On 14 July 2002, during Bastille Day celebrations, a man tried to shoot Chirac. A lone gunman with a rifle hidden in a guitar case, the would-be assassin fired a shot toward the presidential motorcade. He was then overpowered by bystanders. The gunman, Maxime Brunerie, had psychiatric testing.
On 15 December 2011, Chirac was found guilty for corruption. He was allowed to serve his 2-year sentence without prison time.
As a former president, Chirac had a lifetime pension. He was a member for life of France's Constitutional Council.
In 1956, he married Bernadette Chodron de Courcel. They had two daughters: Laurence (1958–2016) and Claude (born 1962). Claude is a public relations assistant and personal adviser. Laurence, who had anorexia nervosa in her youth, did not participate in the political activities of her father.
In 2005, Chirac suffered a stroke. He had been suffering from memory loss and was in ill health. On 10 December 2015, Chirac was hospitalized in Paris for undisclosed reasons, and although his state of health did not "give any cause for concern", he would remain under the intensive care unit.
Chirac died at his home in Paris on 26 September 2019 at the age of 86.
Mario Jeckle
Mario Jeckle (25 August 1974 – 11 June 2004) was a German computer scientist.
From 1997 to 2003, Jeckle attended the University of Applied Science in Augsburg. In 1998, he received his computer science degree for his thesis (paper) "Prozeßkettenmodellierung am Beispiel der Gießwerkzeugentwicklung und prototypische Implementierung auf Basis des EDM/PDM - Systems Metaphase" (An example of process chain modelling in casting tool development and prototype implementation on basis of the EDM/PDM - Systems Metaphase). At Augsburg, he taught Java, Java Threads, XML and software engineering.
In 2003, Jeckle became a professor at the University of Applied Sciences in Furtwangen. He taught about XML, databases, software engineering, and "eBusiness" (electronic business).
Jeckle was also a W3C and OMG representative of DaimlerChrysler Research and developed technical standards for XML, UML 2.0, and others. At the beginning of 2004, he was a member of the Technical Architecture Group of the World Wide Web Consortium (W3C). Jeckle was also an author of books and a well-known speaker at conferences and seminars (information groups).
Jeckle was a member of the International Red Cross. On 11 June 2004, he died while giving aid to others who had a car accident on a German highway. While helping, another driver lost control and hit Jeckle and another man.
Eighty Years' War
The Eighty Years' War, also called Dutch Revolt, was a war between the Netherlands and Spain. It lasted from 1568 to 1648 but was interrupted by a truce between 1609 and 1621. The Peace of Westphalia ended the war. The Netherlands achieved independence in the Eighty Years' War.
Exponent
In mathematics, an exponent indicates how many copies of a number (known as the base) is multiplied together.
For example, in the number formula_1, 5 is the base and 4 is the exponent. This can be read as "5 to the power of 4". Therefore, in this example, four copies of 5 are multiplied together, which means that formula_2.
In general, given two numbers formula_3 and formula_4, the number formula_5 can be read as "formula_3 raised to the power of formula_4" or "formula_3 to the formula_4th power", where formula_4 is the exponent.
Bases raised to the power of two, formula_11, can be read the base "squared", because the area of a square with a side length of x is formula_12.
Similarly, bases raised to the power of three, formula_13, can be called the base "cubed", because the area of a cube with a side length of x is formula_14.
Exponents are used in algebra. Exponentiation is a way of shortening the process/equation of repeatedly multiplying many copies of a number.
An exponent is a number or symbol, placed above and to the right of the expression to which it applies, that indicates the number of times the expression is used as a factor. For example. the exponent "x" in formula_15 indicates x copies of formula_16 are multiplied together.
There are some basic rules used in exponentiation
1. Product Rule:
Proof - When multiplying a base term by two different exponents formula_18, there are "m" number of "a'"s and "n" number of "a"'s.
Since exponentiation is simply repeated multiplication of a base term, we get that:
therefore formula_20.
For example, formula_21. This is true because at first, we had 2 threes times 4 threes which, when multiplied together, gives us 3 multiplied by itself 6 times, or 4+2 times.
2. Quotient Rule:
Proof (when m > n) - formula_23
By grouping the corresponding "a"’s, we get:
Since corresponding "a"’s become 1, we are then left with
therefore, formula_26.
For example, formula_27. Notice that formula_28, which confirms the proof.
In fact, this property holds when m < n as well, but in that case, we get negative exponents instead. When m < n, the format of the problem becomes formula_29.
For more information on negative exponents, see § Negative exponents below.
3. Zero Rule:
Proof: we showed in #1 that formula_31.
This equation holds true for formula_32 as well, so
If this statement is true where formula_34, then formula_32 must equal one. That is:
A negative exponent is the reciprocal of a number with a positive exponent which can be mathematically represented as formula_37 In the same way that regular exponents are considered repeated multiplication, negative exponents can be considered repeated division.
Negative exponents can also be different from -1. In this case the negative exponent can be separated from the positive exponent, so formula_38
The formula for a negative exponent in terms of a whole number exponent is formula_39 There are more than one ways to prove this. The first proof involves a pattern form. It is not always a good idea to use patterns because in cases like exponents they may not be complete patterns for every case, but the proof still holds.
Proof #1: Why is formula_40 ?
divide both sides by formula_44
The second proof proves exactly the same thing but dives further into matching equalities
Proof #2: formula_47
everything in the next row is proved by the rules in the exponent section
therefore formula_52
If we also have a multiplication, like in formula_53, it would be written as formula_54
A complex number can also be taken to the power of another complex number. The relationship between imaginary numbers and trigonometry, which produces the multiplication formula
This formula basically states that the multiplication of complex numbers is related to addition of angles.
There are some relationships between functions and “infinite degree polynomials”, that are called series representations of the functions. The important series representations that we will focus on are:
From these representations, you should notice that the terms of the sine and cosine series combine, along with some sign changes, to make the exponential series. By introducing the number i into the equation, we get
This result is often used as the beginning of the definition for complex number exponents.
There are some cases where we are raising other numbers to the power i. For example, we can use the following relationship formula_60. Here using base 2 and exponent i, we get
When using base i and exponent i, we get
Now we can look at a general formula.
We can rewrite the formula_64 in the trigonometric form formula_65.
By the relationship we found between trigonometry and complex exponent, we can rewrite the equation in complex exponential form, formula_66.
Substituting this equation in the exponent in our original equation, we get formula_67
This can be simplified to be formula_68
The real and imaginary parts of the exponent can be simplified separately to obtain to result formula_69