Recent content by lpetrich

This piecewise linear version is equivalent to x ⊕ y = min( max(x,y), max(1-x,1-y) ) = max( min(x,1-y), min(y,1-x) ) and all three definitions can be extended to alternative negations, by replacing 1-x with some other ¬ x.

Exclusive or again. The previous versions: Quadratic (simple negation): x ⊕ y = x + y - 2*x*y Biquadratic (bilinear negation): x ⊕ y = (x + y - 2*x*y) / (1 + p*x*y) I have discovered a piecewise linear version that satisfies the axioms with simple negation. For (condition 1) and (condition 2)...

As I'd mentioned earlier, (distributiveness) > (absorption) > (idempotence) > (Gödel-Zadeh minmax and-or) > (Idempotence), (absorption), (distributiveness) -- one can prove idempotence, absorption, and distributiveness from minmax. Since these implications go in both directions, that means that...

These functions can be used to set up an exclusive-or function that satisfies the xor-inversion axioms: fxor(x) = - fneg(x), fixor(x) = fineg(-x) x ⊕ y = fi( f(x) * f(y) ) For plain reflection, x ⊕ y = x + y - 2*x*y For bilinear reflection, x ⊕ y = (x + y - 2*x*y) / (1 + p*x*y)

Let's look at negation more closely. Most fuzzy-logic work assumes simple reflection: ¬ x = 1 - x It is easy to show involution, self-inversion, that ¬ (¬ x) = x. But Wikipedia's pages contain bilinear reflection, with parameter p: ¬ x = (1 - x) / (1 + p*x) It is also an involution. In...

Having found exactly one conjunction-disjunction set that is distributive, let us see what one can find of complementation. Here, however, there is more than one conjunction-disjunction set that satisfies this property. For example, Łukasiewicz: x ∧ y = max(x+y-1, 0) and x ∨ y = min(x+y, 1)...

Continuing further, let us consider the consequences of fuzzy logic having the distributive property. x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and likewise for ∧ and ∨ interchanged. Set z = 1 (interchanged: z = 0). This reduces distributiveness to absorption: x = (x ∧ y) ∨ x = (x ∨ y) ∧ x Set y = 0...

Boolean algebra, or crisp logic, involves functions of two truth values: true (T) and false (F). These functions satisfy various interrelationships: Boolean algebra (structure) - Wikipedia Some of these functions: Negation: not ¬ -- conjunction: and ∧ -- disjunction: or ∨ -- exclusive or: xor ⊕...

The combined tagging may be expressed as x = {i,j} where i is the left-side quotient-group element and j is the right-side quotient-group element. For the product of x(xtag) and y(ytag), the result's tag is B(xtag,ytag) = Bval(xtag(1),ytag(2)) From the associativity of the product, B(B(x,y),z)...

That's what I showed earlier. A ring without unity is sometimes called a "rng" ("rung"): Rng (algebra) - Wikipedia -- "ring" without "i" (identity). Let's now consider some further features. Consider multiplication x*a for all x in R for some a. That is a homomorphism of (R,+) onto some abelian...

This suggests a simple special case that we can examine. The additive group has a subgroup with index 2, with a quotient group with size 2. It thus has one coset. Call the subgroup members "even" and the coset members "odd". Let R*R be some order-2 additive subgroup of R: {0,a}, where a+a = 0...

As an introduction, an abstract-algebra ring generalizes integers with addition and multiplication. The multiplication operation is distributive over the addition operation, this multiplication forms a semigroup over set R of its elements, and this addition forms an abelian group over R. But can...

This question presumably refers to how much energy has been put into these particles in an attempt to break them apart. Entity Energy / mass Atoms - ionization of hydrogen (worst case) ~ 10-8 Nuclei - 1 MeV / nucleon ~ 10-3 Hadrons ~ 1 Let's now do this calculation for leptons and...

Doing #124

#101