Patent Application
20170220326
One of classic problems in computing is constructing abstract structures such as groups, rings, modules, fields etc. This invention proposes a method to construct a structure of iNecklace which composes iAlphabet and iUniverse.
In general, in one aspect, the invention relates to a method for iNecklace-iAlphabet-iUniverse comprises receiving a request for intuitive structures from the environment, constructing the iAlphabet, computing the identity, performing algebraic, categorical, and homotopy type constructions, performing constructions with measures, recalling relevant instances with reconstructions, identifying the analytic device, enabling composability to construct iUniverse.
Other aspects of the invention will be apparent from the following description and the appended claims.
(1) Exemplary embodiments of the invention will be described with reference to the accompanying drawings. Like items in the drawings are shown with the same reference numbers.
(2) In an embodiment of the invention, numerous specific details are set forth in order to provide a more thorough understanding of the invention. However, it will be apparent to one of ordinary skill in the art that the invention may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the description.
(3) In general, embodiments of the invention relate to a method and apparatus for for computing. More specifically, embodiments of the invention enable iNecklace-iAlphabet-iUniverse including information and structures proceeded through a series of steps. The structures specify and characterize iAlphabets and iUniverses by producing the collection of all algebras, geometry, topologies, categories, homotopy types, toposes, and measures.
(4) In one embodiment of the invention, initially, a request starts to construct intuitive structures. Take a set of α colors as an input, the intuitive structure is obtained by placing n colored beads around a circle. The result of necklace represents a structure with n circularly connected beads of up to α different colors. In one embodiment of the invention, a determination is made about the intuition to construct intuitive structure with discrete sums M(a,n) (N100).
(5) In one embodiment of the invention, a determination is made about to construct iAlphabet with the above intuitive structures. Take a set A as an alphabet. Set the characters to be the elements of A. Construct a word by a finite juxtaposition of characters of the alphabet A, which is an element of the free monoid generated by A. Take two words u and v as input, w and w′ are conjugate when w=uv and w′=vu. The empty word is the identity in the monoid of words. The conjugate relation is an equivalence relation. An equivalence class of words forms a necklace. Construct iAlphabet as S(a;n) (N101), which is the number of aperiodic words of length n out of an alphabet A containing a letters, and by M(a,n) the number of necklaces of length n.
(6) In one embodiment of the invention, a determination is made about to construct the identity (N102). Construct an isomorphism between the necklace ring and the ring of Witt vectors. Witt vectors is viewed as numbers which have several representations by digits. Construct the real field, when one uses the binary symbols to represent real numbers, a rational number has two distinct binary representations. The construction also satisfies the explicit digital representation of other fields.
(7) In one embodiment of the invention, a determination is made to perform algebraic construction (N103). Taking a commutative ring with identity A as an input, one defines a commutative ring over A on all infinite vectors a=(a,a2, . . . ); b=(b1,b2, . . . ), ai,bjεA. Addition is performed componentwise. Multiplication c=a*b is performed as the arithmetic convolution cn=Σ(i,j)=n(i,j)aibj. The identity of the ring structure is the vector (1,0,0, . . . ). The necklace ring M(A) over A constructed above has the underlying additive Abelian group.
(8) In one embodiment of the invention, a determination is made to measure the iNecklace (N104). The measure can be constructed in 1-dimension and 3-dimension respectively.
(9) In one embodiment of the invention, a determination is made to measure the iNecklace in 1-dimension (N107). Take the Antoine's necklace, which is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected as an input, the Antoine's necklace is constructed iteratively. The iteration 0 is started with a solid torus A0. The iteration 1 is to construct a necklace A1 of a smaller and linked tori that lie inside A0. Each torus composing A1 can be replaced with another smaller necklace as A0. This yields A2. This construction continues the iterations with countably infinite number of times to create an An for all n. Since the solid tori are chosen to become arbitrarily small when the number of iterations increases, the connected components of A are single points so A is closed and dense in itself. A is also totally disconnected with the cardinality of the continuum. Hence, A is homeomorphic to the Cantor set.
(10) In one embodiment of the invention, a determination is made to measure the iNecklace in 3-dimension (N108). The above construction of Antoine's necklace yields a Cantor of 1-dimensional Hausdorff measure in 3. One can adapt the construction to construct a larger Cantor set, Antoine's necklace with positive 3-dimensional Hausdorff measure or Lebesgue measure. Given a torus, one can find four linked tori of arbitrarily large relative measure inside the torus. One can proceed the construction by fixing a sequence rnε(0,1) such that
One can start with a torus T0⊂3 of unit volume and replace T0 by the union T1 of four linked tori T1,1 . . . , T1,4 whose union has measure r1. Then replace each T1,i by four linked tori T1,i,j contained in T1,i whose union has measure r1r2; let T2 be the union of the 16 tori. When the construction continues inductively, the intersection ∩nTn is an Antoine necklace of measure 1/2. One needs to ensure that the diameters of the tori that make up Tn go to zero as n→∞ so that the resulting set is totally disconnected.
