files / problems / hilbert16-limit-cycles / compute / q3 / rr-kolmogorov / certs / identities.json

5.4 KB · on GitHub

{
  "axes": {
    "P": [
      {
        "coeff": "1",
        "x": 1
      },
      {
        "a": 1,
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 2
      }
    ],
    "P_at_x0": [],
    "Q": [
      {
        "coeff": "1",
        "y": 1
      },
      {
        "coeff": "-1",
        "y": 2
      },
      {
        "b": 1,
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "c": 1,
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "Q_at_y0": [],
    "variables": [
      "x",
      "y",
      "a",
      "b",
      "c"
    ]
  },
  "beats_HK5": false,
  "beats_MK3": false,
  "claim": "named cubic Kolmogorov family dx/dt = x(1-x-y), dy/dt = y(1-b x-y-c x^2) has weighted Dulac divergence -1, hence 0 isolated cycles in the open first quadrant; not a bound on H(n)",
  "dulac_constant": -1,
  "general": {
    "P": [
      {
        "coeff": "1",
        "x": 1
      },
      {
        "a": 1,
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 2
      }
    ],
    "Q": [
      {
        "coeff": "1",
        "y": 1
      },
      {
        "coeff": "-1",
        "y": 2
      },
      {
        "b": 1,
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "c": 1,
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "claimed": [
      {
        "beta": 1,
        "coeff": "1"
      },
      {
        "alpha": 1,
        "coeff": "1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "beta": 1,
        "coeff": "-1",
        "y": 1
      },
      {
        "a": 1,
        "alpha": 1,
        "coeff": "-1",
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 1
      },
      {
        "alpha": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "b": 1,
        "beta": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "beta": 1,
        "c": 1,
        "coeff": "-1",
        "x": 2
      }
    ],
    "diff": [],
    "p": [
      {
        "coeff": "1"
      },
      {
        "a": 1,
        "coeff": "-1",
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 1
      }
    ],
    "q": [
      {
        "coeff": "1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "b": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "c": 1,
        "coeff": "-1",
        "x": 2
      }
    ],
    "variables": [
      "x",
      "y",
      "a",
      "b",
      "c",
      "alpha",
      "beta"
    ],
    "weighted": [
      {
        "beta": 1,
        "coeff": "1"
      },
      {
        "alpha": 1,
        "coeff": "1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "beta": 1,
        "coeff": "-1",
        "y": 1
      },
      {
        "a": 1,
        "alpha": 1,
        "coeff": "-1",
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 1
      },
      {
        "alpha": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "b": 1,
        "beta": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "beta": 1,
        "c": 1,
        "coeff": "-1",
        "x": 2
      }
    ]
  },
  "hn_moved": false,
  "isolated_cycles_in_Q1": 0,
  "named": {
    "P": [
      {
        "coeff": "1",
        "x": 1
      },
      {
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 2
      }
    ],
    "Q": [
      {
        "coeff": "1",
        "y": 1
      },
      {
        "coeff": "-1",
        "y": 2
      },
      {
        "b": 1,
        "coeff": "-1",
        "x": 1,
        "y": 1
      },
      {
        "c": 1,
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "claimed": [
      {
        "coeff": "-1"
      }
    ],
    "cleared": [
      {
        "coeff": "-1"
      }
    ],
    "cubic_term": [
      {
        "c": 1,
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "p": [
      {
        "coeff": "1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 1
      }
    ],
    "q": [
      {
        "coeff": "1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "b": 1,
        "coeff": "-1",
        "x": 1
      },
      {
        "c": 1,
        "coeff": "-1",
        "x": 2
      }
    ],
    "variables": [
      "x",
      "y",
      "b",
      "c"
    ],
    "weighted": [
      {
        "coeff": "-1"
      }
    ]
  },
  "named_alpha": -1,
  "named_beta": 0,
  "schema": "hilbert16-rr-kolmogorov/v1",
  "seven_cycles": false,
  "specialization": {
    "one_signed": [
      {
        "coeff": "-1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "a": 1,
        "coeff": "1",
        "y": 1
      }
    ],
    "one_signed_claimed": [
      {
        "coeff": "-1"
      },
      {
        "coeff": "-1",
        "y": 1
      },
      {
        "a": 1,
        "coeff": "1",
        "y": 1
      }
    ],
    "sliced": [
      {
        "coeff": "-1"
      }
    ],
    "variables": [
      "x",
      "y",
      "a",
      "b",
      "c",
      "alpha",
      "beta"
    ]
  },
  "what_this_is_not": [
    "not seven isolated cycles",
    "not a beat of Carvalho-Cruz-Gouveia M_K(3)>=6",
    "not a claim of Gasull-Santana H_K(5)>=28",
    "not a bound on H(3)"
  ]
}