958 Check Completeness of a Binary Tree. Medium. 1967. Given the root of a binary tree, determine if it is a complete binary tree. In a complete binary tree, every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and 2 h nodes inclusive at the last level h. ABlueprint for Managing Supply Chain Disruptions Supply chains are designed to handle changing conditions, but not every supply chain is resilient and agile enough to respond to major disruptions like COVID-19. Find out what you can do now to prepare for the future. One Trusted Partner and Countless Opportunities Let e2open help you resolve complex business, channel, Notresoluce de The Last of Us 2 vous aide également à venir à bout des boss du jeu et vous guide dans The Last of Us Part 2 à chaque moment important de l'aventure. Attention, n'oubliez pas qu'en Vay Tiền Nhanh. Learning Objectives¶ By the end of this section you should be able to Understand the diffrence between unique solutions, no solutions, and infinitely many solutions. Reconize when a matrix has a unique solutions, no solutions, or infinitely many solutions. Reconize when a matrix has a unique solutions, no solutions, or infinitely many solutions using python. Unique Solution¶ The example shown previously in this module had a unique solution. The structure of the row reduced matrix was \[\begin{split}\begin{vmatrix} 1 & 1 & -1 & & 5 \\ 0 & 1 & -5 & & 8 \\ 0 & 0 & 1 & & -1 \end{vmatrix}\end{split}\] and the solution was \[x = 1\] \[y = 3\] \[z = -1\] As you can see, each variable in the matrix can have only one possible value, and this is how you know that this matrix has one unique solution No solution¶ Let’s suppose you have a system of linear equations that consist of \[x + y + z = 2\] \[y - 3z = 1\] \[2x + y + 5z = 0\] The augmented matrix is \[\begin{split}\begin{vmatrix} 1 & 1 & 1 & & 2 \\ 0 & 1 & -3 & & 1 \\ 2 & 1 & 5 & & 0 \end{vmatrix}\end{split}\] and the row reduced matrix is \[\begin{split}\begin{vmatrix} 1 & 0 & 4 & & 1 \\ 0 & 1 & -3 & & 1 \\ 0 & 0 & 0 & & -3 \end{vmatrix}\end{split}\] As you can see, the final row states that \[0x + 0y + 0z = -3\] which impossible, 0 cannot equal -3. Therefore this system of linear equations has no solution. Let’s use python and see what answer we get. import numpy as py from import solve A = [[1, 1, 1], [0, 1, -3], [2, 1, 5]] b = [[2], [1], [0]] x = solveA,b x - LinAlgError Traceback most recent call last in 5 b = [[2], [1], [0]] 6 -> 7 x = solveA,b 8 x C\Users\Said Zaid-Alkailani\Anaconda3\lib\site-packages\scipy\linalg\ in solvea, b, sym_pos, lower, overwrite_a, overwrite_b, debug, check_finite, assume_a, transposed 217 return x 218 elif 0 219 raise LinAlgError'Matrix is singular.' 220 elif info > n 221 matrix detected.' LinAlgError Matrix is singular. As you can see the code gives us an error suggesting there is no solution to the matrix. Infinite Solutions¶ Let’s suppose you have a system of linear equations that consist of \[-3x - 5y + 36z = 10\] \[-x + 7z = 5\] \[x + y - 10z = -4\] The augmented matrix is \[\begin{split}\begin{vmatrix} -3 & -5 & 36 & & 10 \\ -1 & 0 & 7 & & 5 \\ 1 & 1 & -10 & & -4 \end{vmatrix}\end{split}\] and the row reduced matrix is \[\begin{split}\begin{vmatrix} 1 & 0 & -7 & & -5 \\ 0 & 2 & -3 & & 1 \\ 0 & 0 & 0 & & 0 \end{vmatrix}\end{split}\] As you can see, the final row of the row reduced matrix consists of 0. This means that for any value of Z, there will be a unique solution of x and y, therefore this system of linear equations has infinite solutions. Let’s use python and see what answer we get. import numpy as py from import solve A = [[-3, -5, 36], [-1, 0, 7], [1, 1, -10]] b = [[10], [5], [-4]] x = solveA,b x C\Users\Said Zaid-Alkailani\Anaconda3\lib\site-packages\scipy\linalg\ RuntimeWarning Ill-conditioned matrix detected. Result is not guaranteed to be accurate. Reciprocal condition number ' condition number {}'.formatrcond, RuntimeWarning array[[-12.], [ -2.], [ -1.]] As you can see we get a different type of error from this code. It states that the matrix is ill-conditioned and that there is a RuntimeWarning. This means that the computer took to long to find a unique solution so it spat out a random answer. When RuntimeWarings occur, the matrix is likely to have infinite solutions. A When the US explorer and academic Hiram Bingham arrived in South America in 1911, he was ready for what was