(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field.

(4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

Questions in Graph Theory:

In the subject of the degree sequence of graph, answer the following:

• When does a d-regular graph have an Eulerian trail? and When does it have an Eulerian circuit? Note: a d-regular graph is one with degree sequence (d, d, d, . . . , d) for example.
• Can a tree be a regular graph? Why or why not

Use Heun's method to find the value of x at t=3 (assume the change in t=3).

The ode is:

(d^2x)/(dt^2)+(.3dx/dt)-4x=0.

And the initial conditions are:

x(0)=1, dx/dt (0)=0.

1) Find the Laplace transform of f(t)=−(2u(t−3)+4u(t−5)+u(t−8))

F(s)=

2) Find the Laplace transform of f(t)=−3+u(t−2)⋅(t+6)

F(s)=

3) Find the Laplace transform of f(t)=u(t−6)⋅t^2

F(s)=

You are in your first month as an internal auditor in the corporate offices of Cover-Up Fraud-Mart, a large regional variety store chain based in Los Angeles. Your manager has just given you a general overview of the company’s problems with fraud. In fact, losses from fraud exceed losses from shoplifting by tenfold, and management wants your perspective on what it can do to proactively detect fraud. From your fraud auditing class, you know that the data-driven approach is one of the most effective detection methods.

Questions

1. Prepare a project plan for implementing the six-step data-driven approach. List the types of team members who should be involved in each step, how long each step will take, and cost estimates for each step.

2. What software package will you need to purchase to complete the process? Provide arguments for your decision.

3. What techniques should be run in steps 4 and 5?

java programming

You will be given two interfaces and two abstract classes, FileTextReader, FileTextWriter, AbstractFileMonitor, and AbstractDictionary. Your job is to create two classes the first class should be named FileManager, the second class should be named Dictionary. The FileManager will implement the interfaces FileTextReader and FileTextWriter and extend the clas AbstractFileMonitor. Your class signature would look something like the following:

public class FileManager extends AbstractFileMonitor implements FileTextReader, FileTextWriter{...

The constructor signature of the FileManager should look like the following:

public FileManager(String filePath){...

The Dictionary will extend the abstract class AbstractDictionary. Your class signature would look something like the following:

public class Dictionary extends AbstractDictionary {...

The constructor signature of the Dictionary should look like the following:

public Dictionary(String path, FileManager fileManager) throws IOException {...

Please read the programmatic documentation of the two interfaces and abstract classes to gain an understanding of what each function should do. This assignment will test your knowledge of reading and writing to a text file, implementing interfaces, extending abstract classes, throwing exceptions and working with collections specifically sets.

This project is accompanied by two test classes, ManagerTest, and DictionaryTest. They will test the functionality of the functions in the interfaces and the abstract methods of classes. Its recommended that you implement the FileManager before working on the AbstractDictionary.

Abstract Dictionary

package homework;

import java.io.IOException;
import java.util.Objects;
import java.util.Set;

* A class that holds a collection of words read from a text file. The collection of words is used in the methods of this class
* methods provided in this class.

public abstract class AbstractDictionary {
  
   private final FileTextReader FILE_READER;
   private final Set ALL_WORDS;
  
   /**
   * Creates an abstract dictionary of words using the text file at the specified path.
   * @param path a path to a text file containing a dictionary of words
   * @param dictionaryFileReader the FileReader used to read from the text file at the specified path
   * @throws IOException thrown if there is a problem reading the file at the path
   */
  

public AbstractDictionary(String path, FileTextReader dictionaryFileReader) throws IOException {
       Objects.requireNonNull(dictionaryFileReader, "The File reader can not be null");
       Objects.requireNonNull(path, "The path can not be null");

      FILE_READER = dictionaryFileReader;

       ALL_WORDS = FILE_READER.getAllLines(path);
   }
  
   /**
   * Returns a set of all the words contained in the dictionary text file.
   * @return a set containing all the words in the dictionary file.
   */
  

public Set getAllWords(){
       return ALL_WORDS;
   }
   /**
   * Counts the number of words in this Dictionary that start with the specified prefix and have a length that is equal or greater
   * than the specified size. If size the specified size is less than 1 then word size is not taken into account.
   * @param prefix the prefix to be found
   * @param size the length that the word must equal or be greater than. If a value less than 1 is specified, all words regardless of their
   * characters size should be considered. In other words if the size parameter is < 1 word size is disregarded in the calculations.
   * @param ignoreCase if true this will ignore case differences when matching the strings. If false this considers
   * case differences when matching the strings
   * @return The number of words that start with the specified prefix
   * @throws IllegalArgumentException if the specified string is null or empty (Meaning contains no characters or only white space or blank)
   */

public abstract int countWordsThatStartWith(String prefix, int size, boolean ignoreCase) throws IllegalArgumentException;

   /**
   * Tests if this Dictionary contains at least one word with a length equal to or greater than the specified size that starts with the specified prefix.
   * If size the specified size is less than 1 then word size is not taken into account.
   * @param prefix the prefix to be found
   * @param size the length that the word must equal or be greater than. If a value less than 1 is specified, all words regardless of their
   * characters size should be considered. In other words if the size parameter is < 1 word size is disregarded in the calculations.
   * @param ignoreCase if true this will ignore case differences when matching the strings. If false this considers
   * case differences when matching the strings
   * @return The number of words that start with the specified prefix
   * @throws IllegalArgumentException if the specified string is null or empty (Meaning contains no characters or only white space)
   */
  

public abstract boolean containsWordsThatStartWith(String prefix, int size, boolean ignoreCase) throws IllegalArgumentException;
   * Returns a set of all the words within in this Dictionary that start with the specified prefix and have a length that is equal or greater
   * than the specified size. If size the specified size is less than 1 then word size is not taken into account.
   * @param prefix the prefix to be found
   * @param size the length that the word must equal or be greater than. If a value less than 1 is specified, all words regardless of their
   * characters size should be considered. In other words if the size parameter is < 1 word size is disregarded in the calculations.
   * @param ignoreCase if true this will ignore case differences when matching the strings. If false this considers
   * case differences when matching the strings
   * @return A list of all strings that start with the specified prefix
   * @throws IllegalArgumentException if the specified string is null or empty (Meaning contains no characters or only white space)

   public abstract Set getWordsThatStartWith(String prefix, int size, boolean ignoreCase) throws IllegalArgumentException;
}

