In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let \scriptstyle M,\,N be smooth Riemannian manifolds of respective dimensions \scriptstyle m\,\geq\, n. Let \scriptstyle F:M\,\longrightarrow\, N be a smooth surjection such that the pushforward (differential) of \scriptstyle F is surjective almost everywhere. Let \scriptstyle\varphi:M\,\longrightarrow\, [0,\infty) a measurable function.
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The definitions of strict positive homogeneity that was given for -valued functions on immediately extends, without change, to functions that are valued in other codomains. :Definition: Let be a function on valued in (or even in ). We say that is strictly positively homogeneous if for all and all positive real . If never takes the value then we say that is non-negative homogeneous if for all and all non-negative real .
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As in animals, the plant homeobox genes code for the typical 60 amino acid long DNA- binding homeodomain or in case of the TALE (three amino acid loop extension) homeobox genes for an "atypical" homeodomain consisting of 63 amino acids. According to their conserved intron–exon structure and to unique codomain architectures they have been grouped into 14 distinct classes: HD-ZIP I to IV, BEL, KNOX, PLINC, WOX, PHD, DDT, NDX, LD, SAWADEE and PINTOX. Conservation of codomains suggests a common eukaryotic ancestry for TALE and non-TALE homeodomain proteins.
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Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction =A instead of =.
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A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and F(X)\cong F(Y) then X \cong Y.
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A unital subring B \subseteq A has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from A^n \rightarrow A \otimes_B A for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two. Here we use the usual notation A^n = A \times \ldots \times A (n times) as well as the common notion, p is a split epimorphism if there is a homomorphism q in the reverse direction such that pq = identity on the image of p. (Sometimes the subring B in A is referred to as the ring extension A over B; the theory works as well for a ring homomorphism B into A, which induces right and left B-modules structures on A.) Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed. For example, let A be the group algebra of a finite group G (over any commutative base ring k; see the articles on group theory and group ring for the elementary definitions).
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