DS Lecture-1 about discrete structure .ppt

Editor's Notes

• #16: Analog Computers: Characteristics: Analog computers use continuously varying physical quantities, such as electrical voltages or mechanical rotations, to represent and manipulate data. Applications: Analog computers were often used for specific scientific and engineering calculations, simulations, and control systems. Examples: Differential analyzers and slide rules were early examples of analog computing devices.
• #22: http://www.mathily.org/dm-rw.html#:~:text=Google%20Maps%20uses%20discrete%20mathematics%20to%20determine%20fastest%20driving%20routes%20and%20times.&text=An%20analog%20clock%20has%20gears,minimum%2Dweight%20spanning%20tree%20problem.
• #26: One solution is for Alice and Bob to exchange a digital key, so they both know it, but it's otherwise secret. Alice uses this key to encrypt messages she sends, and Bob reconstructs the original messages by decrypting with the same key. The encrypted messages (ciphertexts) are useless to Eve, who doesn't know the key, and so can't reconstruct the original messages. With a good encryption algorithm, this scheme can work well, but exchanging the key while keeping it secret from Eve is a problem. This really requires a face to face meeting, which is not always practical.
• #27: Public key encryption is different, because it splits the key up into a public key for encryption and a secret key for decrpytion. It's not possible to determine the secret key from the public key. In the diagram, Bob generates a pair of keys and tells everybody (including Eve) his public key, while only he knows his secret key. Anyone can use Bob's public key to send him an encrypted message, but only Bob knows the secret key to decrypt it. This scheme allows Alice and Bob to communicate in secret without having to meet. RSA rivest, shamir and adleman
• #28: Number theory, one important part of discrete math, allows cryptographers to create and break numerical passwords. Cryptographers must first have a solid background in number theory to show they can provide secure passwords and encryption methods.
• #29: https://ima.org.uk/779/internet-shopping-stopping-the-scammers/
• #30: http://www.ams.org/publicoutreach/feature-column/fcarc-stern-brocot
• #31: https://www.vice.com/en_us/article/4x3pp9/the-simple-elegant-algorithm-that-makes-google-maps-possible There are three major shortest path algorithms: Bellman Ford’s Algorithm, Dijkstra’s Algorithm, and Floyd–Warshall’s Algorithm.
• #32: https://ima.org.uk/3414/right-track-optimisation-models-railway-planning/
• #34: https://metalbyexample.com/linear-algebra/https://gamedevelopment.tutsplus.com/tutorials/lets-build-a-3d-graphics-engine-linear-transformations--gamedev-7716
• #35: https://ima.org.uk/640/smarter-phones-for-all/MIMO Systems, Claude Shannon (use a code with some redundancy), Turbo Codes (use pairs of encoders and decoders to boost transmission speeds)
• #38: Networked or linked structures are ubiquitous. Examples range from the World Wide Web, large social networks such as the network of board of directors or the network of Hollywood actors, the distribution network of Wal-Mart, the power grid of the US, or even the neurological network of our brain. August 96 - Major power outage that affected the entire west coast of the US from Alaska to California for over 24 hours. This power outage was caused by a single transmission line, That was knowcked down by a tree. More interestingly, the accident Was detected right away and as standard procedure the load of the line Was transferred to the other lines of the set; but they were already overloaded And the extra overload caused the lines to heat up stretch. Well, this was a hot Summer, the trees had grown dramatically and a few lines were knocked down, Causing the domino effect that brought down the entire power grid of The west coast of the US. How did it happened? Key issues: How can we design more robust netwroks? How does disease spread in a ntewrokd of people Graphs that occur in many biological, social, and man-made systems Are often neither completely regular nor completely random – Instead they have a small world topology  nodes are highly clustered And yet the path length between them is small
• #40: http://www.ams.org/publicoutreach/mathmoments/mm99-facebook-podcastFacebook has over 700 million users with almost 70 billion connections.
• #41: graphs, trees, and strings. DNA sequencing involves determining the order of nucleotides (adenine, thymine, cytosine, and guanine) Graph theory is often employed to model relationships between DNA fragments, helping to identify overlaps and assemble the entire sequence. String matching algorithms, a part of discrete mathematics, are used to find patterns or matches within these sequences.
• #42: Bubble Sort: Bubble Sort is a simple sorting algorithm that repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order. The pass through the list is repeated until the list is sorted. The algorithm gets its name because smaller elements "bubble" to the top of the list. Insertion Sort: This algorithm builds the sorted list one element at a time by repeatedly taking elements from the unsorted part and inserting them into their correct position in the sorted part. Selection Sort: This algorithm divides the list into two parts: a sorted and an unsorted region. It repeatedly selects the smallest (or largest) element from the unsorted region and moves it to the sorted region. Merge Sort: Merge Sort is a divide-and-conquer algorithm that divides the list into two halves, recursively sorts each half, and then merges the two sorted halves. Quick Sort: Quick Sort is another divide-and-conquer algorithm that selects a "pivot" element from the array and partitions the other elements into two sub-arrays according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively.
• #46: Logic gates or circuits are electronic devices that implement Boolean functions, i.e. it does a logic operation on one or more bits of input and gives a bit as an output. The relationship between the input and output is based on a certain propositional logic.
• #60: magine you have two friends, Alice and Bob, and you know two things about them: Alice always brings her umbrella when it's raining. Bob only brings his umbrella when it's windy. Now, you wake up in the morning and see that it's raining, but you don't know if it's windy. If you apply logic: Since Alice always brings her umbrella when it's raining, she'll bring it today. Bob, however, only brings his umbrella when it's windy, so you can't be sure if he'll bring it or not. So, based on logic, you can conclude that Alice will bring her umbrella, but you're not sure about Bob because you don't know if it's windy. In this example, logic helps you make reasonable predictions and decisions based on what you know about your friends' habits. It's about using clear and sensible thinking to understand and navigate the world around you.
• #61: A propositional statement is a declarative statement that makes a clear assertion or expresses a fact that can be either true or false, but not both. It provides a straightforward, definite piece of information. Here are a few examples of propositional statements: "The sky is blue." (This statement is either true or false.) "2 + 2 equals 4." (This statement is either true or false.) "It is raining outside." (This statement is either true or false.) In each case, the statement is making a clear declaration or assertion, and we can evaluate its truth value. Propositonal statements are fundamental elements in logic and form the basis for reasoning and logical analysis.
• #79: Example for OR (||): Sentence: "You can choose either tea or coffee for your beverage." Explanation: In this case, the person can choose either tea or coffee, or even both. The use of "or" implies that either option, or both options, is acceptable. Example for XOR (^): Sentence: "You can either take the bus xor ride your bike to school." Explanation: Here, the use of "xor" (exclusive or) indicates that the person has to choose one option but not both. It's either taking the bus or riding a bike, but not a combination of both.
• #89: That it is below freezing is necessary and sufficient for it to be snowing: p↔q