Ion d0 . . . ( k 1) d k = r – s .Homogeneous Function Theorem. Let Ei iN be finite-dimensional vector spaces. Let f : i=1 Ei R be a smooth function such that there exist optimistic true numbers ai 0 and w R satisfying: f ( a1 e1 , . . . , a i e i , . . .) = w f ( e1 , . . . , e i , . . .) (4)for any constructive actual number 0 and any (e1 , . . . , ei , . . .) i=1 Ei . Then, f is determined by a finite number of Ulixertinib web variables e1 , . . . , ek , and it truly is a sum of monomials of degree di in ei satisfying the relationa1 d1 a k d k = w .(5)If there are no natural numbers d1 , . . . , dr N 0 satisfying this equation, then f will be the zero map. Proof. Firstly, if f is not the zero map, then we observe w 0 since, otherwise, (4) is contradictory when 0. As f is smooth, there exists a neighbourhood U = 1 , . . . , i=1 Ei from the origin and also a smooth map f : k (U) R such that f |U = ( f k)|U . As the a1 , . . . , ak are constructive, there exist a neighborhood of zeros, V 0 R, plus a neighborhood on the origin V k (U) such that, for any (e1 , . . . , ek) V and any V 0 which are optimistic, the vector ( a1 e1 , . . . , ak ek) lies in V. On that neighborhood V, the function f 2-NBDG web satisfies the homogeneity situation: f ( a1 e1 , . . . , a k e k) = w f ( e1 , . . . , e k) (6)for any optimistic true quantity V 0 . Differentiating this equation, we acquire analogous conditions for the partial derivatives of f ; v.gr.: f f ( a1 e1 , . . . , a k e k) = w – a1 ( e , . . . , ek) . x1 x1 1 In the event the order of derivation is massive adequate, the corresponding partial derivative is homogeneously of negative weight and, hence, zero. This implies that f is a polynomial; the homogeneity condition (6) is then satisfied for any optimistic V 0 if and only if its monomials satisfy (five). Lastly, offered any e = (e1 , . . . , en , . . .) i=1 Ei , we take R such that the vector ( a1 e1 , . . . , ak ek , . . .) lies in U. Then: f ( e) = – w f ( a1 e1 , . . . , a n e n , . . .) = – w f ( a1 e1 , . . . , a k e k) = f ( e1 , . . . , e k) and f only is dependent upon the very first k variables.Mathematics 2021, 9,12 ofThis statement readily generalizes to say that, for any finite-dimensional vector space W, there exists an R-linear isomorphism: Smooth maps f : Ei W satisfying (4)i =(7)d1 ,…,dkHomR (Sd1 E1 . . . Sdk Ek , W)exactly where d1 , . . . , dk run more than the non-negative integer solutions of (5). 5. An Application Finally, as an application of Theorem eight, within this section, we compute some spaces of vector-valued and endomorphism-valued natural forms related to linear connections and orientations, therefore getting characterizations of the torsion and curvature operators (Corollary 13 and Theorem 15). five.1. Invariant Theory of your Unique Linear Group Let V be an oriented R-vector space of finite dimension n, and let Sl(V) be the real Lie group of its orientation-preserving R-linear automorphisms. Our aim should be to describe the vector space of Sl(V)-invariant linear maps: V . p . V V . p . V – R . . . For any permutation S p , there exist the so-called total contraction maps, that are defined as follows: C (1 . . . p e1 . . . e p) := 1 (e(1)) . . . p (e( p)) . Additionally, let n V be a representative with the orientation, and let e be the dual n-vector; that is definitely to say, the only element in n V such that (e) = 1. For any permutation S pkn , the following linear maps are also Sl(V)-invariant:(1 , . . . , p , e1 , . . . , e p) – C ( . k . 1 . . . p e . k . e e1 . . . e p) . . .Classical invariant theory proves t.