Engel’s theorem for malcev superalgebras

A version of Engel’s theorem for Malcev superalgebras is proved in the spirit of theJacobson-Engel theorem for Lie algebras. Some consequences for the structure of Malcev superalgebras with trivial Lie nucleus are derived.     

Menger’s theorem for infinite graphs

We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set of disjoint A–B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in . This settles an old conjecture of Erdős.     

Moufang’s theorem for non-Moufang loops

We introduce a class of non-Moufang loops satisfying Moufang’s theorem.     

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

Asymptotic methods for spherically symmetric MHD α2-dynamos

We consider two models of spherically-symmetric MHD α²–dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α₀ are linear functions of α₀ and form a mesh in the (α₀, λ)–plane. The nodes of the spectral mesh correspond to double-degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We briefly discuss the spectral behavior of the α²–dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Ando’s theorem for nonnegative forms

In this paper, we present a generalization of Ando??s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et?al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1?C12, 1999). That is, [A]B??? [B]A or [B]A??? [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et?al. in J. Funct. Anal. 257(12), 3858?C3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.     

Alperin’s fusion theorem for localities

In these notes we give a version of the Alperin–Goldschmidt fusion theorem for localities.     

Clifford’s theorem for coherent systems

Let C be an algebraic curve of genus g ≥ 2. We prove an analogue of Clifford’s theorem for coherent systems on C and some refinements using results of Re and Mercat. Received: 27 July 2007     

Beurling’s analyticity theorem for quantum differences

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ _n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.     

Naishul’s theorem for fibered holomorphic maps

We show that the fibered rotation number associated to an indifferent invariant curve for a fibered holomorphic map is a topological invariant.     

Lazard’s theorem for S-pure flat modules

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.     

Shcherbina’s theorem for finely holomorphic functions

We prove an analogue of Sadullaev’s theorem concerning the size of the set where a maximal totally real manifold M can meet a pluripolar set. M has to be of class C ¹ only. This readily leads to a version of Shcherbina’s theorem for C ¹ functions f that are defined in a neighborhood of certain compact sets ${K\subset\mathbb{C}}We prove an analogue of Sadullaev’s theorem concerning the size of the set where a maximal totally real manifold M can meet a pluripolar set. M has to be of class C ¹ only. This readily leads to a version of Shcherbina’s theorem for C ¹ functions f that are defined in a neighborhood of certain compact sets K ì \mathbbC{K\subset\mathbb{C}}. If the graph Γ _f (K) is pluripolar, then \frac?f?[`(z)]=0{\frac{\partial f}{\partial\bar z}=0} in the closure of the fine interior of K.     

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].     

Hechler’s theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechlers classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by Bartoszyski and Kada.Research supported by NSERC. The first author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when the present paper was completed.The second author was supported by Grant-in-Aid for Young Scientists (B) 14740058, MEXT.Mathematics Subject Classification (2000): 03E35, 03E17Revised version: 16 February 2004     

Berkson’s theorem

We give a new proof of the theorem that Amitsur’s complex for purely inseparable field extensions has vanishing homology in dimensions higher than 2. This is accomplished by computing the kernel and cokernel of the logarithmic derivativet →Dt/t mapping the multiplicative Amitsur complex to the acyclic additive one (D is a derivation of the extension field). This research was supported by National Science Foundation grant NSF GP 1649.     

Sch’nol’s theorem for strongly local forms

We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ- or Kirchhoff boundary conditions.     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.