Question Quantifiers are called quantifiers because they express somethingabout the number of elements that satisfy a certain property. Showhow to express the following ‘quantified’ statements in predicatelogic with a unary relation symbol R. You don’t have to introduceany new quantifiers to do this, although you will need to keep inmind that the atomic formula x=yis always considered to be partof the language.(a) Exactly one x satisfies R(x).(b) At least one x satisfies R(x).(c) No more than three x satisfy R(x).(d) All but one x satisfy R(x).

C++ programming with own templates, please add clearexplanations.Integer.h#ifndef INTEGER_H#define INTEGER_H #include #include #include using namespace std;class Integer{private: vector num; //store thenumberpublic: //constructor Integer(string number); //default constructor //operation overload Integer operator +(const Integer& b); Integer operator -(const Integer& b); //output overload friend ostream& operator <<(ostream& out, const Integer& i);};#endifInteger.cpp#include "Integer.h"Integer::Integer(string number) { //TODO: read the string and construct the hugeinteger}Integer Integer::operator +(const Integer& b) { //TODO: perform + opearion overloading}Integer Integer::operator -(const Integer& b) { //TODO: perform - opearion overloading}ostream& operator << (ostream& out, constInteger& i) { //TODO: perform output overloading} main.cpp#include #include #include "Integer.h"using namespace std;//read the data from the filevoid readFile(const char* fileName);int main(){ readFile("input1.txt"); return 0;}void readFile(const char* fileName) { ifstream inputFile; //TODO: Read the input file here. //int lines = ... for (size_t i = 0; i < lines; i++) { Integer a, b, answer; //TODO: Implement your add orminus //answer = a + b; //answer = a - b; cout << i + 1 <<endl; cout << "num1: "<< a << endl; cout << "num2: "<< b << endl; cout << "result: "<< answer << endl << endl; } inputFile.close();} interger.hinterger.h Lab5 Huge Integer (Operator Overloading) For this problem you need to read data from the text file and implement the addition and subtraction of huge integers by using operator overloading. You should implement 3 functions by yourself in class: 1. Open the file and read all the operations from the file. 2. Add the two huge integers. 3. Minus the two huge integers. You need to consider that all the huge numbers are represented as hexadecimal numbers. Hint: The input and output number may be negative Input Format The number in the first line is the number of test cases Then, each following line contains an operator and two big hexadecimal numbers. Output Format Print two numbers and result of the calculation in each test case. Sample Input & Output INPUT: 6 123+789 aaa-bbb 18446744073709551614+18446744073709551614 9223372036854775807-2147483647 c2a3cc35f96982fb018a-666afd613073f3eb8620 -0a4c936b2600424e136+0a4c936b2600424e136 OUTPUT: 1 num1: 123 num2: 789 result: 8ac 2 num 1: aaa num2: bbb result:-111 3 3 num1: 18446744073709551614 num2: 18446744073709551614 result: 3088ce880e6e12aa2c28 4 num1: 9223372036854775807 num2: 2147483647 result: 922337203470d2f21c0 5 num 1: c2a3cc35f96982fb018a num2: 666afd613073f3eb8620 result: 5c38ced4c8f58f0f7b6a 6 num1: -0a4c936b2600424e 136 num2: 0a4c936b2600424e136 result: 0

Vibration assignment 3 Problem 3 A machine is subjected to the motion x(t) = A cos(50t + a) mm. The initial conditions are given by x(0) = 3 mm and x(0) =1 m/s. Answer to the following questions. (a) (2 points) Find the constants of A and (unit: mm and rad) (b) (1 point) Express the motion in the form of x(t) = A, cos(ot) + A, sin(ot) mm,

