Class U1Group

java.lang.Object
org.episteme.core.mathematics.algebra.groups.U1Group
All Implemented Interfaces:
Group<Complex>, Magma<Complex>, Monoid<Complex>, Set<Complex>
public class U1Group extends Object implements Group<Complex>
Represents the Unitary Group U(1).

The group of all complex numbers with absolute value 1 under multiplication. Isomorphic to the circle group (rotations in 2D). Important in electromagnetism (gauge group).

Since:
1.0
Author:
Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

  • Method Details

    • getInstance

      public static U1Group getInstance()

    • operate

      public Complex operate(Complex left, Complex right)
      Description copied from interface: Magma
      Performs the binary operation on two elements.

      This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

      Properties: None required (not necessarily associative or commutative)

      Specified by:
      operate in interface Magma<Complex>
      Parameters:
      left - the first operand
      right - the second operand
      Returns:
      the result of a ∗ b
      See Also:

    • identity

      public Complex identity()
      Description copied from interface: Monoid
      Returns the identity element of this monoid.

      Satisfies: e ∗ a = a ∗ e = a for all a.

      Specified by:
      identity in interface Monoid<Complex>
      Returns:
      the identity element

    • inverse

      public Complex inverse(Complex element)
      Description copied from interface: Group
      Returns the inverse of the given element.

      For element a, returns a⁻¹ such that: a ∗ a⁻¹ = a⁻¹ ∗ a = e (identity).

      Specified by:
      inverse in interface Group<Complex>
      Parameters:
      element - the element to invert
      Returns:
      the inverse element

    • isCommutative

      public boolean isCommutative()
      Description copied from interface: Group
      Tests whether this is an abelian (commutative) group.
      Specified by:
      isCommutative in interface Group<Complex>
      Specified by:
      isCommutative in interface Magma<Complex>
      Returns:
      true if this group is abelian

    • description

      public String description()
      Description copied from interface: Set
      Returns a human-readable description of this set.

      Examples:

      • "ℝ (Real Numbers)"
      • "ℤ/12ℤ (Integers modulo 12)"
      • "{1, 2, 3, 4, 5}"

      Specified by:
      description in interface Set<Complex>
      Returns:
      a description of this set

    • isEmpty

      public boolean isEmpty()
      Description copied from interface: Set
      Returns true if this set contains no elements.

      The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

      Specified by:
      isEmpty in interface Set<Complex>
      Returns:
      true if this set is empty

    • contains

      public boolean contains(Complex element)
      Description copied from interface: Set
      Tests whether this set contains the specified element.

      This is the fundamental operation of a set - membership testing.

      Specified by:
      contains in interface Set<Complex>
      Parameters:
      element - the element to test for membership
      Returns:
      true if this set contains the element, false otherwise
      See Also:

    • fromPhase

      public Complex fromPhase(double theta)
      Creates a U(1) element from an angle (phase).
      Parameters:
      theta - the angle in radians
      Returns:
      e^(i*theta)