Group Theory: Cocycles and Extensions

Cocycles

• For group G and abelian group A, a 2-cocycle is map α: G × G -> A
• 2-Cocycle condition is α(x, y) + α(x y, z) = α(x, y z) + α(y, z) ∀ x, y, z ∈ G
• Z^2(G, A) denotes the set of 2-cocycles.
• A 1-cochain is a map β: G -> A.
• δ: (1-cochains) -> (2-cochains) is defined by δ β(x, y) = β(x) - β(x y) + β(y).
• δ β is a 2-cocycle.
• B^2(G, A) denotes 2-cocycles of the form δ β for some 1-cochain β.
• The second cohomology group is H^2(G, A) = Z^2(G, A) / B^2(G, A).
• Similar definitions exist for Z^n(G, A), B^n(G, A), and H^n(G, A).

Group Extensions

• An extension of G by A is a group E where 1 -> A -> E -> G -> 1 is a short exact sequence.
• A -> E is represented by a -> ā, and E -> G as x ->underline{x}
• This implies underline{ā} = 1, and every element of G has the form underline{x} for some x ∈ E.
• A section s: G -> E is a map where underline{s(x)} = x ∀ x ∈ G.
• e: G × G -> A is defined by s(x) s(y) = ē(x, y) s(x y).
• e(x, y) determines the multiplication in E.
• e ∈ Z^2(G, A).
• Evaluating s(x) s(y) s(z) in two ways yields e(x, y) + e(x y, z) = e(x, y z) + e(y, z).
• E1 and E2 are equivalent extensions of G by A if ψ: E1 -> E2 exists, making a diagram commute.
• If E1 and E2 are equivalent extensions, cocycles e1, e2 differ by a coboundary.
• There is a 1-1 correspondence between equivalence classes of extensions of G by A and H^2(G, A).

Schur Multiplier

• Schur Multiplier of G is H2(G, Z).
• H2(G, Z) is the second homology group of G with coefficients in Z.
• For a free presentation 1 -> R -> F -> G -> 1, H2(G, Z) ≅ (R ∩[F, F])/[R, F].
• Additionally, H2(G, Z) ≅ H^2(G, C*), where C* is the multiplicative group of nonzero complex numbers.

Caching

• Caching stores data in local, fast memory for later use, with RAM as a cache for disk, and L1 cache for L2 data.
• Caching leverages temporal locality (accessing data again soon) and spatial locality (accessing nearby data soon).
• Key questions in caching: placement (where to put new data), identification (how to find data), replacement (what data to remove), and write policy (how to handle writes).

Simple Cache

• A small cache with 8 slots, each holding one word (4 bytes), where the address space is 64 words (256 bytes), needs 6 bits to address.
• 6 bits are divided into 3 bits for the index and 3 bits for the tag.
• Each cache slot has a valid bit (1 if data, 0 if empty).
• To read, with the example word 22 (binary 010110), cache index is 110 (6), and tag is 010 (2).
  • First go to set 6, check if the valid bit is 1 and tag is 2 for a hit, or miss.

Cache Organization

• Direct-mapped cache has each address mapping to one slot.
• Cache size = C, block size = B, number of sets = S, then direct-mapped has S = C/B.
• N-way set-associative cache has S = C/(N*B).
• Fully-associative cache has S = C/B.

Cache Replacement policies

• Direct-mapped cache replacement is trivial.
• N-way set-associative cache replacement policies include:
  • Least Recently Used (LRU): replaces least recently used data.
  • First-In, First-Out (FIFO): replaces the oldest data.
  • Random: replaces a random piece of data.
• LRU is most common but expensive, FIFO is simpler but performs worse, and random is simplest and performs well.

Cache Write Policy

• Write-Through writes to both cache and main memory simultaneously, is simple but slow.
• Write-Back writes only to the cache, marks the line as dirty, and writes to main memory on eviction, is faster but complex.
• Handling write misses:
  • Write-Allocate allocates a cache line and writes there.
  • Write-No-Allocate writes directly to main memory, is simpler but slower.
• Write-allocate is common, and write-no-allocate is simpler

Real Numbers

• The axiom of addition includes closure, associativity, commutativity, and the identity and inverse properties
• Multiplication axioms include closure, associativity, commutativity, identity, and inverse existence for nonzero elements
• Distributive axiom states a ⋅ (b + c) = a ⋅ b + a ⋅ c.

Axioms of Order

• There exists a set P of positive real numbers.
• The trichotomy law: for any real number a, either a is in P, a = 0, or -a is in P, exclusively.
• Closure axioms specify if a, b ∈ P, then a + b ∈ P and a ⋅ b ∈ P

Definitions

• a > b if and only if a - b ∈ P.
• a < b if and only if: b > a.
• a ≥ b if and only if a > b or a = b.
• a ≤ b if and only if a < b or a = b.

Theorems

• a > b implies a + c > b + c.
• a > b and c > 0, then ac > bc.
• a > b and c < 0, then ac < bc.
• If a > b, then -b > -a.
• If a ≠ 0, then a² > 0.
• The number 1 is greater than 0 (1 > 0).
• If a > 0, then 1/a > 0.
• For positive a and b, if a > b, then 1/a < 1/b.

Partial Differential Equations

• PDEs are differential equations (DE) with multiple independent variables
• For u = f(x,y,z,t), a PDE might be A ∂u/∂x + B ∂u/∂y + C ∂u/∂z + D ∂u/∂t = 0
• Contrast with ODEs, such as du/dt = -ku
• PDEs are generally more difficult to solve than ODEs.
• Focus is on linear, 2nd order PDEs common in physics.
• u_x ≡ ∂u/∂x, u_{xx} ≡ ∂²u/∂x², u_{xy} ≡ ∂²u/∂x∂y

Wave equation

• u_{tt} = c² u_{xx}, with constant speed c
• Describes wave motions, such as vibrations of a string, sound, and light waves

Diffusion Equation

• u_t = k u_{xx}
• Describes diffusion, such as a chemical in a liquid

Laplace's Equation

• u_{xx} + u_{yy} = 0
• Describes steady-state potential in a region with no charge, or steady-state temperature distribution

Solving PDEs

• There is no universal method for solving all PDEs
• The solution depends on the equation and boundary conditions
• Solution methods:
  • Separation of variables
  • Fourier transforms
  • Numerical methods

Boundary conditions

• PDEs require these for unique solutions, specifying the solution's value at the domain's boundary
• Dirichlet boundary condition: u = f(x) on the boundary
• Neumann boundary condition: ∂u/∂n = g(x) on the boundary, where n is the normal vector. For example, the 1D heat equation may have u(0,t) = 0 and u(L,t) = 0.

Wave equation example

• String of length L with fixed ends, displacement u(x,t)
• The wave equation is u_{tt} = c² u_{xx}
• Boundary conditions: u(0,t) = u(L,t) = 0 for all t
• Initial conditions: u(x,0) = f(x) and u_t(x,0) = g(x)
• This can be solved using separation of variables