When landing on a product webpage at Amazon.com, customers will find a section called "Customers Who Bought This Item Also Bought" for recommending books, CDs, and other products that are commonly co-purchased with the current product, a feature designed to boost cross-selling. Task 1 : Here we're going to write Python code to process historical transaction data to implement the feature. Suppose that we are using a Python list to maintain all products purchased in different transactions: transactions = [[2, 8, 3, 6, 1, 9], [0, 5, 9], [0, 9], [4, 7, 0, 5, 9], [8, 3], [1, 6, 3, 8], [9, 0, 5], [3, 8], [5, 7, 0, 4], [3, 1, 2, 6, 0]] Elements in each sublist represent product IDs co-purchased in a transaction. Our goal is, for each product in this tiny transaction dataset, recommend products that are most frequently purchased together with the current product. Please complete the following tasks based on the problem description. For each transaction, enumerate all pairs of products that co-occur in the corresponding sublist. For example, processing the sublist [9, 0, 5] should produce three pairs (0, 5), (0, 9), and (5, 9). Pay attention to the order of product IDs in each pair. The same product pair (e.g., product 5 and product 9) can have two different representations (i.e., (5, 9) and (9, 5)). To facilitate the subsequent processing, make sure all pairs each are represented by its ascending representation (e.g., (0, 5) rather than (5, 0)). Moreover, make sure product pairs created from the same sublist are sorted first by the 1st element and then by the 2nd element. E.g., the list of pairs created from the sublist [1, 6, 3, 8] should be presented as [(1, 3), (1, 6), (1, 8), (3, 6), (3, 8), (6, 8)]. Produce a Python list, product_pairs, whose values each are a list of such pairs. The expected output of print(product_pairs) is as follows: [[(1, 2), (1, 3), (1, 6), (1, 8), (1, 9), (2, 3), (2, 6), (2, 8), (2, 9), (3, 6), (3, 8), (3, 9), (6, 8), (6, 9), (8, 9)], [(0, 5), (0, 9), (5, 9)], [(0, 9)], [(0, 4), (0, 5), (0, 7), (0, 9), (4, 5), (4, 7), (4, 9), (5, 7), (5, 9), (7, 9)], [(3, 8)], [(1, 3), (1, 6), (1, 8), (3, 6), (3, 8), (6, 8)], [(0, 5), (0, 9), (5, 9)], [(3, 8)], [(0, 4), (0, 5), (0, 7), (4, 5), (4, 7), (5, 7)], [(0, 1), (0, 2), (0, 3), (0, 6), (1, 2), (1, 3), (1, 6), (2, 3), (2, 6), (3, 6)]] Full credits will be awarded only if your answer is a comprehension implemention. Loop implementations can only get at most 80% of full marks. Task 2 : Next, we will process product_pairs to create a dictionary cooccurences, whose keys are product pairs and whose values represent the number of their co-occurrences across transactions. The expected output of print(cooccurences) should look like the following (the output formatting can be different): {(0, 1): 1, (0, 2): 1, (0, 3): 1, (0, 4): 2, (0, 5): 4, (0, 6): 1, (0, 7): 2, (0, 9): 4, (1, 2): 2, (1, 3): 3, (1, 6): 3, (1, 8): 2, (1, 9): 1, (2, 3): 2, (2, 6): 2, (2, 8): 1, (2, 9): 1, (3, 6): 3, (3, 8): 4, (3, 9): 1, (4, 5): 2, (4, 7): 2, (4, 9): 1, (5, 7): 2, (5, 9): 3, (6, 8): 2, (6, 9): 1, (7, 9): 1, (8, 9): 1} Full credits will be awarded only if the order of keys in your result matches exactly that in the expected output. Task 3 : Process cooccurences to output a new dictionary recommendation whose entries each are a list of recommended products associated with the number of transactions where they were co-purchased. The expected output is as follows: {0: [(5, 4), (9, 4), (4, 2), (7, 2), (1, 1), (2, 1), (3, 1), (6, 1)], 1: [(3, 3), (6, 3), (2, 2), (8, 2), (0, 1), (9, 1)], 2: [(1, 2), (3, 2), (6, 2), (0, 1), (8, 1), (9, 1)], 3: [(8, 4), (1, 3), (6, 3), (2, 2), (0, 1), (9, 1)], 4: [(0, 2), (5, 2), (7, 2), (9, 1)], 5: [(0, 4), (9, 3), (4, 2), (7, 2)], 6: [(1, 3), (3, 3), (2, 2), (8, 2), (0, 1), (9, 1)], 7: [(0, 2), (4, 2), (5, 2), (9, 1)], 9: [(0, 4), (5, 3), (1, 1), (2, 1), (3, 1), (4, 1), (6, 1), (7, 1), (8, 1)], 8: [(3, 4), (1, 2), (6, 2), (2, 1), (9, 1)]} It means that, e.g., for product 5 (corresponding to the entry 5: [(0, 4), (9, 3), (4, 2), (7, 2)]), a list of products (i.e., product 0, 9, 4, and 7), each being represented by a tuple with the 1st element being the ID and the 2nd the number of transactions where it has appeared together with product 5, are recommended. The recommended products are also sorted in descending order by the the number of cooccurences with product 5 (i.e., the 2nd element of inner tuples). # starter code​ transactions = [[2, 8, 3, 6, 1, 9], [0, 5, 9], [0, 9], [4, 7, 0, 5, 9], [8, 3], [1, 6, 3, 8], [9, 0, 5], [3, 8], [5, 7, 0, 4], [3, 1, 2, 6, 0]] Please do not answer the question if u don't really know it.

