"exact cover" meaning in All languages combined

{
  "forms": [
    {
      "form": "exact covers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "exact cover (plural exact covers)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Combinatorics",
          "orig": "en:Combinatorics",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Topology",
          "orig": "en:Topology",
          "parents": [
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      "examples": [
        {
          "ref": "2000, R. Tijdeman, “Exact covers of balanced sequences and Fraenkel's conjecture”, in F. Halter-Koch, Robert F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference, Walter de Gruyter, page 468",
          "text": "A finite set #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;vert 1#x5C;lei#x5C;lem#x5C;#x7D; is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i). If #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;#x7D;#x5F;#x7B;i#x3D;1#x7D;ᵐ is an eventual exact cover, then #x5C;textstyle#x5C;sum#x5F;#x7B;i#x3D;1#x7D;ᵐ#x7B;#x5C;alpha#x5F;i#x7B;-1=1. […] In 1926, Beatty [Bea] studied the case β=0 and published the following result as a problem: S(α₁,0) and S(α₂,0) form an exact cover if and only if α₁ is irrational and α₁²+α₂²=1.}}",
          "type": "quotation"
        },
        {
          "text": "2011, R. Lu, S. Liu, J. Zhang, Searching for Doubly Self-orthogonal Latin Squares, Jimmy Lee (editor), Principles and Practice of Constraint Programming: 17th International Conference CP 2011, Proceedings, Springer, LNCS 6876, page 542,\nIt is straightforward to use clique algorithms to construct a (partial) solution of a given combinatorial problem which is represented as a set system. If the solution of the combinatorial problem corresponds to the exact cover of the set system, a substantially more efficient algorithm can be utilized because of this property."
        }
      ],
      "glosses": [
        "Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*."
      ],
      "id": "en-exact_cover-en-noun-yGCC098u",
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          "collection",
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        ],
        [
          "subset",
          "subset"
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        [
          "set",
          "set"
        ],
        [
          "subcollection",
          "subcollection"
        ],
        [
          "element",
          "element"
        ]
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      "raw_glosses": [
        "(topology, combinatorics) Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*."
      ],
      "related": [
        {
          "topics": [
            "geometry",
            "mathematics",
            "sciences"
          ],
          "word": "tiling"
        }
      ],
      "topics": [
        "combinatorics",
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "exact cover"
      ]
    }
  ],
  "word": "exact cover"
}

{
  "forms": [
    {
      "form": "exact covers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "exact cover (plural exact covers)",
      "name": "en-noun"
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "topics": [
        "geometry",
        "mathematics",
        "sciences"
      ],
      "word": "tiling"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Combinatorics",
        "en:Topology"
      ],
      "examples": [
        {
          "ref": "2000, R. Tijdeman, “Exact covers of balanced sequences and Fraenkel's conjecture”, in F. Halter-Koch, Robert F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference, Walter de Gruyter, page 468",
          "text": "A finite set #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;vert 1#x5C;lei#x5C;lem#x5C;#x7D; is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i). If #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;#x7D;#x5F;#x7B;i#x3D;1#x7D;ᵐ is an eventual exact cover, then #x5C;textstyle#x5C;sum#x5F;#x7B;i#x3D;1#x7D;ᵐ#x7B;#x5C;alpha#x5F;i#x7B;-1=1. […] In 1926, Beatty [Bea] studied the case β=0 and published the following result as a problem: S(α₁,0) and S(α₂,0) form an exact cover if and only if α₁ is irrational and α₁²+α₂²=1.}}",
          "type": "quotation"
        },
        {
          "text": "2011, R. Lu, S. Liu, J. Zhang, Searching for Doubly Self-orthogonal Latin Squares, Jimmy Lee (editor), Principles and Practice of Constraint Programming: 17th International Conference CP 2011, Proceedings, Springer, LNCS 6876, page 542,\nIt is straightforward to use clique algorithms to construct a (partial) solution of a given combinatorial problem which is represented as a set system. If the solution of the combinatorial problem corresponds to the exact cover of the set system, a substantially more efficient algorithm can be utilized because of this property."
        }
      ],
      "glosses": [
        "Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*."
      ],
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        ],
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        ]
      ],
      "raw_glosses": [
        "(topology, combinatorics) Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*."
      ],
      "topics": [
        "combinatorics",
        "mathematics",
        "sciences",
        "topology"
      ],
      "wikipedia": [
        "exact cover"
      ]
    }
  ],
  "word": "exact cover"
}