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  1. #61
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    Analytic number theory.

    Quote Originally Posted by http://en.wikipedia.org/wiki/Analytic_number_theory
    Depending on the choice of coefficients , this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identityhence the coefficients of the product of two Dirichlet series are the multiplicative convolutions of the original coefficients. Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.
    This is where we find the functions by using this 'thing' or formula. basically, everything is cubed, so the answer is to cube the function.
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    Differential equations.

    A differential equation is a mathematicalequation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering,physics, economics, and biology.In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
    If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
    Quote Originally Posted by http://en.wikipedia.org/wiki/Differential_equation
    Solving differential equations is not like solving algebraic equations. Not only are their solutions oftentimes unclear, but whether solutions are unique or exist at all are also notable subjects of interest.For first order initial value problems, it is easy to tell whether a unique solution exists. Given any point in the xy-plane, define some rectangular region , such that and is in . If we are given a differential equation and an initial condition , then there is a unique solution to this initial value problem if and are both continuous on . This unique solution exists on some interval with its center at .
    However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
    such that
    For any nonzero , if and are continuous on some interval containing , is unique and exists.
    [11]
    So, for the "linear initial value problem of the nth order," we need to find x. so, we need to find the x by observing that every entry with an x is to the power of 0, so, x must be zero. with everything else being multiplied by x^0, this means that everything is zero in the last formula.

    But, it cannot be zero all over, as this is a realistic practical problem. this means that it leaves [n] + [y] equals [x]!
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    Asymptotic theory.

    Quote Originally Posted by http://en.wikipedia.org/wiki/Asymptotic_theory
    Asymptotic theory or large-sample theory is the branch of mathematics which studies asymptotic expansions.An example of an asymptotic result is the prime number theorem: Let π(x) be the number of prime numbers that are smaller than or equal to x. Then the limit
    exists, and it is equal to 1.
    Asymptotic theory ("asymptotics") is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratiostatistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory.
    Quote Originally Posted by http://en.wikipedia.org/wiki/Asymptotic_theory
    Examples of asymptotic expansions[edit]


    where are Bernoulli numbers and is a rising factorial. This expansion is valid for all complex s and is often used to compute the zeta function by using a large enough value of N, for instance .
    As you can see, this all has something to do with infinity, as they all have that lop sided eight thing in them. it is not divided by infinity, but, rather multiplied by it.

    So, if they all have infinity in them, in the end, you could fill your page with nines and stuff and still not have the end answer. the good news is there is no such thing as infinity, because, without a start and end point, there is no value, meaning it is nothing as it is not 'capped,' yes?

    Now, if we want to find x, we need to observe b.o.d.m.a.s, where we need to divide first. this means we can take any of the things with division in them and find x, yes? let's get to it!

    In the gamma function, we can see that there in one are is [x / x 'rooted' x], meaning it is 1, squared rooted by pie and x, well, the one will leave the 1 and the pie will leave, well, pie! but, then it is added to one, so is 2! so, in this thing, x = 2.
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  4. #64
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    Actually let's make that x=5? this would be pie plus x though, so it will not give x, will it?

    Let's say it is x = 3 until that equals, but, then it is [x + 1]. so, now, this on one side is x + 1, so on the other side it needs to be x + one, as they are equal, yes?

    For the exponential integral it would be, seeing as how there is an infinity of nothing, minus 1 = -1!

    For the riemann zeta function, x would be 2m-1s.

    Error function would be the same as the gamma function
    Last edited by Brett Nortje; 05-28-2015 at 07:12 AM. Reason: Corrections.
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    Calculus of infinitesimals.

    Quote Originally Posted by https://en.wikipedia.org/wiki/Calculus
    Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves),[1] and integral calculus (concerning accumulation of quantities and the areas under and between curves);[2] these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions ofconvergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering andeconomics[3] and can solve many problems that algebra alone cannot.
    Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.
    But his is about infinitesimals, so, it is about markers without values, as, we all know, something that is infinite has no value, as, it does not have an end point, so it impossible to measure, yet, it is a workable figment.

    [1] Differential calculus



    So, if they are basically repeating over and over, then ignore everything that equals everything else and find the difference, yes? this would mean, the only equation you need to do is a + h - a, which means that they really do equal each other. this means it is f times a minus f times a, leaving h to divide by itself, leaving one. this means, m equals one always!

    [2] Fundamental theorem.



    So, [b + a] times by [function] times by [x] times by [d times by x] = [F(b) - F(a)] means that [x] must be a minus number, of course.

    This means that [d] equals [-a^2] and [x] equal [-b^2], or, the amounts needed to make sure that and that f must be one or something. this is because they are supposed to equal each other, yet, they are made of the same amounts, of course.

