(Of two or more integers) the largest integer which divides each number of a set of integers
- -When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are -$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
- -
When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are -$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
-(Of two or more integers) the largest integer which divides each number of a set of integers
+ +When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are +$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
+ +
When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are +$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
+An infinite series of points extending infinitely in two directions.
-
- An infinite series of points extending infinitely in two directions.
+
+ 
-
-
- An equation giving the valuse for which a quadratic polynomial is equal to 0
- -When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
-
+
+
+ An equation giving the valuse for which a quadratic polynomial is equal to 0
+ +When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
+
A value representing a quantity along a line. -
The set of all real numbers is usually denoted $\mathbb{R}$
- -
- A value representing a quantity along a line. +
The set of all real numbers is usually denoted $\mathbb{R}$
+ +
+ A space $X$ is said to be compact if every open covering $A$ of $X$ contains a finite subcollection that also covers $X$
- -A space $X$ is said to be compact if every open covering $A$ of $X$ contains a finite subcollection that also covers $X$
+ +Of a set, a collection $A$ of subsets of a space $X$ where the union of element of $A$ is equal to $X$.
- -Of a set, a collection $A$ of subsets of a space $X$ where the union of element of $A$ is equal to $X$.
+ +
-
+ - if($el.parent().is("td") || $el.parent().is("th") ) { - // assume we are in a td or th tag, go 2 levels up - var cols = $el.parent().parent().children().length; - $el.parents().eq(1).after( - // .parents().eq(1) was formerly written as .parent().parent() - "
+ if($el.parent().is("td") || $el.parent().is("th") ) { + // assume we are in a td or th tag, go 2 levels up + var cols = $el.parent().parent().children().length; + $el.parents().eq(1).after( + // .parents().eq(1) was formerly written as .parent().parent() + "
When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are -$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
- -When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are -$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
- - - - - - + + + + +When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are +$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
+ +When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are +$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
+ + + + + + \ No newline at end of file diff --git a/Euclids Library- Main/Euclids Library/linesonplane.png b/www/linesonplane.png similarity index 100% rename from Euclids Library- Main/Euclids Library/linesonplane.png rename to www/linesonplane.png diff --git a/Euclids Library- Main/Euclids Library/ordertopology.html b/www/ordertopology.html similarity index 97% rename from Euclids Library- Main/Euclids Library/ordertopology.html rename to www/ordertopology.html index 1be65ecf..b4a28fbb 100644 --- a/Euclids Library- Main/Euclids Library/ordertopology.html +++ b/www/ordertopology.html @@ -1,37 +1,37 @@ - - - - -A topology on an ordered set $X$ generated by the basis: $\\$ 1. all intervals $(a, b)$ $\\$ 2. $[a_0, b)$ if a least element $a_0$ exists $\\$ 3. $(a, b_0)$ if a greatest element $b_0$ exists
- -A topology on an ordered set $X$ generated by the basis: $\\$ 1. all intervals $(a, b)$ $\\$ 2. $[a_0, b)$ if a least element $a_0$ exists $\\$ 3. $(a, b_0)$ if a greatest element $b_0$ exists
+ +
-
-
- Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact.
- - -
+
+
+ Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact.
+ + +
- In the mid-to-late 1930s, Japanese forces in Manchukuo had sporadic border clashes with the Soviet Union and Mongolia. The Japanese doctrine of Hokushin-ron, which emphasised Japan's expansion northward, was favoured by the Imperial Army during this time. With the devastating Japanese defeat at Khalkin Gol in 1939 and ally Nazi Germany pursuing neutrality with the Soviets, this policy would prove difficult to maintain. Japan and the Soviet Union eventually signed a Neutrality Pact in April 1941, and Japan adopted the doctrine of Nanshin-ron, promoted by the Navy, which took its focus southward, eventually leading to its war with the United States and the Western Allies.