(11) In one embodiment of the invention, a determination is made to perform the categorical construction (N105). Taking a functor :sSet→sCat from the category of simplicial sets to the category of simplicially enriched categories, one constructs
:sSet→sCat to be the left Kan extension of the cosimplicial object
Δ•:Δ→sCat along the Yoneda embedding. The objects of
X are the vertices of X. Every edge of
X (x,y) corresponds to a necklace Δn
. . .
Δn
Δk means that the final vertex of the n-simplex is identified with the initial vertex of the k-simplex. A necklace is constructed with a sequence of beads Δn
X(x,y). By tracking additional vertex data, necklaces characterize the higher dimensional simplices of the hom-spaces
(x,y).
(12) In one embodiment of the invention, a determination is made to perform the homotopy type construction (N106). The homotopy types can be constructed by collapsing subspaces and attaching spaces.
(13) In one embodiment of the invention, a determination is made to perform the homotopy type construction by collapsing subspaces (N109). Taking X to be the union of a torus with n meridional disks as an input, one obtains a CW structure on X by choosing a longitudinal circle in the torus, intersecting each of the meridional disks in one point. These intersection points are the 0 cells. The 1 cells are the rest of the longitudinal circle and the boundary circles of the meridional disks. The 2 cells are the remaining regions of the torus and the interiors of the meridional disks. Collapsing each meridional disk to a point creates a homotopy equivalent space Y consisting of n 2-spheres, each tangent to its two neighbors. This is a necklace with n beads. A strand of n beads with a string joining its two ends, collapses to Y by collapsing the string to a point. The collapse is a homotopy equivalence.
(14) In one embodiment of the invention, a determination is made to perform the homotopy type construction by collapsing subspaces (N110). The necklace can be constructed from a circle S1 by attaching n 2-spheres S2 along arcs. The necklace N(n, S1,ai) is homotopy equivalent to the space constructed by attaching n 2-spheres S2 to a circle S1 at points ai. The drawing of the construction is shown in figure ??.
Inductive Necklace (n:nat):=
|strand:Circle→Necklace
|bead:forall(k:nat), k<n→Sphere2→Necklace
|attach:forall(k:nat)(p:k<n), strand base=bead k p base2.
(15) In one embodiment of the invention, a determination is made to perform the reconstruction (N111). Taking partial information as an input, the reconstruction of a necklace needs n beads, each of which is either black or white. A k-configuration in the necklace is a subset of k positions in the necklace. Two configurations are isomorphic when one is obtained from the other by a rotation of the necklace. At stage k of the reconstruction for each k-configuration, the partial information at the stage is a count of the number of k-configurations that are isomorphic to it and that contain only black beads. Given n, with finite number of stages in the worst case, one can reconstruct the precise pattern of black and white beads in the original necklace.
(16) In one embodiment of the invention, a determination is made to construct the analytic device (N112). There are distribution, convergence, stability, and asymptotic evaluations.
(17) In one embodiment of the invention, a determination is made to construct the distribution analytics (N113). Applying the analysis to the the dinner table problem, the nth and 1st distinguishable things are nearest neighbors. Analysis proves that the limiting result in the case of large n. If n goes to infinity, then the number of ways to rearrange the n objects while preserving k nearest neighbors falls on a Poisson distribution, whose mean satisfies k=2.
(18) In one embodiment of the invention, a determination is made to construct the convergence analytics (N114). Take a very large necklace with length n and n+1 as an input. When n→∞, the limit of M(n+1,a)/M(n,a) is bounded. The limit for n→∞ of both bounds is a, so the ratio tends to a for n→∞.
(19) In one embodiment of the invention, a determination is made to construct the convergence analytics (N115). Take M(αβ,n) expressed as a quadratic polynomial in M(α,i) and M(β,j) as an input, one generalizes it to more variables. M(ar,n) can be constructed linearly and integrally in terms of M(a,i).
(20) In one embodiment of the invention, a determination is made to construct the asymptotics analytics (N116). Take W to be a finite word on a two symbol alphabet {0, 1} as an input. If it is the last item in the list of all its cyclic permutation (ordered lexicographically), then W is maximal. The number w(n) of maximal words of length n can be constructed with the aid of Eulers' totient function. The asymptotics of w(n) is lim(1/n)log w(n)=h for some positive value h and h=1. That is because
by just considering the first summand, and
by upper-bounding the number of summands to be n each at most n2n.
(21) In one embodiment of the invention, a determination is made to construct composability (N117). A necklace in the nerve of a category is uniquely constructed by its spine and the set of joins. A necklace is constructed as a sequence of composable non-identity morphisms that each contained in one set of parentheses, which indicates such morphisms are grouped together to form a bead.
(22) In one embodiment of the invention, a determination is made to construct iUniverse (N118). The structures specify and characterize iAlphabets and iUniverses by producing the collection of all algebras, geometry, topologies, categories, homotopy types, toposes, and measures.
Antoine's Necklace. In Wikipedia. Retrieved Dec. 31, 2015, from
Lyndon word. In Wikipedia. Retrieved Dec. 31, 2015, from
Metropolis, N.; Rota, Gian-Carlo (1983). Witt vectors and the algebra of necklaces.
Necklace. In Wikipedia. Retrieved Dec. 31, 2015, from