to be the greatest achievement of his life the exploration of the remote hinterland to the west of Cusco, the old capital of the Inca empire in the Andes mountains of Peru. His goal was to locate the remains of a city called Vitcos, the last capital of the Inca civilisation. Cusco lies on a high plateau at an elevation of more than 3,000 metres, and Bingham’s plan was to descend from this plateau along the valley of the Urubamba river, which takes a circuitous route down to the Amazon and passes through an area of dramatic canyons and mountain ranges. B When Bingham and his team set off down the Urubamba in late July,they had an advantage over travellers who had preceded them a track had recently been blasted down the valley canyon to enable rubber to be brought up by mules from the jungle. Almost all previous travellers had left the river at Ollantaytambo and taken a high pass across the mountains to rejoin the river lower down, thereby cutting a substantial corner, but also therefore never passing through the area around Machu Picchu. C On 24 July they were a few days into their descent of the valley. The day began slowly, with Bingham trying to arrange sufficient mules for the next stage of the trek. His companions showed no interest in accompanying him up the nearby hill to see some ruins that a local farmer, Melchor Arteaga, had told them about the night before. The morning was dull and damp, and also seems to have been less than keen on the prospect of climbing the hill. In his book Lost City of the Incas, he relates that he made the ascent without having the least expectation that he would find anything at the top. DBingham writes about the approach in vivid style in his book. First, as he climbs up the hill, he describes the ever-present possibility of deadly snakes, capable of making considerable springs when in pursuit of their prey’; not that he sees any. Then there’s a sense of mounting discovery as he comes across great sweeps of terraces, then a mausoleum, followed by monumental staircases and, finally, the grand ceremonial buildings of Machu Picchu. 'It seemed like an unbelievable dream the sight held me spellbound ’, he wrote. E We should remember, however, that Lost City of the Incas is a work of hindsight, not written until 1948, many years after his journey. His journal entries of the time reveal a much more gradual appreciation of his achievement. He spent the afternoon at the ruins noting down the dimensions of some of the buildings, then descended and rejoined his companions, to whom he seems to have said little about his discovery. At this stage, didn’t realise the extent or the importance of the site, nor did he realise what use he could make of the discovery. F However, soon after returning it occurred to him that he could make a name for himself from this discovery. When he came to write the National Geographic magazine article that broke the story to the world in April 1913, he knew he had to produce a big idea. He wondered whether it could have been the birthplace of the very first Inca, Manco the Great, and whether it could also have been what chroniclers described as the last city of the Incas’. This term refers to Vilcabamba the settlement where the Incas had fled from Spanish invaders in the 1530s. Bingham made desperate attempts to prove this belief for nearly 40 years. Sadly, his vision of the site as both the beginning and end of the Inca civilisation, while a magnificent one, is inaccurate. We now know, that Vilcabamba actually lies 65 kilometres away in the depths of the jungle. G One question that has perplexed visitors, historians and archaeologists alike ever since Bingham, is why the site seems to have been abandoned before the Spanish Conquest. There are no references to it by any of the Spanish chroniclers - and if they had known of its existence so close to Cusco they would certainly have come in search of gold. An idea which has gained wide acceptance over the past few years is that was a , a country estate built by an Inca emperor to escape the cold winters of Cusco, where the elite could enjoy monumental architecture and spectacular views. Furthermore, the particular architecture of Machu Picchu suggests that it was constructed at the time of the greatest of all the Incas, the emperor Pachacuti 1438-71. By custom, Pachacuti’s descendants built other similar estates for their own use, and so Machu Picchu would have been abandoned after his death, some 50 years before the Spanish Conquest. - Great thanks to volunteer Lan Nguyen has contributed these explanations and question markings. If you want to make a better world like this, please contact us Questions 1-7 Reading Passage has seven paragraphs, A-G. Choose the correct heading for each paragraph from the list of headings below. Write the correct number, i-viii, in boxes 1-7 on your answer sheet. 