Generalization in the context of N-ellipse

In scholarship, a secondary source is a document or recording that relates or discusses information originally presented elsewhere. A secondary source contrasts with a primary, or original, source of the information being discussed. A primary source can be a person with direct knowledge of a situation or it may be a document created by such a person.

A secondary source is one that gives information about a primary source. In a secondary source, the original information is selected, modified and arranged in a suitable format. Secondary sources involve generalization, analysis, interpretation, or evaluation of the original information.

The term conceptual model refers to any model that is the direct output of a conceptualization or generalization process. Conceptual models are often abstractions of things in the real world, whether physical or social. Semantic studies are relevant to various stages of concept formation. Semantics is fundamentally a study of concepts, the meaning that thinking beings give to various elements of their experience.

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.

The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.

Ad hoc is a Latin phrase meaning literally 'for this'. In English, it typically signifies a solution designed for a specific purpose, problem, or task rather than a generalized solution adaptable to collateral instances (compare with a priori).

Common examples include ad hoc committees and commissions created at the national or international level for a specific task, and the term is often used to describe arbitration (ad hoc arbitration). In other fields, the term could refer to a military unit created under special circumstances (see task force), a handcrafted network protocol (e.g., ad hoc network), a temporary collaboration among geographically-linked franchise locations (of a given national brand) to issue advertising coupons, or a purpose-specific equation in mathematics or science.

In social psychology, a stereotype is a generalized belief about a particular category of people. It is an expectation that people might have about every person of a particular group. The type of expectation can vary; it can be, for example, an expectation about the group's personality, preferences, appearance or ability. Stereotypes make information processing easier by allowing the perceiver to rely on previously stored knowledge in place of incoming information. Stereotypes are often faulty, inaccurate, and resistant to new information. Although stereotypes generally have negative implications, they aren't necessarily negative. They may be positive, neutral, or negative. They can be broken down into two categories: explicit stereotypes, which are conscious, and implicit stereotypes, which are subconscious.

In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.

More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension D + 1) is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D=1 is the case of conic sections (plane curves). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If B is true, one can immediately deduce that A is true as well, and if B is false, A can also be immediately deduced to be false. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."

In mathematics, an open set is a generalization of an open interval in the real line.

In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

A faulty generalization is an informal fallacy wherein a conclusion is drawn about all or many instances of a phenomenon on the basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an example of jumping to conclusions. For example, one may generalize about all people or all members of a group from what one knows about just one or a few people:

• If one meets a rude person from a given country X, one may suspect that most people in country X are rude.
• If one sees only white swans, one may suspect that all swans are white.

Expressed in more precise philosophical language, a fallacy of defective induction is a conclusion that has been made on the basis of weak premises, or one which is not justified by sufficient or unbiased evidence. Unlike fallacies of relevance, in fallacies of defective induction, the premises are related to the conclusions, yet only weakly buttress the conclusions, hence a faulty generalization is produced. The essence of this inductive fallacy lies on the overestimation of an argument based on insufficiently large samples under an implied margin of error.

Cartographic generalization, or map generalization, includes all changes in a map that are made when one derives a smaller-scale map from a larger-scale map or map data. It is a core part of cartographic design. Whether done manually by a cartographer or by a computer or set of algorithms, generalization seeks to abstract spatial information at a high level of detail to information that can be rendered on a map at a lower level of detail.

The cartographer has license to adjust the content within their maps to create a suitable and useful map that conveys spatial information, while striking the right balance between the map's purpose and the precise detail of the subject being mapped. Well generalized maps are those that emphasize the most important map elements while still representing the world in the most faithful and recognizable way.