AbstractFileMonitor

package homework;

import java.io.File;
import java.io.FileInputStream;
import java.io.IOException;
import java.security.DigestInputStream;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
* A class with methods to monitor a text file.

public abstract class AbstractFileMonitor {

   private MessageDigest messageDigest;
   private int currentCheckSum;
   private boolean hasChanged = false;
   private long nextUpateTime = 0;
  
   public AbstractFileMonitor(String path){
setFilePath(path);
}
   * Updates the variables that correspond to the file being monitored
   * This function has been throttled such that it will only update values every 250 ms. In other words successive calls to this
   * function in time intervals that are less than 250 ms will yield no change.
   * @throws IOException Thrown if any type of I/O exception occurs while writing to the file
   * @throws IllegalStateException If the {@link #setFile(String)} method in not invoked with a proper file prior to invocation of this
   * @throws NoSuchAlgorithmException If the computer's OS is missing the SHA-256 hashing algorithm
   * method. In other words if no file is currently set to be monitored.

   public void update() throws IOException, IllegalStateException, NoSuchAlgorithmException {

      if(messageDigest == null){
           messageDigest = MessageDigest.getInstance("SHA-256");
       }
      
       if(nextUpateTime > System.currentTimeMillis()) {
           hasChanged = false;
          
          return;
       }
      
      nextUpateTime = System.currentTimeMillis() + 250;
      
       File file = new File(getFilePath());
      
       if(!file.exists()) return;
      
try (DigestInputStream dis = new DigestInputStream(new FileInputStream(getFilePath()), messageDigest)) {
while (dis.read() != -1) ;
messageDigest = dis.getMessageDigest();
}

StringBuilder result = new StringBuilder();
for (byte b : messageDigest.digest()) {
result.append(String.format("%02x", b));
}
  
hasChanged = currentCheckSum != result.toString().hashCode();
  
currentCheckSum = result.toString().hashCode();
   }
   * Tests if the file being monitored has changed since the last time {@link #update()} was invoked
   public boolean hasChanged(){
       return hasChanged;
}
   public abstract void setFilePath(String path);
   public abstract String getFilePath() throws IllegalStateException;
}

FileTextReader

package homework;

import java.io.IOException;
import java.util.Collection;
import java.util.List;
import java.util.Set;

public interface FileTextReader{
   public String readText(String path) throws IOException;
   * @throws IOException Thrown if any type of I/O exception occurs while writing to the file

public Set getAllLines(String path) throws IOException;
   * @throws IOException Thrown if any type of I/O exception occurs while writing to the file
public String getLastLine(String path) throws IOException;
}

FileTextWriter

package homework;

import java.io.IOException;

public interface FileTextWriter {
   * @throws IOException Thrown if any type of I/O exception occurs while writing to the file
   * @throws IllegalArgumentException if the specified string is null or empty (Meaning contains no characters or only white space)
   public void writeToFile(String string, String path) throws IOException, IllegalArgumentException;
}

22.7. Services that require a password often impose constraints to prevent the user from choosing a password that is easy to guess. Consider a service that requires an 8-character password, using the 26 Roman letters (both lowercase and uppercase) and the 10 Arabic numerals.

(a) How many possible passwords are there?

(b) How many such passwords use no numerals? No lowercase letters? No uppercase letters?

integrating factor and the ED solution y (12 + 8x + 6y ^ 2) dx + x (16 + 8x + 12y ^ 2) dy = 0

___

Indicate the option that contains the general solution to: (-4 + 6x + 2y) dx + (1 + 2x + 8y) dy = 0

__

y+xy`=cos(x)

__

Using the integrating factor method, solve and select the option that contains the general solution to the DE: (1 + x ^ 2y) dy = 2xy ^ 2dx

__

Using the integrating factor method, solve and select the option that contains the general solution to the DE: (1 + x^2y)dy=3xy^2dx

Find vectors of the Frenet frame of the curve at any point of the curve x = a(t − sin t), y = a(1 − cos t), z = 4a cos t , where a is a positive constant

Solve the given initial-value problem.

X' =

X +

, X(0) =

X(t) =

You may have used shortcuts to see if a number is divisible by another. Where do they come from? Why are some numbers left out? Could you prove these shortcuts?

Math201:

Directions: Annual Percentage Yield (APV). Find the annual percentage yield (to the nearest 0.01%) in each case.

1.) A bank offers an APR of 3.2% compounded monthly.

Directions: Continuous Compounding. Use the formula for continuous compounding to compute the balance in each account after 1,5, and 20 years. Also, find the APY for this account.

1.) A $2,000 deposit in an account with an APR of 3.1%

Solve the following IVPs:

A. y"" +2y′′+y = 4et, y(0)=1,  y′(0)=1, y′′(0)=1, y′′′(0)=1.

B. y′′′+25y′ = 325e−t, y(0) = 0, y′(0) = 0, y′′(0) = 0.