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.     

Hindman’s theorem and idempotent types

Motivated by a question of Di Nasso, we show that Hindman’s theorem is equivalent to the existence of idempotent types in countable complete extensions of Peano Arithmetic.     

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.     

The degenerated second main theorem and Schmidt’s subspace theorem

In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.     

On Turner’s theorem and first-order theory

A theorem of E.C. Turner states that if F is a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner’s characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner’s theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.     

A Brauer’s theorem and related results

Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations. An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem.     

Oriented matroids and Ky Fan’s theorem

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb. 1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C ^m,r .     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

On Jordan’s theorem

In the present paper, we give some remarks on the well-known Jordan theorem and Hamiltonians.     

Funayama’s theorem revisited

Funayama’s theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama’s original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer’s proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama’s theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer’s proof, there is no obvious way to generalize Johnstone’s proof to the non-complete case.     

Extending Kotzig’s theorem

The weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and Shephard proved that every graph triangulating the torus has weight at most 15. We extend these results for graphs, multigraphs and pseudographs “triangulating” the sphere withg handlesS _g ,g≧1, showing that the corresponding weights are at most about and 24g−9, respectively; if a (multi, pseudo) graph triangulatesS _g and it is big enough, then its weight is at most 15.     

Connected Baranyai’s theorem

Let K _n ^h = (V, ( _h ^V )) be the complete h-uniform hypergraph on vertex set V with ¦V¦ = n. Baranyai showed that K _n ^h can be expressed as the union of edge-disjoint r-regular factors if and only if h divides rn and r divides \((_{h - 1}^{n - 1} )\) . Using a new proof technique, in this paper we prove that λK _n ^h can be expressed as the union \(\mathcal{G}_1 \cup ... \cup \mathcal{G}_k \) of k edge-disjoint factors, where for 1≤i≤k, \(\mathcal{G}_i \) is r _i -regular, if and only if (i) h divides r _i n for 1≤i≤k, and (ii) \(\sum\nolimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )} \) . Moreover, for any i (1≤i≤k) for which r _i ≥2, this new technique allows us to guarantee that \(\mathcal{G}_i \) is connected, generalizing Baranyai’s theorem, and answering a question by Katona.