The lengths of proofs: Kreisel’s conjecture and Gödel’s speed-up theorem

We collect and compare several results that have been obtained so far in the attempts to prove a statement conjectured by Kreisel about the lengths of proofs. We also survey several results regarding a speed-up theorem announced by G?del in an abstract published in 1936, Finally, we connect this to Kreisel’s conjecture. Bibliography: 63 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 153–188.     

Semi-invariants of binary forms and Sylvester’s theorem

The Ramanujan Journal - We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in...     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Implementation of Pellet’s theorem

Pellet’s theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.     

Posner’s second theorem for Jordan ideals in rings with involution

Posner’s second theorem states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend this result to Jordan ideals of rings with involution. Moreover, some related results are also discussed.     

Pontryagin’s theorem and spectral stability analysis of solitons

The main result of the present paper is the use of Pontryagin’s theorem for proving a criterion, based on the difference in the number of negative eigenvalues between two self-adjoint operators L _? and L ₊, for the linear part of a Hamiltonian system to have eigenvalues with strictly positive real part (unstable eigenvalues).     

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.     

Birkhoff’s theorem and viscosity solutions of Hamilton-Jacobi equations

We obtain a partial generalization of Birkhoff’s theorem of invariant curve to higher dimesional case in the context of viscosity solutions of Hamilton-Jacobi equations, or weak KAM theory. This is a new approach after Herman’s proof. This work was supported by the National Basic Research Program of China (Grant No. 2007CB814800) and National Natural Science Foundation of China (Grant No. 10301012)     

Hardy’s theorem and rotations of several complex variables

In this paper, Hardy’s theorem and rotations characterized by complex Gaussians in the complex plane due to Hogan and Lakey are extended to complex spaces of several variables. We point out that conditions under which a function on the n-dimensional real Euclidean space has an analytic extension to the complex space. Moreover, we prove that the function is a rotation of a multiple of real Gaussians through some angle if the extension satisfies certain assumptions.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Poncelet’s theorem and billiard knots

Let D be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in D. This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi??s proof of Poncelet??s theorem by means of elliptic functions.