Problem 3 (Interruptable system logger) Design and implement a logger support the two operations wrlog() and flushlog() to manipulate the log data buffer ”logbuf” #define MAX LOG LENGTH 10 #define MAX BUFFER SLOT 5 char ∗∗ logbuf ;   int wrlog (char∗∗ logbuf , char∗ new data ); int flushlog (char∗∗ logbuf );   In this problem, there are many programs or actors that call wrlog() to append the log data to a shared buffer (i.e., log-file cache) which can be flushed to disk eventually. For simplicity, we assume the buffer contains 5 (= MAX BUFFER SLOT) data slots and the flush event occurs periodically at time out. We also assume a LOG is a fixed length string, i.e. char new log[MAX LOG LENGTH]. In practical point of view, the behavior of the system can be illustrated as a sequence of writing log data and the flush will be periodically triggered when a timeout is reached. int main() { wrlog ( data1 ); wrlog ( data1 ); wrlog ( data1 ); . . . wrlog (datan ); } In summary, the draft operations of the logger are: wrlog() append data to shared buffer but not exceed the buffer size. If it reaches the limits, it needs to wait until the buffer is flushed. flushlog() clean buffer aka. move all the stored items to somewhere (in here is printing to screen) and then delete all of them. This action runs eventually; for simplicy we just make a periodical call here. Further development: The interruptable mechanism of flush log can (further) support more unpredict events, i.e., it can handle a signall SIGUSR1, SIGUSR2 which are introduced in Lab 1 appendix. You are provided a referenced code with non-protected buffer by default, the program’s output is: $./ logbuf flushlog () wrlog (): 0 wrlog (): 1 wrlog (): 2 wrlog (): 3 wrlog (): 4 wrlog (): 7 wrlog (): 12 wrlog (): 13 wrlog (): 16 wrlog (): 17 wrlog (): 21 wrlog (): 24 wrlog (): 11 wrlog (): 26 wrlog (): 14 wrlog (): 6 wrlog (): 27 wrlog (): 28 wrlog (): 8 wrlog (): 20 wrlog (): 9 wrlog (): 22 wrlog (): 10 wrlog (): 19 wrlog (): 25 wrlog (): 29 wrlog (): 18 wrlog (): 23 wrlog (): 5 wrlog (): 15 flushlog () Slot 0: 0 Slot           1: 1 Slot           2: 2 Slot           3: 3 Slot           4: 4 Slot          5: 7 Slot              6: 12 flushlog () flushlog () flushlog () TODO: Implement the protection mechanism for wrlog() and flushlog() routines to make it a safe data accessing (to buffer). If it has a proper configuration then the program behavior is somehow like this illustration (comment out the print function name in wrlog() and flushlog() to make this clean output). $ ./ logbuf Slot           0: 0 Slot           1: 1 Slot           2: 2 Slot           3: 3 Slot           4: 4 Slot           5: 5 Slot           0: 12 Slot           1: 6 Slot           2: 7 Slot           3: 10 Slot           4: 8 Slot           5: 11 Slot           0: 17 Slot           1: 13   Slot 2: 16 Slot 3: 15 Slot 4: 14 Slot 5: 18 Slot 0: 25 Slot 1: 20 Slot 2: 19 Slot 3: 21 Slot 4: 22 Slot 5: 23 Slot 0: 27 Slot 1: 28 Slot 2: 26 Slot 3: 24 Slot 4: 29 Slot 5: 9  

(a) [10 pts) Show that the integrand (differential form belongs to the integral sign) is exact in the domain D. (21,3,7/2) s (y sin z dx + x sin z dy + xy cos z dz) (0,1,0) (b) [5 pts) Evaluate the integral in (a) by showing your works in details.

(a) [10 pts) Show that the integrand (differential form belongs to the integral sign) is exact in the domain D. (21,3,7/2) (y sin z dx + x sin z dy + xy cos z dz) (0,1,0) (b) [5 pts] Evaluate the integral in (a) by showing your works in details.