Mixed Tate motives over Z
... of even Tate twists (since multiple zeta values are real numbers, we need not consider odd Tate twists). In keeping with the usual terminology for multiple zeta values, we refer to the grading on HMT+ as the weight, which is one half the motivic weight. After making some choices, the motivic multipl ...
... of even Tate twists (since multiple zeta values are real numbers, we need not consider odd Tate twists). In keeping with the usual terminology for multiple zeta values, we refer to the grading on HMT+ as the weight, which is one half the motivic weight. After making some choices, the motivic multipl ...
connected
... cut F is a bound if and only if G F has exactly two components. Proof: Suppose G F has more than two component. B, B and A, A are proper subsets of F , so F is not a bound. ...
... cut F is a bound if and only if G F has exactly two components. Proof: Suppose G F has more than two component. B, B and A, A are proper subsets of F , so F is not a bound. ...
Cohomology and K-theory of Compact Lie Groups
... left-invariant differential forms, which in turn have a natural correspondence with skewsymmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole situation is reduced to computing the Lie algebra cohomology. One may further restrict to the bi-invariant differential forms ...
... left-invariant differential forms, which in turn have a natural correspondence with skewsymmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole situation is reduced to computing the Lie algebra cohomology. One may further restrict to the bi-invariant differential forms ...
Foundations of Mathematics I Set Theory (only a draft)
... The set {0, 1, 2, 3, . . .} is infinite. In general, we have x 6= {x} because the set {x} has only one element, whereas the set x could have more than one element. Note that if x = {x}, then x ∈ x, i.e. x is an element of itself, a quite strange and unexpected phenomena, which will be forbidden by a ...
... The set {0, 1, 2, 3, . . .} is infinite. In general, we have x 6= {x} because the set {x} has only one element, whereas the set x could have more than one element. Note that if x = {x}, then x ∈ x, i.e. x is an element of itself, a quite strange and unexpected phenomena, which will be forbidden by a ...
How to Define a Real Number Through the Concept of Dedekind Cut?
... from N we move to the set of integers Z in which we can solve the above type of equations. Again if a, b ∈ Z and a 6= 0 then we cannot always solve the equation ax = b in Z. So from Z we move to the set of rational numbers Q by taking the solutions of the above equations. Consider the equation x2 − ...
... from N we move to the set of integers Z in which we can solve the above type of equations. Again if a, b ∈ Z and a 6= 0 then we cannot always solve the equation ax = b in Z. So from Z we move to the set of rational numbers Q by taking the solutions of the above equations. Consider the equation x2 − ...
contributions to the theory of finite fields
... of the theory of /»-polynomials, which I have developed in a recent paperf On a special class of polynomials. In this special case the /»-polynomials form a commutative ring. However, this paper may be read without reference to the former investigations and one may say that the method applied is the ...
... of the theory of /»-polynomials, which I have developed in a recent paperf On a special class of polynomials. In this special case the /»-polynomials form a commutative ring. However, this paper may be read without reference to the former investigations and one may say that the method applied is the ...
Quasisymmetric rigidity for Sierpinski carpets
... ϕ is (the restriction of) a Möbius transformation. Theorem (B., Lyubich, Merenkov 2016) Let f be postcritically-finite rational map whose Julia set Jf is a carpet. Then QS(Jf ) is a finite group of (restrictions of) Möbius transformations. Corollary No such Jf is a group carpet. ...
... ϕ is (the restriction of) a Möbius transformation. Theorem (B., Lyubich, Merenkov 2016) Let f be postcritically-finite rational map whose Julia set Jf is a carpet. Then QS(Jf ) is a finite group of (restrictions of) Möbius transformations. Corollary No such Jf is a group carpet. ...
STRUCTURE OF LINEAR SETS
... 6.1. Let V2 have the polynomials in two variables x and y with integral coefficients as scalar domain. Let Li be the linear set consisting of all the vectors whose two components have a difference congruent to zero modulo (x2), and L2 the linear set consisting of the vectors whose two components hav ...
... 6.1. Let V2 have the polynomials in two variables x and y with integral coefficients as scalar domain. Let Li be the linear set consisting of all the vectors whose two components have a difference congruent to zero modulo (x2), and L2 the linear set consisting of the vectors whose two components hav ...
SCHOOL OF DISTANCE EDUCATION B. Sc. MATHEMATICS MM5B06: ABSTRACT ALGEBRA STUDY NOTES
... thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields. As we noted above, gr ...
... thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields. As we noted above, gr ...
Non-Homogenizable Classes of Finite Structures
... Here we provide a combinatorial necessary condition for homogenizability (Theorem 3.2 in Section 3). This allows us to prove that certain natural classes of finite structures previously considered in the literature are not homogenizable. Our first example of a non-homogenizable class comes from the ...
... Here we provide a combinatorial necessary condition for homogenizability (Theorem 3.2 in Section 3). This allows us to prove that certain natural classes of finite structures previously considered in the literature are not homogenizable. Our first example of a non-homogenizable class comes from the ...
Intersection homology
... Recall that there is another version of homology called homology with locally finite supports (aka Borel–Moore homology), in which we replace Ci (X) with the larger group Ci ((X)) which consists of possibly infinite linear combinations of simplices that are “locally finite”. Here locally finite mean ...
... Recall that there is another version of homology called homology with locally finite supports (aka Borel–Moore homology), in which we replace Ci (X) with the larger group Ci ((X)) which consists of possibly infinite linear combinations of simplices that are “locally finite”. Here locally finite mean ...
Multiplying Polynomials
... ( x + 5 )( x + 7 ) First – Multiply the first terms Outside – Multiply the outside terms Inside – Multiply the inside terms Last – Multiply the last terms ...
... ( x + 5 )( x + 7 ) First – Multiply the first terms Outside – Multiply the outside terms Inside – Multiply the inside terms Last – Multiply the last terms ...
Banach Spaces
... R defined for L is indeed a norm, except that it may happen that A |f (x)| dx = 0 without f = 0; however from the theory of Lebesgue integration, in this case we get f = 0a.e., so that the failure of this axiom is not drastic. In fact we can identify those functions that are equal almost everywhere ...
... R defined for L is indeed a norm, except that it may happen that A |f (x)| dx = 0 without f = 0; however from the theory of Lebesgue integration, in this case we get f = 0a.e., so that the failure of this axiom is not drastic. In fact we can identify those functions that are equal almost everywhere ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.