todo.org

---

• classification of regular polytope
• holding implements until finite element method and PDE
• example usages of higher order incidence relation
• graph.ts with geometry
• geometry of convex polytope
  • example of give geometry to topological structure
• how to describe edge and face so they are easy to draw ?
• a generic way to specify geometry of cell-complex

graph

• graph without info – for new hackenbush
• reuse cell_complex_t – by `graph_t.as_cell_complex ()`
• find cycles of graph by spanning tree’s complement edges
  • homology of general graph

games – for frontend jobs

• hackenbush
• dots-and-boxes
• connect-four
• go
• hackenbush editor

algebra of cell-complex

• check equality proof
• the game of equality
• 2-dim cell-complex can be expressed purely algebraicly thus classification can also be expressed algebraicly but, for 3-dim cell-complex we do not have the syntax of higher algebra

manifold classification

• another manifold_check
  • any edge occur twice thus when glued we will have not boundary
  • condition on edge_figure_t is weaker then condition on vertex_figure_t
    • every dimension have its element_figure_t the higher dimension the weaker the condition
    • maybe we should use the word “shape” instead of “figure” because of the use of `figure_t`
  • but this is not enough because of pinch points
  • how should we call this kind of weaker manifold_check ? pinchfold_t ? instead of think of name for each dimension we should

    manifold_t	weak_manifold_t (0)
    pinchfold_t	weak_manifold_t (1)
    	weak_manifold_t (2)
    	…

    pinchfold_t ?

• 2-dim manifold classification – zip of john conway
  • can the normalization algorithm works on more than manifold_t ?
• normal forms
  • sphere: a a.rev
  • tori: a1 b1 a1.rev b1.rev a2 b2 a2.rev b2.rev … an bn an.rev bn.rev
  • cross-caps (projective-plane): a1 a1 a2 a2 … an an
    • note that two cross-caps is a klein_bottle
• the normalization algorithm
  1. merge faces at double occuring edge pairs while maintaining homeomorphic to disk
     • only double occuring pairs can be merged without losing information
     • this will reduce the cell-complex to the following state:
       • there are no double occuring edge pairs
       • or merging any more double occuring edge pairs will make it non homeomorphic to disk
         • information of non-disk-ness is encoded by edge pairs if we merge more the information of non-disk-ness will be lost
  2. reduce vertexes to one vertex
  3. make same-direction edges next to each other
  4. make opposite-direction edges ???
  5. cross-cap + torus = three cross-caps
• new im_dic_compatible_p
• new manifold_check
• new vertex_figure_t
• update cell-complex

what knowledge is worth knowing ?

• topological and geometrical modeling
  • mesh
    • polytopal-complex
    • blender
  • clifford algebra
• physics simulation
  • differential equation & difference equation
  • finite element method – PDE
  • direction field – ODE

mesh

• to give geometry to cell-complex, we can
  1. generate mesh for cell-complex
  2. use affine variety of algebraic geometry

polytope

• fourier-motzkin elimination
• simplify inequalities
• double description method

num

• projection matrix for 1-dim subspace
  • rank one matrix
  • P.mul (P) .eq (P)
  • P.transpose () .eq (P)
• projection matrix for m-dim subspace
  • subspace represented by A
    • columns are column vectors of the subspace
  • P = A.mul (A.transpose () .mul (A) .inv ()) .mul (A.transpose ())
    • P.mul (P) .eq (P)
    • P.transpose () .eq (P)
• normal equation
• gram – only gram
• gram-schmidt – with normalization
  • is there a version of gram-schmidt for integer matrix ?
• num.matrix_t.positive_definite_p ()
• abstract/order.ts – for num.ts, for polytope.ts
• use num.ts to re-imp hackenbush

mathematical structures

• ring.cs substructure and ideal_t
• order.ts – lattice_t, poset_t, total_t, heyting_algebra_t

int

• linear diophantine equations with mod – finite field

euclid

• .diag => .main_diag
• .diag .set_diag
• convert invariant_factors to elementary_divisors
• primary_decomposition – [rank, [[p0, n0], [p1, n1], …]]
• chinese_remainder_theorem

computational-science

• stiffness matrix
• circulant matrix

polynomial

• polynomial.ts – symbolic algebra

panel-data

• frame_t.act & series_t.trans
• data_t slice

optimize

• optimize frame_t and series_t by not using data_t but to use matrix_t and vector_t

homology

• what is the meaning of 1 torsion_coefficients ?

homotopy

• presentation of groupoid is the same as 2-dim cell-complex
  • by which we can calculate homology group of groupoid
  • my first aim is to generalize this algebraic structure for 3-dim cell-complex
  • we also want to study group representation i.e. find matrix group iso to given group
• groupoid of 2-dim cell-complex
  • `as_groupoid ()`
  • what is special about manifold’s groupoid
• glob_t
• ht.chain_t
  • `.boundary ()`
  • `.as_group ()` – formalize presentation of group
  • `.as_groupoid ()` – presentation of groupoid with `ht.chain_t`
  • abelianization of `ht.chain_t` to get homology theory abelianization 时如何获得定向 ?
  • `.glue ()` 我们所要处理的代数结构中的元素是 ht.chain_t 这在于 元素是有类型的 (或者说是有边界的) 我们的代数结构类似於 groupoid 而不是 group 元素之间的复合不是简单地左右相乘 而是 沿着边界 glue
    • 我们可以从 presentation of a groupoid 入手 研究 groupoid 对 ht.chain_t 的需要 也就是说
      1. 放宽对元素联通性的要求
      2. 丰富 compose 为 glue
• higher_groupoid_t

cell-complex

• we can fully encode the information of cell-complex by cell-valued incidence matrixes,
  • we can specialize cells for each dimension, for examples:
    • +1,-1 (2-dim rotation) for [2-dim, 1-dim] incidence relation
    • 2-dim rotation for [3-dim, 2-dim] incidence relation
    • 3-dim rotation for [4-dim, 3-dim] incidence relation
• how about adjacency matrix between higher order elements ?
• bounfold_check
• cell_check – is im_dic_compatible_p enough ?
• can we encode cell-complex by graph ?
  • what is “encode something by graph” ? with graph label ?
• product_complex_t
• quotient_complex_t – self-gluing
• vertex_figure_t – 3 dim
• pure_complex_t an n-dimensional complex is said to be pure if each k cell (k < n) is a face of at least one n-dimensional cell
• boundary operator
  • the boundary of the boundary of a cell_complex_t should be zero even if the cell_complex_t is not a bounfold_t

polytopal-complex

• like cell-complex but without self adjacency which simplifies the data structure
• polytopal-complex can be used as basic data structure in meshing

geometry

• quaternion
• clifford-algebra

combinatorial-game

• use go to test game tree searching
• why the games of logic seem like one-player game ?
• aristotle (lukasiewicz) -> de morgan -> peirce
• martin-gardner
• (paper) investigations into game semantics of logic
• surreal – the theory of surreal number
  • theory about two-player normal-ending game

dance

• 3 circle dance
• 4 circle dance