1 Paragraph AAnswer iv Locate 2 Paragraph BAnswer vi Locate 3 Paragraph CAnswer viii Locate 4 Paragraph DAnswer v Locate 5 Paragraph EAnswer i Locate 6 Paragraph FAnswer vii Locate 7 Paragraph GAnswer iii Locate List of Headings i Different accounts of the same journey ii Bingham gains support iii A common belief iv The aim of the trip v A dramatic description vi A new route vii Bingham publishes his theory viii Bingham’s lack of enthusiasm Questions 8-11 Do the following statements agree with the information given in Reading Passage? In boxes 8-11 on your answer sheet, write TRUE if the statement agrees with the information FALSE if the statement contradicts the information NOT GIVEN if there is no information on this 8 Bingham went to South America in search of an Inca TRUE Locate 9 Bingham chose a particular route down the Urubamba valley because it was the most common route used by FALSE Locate 10 Bingham understood the significance of Machu Picchu as soon as he saw FALSE Locate 11 Bingham returned to Machu Picchu in order to find evidence to support his NOT GIVEN Questions 12-13 Complete the sentences below. Choose ONE WORD ONLY from the passage for each answer. Write your answers in boxes 12-13 on your answer sheet. 12 The track that took Bingham down the Urubamba valley had been created for the transportation of Answer rubber Locate 13 Bingham found out about the ruins of Machu Picchu from a in the Urubamba farmer Locate The Last of Us Remastered Complete Walkthrough and Guide – All Artifacts, Optional Conversations, Comics, Shiv Doors The Last of Us, originally released for the PlayStation 3 on June 14, 2013, is widely considered to be one of the best single-player campaigns in PlayStation history. Given that reception, it’s no surprise that the developers at Naughty Dog, as well as the game’s publisher, Sony Computer Entertainment, chose to add a little extra polish to the title and release it on the PlayStation 4. Whether you’re reliving the experience brought to gamers by Joel and Ellie a year ago, or heading out into this post-apocalyptic United States for the first time, our walkthrough will help you with every single detail The Last of Us has to offer. We’ll guide you to all of the Firefly Pendants, Artifacts, Comics, Training Manuals, and Optional Conversations the game has to offer, as well as give you detailed strategies to power through one of the most intense campaigns of all time. Note The collectible guides will follow the completion of the single-player walkthrough and guide. Be sure to check back! Step by Step Guide Chapter 1 Hometown – Escape and Outrun the Infected The first chapter in The Last of Us isn’t so much a gameplay tutorial as it is a true prologue and story introduction. We’ll tell you everything you need to know to complete this tearjerker, including how to fend off the Infected in the town’s streets, as well as outrun them on your way to the highway. Chapter 2 The Quarantine Zone – How to Find Robert The first chapter of the game was all about introducing the story, and while more of that happens here, it’s also an in-game tutorial. You’ll learn all the basics of combat and interacting with objects, as well as what Parts are for and how to collect them. Our walkthrough shows you how to sneak past the Infected, as well as how to reach Robert to get your guns back. Chapter 3 The Outskirts – Safe Location, How to Kill and Evade Clickers The third chapter of the game is where the intensity really gets cranked up. You’ll find yourself faced with a decent amount of the Infected, including both Clickers and Runners. We’ll show you how to sneak right through, or kill everyone and pick the area clean of all items of interest. If you’re looking for the location and combination to the safe, we have that as well. Chapter 4 Bill’s Town – Safe Location, How to Kill the Bloater This