Reverse mathematics and initial intervals

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We show that the left to right directions of these theorems are equivalent to _ACA0${\mathsf{ACA}}_0$ and _ATR0${\mathsf{ATR}}_0$, respectively. On the other hand, the opposite directions are both provable in _WKL0${\mathsf{WKL}}_0$, but not in _RCA0${\mathsf{RCA}}_0$. We also prove the equivalence with _ACA0${\mathsf{ACA}}_0$ of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains.     

A note on monolithic scattered compacta

For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP  , for short) is that the dual unit ball _BE?$B_{E^{?}}$ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X   turns out to be sufficient for _C0(X)$C_0(X)$ to enjoy the CSCP.     

Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak^?${\mathrm{weak}}^{?}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L  -weakly compact sets coincide. In particular, in terms of almost Dunford–Pettis operators into _c0$c_0$, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E   is almost limited if and only if every continuous linear operator T:E→_c0$T:E\to c_0$ is an almost Dunford–Pettis operator.     

Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response

In this paper, we investigate the dynamical behavior of a virus infection model with delayed humoral immunity. By using suitable Lyapunov functional and the LaSalle?s invariance principle, we establish the global stabilities of the two boundary equilibria. If _R0<1$R_0<1$, the uninfected equilibrium _E0$E_0$ is globally asymptotically stable; if _R1<1<_R0$R_1<1<R_0$, the infected equilibrium without immunity _E1$E_1$ is globally asymptotically stable. When _R1>1$R_1>1$, we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity _E2$E_2$. The time delay can change the stability of _E2$E_2$ and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions is also studied. We check our theorems with numerical simulations in the end.     

Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and _Cb(X)$C_b(X)$ be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T   from _Cb(X)$C_b(X)$ to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=_T1+_T2+_T3+_T4$T=T_1+T_2+T_3+T_4$, where _Tk:_Cb(X)→E$T_k:C_b(X)\to E$(k=1,2,3,4)$(k=1,2,3,4)$ are weakly compact operators, and _T1$T_1$ is tight, _T2$T_2$ is purely τ  -additive, _T3$T_3$ is purely σ  -additive and _T4$T_4$ is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

Operator norm attainment and inner product spaces

In this paper we prove that a finite dimensional real normed linear space X$\mathbb{X}$ is an inner product space iff for any linear operator T   on X$\mathbb{X}$, T   attains its norm at _e1,_e2∈_SX$e_1,e_2\in S_{\mathbb{X}}$ implies T   attains its norm at span{_e1,_e2}∩_SX$\mathit{span}\{e_1,e_2\}\cap S_{\mathbb{X}}$.     

Euler class groups and 2-torsion elements

We exhibit an example of a smooth affine threefold A$A$ over a field of characteristic 0$0$ for which there exist non-trivial 2-torsion elements in the Euler class group E(A)$E\left(A\right)$ vanishing in the weak Euler class group _E0(A)$E_0\left(A\right)$. This gives a positive answer to a question of the first author and Raja Sridharan.     

Higher order derived functors and the Adams spectral sequence

Classical homological algebra studies chain complexes, resolutions, and derived functors in additive categories. In this paper we define higher order chain complexes, resolutions, and derived functors in the context of a new type of algebraic structure, called an algebra of left cubical balls  . We show that higher order resolutions exist in these algebras, and that they determine higher order Ext-groups. In particular, the _Em$E_m$-term of the Adams spectral sequence (m>2)$(m>2)$ is such a higher Ext-group, providing a new way of constructing its differentials.     

Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G  . We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group _A5$A_5$, and give various other examples.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

Factorization properties of paratopological groups

In this article we continue the study of R$\mathbb{R}$-factorizability in paratopological groups. It is shown that: (1) all concepts of R$\mathbb{R}$-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G   is R$\mathbb{R}$-factorizable if and only if it is totally ω  -narrow and has property ω-QU$\omega \text{-}\mathit{QU}$; (3) every subgroup of a _T1$T_1$ paratopological group G   is R$\mathbb{R}$-factorizable provided that the topological group G^?$G^{?}$ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space  ; (4) if Π=_∏i∈IGi$\Pi ={\prod }_{i\in I}{G}_i$ is a product of _T1$T_1$ paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π   is R$\mathbb{R}$-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.     

Some observations on compact indestructible spaces

Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelöf _T2$T_2$ indestructible space has the finite derived set property and a compact _T2$T_2$ indestructible space is pseudoradial. Finally, we observe that under CH a compact weakly Whyburn space of countable tightness is indestructible.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.