Got the nice wood grain trim on the doors here and you get the child safety locks on the back doors. Bottoms of the doors look really good. I didn't see any corrosion on them. And we'll take a quick look under the hood. I would personally like to thank you for checking out the video today and hopefully from this HD video. You will have been able to tell just how clean this enclave is all the way around inside and out. Trying to find the hinge here. There it is. Not much to see under here. 3.6 liter. 288 horsepower. Engine bay is very clean. Runs very smooth. This Buick has been fully safety inspected by a service shop as a fresh oil and filter change. All the fluids have been checked and topped off. And this vehicle is 100% ready to go. You can see that that shock is doing a nice job holding the hood up. There's the emission sticker. And we'll do one more final walk around here. Once again I'd personally like to thank you for checking out the video today and hopefully you've been able to tell just how clean this vehicle is all the way around inside and out. And to see more pictures of this enclave or one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers. You name it we got it go to our website www.summitauto.com. Full pictures and descriptions of every single vehicle from two locations all at summitauto.com. And if you'd like to check out more HD videos you can go to youtube.com slash summitauto. Remember to like subscribe and share on this video and all the videos that you see there. In fact in a second you'll see a link to subscribe to our YouTube channel on your left. A link to more enclave videos like this one on your right. If you have not been to our website on the bottom link to this vehicle on our website click those check us out and we're super excited to help you with this extremely clean 2016 Buick Enclave leather all-wheel drive and crimson red tint coat. Thanks again.


Let us go to the next one, continuity. So function, so let us look at this, let us think of let f be this function and I will think of it as a function of n variables taking real values. It can also be a vector function, it is not very hard to extend this to vector function. So it is a function for that takes from Rn to R that means it is a function of n variable and the eventual value that it outputs is a real number. Now we say f is continuous at a point x in Rn if the following holds for every epsilon greater than 0 there exists a delta greater than 0 such that we say f is continuous at a point x in Rn if for every epsilon greater than 0 I can find a delta greater than 0 such that the difference f y minus f x if I look at it in absolute value is no more than epsilon for all y's that are within a delta radius ball of x. So the challenge is thrown in this way you set how close you want your function value to be that is your epsilon you throw me an epsilon and the challenge is that I will be able to find for your delta such that if I look at the function values in the delta radius ball around x look at y is that lie in a delta radius ball look at these y's that lie in a delta radius ball around x then the function values f of y are no more than plus minus epsilon from f of x ok f y minus f x is at most epsilon in absolute value if you can do this alright then we say that the function is continuous at the point x. So we say the function is continuous at x continuous simply so we say f is continuous if it is continuous at all points. So there is another way of now this is a complicated somewhat complicated definition because it is it is posed in terms of this kind of a challenge there is another way of there is another equivalent way of writing this which is the following which is using the language of sequences. So suppose if I have so we say that f is f is continuous we say that f is continuous at x if suppose you take if you take any sequence x k like this that converges to x take any sequence that converges to x then along that sequence you look at f of x k look at the sequence f of x k its limit should be equal to f of x in the language of sequences what the continuity definition is basically saying is you take any sequence of points x k in the in the in the domain of the function domain space of the function and look at the value the sequence of values that gets generated in the ring then that sequence of values is itself a convergent sequence which converges to the value of the function at that. So another in effect what it is saying is limit of f of x k is equal to f of limit of x k. So the limit basically jumps inside the evaluation of the function this is what we mean by continuous. Okay now this then brings us back to optimization. So I had told you that I had told you through the through Queen Dito's problem why it is actually why we cannot take for granted the existence of a solution to an optimization problem because if the solution does not exist you can sort you can get all kinds of absurd conclusions just by arguing inductively without without checking the existence. So we need a way of checking for the existence of a solution that does not entail finding a solution in the first place. Without finding a solution you should be able to be guaranteed rest assured that the solution at least exists then you can go about finding it if you want to. So that is what is guaranteed by this theorem of Weistra. So for me to write Weistra's theorem we should first write out what we mean by an optimization problem formally. So the way I stated the optimization problem previously was this. We are looking for a you have a function like this say a function from R n to R suppose and you have a set S is a subset of R n and what we want is a x star in S such that f of x star is less than equal to f of x for all x in S. Effectively what we want is this if you look at the set of values that f takes over the set S look at the set of values that f takes over the set S so f x such that x belongs to it look at this set what we want is effectively the infimum of that set you want the least we want the greatest lower bound for on that set because we because f of x star is the least possible value of f in this particular set. But then that is this is the but when you think if you look at if you if you think of it this way