- -When civil war broke out in Spain, Hitler and Mussolini lent military support to the Nationalist rebels, led by General Francisco Franco. The Soviet Union supported the existing government, the Spanish Republic. Over 30,000 foreign volunteers, known as the International Brigades, also fought against the Nationalists. Both Germany and the USSR used this proxy war as an opportunity to test in combat their most advanced weapons and tactics. The bombing of Guernica by the German Condor Legion in April 1937 heightened widespread concerns that the next major war would include extensive terror bombing attacks on civilians. The Nationalists won the civil war in April 1939; Franco, now dictator, bargained with both sides during the Second World War, but never concluded any major agreements. He did send volunteers to fight on the Eastern Front under German command but Spain remained neutral and did not allow either side to use its territory
- - - -
- In July 1937, Japan captured the former Chinese imperial capital of Beijing after instigating the Marco Polo Bridge Incident, which culminated in the Japanese campaign to invade all of China. The Soviets quickly signed a non-aggression pact with China to lend materiel support, effectively ending China's prior co-operation with Germany. Generalissimo Chiang Kai-shek deployed his best army to defend Shanghai, but, after three months of fighting, Shanghai fell. The Japanese continued to push the Chinese forces back, capturing the capital Nanking in December 1937. After the fall of Nanking, tens of thousands if not hundreds of thousands of Chinese civilians and disarmed combatants were murdered by the Japanese. - In March 1938, Nationalist Chinese forces won their first major victory at Taierzhuang but then the city of Xuzhou was taken by Japanese in May. In June 1938, Chinese forces stalled the Japanese advance by flooding the Yellow River; this manoeuvre bought time for the Chinese to prepare their defences at Wuhan, but the city was taken by October.
- -
+ In the mid-to-late 1930s, Japanese forces in Manchukuo had sporadic border clashes with the Soviet Union and Mongolia. The Japanese doctrine of Hokushin-ron, which emphasised Japan's expansion northward, was favoured by the Imperial Army during this time. With the devastating Japanese defeat at Khalkin Gol in 1939 and ally Nazi Germany pursuing neutrality with the Soviets, this policy would prove difficult to maintain. Japan and the Soviet Union eventually signed a Neutrality Pact in April 1941, and Japan adopted the doctrine of Nanshin-ron, promoted by the Navy, which took its focus southward, eventually leading to its war with the United States and the Western Allies.
+ +When civil war broke out in Spain, Hitler and Mussolini lent military support to the Nationalist rebels, led by General Francisco Franco. The Soviet Union supported the existing government, the Spanish Republic. Over 30,000 foreign volunteers, known as the International Brigades, also fought against the Nationalists. Both Germany and the USSR used this proxy war as an opportunity to test in combat their most advanced weapons and tactics. The bombing of Guernica by the German Condor Legion in April 1937 heightened widespread concerns that the next major war would include extensive terror bombing attacks on civilians. The Nationalists won the civil war in April 1939; Franco, now dictator, bargained with both sides during the Second World War, but never concluded any major agreements. He did send volunteers to fight on the Eastern Front under German command but Spain remained neutral and did not allow either side to use its territory
+ + + +
+ In July 1937, Japan captured the former Chinese imperial capital of Beijing after instigating the Marco Polo Bridge Incident, which culminated in the Japanese campaign to invade all of China. The Soviets quickly signed a non-aggression pact with China to lend materiel support, effectively ending China's prior co-operation with Germany. Generalissimo Chiang Kai-shek deployed his best army to defend Shanghai, but, after three months of fighting, Shanghai fell. The Japanese continued to push the Chinese forces back, capturing the capital Nanking in December 1937. After the fall of Nanking, tens of thousands if not hundreds of thousands of Chinese civilians and disarmed combatants were murdered by the Japanese. + In March 1938, Nationalist Chinese forces won their first major victory at Taierzhuang but then the city of Xuzhou was taken by Japanese in May. In June 1938, Chinese forces stalled the Japanese advance by flooding the Yellow River; this manoeuvre bought time for the Chinese to prepare their defences at Wuhan, but the city was taken by October.
+ +A set X which has an associated collection of subsests constituting a topology $T$
- -A set X which has an associated collection of subsests constituting a topology $T$
+ +A collection of subsets $T$ over of a space $X$ $(T, X)$ such satisfying: \\ 1. All unions of set in $T$ are in $T$ \\ 2. All finite intersections of sets in $T$ are in $T$ \\ 3. $X$ and $\emptyset$ are in $T$
- -A collection of subsets $T$ over of a space $X$ $(T, X)$ such satisfying: \\ 1. All unions of set in $T$ are in $T$ \\ 2. All finite intersections of sets in $T$ are in $T$ \\ 3. $X$ and $\emptyset$ are in $T$
+ +