Hadwiger’s conjecture for uncountable graphs

We consider the infinite form of Hadwiger’s conjecture. We give a(n apparently novel) proof of Halin’s 1967 theorem stating that every graph X with coloring number \(>\kappa \) (specifically with chromatic number \(>\kappa \)) contains a subdivision of \(K_\kappa \). We also prove that there is a graph of cardinality \(2^\kappa \) and chromatic number \(\kappa ^+\) which does not contain \(K_{\kappa ^+}\) as a minor. Further, it is consistent that every graph of size and chromatic number \(\aleph _1\) contains a subdivision of \(K_{\aleph _1}\).     

On a question of Silver about gap-two cardinal transfer principles

Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel’s constructible universe L, in which the \(\textit{GCH}\) holds and the transfer principles \((\aleph _2, \aleph _0) \rightarrow (\aleph _3, \aleph _1)\) and \((\aleph _3, \aleph _1) \rightarrow (\aleph _2, \aleph _0)\) fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.     

Square principles in ℙ<Subscript>max</Subscript> extensions

By forcing with P_max over strong models of determinacy, we obtain models where different square principles at ω ₂ and ω ₃ fail. In particular, we obtain a model of \({2^{{\aleph _0}}} = {2^{{\aleph _1}}} = {\aleph _2} + {\neg }\square \left( {{\omega _2}} \right) + {\neg }\square \left( {{\omega _3}} \right)\).     

Volume growth of some uniformly contractible Riemannian manifolds

It is shown that if a uniformly contractible Riemannian n-manifold (M,g) is K-quasi-isometric to an n-dimensional normed space\((V^{n},\|\cdot\|)\), (K ≥  1), then\(\liminf_{R\rightarrow \infty}\frac{{Vol}_g( {Ball}_{R})}{R^{n}\omega_{n}}\geq\frac{1}{K^{2n}}\) where ω _n is the volume of the unit Euclidean ball. In particular, if M is uniformly contractible and\(d_{GH}((M,d_g), (V^n,\|\cdot\|)) < \infty \), then M has at least Euclidean volume growth. This corollary covers an earlier result by Burago and Ivanov. Our results are motivated by a volume growth theorem contained in Gromov’s book [Gromov in Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999, p. 256], we give a detailed proof of this theorem. Using the same argument, we also derive a generalization of the theorem which is pointed out by Gromov.     

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in Open image in new window , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.     

More results in polychromatic Ramsey theory

We study polychromatic Ramsey theory with a focus on colourings of [ω ₂]². We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2 ^ω = ω ₂ and \(\omega _2 \to ^{poly} (\alpha )_{\aleph _0 - bdd}^2 \) for every α <ω ₂; (2) 2 ^ω = ω ₂ and \(\omega _2 \nrightarrow ^{poly} (\omega _1 )_{2 - bdd}^2 \).     

How far can we go with Amitsur’s theorem in differential polynomial rings?

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.     

Differentiability of mappings in the geometry of Carnot manifolds

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher’s theorem) and a generalization of Stepanov’s theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis ? the nilpotent tangent cone.”     

Some Structure Theories of Leibniz Triple Systems

In this paper, we investigate the Leibniz triple system T and its universal Leibniz envelope U(T). The involutive automorphism of U(T) determining T is introduced, which gives a characterization of the \(\mathbb {Z}_{2}\)-grading of U(T). We show that the category of Leibniz triple systems is equivalent to a full subcategory of the category of \(\mathbb {Z}_{2}\)-graded Leibniz algebras. We give the relationship between the solvable radical R(T) of T and R a d(U(T)), the solvable radical of U(T). Further, Levi’s theorem for Leibniz triple systems is obtained. Moreover, the relationship between the nilpotent radical of T and that of U(T) is studied. Finally, we introduce the notion of representations of a Leibniz triple system, which can be described by using involutive representations of its universal Leibniz envelope.     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

Groups equal to a product of three conjugate subgroups

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies \(G = {\left( {B{n_0}B} \right)^2} = B{B^{{n_0}}}B\) where n₀ is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.     

On implicit function theorems at abnormal points

We consider the equation F(x, σ) = 0, x ∈ K, in which σ is a parameter and x is an unknown variable taking values in a specified convex cone K lying in a Banach space X. This equation is investigated in a neighborhood of a given solution (x*, σ*), where Robinson’s constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson’s constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone K coincides with the whole space X.     

The Calderón problem for Hilbert couples

We prove that intermediate Banach spaces\(\mathcal{A}\) and\(\mathcal{B}\) with respect to arbitrary Hilbert couples\(\bar {H}\) and\(\bar {K}\) are exact interpolation if and onlyif they are exactK-monotonic, i.e. the condition\(f^0 \in \mathcal{A}\) and the inequality\(K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0\), implyg⁰∈B and ‖g⁰‖B≤‖f⁰‖ _A (K is Peetre’sK-functional). It is well known that this property is implied by the following: for each ?>1 there exists an operator\(T:\bar {H} \to \bar {K}\) such thatTf⁰=g⁰, and\(K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0\). Verifying the latter property, it suffices to consider the “diagonal case” where\(\bar {H} = \bar {K}\) is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ?=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

The probability that a pair of group elements is autoconjugate

Let g and h be arbitrary elements of a given finite group G. Then g and h are said to be autoconjugate if there exists some automorphism α of G such that h = g^α. In this article, we construct some sharp bounds for the probability that two random elements of G are autoconjugate, denoted by \(\mathcal {P}_{a}(G)\). It is also shown that \(\mathcal {P}_{a}(G)|G|\) depends only on the autoisoclinism class of G.     

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f₀, 0} of the utility function (integrand) f₀ is relaxed to the requirement that the integrals of f₀ over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.     

Involution words II: braid relations and atomic structures

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori–Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions x, y in a Coxeter group W with automorphism \(*\), we associate a set of involution words \(\hat{\mathcal {R}}_*(x,y)\). This set is the disjoint union of the reduced words of a set of group elements \(\mathcal {A}_*(x,y)\), which we call the atoms of y relative to x. The atoms, in turn, are contained in a larger set \(\mathcal {B}_*(x,y) \subset W\) with a similar definition, whose elements are referred to as Hecke atoms. Our main results concern some interesting properties of the sets \(\hat{\mathcal {R}}_*(x,y)\) and \(\mathcal {A}_*(x,y) \subset \mathcal {B}_*(x,y)\). For finite Coxeter groups, we prove that \(\mathcal {A}_*(1,y)\) consists of exactly the minimal-length elements \(w \in W\) such that \(w^* y \le w\) in Bruhat order, and we conjecture a more general property for arbitrary Coxeter groups. In type A, we describe a simple set of conditions characterizing the sets \(\mathcal {A}_*(x,y)\) for all involutions \(x,y \in S_n\), giving a common generalization of three recent theorems of Can et al. We show that the atoms of a fixed involution in the symmetric group (relative to \(x=1\)) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the “Chinese relation” studied by Cassaigne et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of “braid relations” spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto’s theorem for involution words in arbitrary Coxeter groups.     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).