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Mathematical Analysis
Mathematical analysis is, simply put, the study of limits and how they can be manipulated. Starting with an exhaustive study of sets, mathematical analysis then continues on to the rigorous development of calculus, differential equations, model theory, and topology. Topics including real and complex analysis, differential equations and vector calculus can be discussed in this category.
2,575 Questions
A Laplace transform is a mathematical operator that is used to solve differential equations. This operator is also used to transform waveform functions from the time domain to the frequency domain and can simplify the study of such functions. For continuous functions, f(t), the Laplace transform, F(s), is defined as the Integral from 0 to infinity of f(t)*e-stdt. When this definition is used it can be shown that the Laplace transform, Fn(s) of the nth derivative of a function, fn(t), is given by the following generic formula:
Fn(s)=snF(s) - sn-1f0(0) - sn-2f1(0) - sn-3f2(0) - sn-4f3(0) - sn-5f4(0). . . . . - sn-nfn-1(0)Thus, by taking the Laplace transform of an entire differential equation you can eliminate the derivatives of functions with respect to t in the equation replacing them with a Laplace transform operator, and simple initial condition constants, fn(0), times a new variable s raised to some power. In this manner the differential equation is transformed into an algebraic equation with an F(s) term. After solving this new algebraic equation for F(s) you can take the inverse Laplace transform of the entire equation. Since the inverse Laplace transform of F(s) is f(t) you are left with the solution to the original differential equation.
What is the theory for a number divided by zero?
The theory is simple: the answer for dividing by zero is NaN, "not a number". Zero cannot divide into any number, including itself. Conceptually, there is no correct finite answer because you can always take away more zeros from the numerator. To get this answer in mathematical terms, we instead look at what is called a limit as a function approaches a point that is NaN. For example, take the function f(x)=1/(x^2). This function has no value at x=0. However, if we look at the value of the function at x values very, very, very (actually, infinitely) close to x=0, we can see a trend and use that to understand what happens at x=0. In the case of 1/(x^2) the function approaches infinity at very small negative and very small positive values, so we conclude that the limit as x approaches 0 of 1/(x^2) is infinite, even though the function has no value at x=0.
The above applies to many, but not all, algebraic structures. Specifically, because it models the world we live in, it's true for fields. The axioms of a field require that any number multiplied by zero is equal to zero and that any number multiplied by its inverse is equal to one. Therefore, 0*0-1 equals both 1 and 0 according to these rules which is a contradiction, therefore making division by zero in general undefinable.
We reach a delema in our understanding of what the results are saying. If we look at the question as we approach zero from a positive side we see our limit is positive infinity and if we approach it for a negative direction we reach a limit of negative infinity. Here we throw up our hands and say this can not be and dismiss the answers as impossibles. What if our understanding of infinity is wrong as in thinking infinity must be a number and thus only one answer is allowed. What if infinity is an area of understanding that we do not yet possess. What if division by zero is just what the answers say. the only understanding we can derive from both being correct is that it represents a joining point of negative and positive numbers.
Let us look at Einstin's theory
A set is a collection of elements usually of the same kind. Each set can be denoted by a capital letter. A set may be defined by listing the members or describing the members. A set is said to be finite when all its members can be listed.
The above is an intuitive definition which seemed sufficient for several years but then contradictions such as Russel's paradox were pointed out, so a great deal of work had to be done to develop axioms that could define sets without allowing any paradoxes. These axioms require considerable mathematical sophistication to understand and often are the basis for a senior level college math
A set is a collection of items.
For example the set of furniture in a bedroom might be {bed, beside cupboard, dressing table, wardrobe, chair}; this is an example of a finite set as there are a finite number of elements in the set.
Another example is the counting numbers {1, 2, 3, ...}; this is an example of an infinite set as there is no end to the members of the set and it is impossible to list everyone of them.
A countable set is a set that can have each of its members put into relation with one of the counting numbers. Both the above example sets are countable.
Explain four ways to prove that two triangle are congruent?
one way is to use the corresponding parts. if they are congruent then the two triangles are congruent. i don't know any other ways without seeing the triangles or any given info. sorry i couldn't help more.
Does the set of all sets other than the empty set include the empty set?
The collection of all sets minus the empty set is not a set (it is too big to be a set) but instead a proper class. See Russell's paradox for why it would be problematic to consider this a set. According to axioms of standard ZFC set theory, not every intuitive "collection" of sets is a set; we must proceed carefully when reasoning about what is a set according to ZFC.
What is the formula for the area of the circle?
r2*pi=A
π ● r²
or pi times the radius squared
If you are given the diameter just divide that by 2 giving you the radius. After that square the radius (multiply by it by itself). Then multiply that answer by 3.14 or pi.
Area of a circle = pi*radius2
How do you find the inverse Fourier transform from Fourier series coefficients?
To find the inverse Fourier transform from Fourier series coefficients, you first need to express the Fourier series coefficients in terms of the complex exponential form. Then, you can use the inverse Fourier transform formula, which involves integrating the product of the Fourier series coefficients and the complex exponential function with respect to the frequency variable. This process allows you to reconstruct the original time-domain signal from its frequency-domain representation.
Is zero divided by zero one or zero?
The answer is neither. You cannot divide by zero at all.
The result of zero divided by zero, as with any other division by zero, is undefined.
How do you take dot product of two vectors?
Multiply the product of their magnitudes by the cosine of the angle between them.
Minuend minus Subtrahend equals Difference
Minuend - Subtrahend = Difference
What is the definition of a Hermitian matrix?
Hermitian matrix defined:
If a square matrix, A, is equal to its conjugate transpose, A†, then A is a Hermitian matrix.
Notes:
1. The main diagonal elements of a Hermitian matrix must be real.
2. The cross elements of a Hermitian matrix are complex numbers having equal real part values, and equal-in-magnitude-but-opposite-in-sign imaginary parts.
Here's what you do you find the smallest number and the larger number then subtract. For EXAMPLE: 12,18,22,24,25,32
32-12=20 so your RANGE is 20.
The possible x values of a function.
What is the dot product of two rectangular components of a vector?
a vector is a line with direction and distance. there is no answer to your question. the dot is the angular relationship between two vectors.
An Axiom is a mathematical statement that is assumed to be true. There are five basic axioms of algebra. The axioms are the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom.
