Bacle H
Posts:
283
Registered:
4/8/12


Re: Maxwell Equations as example of your IDIOCY over all of physics and math #9 Textbook 2nd ed. : TRUE CALCULUS; Something Way above your Head
Posted:
May 26, 2013 1:35 AM


On Saturday, May 25, 2013 10:34:15 PM UTC7, bacle...@gmail.com wrote: > On Friday, May 24, 2013 10:43:44 PM UTC7, Archimedes Plutonium wrote: > > > Alright, I am learning more new things, for in this 2nd edition I have > > > > > > an alternative to the picketfence model. I hacohomology theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel. > > > > > > > > > > > > Contents > > [hide] 1 Definitions > > 2 Examples > > 3 Grothendieck's axioms > > 4 Elementary properties > > 5 Related concepts > > 6 History > > 7 References > > > > > > [edit] Definitions > > > > A category is abelian if > > it has a zero object, > > it has all binary products and binary coproducts, and > > all monomorphisms and epimorphisms are normal. > > > > This definition is equivalent[1] to the following "piecemeal" definition: > > A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all homsets are abelian groups and the composition of morphisms is bilinear. > > A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. > > An additive category is preabelian if every morphism has both a kernel and a cokernel. > > Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. > > > > Note that the enriched structure on homsets is a consequence of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. > > > > The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. > > > > [edit] Examples > > As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. > > If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). > > If R is a leftnoetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. > > As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite > > ve the pure and straight > > > > > > rectangle model and the pure and straight triangle. In the rectangle > > > > > > model we fill the dx of 10^603 width and the height is y itself. In > > > > > > the pure triangle we have a right trcohomology theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel. > > > > > > > > > > > > Contents > > [hide] 1 Definitions > > 2 Examples > > 3 Grothendieck's axioms > > 4 Elementary properties > > 5 Related concepts > > 6 History > > 7 References > > > > > > [edit] Definitions > > > > A category is abelian if > > it has a zero object, > > it has all binary products and binary coproducts, and > > all monomorphisms and epimorphisms are normal. > > > > This definition is equivalent[1] to the following "piecemeal" definition: > > A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all homsets are abelian groups and the composition of morphisms is bilinear. > > A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. > > An additive category is preabelian if every morphism has both a kernel and a cokernel. > > Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. > > > > Note that the enriched structure on homsets is a consequence of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. > > > > The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. > > > > [edit] Examples > > As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. > > If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). > > If R is a leftnoetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. > > As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite > > iangle on the leftside of the > > > > > > point of the graph and the same triangle on the rightside with its > > > > > > hypotenuse in the reverse direction as pictured like this: > > > > > > > > > > > > / > > > > > > /  > > > > > > / __ > > > > > > > > > > > > unioned with thiscohomology theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel. > > > > > > > > > > > > Contents > > [hide] 1 Definitions > > 2 Examples > > 3 Grothendieck's axioms > > 4 Elementary properties > > 5 Related concepts > > 6 History > > 7 References > > > > > > [edit] Definitions > > > > A category is abelian if > > it has a zero object, > > it has all binary products and binary coproducts, and > > all monomorphisms and epimorphisms are normal. > > > > This definition is equivalent[1] to the following "piecemeal" definition: > > A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all homsets are abelian groups and the composition of morphisms is bilinear. > > A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. > > An additive category is preabelian if every morphism has both a kernel and a cokernel. > > Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. > > > > Note that the enriched structure on homsets is a consequence of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. > > > > The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. > > > > [edit] Examples > > As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. > > If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). > > If R is a leftnoetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. > > As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite > > triangle > > > > > > > > > > > > \ > > > > > >  \ > > > > > > __\ > > > > > > > > > > > > is the same area as the rectangle model of the point on the function > > > > > > graph. > > > > > > > > > > > > The problem, though, is that the angle of the hypotenuse does not like > > > > > > like the slope or tangent to the point of that function graph. So I > > > > > > need to see if that hypotenuse is related to the slope or tangent or > > > > > > derivative at that specific point. If it