Connected matroids with a small circumference

Lemos and Oxley proved that if M is a connected matroid with |E(M)|⩾3r(M), then M has a circuit C such that M⧹C is connected. In this paper, we shall improve this result proving that for a simple and connected matroid M, if r(M)⩾7 and |E(M)|⩾3r(M)−3, then M has a circuit C such that M⧹C is connected. To prove this result, we shall construct all the connected matroids having circumference at most five, with the exception of those which are 3-connected and have rank five.     

Power inequalities and spectral dominance of generalized matrix norms

A generalized matrix norm G dominates the spectral radius for all A?M_n(C) (i) if for some positive integer k the rule G(A^k) ? G(A)^k holds for all A?M_n(C) and (ii) if and only if for each A?M_n(C) there exists a constant γ_A such that G(A^k) ? γ_AG(A)^kfor all positive integers k. Other results and examples are also given concerning spectrally dominant generalized matrix norms.     

The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

We define the thin fundamental Gray 3-groupoid S₃(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S₃(M)→C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3-groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.     

Ubiquity of odometers in topological dynamical systems

Let M be the Cantor space or an n-manifold with C(M,M) the set of continuous self-maps of M. We prove the following:

(1)

There is a residual set of points (x,f) in M×C(M,M) all of which generate as their ω-limit set a particular, unique adding machine.

(2)

Moreover, if M has the fixed point property, then a generic f∈C(M,M) generates uncountably many distinct copies of every possible adding machine.

     

Connection-graph and iteration-graph of monotone boolean functions

In this paper, we establish some relationships between the circuits of the connection-graph G_C(F), and the circuits of theiteration-graph G₁(F), of a monotone boolean function F. We first show that if G₁(F) contains an element circuit of length multiple of p? {2,3}, then G_C(F) contains an elementary circuit of length multiple of p; then we prove that if G_C(F) is a subgraph of a caterpillar, then G₁(F) is a subgraph of a tree; at last we exhibit an infinite family of monotone boolean functions {F_n; n = 2 × 5^q, q ≥ 1} such that any G_C(F_n) is a subgraph of a tree, and G₁(F_n) contains a circuit of length 2^q+1, i.e., of the order $\text{n}{}^{\mathrm{<mtext>log</mtext><mtext>2</mtext><mtext>log5</mtext>}}$.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

The convenient setting for non-quasianalytic Denjoy-Carleman differentiable mappings

For Denjoy-Carleman differentiable function classes CM where the weight sequence M=(Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is CM if it maps CM-curves to CM-curves. The category of CM-mappings is cartesian closed in the sense that CM(E,CM(F,G))≅CM(E×F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of CM-diffeomorphisms is a CM-Lie group but not better.     

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M_m(C_2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m?0,t?1, k?1, and n?2k+2. These results have consequences in circular chromatic number.     

Lens spaces and toroidal Dehn fillings

We show that if M is a hyperbolic 3-manifold with ?M a torus such that M(r ₁) is a lens space and M(r ₂) is toroidal, then ??(r ₁, r ₂) ?? 4.     

How to determine the maximum genus of a graph

The maximum genus, γ_M(G), of a connected graph G is the largest genus γ(S) for orientable surfaces S in which G has a 2-cell embedding. In this paper, we define a new combinatorial invariant ξ(G), the Betti deficiency of G, to be ξ(C) = min_C?G{ξ(C) 6 ξ(C) = number of odd components of a cotree C of G (by odd component we mean one with an odd number of edges). We formalize a new embedding technique to obtain the formula:

$\text{γ}{}_{\mathrm{M}}\text{(G)=}\text{1}\text{2}\text{(β(G)?ξ(G))}$

where β(G) denotes the Betti number of G.In a further paper, various consequences will be given.     

Derivations on the algebra of operators in hilbert C*-modules

Let M be a full Hilbert C*-module over a C*-algebra A,and let End*A(M) be the algebra of adjointable operators on M.We show that if A is unital and commutative,then every derivation of End A(M) is an inner derivation,and that if A is σ-unital and commutative,then innerness of derivations on "compact" operators completely decides innerness of derivations on End*A(M).If A is unital(no commutativity is assumed) such that every derivation of A is inner,then it is proved that every derivation of End*A(Ln(A)) is also inner,where Ln(A) denotes the direct sum of n copies of A.In addition,in case A is unital,commutative and there exist x0,y0 ∈ M such that x0,y0 = 1,we characterize the linear A-module homomorphisms on End*A(M) which behave like derivations when acting on zero products.     

Circuit preserving edge maps

It is proved that any one-to-one edge map f from a 3-connected graph G onto a graph G′, G and G′ possibly infinite, satisfying f(C) is a circuit in G′ whenever C is a circuit in G is induced by a vertex isomorphism. This generalizes a result of Whitney which hypothesizes f(C) is a circuit in G′ if and only if C is a circuit in G.     

Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

Two integral operators in Clifford analysis

Segal-Bargmann space F₂(Cn) and monogenic Fock space M₂(Rn+1) are introduced first. Then, with the help of exponential functions in Clifford analysis, two integral operators are defined to connect F₂(Cn) and M₂(Rn+1) together. The corresponding integral properties are studied in detail.     

On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C _M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C _M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C _M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.     

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.     

C (n)-cardinals

For each natural number n, let C ⁽ⁿ⁾ be the closed and unbounded proper class of ordinals α such that V _α is a Σ _n elementary substructure of V. We say that κ is a C ⁽ⁿ⁾ -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C ⁽ⁿ⁾. By analyzing the notion of C ⁽ⁿ⁾-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C ⁽ⁿ⁾-cardinals form a much finer hierarchy. The naturalness of the notion of C ⁽ⁿ⁾-cardinal is exemplified by showing that the existence of C ⁽ⁿ⁾-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et?al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C ⁽ⁿ⁾-extendible cardinals.