On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.     

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.     

The spectrum of a finite pseudocomplemented lattice

Let L be a finite pseudocomplemented lattice. Every interval [0, a] in L is pseudocomplemented, so by Glivenko’s theorem, the set S(a) of all pseudocomplements in [0, a] forms a boolean lattice. Let B _i denote the finite boolean lattice with i atoms. We describe all sequences (s ₀, s ₁, . . . , s _n ) of integers for which there exists a finite pseudocomplemented lattice L with s _i = |{ a ∈ L | S(a) ? B _i }|, for all i, and there is no a ∈ L with S(a) ? B _n+1. This result settles a problem raised by the first author in 1971.     

The fundamental constituents of iteration digraphs of finite commutative rings

For a finite commutative ring R and a positive integer k ? 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a ^k . Let R = R ₁ ⊕ … ⊕ R _s , where s > 1 and R _i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R ₁, …, R _s } (it is possible that N is the empty set \(\not 0\) ). We define the fundamental constituents G _N ^* (R, k) of G(R, k) induced by the vertices which are of the form {(a ₁, …, a _s ) ∈ R: a _i ∈ D(R _i ) if R _i ∈ N, otherwise a _i ∈ U(R _i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* _N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces

Let N be the stabilizer of the word w = s ₁ t ₁ s ₁ ^?1 t ₁ ^?1 … s _g t _g s _g ^?1 t _g ^?1 in the group of automorphisms Aut(F _2g ) of the free group with generators ?ub;s _i, t _i?ub; _i=1,…,g . The fundamental group π₁(Σ_g) of a two-dimensional compact orientable closed surface of genus g in generators ?ub;s _i, t _i?ub; is determined by the relation w = 1. In the present paper, we find elements S _i, T _i ∈ N determining the conjugation by the generators s _i, t _i in Aut(π₁(Σ_g)). Along with an element β ∈ N, realizing the conjugation by w, they generate the kernel of the natural epimorphism of the group N on the mapping class group M _g,0 = Aut(π₁(Σ_g))/Inn(π₁(Σ_g)). We find the system of defining relations for this kernel in the generators S ₁, …, S _g, T ₁, …, T _g, α. In addition, we have found a subgroup in N isomorphic to the braid group B _g on g strings, which, under the abelianizing of the free group F _2g , is mapped onto the subgroup of the Weyl group for Sp(2g, ?) consisting of matrices that contain only 0 and 1.     

Topologies on products of partially ordered sets I: Interval topologies

Given a certain construction principle assigning to each partially ordered setP some topology θ(P) onP, one may ask under what circumstances the topology θ(P) of a productP = ?_j∈J P _j of partially ordered setsP _i agrees with the product topology ?_j∈Jθ(P _i) onP. We shall discuss this question for several types ofinterval topologies (Part I), forideal topologies (Part II), and fororder topologies (Part III). Some of the results contained in this first part are listed below:

1. Let θ_i(P) denote thesegment topology. For any family of posetsP _j ?_j∈Jθ_s(P_j)=θ_s(?_j∈JP_i) iff at most a finite number of theP _j has more than one element (1.1).
2. Let θ_cs(P) denote theco-segment topology (lower topology). For any family of lower directed posetsP _j ?_j∈Jθ_cs(P_i)=θ_cs(?_j∈JP_i) iff eachP _j has a least element (1.5).
3. Let θ_i(P) denote theinterval topology. For a finite family of chainsP _j,P _j ?_j∈Jθ_i(P_i)=θ_i(?_j∈JP_i) iff for allj∈k, P _j has a greatest element orP _k has a least element (2.11).
4. Let θ_ni(P) denote thenew interval topology. For any family of posetsP _j,P _j ?_j∈Jθ_ni(P_j)=θ_ni(?_j∈JP_j) whenever the product space is ab-space (i.e. a space where the closure of any subsetY is the union of all closures of bounded subsets ofY) (3.13).

In the case oflattices, some of the results presented in this paper are well-known and have been shown earlier in the literature. However, the case of arbitraryposets often proved to be more difficult.     

On the enumeration of chains in regular chain-groups

Let N be a regular chain-group on E (see W. T. Tutte, Canad. J. Math.8 (1956), 13–28); for instance, N may be the group of integer flows or tensions of a directed graph with edge-set E). It is known that the number of proper Z_λ-chains of N (λ ∈ Z, λ ≥ 2) is given by a polynomial in λ, P(N, λ) (when N is the chain-group of integer tensions of the connected graph G, λP(N, λ) is the usual chromatic polynomial of G). We prove the formula: P(N, λ) = Σ_{[E′]∈O(N)⁺/～}Q(R_[E′](N), λ), where O(N)⁺ is the set of orientations of N with a proper positive chain, ～ is a simple equivalence relation on O(N)⁺ (sequence of reversals of positive primitive chains), and Q(R_[E′](N), λ) is the number of chains with values in [1, λ ? 1] in any reorientation of N associated to an element of [E′]. Moreover, each term Q(R_[E′](N), λ) is a polynomial in λ. As applications we obtain: P(N, 0) = (?1)^r(N)∥O(N)⁺/～∥; P(N, ?1) = (?1)^r(N)∥O(N)⁺∥ (a result first proved by Brylawski and Lucas); P(N, λ + 1) ≥ P(N, λ) for λ ≥ 2, λ ∈ Z. Our result can also be considered as a refinement of the following known fact: A regular chain-group N has a proper Z_λ-chain iff it has a proper chain in [?λ + 1, λ ? 1].     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Strong Splitter Theorem

The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M ₀, . . . , M _n of 3-connected matroids with ${M_0 \cong N}$ , M _n = M and for ${i \in \{1, \ldots , n}\}$ , M _i is a single-element extension or coextension of M _i?1. Observe that there is no condition on how many extensions may occur before a coextension must occur. We give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, M starting with N and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of thematroids involved is r(M)). Moreover, if two consecutive single-element extensions by elements {e, f} are followed by a coextension by element g, then {e, f , g} form a triad in the resulting matroid.     

On the minor-minimal 3-connected matroids having a fixed minor

Let N be a minor of a 3-connected matroid M such that no proper 3-connected minor of M has N as a minor. This paper proves a bound on |E(M)−E(N)| that is sharp when N is connected.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [? π, π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y:= {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that, on [y _i , y _i?1], f is nondecreasing if i is odd and nonincreasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₂(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

On a family of solutions of a linear second-order differential equation with constant unbounded operator coefficients in a Banach space

In a Banach space E, we study the equation 1 $$ u''(t) + Bu'(t) + Cu(t) = f(t), 0 \leqslant t < \infty $$ , where f(t) ∈ C([0,∞);E), B,C ∈ N(E), and N(E) is the set of closed unbounded linear operators from E to E with dense domain in E. We find a two-parameter family of solutions of Eq. (1) in two cases: (a) the operator discriminant D = B ² ? 4C of Eq. 1 is zero; (b) D = F ², where F is some operator in N(E). We suggest a method for increasing the smoothness of such solutions by imposing more restrictive conditions on the input data W = (B,C,f(t)) and the parameters x ₁, x ₂ ∈ E.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.