is, then, clearly we see how > > > > > > derivative is the inverse of integral, because both have the same area > > > > > > and the triangle hypotenuse would be the derivative. So instead of > > > > > > rectangles forming the integral we can take two triangles. So > > > > > > hopefully I can work this out in the 3rd edition which I plan to start > > > > > > in the next day or so. > > > > > > > > > > > > > > > > > > Alright, this is the 10th page of the 2nd edition and the last page. I > > > > > > want to devote the last page to showing how all this math is begot > > > > > > from the Maxwell Equations. > > > > > > Now on this last page I want to show how Calculus of its empty space > > > > > > between successive numbers is derived from the Maxwell Equations as > > > > > > the ultimate axiom set over all of mathematics. The Maxwell Equations > > > > > > derives the Peano axioms and the Hilbert axioms. But I want to show > > > > > > that the Maxwell Equations do not allow for the Reals to be a > > > > > > continuum of points in geometry but rather, much like the integers, > > > > > > where there is a empty space between successive integers. > > > > > > The Reals that compose the xaxis of 1st quadrant are these: > > > > > > > > > > > > 0, 1*10^603, 2*10^603, 3*10^603, 4*10^603, 5*10^603, > > > > > > 6*10^603 . . on up to 10^603 > > > > > > > > > > > > Pictorially the Reals of the xaxis looks like this > > > > > > ...................> > > > > > > and not like this > > > > > > ____________> > > > > > > > > > > > > So in the Maxwell Equations we simply have to ask, is there anything > > > > > > in physics that is a continuum or is everything atomized with empty > > > > > > space in between? Is everything quantized with empty space in > > > > > > between? > > > > > > > > > > > > I believe the answer lies with the Gauss law of electricity, commonly > > > > > > known as the Coulomb law. The negative electric charge attracts the > > > > > > positive electric charge, yet with all that attraction they still must > > > > > > be separated by empty space. If there was a continuum of matter in > > > > > > physics, then the electron would be stuck to the proton. The very > > > > > > meaning of quantum mechanics is discreteness, not a continuum. > > > > > > Discreteness means having holes or empty space between two particles > > > > > > interacting of the Maxwell Equations. > > > > > > So if physics has no material continuum, why should a minor subset of > > > > > > physics mathematics have continuums. If Physics does not have > > > > > > something, then mathematics surely does not have it. > > > > > > > > > > > > Now I end with reminders for the 3rd edition: > > > > > > > > > > > > REMINDERS: > > > > > > (1) First page talk about why Calculus exists as an operator of > > > > > > derivative versus integral much the same way of add subtract or of > > > > > > multiply divide because in a Cartesian Coordinate System the number > > > > > > points are so spaced and arranged in order that this spatial > > > > > > arrangement yields an angle that is fixed. So that if you have an > > > > > > identity function y = x, the position of points (1,1) from (2,2) is > > > > > > always a 45 degree angle. So Calculus of derivative and integral is > > > > > > based on this fact of Euclidean Geometry that the coordinates are so > > > > > > spatially arranged as to yield a fixed angle. Numbers forming fixed > > > > > > angles gives us Calculus. > > > > > > > > > > > > (2) Somewhere I should find out if the picketfence model is the very > > > > > > best, for it maybe the case that a rectangle model versus a pure > > > > > > triangle model may be better use of the empty space of 10^603 between > > > > > > successive Reals (number points). The picketfence model is good, but > > > > > > it never dawned on me until now that there is likely a better model > > > > > > even yet pure rectangle versus two pure triangles. My glitch is to > > > > > > get the hypotenuse related to the derivative. If I can solve that > > > > > > glitch, I have a crystal clear understanding of the derivative, > > > > > > integral and why they are inverses. > > > > > > > > > > > > (3) I am really excited about that new method of arriving at the > > > > > > infinity borderline of Floorpi*10^603 via Calculus. The first number > > > > > > which allows a half circle function to be replaced by a 10^1206 > > > > > > derivatives of tiny straight line segments and still be a truncated > > > > > > regular polyhedra, is when pi has those 603 digits rightward of the > > > > > > decimal point. The derivative of half circles of any number smaller > > > > > > than Floorpi*10^603 does not form a circle. And is that not what > > > > > > Calculus is all about in the first place taking curves and finding > > > > > > Euclidean straight line segments as derivative and area. Calculus is > > > > > > the interpretation of curved lines into straight line segments. So, > > > > > > onwards to 3rd edition. > > > > > > > > > > > >  > > > > > > More than 90 percent of AP's posts are missing in the Google > > > > > > newsgroups author search archive from May 2012 to May 2013. Drexel > > > > > > University's Math Forum has done a far better job and many of those > > > > > > missing Google posts can be seen here: > > > > > > > > > > > > http://mathforum.org/kb/profile.jspa?userID=499986 > > > > > > > > > > > > Archimedes Plutonium > > > > > > http://www.iw.net/~a_plutonium > > > > > > whole entire Universe is just one big atom > > > > > > where dots of the electrondotcloud are galaxies

