12 Congruent Triangles
... 2. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two lines intersect to form a right angle, then the lines are perpendicular. 4. Through any two points, there exists exactly one line. ...
... 2. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two lines intersect to form a right angle, then the lines are perpendicular. 4. Through any two points, there exists exactly one line. ...
FELL TOPOLOGY ON HYPERSPACES OF LOCALLY COMPACT
... In this paper we shall study the Fell topology of the hyperspace Cld∗F (X) of closed subsets of a regular locally compact non-compact space X. We recall that the Fell topology is generated by the subbase consisting of sets hU i+ = {F ∈ Cld∗F (X) : F ∩ U 6= ∅} and hKi− = {F ∈ Cld∗F (X) : F ∩ K = ∅} w ...
... In this paper we shall study the Fell topology of the hyperspace Cld∗F (X) of closed subsets of a regular locally compact non-compact space X. We recall that the Fell topology is generated by the subbase consisting of sets hU i+ = {F ∈ Cld∗F (X) : F ∩ U 6= ∅} and hKi− = {F ∈ Cld∗F (X) : F ∩ K = ∅} w ...
EXISTENCE AND PROPERTIES OF GEOMETRIC QUOTIENTS
... descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we thus obtain a slightly improved version of Keel and ...
... descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we thus obtain a slightly improved version of Keel and ...
Proving that a quadrilateral is a Parallelogram
... If an angle of a quadrilateral is supplementary to both of is consecutive angles then the quadrilateral is a parallelogram. A ...
... If an angle of a quadrilateral is supplementary to both of is consecutive angles then the quadrilateral is a parallelogram. A ...
Elementary Number Theory
... n · 1 > n · k despite that fact that k is positive and so 1 ≤ k. This is impossible because it violates the same Property of Inequalities. qed 1.3 Definition An integer n is even (or has even parity) if it is divisible by 2 and is odd (or is of odd parity) otherwise. 1.4 Lemma Recall that |a| equals ...
... n · 1 > n · k despite that fact that k is positive and so 1 ≤ k. This is impossible because it violates the same Property of Inequalities. qed 1.3 Definition An integer n is even (or has even parity) if it is divisible by 2 and is odd (or is of odd parity) otherwise. 1.4 Lemma Recall that |a| equals ...
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.