is definitely one of our favorite chapters of the entire game. It features a lot of exciting combat, as well as some of the game’s best dialog. We’ll show you how to find and crack the safe, as well as where to pick up the bow, and Tools Level 1 so you can upgrade your weapons. Finally, if you’re struggling with the Bloater, we have you covered there as well. Chapter 5 Pittsburgh – All Tools, Safe Combination, Kill the Bloater Expect that this chapter will likely be the longest of the game, and it is filled with combat from end to end. There is a lot to do, including upgrading to Tools Level 2, opening Shiv Doors, locating and finding the combination to safes and even another one of those pesky Bloaters. We will guide you through it all, including where to find the keycard and how to start the generator. Chapter 6 The Suburbs – All Tools, Weapons, Safe Combination and Location This chapter isn’t overly long, but it does feature a fair amount of combat and even some good scavenging. We’ll show you how to get through the sewers, including how to find the Shorty and Tools Level 3. When you’re out, we’ve got the location and combination for the last safe in the game, as well as a very well hidden Training Manual. We’ll even show you how to kill the sniper. Chapter 7 Tommy’s Dam – Pick up the El Diablo and Training Manual As far as we know, this is the only major chapter that doesn’t have Infected, but it does have plenty of bad guys just the same. We’ll show you where to get the El Diablo twice, as well as where to find the chapter’s Training Manuals, and even Tools Level 3 if you missed it back in The Suburbs. Finally, we’ll walk you through saving Ellie after she steals a horse. Chapter 8 The University – How to Kill the Bloater, Escape the University This chapter doesn’t feature a crazy amount of combat, but it is some pretty intense fighting when it happens. Our walkthrough will show you the only Shiv Door there is to find, as well as where you can pick up the Flamethrower. With that in hand, we’ll then guide you to how you can kill the Bloater in the dorm rooms, and escape the Hunters to complete the chapter. Chapter 9 Lakeside Resort – How to Beat the David Boss Fight The Last of Us Remastered really only has one boss fight, but it’s an intense battle that can leave your heart pounding. Our guide will show you how you can track and hunt the deer, as well as fend off the waves of Infected afterward. When you are betrayed by your new friends, we’ll show you how to win the boss fight against David. Chapter 10 Bus Depot – How to Kill the Bloaters and Survive the Tunnel The first part of this chapter is one of the most laid back in the game, while the second portion will test your nerves with no less than three Bloaters. Our guide will get you ready for the tunnel, then show you how to take out every single Runner, Clicker and Bloater. We also show you the location of the chapter’s only Training Manual, which is one you’re likely to walk right past without our help. Chapter 11 The Firefly Lab – Reach the Operating Room to Save Ellie At this point the game is winding down, but you still have to survive some of the most deadly enemies of the entire campaign. Our guide will show you how to kill all the Fireflies, as well as get your hands on the Assault Rifle. When you make your way to the top floor, we’ll show you the chapter’s only Shiv Door, then guide you through saving Ellie from a deadly surgical procedure. Chapter 12 Jackson – Game Ending and Comic Location The final chapter of the game sees players return to play as Ellie, although there really isn’t much to do. This one is all about tying off the story, as well as finding the final collectible of the game. You don’t need our help to beat this one, but you might need a hint when it comes to finding the Singularity Comic location. The Last of Us Remastered – All Safe Locations and Combinations In part one of our extras and collectibles guide, we’re going to show you the location of all four safes in the game, as well as where to find the combinations to open them. Safes won’t unlock any trophies for you, but they will give you four Artifacts, as well as a batch of other goodies that will serve you well throughout the game. The Last of Us Remastered – All Comic Book Locations and Endure and Survive This is a game known for its collectibles, and