Page
%P
-
Article
Open AccessFeasibility and beneficial effects of an early goal directed therapy after cardiac arrest: evaluation by conductance method
Although beneficial effects of an early goal directed therapy (EGDT) after cardiac arrest and successful return of spontaneous circulation (ROSC) have been described, clinical implementation in this period see...
-
Article
Extracorporeal apheresis therapy for Alzheimer disease—targeting lipids, stress, and inflammation
Current therapeutic approaches to Alzheimer disease (AD) remain disappointing and, hence, there is an urgent need for effective treatments. Here, we provide a perspective review on the emerging role of “metabo...
-
Book
-
Chapter
Mathematical Induction
Mathematical induction is a technique that is useful for proving many theorems. We describe this technique in detail, and give a number of applications of it. We also explain the well-ordering principle, and s...
-
Chapter
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic is the assertion that every natural number greater than 1 can be uniquely (up to the order of its factors) factored into a product of prime numbers. We present a direct pr...
-
Chapter
Fundamentals of Euclidean Plane Geometry
We describe the fundamentals of Euclidean geometry of the plane. We develop the concepts of congruence and similarity of triangles, and, in particular, prove that corresponding sides of similar triangles are i...
-
Chapter
Sending and Receiving Secret Messages
We describe the RSA method for sending secret messages. This remarkable method allows a person who wishes to receive messages to announce to the world how messages are to be sent and, nonetheless, be the only ...
-
Chapter
Rational Numbers and Irrational Numbers
The “rational numbers” are the fractions; we discuss their basic properties in this chapter. We show that there are “irrational numbers,” including the square root of two. The collection of all rational and al...
-
Chapter
Introduction to the Natural Numbers
The natural numbers are the numbers that we count with; that is, the numbers 1, 2, 3, 4, 5, 6 and so on. We describe the basic properties of the natural numbers. We explain why the product of two negative inte...
-
Chapter
Modular Arithmetic
Modular arithmetic is a way of studying divisibility properties of natural numbers. It provides techniques for easily answering questions such as whether 3 plus 2 to the power 3,000,005 is divisible by 7. It h...
-
Chapter
Fermat’s Little Theorem and Wilson’s Theorem
Fermat’s Little Theorem states that, for every prime number p, if p does not divide the natural number a, then a to the power p − 1 leaves a remainder of 1 upon division by p. This beautiful theorem has a number ...
-
Chapter
An Introduction to Infinite Series
An infinite series is an expression of the form a 1 + a 2 + a 3 + ⋯, where each a i is a real number. We discuss the question of when an infinite series has a “sum” in a precise sense that we will explain. A seri...
-
Chapter
Sizes of Infinite Sets
How many natural numbers are there? How many even natural numbers are there? How many rational numbers are there? How many real numbers are there? How many points are there in the plane? How many sets of natur...
-
Chapter
Constructibility
A straightedge is a ruler-like device that has no measurements marked on it; it is used to construct lines through any two given points. We investigate the famous question of which geometric figures can be con...
-
Chapter
The Euclidean Algorithm and Applications
We describe the Euclidean Algorithm, which provides a way of expressing the greatest common divisor of two natural numbers as a “linear combination” of the numbers. This algorithm has a number of important app...
-
Chapter
Some Higher Dimensional Spaces
Four-dimensional Euclidean space is defined, as is n-dimensional Euclidean space for every natural number n. A few basic properties of n-dimensional spaces are explored. Some infinite-dimensional spaces are also ...
-
Chapter
The Complex Numbers
The polynomial x 2 + 1 does not have any roots within the set of real numbers. A new number, called i, is introduced as a root of that polynomial. The complex numbers are all the numbers of the form a + bi where
-
Book
-
Chapter
Rational Numbers and Irrational Numbers
The “rational numbers” are the fractions; we discuss their basic properties in this chapter. We also show that there are distances that are not rational numbers, which are called “irrational numbers”. In parti...
-
Chapter
Introduction to the Natural Numbers
We describe the basic properties of the natural numbers; that is, the numbers 1,2,3,4,5,6 and so on. We prove that there is no largest prime number, and discuss two famous unsolved problems.
