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L’Hospital’s Rule – maximal filter
L’Hospital’s Rule
If a limit is in one of the indeterminate forms “infinity over infinity” or ”zero over zero,” then we have
That is, the limit is the same after taking the derivatives of both the numerator and denominator. L’Hospital’s Rule may be applied as many times as needed. Students often misapply the Rule by using it when the limit is not in one of the above indeterminate forms. This often results in an incorrect evaluation of the limit.
liar paradox
According to a legend found in many ancient sources (including a letter of St. Paul), there was a certain Cretan named Epimenides who proclaimed, “All Cretans are liars.” This statement involves a contradiction on its face, since if all Cretans are liars and Epimenides is a liar, then the statement must be false, that is, not all Cretans are liars.
There are many versions of the liar paradox, and many puzzles based upon it. The simplest form is, “This statement is false,” a statement that’s true if it’s false and false if it’s true.
limit
ARTICLE
The concept of a limit is central to our modern understanding of the differential and integral calculus. Naively, an expression involving a variable limits to some value L if it becomes arbitrarily close to L for appropriate choices of the variable. See the article for a full exposition.
limit comparison test
A test for the convergence of a series. Refer to the related article for a complete description.
Related article: Series
limit point
Given a sequence of points ai , i = 1, 2, 3, ... , a point L is called a limit point of the sequence if every neighborhood of L contains all but finitely many of the ai .
If X is a metric space, then L is a limit point of X if a sequence from X may be chosen so that L is the limit point of that sequence. Limit points are not in general unique.
See also: accumulation point, perfect set.
line
Naively, “line” is a primitive concept, generally connoting a straight path through space. In mathematics this notion is abstracted formally in different ways depending on the type of mathematics under consideration.
Geometry: Euclid defined the term “line” as a point extended indefinitely in space. However, in modern Euclidean geometry, “line” is a primitive term, left purposely undefined, whose meaning is informed purely by the axioms in which the term appears.
Analytic geometry: Any set of ordered pairs (x, y) of all the points in the Cartesian plane satisfying an equation of the form ax + by + c = 0, where a, b, and c are real numbers. (There are three common and useful forms of this equation; see the entry for linear function.) This definition of the line may be generalized to a Euclidean space of n dimensions to include any set of ordered n-tuples (x1, x2, ..., xn) satisfying a1x1 + a2x2 + ... + anxn + c = 0. In polar coordinates a line is any set of all the ordered pairs (r, q) satisfying an equation of the form r = p sec (q - a), where p is the perpendicular distance from the pole to the line, and a is the angle of inclination of the line to the polar axis.
linear function
A polynomial function of degree one. The graph is a straight line. There are three special forms of the linear equation:
In the first form, called the slope-interecept form, m is the slope of the line and b is the y-intercept. In the second form, called the point-slope form, m is the slope of the line and (x 0, y 0) is any point on the line. In the third form a is the x-intercept and b is the y-intercept.
linear order
A set X is said to be a linear order if there is a relation “<” on its elements such that for all distinct x and y in X, either x < y or y < x, but not both. Sometimes also called a total order.
Cf. partial order, dense linear order.
linear space
See vector space.
locally compact
A topological space is locally compact if every point of the space has a neighborhood whose closure is compact.
locally integrable
A measurable function on a finite-dimensional real space with range in the complex numbers is said to be locally integrable if for all compact sets K we have:
The space of all such functions is denoted L1LOC.
Cf. Lebesgue integral.
locus
A set of points satisfying a (geometric) condition. E.g., a parabola is the locus of points equidistant from a given point and a given line.
log
See logarithm.
logarithm
A logarithm is an expression of the form log bA, where b is called the base of the logarithm and A is called the argument, and it evaluates as the exponent to which the base must be raised to return the argument. Logarithms may be viewed as an alternative form of exponential notation, since we have log bA = x if and only if bx = A. A common logarithm is a logarith with base 10, and a natural logarithm (often written “ln”) is a logarithm with base e.
Cf. laws of logarithms, Euler number.
logarithmic function
A function which acts on its argument with a logarithm.
logic
In the broadest sense, the term “logic” refers to the reasoning processes or forms of argument people use to reach valid conclusions from one or more premises (assumptions). More specifically, any particular logic is a system of inference, by which one reasons from premises to conclusions, or by which one validates one’s conclusions. An incorrect inference, i.e., one which violates the rules of logic, is called a fallacy, and conclusions reached by means of a fallacy are called invalid. (By contrast, when the premises of an argument are false but the argument is formally valid, then the conclusion is called unsound.)
Ordinary logic falls into two broad types; deductive and inductive. Reasoning from general premises to a specific conclusion is deductive. For example, knowing that heavy clouds and a brisk breeze often signal an oncoming storm, one could conclude deductively that it might rain this afternoon. Reasoning from specific premises to a general conclusion, on the other hand, is inductive. For example, observing that every raven one sees is black, one might conclude inductively that all ravens are black. Deductive conclusions are necessarily true if the premises are true and the logic used formally valid (free of fallacy). Inductive conclusions are never certain, but are only more or less reliable. In mathematics only deductive reasoning is used (but see the entry for mathematical induction).
Mathematicians use symbolic logic, that is, logic that can be reduced to a purely syntactical system which disregards the semantic content of any of the symbols in use. The two primary forms of symbolic logic are the propositional calculus and the predicate calculus. The latter augments the former with existential and universal quantifiers (permitting sentences that begin with “there exists ___ such that” and “for all ___”). When quantification is permitted only over elements of the universe of discourse, the logic is called “first-order.” When quantification over classes of objects and/or over predicates is permitted, it is called second-order logic.
Cf. axiom, fuzzy logic, equational logic, syllogism, Hempel's Ravens Paradox.
lower bound
A lower bound of an ordered set is an element which is less than or equal to every element in the set.
Cf. greatest lower bound.
l.u.b
Abbreviation for least upper bound.
Mahlo cardinal
A cardinal k is called Mahlo if it is inaccessible and {a < k:a is inaccessible} is stationary in k.
matrix
A rectangular array of, usually, real or complex numbers, organized into rows and columns. When specifying the size of a matrix, the number of rows is stated first. For example, here is a 2 by 4 real matrix:
The numbers in a matrix are called entries, and are specified by the row and then column in which they appear. In the example above, for instance, A1,2 is the entry in the first row and second column, i.e. 4.98.
Two matrices of the same size (with the same number of rows and columns) can be added componentwise, that is (A + B)i, j = Ai, j + Bi, j. Matrix multiplication is a little more complicated. A matrix M can be multiplied on the right by another matrix T only if T has the same number of rows as M has columns, and the result will have as many rows as M and as many columns as T. The entry of M T appearing in row i and column j is the sum of the pairwise products of the entries in row i of M and column j of T. In other words, if M is an n by m matrix, T must be an m by l matrix, and
Notice that this operation is not, in general, commutative.
A matrix that has the same number of rows and columns is called a square matrix, and these are particularly interesting as they can be added and multiplied, and the results will have the same dimensions.
The most standard use of matrices is to represent linear operators, but they have a wide variety of other uses, and are interesting to study in their own right. Matrices can have a large number of different types of entries, including various kinds of numbers, elements of abstract fields and other algebraic structures, functions, and so forth.
maximal filter
See filter.
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