MathDB

2009 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

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Isometries on the set of non-negative matrices

Denote by Hn H_n the linear space of n×n n\times n self-adjoint complex matrices, and by Pn P_n the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on Hn H_n \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n) and its derived metric. Show that every ϕ:Pn→Pn \phi: P_n\to P_n isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n) or \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n) where U U is an n×n n\times n unitary matrix, X X is a positive-semidefinite matrix, and T ^T and ∗ ^* denote taking the transpose and the adjoint, respectively.
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Langrange function extremals and PDE system

Let U⊂Rn U\subset\mathbb R^n be an open set, and let L:U×Rn→R L: U\times\mathbb R^n\to\mathbb R be a continuous, in its second variable first order positive homogeneous, positive over U×(Rn∖{0}) U\times (\mathbb R^n\setminus\{0\}) and of C2 C^2-class Langrange function, such that for all p∈U p\in U the Gauss-curvature of the hyper surface \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \} is nowhere zero. Determine the extremals of L L if it satisfies the following system \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\}) of partial differetial equations, where y^k(u,v) : \equal{} v^k for (u,v)∈U×Rk (u,v)\in U\times\mathbb R^k, v \equal{} (v^1,\dots,v^k).
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Vector field being weakly tangent to diffeomorphism subgroup

Let H H be an arbitrary subgroup of the diffeomorphism group Diff∞(M) \mathsf{Diff}^\infty(M) of a differentiable manifold M M. We say that an C∞ \mathcal C^\infty-vector field X X is weakly tangent to the group H H, if there exists a positive integer k k and a C∞ \mathcal C^\infty-differentiable map \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M such that (i) for fixed t1,…,tk t_1,\dots,t_k the map φt1,…,tk:x∈M↦φ(t1,…,tk,x) \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x) is a diffeomorphism of M M, and φt1,…,tk∈H \varphi_{t_1,\dots,t_k}\in H; (ii) \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id} whenever t_j \equal{} 0 for some 1≤j≤k 1\leq j\leq k; (iii) for any C∞ \mathcal C^\infty-function f:M→R f: M\to \mathbb R X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}. Prove, that the commutators of C∞ \mathcal C^\infty-vector fields that are weakly tangent to H⊂Diff∞(M) H\subset \textsf{Diff}^\infty(M) are also weakly tangent to H H.
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Steiner triple system and maps into group

A set system (S,L) (S,L) is called a Steiner triple system, if L≠∅ L\neq\emptyset, any pair x,y∈S x,y\in S, x≠y x\neq y of points lie on a unique line ℓ∈L \ell\in L, and every line ℓ∈L \ell\in L contains exactly three points. Let (S,L) (S,L) be a Steiner triple system, and let us denote by xy xy the thrid point on a line determined by the points x≠y x\neq y. Let A A be a group whose factor by its center C(A) C(A) is of prime power order. Let f,h:S→A f,h: S\to A be maps, such that C(A) C(A) contains the range of f f, and the range of h h generates A A. Show, that if f(x) \equal{} h(x)h(y)h(x)h(xy) holds for all pairs x≠y x\neq y of points, then A A is commutative, and there exists an element k∈A k\in A, such that f(x) \equal{} kh(x) for all x∈S x\in S.
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