this creates a the following question how do we know we know that the infimum exists for every bounded set and infimum exists thanks to completeness axiom but the but the completeness axiom does not tell us that the infimum is of the form f of x star for some point x star in the set. So how do we know so this quantity exists as a real number but how do we know does there exists an x star in S such that f of x star equals this particular is clear so so these are two this brings us to two separate questions one is when is the infimum itself finite the second is when is there a matching when is when is that infimum actually achieved by a point in the set. So if there exists such an x star we say the if so we say the infimum is attained by x star in that case it is both so if the infimum is not attained by x star then it is not meaningful to look for such an x star it is not present in the set of alternatives that you are searching for but it is meaningful nonetheless to look for the infimum of f nonetheless that is me that is that is meaningfully although the x star finding the x star itself may not be meaningful. Let me give you an example let us look at this function here. I have this function on real so this is the domain of the function here I am plotting plotting its values. So the function looks like this it is it is decreases up till this point say suppose point A now at point A at point A there is a jump discontinuity at point A its value is actually here not here. So the value of the function at A is this bold dot here and then from here it increases supposedly nearly this. So now this is my function here plotted out here my S is just real numbers. So can someone tell me what is the what is this you are tempted to say f of A but then f of A is this and that is certainly not the minimum value of f. So what is so it is meaningful to talk so what is this particular thing this particular thing is this point here the point to which the function tends to from the left the value of the function to which it tends to from the left but there is no point in the domain where it takes that particular value. The best bet was A itself but for A the point the value is up has been has jumped up. There is no point in the domain for which this thing is attained. So this limiting value is attained. So this is the case where as I said this quantity exists this one exists and is finite it is some finite number it is this this height here but there is no x star. So this is the case where the infimum exists but is not attained. So this is equal to this height. Let us take another example look at the function e to the minus x and suppose S is 0 to infinity e to the minus x how does this look at x equal to 0 this is going to be 1 and then it will decrease exponentially like this. Now what is again question here is again what is this what is this infimum 0 right after all that that is the the greatest lower bound for all the function values of f. So this is equal to 0 but then what is the x star there is there is no there is no finite x star I mean there is no there is no number there is no nothing that we can call as a real number which attains this value x which gives you this particular infimum value. So there is again in this case there does not exist an x star such that f of x star equals this infimum. Now as you can see this is an issue that you are faced with typically you would start with an optimization problem you have a domain that that is specified in us in a sort of implicit way you cannot enumerate every point and check what is happening in the domain. There are a function also whose terrain you complete you do not completely understand the function is specified in some sort of black box fashion for you do not know the full how the undilutations are where the function is higher where it is lower where it may have jumps or any of that with that kind of knowledge you now need to make sure that none of these kind of cases happen that there is actually a that the that the infimum is in fact attained and that is it is meaningful to look for a solution x star for this for the optimizations of right. So that is what Weister's theorem gives you. So now I will state the theorem let s be a subset of Rn that is closed and bounded in other words it is compact let f be a function from Rn to R that is continuous then f attains attains its infimum on x that is there exists an x star in s such that f of x star equals infimum of f of x as x range over there is such a thing called as in this case this case our notation changes we are not then we are not looking that we do not call it the infimum in this case we when the infimum is attained we use the notation minimum is the notation used should denote that the infimum is attained let me end with one small observation which I made last time also but I will just make it more concretely I have been talking about in optimization I have been talking about minimizing function right here you are looking at out here to see I am talking of f x star less than equal to f x. So I am looking at values that are lesser than all other values but that is not so if you are if you are looking for suppose you are you have a function g and you wanted to look for you wanted to find suppose g is a function from Rn to R and you have some subset s of Rn and you are looking for a x star such that g of x star is greater than equal to g of x for all x. In this case the problem you are trying to solve is that of a supremum of this supremum of g of x as x ranges over s and the question is whether this this is attained by some g of x star. Now what this that it is very easy to map this back to the setting that I the setting that we have considered which where we are looking to minimize rather than maximize and that is by simply taking f x as minus g x identically equal to minus g x. If f x is minus g x then the supremum of this is simply the supremum of minus f x as x ranges over s which is it is easy to show is the same as negative of the infimum of f x. So infimizing f is equivalent to supremizing f right. So as a result without loss of generality we will just always work with minimization because that also is very is somehow fits very nicely with the convex shape of functions and convex side.


