Bounded stationary reflection II

Bounded stationary reflection at a cardinal λ is the assertion that every stationary subset of λ reflects but there is a stationary subset of λ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu >{\mathrm{?}}_{\omega }$ and models in which bounded stationary reflection holds at ${\mu }^{+}$ but the approachability property fails at μ.     

The tree property at double successors of singular cardinals of uncountable cofinality

Assuming the existence of a strong cardinal κ and a measurable cardinal above it, we force a generic extension in which κ is a singular strong limit cardinal of any given cofinality, and such that the tree property holds at ${\kappa }^{++}$.     

Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let $E^{(a)}$ denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then $X^{(2)}$ computes $A^{(\omega )}$, we show that there is a ceer R such that $R\ge {\mathrm{Id}}^{(n)}$, for every finite ordinal n, but, for all k, $R^{(k)}?{\mathrm{Id}}^{(\omega )}$ (here Id is the identity equivalence relation). We show that if $a,b$ are notations of the same ordinal less than ${\omega }^2$, then $E^{(a)}\equiv E^{(b)}$, but there are notations $a,b$ of ${\omega }^2$ such that ${\mathrm{Id}}^{(a)}$ and ${\mathrm{Id}}^{(b)}$ are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form ${\mathrm{Id}}^{(a)}$ where a is a notation for ${\omega }^2$.     

Cardinal characteristics at κ in a small u(κ) model

We provide a model where $\mathfrak{u}(\kappa )<2^{\kappa }$ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

Thirteen limit cycles for a class of Hamiltonian systems under seven-order perturbed terms

In this paper we study the existence, number and distribution of limit cycles of the perturbed Hamiltonian system:$\begin{array}{cc}\hfill & x^{\prime }=4y({\mathit{abx}}^2-{\mathit{by}}^2+1)+\epsilon x\left({\mathit{ux}}^n+{\mathit{vy}}^n-b\frac{\beta +1}{\mu +1}{x}^{\mu }{y}^{\beta }-{\mathit{ux}}^2-\lambda \right)\hfill \\ \hfill & y^{\prime }=4x({\mathit{ax}}^2-{\mathit{aby}}^2-1)+\epsilon y({\mathit{ux}}^n+{\mathit{vy}}^n+{\mathit{bx}}^{\mu }{y}^{\beta }-{\mathit{vy}}^2-\lambda )\hfill \end{array}$where μ + β = n, 0 < a < b < 1, 0 < ε ≪ 1, u, v, λ are the real parameters and n = 2k, k an integer positive.Applying the Abelian integral method [Blows TR, Perko LM. Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems. SIAM Rev 1994;36:341–76] in the case n = 6 we find that the system can have at least 13 limit cycles.Numerical explorations allow us to draw the distribution of limit cycles.     

N-cyclic functions and multiple subharmonic solutions of Duffingʼs equation

We introduce, in the abstract framework of finite isometry groups on a Hilbert space, a generalization of antiperiodicity called N-cyclicity. The non-existence of N-cyclic solutions of a certain type for the autonomous ODE $x^{″}+g(x)=0$ implies the existence of N different subharmonic solutions for some forced equations of the type $x^{″}+g(x)+c{x}^{\prime }=\epsilon f(t)$ where c and ε are some positive constants and f is, for instance, a sinusoidal function.     

The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let $G^{\sigma }$ be a weighted oriented graph with skew adjacency matrix $S(G^{\sigma })$. Then $G^{\sigma }$ is usually referred as the weighted oriented graph associated to $S(G^{\sigma })$. Denote by $?(G^{\sigma }\text{;}\lambda )$ the characteristic polynomial of the weighted oriented graph $G^{\sigma }$, which is defined as$?(G^{\sigma }\text{;}\lambda )=\mathit{det}(\lambda I_n-S(G^{\sigma }))=\sum _{i=0}^n{a}_i(G^{\sigma }){\lambda }^{n-i}\text{.}$In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

Hyperations,Veblen progressions and transfinite iteration of ordinal functions

Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions $\mathrm{Hyp}[f]={〈f^{\xi }〉}_{\xi \in \mathsf{On}}$, called its hyperation, in such a way that $f^0=\mathsf{id}$, $f^1=f$ and $f^{\alpha +\beta }=f^{\alpha }°f^{\beta }$ for all α, β.Hyperations are a refinement of the Veblen hierarchy of f. Moreover, if f is normal and has a well-behaved left-inverse g called a left adjoint, then g can be assigned a cohyperation $\mathrm{coH}[g]={〈g^{\xi }〉}_{\xi \in \mathsf{On}}$, which is a family of initial functions such that $g^{\xi }$ is a left adjoint to $f^{\xi }$ for all ξ.