Picard-Lipschitz Theorem: Existence and Uniqueness of Solutions
Given a differential equation (1), if A is an open set, f is continuous in A, and f∈Lloc(A,x), then, for a fixed (t0,x0)∈A, there exists a unique solution of (1) passing through (t0,x0).
Since (t0, x0)∈A and A is open, it is possible to take U1, an open neighborhood of (t0,x0), in which f is bounded. Let H be a bound of f in U1, that is, ||f(t, x)||≤ H, ∀(t, x) ∈ U1. We choose U2, an open neighborhood of (t0, x0), in which f∈L(U2,x), where K is a Lipschitz constant. The set U=U1∩U2
Read MoreMastering English Grammar and Vocabulary: Unit 3
Unit 3: Grammar
Active and Passive Voice
Exercise 1
- Ends
- Will be reduced
- Will be given
- Came
- Were sold
- Don’t miss
Exercise 2
a) Mandarin was taught to the students.
b) The students were taught Mandarin.
a) The phone will be given to her cousin.
b) Her cousin will be given the phone.
a) Some photos are being sent to them.
b) They are being sent some photos.
Exercise 3
Students’ own answers.
Exercise 4
- Decorated
- Spilled
- Checked
- Cut
- Stolen
- Marked
Exercise 5
- Were waiting
- Not to eat
- Is being fed
- Getting
- Was being played
Exercise 6
- Are
Ethical Labor Practices: Tesco’s South African Supply Chain Under Scrutiny
This text from *The Guardian*, dated May 15, 2009, discusses accusations against Tesco for allegedly failing to uphold its commitments regarding the wages and working conditions of agricultural workers in South Africa. It also covers the British chain’s response.
Despite the global crisis, Tesco is making record profits. However, in 2006, the chain was accused of neglecting the living conditions of its workforce. At that time, Tesco committed to paying decent wages that would meet the basic needs
Read MoreJob Application: Editorial Assistant Position
Job Application for Editorial Assistant
I am writing to apply for the position of Editorial Assistant which was advertised in the latest edition of Gulf News. I am currently employed by a Market Research company as a research assistant, but am keen to pursue a career in publishing, because I enjoy reading and write my own poetry.
As you will notice on the attached CV, I graduated in European Literature. At University I gained considerable experience working on the student magazine, so I am familiar
Read MoreCharles Dickens’ Great Expectations: A Detailed Character Analysis
Great Expectations: A Detailed Character Analysis
Great Expectations chronicles the progress of Pip from childhood through adulthood. As he moves from the marshes of Kent to London society, he encounters a variety of extraordinary characters: from Magwitch, the escaped convict, to Miss Havisham and her ward, the arrogant and beautiful Estella. In this fascinating story, Dickens shows the dangers of being driven by a desire for wealth and social status. Pip must establish a sense of self against the
Read MoreSQL Stored Procedures for Driving School Data Management
Exercise 2
Create a Stored Procedure to Retrieve the Number of Practices for a Vehicle in a Given Month
Create a stored procedure that, given a vehicle’s license plate and a month (in numerical format, e.g., 1, 2, 3,…12), returns the number of practice sessions conducted by that vehicle during that month.
Validation Requirements:
- Ensure that the input parameters are not empty.
- Verify that the vehicle exists in the database.
- Confirm that the month is between 1 and 12.
- Display error conditions on the screen.