Welcome to ITU Telecom World 2017 here in Pusan in the Republic of Korea. I'm very pleased to be joined in the studio today by Mr. Amal Akhasa, who is the CEO of Cybersecurity for Rhoda and Schwartz. Mr. Akhasa, thank you very much for being with us. Thank you. Now I'd like to start off by talking about smart digital transformation. It's the theme here of ITU Telecom World 2017. What does it mean to you? I think it's going directly to the heart what is happening worldwide in terms of really transformation in every part of the society, economic and other parts. And yeah, it's a really interesting time now. What do you see as the biggest opportunities of smart technologies in the next five years? I think the smart technologies will give us a lot of new business models, which haven't been possible with all technologies. And if you look especially on the enabler for these kind of technologies, Cybersecurity will get an important part of making this transformation possible. And what are the biggest challenges, especially in terms of cybersecurity and digital resilience? The cybersecurity change was forever to combine, to make the right trade between functionality and security. And a lot of people only look at the functional issues of new technologies without considering the security and the risk issues. And we will see that a lot of these new technologies will only be possible if we consider the security side of these technologies. And in terms of Roder and Schwartz, you're working to meet those challenges. Tell us a little bit in what steps you're taking in that way. So what we are trying to provide is to combine the security issues with functionality, making security as transparent as possible, so that the users do not get notice of the security issues we put. And that makes the deployment of security much easier and much more realistic to implement. Finally, you're no stranger to events such as ITU, Telecom, World. I wanted to ask you, what's the importance of you attending events such as this? You've obviously got a stand here and you're taking some energy and budget, I'm sure, to be here. Why is it important for you to be here? And what's the key message that you might have to participants here? I think the most important thing is to have to meet the people, to meet the network, to see what other people are thinking about the same challenges we're facing and to find solutions together because at the end all this kind of transformation is only possible if we find common standards where all of us adhere to and finding these won't be possible if everyone just sits at home and tries to find the right way to do it. It's only possible at such gatherings like IT World and for that we're really thankful for ITU organizing these and getting all the people together. Well, we welcome you again to this, the first day of ITU, Telecom, World and we hope that the week brings a lot of benefits to you and thank you very much for attending and thanks for being in the studio with us. Thank you very much for you. Thank you. Cheers. Thank you.


Okay, the second pattern we are covering of the behavioral pattern is the so-called strategy pattern. And this is a pattern that is useful in cases where you want to have something like an algorithm, some kind of routine, some filters that are exchangeable, that you might want to choose at runtime or that you envision might change over time as your program evolves and you simply don't want to have a single implementation but you want to be able to choose. So imagine for example a case where you have a navigation system, navigation application like you have in Google Maps or in many cars that tells you where to go. And you of course have different algorithms, different routing algorithms and instead of implementing a single algorithm in the system and you don't have to replace it manually you can implement this as a strategy pattern and simply say I want to have a method with which I can set the routing application, the routing algorithm at runtime. And then the algorithm itself is a different class or an interface that is implemented and you could basically just add another implementation and then change at runtime. So the way this works is you have the context, so that's our application for instance. This application has a strategy, some kind of algorithm, some filter and it has methods to set this strategy. So it has a method, a set method where you provide a strategy and then there is some kind of method that does something and the assumption here is simply that this method is what is actually executing. So you have something like strategy.execute somewhere in there we are executing our algorithm. So this is just where the actual functionality is happening. This context then has a strategy and this strategy has an execute function, execute and it takes some kind of data that depends on your application. If it's the routing application for instance you probably have some kind of source and target or origin and target destination where do you want to go and then the algorithm gives you a way. This is an interface so it's just that we define that we have this method and then a concrete realization of that interface would be a concrete algorithm that implements this method. So for example, well the easiest navigation is just to say go straight. We don't care if there are any houses in between, we assume go straight. Then you can have more sophisticated ones that kind of try to optimize this. First of all have an algorithm that actually routes you in a way that is possible. You have algorithms that might adapt to traffic information and so on. So very different implementations possible and this is indicated here that you can actually have many of them and then you choose at runtime or at compile time or you configure it. This of course is also useful where you might have multiple cases that depend on the situation. So you might have a routing algorithm that's more efficient during rush hour and you might have a routing algorithm that's more efficient in other situations for instance. So it's not only about replacing it over time it might also be applicable in cases where you want to choose depending on the context. So that's the strategy pattern. Context has a strategy and the strategy is executed in a method here. What kind of solid principles do we have? We have again the open-closed idea. So the context here is close to modifications. We're not modifying the algorithm anyhow. The interface itself is close to modification but then we can extend by having concrete realizations and we can basically enrich the behavior of this context by dynamically setting the strategy at runtime, at configuration or at compile time. The other thing we have here is a classic case of dependency inversion and that's because the context does not have a concrete strategy set. It doesn't have a dependency like this to a concrete case. Instead it has a method that allows you to set the strategy at runtime so we basically invert the dependency here. So we decide which concrete strategy we want. The context does not define that. Okay that's it for the behavioral patterns. There are of course more on the slides. There's also the iterator pattern which many of you probably know already from for example Python where it's built in but we'll now go into the next group of object-oriented design patterns.


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