our goal is to help you find every single one of them. We thought that we’d start off small, focusing on something simple like the Comic locations and the Endure and Survive trophy that comes with finding them all. Not only do we give you the chapter each one is found on, as well as a detailed description, we’ll also show you an image of exactly where it can be found. The Last of Us Remastered Beginners Tips Every game has a learning curve, but in the case of The Last of Us, it’s more like a continuous learning journey. Even on our fourth play through two on PS3 and now two on PS4 we’re still figuring things out. Rest assured, we’ll share them all, but for now we’ve compiled some tips that will help you get started on your post apocalyptic journey. The Last of Us Remastered – All 85 Collectible Artifact Locations On every player’s quest to get all the collectibles in The Last of Us Remastered will be the daunting task of finding all 85 Artifacts. Luckily for you, we’ve already been down that road, and we’re now bringing that knowledge to you in hopes of making your life easier. While a lot of these can be found by thoroughly exploring, some are just downright impossible to find without help. Don’t believe us, just take Artifact 28 as an example. Oh, and when you finally do get all 85, you’ll unlock the It Was All Just Lying There trophy for your troubles. The Last of Us Remastered – All Optional Conversations – I Want to Talk About It The characters in The Last of Us Remastered sure do enjoy talking about their feelings. In fact, aside from the non-stop dialog that the game is known for, there are 37 Optional Conversations that players can engage in as well. Of course, that’s assuming they can figure out how to trigger them. While some are fairly easy to find, others are nearly impossible. That’s where we come in. We’ll show you how to get all 37 and unlock the I Want to Talk About It trophy. The Last of Us Remastered – All Shiv Door Locations – Master of Unlocking When it comes to The Last of Us Remastered, players will spend as much time searching for loot as they will killing Hunters. Exploring and gathering up supplies is a large part of staying alive, and the Shiv Doors contain some of the best loot the game has to offer. Our walkthrough will guide you to all 13, as well as provide some visual aids if that isn’t quite enough. When you finally get all 13, we want an honorable mention engraved into your Master of Unlocking trophy. The Last of Us Remastered – All Training Manual Locations – I Got This Not only are the Training Manuals in The Last of Us Remastered helpful, they will also give you a trophy if you manage to collect them all. As with most collectibles, you can find almost all of them by thoroughly exploring your environment, however, there are one or two that are nearly impossible without help. Our walkthrough gives you detailed instructions on where to find every single manual, unlocking the I Got This trophy in the process. The Last of Us Remastered – Top 5 Multiplayer Tips Sometimes trumping the AI just doesn’t give you the same satisfaction as a human counterpart. If you haven’t tried it, The Last of Us Remastered has a very well done multiplayer, and we thought it was best that you show up for battle with the basics locked down. Read on to find out why teamwork really is essential to victory, and why this isn’t another cookie cutter online experience. The Last of Us Remastered – All Weapon Locations When you’re fighting a hostile group of Infected, every weapon you can carry could be the difference between living and dying. While most weapons in The Last of Us Remastered are fairly easy to find, some can be overlooked if you aren’t careful. Just in case you don’t have as many guns as you do holsters, check out our guide to all weapon locations. The Last of Us Remastered – Advanced Tips Not too long ago, we brought you some beginner’s tips to help you get started with The Last of Us Remastered, but now that we all know the basics, it’s time to up our game. Today, we’re going to bring you some advanced techniques that we have perfected over the course of our four complete campaign play throughs. We think you’ll be particularly impressed with our first tip. We’ll post more coverage for The Last of Us Remastered in the coming days!

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