    Or, maybe it is [x * d x + a + b] = [b - a]. For [ab] to equal - [ba] then it would have to be [x = - (a^3)] more or less, or, something nearly exactly like that, leaving room for d and x to have values linked to the a and the b. so, we would say that x an d have negative values which i cannot yet find.
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    Analytic number theory

    Quote Originally Posted by https://en.wikipedia.org/wiki/Analytic_number_theory
    In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.[1] It is often said to have begun with Peter Gustav Lejeune Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.[1][2] It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
    Dirichlet series;

    Depending on the choice of coefficients , this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identity



    It is safe to say that the cursive l squared equals infinity, but, anything of infinity halved would also equal infinity, yes? Obviously, [n^2] equals [k] plus l, yes?
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    Character theory.

    Quote Originally Posted by https://en.wikipedia.org/wiki/Character_theory
    In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.
    Now, before we get to that, we need to know what alternate interior angles are;

    alternate inerior angles.jpg

    It is safe to say, the greater one of the angles, the greater the other angle! this follows that one of them are 'facing' as much as the other, so, the less one of them is, the less the other too. in fact the two angles are the same! It is also said that interior means inside the two parallel lines, of course.

    character theory.png

    Now, the xp is squared along with the g in the first one, then halved. in the second one, to the right of it, it is only the g that is squared then multiplied by the xp. so, it follows that it is like taking [x squared times p squared times g squared] minus [g squared times by x times by p].

    So, it is actually half xp!
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    Effective descriptive set theory.

    Quote Originally Posted by https://en.wikipedia.org/wiki/Effective_descriptive_set_theory
    Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary realparameter (Moschovakis 1980). Thus effective descriptive set theory combines descriptive set theory with recursion theory.
    Quote Originally Posted by https://en.wikipedia.org/wiki/Effective_descriptive_set_theory
    The arithmetical hierarchy, arithmetic hierarchy or Kleene-Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical.
    Quote Originally Posted by https://en.wikipedia.org/wiki/Effective_descriptive_set_theory
    More formally, the arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted and for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters.If a formula is logically equivalent to a formula with only bounded quantifiers then is assigned the classifications and .
    The classifications and are defined inductively for every natural number n using the following rules:
    • If is logically equivalent to a formula of the form , where is , then is assigned the classification .
    • If is logically equivalent to a formula of the form , where is , then is assigned the classification .

    [1] This would seem to be zero plus one on a natural number, or, prime? this would mean that it is one, as, anything to the power of zero is one, yes? now, prime to the power of one is prime, prime to the power of two is not prime! aha! this means, that, there is only one prime, as, any prime multiplied by anything will result in a prime number, because, it is going into it's own number!

    [2] one to the power of prime plus one?
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    Elliptic geometry.

    Quote Originally Posted by https://en.wikipedia.org/wiki/Elliptic_geometry
    Elliptic geometry, a special case of Riemannian geometry, is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p, as all lines in elliptic geometry intersect. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.
    This means, of course, that there is a 'plummet' or 'plunge' in the angles, so that all angles are 'curved.' this also means, that, you can work out what the angles are by using a less then 'equation sign.' it is less than 180 degrees, i think, as that is to me anyways, like a straight line outside the line, of course.

    Seeing as how it is never a straight line, this means that it is also less than 90 degrees at the corners of the diagram. of course, seeing as how it is so 'complex,' and it has values, this means the lines end, somewhere, somehow. if the lines were to meet, then they have definitely a given less than 180 degrees, and, more than ninety degrees, of course.

    Now, if we were to want to find a 'formula for the degrees,' we could say that [degrees] = [length / height * length plus height].
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    Vector calculus.

    Quote Originally Posted by https://en.wikipedia.org/wiki/Vector_calculus
    Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject ofmultivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.
    This is more advanced calculus than we are used to, but, is used in engineering and physics in high end fields, as, they will want to know the resistances of materials and shapes and angles. the people that do this make text books for those that just use it, i think. We cannot work this out the same we worked out the classical calculus i found a easy way to do, because there is more than one value in the brackets, of course.

    Now, this deals with scales and points for vectors, and, 'supposed vector placements.'

    For gradient theorems it is [p] - [q], of course.

    For green's theorem, it is natural to see that the whole quadratic equation concerns the squiggle, and, that means we can simply work it out without the squiggle and times it by the squiggle at the end. as another quick snap shot of what it should be close to, a is to the power of a.

    For stokes theorem, we can cross out everything with a repeat on the next side of the equals sign, and make it much easier.

    For divergence theorem, we could find the values for all the right side of the equals sign, and, then easily work out the rest of the equation.
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